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 <~

d f> = ~(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 <~pV,W,~ ~ ~ > =
For r e p r e s e n t a t i o n s

ooordinate

÷ Mp

>

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

<(B)

÷

are open

s m o o t h maps. sentation

functions

of B.

x

x i = xi(xA), i

,

since both

~xAsK(

B).

% and ~ are repre-

x ~ = xi(xA,t), i = 1,2,3. configuration

of B and ~(t)

Then <~p is a local r e f e r e n c e

to <~p is Fp(t)

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

deformation

bases

bases

:

:

dealing

with k l n e m a t i c s

.

This

3} and

we have

,~p-i o [p~a ra*~p~bb -l tab ~×b h F Ab = ~

We can char-

of <~p and ¢,p, w h i c h

{£--A = 1,2 ~X A '

respectively,

so that

at p r e l a t i v e

b form via Fpi ~ A = FAib, A = 1,2,3.

of the r e f e r e n c e

are just the n a t u r a l

configuration

-i = (~o -l),<(p). o <~p

= ~,p(t)

F~p in c o m p o n e n t

But in terms

:

in R 3, can

of B then the above

of peB for all such p and the local

acterize

(X A)

are smooth

Now let < be a r e f e r e n c e a motion

sets

functions

¢ o <-i

If ~ is a m o t i o n

becomes

connected

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

1,2,3, in such a way that

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

¢(B)

completes

-

K

1

*p~A

:

~

as much of the m a t e r i a l

of a c o n t i n u a

as we shall need

for

now. 3.

Force

and Stress

in C o n t i n u u m M e c h a n i c s

In order to f o r m u l a t e elastic

point we must

we talk about

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

first

the concept

understand

of stress.

equation

of an

what we mean w h e n To this

end,

let

C be

23

a part of a b o d y m a n i f o l d <.

B which

We a s s u m e h e r e that b e s i d e s

differentiable

manifold,

being

distribution

of B is t h e n any m e a s u r a b l e onto a r e g i o n

in R 3.

a three-dimensional

B is also a m e a s u r e

it is e n d o w e d w i t h a n o n - n e g a t i v e is c a l l e d the mass

is in the c o n f i g u r a t i o n

space,

scalar measure

of the body;

theorem)

It is also a s s u m e d that for any con-

continuous

has a d e n s i t y

and thus

p~ w h i c h

of B in the c o n f i g u r a t i o n

m(C)

:

<.

w h e r e ~ is the E u c l i d e a n

(by the R a d o ~ - N i k o d y m

Thus

if CoB is a part

in m o d e r n

continuum mechanics the a c t i o n of the

o u t s i d e w o r l d on a body in m o t i o n and, also, parts

of the body.

a s s u m e d that m u t u a l b o d y forces forces

are c o n t i n u o u s l y

in B

on R 3.

is the c o n c e p t of force w h i c h d e s c r i b e s

b e t w e e n the d i f f e r e n t

density

integral)

volume measure

importance

over K(B)

is c a l l e d the mass

f p d~(f ~ L e b e s q u e K(C)

Of p a r a m o u n t

C

can be m a p p e d

f i g u r a t i o n of B, such as <, the i n d u c e d m e a s u r e is a b s o l u t e l y

which

a part

s u b s e t of B w h i c h

i.e.,

It is u s u a l l y

are absent

distributed.

and N o l l

[ 5 ] we n o w list b e l o w t h o s e

acterize

a s y s t e m of forces

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

and t h a t all

Following Truesdell conditions which

for the b o d y B in any m o t i o n

~(t): (i)

At each time t, a v e c t o r f i e l d b(x,t),

d e n s i t y of the e x t e r n a l b o d y

char-

c a l l e d the

force per unit mass

acting

24

on B in the motion % i s we indicate the motion f#t(C)

(ii)

the configuration ~(t).

b(x,t)dm

exerted

defined

occupied

The vector fb(C),

by %t(B)

by B at time t in

defined to be the integral

is called the resultant

external

body force

on the part C at time t.

At any time t, there

vector

for each xe~t(B);

field t(x;

is defined

to each part CoB a

C) acting on C, called the stress,

for all points

ure of the density contact

corresponds

xs~t(~C).

of the contact

force f (C) exerted

The stress

which

is a meas-

force and the resultant

on C at time t is given by the

~C

surface (iii)

integral

~ t(x; ~t(~C) ~

The total resultant

by: f ( c ) (iv)

C)dS force f(C) acting

~ {b(C) + I t ( C ) .

There is a vector-valued

all points stress

on CoB is given

function

t(x,n)

x in St(C) and all unit vectors

defined

for

n such that the

acting on C is given by t(x;

C) = t(x,n)

where n is the exterior boundary

unit normal

of C in the configuration

stress

vector at x and the above

stress

principle

at the point x on the ~t"

We call t(x,n)

concept

the

is known as the

of Cauchy.

Any system of forces

such as that specified

must obey the fundamental

balance

the principles

of.momentum

of balance

above

laws of mechanics, and balance

i.e.,

of moment

25 of momentum.

If the momentum M(C) of C in the configuration

~t is given by

/ x dm then the principle of balance of ~t(C ) ~ momentum requires that f(C) = M(C) and implies, under suitable continuity conditions,

the existence of a tensor field

~S such that t(~,n) = Ts(x)n for all xs~t(~C); the superposed dot above indicates, respect to time.

of course, differentiation with

We call Ts(X) the stress tensor at the

point x and it is relatively

simple to show that balance of

momentum then implies the Cauchy law of motion: div TS + pb = p~~ while balance of moment of momentum implies the symmetry of the stress tensor, i.e., ~S = T~" 4.

The Constitutive

Material Isomorphisms

Equation of a Simple Elastic Point; and Materially Uniform Elastic Bodies.

By a constitutive e~uation in continuum mechanics we understand a relation between the contact forces, acting on parts of the body B,which are specified by the stress tensor, and motions of B; constitutive as representations All constitutive

equations thus serve

of various classes Of ideal materials.

equations which are dealt with in modern

continuum mechanics must satisfy three basic principles, i.e., the principle of determinism,

the principle of local

action, and the principle of material frame-indifference; the later principle asserts,

essentially,

that the material

properties of a body, i.e., its response to the application

26

of given

forces,

are i n d i f f e r e n t

used by an observer. matically

satisfied

lastic p a r t i c l e s

The first

we s h a l l h a v e sent work.

(points)

either

above

For e x c e l l e n t

on R3; this t e n s o r ~(~C),

where

stress

vector

by T$(x) usually particle

which

discussions

are implied by the and i n t e r e s t i n g

of these within

principles

the scope of

to be found

contact

forces

in [ 5 ] or [27]. at a point x = ~(pl,

are c h a r a c t e r i z e d

is a s y m m e t r i c

tensor

acts on the unit normal

at that point.

Instead

as we have

Ts(P,t)(for

at points

in the last

p at time t in the m o t i o n

of

it into the

of d e n o t i n g

the stress

by the

of order two

CoB is a part of B, and t r a n s f o r m s

just w r i t e

on the

we r e f e r the r e a d e r to

~ of B , p~B,

~ Ts(~t(p))

e-

to make use of t h e m in the pre-

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

~S' w h i c h

of simple

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

important

of simple m a t e r i a l s ,

tensor

consists

particularly

Recall now that the

stress

are auto-

and while

are both

no o c c a s i o n

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

two p r i n c i p l e s

function

and their r a m i f i c a t i o n s , the t h e o r y

of r e f e r e n c e

by a body w h i c h

form of the c o n s t i t u t i v e third principle

to the frame

this tensor

section we shall

tensor

at the

~) if the m o t i o n

~(t)

is

understood. Our basis Definition p is called

definition

II-2.

is the one given by Noll

If peB and ~(t)

a simple

elastic

at p at time t in the m o t i o n

is any m o t i o n

particle ~ depends

[ 6 ]:

of B then

if the stress

tensor

only on %,p(t),

i.e.,

27

there

exists

a tensor

T(p,t)

f u n c t i o n E such that

= E(~,p(t),

p)

We call the t e n s o r f u n c t i o n E the e l a s t i e r e s p o n s e function.

If r is a fixed ~p

of p t h e n we h a v e

T(p,t) . .

~,p(t)

= E(F(t) . .

local r e f e r e n c e

= F(t)

o r

p

configuration

o ~ rp and

(II-l)

,p) z S (F(t) p) ~r~p '

where

S is the e l a s t i c r e s p o n s e f u n c t i o n r e l a t i v e to r ; ~r~p ~p in w h a t follows we shall s u p p r e s s the d e p e n d e n c e of S on ~

r~p .

Following Noll

Definition

II-3.

called materially configurations

[ 6 ] we now make

Two simple

elastic particles

isomorphic

if t h e r e exist

p , q e B are

local r e f e r e n c e

r and r such that ~p ~q (Z-2)

F r o m the d e f i n i t i o n T h e o r e m II-i.

If p , q ~ B are simple

t h e y are m a t e r i a l l y p h i s m r(p,q): ~

it is a simple m a t t e r to d e r i v e

B

p

÷ B

isomorphic

elastic particles

iff t h e r e

exists

E(~,q,q)~ ~ E(~,q~ o r(p,q),~ p)

for all c o n f i g u r a t i o n s

a material

(II-3)

~: B ÷ R 3.

An i s o m o r p h i s m r(p,q): is t e r m e d

an i s o m o r -

sueh that

q

~

then

B

p

~ B

q

w i t h the a b o v e p r o p e r t y

i s o m o r p h i s m of p and q.

We also h a v e

28

the following

result

Theorem

If rp, rq are local reference

11-2

of p,qeB,

of Noll

respectively,

such that

~r(p'q) ~ rq-i

o ~rp: Bp ÷ Bq

is a material p,q relative

isomorphism,the

~

of B

p

elastic

uniform

exactly

If (11-2)

defines

a material

body materially isomorphic

isomorphism

uniform

if it Thus,

each particle

spaces

sense different

If B is materially

whose

com-

isomorphisms

at the particles

of

(points).

of the body are comprised

its particles

are pairwise

the body is materially uniform we define

(U , r e ) where

specified

(pairwise)

material

particles

i.e.,if

isomorphic,

of these

into configurations

comprise

tangent

for B to be a pair

neighborhoods

configurations

of the same material,

uniform.

a reference

chart

U cB is an open set and

~

r

of

is valid

particles.

elastic body,

provided

a la (II-4)

the respective

materially

simple

are first brought

by local reference positions,

functions

in the same way as every other particle,

to a given deformation, particles

response

(11-2).

of pairwise materially

responds

(II-4)

q

in a materially

In this

(11-4)

configurations

and B .

We call a simple consists

elastic

to rp, rq satisfy

for some r and r then ~p ~q r(p,q)

[ 6 ]:

.

is a smooth field of local reference

configurations

on

~

U , which has the property

that relative

elastic response

S are independent

functions

to rp, p~U

, the

of p, i.e.,

2g

there

exists

a tensor

S (F) = S(F,p), ~e

~

YpeU

~

hood,

re(p,q)

to

~ re-i ~q

Now let

{Ue, ael}

funetion # we will

elastic

body.

was r e q u i r e d hood

atlas

call

is an open

response

to ~.

B have

the more

above

When

eel}

is, in general,

S

the smoothness a local

reference is

It is triv-

a reference

uniform

(simple)

[5 ], [6 ] it reference

atlas

neighbor-

of the form of reference

[7 ] and brings of the

response

are independent

B possesses

definition

on

call S~_ the re-__

by a single

is due to Wang

functions

charts

is a reference ~e

materially

a reference

peUe.

collection

of B.

work of Noll

general

r~,

=

and term a collec-

functions

it a smooth

fact that

B.

~p

this

set ~eS = S~ and will

important

on

not n e c e s s a r i l y

compatible

covering

if ~ = {(Us ' e ) ,

B be covered

that

on UenU6

of B p r o v i d e d

relative

that

r)};

given

does

of p and

if the induced

of m u t u a l l y

In the original

U, i.e.,

= {(U,

function

isomorphism

that there

compatible

atlas

of e, ~el, so we will

atlas

be a m a t e r i a l

eel}

for B the induced

sponse

nei~hbor-

map then

9: Ue => R3 such that

charts

ial to see then that atlas

the response

~eS and ~8 are identical

and

a reference

(Us, r) ~e , (Ue, ~r8) be two reference

a reference

maximal

and S

we also note

~ = {(Us ' e ) ,

charts

U

If r e is a r e f e r e n c e

o r~pe must

B; we eall these

tion

We call

ma~,

a configuration

functions

such that

e

(U , re).

q for all p,qeUe; exist

.

S

~e

e

r e a reference

relative

function

out the

field of r e s p o n s e

and not a global

property

BO

Remark.

Reference

atlases

shown

(i.e.,

Noll

atlas

on B so is KS = {(U

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

[ 6 ], Wang

satisfying

of B are r e l a t e d

also exist r e f e r e n c e of R 3 w h i c h reference G(~)

satisfy

which 5.

Elastic

Symmetry

det K>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}
= G.

~n

8.

Material

Tangent

Charts and Material Atlases.

The Material

Bundle T(8~%) and the Bundle of Reference

Frames

E(B,}). For an arbitmamy differentiable which the tangent

spaces

the tangent bundle T(M).

manifold M, the way in

fit together

is characterized

If we now specify that M E B

(a simple elastic body) then we note that, independently functions

of the distribution

structure

as T(B) is defined

of the elastic response

on B~ it can not possibly

structure of the body~

by

characterize

in order to represent

of B in differential

the material

the material

geometric terms we follow

3B

Wang

[ 7 ] and i n t r o d u c e

material follows

atlas,

the concepts

and m a t e r i a l

of m a t e r i a l

tansent

bundle.

B is takan to be a m a t e r i a l l y

chart,

In all that

uniform

simple

elastic

body. Definition

11-7

(Wang

[7 ])

T(B) is called a m a t e r i a l

A bundle

chart

: B

isomorphisms

% ) of

Vp,qeU

(If-12)

÷ B P

are m a t e r i a l

(Us,

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

-i ~ ~a,q o ~ , p

~(P'q)

chart

•

q Two m a t e r i a l

charts

(Us, ~B), (U6, ~B) are termed compatible if r~B(p,q)

~ ~ , p ° ~ B , -iq : B q ÷ B P is a m a t e r i a l

for all peU~ and qsU B. compatible

material

From the above

A maximal

charts

collection

is a m a t e r i a l

definition

atlas

are tied t o g e t h e r

isomorphism.

Also,

if

material

atlas

on U

of B. charts

via the rule of m a t e r i a l

~ ) is a m a t e r i a l

chart

~ ra : B ~ R3 ~P P

is a smooth and,

atlas

in a

of B and we set

~ , -pI

then r

of p a i r w i s e

we see that the b u n d l e

in a m a t e r i a l

(Us,

isomorphism

field of local

(11-13)

configurations

defined

furthermore,

a-i r

~q

is a m a t e r i a l

o

r

~p

= ~,q

isomorphism

o %~,p-i _- r( p , q ) ~

of p and q.

Therefore,

(Us, r~)

36

is a r e f e r e n c e the c o n v e r s e

chart on B w h e n r ~ is d e f i n e d by

is also e a s i l y

simple task to d e m o n s t r a t e for the c o m p a t i b i l i t y atlas yields, previous erence

of two m a t e r i a l

for c o m p a t i b i l i t y

i.e.,

eel}; n o t e that

g r o u p of B r e l a t i v e the s t r u c t u r e atlas

group of

T(B).

(i.e., the one g e n e r a t e d

@eB a b o v e

of charts w i t h i n a ref-

G(¢),~ w h i c h

atlas w h i c h via

and the s u b - b u n d l e

is the i s o t r o p y

Lie s u b g r o u p of GL(3),

If ¢ is a g i v e n r e f e r e n c e

(11-13)

transformations

to be ~(¢) and w h o s e by

dharts on B, the

in G(¢) w h e r e

to ¢, is a c l o s e d

for B, the m a t e r i a l

The c o o r d i n a t e

charts w i t h i n a m a t e r i a l

z r~pe o r ~pB-I

are s m o o t h and take t h e i r v a l u e s re),

g i v e n above

the fields

-i o ¢~,p ~eB (p) : Ce,p

: {(Ue,

and

It is a

that the d e f i n i t i o n

for the i n d u c e d r e f e r e n c e

definition

atlas,

seen to be valid.

(11-13)

to

is d e n o t e d by ¢(¢).

for ~(¢) are g i v e n by the

T(B) w h o s e atlas is t a k e n

of

structure

corresponds

group

is G(¢)

is now d e n o t e d

T(B,¢) and is c a l l e d the m a t e r i a l t a n g e n t b u n d l e of B

relative

to ¢.

We say that we h a v e

T(B) to the b u n d l e T(B,¢). admits

a reference

correspondence B, i.e.,

between

material

tangent

bundle

the b u n d l e

B is m a t e r i a l l y

¢ and t h e n

reference

every materially

a well-defined material

atlas

As

"reduced"

(11-13)

u n i f o r m it

sets up a i-i

and m a t e r i a l

atlases

on

u n i f o r m s i m p l e e l a s t i c b o d y admits

atlas,

say ~(~) ÷+ ~, so that the

T(B,¢)

of B r e l a t i v e

to ¢ is also

37

well-defined. Exercise

i

reference

Using the relationship atlases

material

fibre

space

fibre bundle

coincides

bundle

there

space,

structure

as indicated

is associated group,

in Chapter

a principal

E(B), which

in Chapter

be constructed;

transformations

as presented

E(B,~)

to a reference

principal

all, we define relative

bundle

bundle

of T(B),

atlas

of reference

ep(e)

=

of all reference

as follows:

a material

= ~e,p(i)~ where basis vectors

frames

~ peB

E (~). P

chart

frame at p

(U ,~ )e~(~)

~i is the frame consist-

of R 3.

at p relative

The projection

frames

first of

We denote the set

to } by E (~) and take p

~

could

~ of B) as the as-

a linear frame e to be a reference ~p

ing of the standard

E(B,~) =

frames

the same type of construction

of T(B,~),

to ~ if there exists

such that ep

principal

coin-

In particular,

of linear

there we can form the bundle

(relAtive

sociated

I how the bundle

using precisely

of left-multi-

bundle whose base

and coordinate

is the associated

group,

I, to each fibre

cide withe those of the given fibre bundle. we indicated

is a fibre

with its structure

which then aets on itself via the operation Also,

that holds

atlases.

Recall now that a principal

plication.

among all

for B derive the relationship

among all corresponding

bundle whose

which exists

~

map ~ is~ of course,

a

~

map 7: E(B,~) + B such that - l ( p ) the fibre of E(B,¢)

at p.

= Ep(¢),~ i.e.,

Ep(¢)~ is

We now define the bundle

~(~) = {(U ,~e), e~I} by ~e(p,G)

= RG(~p)

atlas

for all ep s Ep(~)

38

and GeG(@), they

The bundle

are isomorphisms

6e :

Exercise --

2

÷

~-l(u

)

= RG

~aB(p)

the coordinate

}(~).

maps

formations

of %(¢)

As with

chosen

~e above

e p (e) satisfy

(e ( ~ ) )

for

up

so that

result

implies

and 6(¢)

of the bundle

and the d e f i n i t i o n

that

(2)

of

the coordinate

trans-

are the same.

T(B,¢) and T(B),it

E(B,%) is a subbundle

the trans-

peUanUB w h e r e

all

transformations

Show that this

the bundle

been

E(B,@)

c

Show that the frames

9~8 denotes

that

~a have

of the form

G(@)

×

law e p ( B )

formation

atlas

U

maps

is r e l a t i v e l y

of the bundle

easy to see

of linear

frames

E(B). Exercise bundle

3.

Let ~ be a reference

of reference

frames

atlas

relative

for B and

to }.

E(B,~)

If K~GL(3)

the

then

show that

and that above

E(B) = U¢ E(B,~). ~

it follows

that the bundle

E(B,~)~ are related where

CK(9)

in turn, We will

(2)

,

~

implies

close

R g~p e denotes

this

atlases

of E(B,K~)

~(K~)~~ = {U~, RK_I~

VGsGL(3)

]](p,G)

K--

This,

via:

= KGK-I

IRK_ 1 o ~a o C

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

°

= RK_I(~

Therefore .

that-K-IGKsG(K~), section

with

right-multiplication

and

CK_ I, ~sl}

,

(K-1GK)),

'P .

rule

.

.

peU , G s G ( ~ ) .

GsG(~).

some

observations

via G.

39 concerning

the Lie algebras

fundamental

fields

the isotropy

of the isotropy

on the bundles

groups

E(B,~).

G(~) are closed

they are also Lie groups

and have,

groups

First of all, as

Lie subgroups therefore,

g(~); recalling

our definition

the elements

of g(%)

of all left-invariant

fields

on G(~) and the bracket

defined by [u,v]

= Lv,

Vu,v

operation

of SL(3)

associated

Lie algebras

consist

G(}) and

in Chapter

I,

vector

on the algebra

g(9), where

L denotes

is

the Lie

m

derivative.

By the standard

Lie algebra

is isomorphic

the identity (i.e.,

element

see Wang

representation

to the tangent

of the group.

[7 ]) establishes

algebra

g(~) of

algebra

of SL(3) which,

in turn,

gl(3).

As the isotropy

groups

tion law:

(refer to Chapter GCG(}),

reference Remark which

: KG(~)K -I, !, i.e.,

Veg(~))

g(~) transform

A simple argument the fact that the Lie

G(})

satisfy

VKcGL(3),

exp

of sl(3),

the Lie

is a Lie subalgebra

of

the transforma-

the exponential

map

(GVG -I) = G exp (V)G -I

can be used to show that the Lie algebras

as g(K%)

= Kg(})K -I under a change

of the

atlas.

The transformation is given above,

law for the Lie algebras

defines

an equivalence

on the class of all Lie subalgebras alence

space to G(~) at

G(~) is a Lie subalgebra

G(K~)

of g(~), this

class relative

type of Lie subal~ebra the Lie subalgebras

of gl(3)

to this relation (of gl(3)).

of sl(3)

relationship and an equiv-

is then called a

A classification

and their

g(~),

corresponding

of

40 (connected) Further

subgroups

discussion

be a s s o c i a t e d

[31]; we note,

approach

eonnceetion

(and hence tangent

differs

vector

fields as G(%)

and ~(¢)~ : {(U~,~

~~ (p) ~ ~ , p ~

in Nono that

from the one taken here

a bracket

structure)

in ma-

operation

on the space of smooth

is the structure

group

is the bundle

for

atlas,

I (where the c o n s t r u c t i o n

~:

fields g on E(B,¢),

(v)~, for p e U~, where E(B,~) x~

÷ B P , where

= 0, so that g(¢)

collection

of f u n d a m e n t a l

is easily

E(B,~),

we may follow is carried

~(x)

4.

is i s o m o r p h i c

fields

If 7:

= p, then

E(B,~)÷B,~

it follows

to ker 7,.

~ on E(B,@), --

i.e.,

of g = gl(3),

g(¢) ~

~

the

of the choice

fields

set

of the bundle

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

~ on E(B,¢)

charts

in ~(¢).

field on E(B,¢) show that

~

RK_I

The

on E(B).

Show that the f u n d a m e n t a l

are i n d e p e n d e n t 5.

fields

via

v~g(~).~

seen to be a Lie s u b a l g e b r a

of f u n d a m e n t a l

Exercise

may be found

may

t e n s o r of a ( c u r v a t u r e - f r e e )

that w,(~)

Exercise

Lie algebras

E(B)) and define the Lie algebra g(¢), c o n s i s t i n g

of the f u n d a m e n t a l

so that

[7 ].

on B.

), ~ I }

the lead of C h a p t e r

in Wang

in particular,

on B to define

a Lie a l g e b r a

Finally,

out for

in w h i c h

w i t h simple m a t e r i a l s

that he uses the t o r s i o n terial

can be found

of the ways

[301 and B e l i f a n t e Belifante's

of SL(3)

(~) is a f u n d a m e n t a l

field on E(B,Kg), ~KsGL(3).

41

7.

Material

Connections

on Simple E l a s t i c

Let ~ be a given r e f e r e n c e uniform

simple

of r e f e r e n c e bundle

elastic

frames

body,

and let

relative

÷ ~-I(u~)

for B, a m a t e r i a l l y

E(B,@) be the bundle

to ~; as b e f o r e we denote

atlas by ~(~)~ = {(U~,~

~ : U~ x G(~)

atlas

Bodies

the

), ~EI} where

c E(B,@).~

A material

connection

Definition

II-8

nection

E(B,@) w h e r e @ is any r e f e r e n c e atlas for B.

on

on B is then d e f i n e d

A material

From C h a p t e r

connection

as follows:

H on B is a "G" con-

I (§5) we see that a "G" c o n n e c t i o n

E(B,@) must s a t i s f y the f o l l o w i n g condition: smooth

curve

parallel Pt,e:

in U~cB,

transports

G(@) + G(9)

O~tsT,

along

must be elements G(@), leU

passes

through

at t = O. integral tangent

in turn,

to p t(~)

element

implies

of

curve

E(B,@), i.e., of

that

pt(~),

a restriction

group

O~t
of G(@),

at t = O must b e l o n g in terms

curve

in G(~) w h i c h

of the i s o t r o p y

Lie a l g e b r a

To state this r e q u i r e m e n t on B, and thus o b t a i n

(I!-14)

group

must be a smooth

curve of g(@) the vector

are the

to H then the maps

In o t h e r words, for a given

the i d e n t i t y

This,

÷ El(t)

by

of the s t r u c t u r e

O~t
El(O)

is a

Pt o ~ , l ( O )

for each t~[O,T].

, pt(~),

I relative

defined

-I Pt,~ -- ~ , l ( t ) 0

and Pt:

if l(t)

H on

i.e.,

the

to g(@).~ ~

of local

coordinates

on the c o n n e c t i o n

symbols

42

of a material connection H on B, let ~: B ÷ R 3 be a configuration of such B that %(p) = (xl(p), x2(p), x3(p)).

If

(U ,~e) is in ~(9) we can represent ~e by the component form ~

: y~ 9i

O ~---~ ~x ]

for R 3.

where {e i} ~ i is the standard basis '

~

~

For the component form of the parallel transports

Pt along I from I(0) to l(t) we have

Pt

p}(t)dl(o)Xi] x ~ ~x I 1 (t)

where the functions p~(t) must satisfy the equations of parallel transport, i.e.,

i p~ih = 0 ' i,j : 1,2,3

gi

~j

+ r~k

and the initial condition p~(O) = 6~; a unique solution ] ] 0~(t) of (11-14) satisfying this initial condition is guranteed to exist by virtue of the standard existence theorem for ordinary differential equations and the assumed smoothness of I and ~r.

Finally, we will represent the

"induced parallel transports" pt(~) relative to ( U ~ , ~ ) pt(e) = ~(t)e.~l ~ representations

~(t)

~ei.

by

Combining the various component

given above we deduce easily that

= (y-l)k(l(t))p~(t)y~(l(O)

(II-15)

The tangent vector to pt(~), at t : O, thus has the component form ~i(O)eij ~ ~ej where

43

i Ira(O) + (y-l)k(l(O))~(O)~(y~)(l(O))

(II-16)

I(0) But, as pm.(t) is a solution of the equations of parallel 3 transport, i.e., p~(O)=-r~m(X(O))Im(o), substitution in (11-16) then yields

"i(o) : (y~)C~(o))I (y-l)~ ~J

~xm

Fkm( 1 (O))(y_l)k( 1 (0) I ~m(o) I(O)

Since I(0) and i(O) are both arbitrary we have the following major result (Wang [ 7 ], [32]): Theorem II-3.

Let ¢ be a reference basis for B, G(¢) the

isotropy group relative to ~, E(B,¢) be the bundle of reference frames relative to ~ and ~(~) = {(U ,~ ), ~el} the bundle atlas.

If the maps $~,p: G(¢) + Ep have the re-

presentation ~ , p

= Yj~i i

O

~o~xll p relative to a local co-

ordinate system (x i) on Uegp then the functions F ijk(P) are the connection symbols of a "G" connection H on E(B,#) iff the matrices { ~(y-l)~y£~ _ r~m(y_l)kl, -L m

m : 1,2,3 1

belong to g(~) at each point psU . Remarks.

The above proof follows the argument of Wang [32]

except that our maps ~ , p

are inverse to the corresponding

(II-17)

44

ones w h i c h he employs. same r e s u l t

For a d i f f e r e n t

d e r i v a t i o n of this

one c o u l d also look up the p r o o f in [ 7 ] (which

uses the c o n c e p t of a c o n n e c t i o n f o r m w h i c h has b e e n a v o i d e d here). on

It is not too d i f f i c u l t to show that

E(B,~)

c o r r e s p o n d to those

affine

connections

w h i c h h a v e the p r o p e r t y that the p a r a l l e l tangent

spaces

isomorphisms.

Exercise

6.

V e r i f y the above

Remark

of the

0t) must be

concerning affine

on T(B,~).

For an i m p o r t a n t

class of simple

k n o w n as s o l i d c r y s t a l b o d i e s w i t h the i s o t r o p y is always

statement

T(B,~)

on

transports

(which are i n d u c e d by the maps

material

connections

"G" c o n n e c t i o n s

trivial,

see that m a t e r i a l

the

group r e l a t i v e i.e.,

g(~)

connections

elastic materials

Lie a l g e b r a a s s o c i a t e d to any r e f e r e n c e

= {O}; f r o m T h e o r e m on such b o d i e s

atlas II-3 we

are c h a r a c -

t e r i z e d by the c o n d i t i o n s

Fj -i ~ jm ik = - ~ m

(II-18)

i

-]

where

~ = y -.

It is a s i m p l e m a t t e r to c o m p u t e that

R = 0 (the c u r v a t u r e (II-18))

t e n s o r b a s e d on the F symbols

so that m a t e r i a l

are b o t h u n i q u e w o r k of N o l l

connections

on s o l i d c r y s t a l b o d i e s

and c o m p l e t e l y i n t e g r a b l e .

[ 5 ], [ 6 ], all m a t e r i a l

d e f i n e d so as to be c o m p l e t e l y

in

In the e a r l i e r

connections were

integrable;

a simple elastic

4B

body, however, material Wang's,

need not admit any completely

connections

and some non-trivial

to this effect, will be presented

integrable

examples

of

at the end of

this chapter. A very interesting

class of simple elastic bodies

which can be equipped with a torsion-free is the class of solid bodies.

material

A simple elastic material

B is said to be a solid body if, for some reference ~, G(~)c0(3),

the orthogonal

connection

group on R 3.

to see that this later definition

atlas

It is trivial

is consistent wi~h de-

finition

11-6 if we define a solid body to be one which

consists

of solid particles.

property that G(~)c0(3)

A reference

atlas

~ with the

is called undistorted.

To proceed

we need the following Definition body.

11-9

(Wang [7 ]).

Let peB,

A tensor t belonging to Tr'S(p)

if it is invariant elements

of g(p),

a simple elastic is called intrinsic

under all transformations i.e., by material

If we choose a chart

induced by

isomorphisms

of p.

(U , r ~) in the reference

atlas

~

~, such that peU

, and then define

[g~(p)](u,v) ~ ~

= r~(u).r~(v) ~p ~ ~p ~

~

then it is easy to show that g~ is an intrinsic tensor in T0'2(p) which can be extended so as to yield an intrinsic Riemannian metric on B; we call this metric the induced metric of the undistorted intrinsic

reference

is a direct eonsequence

faet that the values

of Euelidean

atlas ~.

That g~ is

of its definition and the dot products

are preserved

46

under transformations versely,

Wang

B possesses

i n d u c e d by e l e m e n t s

[ 7 ] has

shown that

Remark

atlas

Con-

elastic body

g then

and there m u s t e x i s t

B must

an u n d i s t o r t e d

~ such that g = g~.

