North holland series in applied mathematics and mechanics 21 an introduction to thermomechanics

NORTH-HOLLAND SERIES IN

APPLIED MATHEMATICS
AND MECHANICS
EDITORS:

E. BECKER
Institut fur
Technische

Mechanik

Hochschule,

Darmstadt

B. B U D I A N S K Y
Division

of Applied

Harvard

Sciences

University

W.T. KOITER
Laboratory

of Applied

University

H.A.
Institute

Mechanics

of Technology,

Delft

LAUWERIER

of Applied

University

of

Mathematics
Amsterdam

V O L U M E 21

N O R T H - H O L L A N D PUBLISHING C O M P A N Y - A M S T E R D A M · NEW YORK · O X F O R D


AN INTRODUCTION TO
THERMOMECHANICS

Hans ZIEGLER
Swiss Federal Institute of Technology,

Zurich

and
University of Colorado,

Boulder

Second, revised edition

1983
N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y - A M S T E R D A M · N E W YORK · O X F O R D


© N O R T H - H O L L A N D PUBLISHING COMPANY—1983

All rights reserved. No part of this publication may be
reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying,
recording or otherwise, without the prior
permission of the Copyright
owner.
First printing 1977
Second, revised edition 1983

PUBLISHERS:

N O R T H - H O L L A N D PUBLISHING C O M P A N Y
A M S T E R D A M OXFORD NEW YORK

SOLE DISTRIBUTORS FOR T H E U . S . A . A N D C A N A D A :

ELSEVIER SCIENCE PUBLISHING C O M P A N Y , Inc.
52 VANDERBILT A V E N U E
NEW YORK, N.Y. 10017, U . S . A .

Library of Congress Cataloging in Publication Data
Ziegler, Hans, 1910 - An introduction to
{North-Holland
series in applied mathematics
Bibliography: 2 pp.
Includes index.
1. Thermodynamics.
2. Continuum
I. Title.

thermomechanics.
and mechanics, 21)

mechanics.

QC311.ZE
531
76-973
ISBN 0-444-86503-9

P R I N T E D IN T H E N E T H E R L A N D S

PREFACE
C o n t i n u u m m e c h a n i c s d e a l s w i t h d e f o r m a b l e b o d i e s . I n its e a r l y s t a g e s it
w a s c o n f i n e d t o a few s p e c i a l m a t e r i a l s a n d t o p a r t i c u l a r s i t u a t i o n s , n a m e l y
t o ideal liquids or t o elastic solids u n d e r i s o t h e r m a l or a d i a b a t i c c o n d i t i o n s .
I n t h e s e s p e c i a l c a s e s it is p o s s i b l e t o s o l v e t h e b a s i c p r o b l e m , i . e . , t o
d e t e r m i n e t h e flow a n d p r e s s u r e distributions or the d e f o r m a t i o n a n d stress
fields in p u r e l y m e c h a n i c a l t e r m s . T h i s is d u e t o t h e fact t h a t t h e s o l u t i o n
c a n b e d e v e l o p e d f r o m a set o f d i f f e r e n t i a l e q u a t i o n s w h i c h d o e s n o t
contain the energy balance.
From

the

viewpoint

of

general

continuum

mechanics,

however,

p r o b l e m s o f t h i s t y p e a r e s i n g u l a r . A n y o n e w o r k i n g in t h i s field k n o w s t h a t
s o o n e r o r l a t e r h e g e t s i n v o l v e d in t h e r m o d y n a m i c s . T h e r e a s o n f o r t h i s is
t h a t in g e n e r a l a c o m p l e t e set o f d i f f e r e n t i a l e q u a t i o n s c o n t a i n s t h e e n e r g y

balance. Since p a r t of the energy exchange takes place as heat flow, the
a p p r o p r i a t e f o r m o f t h e e n e r g y b a l a n c e is t h e first f u n d a m e n t a l l a w o f
thermodynamics,

and

it b e c o m e s

clear

therefore

that

it is

impossible to separate the mechanical aspect of a p r o b l e m

generally
from

the

t h e r m o d y n a m i c processes accompanying the motion. T o obtain a solution,
the f u n d a m e n t a l laws of b o t h mechanics a n d t h e r m o d y n a m i c s must be
applied.

In gas dynamics and

in t h e r m o e l a s t i c i t y

this has long

been

recognized.
T h i s s i t u a t i o n h a s its c o u n t e r p a r t in t h e r m o d y n a m i c s . U n t i l r e c e n t l y t h e
i n t e r e s t in t h i s field w a s a l m o s t e x c l u s i v e l y f o c u s e d o n p a r t i c u l a r l y s i m p l e
b o d i e s , mainly o n inviscid gases, characterized by certain state variables as,
e . g . , v o l u m e , p r e s s u r e a n d t e m p e r a t u r e . I n o t h e r b o d i e s , h o w e v e r , o r if
v i s c o s i t y is t o b e t a k e n i n t o a c c o u n t , o n e is c o m p e l l e d t o u s e c o n c e p t s f r o m
continuum mechanics, replacing the volume by the strain tensor and the
p r e s s u r e b y t h e s t r e s s t e n s o r . It m a y e v e n b e n e c e s s a r y t o h a v e r e c o u r s e t o
the m o m e n t u m

theorems,

and

to account

for

the

kinetic energy

in

f o r m u l a t i n g t h e first f u n d a m e n t a l l a w . I n s h o r t , t h e r m o d y n a m i c s c a n n o t

be separated from continuum mechanics.
I n v i e w o f t h e s e s t a t e m e n t s it b e c o m e s c l e a r t h a t c o n t i n u u m m e c h a n i c s
and

thermodynamics

are inseparable: a general theory of

continuum


vi
m e c h a n i c s a l w a y s i n c l u d e s t h e r m o d y n a m i c s a n d vice v e r s a . T h e e n t i r e field
is t r u l y i n t e r d i s c i p l i n a r y a n d r e q u i r e s a u n i f i e d t r e a t m e n t , w h i c h
p r o p e r l y b e d e n o t e d a s thermomechanics.

may

S u c h a u n i f i e d t r e a t m e n t is t h e

topic of this b o o k .
In order to a m a l g a m a t e t w o b r a n c h e s of science, o n e needs a c o m m o n
l a n g u a g e . C o n t i n u u m m e c h a n i c s h a s a l w a y s b e e n a field t h e o r y , e v e n in its
r u d i m e n t a r y f o r m s like h y d r a u l i c s o r s t r e n g t h o f m a t e r i a l s . T o t r e a t e v e n
such a simple p r o b l e m as b e n d i n g of a b e a m , o n e m u s t recognize t h a t the
states of strain a n d stress d e p e n d o n position a n d possibly o n t i m e . T h e
o b j e c t o f t h e r m o d y n a m i c s , o n t h e o t h e r h a n d , h a s a l w a y s b e e n a finite
v o l u m e , e.g., a m o l e , a n d the state within the b o d y has been

tacitly

a s s u m e d t o b e t h e s a m e t h r o u g h o u t t h e e n t i r e v o l u m e . It is s u r p r i s i n g t h a t
this p h i l o s o p h y h a s been m a i n t a i n e d even at t h e age of statistical a n d
quantum

mechanics, although

it is c l e a r l y i n c o n s i s t e n t

with the

first

f u n d a m e n t a l l a w in its c o m m o n f o r m : A t least p a r t o f t h e h e a t s u p p l y
a p p e a r i n g in t h i s l a w is d u e t o h e a t flow t h r o u g h t h e s u r f a c e o f t h e b o d y .
A s long as this process goes o n , t h e t e m p e r a t u r e of t h e elements n e a r t h e
surface differs from the o n e of the elements further inside t h e b o d y ; the
s t a t e o f t h e b o d y is t h e r e f o r e n o t h o m o g e n e o u s .
There are t w o ways out of this dilemma.
T h e h i s t o r i c a l w a y , still d o m i n a t i n g v a s t a r e a s o f t e a c h i n g in t h e r m o d y ­
n a m i c s , c o n s i s t s in t h e r e s t r i c t i o n t o i n f i n i t e l y s l o w p r o c e s s e s . I n p l a c e o f
actual processes o n e considers sequences of (homogeneous)

equilibrium

s t a t e s . E x c e p t f o r a few s p e c i a l c a s e s , s u c h i d e a l i z e d p r o c e s s e s a r e p r a c t i ­
cally r e v e r s i b l e , a n d t h i s e x p l a i n s w h y in classical t h e r m o d y n a m i c s

(or

rather thermostatics) the limiting case of reversibility plays such a d o m i ­
n a n t r o l e . H o w e v e r , t h e e n g i n e e r e n g a g e d in t h e c o n s t r u c t i o n o f t h e r m o m e c h a n i c a l m a c h i n e r y c a n n o t limit h i m s e l f t o i n f i n i t e l y s l o w p r o c e s s e s a n d
hence has never t a k e n this restriction seriously. T h e situation strongly re­
s e m b l e s t h e o n e in p r e - N e w t o n i a n m e c h a n i c s w i t h its a t t e m p t s t o d e v e l o p
dynamics from purely static concepts.
T h e m o d e r n w a y o u t o f t h e d i l e m m a is d i f f e r e n t b u t s u r p r i s i n g l y s i m p l e :
instead of infinitely slow processes o n e considers infinitesimal elements of
t h e b o d y in w h i c h a p r o c e s s t a k e s p l a c e , a d m i t t i n g t h a t t h e s t a t e v a r i a b l e s
differ

from

element

to

element.