If ~ is an u n d i s t o r t e d

obviously,

if a s i m p l e

an i n t r i n s i c R i e m a n n i a n m e t r i c

a c t u a l l y be a s o l i d b o d y reference

of 0(3).

so is

O~ w h e r e

reference

0 c 0(3).

atlas

for B then,

It is t r i v i a l

to show

t h e n that g~ = g0~" ~ ~

Remark

It can be shown that all u n d i s t o r t e d

for B are of the f o r m K~ w h e r e

reference

atlases

the s y m m e t r i c p o s i t i v e - d e f i n i t e

~ ~

part of K, in its p o l a r d e c o m p o s i t i o n , elements

in G(~).

This fact,

c o m m u t e s w i t h all

in c o m b i n a t i o n w i t h the p r e v i o u s

~

remark,

leads to the c o n c l u s i o n that

metrics

on B have the f o r m gu~ w h e r e U is any s y m m e t r i c

positive

definite

elements

of the i s o t r o p y g r o u p

t e n s o r in GL(3) w h i c h

Now let H be any m a t e r i a l body,

all i n t r i n s i c R i e m a n n i a n

c o m m u t e s w i t h all

G(~). c o n n e c t i o n on B,a s o l i d e l a s t i c

and let g be any i n t r i n s i c R i e m a n n i a n m e t r i c

As g is i n t r i n s i c , to H are m a t e r i a l it follows

and as the p a r a l l e l isomorphisms

that the c o v a r i a n t

transports

of the t a n g e n t derivative

on B. relative

spaces

on B,

of g r e l a t i v e

to

H m u s t v a n i s h , i.e.,

"~ gijlk

where

= 0x

I - rikglJ

I - rjkgil

the F~k7 are the c o n n e c t i o n

= 0

s y m b o l ~ of H.

(II-19)

If we now

47

restrict H to be torsion

free and solve

(11-19)

variants which are obtained by permuting the fact that gij = gji' symbols

(and its

indices

and using

F~j = -F~i)_ for the connection

F we find that

Fijk : {jk i}

" ~ g k m .+ ~ = %glm( ~x ]

where the

{j~} are the classical

other words,

_ ~g ~k)

dx

Christoffel

if H is torsion-free,

the Riemannian

connection

(II-20)

~x

relative

symbols.

In

it must coincide with to g.

We have thus

proved the following Theorem 11-4

(Wang [7 3).

A solid body

with at most one torsion-free material if it exists, relative

connection which,

must coincide with the Riemannian

connection

at once that no

on a solid elastic body B can be torsion

free if the Riemannian

connection relative

Riemannian metric on B is not a material problem which then arises,

of course,

to any intrinsic

connection;

is to determine

which classes of solid bodies the Riemannian relative to intrinsic Riemannian metrics nections.

connection

to any intrinsic Riemannian metric on B.

From the theorem above it follows material

B can be equipped

If B is an isotropic

for

connections

are material

solid body,

the

con-

i.e., G(~) = 0(3),

then it can be shown that the most general type of symmetric positive

definite tensor U which commutes with all elements

of G(@) is of the form U = cl, where e>0; but, this implies

48

that the i n t r i n s i c to a c o n s t a n t

Riemannian metrics

factor.

This,

is a u n i q u e R i e m a n n i a n Riemannian metrics g(p)

implies

connection relative

on an i s o t r o p i e

that t h e r e

to all i n t r i n s i c

solid body.

But

O(B ), r e l a t i v e to any i n t r i n s i c m e t r i c on B, and we

=

have,

in turn,

on B are u n i q u e up

P therefore,

Theorem

11-5

the f o l l o w i n g

(Wang

[ 7 ]).

The

(unique)

Riemannian

a s s o c i a t e d w i t h the i n t r i n s i c R i e m a n n i a n m e t r i c s tropic

solid b o d y is a m a t e r i a l

tropic

solid b o d i e s

w h i c h this Remark

Moreover,

isofor

is the case.

For s o l i d bodies,

Riemannian

o t h e r t h a n the i s o t r o p i c

and,

connection;

in fact,

ones, the

do not u s u a l l y give rise to a

none of the i n d u c e d R i e m a n n i a n connection

on an iso-

are the only type of solid b o d i e s

intrinsic Riemannian metrics unique

connection.

connectior

by T h e o r e m 11-4, connections

a torsion-free

therefore,

can be a m a t e r i a l

material

connection

does not exist on B.

We w i l l h a v e m o r e to say about this

s i t u a t i o n in the next

section.

(11-19) may be s o l v e d

for F~k so as to y i e l d the f o l l o w i n g

relationship material

For now we note only that

b e t w e e n the c o n n e c t i o n

symbols

of an a r b i t r a r y

c o n n e c t i o n H on B and the C h r i s t o f f e l

any i n t r i n s i c m e t r i c

symbols

g:

i il(T _ Fijk = {'~}] + ~g jkl Tljk + Tklj) where

~T is the torsion

tensor of H and Tjk I = T ijk gil"

of

49 Finally,

let us note that,

in general,

if g~ is an ~

intrinsic

Riemannian

the c u r v a t u r e

metric

tensor

g~ need not vanish,

on B, a solid

of the R i e m a n n i a n i.e.,

to g~ is not n e c e s s a r i l y

connection

the R i e m a n n i a n a flat

elastic

body, based on

connection

connection.

relative

It is a w e l l - k n o w n

~

classical

result

in d i f f e r e n t i a l

curvature

tensor

of the R i e m a n n i a n

metric

geometry

connection

such as g~ is the zero tensor,

a local

coordinate

relative

Riemannian

components

the m e t r i c

connection

based

on a exist

of any p~B

of g~ reduce

g on B is l o c a l l y

relative

if the

then there must

s y s t e m in a n e i g h b o r h o o d

to w h i c h the

other words,

that,

to Bij.

Euclidean

to g is flat.

In

if the

We now make

the f o l l o w i n g Definition

II-i0

Let ~ be an u n d i s t o r t e d

for the solid e l a s t i c

body

reference

B and g~ the i n t r i n s i c

atlas

Riemannian

~

metric

on B i n d u c e d

if g~ is l o c a l l y

by ~.

Then

~ is c a l l e d a r e g u l a r

atlas

Euclidean.

~

The f o l l o w i n g reference Theorem

atlases

11-6

is r e g u l a r reference

characterization

(undistorted)

has been given by Noll in [ 6 ]:

An u n d i s t o r t e d

reference

iff every r e f e r e n c e neighborhood

has

r~p = ~Q (p) o ~ , p ,

where

of r e g u l a r

atlas

~ = {(U

map r ~ on a simply

a representation

of the

,r ~)

connected form

VpeU

~ : U s ~ R 3 is a c o n f i g u r a t i o n

~el}

(II-21)

of U~ and Qe_ is a

50 smooth field of orthogonal

8.

Homoseneity,

Local-Homoseneity

Once again, we take definition

tensors

on 9a(U~).

and Material

Connections

B to be a simple elastic body.

given below corresponds

to the intuitive picture

that we have of the body's being either homogeneous particles

of B may be brought

which each responds

exactly

into one configuration

brought

(all ~ from

as any other to a given defor-

mation from this configuration) psB, then all particles

The

or locally homogeneous

in some neighborhood

into one configuration

(if

N

of p can be P Such that each particle in

N

responds exactly as p does to deformations from this P configuration.) In mathematical terms we have Definition (i)

II-ll A simple elastic body B is called homogeneous

it can be equipped with a global reference the property that rp = ~ p

chart

for some configuration

if

(B,r) with 4: B + R 3

and all pcB (ii) eneous

A simple elastic body B is called locally homo$-

if it can be equipped with a reference

atlas

= {(U ,re), ~el} with the property that to each ~EI, there corresponds

a configuration

~:

C~ ÷ R 3 such that rp : ~ p

for all psC . Remark

The definitions

of the relationship

given above are equivalent,

between reference

their induced material atlases

atlases

in view

¢ on B and

¢(¢), to the ones given by

51

Wang

[ 7 ]; these

of m a t e r i a l

later d e f i n i t i o n s

charts

belonging

It s h o u l d be obvious body B is l o c a l l y bundles

that

by just one c o o r d i n a t e Clearly, erence

to a p r e - b u n d l e

every h o m o g e n e o u s

homogeneous

are all trivial,

are p r e s e n t e d

and that

i.e.,

the base

neighborhood

of the form ~ = {(B,r)}

that B is h o m o g e n e o u s ,

i.e.,

atlases

were

uniform

kind of m a t e r i a l

body B admits

exists

simple

connections

(Wang

Proof

is c o n t a i n e d to ~.

i.e.,

global

etc.,

in terms

to charof a of the

If B admits

Our

a flat

then it is locally h o m o g e n e o u s .

atlas

in E(B,~),

relative

admit

it is p o s s i b l e

body B,

[ 6 ])

c o n n e c t i o n (3) on B w h i c h

~; then each h o r i z o n t a l the b u n d l e

If ~ is such a h o r i z o n t a l

coordinates

(3)

[ 7 ], Noll

Let H be a m a t e r i a l

to the r e f e r e n c e

a configuration

w h i c h the body admits.

Theorem

connection

that

elastic

is the f o l l o w i n g

material

a ref-

In the work of

local h o m o g e n e i t y ,

first result 11-7

atlas.

considered.

the h o m o g e n e i t y ,

materially

tangent

it does not f o l l o w

that there

We now want to d e m o n s t r a t e acterize

elastic

space B can be c o v e r e d

[ 5 ], [ 6 ], only simple bodies B w h i c h

reference

simple

its m a t e r i a l

~: B ÷ R 3 such that r~P = ~,p for all pEB. Noll

of T(B).

atlas

of the b u n d l e

even if a l o c a l l y h o m o g e n e o u s

atlas

in terms

to

(U~,~)

a "G" c o n n e c t i o n

curve

of r e f e r e n c e

on E(B,}).

in E(B)

frames

curve then in terms it is d e t e r m i n e d

corresponds

relative

of local

by the frames

B2

{e~ ( t ) ~ I ~

where

~(~(t))

of p a r a l l e t

= ~(t)

i The e. must ]

e U .

satisfy

the e q u a t i o n s

transport

~(t)]

and an initial

+ Fkli (~(t))e~(t)~l(t)

condition,

loss of g e n e r a l i t y , the c o n n e c t i o n

of H.

a local

ri

jk ~ O; it then follows

(11-22)

= O

w h i c h we may take,

to be e~(O) ]

symbols

then we may choose

: 6~; the F Ijk are,

Now,

any

of course,

if H is a flat c o n n e c t i o n

doordinate

that

without

system

TJI

{~

(~l) in w h i c h

, i = 1,2,3},

the

~(t)

natural

frames

reference pcU

(t) , j = 1,2,3}

of the local

frames.

But,

, is a r e f e r e n c e

(U ,# ) ~ %(~) ~

a reference It follows duces the Remark general,

by definition,

frame

iff there

such that e (~) = # ~p

a frame

exists

~,p

(~l),

are

e ~p (e), at

a material

(i) or, ~

chart

equivalently,

(U ,r ~) s ~ such that e (e) : r~-l(i). ~ p ~P ~ i m m e d i a t e l y that r ~ = e where @~: U ÷ R 3 in~p ~*p coordinates

While the converse false

a partial

can be shown that

connection

vanish

~i on U .

Q.e.D.

of the above t h e o r e m

converse

does exist,

(Wang [ 7 ]) if a simple

B is localy h o m o g e n e o u s

of H(p)

system

chart

local

a material

coordinate

in some

i.e.,

elastic

then for every pEB there

H(p)

is, in

body exists

such that the c o n n e c t i o n

local

coordinate

it

symbols

s y s t e m near p.

53

R e c a l l now that a "G" c o n n e c t i o n on E(B) is flat iff b o t h the

curvature

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

c o n n e c t i o n vanish.

Also,

and the above remark,

a s s o c i a t e d w i t h the

as is e v i d e n t

the e x i s t e n c e

from T h e o r e m 11-7

of a flat

"G" c o n n e c t i o n

on E(B,~) is r e l a t e d to the local h o m o g e n e i t y we may use the t o r s i o n material

connections

g e n e i t y of B.

and c u r v a t u r e

tensors

on B to c h a r a e t e r i z e

Actually,

11-18,

as a " m e a s u r e "

In fact, tions

connection,

whose

of the

in m o s t of those

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

was p r e s c r i b e d

theories

in c r y s t a l

a priori,

symbols

w i t h no r e f e r e n c e

the p r e s c r i b e d

c o n n e c t i o n was

integrable

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

we may

integrable

density.

of c o n t i n u o u s

lattices,

inhomo-

are g i v e n v i a

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

equations,

to c h a r a c t e r i z e

the local

completely

connection

Thus,

associated with

for s o l i d c r y s t a l b o d i e s ,

use the t o r s i o n t e n s o r of the u n i q u e material

of B.

distribu-

where

the g e o m e t r y

to c o n s t i t u t i v e

always

completely

t o r s i o n t e n s o r was

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

always u s e d

distribu-

tion. On the o t h e r hand,

Wang

[ 7 ] has p r o v e n that

a simple

e l a s t i c b o d y B can be e q u i p p e d w i t h a t o r s i o n - f r e e connection H(U)

iff t h e r e

exists

on some n e i g h b o r h o o d

follows

f r o m the r e m a r k

a torsion-free material

U of pcB,

above that

for each pEB.

material connectiol It t h e n

every locally-homogeneous

simple e l a s t i c b o d y can be e q u i p p e d w i t h a t o r s i o n - f r e e material

connection.

We can s u m m a r i z e

our r e s u l t s

in the

B4

scheme

shown b e l o w

~flat

material

connection

=>

B is locally h o m o g e n e o u s

~torison-free material connection

Now,

if B is a solid body,

Theorem

11-7

Theorem

II-8

exists

and,

A solid

then a genuine

in fact,

elastic

body

B is locally h o m o g e n e o u s

with

Proof:

of the t h e o r e m

of T h e o r e m sity.

As

equipped

11-7

connection

relative

As

for each p~B

~el

[g~(p)](u,v)

of g~ are just g~ is locally

however, material

the R i e m a n n i a n

Riemannian

be an u n d i s t o r t e d

metric

g on B.

reference

we can choose

÷ R 3 such that p c U

con w

atlas

~ so that and

If we define

= ~,p(U)

coordinates

it can be

connection;

on B, namely,

and ~ : U

r~p = Oa~p, VpEU a •

local

we know that

B is locally h o m o g e n e o u s ,

H.

to the proof of neces-

one such t o r s i o n - f r e e

to any i n t r i n s i c ,r~),~el~

connection

is just a r e s t a t e m e n t

material

at most

can be d e f i n e d

So, let ~ = ~(U for B.

homogeneous

with a torsion-free

as B is a solid body,

nection

a flat m a t e r i a l

so we may go d i r e c t l y

B is loeally

to

we may state

iff it can be e q u i p p e d the f i r s t - p a r t

converse

• @~,p(V),

(x m) i n d u c e d

6ij.

Thus

Euclidean,

Vu,vEBp,

on U

by ~

~ is r e g u l a r i.e.,

then r e l a t i v e

to

the c o m p o n e n t s

and the i n d u c e d m e t r i c

the R i e m a n n i a n

connection

55

d e r i v e d f r o m g~ is flat. Remarks

Following

Q.E.D

T h e o r e m 11-5 we i n d i c a t e d that,

the i n t r i n s i c R i e m a n n i a n m e t r i c s rise to a u n i q u e R i e m a n n i a n free m a t e r i a l indicated,

connection

connection,

that

connection.

For i n s t a n c e ,

case w i t h

a constant

connections

it can be shown

symmetric

positive-

atlas

for B; thus,

the

and this

Riemannian

implies

connection.

of G(~),

intrinsic

are u n i q u e up to that they give

However,

all m a t e r i a l

on s o l i d o r y s t a l b o d i e s must be c u r v a t u r e - f r e e

a material

(II-18)).

connection

Riemannian metrics a unique Riemannian

If our R i e m a n n i a n

in turn,

Therefore,

on a c u b i c

even t h o u g h

[ 7 ],

that

B is

all i n f r i n s i c

that c o n n e c t i o n

can not be

if B is not l o c a l l y h o m o g e n e o u s .

i l l u s t r a t i n g the r e s u l t s

found in W a n g

implies

c r y s t a l b o d y B give r i s e to

connection,

connection

connection

t h e n it w o u l d have to be a flat

c o n n e c t i o n and this,

locally homogeneous.

examples

solid b o d i e s ,

on cubic c r y s t a l b o d i e s

of the f o r m

a material

on

c r y s t a l b o d y then,

c o m m u t e s w i t h all e l e m e n t s

scalar factor

rise to a u n i q u e

if B is a cubic

isotropic

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

material

[7 ] has

still n e e d not be a m a t e r i a l

is the m o s t g e n e r a l

Riemannian metrics

were

As W a n g

b o d y do give rise to a u n i q u e R i e m a n n i a n

definite tensor which

(i.e.,

and thus a t o r s i o n -

can not exist.

connection

that U = cl, c~O,

where

connection

do not give

even if all the i n t r i n s i c R i e m a n n i a n m e t r i c s

a solid c r y s t a l l i n e

as is the

on solid b o d i e s

in g e n e r a l ,

§i0-ii.

of this

s e c t i o n may be

Some

58

9.

Field Equations

of M o t i o n

We w i s h to i n d i c a t e

in this

s e c t i o n how exact

equations

of m o t i o n may be d e r i v e d

possesses

a continuous

for an e l a s t i c body w h i c h

distribution

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

homogeneities).

Our d e r i v a t i o n ,

approach

[ 7 ] and the r e s u l t i n g

of W a n g

more g e n e r a l than those the d e s c r i p t i o n

first

of Noll's

c o n t a i n e d in T r u e s d e l l

follows

early w o r k

on the r e s p o n s e material

[ 6 ] (see also

structure

n e c t i o n on

of the body.

of H.

u n i f o r m e l a s t i c body and

connection

a configuration

a global

the s y s t e m

and the

on B i.e.,

a "G" con-

E(B,~) w h e r e ~ is some r e f e r e n c e atlas for B.

We i n t r o d u c e induces

no

of any k i n d and is b a s e d soley

Let B be a s m o o t h m a t e r i a l l y let H be a fixed m a t e r i a l

is

As has been p o i n t e d

f u n c t i o n of the e l a s t i c points

geometric

are

in this a r e a w h i c h

[ 5 ], §34).

or l i n e a r i z a t i o n

the

field e q u a t i o n s

out in [ 7 ], the d e r i v a t i o n to be g i v e n here employs approximation

(in-

essentially,

found by Noll

& Noll

field

coordinate

(x ±) we take

~: B + R 3 and assume that system

F~k(X) ~

(x i) on B.

Relative

as the c o n n e c t i o n

Recall that the c o n s t i t u t i v e

it to

symbols

e q u a t i o n of an e l a s t i c

p o i n t x is of the f o r m

~s (x) : ~ ( ~ x ' where

Ts(X)

t i o n ~.

is the stress

x)

at the p a r t i c l e

We may also w r i t e this

x in the c o n f i g u r a -

in the f o r m

Ts(X) : ~(~(x))

57

where

S is the e l a s t i c r e s p o n s e

Of course, F(x)

if x~U

ordinate

system

~ = {(U s ,r ~e ) , ~I].

, where

e-i = ¢~'~x o r~x

function relative

In c o m p o n e n t

(x 1), we may

form,

to %.

then

relative

to the co-

drop the S s u b s c r i p t

on T and

write T}(x) 3 Note

The m a p p i n g

~(x)

is e a s i l y coordinate Now, (refer to

= S~([FP(x)]) 3 q ~:

U

+

e

E(B,U

: {F~(x) ~ 3 Z-~x

(II-23)

e

) given

via

, j : 1,2,3},

e

seen to d e f i n e a c r o s s - s e c t i o n neighborhood in c o m p o n e n t

U

xsU

in E(B,~)

above the

e

f o r m the C a u c h y e q u a t i o n s

of m o t i o n

§2)

div TS + pb_ : ~8 assume the f o r m ~T~ 3. + pb i

= p~i

(II-24)

~x ] w h e r e the b i are the c o m p o n e n t s in the

configuration

the a c c e l e r a t i o n From

(11-23),

¢(B)

of the e x t e r n a l b o d y force ..i

and the x

are the c o m p o n e n t s

f i e l d in some m o t i o n of B, say,

¢(t).

(II-24), we e a s i l y o b t a i n

Hil

~Fk " ..i - + pb z : px jk ~x 3

where

of

j k [FP]) q

- ~S /~F

is the g r a d i e n t

(11-25)

of ~S.

Now the

58

response

function

S satisfies S!([FP]) 3 q

If in

(II-26)

group which

passes

(11-26)

= S(FG),

~GeG(@),

so

= S!([FPGr]) ] rq

we let G(t)cG(¢) ~ through

at t = O, then G!(O) ] tion of

S(F)

(II-26)

be a curve

the identity

~ H~eg(¢) ] ~ ~

in the

element

isotropy of the group

and s t r a i g h t f o r w a r d

differentia-

yields

H i I ( [ F P ] ) F k H ~ : O, VFEGL(3) jk q r ± after

setting

E(B,~)

over

t : O.

As

[F}] ]

U mx we may apply

(II-27)

defines

a cross-section

the condition

in

of T h e o r e m

II-3

~

with - I

replaced

symbols

by F,~ i.e.,

of a material

H I 5 {F

the

connection

(x)[ xml x + F m(X)F

F~k(X)j

are the

connection

H on B iff the matrices

], m

1,2,3} A

belong

to g(@) ~

(II-27),

at each point

xeU

.

Replacing

H by H in

~

after

setting

m = j, yields

Hi I ~F~ _ Hi I Fk F n jk([F~ ])~x ]" = jk ([Fp])q nj i which,

in turn,

may be used to reduce

(11-27)

(II-25)

To

_ H i l ( [ F P ] ) r k . F ~ + pb i = p~i jk q n] ± which hood

are the equations U .

of motion

It can be shown

(i.e.,

(II-28)

on the coordinate Wang

[7"],

neighbor-

§9) that the

59 above

equations

hood U ~x. symbols

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

Of course,

Fi jk as well

time-dependent.

as ~ = $(t)

the local

To r e n d e r

which

a global

coordinate

system

for the F ijk and the F pq (from

F~k(~)

=

the c o n n e c t i o n F pq are

explicit

we

K: B ÷ R 3

(XA) on B.

in terms

of the form x i = xi(X,t).

neighbor-

gradients

configuration

%(t) is then r e p r e s e n t a b l e

functions

is a motion,

this t i m e - d e p e n d e n c e

a fixed r e f e r e n c e

motion

coordinate

deformation

may i n t r o d u c e induces

of the

The

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

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

laws

(X a) + (xi)) are given by

A

x-W x x--W x

-

K

FP(x)

= F D ~x p

q

q~x-~

K

where

F and F are the r e p r e s e n t a t i o n

coordinate

s y s t e m and x = x(X,t).

relations

above

into

HiJkI ( ~2xk

SX B

KA

(11-28)

of F and F in the

If we

substitute

(X A)

the

we get

~D ~x k ~X C )F

+ pb I" = px..i

(II-20)

8xD~xB ~x ~ - FDe ~XA 3x--'-j- 1 where

Hil jk = H ~ (IFK~ x~D

homogeneous, material

the E u c l i d e a n

connection

(X A) c o o r d i n a t e then have follows

KA FBC

that

7).

special

connection

for an a p p r o p r i a t e

system

(X D)

In the

induced

reduces

B is

in K(B) will be a choice

of ~.

by this p a r t i c u l a r

K ~ 0 and F = I (identity

(11-29)

case where

to the usual

map).

In the

K we w o u l d It then

equations

of m o t i o n

60 for homogeneous

elastic bodies,

iD ~2xk HJ k ~

i.e., to

~XA pb i p~i ~x j + =

(II-30)

where H iD = H iD ([x~A]) jk jk

Remarks

While the equations

(11-29) appear to be of a local

nature only, i.e., valid on the coordinate neighborhood U cB, Wang

([ 7 ] §12) has shown that it is possible to re-

write them in the global form

~iD( ~2xk jk ~xD~x B


where the ~iD ([xnA]) jk

=

(II-31)

Hil( [xnk < F K] )~D. jk q

In terms of the components of the Piola - Kirchhoff
~k = JT I ~X A j= det[xi, A] ~x I ' the Cauchy equations of motion can be cast in the form

sTA + XA

..k

p
p<x

(II-32)

=

where P< denotes the mass density in <(B).

If, following

81 Wmng

([ 7 1, §12) we introduce the response

function A which

is defined for all FeGL(3) via .-i Ai([FP]) 3 q

- det[FP]sk([Fr])Fk q 3

then it is possible to show that

(II-32) leads to the system

of equations

~DF ~2xk ik (~xD~x F

KA ~x k ) 7D~A i ..i FFD ~X--~ - AiTDA + pKb : OKX

(II-33)

where K

BDF(ExlA])ik - ( - - ~ - ~ - ) S i~sI~D~F k~s~l det F sl s k Bjk -: ~Aj/~F I

1 K

i

S

det F and K

KA

_

~BC ~ FBC are the components ordinate

KA FCB of the torsion tensor of H in the co-

system (xA).

If B is a homogeneous

elastic body

then the system (11-33) may be reduced to

BDF z2xk i ..i ik sxDsx F + p
reference

configuration

62

(i.e., if ~ = (B,r) is a global reference atlas for B KA then rp = ~ p for all peB). In this case FBc(XD) ~ 0 and ~DF ik is easily seen to be independent of ( ~ ) .

63

Chapter I.

III.

Generalized

Elastic

Bodies

Introduction In the p r e v i o u s

the t h e o r y which

c h a p t e r we p r e s e n t e d

of c o n t i n u o u s

originated

our results,

distributions

in the works

thus

far,

of i n h o m o g e n e i t i e s

class

elastic

elastic

bodies;

considered simple

bodies w h i c h

are n o n - s i m p l e ,

with

be p r e s e n t e d [32] who,

here

elastic

i.e.,

bodies

in Chapter

are the I as well as

consisting

of

quasi-elastic

particles,

and

state variables.

The concepts

to

are b a s e d on a 1969 p a p e r of C. C. Wang

theory

by recent work on the

of simple m a t e r i a l s

of the g e o m e t r i c

to c e r t a i n

w h i c h may be

bodies

in turn, was m o t i v a t e d

thermodynamieal extension

internal

In this

of m a t e r i a l s

discussed

particles,

bodies.

w h i c h we shall call g e n e r a l i z e d

the kinds

materials

elastic

materials

chapter

among

distributions

so as to cover a much w i d e r

as being g e n e r a l i z e d

elastic

oriented

materials

[ 7 ];

that our d i f f e r e n t i a l -

ideas may be e x t e n d e d

of p h y s i c a l

[ 6 ] and Wang

apply only to static

in simple

of

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

of Noll

chapter we want to d e m o n s t r a t e geometric

a formulation

classes

theory

presented

of n o n - s i m p l e

to c o n s i d e r

an

in the p r e v i o u s

and n o n - u n i f o r m

materials. 2.

Index

Sets and G e n e r a l i z e d

Let B be a m a t e r i a l dimensional

body,

differentiable

Elastic i.e.,

manifold,

Bodies

an o r i e n t e d

three-

whose points we consider

64

to be p a r t i c l e s , scalar m e a s u r e body.

and w h i c h

that we call the mass

Let @(t) be a m o t i o n

that p is t e r m e d a tensor

a simple

function

More generally, is a simple

elastic

particle

parameters gradient

equation Ts(P,t)

to deal with

the heat

neglect

these

stitutive

theory

of simple

(elastic)

p~B at time t>O is d e t e r m i n e d

at p such as the t e m p e r a t u r e at p at time t.

= E (k (t) ~p ~p

bodies not

thermo-

and the

We then have

@(p,t),

as c o n s t i t u t i v e

a con-

for the

from the t h e r m o d y n a m i c a l

extend

our domain

of i n t e r e s t

it is n a t r a l of peB"

equations

and deal

stress

(III-l)

g(p,t)) (4)

and the entropy;

other q u a n t i t i e s

(4) 8 r e p r e s e n t s

of psB then p

k (t) of p but also by various ~p

flux,

equation

configuration

exists

~ E ~ p (k ~ p (t))

stract

particles,

if there

of the form

as well

energy,

Then we recall

if

= E(k ~ ~ p (t),p)

only by local motions

stitutive

particle

of the

: E(~,p(t),p).

at a p a r t i c l e

temperature

of B and let peB.

elastic

in the t h e r m o d y n a m i c a l

dynamical

distribution

if k (t) is any local m o t i o n ~p

TS(P,t)

the stress

with a n o n - n e g a t i v e

E such that

Ts(P,t)

Now,

is e n d o w e d

as in [32] we shall solely w i t h the con-

tensor.

situated

for the i n t e r n a l

In order to aballuded

to,

and

from simple to n o n - s i m p l e

to e x t e n d our d e f i n i t i o n

from simply

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

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

elastic

of "local preserving

and g the t e m p e r a t u r e

gradient

65

isomorphism where

the

particle besides

of Bp with @i(p)

i.e.,

are certain

p (examples

of what

these

is then a o n e - p a r a m e t e r

of all sets sociated

@l(P,t),...,

{@l(p),...,

8n(P,t)}.

with psB the index

[32],

we will

takes

the form

denote

associated

with the be,

above,

will

A local motion

of local We will

of physical

set of p and,

@n(P)}

might

mentioned

section). family

@n(p)}

@l(p),...,

parameters

quantities

at the end of this

{kp(t),

{kp,

parameters

the t h e r m o d y n a m i c a l

be p r e s e n t e d of pEB

R 3 to a set

configurations, call the class

parameters following

as-

Wang

set by N(~ ) If D denotes the class P P of all o r i e n t a t i o n - p r e s e r v i n g maps of B with R 3, then the P constitutive equation of a s e n e r a l i z e d elastic particle p

Ts(P,t) ~

where

kp:

fixed tsR

this

= Ep(kp(t),

R + ÷ Dp and ep:

(III-2)

@ (t))

~p

R+

÷ Np.

In other words,

for each

+

, E : D × N + L (R3,R3), where L (R3,R 3) denotes ~p p p s s the set set of all symmetrie linear maps of R 3 onto R 3. Remark finite

We will require

here

that the index

dimensional

set N

form a P say,

d i f f e r e n t i a b l e m a n i f o l d with, + dim N = n; for each teR the domain of E (the response p ~P function of pEB) is then the finite d i m e n s i o n a l product manix N . If we set N(B) ~ ~.)~ N then the map @ P P Ps~ P ~P introduced above gives rise to a c r o s s - s e c t i o n ~@(to ) of this

fold D

index bundle

N(B) via the map

(5) we have also used Np to denote a neighborhood of pgB but no confusion should arise here as the meaning will be clear from the context.