In

other

words:

one

conceives

t h e r m o d y n a m i c s a s a field t h e o r y in m u c h t h e s a m e w a y a s c o n t i n u u m
m e c h a n i c s h a s b e e n t r e a t e d f o r m o r e t h a n 2 0 0 y e a r s . I n s u c h a field t h e o r y ,

r e a s o n a b l y fast p r o c e s s e s c a n b e t r e a t e d w i t h t h e s a m e e a s e a s s l o w o n e s ,


vii
a n d restriction t o reversible processes b e c o m e s unnecessary. Finally, this
field t h e o r y is t h e p r o p e r f o r m in w h i c h t h e r m o d y n a m i c s a n d c o n t i n u u m
m e c h a n i c s a r e easily a m a l g a m a t e d .
The

strong

interdependence

thermodynamics

was

generally

of

continuum

recognized

about

mechanics
three

and

decades

ago.

V a r i o u s s c h o o l s h a v e s i n c e c o n t r i b u t e d t o t h e r m o m e c h a n i c s , e a c h f r o m its
p o i n t o f v i e w a n d in its o w n l a n g u a g e o r f o r m a l i s m . It is n o t t h e a i m o f t h i s
book to report on the various approaches nor to compare them. The book
is i n t e n d e d a s a n introduction

to this fascinating

field, b a s e d o n

the

simplest possible a p p r o a c h .
E x c e p t f o r a n i n t r o d u c t i o n t o t h e t h e o r y o f c a r t e s i a n t e n s o r s t h e first
three chapters are concerned with the mechanical laws governing
motion

of

a continuum.

They

are based

on

considerations

of

the
mass

geometry, o n the principle of virtual power a n d o n a general form of the
r e a c t i o n p r i n c i p l e . It is well k n o w n t h a t t h e m o s t g e n e r a l a p p r o a c h

to

c o n t i n u u m m e c h a n i c s m a k e s u s e o f t h e d i s p l a c e m e n t field a n d o f m a t e r i a l ,
a n d hence curvilinear, c o o r d i n a t e s . F o r a beginner, however, this a p p r o a c h
presents c o n s i d e r a b l e m a t h e m a t i c a l difficulties t h a t a r e a p t t o o b s c u r e the
p h y s i c a l c o n t e n t s . S i n c e p h y s i c s d e s e r v e s p r i o r i t y in a n i n t r o d u c t i o n o f t h i s
t y p e , a t r e a t m e n t b a s e d o n t h e v e l o c i t y field h a s m a n y a d v a n t a g e s a n d h a s
t h e r e f o r e b e e n p r e f e r r e d . T h i s k i n d o f a p p r o a c h h a s b e e n p r e s e n t e d in a
masterly

fashion

by

Prager

in

his

"Introduction

to

Mechanics

of

C o n t i n u a " , a n d s i n c e t h e r e is n o t m u c h p o i n t in m a k i n g c h a n g e s j u s t f o r
t h e s a k e o f o r i g i n a l i t y , t h e first t h r e e c h a p t e r s a n d c e r t a i n p o r t i o n s o f t h e
subsequent applications are similar to the corresponding parts of P r a g e r ' s
book.
C h a p t e r 4 deals with thermodynamics.
representation,

familiar

from

textbooks

It s t a r t s

from

in t h i s field,

the

classical

introduces

and

discusses the concept of (independent a n d d e p e n d e n t ) state variables, a n d
s h o w s h o w t h e f u n d a m e n t a l l a w s c a n b e f o r m u l a t e d in t e r m s o f a field
t h e o r y . A c h a r a c t e r i s t i c p o i n t o f t h e p r e s e n t t r e a t m e n t is t h e f a c t t h a t t h e
stress a p p e a r s as t h e s u m of a quasiconservative a n d a dissipative stress.
T h e first is a s t a t e f u n c t i o n , d e p e n d e n t o n t h e free e n e r g y , t h e s e c o n d is
c o n n e c t e d with t h e dissipation function. In view of later

developments

( C h a p t e r 14) t h e r o l e o f t h e t w o f u n c t i o n s is e m p h a s i z e d . T h e d e f o r m a t i o n
h i s t o r y is r e p r e s e n t e d in t h e s i m p l e s t p o s s i b l e m a n n e r , n a m e l y b y i n t e r n a l
parameters.
C h a p t e r 5 deals with the characteristic properties of various materials. A


viii
r o u g h c l a s s i f i c a t i o n o f b o d i e s is p r e s e n t e d , a n d t h e c o n s t i t u t i v e e q u a t i o n s
o f s o m e c o n t i n u a a r e d i s c u s s e d . T h e g e n e r a l t h e o r e m s e s t a b l i s h e d in t h e
preceding chapters, supplemented by the proper constitutive
determine

the

thermomechanical

behavior

of

a

given

relations,

body.

This

is

i l l u s t r a t e d in C h a p t e r s 6 t h r o u g h 1 1 , w h i c h d e a l w i t h t h e a p p l i c a t i o n o f t h e
theory to various types of c o n t i n u a .
C h a p t e r s 12 a n d 13 c o n t a i n a s h o r t o u t l i n e o f g e n e r a l t e n s o r s a n d t h e i r
a p p l i c a t i o n in t h e s t u d y o f l a r g e d i s p l a c e m e n t s . T h e r e p r e s e n t a t i o n f o l l o w s
t h e lines o f G r e e n a n d Z e r n a in t h e i r e x c e l l e n t b o o k o n

"Theoretical

E l a s t i c i t y \ T h e i n c l u s i o n o f t h i s m a t e r i a l m a k e s it p o s s i b l e , in p a r t i c u l a r ,
,

t o p o i n t o u t (a) t h e i m p o r t a n c e o f a p r o p e r c h o i c e o f t h e s t r a i n m e a s u r e a n d

o f t h e c o r r e s p o n d i n g s t r e s s , a n d (b) t h e d i f f e r e n c e b e t w e e n c o v a r i a n t a n d
contravariant

c o m p o n e n t s of a tensor, essential for t h e p r o o f of

the

o r t h o g o n a l i t y c o n d i t i o n in C h a p t e r 14.
U p t o a n d i n c l u d i n g C h a p t e r 13 t h e s u b j e c t m a t t e r , in s p i t e o f a p e r s o n a l
t i n g e in t h e p r e s e n t a t i o n ,

remains within confines

that appear to

be

generally accepted by n o w . T h e remainder of the b o o k transgresses these
t r a d i t i o n a l l i m i t s . It m a y b e c o n s i d e r e d , t o g e t h e r w i t h C h a p t e r 4 , a s a
synopsis of the a u t h o r ' s c o n t r i b u t i o n s to t h e r m o m e c h a n i c s ,

published

f r o m 1957 o n w a r d s , o c c a s i o n a l l y w i t h t h e a s s i s t a n c e o f D r . J u r g N a n n i a n d
P r o f e s s o r C h r i s t o p h W e h r l i . It is c l e a r t h a t in a s y n o p s i s o f t h i s t y p e m a n y
p o i n t s w h i c h o n c e s e e m e d e s s e n t i a l b u t h a v e lost t h e i r i m p o r t a n c e c a n b e
dropped,

and

it is e q u a l l y

obvious

that

many

thoughts

which

once

a p p e a r e d v a g u e h a v e s i n c e a s s u m e d a m o r e c o n c i s e f o r m . I n c i d e n t a l l y , in a
field

which

is still in

a

state

of

development

a

certain

amount

of

c o n t r o v e r s y c a n n o t b e a v o i d e d ; in t h i s r e s p e c t I a s s u m e full r e s p o n s i b i l i t y
for t h e final c h a p t e r s .
C h a p t e r 14 r e t u r n s t o t h e b a s i s o f t h e r m o d y n a m i c s . T h e c l a s s i c a l t h e o r y ,
restricted to reversible processes, tacitly excludes gyroscopic forces. W i t h
e x a c t l y t h e s a m e r i g h t t h e y m a y b e e x c l u d e d in t h e i r r e v e r s i b l e c a s e . T h e
o b v i o u s w a y o f d o i n g t h i s is t o a s s u m e t h a t t h e d i s s i p a t i v e s t r e s s e s a r e
d e t e r m i n e d b y t h e d i s s i p a t i o n f u n c t i o n a l o n e m u c h in t h e s a m e w a y a s t h e
q u a s i c o n s e r v a t i v e f o r c e s d e p e n d o n t h e free e n e r g y . F o r c e r t a i n s y s t e m s , t o
be called e l e m e n t a r y ,

the connection

between dissipative stresses

dissipation function then turns out to have the form of a n
condition,

and

orthogonality

a n d it f o l l o w s t h a t t w o s c a l a r f u n c t i o n s , t h e free e n e r g y a n d t h e

dissipation function (or the rate of e n t r o p y p r o d u c t i o n ) completely govern
a n y kind of process.