66

[@(t we will

o

call

)](p)

~ @ (t); p o

@ (') the index ~p

the index section. the @i(p)

general, further

be d e t e r m i n e d

the stress

tensor,

tensor

indicates, elastic

values

the index

equations,

case where

parameters

associated

that

geometric

ourselves

to r e s p o n s e

in

alone a n d

laid down for

particle

however,

structure

to this

cannot,

We note also that in

elastic

need not be symmetric;

the m a t e r i a l

@p(.)

principles

beyond

the o r i e n t e d

@(t ): B ÷ N ( B ) ~ o

to in the

function

by m e c h a n i c a l

body is i n s e n s i t i v e

restrict

alluded

must be introduced.

certain cases, i.e., stress

and

thermodynamical

pEB,

constitutive

(III-3)

function

As a l r e a d y

are certain

with the p a r t i c l e

Vp~B

p, the

as Wang

[32]

of the g e n e r a l i z e d

fact and thus we will

functions

E~p w h i c h

take t h e i r

in L (R3,R3). s

Some examples in order;

of specific

types

of index

the most

important

ones

is either P (@l(p) ,. .., 8n(p))

a trivial

set c o n s i s t i n g

(i)

N

constant

value.

the stress elastic

tensor

particle

or the index

In both cases is t r i v i a l l y

sets are now

are

function

@p

of just one point R+

:

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

÷ NP has a equation

seen to be that of a s i m p l e

p, i.e., we may define

E~p(k~p(t)) ~ E~p(k~p(t),

81( p),...,

@n(P))

for any k : R + ÷ D • ~p P (ii)

p is a simple

for

thermoelastic

particle,

i.e.,

67

@ (t) = (@(p,t), ~p spectively,

g(p,t)) ~

where

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

@(p,t)

and t e m p e r a t u r e

local m o t i o n k (t) of p, at time t. ~P for each pEB. (iii)

p is a q u a s i - e l a s t i c

e f f e c t of the p a r t i c l e dependence

and g(p,t) ~

Thus,

are , re-

gradient, N

in some

~ R+ × R 3 p

particle,

i.e.,

the

"memory"

is r e p r e s e n t e d by an e x p l i c i t

of E ~p on t; the i n d e x set h e r e

is N p ~ R+× R 3 × R +

and s p e c i f i c a l l y we h a v e @p(t)

= (@(p,t),

g(p,t),

t)

Q u a s i - e l a s t i c r e s p o n s e w i l l be of c o n s i d e r a b l e us in the next c h a p t e r w h e r e we d i s c u s s

i n t e r e s t to

anelastic behavior

and d i s l o c a t i o n motion. (iv)

p is

more

general

Cartesian over

an oriented cases

product

R3 "

The

value

particle,

can of

be

considered

R3 w i t h of

i.e.,

itself

@ (t ) at p o

for in and

any

teR +,

which

is P some tensor

fixed

t

o

N

is

R3

Op(t) the spaces

called

the

d i r e c t o r of p at t • o (v) i.e.,

p is a simple N

elastic particle with internal

~ R +× R 3 × R n so that P ~P0(t) = (0(p,t),

where

the @~(p,t)

@ is of the f o r m ~P

~g(p't)' @l(P,t),...

are t e r m e d

variables,

@n(P,t))

i n t e r n a l variables.

68

3.

Local M a t e r i a l

the Phase

Automorphisms,Phase

Isotropy

simple

elastic

isomorphism structures elastic

which

involved

particles,

gives

however,

in C h a p t e r

this

I.

where we dealt

it is the concept

elastic

than that which was

encountered

chapter,

rise to the m a t e r i a l

on g e n e r a l i z e d

particles

Points,

Group.

As was the case in the previous with

and T r a n s i t i o n

bodies;

concept

appropriate

of m a t e r i a l

geometric

for g e n e r a l i z e d

is slightly

more

for the s i t u a t i o n

We shall begin with a series

of

four definitions: Definition

III-i

Let

A local m a t e r i a l m o r p h i s m A:

B

~ E

automorphism

÷ 6 p

(<,e)

(K,e)

Definition

Definition p at

then we will

in ~ P

(111-4)

The c o l l e c t i o n automorphisms

group of p at

N(~o,eo).

G p (Ko,e o) c o n s i s t i n g ~ of p at

isotropy

group Gp(Ko,e o) of

on some n e i g h b o r h o o d

call

a phase

x N . P

exists

the

(~o,0o).

If the local

(~o,eo)

of

(K ,8) is called ~o

is i n v a r i a n t

such n e i g h b o r h o o d point

(KoA,8)

in some n e i g h b o r h o o d

111-3

(Ko,eo)

x N . p

(K ,e ) is an auto~o o

~p ~ ~

!II-2

isotropy

p

such that

~ E

all local m a t e r i a l local

of p at

in D

P

~p ~

for all

(
point

we say that

N(~o,e o)

in Dp x Np; if no

(Ko,eo)

is a t r a n s i t i o n

69

Definition

III-4

Dp x Np have equivalence a phase

class

if psB and p in

induced

(K~o,@o)

by this

(~o,8o)

to the

last

is any phase

simple open

Using

definitions

exercise

definition

s Pp. given

above,

in Np x N p,

point

llI-i through

to show that

An

say P , is called P group of Pp, is

(K~o,@o)

-- {(K,@) s D x N I (K,@) is a phase ~ p P Dp x Np and Gp(~,@)~ = Gp(K~o,@o )}

Remarks

(~l'~eI ) in

relation,

to be Gp(K~o,~@o ) for any

according

and

if Gp(Ko,8~ o) = Gp(KI,81).

G(Pp),the phase isotropy

of p and

Thus,

points

the same phase

then defined

P

Two phase

any phase

point

III-4

it is a

of psB must be an P is closed under the action

set in D

P

x N and that P P P P of the isomorphisms in G(Pp ,) i.e. , if

(~~o,8o) s Pp and

A ~ G(P

[32] notes

~

) then p

(K oA,0) ~o

turn out that

e P •

~

for an arbitrary

psB all points

(K,8)

E D

~

type

of t r a n s i t i o n

a multiple belongs phases that

As Wang

point;

it may

p

× N p

point this

generalized

elastic

particle

are t r a n s i t i o n

points.

One

can be singled

out is called

P

which

is a t r a n s i t i o n

point

([o,eo)

which

to ~pl n...n~pN where pl . pN are N n o n - e m p t y P P p''" , p of p. For a m u l t i p l e point (~o,@o) it can be shown

G (K ,8 o ) c p ~o The physical

says that

G(p I ) n...n G(pN). p interpretation

an a u t o m o r p h i s m

A: ~

automorphism figurations

of p at K ~O

(111-4)

+ B p

~

o

) ~ D

is clear:

it

is a local material P

x N if the local conp p and K oA can not be d i s t i n g u i s h e d (via ~O

(~o,e

B

p of

70

measurement

of the stress

superimposed

on

K

at p) in any small d e f o r m a t i o n

and any small

o

change of the i n d e x

Note that the c o n d i t i o n w h i c h w o u l d define automorphism,

say A

, of p in the p r e s e n t

@ . o

a material s i t u a t i o n is

that

E (K,@) ~p ~ for all

(<,8)

- E (
e D

~

p

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

(111-5)

(and not just

(111-5)

in

of a fixed point

(~o,@

o

) e N

p

× N ). In p

c o n f o r m w i t h that

introduced

II, we shall d e n o t e the group of all m a t e r i a l

automorphisms if

(<,@) ~

o r d e r to have our n o t a t i o n here in C h a p t e r

for all

P

of p by g(p);

if the index is fixed,

i.e.,

is r e q u i r e d to h o l d for all KeD

and a g i v e n P s i m i l a r to the group g(p) intro-

@sN

then g(p) is f o r m a l l y P d u c e d in the p r e v i o u s c h a p t e r w i t h the e x c e p t i o n that now varies,

in g e n e r a l , w i t h the i n d e x 8.

Exercise

7.

particle

then

Show that

if peB is a g e n e r a l i z e d

~(p) c

~7 (K,@)

boundary

8

x~N p

P

If p has a n o n - e m p t y p h a s e

~P

of P P

in D P

× N P

elastic

Gp(<,0) @ D

~

Exercise

must

P

P consist

prove that the soley of t r a n s i -

P

t i o n points

4.

it

Material

I s o m o r p h i s m and G e n e r a l i z e d E l a s t i c i t y

We b e g i n w i t h the f o l l o w i n g

71 Definition particles.

III-5

Then p and q are c a l l e d m a t e r i a l l y

if t h e s e exists r(p,q):

Let p , q s B be two g e n e r a l i z e d

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

preserving

Bp ÷ Bq and a d i f f e o m o r p h i s m

elastic isomorphic

isomorphism

~i ( p , q ) :

Np ÷ Nq such

that Ep(~,O) ~ E q ( ~ O ~ ( p , q ) - l , for all

Ei(p,q)](@))

(11I-6)

(<,O) e D p x N p"

U s i n g the d e f i n i t i o n

above we can now d e f i n e

I: Dp x Np ~ ~q × Nq via I(~,8) we call ~I a m a t e r i a l

= (~or(p,q) -I,

a map

[i(p,q)](8));

i s o m o r p h i s m of p and q if E~ p

= E~ q oI~

on D

× N • By u s i n g the d e f i n i t i o n of local m a t e r i a l P P a u t o m o r p h i s m it is a r e l a t i v e l y simple m a t t e r to p r o v e the f o l l o w i n g exits

two r e s u l t s

a m o n g the

a m o n g the p h a s e T h e o r e m Ill-I

c o n c e r n i n g the r e l a t i o n s h i p w h i c h

local i s o t r o p y isotropy groups

Gp(~,8)~ = r ( p , q ) - i

o Gq(I(K,8))~ ~

A point

(<,8)

( t r a n s i t i o n point)

iff

( t r a n s i t i o n point) w h e r e i s o m o r p h i s m of p,qeB. : I(P q

) c D P

e N

G(P

p

× N p

e D

I: D

x N

~

Also,

P P

iso-

p,qeB then (111-7)

is a p h a s e p o i n t P

q

P c D

x N

q

+ N

is a p h a s e p o i n t × N

q × N

is a m a t e r i a l q

is a p h a s e of p P P P is a p h a s e of q and

× N q

0 r(p,q)

I(<,@)

~

and

on the o t h e r

elastic particles

~

iff P

on the one hand,

If I: Dp × Np ÷ D q × N q is a m a t e r i a l

m o r p h i s m of the g e n e r a l i z e d

T h e o r e m III-2

groups,

q

) = r(p,q) -I o G(P ) o r(p,q) ~ q ~

(111-8)

72

ized

If each

pair

of p a r t i c l e s

elastic

body

B are m a t e r i a l l y

of D e f i n i t i o n form

III-5

generalized

then

elastic

M(B)

and

define

via

:

a phase

p(B)

where

U

(¢~p(t), ,, As single

in [32]

elastic

body

Now,

B is a m a t e r i a l l y

such

a b o d y we

uni-

set

P

c M(B)

and P in the above u n i o n are r e l a t e d P q I a m a t e r i a l i s o m o r p h i s m of p and q. ~

A motion

¢(t):

we

~t ,

shall

B ÷ R 3 is said

local

~

R+

to lie

in

motion

and all p~B.

now restrict

P(B) of the m a t e r i a l l y

phase

sense

P

~ P(B) ,

@ (t))

in the

P

P(B) if the i n d u c e d ~p

isomorphic

general-

× N P

P

peB

P p = ~I(P q ) ' w i t h

the p h a s e

~

for

to the

of B via

=

III-6

belonging

say that

body;

t] psB

any two p h a s e s

Definition

we

p,q

our a t t e n t i o n

uniform

to a

generalized

B.

let

P

be the

phase

of psB w h i c h

belongs

to P(B).

P Then,

following

Wang

[32]

we

can w r i t e

P

as the

disjoint

P union

P

=

~

@sV P

P

where

V

each

P 8sV

is open

in N

W P

and w h e r e

P We r e f e r to

. P (8) the d e f o r m a t i o n

W (e) P

V

P range

(8) is o p e n in N for P P as the i n d e x r a n g e and call at

W

8 for the

phase

P . P

73 Definition

111-7

A diffeomorphism

:~p: Vp ÷ Vp is called

an index a u t o m o r p h i s m of psB in the p h a s e

P

if for all P

a P

(K,e)

P

•

E~p(K,8)~ ~ ~Ep(~'~ ~p(8))) Definition

III-8

(III-9)

The c o l l e c t i o n

morphisms

I ( P ) of all i n d e x autoP P is c a l l e d the index i s o t r o p y group of

Remark

III-5 we p r e s e n t e d one c o n c e p t of

: V ÷ V ~P P P psB in the phase P . P In D e f i n i t i o n

material

isomorphism which

b e l o n g i n g to a g e n e r a l i z e d a material

~

i(~,@)

i(p,q): ~

÷ P

p

V

+ V p

more restrictive

of the

We may also d e f i n e

form

[i(p,q)]@),

and the c o n d i t i o n q

E

= E oi is r e q u i r e d

~P

P

only;

~q

such a d e f i n i t i o n

P t h a n the one p r e s e n t e d

If I is a m a t e r i a l

law

q

= (~o~(p,q)-l,

to h o l d on the p h a s e

phase

e l a s t i c body.

i s o m o r p h i s m of p , q s B w i t h r e s p e c t to P(B) as an

i s o m o r p h i s m I: P

where

is a s s o c i a t e d w i t h p a r t i c l e s

~

is c l e a r l y

in D e f i n i t i o n

i s o m o r p h i s m of p , q s B w i t h r e s p e c t

II!-5. to the

P(B) then it is easy to see that the t r a n s f o r m a t i o n

(111-8)

moreover,

for the p h a s e

istropy

group

the i n d e x i s o t r o p y groups

is still v a l i d and,

transform

a c c o r d i n g to

the rule I(P

) = i(p,q) -I o I(P P

q

) o i(p,q) ~

(III-i0)

74

Exercise

9

isotropy

group w h i c h

Remark

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

R e c a l l that

isotropy

the

in C h a p t e r

and index

1 we r e q u i r e d

with the r e f e r e n c e

In the p r e s e n t

both the phase

situation

isotropy

i d e n t i t y map of V

5.

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

The M a t e r i a l

groups

space

Definition

fibre

111-9

Let peB

a differentiable

manifold

atlas

~, be a that

be Lie groups.

the concepts

space;

to this

(a g e n e r a l i z e d Y which

of index

end we make

elastic

body);

is d i f f e o m o r p h i c

According

to d e f i n i t i o n

a diffeomorphism

called

defined ~q

~p:

uniform

o i(p,q): ~

exists,

for each

Vp ÷ Y~ such a d i f f e o m o r p h i s m configuration

generalized

via the r e m a r k

fibre

Y is an index fibre

to P(B) iff there

an index r e f e r e n c e

materially

111-9,

elastic

following

of p. body,

Definition

V ~ Y is an index r e f e r e n c e p

p if ~q is an index r e f e r e n c e

then,

to the

V

space of B with r e s p e e t p~B,

As

element

of any p a r t i c l e pEB is called an index P space of B w i t h r e s p e c t to the given phase P(B).

Remarks

the

- Index Atlas.

and m a t e r i a l

index range

G(~),

of I(P ) it P group I(P ) acts as P group on V . P

We want to begin by i n t r o d u c i n g fibre

that

we will r e q u i r e

is the i d e n t i t y P is easy to see that the index i s o t r o p y an (effective)

for the index

is given by Iii-I0.

group a s s o c i a t e d

Lie group.

rule

If B is a and I is

III-8,

then

configuration

configuration

is

of q.

If

of

75 ~p:

Vp ÷ Y is an index reference Ip ~ ~p o I(Pp)

is called the relative

o n~l:

configuration

then

y + y

index isotropy

group of p (with re-

spect to n ). Clearly, I acts as an effective ~P p formation group on the index fibre space. Definition

III-10

Any diffeomorphism configuration

Let psB

(a generalized

Lie trans-

elastic body).

~

of p and R 3, the diffeomorphic

copy of B P

for any psB, Remark

is

As in Chapter

an e f f e c t i v e

the relative to

is called the material

fibre space.

I,

Lie transformation

phase i s o t r o p y

group on R3; we c a l l

group o f p w i t h

G P

respect

~p L e t ~p,

(local)

~p be an i n d e x r e f e r e n c e

reference

we may define which,

the concept

in this

Our defining

configuratlon

case,

equation

configuration

o f p~B.

of a relative

and a

As i n C h a p t e r I response

is taken with respect

S of p ~p

to ~p and ~p.

is

Sp([,y) : for all F e GL(3) S : GL(3) P

(IIl-ll) and all y e Y.

× Y ÷ L(R3,R3),we

We note that while

are really

interested

in the

78

values

of S * f o r p

Y : ~p(8),

Exercise

(F,y) ~

Wp(y)

9

: Wp(@)

p

E yUy

Wp(y)

where

for e a c h

o ~i

Show that

S (FK,y) p ~~

-S

S (F,~(y)) p ~

f o r all

s P

F s W (y), ~

(F,y) p ~ - S'{(F,y) p ~

K E G

, yeY and

p

P

We c a n n o w d e f i n e

the

chart

in B w i t h

index

bundles

respect

of

U

~sI

. P

concept

of a m a t e r i a l

to the p h a s e

are d e n o t e d

P(B);

respectively

- index

if the p h a s e by T ( U

) -

~ and

V(U ) =

~,, V

and B P~U~ P

t h e n we h a v e

p s u c~ p

Definition chart

III-ii

in B - w i t h

A triple respect

open

set

(i)

~ : T(U ) ÷ U

•~

(B) p

(ii)

(U

,}e,~

) is a m a t e r i a l - i n d e x

to the p h a s e

P(B)

- if U

is an

in B a n d

~ {p}

x R3 i s

a diffeomorphism

such

that

x R 3 for each p~U

~ : V(Ua ) ~

D~~ ( V p)

~ {p}

(iii)

If ~ a , p

Bp a n d

Vp,

Ua

x

x Y for

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

each

and ~ a , p

respectively,

such that

p~U a

denote

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

of ~

and ~a to

a n d we s e t

^ r(p;q) -= ~ ,-i q o ~,p (III-i3) ^

i(p,q)

-i - Be,q

o ~(~,p

77

then

~I:

Pp ÷ Pq,

as

defined

A

III-8

with

o f p and

r ÷ r and

q with

Theorem

III-3

B with

respect

in the

remark

following

Definition

A

i ÷ i, m u s t

respect If

(U

to

the

a material

phase

,~,~ )_

is

phase

P(B)

to the

be

isomorphism

P(B).

a material-index then

S

chart

, taken

with

in

respect

P to

the

~a,p:

reference

Vp ÷ Y, i s

Proof

The

proof

A

r(p,q),

the

material e E ~q As

independent follows

A

that

E up

configurations

i(p,q),

as d e f i n e d

isomorphism

is

o f p.

directly

from

in

I which

o I identically S

~ : B + R 3 and ~~,P P

on

independent

(III-11)

(111-13)

has

the

and

give

the

rise

property

fact

to

that

P . p of p, w h e n

it

is

computed

relative

P to

the

the by

reference

response S

~ S

configurations

function

,

pEU

~a,p

S~ r e l a t i v e _

; the

domain

to the

of

S

P

Following group of

Wang [ 3 2 ] we now d e n o t e

o f the

the

chart

chart

and

Remark

peU Note

P ~

the

relative

index

o

I(P

)

P

(U

~ P

~

relative

o G(P

-- n ~~,P

I

all

is

chart

by Ga and I , r e s p e c t i v e l y ,

=

for

the

and ~ a , p we c a n d e f i n e ,~,~

, Vp~U

G

~ G

phase

isotropy

isotropy

group

where

o

(III-14)

) o n -i ~e,P

and P

•

p

• that

)

I

~ I ~

for p

all

pcU

.

78

The f o l l o w i n g material-index

charts

of the d e f i n i t i o n in C h a p t e r

definition

of the c o m p a t i b i l i t y

in B is the n a t u r a l

of c o m p a t i b i l i t y

I for r e f e r e n c e

charts

of two

generalization

w h i c h was on simple

introduced elastic

bodies,

i.e., we make Definition

III-12

(Ua,~,~a)

and

Two m a t e r i a l - i n d e x

(UB,~,~B),

charts

in B, say,

w i t h U~ n UB ~ @, are called

^ -i c o m p a t i b l e if GaB( p ) ~ ~B,p o ~ a , p e G(P ) a n d ^ -i P laB(P) ~ ~B,p o ~a,p ~ I ( P ) for each p~U a n U B. P It follows at once from this last d e f i n i t i o n S

= S , G

: G 8 and

which prevailed (i.e.,

is false.

elastic

= IB; h o w e v e r ,

unlike

for the case of simple

see the r e m a r k

statement simple

I

on page

11-15)

that

the s i t u a t i o n

elastic

particles

the c o n v e r s e

of this

By analogy with the p r e s e n t a t i o n

particles,

we call the

fields

defined

,

the

coordinate

chart

transformations

of the d e f i n i t i o n

above, and

(lll-4),we

laB( p )

Ia ~

~ (~)

~

'

chart

(UB,~g,~8) given

see that GaB( p ) ~ Ga ~ GB and

I B for

all p~Ua

GaB and l a b

are

n U B.

Also,

because

the c o o r d i n a t e

smooth fields

We are now in a p o s i t i o n atlas

a

of c o m p a t i b i l i t y

and ~a(n B) are d i f f e o m o r p h i s m s

formations

via

from the m a t e r i a l - i n d e x

(Ua,~a,~ a) to the m a t e r i a l - i n d e x

and by virtue

for

to define

trans-

on Ua n UB a material-index

for B w h i c h we take to be a c o l l e c t i o n

79

T

= {(Ua,~a,~a)

charts

, eel}

in B, where

we also require

of m u t a l l y

{U

, ael}

compatible

is an open

that T be maximal

conditions.

If B can be equipped

then we will

call

elastic

that

with respect with

of B;

to these

a material-index

materially

as all the m a t e r i a l - i n d e x

atlas

are m u t u a l l y

the relative

(U , ~ , ~ define

covering

uniform

two atlas

generalized

body.

Now, index

B a smooth

material-index

compatible,

response

function

by ~T ~ ~e for all esl;

in a m a t e r i a l -

it should

function

) e T, is i n d e p e n d e n t a response

charts

S

with respect

of ~.

We can, therefore,

~T with respect the domain

be clear

to the atlas

of ~T is denoted

by

~

PT ~ P~'

~aEl.

In a similar

fashion

the relative

phase

~

isotropy

group

G T and the relative

I T are defined spectively;

by G T = G

they

are,

for the response

Remark

It is important

K:

re-

and index

isotropy

Wang

In fact

[32],

atlases

suppose

§2) that T and

that

~ T

Y ÷ ~ Y are~

~:

(FK,.~(y))

diffeomorphisms

¢ PT a n d

~

~

([,y)

e PT"

on B but,

with

K ~ GT

= ST(F,y)

ST(FK,~(y))

for

~

Then T z {(Ua,~a,n

~ {(Us, KoCh, ~~-l°n ,~ ) atlases

(i.e.,

to have two m a t e r i a l - i n d e x

R 3 ÷ R 3 and

for all

group

~T

to note

~

¢ I T but

the phase

function

with T ~ T and yet ~T ~ S-. ~

isotropy

and I T = I , ~ I

of course,

groups

it is possible

~EI

index

~el}

by virtue

) , a~l]

are distinct

and

material-index

of the d e f i n i t i o n

of S , for ~P

80

any peU

(i.e.,

ST(F,y) to-one

(III-Ii)),

= ST(FK,~(y))

: ST(F,y).

correspondence

and t h e i r r e l a t i v e

it is c l e a r that

between

response

Thus, we lose (6) the one-

(material-index)

functions

atlases

T

~T w h i c h p r e v a i l e d ~

for the a n a l o g o u s

atlases

and r e s p o n s e

functions

on simple

e l a s t i c bodies. Exercise

i0

If L: R 3 ÷ R 3 and I: Y ÷ Y are a r b i t r a r y A

diffeomorphisms,

show that T : {(C

a material-index

atlas

is.

for B w h e n e v e r

S h o w also that the f o l l o w i n g

for the r e s p o n s e groups

relative

([L,~(y))

,)L o ~ , l ~- l~o n ~ T : {(U

~I} ,~,~

transformation

is

), aeI}

laws h o l d

f u n c t i o n s , t h e i r d o m a i n s , and t h e i r i s o t r o p y ^ to T and T: if ([,y) a PTA then

s PT and

S}(F,y)

-= S T ( F L , k ( y ) )

(III-i5)

^ = ~l-lolTO ~l IT Finally,

if K and $ are d i f f e o m o r p h i s m s

spectively, remark

w h i c h have the p r o p e r t y

above,

of R 3 and Y, re-

delineated

in the

then

KGTK-I

tO TO

= GT

-1

(III-16)

= IT

(6) this correspondence can be assumed valid when we derive the field equations of motion~as the exception noted in the remark above does not come into play.

81

6.

Material

Tangent

and Index Atlases,

In Chapter material

Homo@eneity

material-index

however,

atlas

and by d e f i n i t i o n for the m a t e r i a l V(B,T), {U

forms

atlas

e {(U

there

~(T)

,~e)

bundle

cover

(ii) of D e f i n i t i o n

(ii') the

coordinate

transformations

following

Definition

111-12)

and,

with

in so doing,

atlases

R e c a l l now that

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

satisfying

III-II G 8 and I B (defined

are smooth

fields w h i c h

take

IT , r e s p e c t i v e l y .

maximize

respect

to T via

for B and that

(i) and

in G T and

both a m a t e r i a l

and the index bundle

to T.

conditions

and ~ ( T )

and ~

is a

(III-17)

T(B,T)

with respect

an open

), ~ I ]

sets now form the p r e b u n d l e

tangent

therefore,

,~,~

, ~eI}

~

We can,

If B is a general-

relative

(i ~) for each ~EI,

their values

%.

for B then we can define

respectively,

, ~sl}

to deal only with the

and T ={(U

and an index atlas

~'(T)

the M a t e r i a l

T(B,~), of a simple e l a s t i c body

bundle

body,

Bundles;

and Local Homogeneity.

to a given r e f e r e n c e

ized e l a s t i c

atlas ~ ( T )

and Index

I it was n e c e s s a r y

tangent

B, r e l a t i v e

Bundles

the p r e b u n d l e

to c o n d i t i o n s

atlases

(i ~) and

we are led to the f o l l o w i n g

~(T)

(ii ~) above

82

Definition

obtained

III-13

The unique

C(T)

= {(U

,~ ), acJ}

n(T)

: {(U

,n

by m a x i m i z i n g

conditions

(i') and

T(B,T) ~

V(B,T),

and

group

of T(B,T)

Y and

IT. ii

and n'(T),

(ii') above

w i t h respect

are the bundle

respectively.

are

sets

e~K}

),

C'(T)

The fibre

R 3 and G T while

A

Exercise

maximal

to

atlases

for

space and s t r u c t u r e

those of

V(B,T)

are

A

If ~'(T)

and ~'(T)

atlas

and the index atlas

atlas

T, which

is d e f i n e d

relative

respectively,

atlases

the m a t e r i a l

to the m a t e r i a l - i n d e x

in the s t a t e m e n t A

show that the bundle

are,

A

A

of exercise

A

~(T) and n(T)

i0,

A

of T(B,T)

and

of T(B,T)

and

A

V(B,T), V(B,T)

respectively,

are r e l a t e d

to those

via

A

A

~(~)

: {(%,

no~ ), ~ J } (III-18)

A A

n(T)

= {(U

,

l_lo q

As was the case for simple define,

in an a n a l o g o u s

and locally h o m o g e n e o u s we say that

manner,

chart,

~: B ÷ R 3 such that

elastic

say

elastic

if there

(B,~,~),

~p = ~,p,

bodies,

the notions

generalized

B is h o m o g e n e o u s

material-index

), ~eK}

exists

we can

of h o m o g e n e o u s bodies,

i.e.,

a global

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

VpsB.

Similarly,

B is said

83

to be locally homogeneous index chart,

say

such that psU Definition

exists

a (local)

(U ,~~ ,~~ ) and a configuration

and }ap =

111-14.

homogeneous

if there

VpeU

The material

if the bundle

atlas

.

similarly,

for each psB there peU

and ~ p

% : U~ ÷ R 3. there

exists

Exercise

12

=

exists

tangent

bundle

~(T) has a global

13

Also,

we say that

a global

chart

V(B,T)

Show that the conditions of T(B,T)

(Wang

[32];

§3).

atlas

V(B,T)

if

for the homogeneity

Prove that

iff both T(B,T)

without relative

of the

T.

(or locally homogeneous).

is homogeneous

is homogeneous

are independent

homogeneous

(7)

if

(B,n) in ~(T). (7)

homogeneous)

index bundle

~: B + R 3

(U ,~ ) ~ ~(T) such that

(or locally

T(B,T)

is

chart

is locally homogeneous

a chart

of the material-index

Exercise

T(B,T)

~e,p for all psUe and some configuration

and local homogeneity choice

T(B,T)

~ : U~ ÷ R 3

We can also make

(B,~) such that ~p = ~,p for some configuration and all peB;

material

B is homogeneous and V(B,T)

are

Can it be true that

the some being true for the to T?

every fibre bundle is (locally) trivial is, a priori, locally homogeneous.

so V(B,T)

84

7.

Material

and Index C o n n e c t i o n s

In C h a p t e r

II we i n t r o d u c e d the concept

c o n n e c t i o n on a simple e l a s t i c body b e i n g a "G" c o n n e c t i o n atlas

for B.

of a m a t e r i a l

B~ this was

d e f i n e d as

E(B,~) w h e r e ~ is some r e f e r e n c e

on

In e q u i v a l e n t

terms,

on B, a simple e l a s t i c body,

a material

is an affine

connection H

c o n n e c t i o n of

T(B,@) w h o s e i n d u c e d p a r a l l e l t r a n s p o r t s of the t a n g e n t spaces on B are m a t e r i a l ports

Pt:

isomorphisms.

E l ( O ) ÷ El(t), w h e r e

X is a smooth

be e x t e n d e d to w e l l d e f i n e d maps selves

induce maps

(11-14) w i t h ~ : U

Pt,a:

~ c U s and

× G(@) + ~-I(u

The p a r a l l e l

^Pt:

curve

trans-

B, may

BX(O) ÷ Bl(t) and them-

R3 ÷ R3 w h i c h

are d e f i n e d via

(U ,~ ) ~ ~(%).

(Recall that

) and that the fibre space of

~

E(B,@), i.e., R 3 c o i n c i d e s w i t h the s t r u c t u r e group G(@)) ~

The c o n d i t i o n that the maps

~t be m a t e r i a l

in the sense of C h a p t e r

is then e q u i v a l e n t

II,

q u i r e m e n t that the t r a n s f o r m a t i o n s

isomorphisms, to the re-

Pt,~ f o r m s m o o t h curves

in G(}) w h i c h pass t h r o u g h the i d e n t i t y e l e m e n t indeed,

this

striction

'

is the c o n d i t i o n w h i c h

in T h e o r e m 11-3

'

led to the

on the c o n n e c t i o n

of H and it is this r e s t r i c t i o n ,

connections

Now, materially

on simple

s t r u c t u r e on the m a t e r i a l

symbols

F jik

characterizes

e l a s t i c bodies.

let T be a m a t e r i a l - i n d e x uniform generalized

stated re-

g i v e n in terms of the

Lie a l g e b r a g(~) a s s o c i a t e d w i t h G(9), w h i c h material

at t = O;

atlas

for B, a s m o o t h

e l a s t i c body.

tangent bundle

B e c a u s e the

T(B,T)

is e s s e n t i a l l y

8B the same

as that

on T(B,~),

and ~ a reference

atlas

we define

a material

on T(B,T)

a condition

Theorem

11-3

a fixed

global

with

B a simple

elastic

on B, it is easy to see that

is applicable coordinate

equivalent

here.

to that

in other words,

system

%: B ÷ R 3 then any chart

if

on T(B) to be a ~'G" connectio~

connection formally

body

on B induced

(U ,~ ) in ~(T)

stated if

in

(x i) is

by

can be c h a r a c t e r i z e d

by ~ and the parallel any arbitrary

i

"

= ~j dxl ~ ~i

transport

point

~(t)

from a reference

on ~ c U

~t = ~ ' ( t ) d l ( o ) X l ~here

i(o)

~t(e)

the chart written

:

has the component

form

l(t)

O,

i,j

: 1,2

o ~t o ~ -i ,~(O)'

~ ~,l(t)

(U ,} ), is an i s o m o r p h i s m

in component

~t(~)

In order

~(O) to

= 6i. and

~i + i ol~k Z Flk j

The map

® ~

point

curve

of R

3

relative

which

to

can be

~ ej~

connection

to be a "G" connection be an integral

taken

form as

= ~(t)ei

for H, w i t h

3

symbols

on T(B,T),

Fijk relative

we require

of the Lie algebra

g(T)

that

to

~t(e)

of GT; this

(xl),

88 then

leads

to the c o n d i t i o n

r

J

{(~-l)i ~{k ] [$X m belong

to g(T)

Thus in C h a p t e r

everything

II; however,

to g e n e r a l i z e d general

picture

can now define

an index

on

T.

connection

~T a s s o c i a t e d induced

bodies

with

by ~T"

from simple

V(B,T)

So, f o l l o w i n g connection

the

Wang

group on

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

infinitesmal

generator

group

atlas

which

§3), we

T.

characterize the Lie a l g e b r a

IT of Y w h i c h

and c o n s i d e r

%u(S) : exp(su),

([32],

consider

Y which

to a

V(B) as a "G" con-

Lie a l g e b r a

let usiT

bodies

into the

relative

V on

conditions

~T and the

elastic

B, we i n t r o d u c e

on V(B) we must

Thus,

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

This

much the same as it was

V(B,T), for any m a t e r i a l - i n d e x

In order to deduce an index

is p r e t t y

the index bundle atlas

m = 1,2,3}

.

as we pass

elastic

material-index

nection

F~m{i

at each psU

far,

that the m a t r i c e s

is

the o n e - p a r a m e t e r

is given by

ssR. can be c h a r a c t e r i z e d

u, i.e.,

by the v e c t o r

by its

field defined

via ~(Y) We note that

d - ~

(III-19)

vector

fields

on

it the

Lie a l g e b r a

[~u (s)](y) defines

Y; we denote

s=O'

VYeY

(III-19)

a map of iT onto a set of this

of Y (induced

set by ~T and we call

by IT ).

The b r a c k e t

opera-

87

tion on this

Lie algebra

the Lie derivative,

is just the usual one induced

i.e.,

Once again we take

if [,~ E ~T then

(x I) as a global

Dn B and let (y6, ~=l,2,..n) in Y. as ~

Also, : U

= L~

coordinate

system

be a local coordinate

system

let (U ,~ ) c ~(T)~ be an index chart.

Then,

x Y ÷

), this

V(U

of the local coordinate a local coordinate each p s U

[u,v]

by

, (xl(p),

diffeomorphism

system

(xm,y 6) ÷ (xl,y~)

system in V ( U

is a local coordinate

Y~)

Now let ~: V(B)

map so that

= Vp,

pEB.

a map

, which

In other words,

).

the fibre over p. -l(p)

induces

+ B denote

Then,

for

system in Vp, the projection

~,(-~)

= O, 6=1,2,..

{~---f}~ are vertical vectors in ~y In terms of a connection V on V(B) and the local

so that the basis vectors V(B).

coordinate

system

(xl,y~)

, horizontal

subspaces

relative

to V are spanned by sets of the form

{~

.

v~(×,y)i-~,

_

~x m

and we call the tensor

V ~

the connection (U ,n ). parallel

=

:

1,2,3}

field

V ~. d x ]" (~ Z ] ~y6

form of V with respect

If I c U transport

determined

i

~yO

m

is a smooth yt(~)

by solving

along

to the index chart

curve then the induced I with respect

the equations

is

to V is

of parallel

transport

88

which,

in this

y

+ V

case,

1

In other words, solution

of

form a smooth

smooth

x,y) [yt(~)]

yt(e)

curve

(y(O)

of R 3

the fibre

of Y, the fibre

in I T ; this

Lie algebra

each t.

A simple

parallel

transport

(II!-20)

where

y(t)

space

in GT, We now require

ing that the infinitesmal of the

= y(t),

n.

So, just as we required

at(e) curve

the form

: O, ~ : 1,2,...,

(111-20).

transformations

formations

assume

space

~T of W, which computation 111-20

d d--£ [Yt (~)](y(O))

that

that the trans-

of

V(B,T),~

of yt(@)

is induced

employing

the

of T(B,T)

is, however, equivalent generator

is any

form a to requir-

be a member by

IT, for

the equations

of

yields

: y(t) :

~6(t)~ ~y-~y(t)

= -V.6(l(t),y(t))~i(t)~

l and leads Theorem (i.e., atlas with

y(t)

us to the following

111-4.

A connection

a "G" connection ~

respect

to the index

system

V on V(B) is an index

on V(B,T)

T) iff the components

coordinate

~y

relative

V~ of the ]

chart

connection

to any m a t e r i a l -

connection

form of V

(U ,H e) e T ( and the local

(xi,y B) on U × Y) satisfy,

89

Vj(p,.)~

Remark

~ yT,

Vp~U

; j=1,2,3

It is a simple matter to prove that the tensor

V (the connection dependent

form of V with respect

of the choice

same sort of invariance

of coordinate

to (U , ~ ) )

systems

field

is in-

(xi,y6)~

does not hold, however,

the

with regard

to a change of the index chart. 8.

Field Equations Following

derivation elastic

Wang

of Motion ([32],

in Generalized

§4) we will now go through

of the field equations

bodies

of motion

B in a single phase,

obtain the equations

for motion

as we did in Chapter

ZI making

however,

of the concepts

Elasticity

say,

use,

for generalized

P(B).

in a global

In order to

form we proceed

in the present

of both material

the

situation,

and index connec-

tions. So, let B be a smooth materially

uniform generalized

elastic body~ we take T as our fixed material-index for B in the fixed phase T are of the form sponse

P(B).

(U ,~e,~

Material-index

atlas

charts

), ~el, and the relative

in

re-

function with respect

possibility response

to T is ~T" As there is no ~~ here of confusion between ~T and the relative

function

of simplicity subscript

S~ of Chapter

in writing,

II, we shall,

for the sake

drop both the asterisk

T on the response

function

and the

~T; the eonstitutive

90

equation

for the

stress

tensor

~S then has the form

~S : ~(['Y) and r e l a t i v e ordinate

to the s t a n d a r d

system

(III-23)

basis

for R 3 and a local

(y6) in V we have the c o m p o n e n t

co-

form

T I] = Sl3(F,y)

where we have Of great

dropped

importance

sults w h i c h

the s u b s c r i p t

to us in what

are o b t a i n e d

in w h i c h we d e r i v e d under the usual

bases

(II-27)

smoothness

ij Skl=

Recall

now

elastic

re-

to the way bodies.

So,

on the functions

the functions

ij Skl and S~3~ as the

of S w i t h respect

to the n a t u r a l

~ ~siJ/zF kl

s~J(F,y)

(III-25)

~ ~siJ/~y 6

that

S(FK,y)

for all KeG T and all sides of

(III-26)

= S(F,y)

S(F,[(y))

both

are several

analogous

for simple

tensor.

Y, i.e.

ij F,y) Skl(-

S ij :

follows

assumptions

of the g r a d i e n t

in R 3 and

on the stress

in a m a n n e r

S 13, we b e g i n by d e f i n i n g components

(III-24)

= S(F,y)

~el T.

(111-267),

If we take the gradient

with respect

to the n a t u r a l

on

91

basis in R 3, we easily obtain

""

ij

(III-27)

S = Skl(F,y )

On the other hand, if we take the gradient on both sides of (111-262) , with respect to the natural basis in Y, we get

s~J([,~(y)) ~ (y) = s~J([,y) ~yV

(III-28)

Now let K = K(t) and 6 = 6(t) be smooth curves in G T and IT ~

which pass through the respective isotropy groups at t = O.

identity elements of these

Then differentiating

(III-26 I) through with respect to t yields

equation

(after setting

t : 0): ij k sl SkI(F,y)FsD = 0

(III-29)

where D = ~K(O) e g~ (T), the Lie algebra of the isotropy group G T.

Finally,

differentiating

equation

(III-262)

~

with respect to t (and then setting t = O) yields

s~J ( F , y ) ~ (y) = 0

(III-30)

d where ~-[ [~(t)yllt=O ~ u(y)~ c ~T' the Lie algebra of Y, which is induced by IT •

(Note that ~(t)y c y is a smooth

curve, for each y~Y, which passes through the element

92 y s V at time t = 0).

Obviously,

(111-29)

and

(III-30)

are valid for all D s ~g(T)~ and u~ E YT' r e s p e c t i v e l y . Now, global

let $ be a c o n f i g u r a t i o n

coordinate

and suppose chart

in T.

determined

that

system (U , ~ , q

via

(111-23)

in c o m p o n e n t

equation (111-31)

: siJ((~-l)l'k

Chapter

+

is

and

As in C h a p t e r

equation

=

(11-24)

of the g r a d i e n t (111-31).

yields

p~i

(111-32)

functions

the e q u a t i o n

is valid e v e r y w h e r e

only.

are the same

As was the case in

is a local one valid U

of m o t i o n for points

To obtain

(11-25), in the

an e q u a t i o n

on B, we will follow the lead

in Chapter

II,

so sub-

~x j

neighborhood

of the d e r i v a t i o n

dropped.

S~ 3 $@@ + pb I

II, when we d e d u c e d

coordinate

at $(x)

(III-31)

is given by

into this

of the S ij in

the above e q u a t i o n

which

has been

~x 3

the a r g u m e n t s

as those

tensor

e6)

..

•

where

material-index

F ~ $*x o }~l(x)

of m o t i o n

ij ~( -l)kl Skl

section,

form

where the ~ s u b s c r i p t

of

stress

the

§($*x o ~[l(x), ~ (e(x))

:

TiJ(%(x))

sitution

e be an index

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

with

induces

i.e.

T ($(x))

the Cauchy

let

If x E U , then the

-i y s qe (8(x)),

or,

(xi),

of B w h i c h

II and make use of the

93 conditions nections

which characterize,

respectively,

on V(B).

on T(B) and index connections

be a fixed material

connection

connection

on V(B); as in

connection

symbols

the coordinate

material

§7, we represent

(x i) and

So, let H

on T(B) and V a fixed index H and V by the

Fjk i and Vi, ~ respectively

systems

con-

(yB).

Since

relative to (III-29) must

hold for each D s g(T) and the matrices

{(~-I)~

[~--~m - rk i ~x im ~k ]' m = 1,2,3}

must belong to g(T) at each point psU be the connection

symbols

we should obtain from

ij [~(%-l)kl Ski ~x ~

of a material

(III-29)

+

i , if the Fjk are to connection

on T(B),

the equation

Fk -l)sl] ({ sm

(III-33)

= 0

(Note that F ~ %~x 0 ~-l(x) and that i

~(~-i)} i ( ~-~)~ ~ J). ~x m 3 - ~k 3x TM

~l

In a similar vein, we note that

(III-30)

must be valid for

all 9SYT and (by virtue of (III-22) vj~(p~-)-~ ZY to ~T

' at each psUe,

of the connection

6 if the vj are to be the components

form of an index connection

Therefore,combining

must belong

(111-30)

on V(B).

and (111-22) we obtain

94

ssj~ ~;B

=

(III-34)

0

Before continuing the present line of argument we will pause to review certain ideas pertinent to the definition of "covariant derivative."

In Chapter I (§6) we defined the

operation of covariant differentiation by working with the linear isomorphisms of tensor spaces T['S(t), over ~ c B, which are induced by the parallel transports along ~ relative to a given connection.

An alternative,

valent approach is the following Let 8: B ÷ V ( B ) (111-31).

but completely equi-

(Wang, [323, §3):

be the fixed index section which appears in

Then we may define the eovariant derivative of

8 with respect to V as the slope of 8 at any point (p, 8(p)) e V ( B )

relative to the horizontal

(relative to V) at (p, @(p)).

subspace

In other words,

the covariant derivative of @ at p~B and veB

if D @ denotes P is an arbitrary

P vector, then [D 8](v) is the vertical component of p 8~p(V) e V(B)(p,8(p)). In order to obtain a component form for D8 we again take (xi,y ~)

as the local coordinate system

in V ( B ) which is induced by the chart coordinate systems

(U ,~ ) e T and the

(x i) in B and (yB) in Y.

of this chapter) that the horizontal subspace

We recall

(§7

(relative to

V) at (p, @(p)) is spanned by the set

(~x'~ (p,@(p))

- V~(p, @ (p))~-~ , i : 1,2,3} ~y (p,@(p)

(III-35)

95

Let veB ~

have the component

8,p([)

(SB) •

space to

(III-35)

V(B)

components,

at (p,e(p))

if

this

(p,e(p)

vector

in the tangent

component

being

and h o r i z o n t a l

given

by

(III-37)

+ V~(p,e(p))] 33y-~ ( p , e ( p ) ) deduce

(veB

is arbitrary)

P

that

(III-38)

+

V~.( •

(e a)

the

0(-)).

Note

z

~x z

are constants

with

that respect

to

.

Exercise

14

Verify

e,p(V)eV(B)(p,e(p)) To continue of the field

that the vertical is given by

with

equations

of motion

.. and S~ 3 in o ~x j

- - i )k l

component

of

(111-37).

our d e r i v a t i o n

we now be substitute

ij Skl ~x 3

we compute

: (DS)~dx I @ 3____ 3y ~

~e6

V(B)

(III-36)

~x 1 p

we now easily

(De) ~. -- e 6 : z li

8 1i = v fz( . , e )

÷

V

up into its vertical

Jp

De

bodies

+

8: B

and

~ x l IP

8,p (v)eV(B) ~ (p,O(p))

we may break

: v i [ 386

(III-37)

(xi,y6)

for

with the vertical

[Dpe](y)

where

Then

= v i ~ "I ~x ± (p,8(p))

using

From

v = vi ~ • I ~

the components

and,

form

P

of the global

for g e n e r a l i z e d

form

elastic

for the expressions (111-32)

by using

(111-33)

96 and

(III-34),

ariant

respectively,

derivative

the following

and the definition

86tj given

system

above;

of equations

of the cov-

in so doing we obtain for the motion

of points

in B: _SkliJ Fk.( - l ) S~ l s 3

Exercise side of chart

15

(III-39)

that the second

is independent

(III-39)

term on the left-hand

of the choice

of the index

(U ,q ) ~ T. Now,

which

Verify

+ S~ 3 @~j + pbl = p~i

just as in Chapter

the functions

a motion

~(t):

in a motion reference chapter,

B ÷ R3

%(t)

where

of B, x = x(t). <: B =>

the problem

of F arose~

by the deformations system

Suppose

in

we choose

a fixed

R 3, as we did in the last

of r e n d e r i n g

then the motion

x i = xi(~,t),

X A on B.

a situation

will be t i m e - d e p e n d e n t in 3 F = F(x), for instance, and

i.e.

configuration

dependence

ordinate

F=kj and

II, we have

explicit is again

where

the timecharacterized

K induces

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

the co-

from

K

£~k(X)=>

F[c(X)

by the rule

of

as we go from §7, Chapter

(x l) + (X A) is still

I, while

f r o m V~(x) ÷ VA(X) and ~ (x) + ~ ( X )

<~ 8X A V~1 = V A ~~ X 1 Substituting

given

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

are given via

i
for the functions

•

F~k(X) V~(x) 3 ' 3

and

i

~j(x)

97 in (III-39)

then yields

ij.~-l)A1

~X B

Skl

.

8x ]

k)

~2xk

__

FAB 8X c

(3xA3x B

(III-40)

3X B• K0 6 + S~ ] ~x 3 IB

+

pb I

=

p~i

Z8 <~ 861B ~ ~ + VA ~X B

K

Remarks

In (III-40),

general,

on both

changing

during the motion;

time evolution

still depends

(XA) and t as the index section the governing

in

e may be

equation

for the

of the index section has not been treated

in [32 ]. K

Exercise

15

Show t h a t

the

in (111-40)

are independent

index chart

(U , ~ , q e )

K.

terms

and S kijl (~-I)AI ~

S~ j 06

IB

of the choice of the material-

in T.

Thus

if we define

•

_l]A ij .<-I.AI ~k - Ski t ~ )

<.. S~ ] K SB] " ' 06 I B

then the equations

of motion

(111-40)

assume the global

form i~A(

Remarks

~2xk KC ~xA----~B - FAB

As in Chapter

3x k

K.

°

°

13 8X B + pb I = p~i ) + S B 3x ]

II we could present here the global

form which the Cauchy equations

of motion assume

for a

(III-41)

98

generalized elastic body if we employ the components of the K

''A

Piola-Kirehoff

stress tensor,

i.e., I~, instead of the

components T} of the Cauchy stress tensor

(the relationship

between ~S and ~S is given on page II-43 and the equations of motion are again those of equation

(11-32)).

While it

is certainly more convenient in some initial-boundary value K problems to employ ~S in place of TS; the equations of motion, in the present situation,

become quite involved; we remark

only that they may be reduced to the global form

~x k )

~iAB ( 32x k

KC

~k

FAB

~X---A-~ xB

(III-42)

K.

-

~iA KB ~l bi ..i TAB + + PK : PK x

~iAB ~iA ~i where the fields ~k , L , and are defined on S; the ~BC are the components of the ssJne torsion tensor which appeared before in equation

(II-33).

For the formal

definitions of the (global) fields appearing in (III-42) the reader may consult Wang

([32],

§4); his notation differs

only slightly from that which we have employed here. The field equations of motion

(III-41) have been

derived under the restrictive assumption that our generalized elastic body B is in a single phase only, say, P(B).

If

the set M(B) of §4 consists of more than one phase then a phase transition may take place during the course of a

99

motion ¢(t): B + R 3.

Some interesting examples of the kinds

of phase transition which can be dealt with within the theory of smooth materially uniform generalized elastic bodies are discussed in §5 of Wang [32]; in addition, the general theory of this chapter enables us to consider the problem of a change of homogeneity in inhomogeneous example,

simple elastic bodies.

For

suppose that B is a simple thermoelastic body which

is undergoing some thermodynamical process and that @. and i Of are the initial and final index sections for B in such a process

(i.e., we are looking at example

(ii) §2).

Suppose

that when we hold the index section fixed at either 0. or i 6f, the body behaves just like the smooth materially uniform elastic bodies of Chapter II, i.e., if p,qsB then exist isomorphisms r.(p,q): ~l

Bp ÷ B q

rf(p,q) : Bp ÷ B q such that for all <: B

P

+ R 3,

Ep(~,@i(p) ) - E q ( • O

{i(p,q)-i ' @i(ql)

(III-43)

and Ep(~, @f(p)) z Eq(~ o ~f(p,q)-l,

@f(q))

(III-44)

If in the neighborhood of any peB we can find smooth local fields ri(P, ") and rf(p,-)

satisfying

respectively , then, provided

(III-43) and (III-44),

@l. and @f may be held fixed

100

we can suppress the dependence the index sections materially

of the response

and regard B, in each case,

functions

on

as a smooth

uniform elastic body of the kind dealt with in

Chapter II; the inhomogeneity will differ,

in general,

of B, in each of these cases,

as in one case it depends

on the

field --mr'(P'') and in other on the field ~rf(P'')' for points peB.

The fields ~i(p, ") and ~ f ( p , ' )

of course,

in any definite way.

need not be related,

An alternative

the problem of a change of inhomogeneity, of the anelasticity the next chapter.

theory of Eckart

approach to

within the context

[15] is presented

in

101

Chapter i.

IV.

Anelastic

Behavior and D i s l o c a t i o n Motion

Introduction Like the theories

Chapters

of material

uniformity presented

II and III, the geometric

structure theory for

anelastic materials,

which will be presented here,

on the concept of material however,

the inhomogeneity,

permanently

atlas which characterizes functions

completely

anelastic materials

(as we shall soon

the distribution

of the response

is now time-dependent).

in this chapter

as possible,

is not

of the fact that the material

on the body manifold

Our aim, therefore,

is based

For anelastic bodies,

in each given motion,

fixed in the body manifold

see, this is a consequence

[15].

isomorphism.

in

is to describe,

the evolution

as

of inhomogeneity

in

of the type first considered by Eekart

Stated in more direct terms, we will equip the points

of our body manifold

B with constitutive

relations

similar

to those first proposed by Eckart and then study the evolution of the geometric concept of material with the consitutive

structures which arise from the

isomorphism that is naturally relations

of the points

Prior to stating the constitutive

associated

comprising

relations

B.

for points

of an anelastic body B, we need to append to the material on kinematics following

(Chapter II,

§2) thus far presented,

the

(8)

(8) local configurations of peB will be denoted here by ~(p) instead of r~p ,as in chapter II.

102

Definition ~ r(p,t): time.

Ill-i

Let %(t):

Bp ÷ R 3 a local m o t i o n Then the one p a r a m e t e r

is called the h i s t o r y %.

B ÷ R 3 be a m o t i o n

of p, and t o > O a fixed to family %s ~ %(to-S) , se[O,~),

of B up to the time t o in the m o t i o n

In a similar manner,

local h i s t o r y

of B,

we call rt°Cp)~s z r(p,to-S)

of psB,

up to time to,

#(to-S)

is t e r m e d

the

in the local m o t i o n

r(p,t). For s>O, time-lapse motion

s from the p r e s e n t

times

If the m o t i o n

~ has

constant

a rest

history

If ~(t):

history

to ~s such that

(at

~o ) .

configuration

via

we s h a l l

Ft°(s)

~ F(t

~

2.

~

Elastic

denote -s),

the

B + R 3 is a

of B up to the two

%o then

%(t)

Naturally,

local

: ~o'

to ~s = ~o' Vse[O'~)' is

~

2o

3 a local rest history (at ~o(p): g ~ R ) if to P ~s ( P ) = ~o ( p ) ' ~s~[O,~). I f F:~ R3 + R 3 i s gradient

a_~t

to t1 if #s = ~s ' V s e [ O ' ~ ) "

to,t I is the same

and a h i s t o r y

called

time.

of B then we say that the h i s t o r y

distinct

VteR

the past

(p)

is

called

a deformation

deformation

history

~se[O,~).

O

and A n e l a s t i c

Response

Functions;

Anelastic

Transformations The class considered a very

in a s y s t e m a t i c

special

can be t e r m e d material

of a n e l a s t i c

subclass

w h i c h was

way by Eckart

of a larger group

~uasi-elastic.

point was

in an attempt

materials,

The concept

introduced

to f o r m u l a t e

by Wang

&

in 1948

first [15],

of m a t e r i a l s

form which

of a q u a s i - e l a s t i c Bowen

a thermodynamieal

[33]

theory

in 1966 for

103

non-linear

materials

with memory which was more general

than that which was first proposed for simple materials

by Coleman

with fading memory;

[34],

[35]

in this work of

Wang g Bowen the need to trace the local configurations particles specify

pcB back to past infinity

the memory

(i.e.,

effect of the history

variable,

is obviated.

The complete

presented

in [33] shall,

however,

as s+=),and

of to

of each state

thermodynamical

not be needed

structure

in our work

here as we will be dealing,

once again, with a purely me-

chanical

uniformity,

theory.

uniformity,

within

thermodynamics

(Material

the context

of dislocation

the next chapter). Definition

111-2.

Let B be a material

local configuration

and the

motions, will be treated

~

in

begin with the following body.

A particle

if at any lime t o in

any local motion r(p,t) of p we can define to response function E such that the stress ~p present

symmetry

of thermoelasticity,

We can, therefore,

pgB is called a quasi-elastic

actually

an instantaneous tensor ~S in the

r(p,t o) is given by

tO(rtO ~o (P))

to TS = E~p (r(p~ 'ot )) = E~p

Remark

to E at time t o is determined, ~Pt by the past history r O(p), sE(0,~),of p in the

The response

in general,

(IV-I)

function

~S

local motion up to time t • o rt°(p) ~S

and rtl(p)

Thus,

if the past histories

of p up to the distinct

times to, tl, in

~S

the local motion r(p,t),

are the same

(i.e.,

if they agree

104

on

tO(r(p tl E )) = E (r(p)) ~p ~ ~p ~

(0,=), we w i l l h a v e

configurations that

r(p):

B

~

p

÷ R 3 of p; we note also

if r ( p , t o) = r ( p , t I) then the stress t e n s o r at p at

t h e s e two d i s t i n c t o t h e r hand,

times w i l l

if ~(p,t)

local m o t i o n s an a r b i t r a r y

of p~B, toSR

also be the same.

and r(p,t), then,

tsR

different.

We can,

stantaneous

response

On the

, are two d i f f e r e n t

in g e n e r a l ,

; the i n s t a n t e o u s

c o r r e s p o n d i n g to these two

t sR o

for all local

^t o t rs~ (p) ~ r~s°(p) for

response

functions

local m o t i o n s w i l l t h e n also be

therefore,

c o n c l u d e that w h i l e an in-

E t° can be d e f i n e d at each ~P to , in each local m o t i o n of peB, E is not a fixed p

material

function

f u n c t i o n of p at t • o

In o r d e r to single out f r o m the class of q u a s i - e l a s t i c materials points)

(bodies

that i m p o r t a n t

identified

soley of q u a s i - e l a s t i c m a t e r i a l

subclass

as b e i n g a n e l a s t i c ,

we f o l l o w W a n g assumptions: I.

comprised

of m a t e r i a l s

w h i c h may be

in the sense of Eckart

[15],

& B l o o m [19] and lay down the f o l l o w i n g each psB is a q u a s i - e l a s t i c

For e a c h psB,

there

exists

point

and

on e l a s t i c r e s p o n s e

function E d e f i n e d on D (the set of all local c o n f i g u r a t i o n s ~p P r(p) of p) such that the stress t e n s o r T ° in any rest h i s t o r y ~S

at r ED is given by ~o p T ° = E (r ) ~s ~p ~o

(IV,2)

For e a c h peB, E is a s m o o t h f u n c t i o n on D w h o s e v a l u e s ~p P

105

lie in L (R3,R3). s II.

In any local m o t i o n

r(p,t)

of p (i.e.,

p is not

~

at rest) ~(p,t):

there

Bp

BP ,

(i) ~(p,t) linear

exists

an anelastic

VteR

transformation

function

, with the properties

is a smooth

function

orientation-preserving

of t whose

automorphisms

values

of B

are

for each P

tcR (ii) ~(p,t)

÷ id B

as t+-= P instantaneous (anelastic)

(iii) the to E of p at time t is given ~p o t E~°(r(p))~ where

r(p) ~

Remark smooth

As ~Ep is a smooth

via

Also,

(IV-3)

of D p . function

of t, clearly,

both t and r(p).

function

: Ep(r(p)°~(p~ ,to ))

is any element

function

response

on Dp and ~(p,t)~

E~(r(p))

by virtue

depends

of condition

is a

smoothly

on

(ii) above,

we have Et(r(p)) ~p ~ Note

also that,

t, in each

= Ep(~(p)),

configuration

fixed

local

As it is usually stress

tensors

t+-~

as a consequence

E[(~(p)o~(p,t)-l)= in the local

+ E (r(p)), ~p ~

(IV-3),

VteR;

thus

we have

the stress

tensor

r(p)°~(p,t) -I is independent

configuration

assumed

in two

of

(IV-4)

r(p)

in c o n t i n u u m

local

of p, for all tsR

m~chanics

configurations

of

that the

of the same

.

106

material point cannot be the same, unless the densities those local configurations

in

are the same, we will also impose

assumption (iv)

in any local motion r(p,t) of peB an anelastie

transformation function ~(p,t) must be isochoric,

i.e.,

~(p,t) e SL(B ), VteR. P We now choose a fixed local reference configuration for p which we will designate by r(p), i.e., r(p) will no longer denote any arbitrary element of Np but, rather, a fixed element which we have singled out from Dp. Chapters II and IIl, if

As in

~: B ÷ R 3 is any configuration

of B, then we can represent the local configuration

~,p in

Dp by the deformation gradient F(e~,p~ o ~r(P)-I) from r(p) to ~,p.

Obviously, we may define a relative elastic

response function S (p) in a manner similar to that of Chapter II, i.e.,

~F~GL(3) ~

Sr(p)(F,p)~ ~ Ep(For(p))~ ~

(IV-5)

and, if r(p,t) is any local motion of p, we can also define a relative anelastic response function via

St (F,p) ~ Eup t (For(p)) -E(p) ~ ~ ~ ' ~teR, FeGL(3) ~ Now, by putting

(IV-3),

(IV-6)

(IV-5), and (IV-6) together

we obtain, for any toeR, t o

Sr(p)(F)

= Sr(p)(FAr(p)(to)),

VFeGL(3)

(IV-7)

107

to

as the relationship between ~r(p) and ~r(p)' where

A (p)(t) ~ r(p) 0 ~(p,t) o r(p) -I,

~teR

is called the relative anelastic transformation function. Clearly, A (p)(t) is a smooth tensor-valued function which is isochoric,

for each tsR, and satisfies Ar(p)(t)

t÷-~; as a consequence, ly on both F and t and ~

as

t÷-~.

for elements tion

3.

of

,

The a n a l y s i s

any

of

p

P global

Anelastic

we see that S t (F) depends smoothr(p) ~ in addition, ~t )(F) + S (F) ~(p

above

which

+ i, as

are

also not

configuration

goes

the

induced

} of

~

r(p)

through,

of

local

~

course, configura-

B.

Symmetry Groups and Anelastic Inner Products.

The various response functions and relative response functions

introduced in the previous section give rise, of

course, to a variety of symmetry groups and to a set of relationships

among these groups and the anelastic trans-

formations functions ~(p,t) and Ar(p)(t).

To begin with,

we define the elastic symmetry group g(p) via

and the anelastic symmetry group gt(p) by

t

~

Et(6)

V~eP

}

An immediate consequence of these definitions

and (IV-3)

108

is the r e l a t i o n

g t (p) = ~(p,t)

Note also that if function ~(p,t) and

~(p,t)

o g(p)

is an a n e l a s t i c

in the local m o t i o n

provided

o ~(p,t) -I

r(p,t)

(IV-8)

transformation

of psB then so is

that ~(p,t) -I o ~(p,t)

E ~(t)

e g(p),

VtsR,

~(t) + id B

as t÷-~. C o n v e r s e l y , if ~(t) is a smooth P f u n c t i o n with ~(t) e g(p), V t e R and ~(t) ~ id B , as t÷-~ P then ~(p,t) E e(p,t) o ~(t) is an a n e l a s t i c t r a n s f o r m a t i o n function g(p)

in the local m o t i o n

is n o n - t r i v i a l ,

(IV-3)

r(p,t)

if ~(p,t)

is.

and the r e q u i r e m e n t

Thus,

if

that

e(p,t)

÷ id B as t+-~ do not u n i q u e l y d e t e r m i n e an a n e l a s t i c P t r a n s f o r m a t i o n f u n c t i o n in a given local m o t i o n of p. From

(IV-8)

deduce

that gt(p)

t g (p) ÷ g(p) (IV-8)

and the fact that depends

as t~-~.

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

equivalence

class

as t+-~ we P on t and, moreover,

smoothly

The r e a d e r

~ ~(t)

as t÷-~. P The r e l a t i v e elastic

+ id B

can easily

of the choice

of a n e l a s t i c

by ~(p,t) -I o ~(p,t)

e(p,t) ~

check that

of ~(p,t)

within

transformations

~ g(p),

~t~R

the

defined

and

~(t) + id B

as usual,

symmetry

group

Gr(p)

is defined,

by

and the r e l a t i v e

anelastic

symmetry

group

t Gr(p)

is given

109

by Gt St ~(p) ~ {KESL(3)IS ~(p) t (FK ~~' p) = ~(p) (F,p) ~ , VFsGL(3)} ~

Concerning the relationship which exists between Gr(p) and Gt r(p) we note, first of all, that

Gr(p) = r(p) o g(p) o r(p) -I

(IV-f1)

as in Chapter If, and that

Gt

~(p)

= r(p) o gt(p) o r(p) -i

It then follows,

(IV-12)

~

at once, from (IV-8), that

G r(p) t = Ar(p)~ (t) 0 Gr(p)~ o Ar(p) (t)-i

(Iv-13)

so that G t

r(p) ÷ Gr(p) as t÷-~; the relations (IV-II) and

(IV-12) are simple consequences

of the definitions

of the

various symmetry groups involved and the relationships which exist among their associated response functions. Note also that just as ~(p,t) is non-uniquely determined, in general,

if g(p) is non-trivial

uniquely determined,

i.e., if Ar(p)(t)

elastic transformation Ar(p)(t)

~ Ar(p)(t)

so will Ar(p)(t) be nonis a relative an-

function so is

0 K(t) provided K(t) e Gr(p),

VteR,

and K(t) ÷ i as t÷-~. The definitions of the symmetry groups given above

110

enable us to apply Noll's to anelastic

material

an anelastic

fluid point

this

is equivalent

general

points.

classification

For instance,

if g(p)

to stating

we will call p

= SL(B

); in view of (IV-II) P that G (p) = SL(3) relative

to any local reference

configuration

ing theorem

fluid points has an almost

concerning

of materials

r(p) of p.