ix
C h a p t e r 15 s h o w s t h a t t h e o r t h o g o n a l i t y c o n d i t i o n is e q u i v a l e n t t o a
n u m b e r of e x t r e m u m principles, a m o n g t h e m a principle of m a x i m a l rate
of e n t r o p y p r o d u c t i o n . T h i s last principle suggests a generalization of the
o r t h o g o n a l i t y c o n d i t i o n for systems of the so-called c o m p l e x type. This
g e n e r a l i z a t i o n will b e r e f e r r e d t o a s t h e orthogonality

principle,

a n d it is

e a s y t o see t h a t it r e d u c e s t o O n s a g e r ' s s y m m e t r y r e l a t i o n s in t h e l i n e a r
c a s e . F i n a l l y , C h a p t e r s 16 t h r o u g h 18 a r e c o n c e r n e d w i t h a p p l i c a t i o n s o f
the orthogonality condition a n d the orthogonality principle to

various

types of continua.
A s already m e n t i o n e d , I have tried to keep the m a t h e m a t i c a l formalism
a s s i m p l e a s p o s s i b l e . I a s s u m e , h o w e v e r , t h a t t h e r e a d e r is f a m i l i a r w i t h
vector

algebra

and

analysis,

with

the

basic

laws

of

mechanics

and

t h e r m o d y n a m i c s , w i t h t h e e l e m e n t s o f g e o m e t r y in ^ - d i m e n s i o n a l s p a c e
a n d of the theory of functions, a n d with the n o t i o n of convexity.
p r o v i d e t h e r e a d e r w i t h a m e a n s o f t e s t i n g his g r a s p o f t h e

To

matter,

p r o b l e m s have been a d d e d at the end of each section wherever this was
possible.
In

the

second

edition

of

this b o o k

the thermodynamic

aspect

of

c o n t i n u u m m e c h a n i c s h a s been stressed wherever this seemed desirable;
besides, s o m e weak points have been strengthened. In C h a p t e r 1 a section
d e a l i n g m a i n l y w i t h i n v a r i a n t s h a s b e e n a d d e d , a n d in t h i s c o n t e x t t h e b a s i c
i n v a r i a n t s o f s e c o n d - o r d e r t e n s o r s h a v e b e e n r e d e f i n e d . C h a p t e r s 11 a n d
18, d e a l i n g w i t h v i s c o e l a s t i c i t y , h a v e b e e n e x t e n d e d t o i n c l u d e t h e r m a l
e f f e c t s . T h e first o n e a p p e a r s s u p p l e m e n t e d b y a s e c t i o n , t h e s e c o n d o n e
h a s b e e n c o m p l e t e l y r e w r i t t e n . S e c t i o n 14.4 a p p e a r s in a n e w f o r m , a s d o
C h a p t e r 16, o n n o n - N e w t o n i a n l i q u i d s , a n d C h a p t e r 17, o n p l a s t i c i t y . I n
C h a p t e r 15 a s e c t i o n d e a l i n g w i t h t h e d e r i v a t i o n o f t h e s e c o n d f u n d a m e n t a l
law from the orthogonality condition has been a d d e d . O n the whole, the
terminology

has been

simplified,

particularly

in c o n n e c t i o n

with

the

classification of materials (fluids, solids a n d viscoelastic bodies). M a n y
m i n o r c h a n g e s h a v e b e e n m a d e , a n d m i s p r i n t s o f t h e first e d i t i o n h a v e b e e n
e l i m i n a t e d . M o s t o f t h e p r o b l e m s h a v e b e e n r e f o r m u l a t e d in s u c h a w a y
that they n o w show the m a i n results.
I a m greatly indebted to Professors William Prager and W a r n e r

T.

K o i t e r , w h o h a v e b o t h c r i t i c a l l y r e a d t h e m a n u s c r i p t o f t h e first e d i t i o n a n d
p r o v i d e d n u m e r o u s suggestions for i m p r o v e m e n t . I a m also grateful

to

Professors R a l p h C . Koeller a n d William L. W a i n w r i g h t for pointing o u t
t h a t s o m e o f t h e a p p l i c a t i o n s in S e c t i o n 15.3 a n d C h a p t e r

16 l a c k e d


χ
generality.