The followtrivial

proof: Theorem

IV-I

Every anelastic

fluid point

is an elastic

fluid point. Proof:

To begin with let us note that peB is an elastic

fluid point if the symmetry

group g t(p) of the response

function r(p,t)

E t coincides with SL(B ) for each teR. So, let ~p P be a local motion of p for which ~(p,t) is an as-

sociated ~(p,t)

= e(p,t)

formation ~(t)

anelastic

transformation

function.

0 ~(t) will also be an anelastic

function

~ g ~ SL(Bp)

in the local motion

is isochoric,

~(p,t)

~ idBp as t~-~, we may choose ~(p,t)

function by

~ id

r(p,t)

Bp,

i.e.,

e(p,t) ~

~t~R,

in the local motion

if

~ SL(B p ), ~teR,

= E (~(p)), p ~ ~> gt(p)

is an anelastic

transformation

of p and, therefore,

~teR,

~eD

z SL(B

eompletes

the proof.

Q. E. D.

p ),

P whmch

and

r(p,t)

= g(p)

as

~(t) ~ ~(p,t) -I.

(IV-3) Et(B(p)) ~p ~

trans-

for each tER and ~(t) ÷ idBp as t~-~;

~(p,t) ~

Thus,

Then

~t~R

111

In order to define point we must assume

the

that

concept

of an a n e l a s t i c

solid

B

can be e q u i p p e d w i t h an i n t r i n s i c P e l a s t i c inner p r o d u c t m such that g(p) c O(B )(g) In other P words if ~eg(~) imples that

m ( ~.u ,.~ v.) . =. m(u,v) . then g(p)

c 0(Bp)

then p is called in Chapter

II, we w i l l

=>

t

KU .

relationship

m

an i s o t r o p i e

undistorted

K ~ e G ~(p)

gt(p)

and p is called

p

.

(IV-15)

an a n e l a s t i c

solid point.

~ O(Bp), r e l a t i v e to some inner p r o d u c t m on Bp,

If g(p)

r(p)

u,v~B ~ ~

,

• KV = U'V, .

.

.

(IV-8)

solid point.

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

if G (p') c 0(3); .

.

As

configuration

if such is the case then

V U , V e R 3. .

In view of the

.

between

we can define

on B

anelastic

the s y m m e t r y

an i n t r i n s i c

groups g(p)

anelastic

and

inner p r o d u c t

via P mt(u,v)

~ m(~-l(p,t)u,~-l(p,t)v), . . . . .

and each teR in each local m o t i o n is an a s s o c i a t e d follows

that

relative

anelastic

if g
g(p) m

t

(9)

16

c 0(Bp)

If m

t

= m t (u,v), . . . .

relative

(for each teR).

function.

to m, gt(p)

by

Vu,v~B

p

(IV-16)

It

c 0(Bp)

Show also that

• verify

that

O(Bp ) r e l a t i v e to

to m => g t (p) c

if r(p)

O(B ) is the r o t a t i o n group on B P

~(p,t)

i.e.,

.

is defined

(IV-16)

p

for w h i c h

transforamtlon

to m t for each teR,

Exercise

r(p,t)

c 0(Bp) r e l a t i v e

~ g t (p) => m t (~u,~v) . . .

Vu,veB

is u n d i s t o r t e d ,

relative P

to m

112

Gt r(p)

c 0(3) r e l a t i v e

which

is d e f i n e d

(Note that both m defined

by

mt(u,v)

+ m(u,v)

t

(IV-17)

Flow Rules;

(

)t

,

on

R

3

by

~ A~ ~ (-Ip ) ( t ) U

(U,V)t

4.

to the inner p r o d u c t

• A~r(p -I ) ( t ) V~,

as d e f i n e d depend

and

by

(IV-16)

smoothly

(U,V) t + U'V,

Uniqueness

~ U~, V s~R

3

and

(IV-17) t

(

)

as

on t with as t ÷-~.

of the A n e l a s t i c

Transformation

Function Suppose funetion

that ~(p,t)

in some

indicated

in

is an a n e l a s t i c

local m o t i o n

§3,e(p,t)

r(p,t);

which

transformation

as we have

is d e t e r m i n e d

by rt(p)

'

is g e n e r a l l y

non-unique

~S

if g(p)

application

of the theory

that

is g o v e r n e d

~(p,t)

where

~ is some

rt(p)

of the a n e l a s t i e

any special condition motion the

"initial"

local m o t i o n ~(p,t); class

of this

chapter,

functional

of the past

point

p.

Of course, condition

we must

~ is not unique, r(p,t)

our choices

engendered

a value

assume

form & =

local h i s t o r i e s

While we will not employ

function

~(p,t)

'

In any

we w o u l d

by a flow rule of the

that ~ be a smooth

Because

is n o n - t r i v i a l .

form for ~ in this t r e a t i s e

r(p,t).

already

we will

impose the

of t in each local

append to the flow rule

÷ idBp as t÷-~. at each time t in a given

of ~ must be a s s i g n e d

are limited,

by the r e l a t i o n

of course,

at some

to the e q u i v a l e n c e

~(p,t) -I o ~(p,t)

~ ((t)sg(p)

113

VtsR,with

~(t) + idBp as t÷-~.

To make matters

a bit

worse, we note that if ~ and [ belong to the same equivalence ~

class, and ~(t o) = idBp, for some time t o , then generally 9(p,t o) ~ ~~(P'to) , even though e(p,t o ) ~

~(p,t o)

: [(P,to).~

~(p,t o) : ~(p,t ~

~

In fact

)~

(IV-18)

O

where ~Egp, the Lie algebra of g(p).

Thus, ~@(~(n't o ). ~

, to )

is unique only to within an arbitrary additive tensor in

~(P,to)g pThe non-uniqueness

of ~(~(P,to) , t o ) described

above

may be removed in the following way: note first of all that because ~(P,to)-i

~(p,t)sSL(Bp),

VtsR,

we must have

0 ~(P,to)SSL(~ p) or

tr(9(P,to)-i

0 9(~(P,to) , to)) = 0

for each t oeR; but this implies that ~~(~(P't o ) ~ belong to e(p,t o) o sl(B ). P the direct sum sl(Bp)

If we decompose

=

gp

(Iv-19)

, to) must sl(B ) into P

~ h

~p

(IV-20)

then we may define a unique value for ~(~(P,to) , t o ) by imposing the selection ~

~(p,t °

)-i

condition:

o ~(~(P,to),

t o ) ~ hp

(IV-21)

114

Now that we have uniquely

determine

attention solution

given

which

which

enables

of ~a(P,to), guarantees

exists

and the selection

which

condition

by

us that

of g(p),

the solution

the initial selection

for all of the

condition,

condition

Let ~(p,t)

(IV-21),

and {(p,t)

~(t) ÷ idBp as t+-~. and using

the

for elements

the adjoint

as t÷-~)

component

class,

to

and the

determined.

be any two solutions

of ~ =

i.e.,

~(t)E

g(p),

Differentiating

with

~_~(~d-i)

is

a = @, subject

is uniquely

where

fact thatQ0)

which

the identity

+ idBp

equivalence

~ ~(t)

under

flow rule

(~(p,t)

lie in the same

~(p,t) -I o ~(p,t)

~s~(p),

the inital

(IV-21).

if h~p is invariant

ad(~)

a unique

satisfies

Let h be the subspace of SL(B ) ~p P (IV-20) and define [ad(6)]n ~ ~oqo~-i Then

us to

we turn our

IV-2

representation

which

the value

to a t h e o r e m

~6'qeL(Bp'Bp)"

Proof

a condition

of the flow rule

condition Theorem

found

= -a-i

~tsR

and

respect

to t

o ~" o a-I

we obtain -I :

--a

• o

~

-i o

~

-i o

~

+

~

(IV-22)

o

or

~-i

o e

-1

o ~(~,.)o~

--I + e o ~(~,'):

~-i

o [

(IV-23)

~0) to avoid long expressions we now drop the displayed dependence of e on p and t and assume this to be understood.

115

as ~(p,') .

: @(~(p,'), .

.

") and ~(p,.) : ~(~(p,.),

.

.).

~

By using the definition of "adjoint representation" is given in the statement

which

of the theorem, we may rewrite

(IV-23) in the form -[ad(~)](~ -1 o ~(~,

• ))

+ -i

By virtue of the hypothesis condition

o

~(9,.)

=

~-l

o

~[

(Iv,24)

of the theorem and the selection

(IV-21), the left hand side of (IV-24) is contained

in hp; but ~-i o ~ e gP since ~(t)e g(p), VteR. by definition,

the orthogonal

complement

Since hp is,

of gp in sl(Bp),

it must follow that ~-i o ~ = O so ~(t) = ~o' the initial value of ~~, for all tsR.

As ~~(t) ÷ idBp for t+-~, we must

have ~~o = idBp; this, in turn, implies that ~(t) ~ idBp,

VtcR,

or ~(p,t)

= ~~(p,t),

~tcR.

Q. E. D.

For solid and fluid anelastic points it is a relatively simple matter to verify that the orthogonal

complements

in sl(B ) of the Lie algebras of the respective isotropy P groups g(p) are invariant under the action of ad(~) for all ~ ( p ) .

We know, in fact, that for an anelastic fluid

point p, g(p) = sl(B ) so gp = sl (Bp ); this implies that P hp= 0 and that the unique solution of ~(t,p)~ = ~0 satisfying ~(t,p) ÷ idBp as t÷-~ must be, of course, ~(t,p)

~ idBp,

~tsR.

On the other hand,

if p is an an-

elastic solid point then we may choose an intrinsic elastic m on B

P

such that g(p)

c O(B

)

P

relative to m.

Now let

116

and B be two transformations

in L(B ;B ) and let BT denote

~

p

the transpose

P

of B with respect to m.

We define the inner

product on L(Bp;Bp), which is induced by m, via

(e,~) ~ tr(eoBT), . . . . . .

~,8sL(B

Now let hp be the orthogonal

B ) p, P

complement

of gp in

sl(B ), relative to the inner product ( , ) on L(B B ), P P, P i.e., (~,B)~ ~ = O, ~ E h p , ~~6EZp" To show that h~p is invariant under ad(~), for all ~e~(p), we choose ~sL(Bp Bp) such that

~ is an orthogonal transformation

then have ~-i = ~T and for ~ h p ,

([ad(~)]~,

relative to m~ we

~gp

6) = tr(~o~o~-loB T) ~

= tr(~o(~-loso~) T) = (~, [ad(~-l)]B) =

O

since gp is invariant under ad(~ -I) for all ~e~(p).

It

now follows at once, from the definition orthogonal

complement

if ~Shp.~ We deduce, anelastic ~(p,t),

of gp in sl(Bp),

of h as the ~P that [ad(~)]~ ~ hp

from the theorem above, that for an

solid point p the anelastic transformation VtsR,

if ~ satisfies

functiol

is unique in each local motion r(p,t) of p the selection

[~(p,t) -I o {(p,t)] ~ g p , elastic inner product on B

condition,

VtER, P

namely,

if

relative to any intrinsic

117

Now suppose that we define the relative anelastic transformation

Ar(p)(t) by

A r ( p ) ( t ) ~ ~(p)o~(p,t)o~(p) -i , VtsR, where r(p) is our local reference configuration. ~(p,t) = 9(~(p,t), ~r(p)(t)

Then the flow rule

t) has the representation

= ~r(p)(Ar(p)(t),_ t) and the initial condition

involved is Ar(p)(t)

÷ ~ i, as t÷-~.

As in [19] we now

suppress the subscript r(p) in ~r(p)' the local reference ~

configuration r(p) being understood{ equivalence

then within the

class defined by A(t)-iA(t)

~ K(t) c Gr(p), ~ t s R , ~

and the initial condition K(t) ÷ i as t+-~, the value of ~(A(t), t) is unique to within an arbitrary additive tensor belonging to A(t)G~p algebra of Gr(p).

where G~p denotes the Lie

If Hp denotes the orthogonal

com-

plement of G in sl(3), i.e., sl(3) = G ~ H then the ~p -p ~p selection rule, in this case, assumes the form

A(t)-l~(A(t), ~

Remark

t) s H , V t s R

~

(IV-25)

~p

If p is an anelastic solid point and r(p) is an

undistorted local reference configuration of p then the selection rule

(IV-25) becomes

tr(A(t)-l~(A(t),

t)K) : O,

VteR

VKsG ~p

If this condition is satisfied then A(t) is unique in each

118

local motion of p. Now consider S(F,p) and st(F,p), FcGL(3), the elastic and anelastic response functions, to the fixed local reference

respectively,

configuration r(p)

subscripts on S, S t have been dropped).

relative (the r(p)

By (IV-7) the

relationship between S and S t is given by ~ St(F,p)

= S(FA(t), p) which can be written in component

form relative to the standard basis of R 3 as

a

(IV-26)

Sb(F d, t) : Sb(F f Af(t))

where t has been written as an explicit argument and the dependence of the response functions on the point psB has not been displayed.

If we set Ffc = 6cf in (IV-26), differentate

with respect to t, and use the flow rule, we get

ade))~d(Af(t Tb(t) : Sbc(Af(t

where Tb(t)

a(A~(t)) are the components of the stress Sb

tensor in the given local reference _ad._e) ~bct~f

a fe)/~F~, E ~Sb(F

A~(t), therefore, C

e

~d(Af(t), t).

(IV--27)

), t)

yF~GL(3) ~

configuration and At any fixed value of

T~(t) depends linearly on the flow rate

While

(IV-27) represents

a necessary condition

to be satisfied by ~, this condition alone in is not, in general,

sufficient to determine ~.

Exercise 17

Show that when p is an anelastic solid crystal

119

point, i.e., G = {!}, G~p = {O} and H~p : sl(3), relative to an undistorted local configuration r(p),

(IV-27) can

C

not be solved uniquely for ~d in terms of }a b" Before concluding this section, we should point out that there are exceptions

to the type of situation illus-

trated by the example in the above exercise,

i.e., if p

is an isotropic anelastic solid point the system of equations equivalent to (IV-27) may be invertible.

In order to see

this we let r(p) be an undistorted reference

configuration

of p and define an elastic inner product m on B

m(u,v)

Using B

P

: ~(p)~

• r(p)v,

P

via

~2,~SBp

(IV-16) we define the anelastic inner product m

(IV-27) t

on

by

mt(u,v) = m(e(p,t)-lu,~(p,t)-lv) = r(p)o~(p,t)-lu

(zv-28)

• r(p)o~(p,t)-iv ~

= Ar(p)(t)-ir(p)u~~ " Ar(p)(t)-ir(p)v~

= ~(p)u

• ~-l(t)~(p)y

where C(t) ~ Ar(p)(t)A~(p)(t) Green tensor of At(p).

is called the left Cauchy-

If we define y(t) e ~(p,t)o~(p,t) T

to be the left Cauchy-Green tensor of

e(p,t), relative

120

to m, then it is a simple matter to show that

m t( u,v) = m(u, y(t)-iv)

(IV-29)

C(t) = r(p) o y(t) 0 r(p) -I,

i.e., C(t) is the representation local reference

of y(t), relative to the

configuration r(p).

Moreover,

y(t) and

C(t) are smooth functions of t in each local motion r(p,t) and, clearly, y(t) ÷ id B

and C(t)~ ~ ~i' as t÷-~.

It is a

P relatively

simple matter to show that

~(p,t) and ~(p,t)~

are anelastic transformations which belong to the same equivalence

class

(in the sense, of course, that

~(p,t) -I o ~(p,t)~g(p), as t÷-~) iff,

~teR,

V t E R and ~ ( p , t ) -I o ~(p,t) ÷ id B

~(p,t) o ~(p,t) T = ~(p,t) o ~(p, t)T,

Pi.e.,

iff y(t) = ~(t). ~

Exercise 18

Verify the above statement

transformations

concerning anelastic

belonging to the same equivalence

class.

Show also that A (t) and (t) satisy ~r(p) ~r(p) Ar(p)(t)

o Ar(p)(t)T = Ar(p)(t)

o Ar(p)(t) T,

these relative anelastic transformation -i Ar(p)(t) o

Ar(p)(t)

- K(t) ~ Gr(p),

VtER,

iff

functions satisfy

VteR,

and K(t) + i

as t+-~. As the exercise above indicates,

the respective equiv-

alence classes of ~(p,t) and Ar(p)(t)

are completely deter-

121

m i n e d by y(t)

and C(t).

(~ : ~

by flow rules

~ = T,

and

A =

and

C =

the flow rules

of the form

= ~

where

Thus, we may replace

H : H T and

and C(t)

at each ts~)

Exercise

19

(to insure

tr(y-low)

the u n i q u e n e s s

= 0, tr(c-lo~)

Show that ~ and H d e t e r m i n e

of y(t)

= O.

the values

of

and ~ via ~(e(p,t),

~(A(t),

Since function

t) = ½z(t)

t) = ½~(t)

it can be shown

S(F,p)

o [~(p,t)-l] T

o [A(t)-l] T

[ 5 ] that the r e l a t i v e

of an i s o t r o p i c

(IV-30)

solid

response

can be r e p r e s e n t e d

A

by a f u n c t i o n

S(D,p),

of the

F, if the local r e f e r e n c e

left-Cauehy

configuration

Green t e n s o r D of r(p)

is u n d i s t o r t e d ,

we can now reduce the s y s t e m of e q u a t i o n s T~(t) = S~(A~(t)) ~ Aa e to the s y s t e m T (t) = Sb(Cf(t)); d i f f e r e n t i a t i o n with respect to t now yields

T~(t)

A

where

ag ef ) Sbh(C

Aad e )~(t) = Sbe(Cf(t)

A

h

- zsa(ce)IZCgo±

for all p o s i t i v e - d e f i n i t e

(IV-31)

122 symmetric

linear

transformations

of R 3.

of H B e l o n g

to a linear

of T b e l o n g

to a space of d i m e n s i o n

it may now be p o s s i b l e

Because

space of d i m e n s i o n

to invert

5 while those

(not g r e a t e r

(IV-31)

the values

than)

6

and solve for

"a

o in terms Hd which

of Tb; this

exists

belongs

relative

to a space

in sl(3) w h i c h

5.

Material

contrasts

to the

development

is of d i m e n s i o n

Uniformity

let p,qsB.

points;

theory

of e l a s t i c

in general,

Y takes

its values

of A n e l a s t i c i t y

i.e.,

a material

body

as the key to the

for such m a t e r i a l s

and a n e l a s t i c

we need to p r o c e e d w i t h

rests

materials

iso-

some definitions.

So,

Then we can state IV-3

r(p,q): ~

÷ B

p

6 whereas

where,

8.

body,

of a g e o m e t r i c

Definition B

(IV-27)

in the Theory

soley of a n e l a s t i c

w i t h the concepts morphisms,

system

of d i m e n s i o n

Let B be an a n e l a s t i c consisting

sharply w i t h the s i t u a t i o n

q

A linear o r i e n t a t i o n - p r e s e r v i n g is called

an elastic

material

isomorphism

isomorphism

of p and q if

E-p ( .~ ( q ).o r ( p., q ) ).

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

r(p,q)

satisfying

that p and q are m a t e r i a l l y if the points

(IV-32)

= Eq(~(q)) ' V~(q)sD q

isomorphic

of B are p a i r w i s e

we say that our a n e l a s t i c

(IV-32)

body

and,

materially

exists we say furthermore, isomorphic

B is m a t e r i a l l y

then

uniform;

123

this definition of material

is completely

consistent with the definition

uniformity presented

elastic response

in Chapter

11, as the

function E is the response -p

peB in any local rest history

function of

at some ~(p)eD -~ p•

It is very

easy to see now that relative to the elastic response functions

Ep, peB, many of our results

be carried over without groups

change.

g(p) and g(q) associated,

and E are related via ~q elastic material

in Chapter II may

For instance,

the isotropy

respectively,

with E -p

(11-8) where r(p,q)

isomorphism of p and q.

now denotes any Now let us sup-

pose that the anelastic body B is not at rest but, rather, that B is undergoing motion

some motion

%~p(t) of each peB~ then the response

any time t~R,

in the loeal motion

by the anelastic response E by ~p

#(t) which induces

(IV-3) with a(p,t) ~

(for the local motion

%,p(t),

a local

of peB, at

is characterized

function E~, which is related to an anelastic

%~p(t)).

transformation

We can, therefore,

function

make the

following Definition

IV-4

Fix tER.

An isomorphism t ( p , q ) :

is called an anelastic material time t, in the motion

i s o m o r p h i s m of p and q, a t

%(t) of B, if

Et(v(q)ort(p,q)) = Et(v(q)), Vv(q)eD ~p ~

If we compare relationship

Bp ÷ Bq

~q ~

(IV-33) with

(IV-32)

~

(IV-33)

q

and make use of the

(IV-3) between the response

functions

E~p and

124 t

Ep, at time t, we can easily prove Theorem

IV-3

In any motion

elastic material

~(p,q)

~(t) of B, rt(p,q)

isomorphism

of p and q, at time t, iff

z ~(q,t) -I o t ( p , q )

o e(p,t)

is an elastic material

isomorphism

a(.,t)

is an anelastic

transformation

motion

~,(.)(t)).

It is a direct anelastic

consequence

body B is materially

anelastioally materially at all times tsR) ~(t):

uniform

material

isomorphism

Exercise

20

material

isomorphism,

independent respective

rt(p,q)

By making

in the local

IV-3 that an

if and only if it is

at any one time (and hence

(and hence

in all motions

B is anelastically

(at time t in the motion p,qEB,

function

uniform

in any one motion

for any pair of points

(IV-34)

of p and q (note that

of theorem

uniform

B ÷ R3); by saying that

is an an-

materially

%(t) of B) we mean that

there exists satisfying

use of II-8,

an anelastic

(iV-33).

with r(p,q)

an elastic

show that the relationship

of the choice of ~(p,t) equivalence If rt(p,q)

and a(q,t)

(IV-34)

is

within their

classes.

Exercise

21

is any anelastie

material

morphism

of p and q, at time t, in the motion

iso-

%(t) of B,

show that gt(q)

= rt(p,q) N

o gt(p)

o rt(p,q) -I

(IV-35)

125

We will now add the following basic assumption B to those already given Iii.

~D"

the anelastic body B is materially

for each psB there exists a neighborhood field r(p,')

about

uniform and

and a smooth P isomorphisms defined on

of elastic material

N

N ; we do not require that N ~ B for any psB P P Exercise

22

Show that if r(p,q)

is a given elastic material

isomorphism of p and q and r(p,q)

is any linear isomorphism

for B

to B then ~(p,q) is also an elastic material isoP q morphism of p and q iff ~(p,q)-1 o r(p,q)sg(p)(or iff r(p,q) -I o ~(p,q)sg(q)).

Thus,

m o r p h i s m of p and q is unique

an elastic material

iff g(p)

that by virtue of (11-8), with r(p,q)

iso-

~ {id B }; note also P an elastic material

isomorphism of p and q, g(q) is the trivial

group id B q

if g(p) is the trivial group id B . q Remarks

The results

of the above exercise point to the

fact that it is generally

impossible

local field of elastic material

to extend a smooth

isomorphism~

defined on Np, to a smooth global field on B. do exist,

i.e.,

isomorphisms

say r(p,') Exceptions

every smooth local field of elastic material

on a star-shaped body B can be extended to a

smooth global field of material

isomorphisms;

details we refer the reader to the discussion

for further in Wang

[ 7 ].

~ the "smoothness" part of this assumption is consistent with the one set down in Chapter II for (smooth) materially uniform elastic bodies.

126

Now let us note that b e c a u s e (above)

concerning

bodies t h e

stress

smooth m a t e r i a l l y tensor

s m o o t h rest h i s t o r y true

field

uniform

IV.

Zn each

transformation

that we must

smooth m o t i o n

function

anelastic

~(p,')

~(t)

~(t)

of B then it

add a s s u m p t i o n

of B an a n e l a s t i c

can be chosen,

in such a way that e(',t)

in any

the same thing to be

at each time t in any smooth m o t i o n (IV-3),

assumption

~S must be smooth

of B; if we want

is clear, in view of

peB,

of our third

is a smooth

for each

(global)

function

on B for each teR. One very i m p o r t a n t assumption, that

is that

it r e q u i r e s

it be a smooth

a smooth

local

thing to note,

global

concerning

of ~(.,t),

field d e f i n e d

field on some n e i g h b o r h o o d

this

last

for each teR,

on B and not N

just

of each peB. P

Our reasons simple:

for i m p o s i n g

as t÷-~,

In fact,

equation elastic

response

initial

structure

we can c o n s i d e r functions

time via the a s s o c i a t e d

response

functions; functions

E(.)

assumptions

the d i s t r i b u t i o n on B as i n d u c i n g

on B from w h i c h the a n e l a s t i c

on B, in any given m o t i o n

formation

are quite

÷ id B , for each peB, and the iP is c e r t a i n l y a smooth (global) field on

(.) in view of our s m o o t h n e s s

(IV-34)

here

~(p,t)

d e n t i t y map id B B.

such a r e q u i r e m e n t

~(t),

field

are g e n e r a t e d

~(.,t)

in o t h e r words

of the an

structures

smoothly

of a n e l a s t i e

in

trans-

the field of a n e l a s t i c

t will be d i s t r i b u t e d E(.)

at each time teR, in any m o t i o n

and

~(t) of B.

smoothly

on B,

To be more precise,

127

let ~(t) be a m o t i o n

of B and r(p,') defined

field of

elastic

material

hood

of p; then at each time t in the m o t i o n

N

isomorphisms

a smooth

on some n e i g h o r ~(t) the

P field rt(p, ") = ~(',t)

where

e(',t)

denotes

transformation assumption

the

o r(p,')

smooth

functions,

isomorphisms

of a n e l a s t i c

atlases

to d e s c r i b e

II can be carried

In other words,

an e l a s t i c

of an open set

local r e f e r e n c e ative to r(p) independent

by

material

of p,

Sr(F)

reference

chart

VpeU. ~

z Sr(p)(F,p),~

as regards

defined

situation.

is a pair

U c B and a smooth

response

and

know that the t h e o r y

over to the p r e s e n t

configurations

the e l a s t i c

how e l a s t i c

for B are c o n s t r u c t e d ;

the former kind of atlas we a l r e a d y

consisting

is g u a r a n t e e d

defined

reference

of C h a p t e r

(IV-36)

field of a n e l a s t i c

existence

field

on N . P We are now in a p o s i t i o n

anelastic

global

whose

IV, is a smooth

o e(p,t) -I,

(U,r)

field r(')

of

on U such that rel-

function

S (p)(F,p)

is

We set

VFeGL(3),

pen

(IV-37)

~

and call

S

the elastic

response

function

relative

to

~r

~2) in B l o o m & W a n g [19] such charts were called m a t e r i a l charts; we r e n a m e them r e f e r e n c e charts here so as to be c o n s i s t e n t with the d e f i n i t i o n s p r e s e n t e d in C h a p t e r II and in the o r i g i n a l paper of Wang [ 7 ].

128

(U,r);

the r e l a t i v e

elastic

Gr - Gr(p),

of course,

two elastic

reference

Vp~U. charts

S (F) = S-(F), V F s G L ( 3 ) r

~

r

¢ = {(U erence

charts

erence

atlas

metry

that

is

II, we call

(U,r)

and a m a x i m a l

compatible

if

collection

compatible

elastic

ref-

B : ~U is called an elastic refI ~ If ~ is an e l a s t i c r e f e r e n c e atlas on

on B.

= Sr~(F) , ~ I

and G~ - G r ~ , ~ I

the elastic

to @ and G¢ is t e r m e d

response

the r e l a t i v e

;

function elastic

sym-

of ~.

From the theory

every

and

(U,r)

such that

S¢ is called

group

Remarks

of

in Chapter

(U,r)

~sl} of p a i r w i s e

B then we set S¢(F)

relative

As

group

~

,r~),

naturally,

symmetry

presented

smooth m a t e r i a l l y

be e q u i p p e d

with

an elastic

in C h a p t e r

uniform

II it follows

anelastic

reference

atlas

body

}.

B can

The r e l a t i o n A

ship b e t w e e n B, as well

any two e l a s t i c

as the various

the c o r r e s p o n d i n g relative

elastic

set-theoretic

symmetry

we r e c a l l

II also

we say that

w h i c h hold among

response

functions

are d e l i n e a t e d

and

in C h a p t e r

KeG} iff KG} = G} in the

at this point,

and local h o m o g e n e i t y

carry over to the p r e s e n t

B is locally

each peB there

that

~ and ~ on

sense.

of h o m o g e n e i t y

Chapter

elastic groups

It should be obvious, tions

atlases

relationships

relative

II; in p a r t i c u l a r ,

reference

exists

elastically

a neighborhood

that the definipresented situation,

homogeneous N

P

in i.e.,

if for

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

129

~K: Np ÷ R 3 such that chart.

Similarly,

exists

elastic

Having

in a p o s i t i o n reference

Et(")

which

we fix teR,

chart

let %(t):

is

defined

then a pair

of an a n e l a s t i c materially

B ÷ R 3 be a m o t i o n

on Dp,

response

VteR

(U,r(',t))

on U (at time t) is called

t, i.e. , S tr(p,t)(F,p),

anelastic

of

of any p~B in the m o t i o n function

and each peB.

consisting

configurations

an a n e l a s t i e

response

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

reference

function

of p,

If

of an open

field of local r e f e r e n c e

if the r e l a t i v e

w h i c h go

atlas we are now

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

by the a n e l a s t i c

U c B and smooth

defined

ingredients

reference

that the r e s p o n s e

is e h a r a c t e r i z e d

is a

chart.

of an e l a s t i c

Thus,

if there

(B, ~ )

for the smooth a n e l a s t i c a l l y

u n i f o r m body B. B and suppose

homogeneous

the e s s e n t i a l

to d e s c r i b e

atlas

reference

~: B ÷ R 3 such that

referenee

reviewed

into the m a k e - u p

is an elastic

B is e l a s t i e i t y

a configuration

(global)

set

(Np, K~)

at time

~pEU.

In

such a case we set S~(F) ~ ~

~ S~ tr ( p , t ) ( F , p ) ,

and we call S t the r e l a t i v e

(iv-38)

VpsU

anelastie

response

function

anelastic

symmetry

group

~r

of

(U,r(.,t));

(U,r(.,t))

is defined,

G t ~ Gt ~(p,t)' elastic

the r e l a t i v e

VpeU.

and a n e l a s t i c

of course,

by the c o n d i t i o n

Concerning

the c o n n e c t i o n

reference

charts

of

that

between

on B, and their

130

corresponding

relative

T h e o r e m IV-4

If

relative

elastic response

St ~ S ~ r

functions,

we can now state

(U,r) is an e l a s t i c r e f e r e n c e

anelastic reference function

response

function

~r t h e n

chart w i t h r e l a t i v e

(on GL(3))

chart w i t h

(U,r(-,t))

anelastic

iff the s m o o t h field

r ~

is

response ~(-,t) ~

d e f i n e d on 8 by

~(p,t)

e r(p,t) -I o ~(p),

Vpeu

(iv-39)

is a field of a n e l a s t i e t r a n s f o r m a t i o n s Proof

at time t.

We w i l l p r o v e the t h e o r e m g o i n g in just one d i r e c t i o n ;

the p r o o f

in the o t h e r d i r e c t i o n w i l l be left as an e x e r c i s e

for the reader. erence

So,

suppose

chart w i t h r e l a t i v e

that

(U,r)

is an e l a s t i c ref-

elastic response

function

S ~ r ~

and that ~(p,t)

E r(p,t) -I o r(p), ~ p s U ,

of a n e l a s t i c t r a n s f o r m a t i o n s that

(U,r(',t)) ~

at time t.

is a n e l a s t i c r e f e r e n c e

anelastic response

function

defines

We w a n t to show chart w h o s e r e l a t i v e

St coincides with r ~

So, e m p l o y i n g

(IV-3), w i t h

r(p)

a field

S

on GL(3).

~ r

÷ For(p,t),

F an a r b i t r a r y

e l e m e n t of GL(3), we get

Et(For(p,t)) p ~ ~

: E (For(p,t)oe(p,t) p ~ ~

: E (For(p,t)or(p,t)-ior(p)) p ~ ~ : Ep(For(p))~ ~ d~f ~r(p)S (F,p)~ ~

: S (F)

(IV-40)

131

But Et(For(p,t)) d~f S t p ~ ~ ~(p,t)(~,p)

S t ~r(p,t)(F,p)

= Sr(F),

from which the required The compatibility erence

charts

and

(U,r(',t))

that

motion ~(t)

reference

atlas

= S~(F),

~

,r~(',t)),

~I)

in

reference VFeGL(3).

r

~

~(t) of B at time t (in the

~(t) of B) is then defined to be a maximal

= ~(U

ref-

is defined

for elastic

St(F) r

An anelastic

Q. E. D.

for two anelastic

condition

we require

(IV-41)

is immediate.

condition

(U,r(-,t))

namely,

VFeGL(3)

result

analogy with the similar charts,

so

of pairwise

collection

compatible

anelastic

~

reference

charts.

G~t = Gr~,t V~E!; anelastic elastic

these

response

symmetry

t V~I we set S~t = Sr~,

Naturally,

are called,

function

respectively,

and

the relative

of ~(t) and the relative

group of ~(t).

The following

an-

set of

~

exercises

concern

different

anelastic

formation

rules which

are the direct

themselves

with the relationship

reference

on B and the trans-

apply when we change

counterparts

when we make a change

atlases

of the results

from one elastic

between

atlases; which

reference

they

apply atlas to

another. Exercise

23

Show that at each time t~R in each motion

of B there exists [hint:

use Theorem

an anelastic

reference

IV-4 and the existence

atlas

~(t).

theorem for

~(t)

132

elastic

reference

Exercise

atlases

~]

24 Show that any two anelastic ^ ~(t) are related by

#(t) and

reference

atlases

A

~(t) for some

= K~(t)

KeGL(3);

deduce

S t ÷ S ~ and g~t ÷ g Now, rules

in the above

exists

each motion

~(t)

respondence

between

GL(3); = {(U

,r~),

corresponding

~(t)

where

~(-,t)

elastic

exists

relative

by virtue

= {(U

,r~( ")~

response

response

will

that

atlas

smooth

S~'

reference

atlas

would

coratlases on

on B, a

be given

~el}

global

under

in

if

by

(IV-42)

field of an-

on 8; of course,

not be invariant

~ on

a one-to-one

reference

o ~(',t)-l),

functions

corre-

atlases

IV-4,

at

functions

S~ and S~t agree

of Theorem

reference

~(t)

functions

and anelastic

is any p a r t i c u l a r

atlases

reference

is an elastic

ane!astic

a one-to-one

such a one-to-one

elastic

elastic

transformation

correspondence

there

of B we can establish

~el}

for

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

relative

via the r e q u i r e m e n t

in fact,

laws

reference

know that

between

associated

that

anelastic

associated

B and their

and ~(t)

consequence

exercise

Since we already

spondence

~I}

as ~ ÷ ~.

between

time t and their

~K°re(''t))'~

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

it is a direct

correspondence

S~.

= {(Ca,

this

a change

of the

133

f i e l d e(',t).

Note also that b e c a u s e

of the i n i t i a l c o n d i -

t i o n s a t i s f i e d by all a n e l a s t i c t r a n s f o r m a t i o n r~(.,t) eel;

E re(.)

o e(',t) -I + re(.),

in this sense, we h a v e

point

on,

if we use the same n o t a t i o n s

the a s s u m p t i o n that the charts are r e l a t e d via Elastic Let

~ and

atlases in t h e s e

F r o m this

~(t) to d e n o t e

on B we do so w i t h respective

atlases

(IV-42).

and A n e l a s t i c M a t e r i a l

B be a s m o o t h m a t e r i a l l y

By an e l a s t i c m a t e r i a l

connection

Connections u n i f o r m a n e l a s t i c body. on B we m e a n

c o n n e c t i o n H on B such that the p a r a l l e l tangent

for each

~(t) ÷ ~, as t÷-~.

e l a s t i c and a n e l a s t i c r e f e r e n c e

6.

as t÷-~,

functions,

spaces r e l a t i v e

an a f f i n e

transports

to H, a l o n g any curve

of the

~ c B, are

elastic material

isomorphisms.

all the m a t e r i a l

of C h a p t e r II d e a l i n g w i t h m a t e r i a l

nections

particular,

connections

on an a n e l a s t i c

we can state that every

form anelastic body

B can be e q u i p p e d w i t h

connection.

Also,

of h o m o g e n e i t y c e r t a i n kinds

connections

B to be l o c a l l y

condition

of

§6 of

between

the p r o p e r t i e s

and the e x i s t e n c e

of

on an e l a s t i c b o d y

B, may be c a r r i e d o v e r to a n e l a s t i c b o d i e s e s s a r y and s u f f i c i e n t

in

an e l a s t i c

all the r e s u l t s

and l o c a l - h o m o g e n e i t y of m a t e r i a l

body;

s m o o t h m a t e r i a l l y uni-

C h a p t e r II, c o n c e r n i n g the r e l a t i o n s h i p

body

con-

on an e l a s t i c b o d y may be c a r r i e d o v e r h e r e to

elastic material

material

It s h o u l d be c l e a r that

(e.g.,

for an a n e l a s t i c

elastically-homogeneous

a necsolid

is the

134

existence

of a flat e l a s t i c m a t e r i a l

is an e l a s t i c r e f e r e n c e

atlas,

connection).

(U,r) a r e f e r e n c e

in ~, and F = ~,o r -I the d e f o r m a t i o n K~

(where

~: B + R 3 i n d u c e s

B) then, A

FBC(X D)

the c o o r d i n a t e

by v i r t u e of T h e o r e m 11-3,

are the c o n n e c t i o n

gradient

symbols

If chart

f r o m r to

system

(XA) on

the f u n c t i o n s of an e l a s t i c m a t e r i a l

c o n n e c t i o n H on B iff the m a t r i c e s C _-IA.__~FB + C E [Y C t~xD FEDFB)],

are c o n t a i n e d

D : 1,2,3

in g~, the Lie a l g e b r a of G%.

We now w a n t to turn our a t t e n t i o n an a n e l a s t i c m a t e r i a l

connection

an a f f i n e c o n n e c t i o n H(t) transports

in each m o t i o n

connection

I, f r o m I(O) to I(T),

tions

on B, at each time

c o n n e c t i o n H.

H(t);

Thus,

Also,

let

the c o n v e r s e

let I = I(T) (13) be a

d e n o t e the p a r a l l e l

TsT, w h i c h

that

H on B gives r i s e to

connection

is also true.

curve in B and let p(T)

and ~(.,t)

To show that t h e r e exists

c o n n e c t i o n H(t)

unique anelastic material

material

I ~ B, are an-

~(t) of BD we may d e m o n s t r a t e

each elastic material

statement

is d e f i n e d to be

on B such that the p a r a l l e l

isomorphisms.

an a n e l a s t i c m a t e r i a l

of this

on B; this

d e f i n e d by H(t), a l o n g all curves

elastic material

tER,

to the c o n c e p t of

transport

is i n d u c e d by some e l a s t i c %(t) be any m o t i o n of B

a s m o o t h field of a n e l a s t i c t r a n s f o r m a t i o n

d e f i n e d on B in the m o t i o n

(13) 0 -< T -< T

for some T > O.

along

%(t).

func-

If for e a c h tsR

135

we define a map pt(r):

BI(0) + ~l(T) by

pt(T) ~ e(l(T), t) 0 p(T) o ~(~(0), t) -I then, by virtue of (IV-36), isomorphism~

conversely,

(IV-43)

pt(T) is an anelastic material

if H(t) is an affine connection on

B such that the parallel transports along l, from l(0) to l(T),

TsT, are given by anelastic material isomorphisms

pt(T) then the maps p(T) defined by (IV-43) are elastic material isomorphisms.

Thus,

(IV-43) establishes

a one-to-

one correspondence between elastic and anelastie material connections on B. In order to determine the connection

symbols of H(t)

in terms of those of an associated elastic material connection H (and the components anelastic transformation

of the particular field of

functions ~(.,t) which appears in

(IV-43)), we choose a reference configuration

K : (XA) and

note that the maps p(T) and pt(T) are characterized by the equations of parallel transport dvA(T) dT

+

A ID(T))vB(T) dlC(T) FBC( dT

= 0

(IV-44)

and duA(T) + A )dlC(T) dT FBc(ID(T)' t)uB(T d-}

where the components of l(T) in the coordinate

0

(IV-45)

system ( ~ )

have been denoted by IA(T); the components of the connection

136

symbols FB CA

of H and H(t) have been denoted,

A D (X D) and rBc(X , t) respectively.

v(T)

is a vector

transport

= p(T)v(0)

field u(T)

because,

: ~(I(T),

= p

:

Conversely,

fore,

via parallel

t)v(T)

space

of (IV-44);

is a solution

BI(0))

then the of (IV-45)

by (IV-43),

u(T)

v(T)

is obtained

is a solution

= e(l(T),

by

Now suppose that

(from the vector v(0) in the tangent

so that v(T) vector

field on I which

of course,

if u(T)

= e(l(T),

o

t

(T)

t

t)op(T)v(0)

o ~(~(0),

(IV-46)

t)v(0)

(~)u(O)

= 0t(T)u(0)

t)-lu(T)

satisfies

is a solution

if we denote the components

(IV-45)

of (IV-44).

of ~ relative

then There-

to (XA)

A D b y - ~B(X , t), then

uA(T)

will be a solution of (IV-44).

of (IV-45)

Assuming

we now substitute

E~dv A ~ALdT

: ~[(ID(T),

that

(IV-47)

t)vB(T)

whenever

vA(T) into

(IV-47)

vA(T)

is a solution

is a solution

(IV-45)

of (IV-44)

and obtain

-IA $~[ F D B dl e + ~ F (~-~ + FDC ( -,t)~B)V dT- = 0

If we now compare

(IV-44)

with

(IV-48)

we easily get

(IV-48)

137

FAc(.)

= ~FIA(.,t)(

~F(''t) ~X C

A which we may solve for FBC(.,t) Theorem

IV-5.

connection nection

The connection

F D + FDC(',t)aB(',t)

so as to obtain

symbols

H on B and its associated

H(t) are related

A F B C ( ,• t )

of an elastic anelastie

terms

A ~-IF = ~F ( • ,t)( B (',t) ~X C

Not only do (IV-49) of H and vice versa

formation

function

material

+

F FDC

and

(IV-50)

(relative

~(',t))

(.)~-

3X C

determine

to the anelastio

to-one

correspondencd

a change Now, tions

(IV-50)

H and H(t), which

is not, of course,

of the a n e l a s t i c t r a n s f o r m a t i o n it is possible

FBcA(-,t),which

tion symbols the functions

to verify

connection

reference

(U,r) and

charts

material

on B.

con-

is given

function

under

~(.,t).

that the func-

(IV-50),

are the con-

connection

are the connection

elastic material

trans-

The one-

invariant

directly

are defined via

of an anelastic A FBC(')

H(t) in

but in view of the intial

between

symbols

on B iff of an

To do this we select

(U,r(-,t))

(IV-50)

3e~(',t)

-

A t) + FBC A (.) as t+-~. for ~ we have FBC(.,

and

con-

ID (.,t)) B

dition

by (IV-49)

material

via

-iD(.,t)( F ")~9(-,t) = ~ B FDC(

Remark

(IV-49)

in ¢ and ¢(t),

138

respectively,

so that r(.,t) = r(.) o ~(',t) -I

where e(-,t) is a smooth field of anelastic transformation functions defined on B.

If F(.) ~ ~,or

--1

•

is the deforma-

tion gradient from r to K, (where K: B + R 3) then the deformation gradient F(.,t) from r(.,t) to K, is given by

F(',t) = F(') o r(t,') o ~(',t) o r(-,t) -I

(IV-51)

or, in component form (relative to the coordinate system (XA) which is induced on B by K)

(IV-52)

FB

Using

(IV-52) and (iV-50) we now compute that

-IA ~F~(.,t) C )F~(-,t)) F C (.,t)( + FED(.,t ~X D

_ =

$F~(.)

FcIA ( ) ( ~ .

C )F~()) + FED(•

so that the matrices

-IA ~F~(.) + FCD(.)FE(.)), [Fc ( 7

are contained in g~ iff the matrices

D = 1,2,3]

(IV-53)

139

D = 1,2 3] [FcIA(- • ,t) ( 8F~(.,t) + p~D(.,t)F~(.,t)) 8X D ' ,

are contained in g~(t)(~g~; by virtue if Theorem IV-4).

note that G~ = G~(t),

VtcR,

But this last condition,

relative to the Lie algebra g~(t) of G~(t), is precisely ~

~

the one which guarantees that the functions

A

FBC(-,t) are

the connection symbols of an anelastic material connection on B. Now let v(.) be any smooth vector field defined on B. A direct computation,

using

(IV-50) and the definition of

covariant differentiation,shows

that the eovariant deriv-

ative of v(.), relative to H, is related to that of the vector field ~(.,t)v(.), ~(.,t)(svB(-) 8X--~

relative to H(t), via B .)vC()) + FCD( •

8 A(.,t)vB(.)) = 8_~(~B

+ FCD( A .,t)~(

(TV-54)

,t)vB(°)

A B A B if we hold t fixed and denote eovaror ~BV,D : (~B v )ID' iant differentiation with respect to H and H(t) by "," and "I", respectively;

an equivalent form of (IV-54) would

B = (~ ~IAvB), D be ~BIAvlDExercise 25

Verify the relations

given above for covariant

differentiation of the contravariant vector field v('). Show that the analogous results for a covariant vector

140

field w(') are e~IB WB,D : ( ~ I B w B ) ID and ~AWBIDB = (~AWB)B'D" Verify that for an arbitrary tensor field V with components V AB C the appropriate formulas

are

A -IQ B,PR , A -IQ B,,PR, ~P~C ~RVQ,D = ~ P ~ C ~RVQ )]D

(IV-55)

-IA Q -IB. PR , -IA Q -IB. PR, ep eC~R VQl D = <~p eCeR VQ ),D By using

(IV-54) and its variant,

it is a simple matter

to prove the following theorem, which relates the torsion tensors of H and H(t): Theorem IV-6

Let e and e(t) denote, respectively,

the torsion

tensors of H and H(t); then

A e c(.,t)

:

eBC(-)

D -!A

D -IA

+ ~B ~ DIC -

~C a DIB

A ) _ ~-IDA : OBC(" B aD,C Proof

(IV-56)

+ ~-IDA C ~D,C

In terms of the covariant differention operation

with respect to H and H(t), we may rewrite the two equations appearing in (IV-50) in the form

A FBC(''t)

A -IDA = FBC(') - ~ B eD,C

A D -IA : FBC(') + ~B (~ D C

(IV-57)

141

The d e s i r e d

results,

i.e.

(IV-56),

now follow

from the d e f i n i t i o n

A ~ FBC A A eBC - FCB.

that

smoothly

e(.,t)

depends

We note

directly in p a s s i n g

on t in each m o t i o n

¢(t) of

8 and that we also have

e(',t)

Exercise sults

26

of the

smooth

+ e('),

Prove

as t ÷-~

Theorem

IV-6 by first

last e x e r c i s e

function

f(')

B ~Af[BD

= ~f

to the g r a d i e n t

+

27

the Ricei

Apply

of an a r b i t r a r y

B ~Af,BD

-IB -IB A f,BD : ~ A I D f l B

Exercise

the re-

so as to obtain

,B

and then a p p l y i n g

applying

+

to the c o v a r i a n t

vector

field v(.)

-IB e A flBD

identity

the g e n e r a l

fields

(IV-58)

to

result

derivative

so as to o b t a i n

(IV-58)

(IV-55)

for t e n s o r

of an a r b i t r a r y

smooth

the result

vB : C -IGIEVB + ~ -IB, A ~ BAvG.JIDE ,DE ~D ~ C ,G

Using the result state

following

Theorem

IV-7

the c u r v a t u r e

of E x e r c i s e

natural

Let ~(.) tensors

A ~BCD (''t)

companion and ~(',t)

(IV-59)

27, above,

we can now

to T h e o r e m

IV-6,

denote,

of H and H(t);

A -IF E . = ~E e B ~FCD ~')

i.e.,

respectively,

then

(IV-60)

142

w h e r e the c o m p o n e n t s

of Q are g i v e n by

A A SFBD ~ B C D - ~X C

Proof

~FABC + A E ~X D FECFBD

the p r o o f may be o b t a i n e d

A

E

- FEDFBc

via a d i r e c t

computation

b a s e d on the above

A and f o r m u l a for 9BCD

(IV-57)

a p p l y i n g the Ricci

i d e n t i t y to

the f o r m e r p r o c e -

dure gets q u i t e i n v o l v e d Two c o r o l l o r i e s First

@BC ( A ") ~ 0 i t

therefore,

~ABCD(',t)

curvature-free dependent

= v

.

A

O B C ( ' , t ) ~ O;

in g e n e r a l ,

H(t)

by v i r t u r e of

~ 0 iff ~ (') ~ O, so that H(t)

is

iff H is; this can also be d e t e r m i n e d

c o n d i t i o n for the

v(p)

then,

On the o t h e r hand,

of the r e l a t i o n

= 0 which

it s h o u l d be c l e a r that

does n o t f o l l o w t h a t

we n e e d only note that

r V,B

of (IV-56)

if H is t o r s i o n - f r e e

is not t o r s i o n - f r e e . (IV-60),

and is not r e c o m m e n d e d .

f o l l o w f r o m the last two theorems.

of all, by v i r t u e

e v e n if

(IV-59);

or by

(IV-60)

~ and ~(.,t),

~ = 0 is the n e c e s s a r y

local e x i s t e n c e

s a t i s f y an i n i t i a l

However,

between

we h a v e

in-

of s o l u t i o n s

i.e.,

and s u f f i c i e n t to the s y s t e m

c o n d i t i o n of the f o r m

already

shown

(i.e.

(IV-46))

~O

that v

is a s o l u t i o n of v A

A of UjB = O. for some of

It f o l l o w s

: 0 iff u = ~v is a s o l u t i o n

i m m e d i a t e l y that ~(.)

(and h e n c e all)

teR.

As a f u r t h e r c o n s e q u e n c e

(IV-50) we also have

T h e o r e m IV-7

A FAB(.,t)

~ 0 iff ~ Q(-,t)

A (.), = FAB

VtcR

= ~ 0

143

Exercise

28.

Prove Theorem

IV-7 by establishing

the

formula A FAB(',t)

A - FAB(')

~ [C( (log det D " ,t)]) ~X B

-

and then using the fact that ~(',t) det[~[(-,t)] 7.

i.e.,

= I,

Anelastic

Solid

Bodies;

Dislocation

Let B be a smooth materially body;

is isochoric,

the results

of §6, Chapter

Motions

uniform anelastic

II lead one to believe

that some very interesting

results

may be obtained

special

solids,

namely,

kinds of anelastic

and isotropic

solids

begin with suppose

and, indeed,

that

~(p,t)

B is an anelastie

must be unique

since the identity g(p)

component

is the trivial

for two

solid crystals

such is the case.

body and that pEB; then the anelastie function

solid

solid crystal

transformation

in each motion

~(t) of B

~(p) of the symmetry

group consisting

To

group

soley of the map id B P

(Recall that if ~(p,t) function formation

is ~n~ anelastic

for p in the motion functions

form ~(p,t)

~(t) ÷ idBp as t÷-~). ~

~(t)

Also,

tion H on B must be unique in fact,

determined

anelastic

material

mined by (IV-50);

%(t) then all anelastic

for p in the motion

o ~(t) where

via

transformation

c g(p),

%(t) are of the ~tER,

and

the elastic material

(where the connection

(II-18))

connection of course,

trans-

connecsymbols

are,

so that the associated

H(t) is then uniquely both the elastic

deter-

and anelastic

144

material

connections

symbols

of the unique

anelastic (IV-53)

solid

are c u r v a t u r e - f r e e . anelastic

crystal

body

and the fact that

[ F - c A ( - , t ) ( -~F~(',t) -~ aX D is the

zero m a t r i x

Because

both H and H(t)

may use the t o r s i o n respectively, anelastic Definition

IV-5

with the unique

Let

solid

dispensed

~(t):

@(.,t)

be the t o r s i o n

material

solid

with the case where

B ÷ R3, we may define

and an i n t r i n s i c

anelastic

anelastie

solid point;

isotropie

anelastic

elastic

referenee

in

isotropic

H(t)

crystal motion

to

associated on a body

B.

in B if

B is an a n e l a s t i c at the special solid body.

As

§3, at each time t in a m o t i o n

an i n t r i n s i c metric

elastic

metric

m

m t on B

for the special

solid point we may

atlas

on an

therefore, tensor

a dislocation

we

to c h a r a c t e r i z e ,

connection

anelastic

exists

indicated

on page IV-38.

B; we are led,

B is an a n e l a s t i c

we have a l r e a d y

for D = 1,2,3

,t)]

crystal we now want to look b r i e f l y

case in w h i c h

via

and a n e l a s t i c i n h o m o g e n e i t y

body

uniform

on an

and c u r v a t u r e - f r e e

@(.) and

@(.,t)

Then we say that there

Having

are unique

anelastic

smooth m a t e r i a l l y

= {0}~ so that

see the a r g u m e n t

tensors

crystal

connection

can also be d e t e r m i n e d

~(t)

the e l a s t i c

solid

material

+ FED('C,t)F~(

(i.e.

The c o n n e c t i o n

w h e r e p is any P case where p is an choose

an u n d i s t o r t e d

~ and its c o r r e s p o n d i n g

anelastie

145

reference

atlas

¢(t) and t h e n

set

m(u,v) ~ r(p)u • r(p)~, ~u,vEBp

(IV-61)

m t ( u., v ) . = r (. p , t ).u

(IV-62)

and

where

(U,r)

and

(U,r(-,t))

•. r ( p ,.t ) v ,

are any r e f e r e n c e

in ¢ and ¢(t), r e s p e c t i v e l y , Remark

on U it f o l l o w s (IV-62),

functions

= r(-)

(IV-16).

[19],

IV-8

are r e l a t e d via

t

, as d e f i n e d by

of T h e o r e m s

The R i e m a n n i a n

(IV-61)

b o t h m and m t are s m o o t h and r(.,t)

are smooth.

w h i c h we carry o v e r d i r e c t l y

is a c o n s e q u e n c e

Theorem

Also,

as the r e f e r e n c e m a p s r ( . )

The f o l l o w i n g r e s u l t ,

contained

0 e(',t) -I

at once that m and m

satisy

charts

such that peU.

Note that since r and r(.,t)

r(-,t)

and

Vu,veB p

from

II-4 and 11-5 of C h a p t e r II:

connections

H R and HR(t)

as-

s o c i a t e d w i t h the m e t r i c s m and m t are the u n i q u e t o r s i o n free e l a s t i c m a t e r i a l

connection

and a n e l a s t i c m a t e r i a l

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

solid body

homogeneous

B; the b o d y

F r o m the s t a t e m e n t obvious

B is l o c a l l y e l a s t i c a l l y -

iff H R is c u r v a t u r e - f r e e

elastically-homogeneous

that,

iff HR(t)

and B is l o c a l l y an-

is c u r v a t u r e - f r e e .

of the t h e o r e m above it s h o u l d be

just as we u s e d the t o r s i o n t e n s o r s

w i t h the u n i q u e

an-

e l a s t i c and a n e l a s t i c m a t e r i a l

associated

connections

146

on an a n e l a s t i c local the

elastic

solid

crystal

and a n e l a s t i c

curvature

tensors

to c h a r a c t e r i z e

anelastic

inhomogeneity

B.

Unfortunately,

equations

(IV-50)

the c o n n e c t i o n

~(t):

dependent

of H R and those

lose the

Exercise

29

which

We now want

elastic

the s t a t e m e n t

charts

function

to HR(t)

via

to d e t e r m i n e

symbols

(U,r) and

In fact,

in a given and in-

we t h e r e f o r e

and a n e l a s t i c

e(-,t)

material

that

can exist

relative

if

no an-

to w h i c h

(IV-43). the r e l a t i o n s h i p of HR(t).

(U,r(',t))

= K, or

of

above by s h o w i n g

of H R and those

F(')

solid

(IV-43).

spectively, (with ~ u n d i s t o r t e d )

where

quantities;

and

between

are unique

while m t is not then there

H R is i s o m o r p h i c

we choose

of these

between

transformation

connection

are not all unique

is given by

Verify

m is E u c l i d e a n elastic

choice

isomorphism

connections

anelastic

of HR(t).

H R and HR(t)

HR

of e l a s t i c

we may not avail o u r s e l v e s

B ÷ R 3 while

of the

we may use

when we seek the r e l a t i o n s h i p

symbols

the

connections

the d i s t r i b u t i o n

on an i s o t r o p i c

~, %(t), m, m t, and e(.,t) motion

inhomogeneities,

of the R i e m a n n i a n

and HR(t)

body

body to c h a r a c t e r i z e

between

the

Once again

in ~ and ~(t),

re-

and set

-i

F(',t)

= K, or(-,t)

K: B ÷ R 3 induces

coordinates

-I

(XA) on B; the r e l a t i o n

147

ship between F(') and F(',t) is given by (IV-S2). the connection symbols

Now,

of H R are, of course, the Christoffel

symbols of m, i.e.,

~mBD + ~mCD {~x---C~x B

{B~} = ½m AD

where m

AB

~mBC

and mAB are the contravariant

ponents of m in the reference the connection t of m , i.e.,

and covariant

configuration

K.

~mBD(',t) ~X c

It is a simple matter to show that

~mCD(',t) +

~X B

com-

Similarly,

symbols of HR(t) are the Christoffel

{ A c} (.,t) : ½mAn(-,t)(

the following

(IV-63)

~x D }

symbols

~mBC(',t) )

~X D

(IV-61) and (IV-62) have

covariant and contravariant

component re-

presentations:

AB m

CD_A~B =

~

YC~D

-IC -ID mAB = ~CD F A F B

mAB(.,t)

A t)F~(,t) = ~ CD FC(',

m A B ( ,•t )

= 6CD F -IC A (-,t )F-~D(.,t)

which are related to each other via the transformation

rules

(IV-64

148

( r e l a t i v e to some c h o i c e of a s p e c i f i c formation function

m

AB

(',t)

anelastie trans-

~(',t))

A

= ~C(',t)~

(',t)mCD(.) (IV-65)

mAB(''t)

Note that

-IC -ID : ~ A (',t)~ B (''t)mcD(')

(IV-65)

and the i n i t i a l

that m t + m as t÷-~; that HR(t)

(IV-65)

it then f o l l o w s

÷ H as t ÷-~.

relation between into

approach becomes

from

(IV-63)

and

implies (IV-64)

Now, we can o b t a i n the d e s i r e d

{~C } and

(IV-64)

c o n d i t i o n on ~(',t)

{B~}(',t)

if we s u b s t i t u t e

and t h e n m a k e use of

extremely

involved;

may be b a s e d on the p r o p e r t i e s

from

(IV-63) but this

an a l t e r n a t e a p p r o a c h

of the c o v a r i a n t

derivative

and

leads to the f o l l o w i n g T h e o r e m IV-9 connections m

t

The C h r i s t o f f e l

of the R i e m a n n i a n

H R and HR(t) , a s s o c i a t e d w i t h the m e t r i c s

, respectively,

{BC A } (.,t)

symbols

are r e l a t e d by

= {A } + ~ ~HA ~ J D~L(~ -1E B a - 1DY ,;,C

+

m and

-IE -IF C e D

),B

, -IE -IF~ -

~

B

a

C

;

(zv-66)

,D]mHJmEF

A

where

~(.,t)

are the

formation function Proof:

components

of some a n e l a s t i e t r a n s -

for B in some m o t i o n

Following Wang

& Bloom

%(t).

[19] we d e f i n e a t e n s o r f i e l d

149

£(',t)

on <(B)

via

AAc(.,t)

Clearly,

- {Ac}(.,t)-

A A ABC = ACB.

Once again we hold

covariant

differentiation

" , " and

"l", respectively.

on B and w is a covector

with

respect Then

field

(IV-67)

{BAc}(-)

t fixed

and denote

to H R and HR(t)

if v is a vector

simple

computations

by

field produce

A _ vA A C VlB ,B = ACBV

(IV-68)

and WAIB

These

results

theorem; and

will

however,

(IV-69),

C = -AABW c

not be needed a result,

which

mABiC(.,t)

WA,B

is needed

- mAB,C(',t)

(IV-69)

until we come to the next

in the same here

direction

as

(IV-68)

is

= -mDB(',t)A~C

(IV-70)

D

which

is equivalent

mDA(',t)ABC to

mDB(.,t)A~C(.,t)

D + mDA(.,t)ABC

(IV-71)

: mAB,C (.,t)

since m t is h o r i z o n t a l

with

respect

to HR(t)(i.e.

mABjC(',t)

= 0

150

because m t, being an intrinsic must be invariant anelastic variant

inner product,

under all induced transformations

material

isomorphisms).

derivative

(IV-652)

anelastic

with respect

If we now take the coto H

R

on both sides of

and use the fact that m is horizontal

to HR, i.e., that mAB,C(.) ,

mAB,C (''t) = ~ Combining

(IV-71) and

mDB(-,t)A~c

-IE

via

with respect

= O, we get -IF)

A ~ B

)

(IV-72)

,CmEF ("

(IV-72) we have

(IV-73)

+ mDA(.,t)ATc

-IE -IF, = (~ A ~ B J,CmEF (') and so if we use the symmetry

condition

on A and solve for

A ABC we find

Aic(''t)

• -IE

= ½mAD(''t)[t~

-IF,

B e D ~,C

(IV-74)

+ (~-IEc ~-IF'D),B _ ( -IEB -IF),D]mEF(')C A Dr, -IE -IF = 2C~H(~J L ~e B ~ D ) ,C , -IE -IF, , -IE -IF ,D]mHJ( + t~ C ~ D ),B - te B ~ C ) ")mEF(') from which the required result follows. Exercise

30

Verify that the right-hand

Q. E. D. side of (IV-742)

is

151

independent

of the choice

m and the a n e l a s t i o Now,

of the i n t r i n s i c

transformation

as the d i s t r i b u t i o n on an i s o t r o p i c

characterized

by the r e s p e c t i v e

and HR(t)

the f o l l o w i n g

cise r e l a t i o n s h i p of c o n s i d e r a b l e Theorem

iV-10

associated,

these

The c u r v a t u r e

respectively,

the pre-

tensors,

is

~(.,t)

and ~ w h i c h

and H R, are r e l a t e d A

are by

(!V-75)

components

field on B, we easily

vAIB,

where

compute

A D D A = ADCVlB - A B C V L D

derivative

of

(IV-76)

(IV-68)

with

to H R we obtain

A VIB,C

and a d d i t i o n

-

delineates

of H R

A D - ADBAEC)

field w i t h

If we now take the c o v a r i a n t

A

tensors

AEB,C- AEC,B

A A V[B c - VlB,c

vIBC

tensors

A A = QECB + (

For the t e n s o r

is a v e c t o r

respect

in-

solid body may be

two c u r v a t u r e

with HR(t)

A D + ADCAEB

v(.)