A

Hansheinrich

special

word

of

thanks

Ziegler,

for

his

valuable

is d u e

to

my

linguistic

son,

Professor

assistance

in

the

p r e p a r a t i o n o f t h e t e x t . I finally e x p r e s s m y g r a t i t u d e t o D r . C a r l o S p i n e d i
for

his h e l p ,

particularly

in p r o o f r e a d i n g ,

and

to the Daniel

Jenny

F o u n d a t i o n for s u p p o r t in t h e p r e p a r a t i o n o f t h e d r a w i n g s .
Z u r i c h , J u l y 1982

H a n s Ziegler


CHAPTER 1

MATHEMATICAL PRELIMINARIES
I n o r d e r t o d e s c r i b e t h e configuration
reference

system,

of a n arbitrary b o d y , we need a

e.g., a rigid b o d y o r f r a m e serving as a basis for the

o b s e r v e r . A n y q u a n t i t a t i v e t r e a t m e n t r e q u i r e s a coordinate

system

fixed t o

t h i s r e f e r e n c e f r a m e . O u r first t a s k is t o d e v e l o p t h e m a t h e m a t i c a l t o o l s
needed for the description of the m o t i o n or, m o r e generally, of a n y process
in

which

framework

the

body

must

in

consideration

be consistent

with

takes
the

fact

part.
that

The

mathematical

the choice

of

the

c o o r d i n a t e s y s t e m is a r b i t r a r y . I n c o n s e q u e n c e , o u r s t a r t i n g p o i n t m u s t b e
the study of coordinate t r a n s f o r m a t i o n s .

R e s t r i c t i n g o u r s e l v e s in

this

c h a p t e r t o c a r t e s i a n c o o r d i n a t e s y s t e m s , w e will d e v e l o p t h e c o n c e p t o f t h e
cartesian tensor.

1 . 1 . Cartesian t e n s o r s
L e t u s r e f e r ( F i g . 1.1) t h e t h r e e - d i m e n s i o n a l p h y s i c a l s p a c e t o a g i v e n

Fig. 1.1. Cartesian coordinate systems.


2
r e f e r e n c e f r a m e a n d h e r e t o a cartesian,
coordinate

system

x x ,x
u

axes. T h e axes X

2

i.e., rectangular a n d rectilinear,

w i t h u n i t v e c t o r s #Ί, ι , 1 3 a l o n g t h e c o o r d i n a t e

3

2

(p= 1 , 2 , 3 ) w i t h u n i t v e c t o r s I

p

define a n o t h e r cartesian

p

coordinate system with the s a m e origin O . Denoting t h e cosines between
the axes X

p

and x by c
t

we have, for arbitrary indices ρ a n d / between 1

p h

and 3,
c

= c o s (X , x ) = I -i .

pi

p

i

Let Ρ be a point with coordinates

p

(1.1)

i

i n t h e first s y s t e m . I t s c o o r d i n a t e s i n

the second system are the projections of the radius vector (or, equivalently,
o f t h e s e q u e n c e o f s t r a i g h t s e g m e n t s r e p r e s e n t i n g t h e xi) o n t o t h e a x e s

X.
p

M a k i n g use of (1.1), we obtain
X\=c x +c x
u

x

n

+ c x ,

2

n

X = c \X\ + c x
2

2

22

3

+ C23X3,

2

^3=^3,^+032^2 +

(1.2)

^ 3

a s c o o r d i n a t e t r a n s f o r m a t i o n s b e t w e e n t h e t w o c o o r d i n a t e s y s t e m s . It is
e a s y t o see t h a t t h e i n v e r s i o n s a r e
x =c X +c \X
l

u

l

2

x = c X\
2

+

2

+ c X

n

22

cX
3l

39

+ c X

>

(1-3)

^ = Σ c X,

(1.4)

2

32

3

3 - \?>X\ + 2 3 - ^ 2 + 3 3 - ^ 3 ·

x

C

c

c

A m o r e c o m p a c t w a y t o w r i t e ( 1 . 2 ) a n d ( 1 . 3 ) is
3

3

Χ =Σ

Cpi*b

Ρ

pi

p

p=\

1=1

w h e r e ρ is free i n t h e first e q u a t i o n , a n d / i n t h e s e c o n d o n e . W e m a y e v e n
dispense of the s u m m a t i o n symbol by a d o p t i n g , once a n d for all, t h e soc a l l e d summation

convention

stipulating that whenever

a letter

index

a p p e a r s t w i c e in a p r o d u c t t h e s u m is t o b e t a k e n o v e r t h i s i n d e x . W e t h u s
w r i t e , in p l a c e o f ( 1 . 4 ) ,
Xp

=

pi i>

c

x

i

x

=

CpiXp ·

(1 · 5)

It is c l e a r t h a t a n i n d e x a p p e a r i n g o n c e i n a t e r m o f a n e q u a t i o n l i k e ( 1 . 5 )
m u s t a p p e a r in every single t e r m . O n t h e o t h e r h a n d , t h e s u m m a t i o n index
is s o m e t i m e s c a l l e d a dummy

index

letter.

may become

Such

a

replacement

s i n c e it m a y b e r e p l a c e d b y a n y o t h e r
necessary

to avoid

indices


3
a p p e a r i n g m o r e t h a n t w i c e . T o i n s e r t ( 1 . 5 ) i n t o ( 1 . 5 ) i , e . g . , it is n e c e s s a r y
2

t o w r i t e ( 1 . 5 ) in t h e f o r m
2

ΛΓ/ —

(1.6)

CqiXq.