~(.,t).

and a n e l a s t i c

curvature which

metric

interest:

A ~ECB ( ,•t )

Proof:

anelastic

theorem,

between

function

of e l a s t i c

homogeneities

elastic

of

_ vA = AA D AA D ,BC DB,C v + DBv,C

(IV-76)

and

(IV-77)

(IV-77)

then p r o d u c e s

A = AA vD + A D + A D v,BC DB,C ADBV,c ADcVIB

D A - ABCVID

(IV-78)

152

The proof is now completed by using

(IV-68) to rewrite

(IV-78) in the form

V

A A -A E IBC - v,BC = AEBCV

(IV-79)

A D + A D + (ADBV,c ADCV,B-

D A ABCV,D )

A + ADCAEB A D - ABCAED, D A and then taking the where -A AEB c ~ AEB,C skew symmetric part of (IV-79), with respect to the indices B,C and using Ricei's identity. Exercise 31 the formula Exercise hint:

32

Follow the directions

given above and verify

(IV-75). Show that

(AB)

(',t)

= (~B)

('),

VtsR.

from (IV-65) and the fact that ~(',t) is isochorie,

taR, if follows that det[mAB(-,t)]

= det[mAB(')] , V t a R

so that the result follows immediately if we can verify the formula

~X B

8.

(logJdet[mcD])

= {AAB}

Equations of Motion for Anelastic

Bodies

The derivation of the field equations of motion for smooth materially uniform anelastic bodies is a very simple matter once we have, at hand, the corresponding equations

153

of motion

for smooth materially

On the other hand, anelastic

bodies

equations

(11-24)

chart

(U,r(.,t))

to derive

digectly,

uniform elastic

the equations

bodies.

of motion

for

we would begin with the Cauchy

and then select

an anelastie

~ %(t) and a global

reference

configuration

K: B ÷ R 3.

In place of (11-23) we would have

T}(x) ] as F(.,t)

is the deformation

the F(-) appearing gradient

in (11-23),

atlas which

that by virtue St r(.,t)(F,.)

on U coincides

and all FsGL(3)

'

To complete

the steps analogous through~l-2~; replacing

the response

the major difference

material

connection

H(t).

is not required

for all teR

S$] appearing

in

appear

we need only follow from

(11-24)

is that in addition the

in Chapter

to

(elastic)

II by an anelastic

As it turns out, a direct

deri-

for we can simply make the required

~ ~ ~(t),

field equations

Note also

function

to those used in proceeding

H employed

global

¢(t).

~ by ~(t) we would now replace

i.e.,

the deformation

the same as those which

connection

changes,

with

the derivation

to <,;

e ~, the elastic

with Sr(.)(F,-),

material

vation

(U,r('))

so that the functions

are, essentially,

in 11-23.

IV-4,

from r(.,t)

is, of course,

is associated

of Theorem

(IV-80)

gradient

from r(.) to <, where

reference

(IV-80)

: S}(FP(x,t)) ] q

F(.) ÷ F(.,t),

of motion(II-39;

and H + H(t),

in the

for the sake of

154

convenience

we rewrite

these

equations

below

in the form

~iA k ,B " ..i jkXA,B~j + pb I : px where

~(t):

B ÷ R 3 is r e p r e s e n t e d

tions

xi(xA,t) and i ~xi(xB;t) xA : 3X A --

In (IV-81) rel~tive

k B denotes XA,

to some

a global

the

covariant

material

C FAB

field

defined

func-

derivative

connection

of x~

H, i.e.,

~x k

A FBC are the connection

the

by the d e f o r m a t i o n

X~ 3xA(xJ~t) l : ~x I.

elastic

k ~2xk XA, B -- 3xA~xB

where

'

(IV-81)

symbols

-iA of H; also Hjk is

by

~iA = iB i G~)F ~ jk Hjk([XGFH] ' where

H jk iB = ~Sj/3FA" i k

formation vative

of x~ with respect

derivative from

from

If in (IV-81)

A ) A FBC(. ~ FBC(-,t)

F(-) + F(-,t)

in c o m p o n e n t

= F(')

then

respect

out the trans-

the covariant

to H is replaced

k k XAi B (of x A with

~ ÷ ~(t) we also make

we carry

by the

to H(t)).

deri-

covariant

As we go

the t r a n s f o r m a t i o n 0 r(',t)

0 e(',t)

=

0 r(-,t) -I

making

155

these various changes in (IV-81) we get the global field equations of motion for a smooth materially uniform anelastic body, i.e.,

~iA k X B " ..i jk XAIB j + pbz = px

(IV-82)

where ~iA HiB( I F G A C jk E jk [XF~GFH])~cFB

Recall now (i.e. , eq. k = 1,2,3.

(IV-83)

Ak Ak (IV-51))that ~CXA]B = (C~cXA) ,B for

If we substitute this result into (IV-82)

then we get ~iA k D X~ + pb i pii jk(XD~A),B ~ =

(IV-84)

~iA iB ~ I F G )~B jk = Hjk(LXF~GFH ]

(IV-85)

where

A quick glance at (IV-84),

(IV-85) shows that this system

of equations is formally the same as those for a smooth materially uniform elastic body

(i.e.,

(IV-81)) except that

the variable x~ has been replaced throughout the system by IF the variable XF~ G. Exercise

33

Derive the variant of the system of equations

(II-33) which is applicable for a smooth materially uniform anelastic body B.

156

Chapter ~. i.

Thermodynamics

and Dislocation Motion

Introduction In this final chapter we want to generalize

chanical model of anelastic response allow for thermodynamical

the thermodynamics

bodies,

[20].

points

chapter.

(alternatively,

we introduce

and their associated

global structures Finally,

the "entropy

the respective

(inhomogeneous)

field equations

thermoelastic

bodies with uniform symmetry. discern,

in-

and anelastic

geometric structures

of balance

structures. to derive for both

bodies and thermo-anelastic As the reader will soon

by replacing the previous

concept of material

uniformity with the concept of symmetry uniformity, be substituting

Secondly,

isomorphism and use

thermoelastic

we use the induced geometric

for the assumption that the material

of our body have the same material

of

flow rules

inequality").

and their associated

global thermodynamic

functions

Clausius-Duhem

the concept of symmetry

it to generate

First of all,

on the constitutive

which are implied by the well-known equality

and on a re-

Three kinds of basic

results may be found in the present we derive those restrictions

our work here is

[36], which dealt with

of inhomogeneous

cent paper by Bloom and Wang

thermo-anelastic

in such a way as to

influences;

based on an earlier paper of Wang

the me-

we will points

response the weaker as-

sumption that they have the same material

symmetry.

157

2.

The Concept

of a Thermoelastic

Point and the Entropy

Inequality. Let B be a material is a thermoelastic

point ~

local configuration function

body and let psB; we say that p

~(p)e~p,

s, the entropy

~(p) are determined

if the stress

at p, in any

as well as the internal

energy

n and the heat flux vector q at

by constitutive

equations

which are

of the form

~S = ~p(~(P)'

O(p), g(p))

(V-I)

0(p),

g(p))

(V-2)

n : ~p(~(P),

8(p), g(p))

(V-3)

9 : [p(~(P)'

e(p), g(p))

(V-4)

: ep(~(P),

where

8(p) is the local temperature

local temperature ~(p);

gradient

at p in the local configuration

in a local motion we would,

~(p) ÷ ~(p,t), subscript

8(p) ÷ 8(p,t),

on the constitutive

(V-4) indicates

at p and g(p) is the

the explicit

of course,

replace

and g(p) ÷ g(p,t). functions dependence

on the point p; of the new functions

The p

in (V-I) through of these functions

appearing

and h

above

are scalar-valued for each pcB while £ takes p ~P values in R 3 for each psB.

~

see, also,

example

(ii) on page III-4.

~p its

158

As we have done in the previous write our constitutive

equations by introducing a preferred

local configuration r(p)¢D ~

chapters we may re-

p

which we will call, as usual, a

local reference configuration.

With F = ~(p) o r(p)

-i

as

the local deformation from r(p) to v(p) we may rewrite (V-I) through

(V-4) in the form

~S = ~r(p) (F'e(p)'~ g(P)'~ p)

e = Er(p)([,6(p),

(V-I')

g(p), p)

(V-2')

n = Hr(p)(F,e(p),~ g(p),~ p)

_

= Lr(p)(F,e(p),~ g(p),~ p)

(V-3')

(V-4')

where Sr(p)(F,~(p), . .

g(p), . p) .E Ep(For(p), .

for all FeGL(3) and the other functions through

e(p), g(p))

appearing in (V-I')

(V-4') are defined in a similar manner; note that

the dependence of the response functions on p has been shown explicitly by grouping p with the arguments of these functions. When considering thermoelastic bodies

(i.e., material

bodies composed of thermoelastic points) it is not sufficient to restrict ones attention to mechanical balance laws such as the principle of balance of momentum, which yields, as

159

a consequence

(given

the q u a n t i t i e s

sufficient

involved)

or the p r i n c i p l e

as a c o n s e q u e n c e

tensor

~S"

We must

second laws principal

Cauehy's

of b a l a n c e

yields

also c o n s i d e r

of energy b a l a n c e inequality;

two p r i n c i p l e s

can be shown

- tr(Ts

demonstrated

duced e n t r o p y strictions through

consistent I.

of m o t i o n

of momentum, of the Cauchy

the s o - c a l l e d

which

11-24,

which Stress

first

and

are also t e r m e d the

and the entropy a combination (i.e.,

on

see

inequality

of these

or

latter

[ 5 ]) to lead to the

grad v)~ + q'ge -< 0

by Coleman

inequality

on the

(V-4')

conditions

inequality ~

p(~+ne)

It was

of moment

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

entropy

equations

the s y m m e t r y

Clausius-Duhem

reduced

smoothness

(V-5)

constitutive

if these

leads

[37] that the re-

to the

functions

constitutive

with the second

the v e c t o r

and Noll

(V-5)

following

appearing

relations

re-

in (V-I')

are to be

law of t h e r m o d y n a m i c s (16)"

function

~r(p) must

satisfy

the in-

equality Lr(p)(F,e,g,p)~

• ~g ~ 0

~

(V-6)

v r e p r e s e n t s the v e l o c i t y in some motion ¢(t) of B and ~ e~q8 is called the f r e e - e n e r g y f u n c t i o n ; p is the d e n s i t y in the c o n f i g u r a t i o n ¢(t) at time t. (i@

for a d e t a i l e d

proof

see

[37].

160

for each psB and all (F,e,g). II.

the functions S r ( p ) ,

E (p) and Hr(p) a r e a l l

in-

dependent of the variable g(p) for each p~B. III.

there exists a function ~r(p)(F,8,p) called the

free-energy

(see footnote (15) below) such that for each

psB Sr(p) (F,e,p) = pF ~ ~

~r(p) (F,e,p)

Em(p)(F,e,p) = ~r(p)(F'e'P)~

T

(V-7)

(V-8)

~

+ e ~

Cr(p)(F,e,p)_

and Hr(p)(['e'P)

= ~

Cr(p) (F,e,p) ~

(V-9)

The symmetry group Gr(p)(p) is defined to be the set of all maps KeSL(3) such that for all (F,e)

~(p)(FK,e,p)

s

Exercise 34.

Show that

= 9{(p)(F,e,p)

(V-10)

,

Gr(p)(p) is contained in the sym-

metry groups of Er(p) and Hr(p). Remark

It has been proven by Truesdell [38] that KeGr(p)(p),

the symmetry group of Sr(p) iff

161

Sr(p)(FK,e,p)

+ ~r(p)(K,e,p)

so that, in general,

(V-ll)

= ~r(p)(F,0,p)

- Cr(p)(l,e,p)~

Gr(p) ~ Gr(p)(p).

We now make a basic assumption which will be crucial for the development

of the material

theory for thermoelastic with uniform symmetry,

geometric structure

and thermo-anelastic

material bodies

i.e., we will assume that the sym-

metry group of L (p), for any peB, coincides with G (p)(p), the symmetry group of ~r(p)

(we note here that the results

of Coleman and Noll [37] do not relate,

in any definite

way, the symmetry group of ~r(p) to those of either ~r(p) or Sr(p).)

Under this assumption

the symmetry groups of

mr(p) and ~r(p) will be closed Lie subgroups

of SL(3) since

the free energy function ~r(p) is usually assumed to be a smooth function for each peB. The transformation introduced

laws for the various symmetry groups

above closely resemble those which we have met

in the previous

chapters,

i.e., if ~(p) e 0

local reference

configuration

P

is any other

of peB and we transform from

r(p) ÷ ~(p) then G~(p)(p)

= LGr(p)(p)L-I

(V-12)

where L E ~(p) o ~(p)-i and Gg(p)(p)

= LGr(p)(pgL-I

(V- 1,3)

162

Both (V-12) and (V-13) may be proved by introducing the symmetry groups g(p) and g (p) which are defined via

g(p) = {AeSL(Bp)

I Ep(v(p) o A, @(p))

E (v(p) s D p (at each fixed ~p ~ , 6(p)) VV(P) ~ value of 8(p))} and g (p) = {AESL(B p ) I ~(v(p) o A, ~

~(v(p), ~

8(p), p),

Vv(p) ~

@(p), p) =

E Dp , at each @(p)}

where ~(v(p),

-i @(p), p) a ~r(p)(F,8,p),~ F ~ v(p)~ 0 r(p)~

The relations

(V-12) and (V-13) now follow, respectively,

from the results Gr(p)(p)

= r(p) o g(p) o r(p)-i

(V-14)

Gr(p)(p)

= r(p) o g (p) o r(p) -I

(v-15)

which are a direct consequence of the definitions

of the

various symmetry groups involved and the relations which exist among their associated response functions. 3.

Geometric Structures on Thermoelastic

Bodies with

Uniform Symmetry In Chapters !I-IV we have based the construction of

163

different

kinds

of material formity;

of geometric

bodies,

in this

on various

concept

for several

concepts

of material

kinds uni-

section we want to show how a geometric

theory may be constructed exhibits

structures,

uniform

for a thermoelastic

symmetry.

body which

In order to make this latter

precise we first state

Definition

V-I

Let p and q be any two points which belong

to the thermoelastic s(p,q):

body B.

Then a linear isomorphism

Bp ÷ B q is called a symmetry

q if the symmetry

groups

g(q)

isomorphism

of p and

g(p) and g(q) are related

= s(p,q)

via

(v-16)

o g(p) o s(p,q) -I ~

We may note, consequence s(p,q)

first of all, that it is an immediate

of (V-14)

and the definition

E r(q) -I o r(p) defines

and q whenever

r(p)

a symmetry

and r(q) are local

above that isomorphism

(reference)

of p

con-

~

figurations Gr(p)(p)

of p and q, respectively,

= Gr(q)(q).

isomorphism

Conversely,

which

if s(p,q)

of p and q of the form s(p,q)

for some r(p)

£ Dp, r(q)

of Sr(p)(F,@,p)

satisfy is a symmetry

= r(q)

£ Dq, then the symmetry

and Sr(q)(F,@,q)~

must coincide,

-i

o r(p),

groups that is,

Gr(p)(p) = Gr(q)(q). Remarks

Recall that a material

(in the present

situation)

isomorphism

of p and q

would be a linear isomorphism

164

l(p,q): ~

B

p

÷ B

q

which

Eq(~(q),

satisfies

(i.e.,

@) = Ep(~(q)

~ for all ~(q) s Dq and each fixed 8 supscript

on I, in this case,to

it will depend,

in general,

isomorphic then we know (V-16)

on 8.

see

o ~ I(p,q),

equation

(II-3))

8)

(V-17)

8; we might even place a indicate the fact that If p and q are materially

(i.e., see equation

(11-8)) that

is satisfied with s(p,q) replaced by l(p,q);

fore, every material isomorphism. symmetry

isomorphism is, a priori,

Clearly,

same material

the fact that material

response

of different

points of B have the

are all the same,

classes

of the relative

symmetry

in the sense symmetry groups

point of B coincide with one another;

converse of this statement

is, in general,

we say that B is a thermoelastic

isomorphism satisfying

false.

the Naturally

body with symmetry uni-

formity if for each pair p,qsB there exists (symmetry)

isomorphism.

implies that the relative

groups of the body points that the conjugate

a symmetry

there is no reason why an arbitrary

isomrophism should also be a material

In other words,

there-

a linear

(V-16).

In order to the develop a geometric theory for thermoelastic bodies with uniform symmetry we introduce cept of a symmetry

chart

given the development symmetry

(U,r(.)) which,

in Chapter

i s o m o r p h i s m above,

the con-

as one would expect,

II and the definition

of

is a pair consisting of an open

165

set UcB and a smooth figurations

on U such that the symmetry

are independent (U,r('))

field r(') of local reference

of p,

~psU

were an elastic

that the relative dependent

of p, ~psU;

is a symmetry is usually

chart but,

not true).

as already noted,

If (U,r(-))

chart can not,

smooth global

field on B.

two symmetry

the converse chart

and we call Gr the symAs was the case in

maps r(.) defined on ref-

in general,

the lead of the previous

charts

(U,r(.))

UcB, the field of local configurations

in a symmetry

Following

be in-

is a symmetry

to r(').

II with smooth reference

erence neighborhoods

Sr(p)(F,@,p)

this would also imply that

group of U relative

Chapter

if

chart we would require

functions

then we set Gr = Gr(p)(p) , ~ p e U metry

G (p)(p)

(Note the difference:

reference

response

groups

con-

(U,r('))

and

be extended

chapters,

(U,r('))

to a

we call

compatible

if for

all peUnU Sr(p)([,O,p)

Clearly

(V-18) implies

and also that G fine a symmetry = {(U

,ra(')),

= S~(p)([,@,p),

that r(p)-i

0 ~(p)

~[,8

e g(p),

E G~; as should be expected atlas to be a maximal ~sl} ~

of mutually

(V-18)

VpsUnU

by now we de-

collection compatible

symmetry

~W we have repeatedly used the same symbol I in specifying the index set in each of the atlases used in this treatise; in general, these index sets will be different of course.

166

charts

such that ~ U e = B.

formation

Note that the c o o r d i n a t e

G ~ ~ r e o r~ B-I,

from r e to r~ B , is a smooth

on Ue n U6, for all e,Bsl, w h i c h G A ~ Gre,

response

takes

its values

ecl; we call G A the s y m m e t r y

to the atlas

A.

Obviously,

function

sA(~,e,p)

trans-

SA(F,e,.)

field

in

group of B r e l a t i v e

we can also define

a relative

on B w i t h

= Sre([,e,p),

Vp~U e, ~Ez

(v-19)

~

however,

this

now depends, relative

smooth

in general,

weakness

of d i f f e r e n t

distribution

of the

points

on peB;

instead

necessary

and s u f f i c i e n t

condition

of m a t e r i a l

condition

atlas

response

~A be a fixed

stated

of the m a t e r i a l

above

the c o o r d i n a t e

conditions

delineated

Exercise

35

. . = .{(Ue, KoA .

points

Show that Kore(-)),

function which peB.

a

atlas of r e l a t i v e

is in-

that G A is the s y m m e t r y functions G

group

relative

to

satisfy the

11-16.

for each KeGL(3), ~el}

is a s y m m e t r y

A is and verify the t r a n s f o r m a t i o n ~

iso-

From the d e f i n i t i o n s

transformations

on page

spaces

In fact,

for a s y m m e t r y

S A of r e s p o n s e

A and that

via s y m m e t r y

~A'

up the

that the t a n g e n t

is that the d i s t r i b u t i o n

it is i m m e d i a t e

for the d i s t r i b u t i o n

functions,

again points

isomorphisms.

A to be a m a t e r i a l

dependent

this

of B be c o n n e c t e d

morphisms

functions

of r e s p o n s e

rules

atlas

for B if

187

= GkA :

so t h a t

Vr,o,p

-1

G A is c h a r a c t e r i z e d by the

condition

and A is c h a r a c t e r i z e d by the d i s t r i b u t i o n response

functions

~A' i.e.,

KeGA<--> KoA =

of the r e l a t i v e

A~ = ~A <=> ~A = S~.

Now, as has b e e n p o i n t e d out by W a n g in [36], apart f r o m the fact that the d i s t r i b u t i o n of r e l a t i v e ~A d e p e n d s

on the m a t e r i a l p o i n t s

psB,

response

a s y m m e t r y atlas

is f o r m a l l y the same as the m a t e r i a l

atlases

treated

structure

in C h a p t e r

a thermoelastic thermoelastic

II; the g e o m e t r i c

b o d y by a s y m m e t r y

V-2.

atlas.

A symmetry

connection whose

# which were

(i.e.,

i n d u c e d on B is a s m o o t h

is, t h e r e f o r e ,

is i n d u c e d on a s m o o t h m a t e r i a l l y

e l a s t i c b o d y by a m a t e r i a l Definition

atlas

body with uniform symmetry)

same as that w h i c h

functions

the

uniform

In p a r t i c u l a r we make

c o n n e c t i o n on B is an a f f i n e

induced parallel

transports

are s y m m e t r y

isomorphisms. Exercise

36

(U,r(-))

£ A be a s y m m e t r y

ordinates

Let A be a s y m m e t r y chart.

for B and let

If K:

B + R 3 induces

(x i) on B and F = K~or - i is . the d e f o r m a t i o n

f r o m r~ to <~, p r o v e t h a t the symbols

atlas

of a s y m m e t r y

3

~x m

functions

co-

gradient

jk are the c o n n e c t i o n Fi

c o n n e c t i o n on B iff the fields of m a t r i c e s

+ FlmFk)]

m = 1,2,3

168

belong to gA' the Lie algebra of the symmetry group G A. hint:

follow the same argument

as that u s e d in proving

Theorem II-3. 4.

Thermodynamics

and Anelastic

Response

Let B be a material body and p a point of B. that,

in the previous

chapter,

Recall

we have defined p to be an

anelastic point if it is a quasi-elastic the instanteous

(anelastic)

by t r a n s f o r m i n g

a fixed elastic response

ing to the rule

(IV-3) where toER, r(p) is any element of

N

p

, and e(p,t) ~

is an isochoric

anelastic transformation process

response

point such that

function

E t is given ~p

function Ep accord-

a u t o m o r p h i s m of B

function;

p

called an

in any ri$id or rest

of p we assumed that ~(p,t)

= id

and that

Bp

~(p,t) ~ id B , as t÷-=, in any process of p. We also asP sumed in Chapter IV that the evolution of an anelastic transformation

function

e(p,t)

is governed by a flow rule of the

~

form

~(p,t)

= $(e(p,t),

~

t) and we examined

related to the uniqueness

of such t r a n s f o r m a t i o n

In order to generalize elasticity, mechanical

certain questions

~

the mechanical

which was presented

model of an-

in Chapter IV, to a thermo-

one we begin with the list, i.e.

of constitutive

relations

regard the constitutive elastic response

functions.

(V-I) - (V-4),

for a thermoelastic

functions

point p and

Ep, ep, hp, and lp as the

functions which hold for a thermoanelastic

point p in any rest process with constant

(9(p),

@(p), g(p)). ~

169

If the process is not a rest process then we assume that the instantaneous

response functions

related to ( E p , e p , h p ~ p )

(Ep ,ep,hp,£ t t t ) are

via

(v-2o)

(Et,e t ~ p p,hpt~p)(~(p)~ ~ , 6(p), ~g(P))

=

(~p,ep,hp~p)(~(p) o 9(p,t), e(p), ~(p))

where e(p,t), the anelastic transformation

function, is

governed by the flow rule associated with the thermodynamic process to which B has been subjected. We now want to determine the thermodynamical

restrictions

on the constitutive relations (V-20) which follow from the reduced entropy inequality

(V-5):

In order to proceed we

again choose a fixed local reference configuration r(p) in D

P

and rewrite the system (V-20) in the form

St et ht It (F,0,g,p) (~r(p)' ~(p)' ~(p)' ~~(p) ~ ~

(V-20')

-- (Sr(p), er(p), hr(p)' !r(p))(FAr(p) (t),@,g,p)

where F = ~(p) o r(p) -I is the deformation gradient relative to r(p) and Ar(p)(t)

= r(p) o e(p~t) o r(p)-i is the an-

elastic transformation

function relative to r(p); as r(p),

the preferred local reference configuration of p which we have chosen, will be held fixed in the analysis to follow, we will now suppress the r(p) subscripts which appear in

170

the response formation

functions

function.

and the relative

The free energy

anelastic

function

trans-

~ ~s-~0

then has the form = F(F,O,g,p) in any rest process

(V-21)

and

= Ft(F,@,g,p)

at any tsR in a general we easily

~ F(FA(t),@,g,p)

thermodynamic

A~(t),

a ] as the relations F and

(V-23)

process.

From

(V-22)

get

aFF~ = a;!

of

(V-22)

Ft .

which

A direct

exist

aFt - aF aF t a@ a@ ' ag i among

computation

= aF ag i

the various employing

(V-23)

gradients

(V-22)

and

now yields

•

•

(v-24)

+ ~F

= a F t .a ~F t Fa~CA-id aF b F b + aF b o d c

+

~F t

~ + aF t

ga

171

By substituting

(V-24) into (V-5) we easily obtain the

following results I.

both F and Ft are independent of g, i.e.,

= Ft(F,e,p) = F(FA(t),@,p) II.

both h and h t are independent of g and are related ~

to F and Ft via

ht(F,@,p)

= h(FA(t),@,p)

~(FA(t),

@,p)

Ft

De III.

(F,e ,p)

S and S t are both independent of g and are related

to F and Ft via

Sba(F,@,p)

= Sb(FA(t),@,p) = pF~A~(t)

=

~F(FA(t),e,p)b ~F d

a ~Ft(F,O~p) oF e ~F b C

IV.

I and it satisfy the inequality

£J(FA(t),8,g,p)gj

= /tJ(F,@,g,p)gj

and V.

A(t) satisfies the inequality F b-la stb(F'8'g)FCAd(t)A'lee - ~~ a e a

< 0 -

_< 0

172

Exercise

37.

Preform the substitution

(V-5) and verify the results Remarks with

If we compare the results

(V-6) through

elastic response anelastic

given above,

i.e.,

I-V,

(V-9) of §2, we easily see that the

functions

S, e, h, and £ of a thermo-

as those satisfied by the response

thermoelastic

point.

for a thermoanelastic relations

V above.

point p satisfy exactly the same thermodynamic

restrictions a

I through

of (V-24) into

(V-20)(or

restrictions

The complete

theory

of the constitutive

the flow rule,

functions which are

and the restriction

the flow rate which is contained

in condition

V above.

5.

Isomorphisms

in Thermo-

Symmetry Groups

and Symmetry

on

Anelasticity. Because the elastic response anelastic

functions

point obey the same restrictions

thermoelastic

of a thermoas those of a

point we define the elastic symmetry groups

by

g(p) = {AeSL(Bp) I (Ep,lp)(V(p)oA,

e(p), g(p))

= (Ep, lp)(V(p), e(p), g(p)), ~v,e,g} and

g (p) = {AESL(Bp) I (e,h,@)(~(p)oA, = (e,h,~)(v(p),

of

(V-4)), the thermodynamic

on the anelastic response

given by I-IV above,

constitutive

point p now consists

(V-I) through

functions

e(p))

O(p)), ~v,O}

173

assuming, as we have already indicated, that the symmetry group of lp, for each peB, coincides with that of Ep.

The

elastic symmetry groups g(p) and g (p) give rise, of course, to the anelastic symmetry groups gt(p) and g*t(p) via gt(p) = ~(p,t) o g(p) o ~(p,t) -I

(V-25)

and g*t (p)

Exercise 38

:

~(p,t) og* (p)

o e ( p , t ) -I

(V-26)

Use (V-25) and (V-26) to prove that

Asgt(p)<=> AeSL(Bp) and

: (Et,lt)(~(p) @(p), g(p)) ~p ~p ,

and that Aeg *t (p)<=> AsSL(B ) and ~ p {(e~,h tp,fpt)(~(p)~ o A, 8(p)) t : (e~, htp,fp)C~(p), e(p)),

V~,0}

where fp(~,8) = F(~(p) o r(p)-l, 8(p), p) and f~(~(p), e(p)) ~ fp(~(p) o ~(p,t), @(p)).

174

Relative r(p)

to the preferred

local reference

e Dp which we have singled

g(p),

g ~ (p), gt(p),

metry

groups

and (V-15),

out, the groups

and g ~t (p) are represented

* Gr(p)(p) , Gr(p)(p),

the first two of these

symmetry

respectively,

configuration

~

*t

G (p)(p), groups

by the sym-

and Gr(p)(p);

are given by (V-14)

while

Gt (p) = r(p) r(p)

0 gt(p)

0 r(p) -i ~

~

*t Gr(p)(p)

= r(p)

" o g~t(p)

o r(p) -I

so that KeGtr(p)(p)

<=> KeSL(3)

and

{(st,lt)(FK,8,g) = (st,lt)(F,@,g), V(F,@,g) *t for Gr(p)(p)

with a similar result Having metry

considered

groups

the constitutive

of an arbitrary

relations

thermoanelastic

will now assume that all the body points anelastic

and that B satisfies

stitutive

assumptions

A.

the elastic

are distrubuted thermoelastie

the following

functions

point peB we

of B are thermo-

which were delineated response

and sym-

global

con-

in [20]:

of the body points

on B in exactly the same way as those of a

body with uniform symmetry,

i.e.,

in all rest

175

processes

the global

as that w h i c h was B.

structure

formulated

in a general

evolves

smoothly

way as in the

(mechanical)

A and B above exists

context

imply

an e l a s t i c

A : {(U

B is p r e c i s e l y

§3 of this

,r~), ~ ! }

global

functions

anelasticity

with

structure,

theory

First

A for B w h i c h

Us, an open

same

structure through the same

of C h a p t e r

situation,

all of the following: atlas

global

e, in e x a c t l y

of the p r e s e n t

symmetry

the

chapter.

of B the e l a s t i c

into the a n e l a s t i c

transformation

the

in

procsss

the a n e l a s t i c

Within

of

IV.

hypotheses of all,

there

is a c o l l e c t i o n

set in B, for each eeI,

such that

U = B, and re(.) a smooth I c o n f i g u r a t i o n s on U w h i c h satisfies

field of local r e f e r e n c e

~

(i) sponse

there

exists

functions

a smooth

(SA,IA)

distribution

of r e l a t i v e

re-

on B such that

(SA,IA)(F,e,g,P) (V-27)

= (St e, lr~)(F,e,g,P) VpeUe , (ii)

Ve~I. the distribution

(SA,£ A ) ~~

has a u n i f o r m ~ e l a t / v e

symmetry group GA, . i . e . ,

GA = G r a ( p ) ( p ) , Secondly,

the a n e l a s t i c

in a g e n e r a l

process

~psU , global

VasI

structure

is o b t a i n e d

(V-28) of 6" at any time t

from the e l a s t i c

global

176

structure as follows:

there exists a smooth field of global

anelastic transformation

functions

e(',t) on B such that an ~

anelastic symmetry atlas A(t) may be defined by A(t) = {(Us, rS(.,t)), s¢l, rS(',t)~ = rS( ")~

ssI} where for each

o ~a(''t) -I on U s,

VteR;

this anelastic

symmetry atlas satisfies (i') response

there exists a smooth distribution functions

of relative

(§~(t)'- ~](t) ) on B such that

(S~(t),l~(t))(F,@,g,p)

(V-29)

= (S~s(p,t) ,l~s(p ,t) ) (F, @,g,P)

Vp~us,

V~I.

(ii')

t

t

the symmetry group of (~A(t)' ~A(t) ) is

uniform on B, i.e.,

t = Gr~(t,p) t (P) , V PeU s GA(t) Remarks

(v-3o)

sel

Note that, just as in Chapter IV, the relative

anelastic response elastic response and, moreover,

functions

functions,

coincide with the relative i.e.,

t t (§A(t)' ~A(t) ) ~ (~A'~A)

the relative anelastic

symmetry group G~(t)

coincides with GA; these facts are, of course, a simple consequence

of the fact that the elastic and anelastic

symmetry charts

(Us,r~( ~ -)) and (Us, ra(-,t)), ~

respectively,

177

are connected Note

via re(',t)

also that,

sponse

as in §3 of this t ~A and SA(t)

functions

pcB since we are working morphism 6.

rather

Structural Just

= re( • ) o e(',t) -I

depend

with

on a t h e r m o a n e l a s t i c

nection

on the tangent

kcB are symmetry transports

spaces

and let

symmetry

isomorphism. Bodies con-

B to be an affine

con-

along

induced

parallel

any smooth

curve

If we denote

(U ,r~(-)) atlas

iso-

symmetrY

of B whose

isomorphisms.

by p(T)

in A, an elastic

body

bundle

of the tangent

an elastic

re-

on the points

of symmetry

on T h e r m o - a n e l a s t i c

nection

transports

explicitly

of material

as in §3 we now define

Vp~U

the relative

the concept

than with that Connections

chapter,

VtER

these

parallel

be a symmetry

for B, with

kcU

chart

, then we

~

know that

p(T)

is a symmetry

mappings

Pt,~ E r (k(t))

G A which

pass

latter

condition

We have

already

(U ,r~(.))~ where

through

implies

that

in A~ by the d e f o r m a t i o n

A FBC

a global

connection

are elements at t=O;

gradient coordinate

~ ~A"

the charts

F~ ~ K~or ~ - I~, ~ system

condition

the connection

(X A)

for the symbols

H on B is that

C -IA-~FB C E [F C ~3X--~ + FEDFB)]

of

this

~-~ d Pt, e t=O

if we represent

and sufficient

(X D) to represent

iff the induced

element

that the vector

demonstrated

on B, then a n e c e s s a r y

a symmetry

o p o r-l(k(O))~a

the identity

K: B ÷ R 3 induces

functions

isomorphism

s ~A~ ' D = 1,2,3.

of

178

Recall now that in Chapter an anelastie connection

material

IV we were able to generate

connection

H by relating

H t from an elastic material

the parallel

pt(T) of H and H t, respectively,

transports

along any p a t h

p(T) and Xc8 from

l(0) to l(T), by pt(T)

where

e(',t)

functions

= e(l(T),t

) O p(T) O ~(X(0),t) -I

is some field of anelastic

(in a general

process

A of the elastic

atlas

symmetry

atlas A(t) of the anelastic

structure

same way, we may use the relation symmetry

connection

transformation

of 8) which maps the elastic

symmetry

anelastic

to the anelastic

structure.

from an elastic

In other words, if H is a structural

relative

to A which

the connection

H t whose

are given via (V-31) to A(t), where

parallel

induced

charts

an

symmetry

con-

connection

transports

parallel

is a structural

corresponding

In the

(V-31) to generate

nection.

induces

(V-31)

p(T),then

transports

connection (U ,re(.))

pt(T)

relative and

~

(U ,r~(.,t))_ (IV-39)

in A and A(t),~ respectively,

with ~(.,t)

functions

the field of anelastic

which appears

in (V-31).

are related by transformation

We may, therefore,

state

the following Theorem

V-I

anelastic (IV-39).

Let A and A(t) be, respectively,

symmetry

atlases

Then the functions

for B which A (xD,t) FBC

elastic

are related

and

via

are the connection

179

symbols

of an anelastic

to A(t),

symmetry

D = 1,2,3,

~F~(X~t)

C

XD

are contained

+ FED(

and the connection symmetry

,t

X,t))],

~ ~A' where the functions

of the deformation

~ K,ore(.,t) -I from r~~ to K,.

A FBC(-,t)

X F )F~( F

in ~A(t)

F (X,t) are the components

elastic

on B, relative

iff the matrices

[F-~A(xF,t)(

F

connection

connection

Moreover, A FBC(')

symbols relative

gradient

the functions of the associated

to A are related

by

(IV-50). 7.

Field Equations

for Thermoelastic

and Thermoanelastic

Bodies. The procedure global

which we will

field equations

form symmetry employed of motion

follow

for thermoelastie

is not very different

in Chapter

in order to derive

uniform

with uni-

from that which was

II, where we derived

for materially

bodies

global

elastic bodies.

equations Of course,

as we are dealing here with a thermodynamical

theory,

must take into account

of motion

not only the equations

which are to be satisfied tion of heat transfer; tions

for thermoelastio

followed global

field equations

by points of B but also the equa-

once the appropriate bodies

in §8 of Chapter

we

have been found the approach

IV will produce

which

global field equa

the corresponding

apply to thermoanelastic

bodies

180

exhibiting uniform symmetry. We begin by considering the equations of motion for a thermoelastic body B and assume that we have singled out a fixed elastic symmetry atlas A = {(U ,re(.)), eel}; in what follows we will suppress the subscript A on the relative response functions but we will retain the A subscript on G A and ~A"

Now let %(t): B + R 3 be an arbitrary motion of

B which induces time-dependent

coordinates

(xm(t)) on B.

If we denote by %t(B) the configuration of B at time t in the motion 4, then the stress tensor in %t(B) is determined, of course, as follows:

for psB we choose

(U ,r~(.)) e A such

that peUe and we set ~PF = @,p(t) o r~(p) -I.

Then, relative

to the standard basis of R 3, the components of the Cauchy stress tensor at p,at time t,are given by

TZ](%t(p))

= SZD(Fp,

@(p),p)

(V-32)

which is, of course, a local formula valid only for points pEU

(note that

Exercise 39

[~t(p)] i = xi(p)).

If, as usual, we set

HkliJ(F,8,P)~ ~ ~siJ/~F kl for all FsGL(3), where the F kl are the components of F relative to (x l) then

ij

Hkl(~'O'P

)rkmKl

m :

o,

181

and :

are both

consequences

S(FG,e,p)

:

of the fact that

S(F,O,p), ~(F,O,p)

if GsG A.

Prove these statements

by following

procedure

as that which was employed

Now, from the s u b s t i t u t i o n equation of motion

in Chapter

§9.

of (V-32) into Cauchy's

"

Hi +

+ pbl:

~S13/~8 and H i = ~si]/3x 3 .

~

II,

(II-24) we get

ij 8F kl "" Hkl 3x. + H~]gj ] where H~ ]

the same

..i px

(V-33)

The arguments

ij , H ~j , and H i are 5' 8(p), and x m , of course, Hkl ij the fields H~ 3 and H i satisfy the symmetry Hkl

of

and like

conditions

and

for

all

Exercise

(F,e,p). 40

dependent

(U ,r~(-))

Prove

of

the

~ A.

that

choice

the of

local the

fields

elastic

H~ j

and

symmetry

Hi chart

are

in-

182

As a d i r e c t . c o n s e q u e n c e follows motion

that

(V-33),

an elastic bols

of the above exercise, it ij ~F kl only the t e r m Hkl 3x 3 , in the equations of is a local

symmetry

relative

expression;

connection

H on B whose

• (x l) are functions

to

however,

i

Fjk(xm)

if we choose

connection then,

sym-

because

the matrices [F-Ii(3F~ • __ + j i)] m = 1,2,3 ] ~x m FlmFk '

must

lie in ~A'

it follows

t~at

ij 3F kl F k Fnl + ) : 0 Hkl(~xm nm We can,

therefore,

rewrite

(V-33)

(V-34)

in the form

ij nl k ij Hi " ..i - Hkl F Fnj + H@ gj + + Pb 1 = Px

Exercise

41

Show that

(V-35)

the field

-'" i J ( F~, 8 , x m ) F n l H~ In ~ Hkl is independent (U ,re(.)) (V-35)

of the choice

e A and thus

is a global

Now,

is not a convenient and x TM varies follow

prove

equation

as in Chapter

of the elastic that

valid

the equation for all points

II, the equation

one with which

with time

in a motion

the lead of Chapter

symmetry

of motion peB.

of motion

to work

since

~(t) of 8.

II and introduce

chart

(V-35)

F jk i = F~k(xm) Thus,we

a fixed reference

183

configuration ~: B + R 3 which induces coordinates B.

(XA) on

The motion ¢(t) is, once again, characterized by the

deformation functions x i = xi(xA,t), while the connection KA symbols FBC of H, taken relative to (XA), and the components K

•

K

~-i

FA] of F

-K,or~

are related, respectively,

tions Fijk and F m] "" by the equations preceding

to the func(11-29); sub-

stituting from these equations into (V-35) we get

~ijA (~2xk

k

KC

~xA~xB •

~xk) ~X B ~x j

(V-36)

tAB ~Xc

°

+ H~]gj + H i + pb I = p~i

where Hi jA Remarks

ij )~AI. HkI(F,8,xD

It might be instructive at this point to indicate

exactly how one would arrive at the appropriate

field equa-

tions of motion in terms of the Piola-Kirchhoff stress tensor K

T (which is related to the Cauehy stress tensor KA We begin by defining T S via T k = JT ~ ~xAI ' J = det[xi'A])" ~ ~x the Piola-Kirchoff response function P relative to A (actually, PA ) by

P(F,8,p)

~ (det

F)S(F,e,p)(F-I)

where the superscript T denotes

T

"transpose"

(V-37)

Solving

(V-37)

for S we obtain i P(F e,p)F T = (det F) - ~' ~

(V-38)

184

whose component form, when differentiated F, yields

with respect to

(i.e., see [363)

HiJ _ 1 j - i m + (~J6im - FiFlkl)pim] kl (det F) [Fmgkl

ij

with Qkl ~ ~pij/~Fkl"

Substituting

(V-39)

(V-39) into (V-36)

then yields ~iA _~2x k QkB ( ~xAsx B


(V-40)

"" + JHiDgjo + JH1" + p b i : PK x.i where ~iA _ QkB

K

i (det


K

ir FAFBI < Qkl (F'o 'XD) r F) KA =

FBC

-


(V-41)

(v-42)

and

~ iA

_

I

< (det F)

• < Ak pIk(F,0,xD)F

= JS ij(F,0,x m) ~xA

Note also that dH%3gj - ~28i A

jHi = I A ~XA

~ ~iA Zxm

< and gA

K : i xm,A where gA gi X,A

(v-43)

185

Now, let us turn our attention to the thermodynamical aspects

of the global field equations

body with uniform symmetry,

for a thermoelastic

i.e., to the derivation of a

global form for the usual e~uation of heat transfer,

div q + pr = p@~

where r respresents Remarks

(V-44)

the energy supply.

the equation of heat transfer

following way:

(V-44) arises in the

first of all we have the principle

of balance

of energy whose local field equation has the form

p~ = tr(Ts grad v) + div q + pr

where v = ~ x is the velocity

field.

determined by the free-energy

(V-45)

Since ~S and ~ are

function ~ via (V-7) and

(V-8)

we have p~ ~ tr(Ts grad v )

- pne

(V-46)

and (V-47)

The desired result,

i.e.

stitution of (V-46) and

(V-44),now

follows

(V-47) into

(V-45).

We now want to proceed with expressing

from the sub-

(V-45) in a

global form which will be valid for all points p in a thermoelastic

body with symmetry

uniformity.

The real

186

key to our work here is the assumption group of the heat flux function

that the symmetry

1 is the same as that of ~

the response

function

some fixed elastic

S, where both are taken relative

symmetry

to A, 1 is a smooth

global

atlas A on B; therefore,

to

relative

field on B with

~

/(F,e,g,p) . . . = lr~(p)(F,0,g) .

for any elastic that

peU

Exercise

symmetry

chart

(U ,r~('))

E A such

.

42

If we set £~k(F,0,p)~

E 9£i/~FJk,

show that

I i (FG O,g,p)G k l~ (F 0,g,p) jk ~~' m = jm ~' ~ '

VGeGA

(V-48)

V(F,O,g), and that VK~gA i ijk(F,0,g,p)F]mK~ hint:

: 0

(V-49)

use the fact that

zi(20,e, and all

(F,0,p),

,p)

=

A,

and then proceed

as in Chapter

If,

§9.

Once again, we denote the motion of B, which induces global time-dependent

coordinates

heat flux in Ct(B) is determined,

qi(¢t(p))

: /i(Fp,

x

i

on B, by ¢(t).

of course,

The

by

e(p), g(p), xm(p))

(V-50)

187

where Fp = ~,p (t) o r~(p) ~ -I.

Substituting

the equation

(V-44) yields the local

of heat transfer

(V-50) into

equation,

li aF jk + i + £~j ag$ + jk ax i £@gi ~ ax I which

is valid

Exercise

43

+ pr = p86

(v-s1)

(only) on U ; in (V-51) we have set

i l@

£

a£ l , liJ =

a9

g ~

al I =

~g. ]

Prove that l@, 1

of the choice of the symmetry

%1 l , l

=

i ax

, and I are all independent chart

(U ,r~(.))_ e A.~

As a result of the above exercise we are again faced with the problem of converting in our field equation,

only a single expression

i •e. I ijk ~aFJk

' from a local to a

0 2 ~

global elastic

form.

In order to do this we again introduce

symmetry

connection

H, whose

relative to (x i) are functions

connection

F ijk(xm);

an

symbols

then, as a con-

sequence of (V-49), we obtain from (V-51) and the condition

_F_I i aF~ [ j (--~ +

i i )] s rlmFk

~A'

m = 1,2,3

the system of equations

^i _nk j Ii + 113 - £jk ~ Fni + 8gi g •

.

agi[ + ax

1 + pr = pS~

(V-S2)

188

Exercise 44

Show that £~J ~ : £~n(F,8,g,xm)FJn4 is a global

field on 6 by using Exercise 45

(V-48).

If we again take K: 8 + R 3 as a fixed reference

configuration which induces a global coordinate system (XA) on B show that, relative to K, the system (V-52) assumes the (global) form

~iA 82xk I k ( ~xA~x --- B


i

+ £~j

~gj ~x i + £ + pr = peq

+ £eg i

~iA 7_" where £k : £fl(F'@'~ 'xm)FAZ'K-

Exercise 46

(V-53)

<

In terms of the heat flux vector @K taken relative

to the reference configuration

<, the equation of heat trans-

fer takes the form

Div q< + p
(V-54)

Show that if we define a heat flux function Q relative to the elastic symmetry atlas A by Q(F,8,g,p)

~ (det F)F -I £(F,8,g,p),

i so that £ = (det F) FQ(F,8,g,p),then

jk = (det F) [Fm~jk 4 i where -30"k ~ sQi/~FJk.

Substitute

-

(V-55) into (V-53) and

(V-55)

189 show that we obtain

as our new set of equations

~AB ($2xk gk 8xA~xB

+ j.Zigj

i + J£egi ~AB £k =

where

KC ~xk) FBA ----C ~X ~

A[B q~AB

;)g~ + j,$,, +

~x I

(v-56)

PE r

: p<8~ I

K < K Zklr (F'O'g'xm)~'AFBI~ r

i

(det F) Note also that

i JZQgi

~qAK -

28

jzij _..~L~g" _ ~qKA ~ZB g ~x I < @X A

< gA'

'

SgB A and JZ - --SqK where - ~X A attention hand

< i gA = gi x ,A.

to the e x p r e s s i o n

side of the equation

virtue

of the constitutive

striction

(v-g) we clearly

n(~t(p))

and two distinct pose

that

fine

a smooth

which

of heat

on the right-

transfer

(V-44).

(V-3')

By

and the re-

have

= Hre(p ) (F,8,p),YpeU

G A = GA; then

let us turn our

appears

relation

possibilities

global

Now,

arise.

(V-54)

,

First

of all sup-

it is easy to see that we may de-

field on B via

(V-58)

-

where

(Ue,£e(-))

s A is any elastic

s/mmetry

chart

such

190

that peC ~8)

If we drop the A subscript on the response ~

function

H and set Hij(F,8,p)

then we have,

~ ~H/~F ij, He(F,8,p)~ ~ ~H/38

~GcGA,

Hij(FG,e,P)G ~ ~ ~

(v-59)

= Hik(F,8,p)~

and

= H0([,e,p)

(v-6o)

for all (F,O,p). Exercise

47

Verify the results

contained in (V-59) and

(V-60). Now, i f

we d i f f e r e n t i a t e the component form o f (V-58)

through with respect to t we easily obtain

= HjkFJk + He@

(V-61)

where the first expression on the right-hand

side is a

local one, valid only for points peCe; in order to convert this expression

~@ peC

into a global one, which will be valid for

this follows, of course, from the fact that if n C8, where (U ,re(.)), (U~,re(.)) are both elastic

symmetry charts in A, then the coordinate transformations ~ 8 , from r6(.) to r~(.) will be elements of G~ which,by virtue of V-9,is the symmetry group of H A . The case G A = G A includes

all solid bodies.

~

191

all points

peB, we need only note that F II

Substitution

= F A3

from

~x 1 ~v 1 ~XA ~ x I

(V-62)

into

(V-62)

(V-61)

then produces

the

equa%ion

6 : H§ ~xl ~vJ + He 6

(V-63)

3 ~X A ~x I

where

H~3 = HJ k ( ~ ' 8 ' p ) F
course,

it may be the case that

not define

a smooth

Even in this (V-II)

case,

global however,

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

Hre(p)(FG,e,p)

field on B.

G A ~ G A so that

field

of response

(V-58)

functions

we know that we may (V-9)

Of does on B.

combine

so as to obtain

(V-64)

= Hra(p)(F,@,p)

+ Hr~(p)(G,@, p) - Hre(p)(l,6,p) ~

for all peU~ and all GeGA;~ if we again HA(F,0,p)

= Hra(p)(F,@,p),

for any

set

(U ,F~(.))

e A such that

~

peU

, then

yields

differentiation

the fact that

of

(V-64) with respect

the gradient

to F

of H A is a global

field

~

on B, i.e.,

Hij(F,e,p) for all

= Hre(p)i j (F,@,p)

(U ,r~( • )) e A such that ~

peU

.

(v-6s) We can now proceed

192

exactly

as we did in the case where

Finally, global

equations

transfer with

we want

to indicate

of m o t i o n

uniform

symmetry;

instead

is e s s e n t i a l l y

in C h a p t e r

global

of m o t i o n

uniform

anelastic

e l a s t i c body. metry

atlas

A = {(U

symmetry

re(.,t)

and re(.)

elastic where

F (.,t)

U •

A(t)

~e

be r e w r i t t e n

uniform

the elastic

= {(U via

,r~(.~t)), (IV-39).

~I}

sym-

an-

where

We also r e p l a c e

H by its a s s o c i a t e d

H t and t r a n s f o r m

an-

F (-) ÷ F (-,t)

o re(.,t) -I on

~

by the c o r r e s p o n d i n g 46.

the

for a smooth m a t e r i a l l y

: F (.) o r~(', t) o ~(.,t)

Finally we replace

Exercise

the same

IV where we d e d u c e d

we r e p l a c e

connection

connection

body

~sI} by its a s s o c i a t e d

are r e l a t e d

symmetry

symmetry ~e

,re(-)),

atlas

of heat

body with

from those of a m a t e r i a l l y

In other words,

elastic

the e l a s t i c

body

equation

of a t h e r m o e l a s t i e

the p r o c e d u r e

equations

obtain the

B is a t h e r m o a n e l a s t i c

one w h i c h was f o l l o w e d field

how we w o u l d

and the global

for the case where

uniform symmetry

GA = GA

the c o n n e c t i o n

connection

symbols

symbols

Show that the equations

A FBC (.) of H

A FBC(.,t)

of m o t i o n

of H t . (V-36)

in the form

ziA k ~i ~X B ~i .. njkXA, B + jB ) ~~x I + Hjsg i + pbj : ~xj

where

k B denotes XA,

relative

can

the c o v a r i a n t

to the c o n n e c t i o n

H and

derivative

(V-66)

~x k of x~ ~ ~

193

$Si(F,8,X K) ~iA : HiA. %,xK)FAB jk jk <x*°F,

jB

HjB(X:'c°F'e'xK)

=

:

j

=

~

A FB

(v-67)

~Si(F,8,X K) ] B ~X

(v-68)

~S!(F,8,X K)

3e : HjS(~*°['8'xK) :

J ~ ~e

If we now make the indicated transformations in the field equaiions of motion (V-66) and take into account the ~iA ~i ~i definitions of the global fields H j k , HjB, and Hj8 t h e n i t is a straightforward matter to show that these balance equations become

KjkXAl B +

~X I• +

(V-70)

]sgi + pb.3 = px.3

where ~iA jk

:

~iB . K. A_C H.. (x~aF,8 3~ .... ,x )aC~ B

(V-71)

~i ~i jB : H.-(x.~F ]S ~~~~' O,X K)

(V-72)

~i = H.~(x~eF,8,X K)

(V-73)

3 e

3 ~

.....

i A~B We recall that the argument x.aF~,~~has components XAeB~ C

and t h a t

FAB(XK,t)

o

we

also

make

note

194

of the relation Ak ~cXAjB

between

Ak = (ecxA),B

the covariant

symmetry

connection

metry connection be rewritten

derivatives

relative

H and the associated

H t then the balance

to the elastic

anelastic

equations

sym-

(V-70)

ean

in the form

~ iA.

C k. jk~AxC),B

~i + KjB)

~X B ~x i

(V-74)

~i = p~. + Kjeg i + pbj 3 In this

case ' the global

~iA :jk

SO that

H~(~*~'8'xK)F~

(V-74) is precisely

the variable equation~ Exercise

that

throughout

the

kB by the variable XBe A. 47

Derive the variant

of the system

function

(V-74) which

for the Piola-Kirchhoff

tensor.

Finally, equation body;

the same as (V-66) except

k • x A zn (V-66) has been replaced,

is based on the response stress

field ~iA jk is given by

we want to consider

of heat transfer

to accomplish

equation

the global

(V-44) assumes

form which the

in a thermoanelastic

this we first rewrite the global balance

(V-53) in the equivalent

form

195

kXA,B

~x m

£@gi

~

~x m

(V-75)

where £k~iA= £kB(x,F,@,g,X K)~

FBA

(V-76)

~i i £B = £B(x*F'@'g'XK)~ ~ ~

(V-77)

£e~i = £i(x,F,8,g,xK ) ~ e~ ~

(V-78)

£$J = £igj(x,F, e,g,xK)

(V-79)

iA ~£i(F,e~g~xK) and £k =
i ~£i(F~e~g,xK) £8 = 28

£ij = ~£i(F,e,~,X K) ~gj

i ~£i(F~6~g~xK) £A = ~XA are the gradients of the response function £ relative to the elastic symmetry atlas A. transformations

Once again we carry out the

A + A(t) and H + H t and in so doing we

196

mk XA] B

ax z

megi

(v-so)

8X I

g

where ml A, m~iB, me, ~i and ~i] "" are global fields given by g ~iA iB K. A~C mk = £'K (x~aF,e,g,X~=~~ )aCZ B

mB~i = £i(x,~F,e,g,X K . .). . 6 ~i i m e -- l e ( x , 2 F , e , g , X ~ ) and

~ij = £ i J ( x ~ F , e , g , x K ) g g .... If we now use the relationship

between the covariant

deri-

vatives with respect to H and H t then we can rewrite (V-80)

as

~iA. C k. + ~B ) ~xB + ~i + ~ij ~ + pr = peh (nk taAxC) ,B ax i mogi g ax z

(V-81)

~iA where the global field n k is given by

• ~iA nk = £kB(x,aF,e,g,X ~

K

A )FB,

i.e.,

(V-81) is the same as (V-75) except that, once again, i B the variable XAk is replaced by the variable XBa A throughout the equation. i.e.

We now rewrite the global expression

(V-63), in the form

for ~,

197

~A i ~v ] = hjx A + bee

where the global fields

A

hj = h

(v-s2)

hj ~A and h0 are given by

A (x.,oF,e,xK)F B~,,~

(v-83)

and

he : he(~*['e'xX) with B _ ~H(F,e,X K) h - ~H(F'e'xK) hj <" ' ~e

We again replace

A by A(t) and H by H t and obtain

Ai A B = ~~jXBC~

$v3

~x i

+ ~

(V-84)

8

where k~ and ke are given by 3

:A3 = hB(x~F,e,X K - ) ~~3

FBA

and k8 = hs(x~F'8'xk)'~"~~ Putting all our results together we have the following equation of heat transfer for a thermoanelastic

body with

uniform symmetry: ~iA. B k. + ~i) ~X c + ~i "" ~g~ nk t~AXB),C c ~x i msgi + ~g3:~ ~x z + pr ~A B i ~v 3 + ~ezS~) = pO(~j~AX B ~x i

+

pr

(V-85)

198

Exercise we

use

48

Derive

the r e s p o n s e

relative

to the

the

variant

function

reference

of

(V-85)

for the h e a t

configuration

~.

which flux

obtains vector

if

199

Chapter VI

Some Recent Directions in Current Research

We indicate here, just very briefly,

some recent trends

in research on anelasticity theory, dislocation motions, relations with plasticity.

and

In [39], Reinicke and Wang have

taken up the study of flow rules associated with anelastie bodies of the type considered by Wang and Bloom [19].

Recall

that the flow rule in the theory of anelasticity is the governing first-order differential equation ~ = ~ where Z(p,t) is the anelastic transformation E(p,t);Z(p,t) as t ÷ ~.

function in some local motion

statisfies the "initial"

condition ~(p,t)÷idBp

Some general results concerning the flow rule have

already been delineated in Chapter IV (§4). type of flow rule which is appropriate plastic response is considered. reference configuration,

In [39] a special

for theories of elastic-

With E(p) a fixed local

E(p,t) a local motion,

Ar(p)=E(p)oz(p,t)o~(p) -I the relative anelastic transformation function,

and F (p)(t)~E(p,t)oE(p) -I the deformation gradient,

Reinicke and Wang identify A (p)(t) -I as the plastic part of F (p)(t) and Er(p)(t)~Fr(p)(t)Ar(p)(t) F (p)(t).

as the elastic part of

At any instant t in the local process E(p,t) the

plastic part of the deformation gradient, A (p)(t)-l, is fixed St is just the and the relative anelastic response function ~E(p) elastic reponse function S (p) evaluated at the elastic part Er(p)(t) ~

for all Fr(p)(t) ~

"

[see IV-6 for the definition of S ~t( p )

200

In m a t e r i a l s

with

that the p l a s t i c process

elastic-plastic deformation

response

remains

and that the d i f f e r e n t i a l is a linear

function

the total

deformation

in a loading

the yield

criterion

anelastic

point

by a

tensor

to be linear

between

being

and Wang define

of change

Then ~(p,t)

configuration

correspond material

is an u n l o a d i n g

to a point

direction

~(p,t)

of a r b i t r a r y

direction

if

> 0

(VI-I) if (VI-I)

range

does not hold

is said to

of the e l a s t i c - p l a s t i c

if

as in this

case all i n c r e m e n t s

= 0 ~

(VI-2)

from E(p,t)

are u n l o a d i n g

incre-

But if ~(E(p,t),~(p,t))

then the subspace corresponds ections

criterion

~(p,t)

direction

in the e l a s t i c

For an

are c o n s i d e r e d

at time t in a process

~(~(p,t),~(p,t))

ments.

values

is a loading

~(~(p,t),~(p,t))[~(p,t)]

A local

the y i e l d

functions

on the rates

processes,

surface.

whose

of

criterion

and u n l o a d i n g

by a y i e l d

de-

increment

A yield

field ~ ( ~ ( p , t ) , ~ ( p , t ) )

local processes.

and ~(p,t)

of the p l a s t i c

process.

loading

assumed

in an u n l o a d i n g

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

defined

p Reinicke

constant

increment

formation

is used to d i s t i n g u i s h

it is u s u a l l y

of all tensors

to the tangent

~(p,t)

plane

on the p o s i t i v e

l(E(p,t),~(p,t))[~(p,t)]

~

% for w h i c h

~(E(p,t),z(p,t))[%]

of the yield

side of this

>0 are loading

surface

subspace

directions

as all dirwith

while

the

= 0

201

subspace

itself

and the n e g a i t v e

~(~(p,t),~(p,t))[~(p,t)] The flow rule

~ 0 are formed

in an u n l o a d i n g

process

~(p,t)

so that ~(p,t) loading

remains

process

the authors

~ is a f o u r t h - o r d e r

transformations cesses

~(p,t)

are c o n s i d e r e d

only i n d e p e n d e n t except

in this

a fixed exist

functional

both

special

relation

rule i n t r o d u c e d body m a n i f o l d

in a

loading

of pro-

and u n l o a d i n g

periods

~ is a field

transformation

case for w h i c h E(p,t)

(VI-4),(VI-5)

above has a global

function;

does not

with the "initial"

in order to d e t e r m i n e in each given process

to analyse symmetry

by showing

in ~(p,t)

is integrable,

and ~(p,t)

Z(p,t)

and Wang then p r o c e e d

and conclude

(VI-5)

together

be solved

function

are linear

Examples

case w h e r e

anelastic

(VI-5)

values

~(p,t).

on the flow rule by m a t e r i a l

frame-indifference

while

as a flow rule

field whose

between

transformation

Reinicke

ion p l a c e d

the r e s t r i c t -

and m a t e r i a l

that the local

representation

flow

on the whole

B.

In [40] R e i n i c k e waves

tensor

Z ( p , t ) + i d B p , t ÷ +~, must

the a n e l a s t i c r(p,t).

[39~ choose

special

and the flow rule

condition

in such a process

of ~ ( p , t ) , t h e

latter

has the form

= ~(~(p,t),~(p,t))[~(p,t)]

contain

as is the

directions.

(VI-4)

on the rate of change

which

by u n l o a d i n g

with

= 0

constant

~(p,t) where

side of the s u b s p a c e

in a m a t e r i a l l y

considers uniform

the p r o p a g a t i o n

smooth a n e l a s t i c

of a c c e l a r a t i o n body using

the

202

special flow rule formulated distinguishes

in [39].

a loading process

from an unloading process there

result two distinct propagation waves

in anelastic

bodies,

As the flow rule itself

conditions

one for loading waves

loading waves.

The propagation

generalizations

of the usual propagation

waves

conditions

and one for un-

turn out to be

conditions

for acceleration

in elastic bodies with the growth and decay of the amplitudes

of acceleration waves surfaces

computed by use of the theory .of characteristic

and bicharacteristics.

Reinicke

port equations for loading and unloading multiplicity

one and these equations,

the wave amplitudes

[40] also derives

acceleration

equations

trans-

waves of

which govern the variation

along the bicharacteristies

of the usual transport body.

in [40] for acceleration

are generalizations

for acceleration

In the course of the presentation

of

waves

in an elastic

in [40] the global equations

of motion for anelastic bodies and bodies with elastic range are derived following derivations

the general presentation

are formally

equivalent

[40] derives

propagating

one-dimensional

Finally we note

the amplitudes

of acceleration waves

opposed to the transport

type obtained by Bloom [41] in his study of

acceleration

[40] indicates

to the assumption

in

equation of B e r n o U l l i

in anelastic material bodies;as

equation of Riccatti

Reinieke

bodies only.

a differential

type~as the equation governing

[42]; these

to the one presented

Chapter IV (§8) for anelastio material that Reinicke

of Wang

waves in anelastic materials;

the difference

in [40] that

which is absent in presentation

in the two results

as is due

[ ~ ] ~ 0 in a loading wavejan assumption given in [41].

203

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(i)

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(2)

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(6)

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(7)

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(8)

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i pp.

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204

(i2)

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(i3)

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(i4)

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(i5)

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(20)

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