Thus,
X

P

= c i CqiXq

a n d similarly

P

*,· =

(1.7)

c c Xj,
pi

pJ

w h e r e t h e r i g h t - h a n d sides a r e d o u b l e s u m s .
I t is o b v i o u s t h a t t h e c o e f f i c i e n t o f X

in ( 1 . 7 ) ! m u s t b e 1 f o r q-p

q

f o r q^p.

and 0

A similar s t a t e m e n t holds for ( 1 . 7 ) . I n t r o d u c i n g the so-called
2

Kronecker

symbol

p q

Π

forp

(0

f o r ρ Φ q,

= q,

(1.8)

we thus have
Cpi^qi ~ 3pq>

pi PJ

C

(1.9)

~ ^U'

C

These equations might be interpreted as o r t h o n o r m a l i t y conditions; they
a r e v a l i d o n l y in o r t h o g o n a l c o o r d i n a t e s y s t e m s .
The c

pi

m a y be written as a m a t r i x ,

C\2

(1.10)

C22
C

J>1\

33.

H e r e t h e first i n d e x i n d i c a t e s t h e l i n e , t h e s e c o n d t h e c o l u m n in w h i c h a
g i v e n e l e m e n t is s i t u a t e d . F o r a n y fixed v a l u e o f ρ t h e c

p h

a p p e a r i n g in t h e

/7-th line of t h e m a t r i x (1.10), a r e , a c c o r d i n g t o (1.1), t h e c o m p o n e n t s of
the unit vector I

p

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

h

T h u s , the d e t e r m i n a n t of the

m a t r i x is t h e t r i p l e p r o d u c t

detc

= I

pi

l

(/2X/3).

(1.11)

It f o l l o w s t h a t
det

c

pi

= ±1,

w h e r e t h e positive sign c o r r e s p o n d s t o t h e case w h e r e b o t h

(1.12)
coordinate

systems a r e right- o r l e f t - h a n d e d , t h e n e g a t i v e sign t o t h e case w h e r e o n e of
t h e m is r i g h t - h a n d e d a n d t h e o t h e r o n e l e f t - h a n d e d . I n t h e first c a s e t h e
s e c o n d c o o r d i n a t e s y s t e m is o b t a i n e d f r o m t h e first o n e b y a r o t a t i o n a b o u t

4
Ο , in t h e s e c o n d c a s e a r e f l e c t i o n o n a p l a n e p a s s i n g t h r o u g h Ο m u s t b e
added.
M a k i n g o n c e m o r e use of (1.1), we o b t a i n
I = (I · */)// = c ii,
P

P

// = {irl )l

pi

p

= c I.

p

pi

(1.13)

p

C o m p a r i n g this to (1.5), we n o t e t h a t the base vectors of the t w o cartesian
coordinate

systems

transform

as

the

coordinates

of

e q u i v a l e n t l y , a s t h e c o m p o n e n t s o f its r a d i u s v e c t o r ) . I n

a

point

(or,

non-cartesian

c o o r d i n a t e systems, this w o u l d n o t be true.
O u r p r e s e n t i n t e r p r e t a t i o n o f ( 1 . 2 ) is t h i s : P i s a p o i n t fixed in s p a c e , i . e . ,
in o u r r e f e r e n c e f r a m e , a n d ( 1 . 2 ) c o n n e c t s its c o o r d i n a t e s in

different

cartesian systems. A n o t h e r i n t e r p r e t a t i o n , to be used later, considers (1.2)
as representing a displacement with respect to the reference frame:
c o o r d i n a t e s y s t e m is fixed a n d t h e X

p

the

are the instantaneous positions of the

p o i n t s w i t h o r i g i n a l p o s i t i o n s x , . T h e d i s p l a c e m e n t is o b v i o u s l y a r o t a t i o n
a b o u t O, possibly c o m b i n e d with a reflection o n a plane passing t h r o u g h
O.
A scalar

A is a q u a n t i t y w h i c h is i n d e p e n d e n t o f t h e c o o r d i n a t e s y s t e m .

D e n o t i n g the c o r r e s p o n d i n g q u a n t i t y in t h e system X

p

b y Λ, w e t h u s h a v e

Λ=λ.
A vector

(1.14)

ν h a s a d i r e c t i o n a n d h e n c e t h r e e c o m p o n e n t s υ,·. T h e v e c t o r

itself is i n d e p e n d e n t o f t h e c o o r d i n a t e s y s t e m ; its c o m p o n e n t s t r a n s f o r m a s
t h e c o o r d i n a t e s o f a p o i n t ( t h e e n d p o i n t o f ν w h e n t h e c o o r d i n a t e o r i g i n is
chosen as the starting point), i.e., according to (1.5),

Vp = c iV
P

i9

Vi = c V .
pi

(1.15)

p

T h u s , a vector might be defined as a triplet of c o m p o n e n t s

transforming

according to (1.15), a n d this definition might be used to o b t a i n s o m e of the
r u l e s o f v e c t o r a l g e b r a , s u p p l y i n g , e . g . , t h e p r o d u c t λν o f a s c a l a r a n d a
v e c t o r o r t h e s c a l a r p r o d u c t u*v

of t w o vectors.

G e n e r a l i z i n g ( 1 . 1 5 ) , let u s d e f i n e a cartesian

tensor

o f o r d e r η a s a set o f

3 " c o m p o n e n t s //,.../ t r a n s f o r m i n g a c c o r d i n g t o
Tpq...s

=

Cpi^qj · · · s/lij.../>
c

Uj pi^qj
c

· · · ^sl^pq...s*

0·16)

N o t e t h a t t h e o r d e r o f t h e t e n s o r is g i v e n b y t h e n u m b e r o f its i n d i c e s . I n
a c c o r d a n c e with this definition, a scalar λ m a y be considered as a tensor of
o r d e r z e r o . A v e c t o r is a t e n s o r o f o r d e r o n e , symbolically

denoted by

v.


5
W e will h e n c e f o r t h p r e f e r t h e index

notation,

representing a vector by the

s y m b o l i>; o f its g e n e r a l c o m p o n e n t a n d k e e p i n g in m i n d , o f c o u r s e , t h a t t h e

c o m p o n e n t s t r a n s f o r m if t h e c o o r d i n a t e s y s t e m is c h a n g e d .
W e will b e m o s t o f t e n c o n c e r n e d w i t h t e n s o r s o f t h e s e c o n d
denoted

symbolically

by

t

and

in

index

notation

by

order,

Here

the

t r a n s f o r m a t i o n s (1.16) are
Tpq — Cpi Cqj tjj ,

Uj

pi qjT .

c

c

pq

(1.17)

T h e 9 c o m p o n e n t s of a s e c o n d - o r d e r tensor m a y b e a r r a n g e d in m a t r i x
form:
tn
hi

(1.18)

hi

hi

hi

'33.

F o r o t h e r t e n s o r s t h i s is n o t t r u e ; t h e c o r r e s p o n d i n g a r r a n g e m e n t o f t h e 2 7
c o m p o n e n t s o f a t h i r d - o r d e r t e n s o r , e . g . , is a t h r e e - d i m e n s i o n a l b l o c k .
W r i t i n g t h e K r o n e c k e r s y m b o l (1.8) as a m a t r i x , w e o b t a i n

(1.19)

4/ =

If w e i n t e r p r e t t h e e l e m e n t s o f t h i s d i a g o n a l m a t r i x a s c o m p o n e n t s in a
coordinate system x

if

(1.19) defines a s e c o n d - o r d e r tensor. O n a c c o u n t of

( 1 . 1 7 ) ! a n d ( 1 . 9 ) , its c o m p o n e n t s in a n a r b i t r a r y c o o r d i n a t e s y s t e m X

p

^pq ~~ Cpi^qjdij

— Cpfiqi "pq>
z

are

(1.20)

i . e . , t h e y a r e t h e s a m e in a n y c o o r d i n a t e s y s t e m . A n a r b i t r a r y t e n s o r w i t h
c o m p o n e n t s t h a t a r e i n v a r i a n t is c a l l e d isotropic.

Examples encountered so

far a r e t h e s c a l a r a n d t h e K r o n e c k e r t e n s o r .

Problem
S h o w ( b y m e a n s o f a few s i m p l e c o o r d i n a t e t r a n s f o r m a t i o n s ) t h a t a n y
isotropic tensor of order t w o has the form

λδη.

1 . 2 . T e n s o r algebra
I n t h i s s e c t i o n w e will b r i e f l y d i s c u s s t h e p r i n c i p a l r u l e s o f t e n s o r a l g e b r a .
I n s o m e c a s e s w e will r e s t r i c t o u r s e l v e s t o t y p i c a l e x a m p l e s w h i c h a r e easily
g e n e r a l i z e d , a n d w e will l e a v e p a r t o f t h e p r o o f s t o t h e p r o b l e m s e c t i o n .


6
Let r

i j k

_

and s

m

i j k

_

be t w o tensors of equal but arbitrary

m

order.

A d d i n g corresponding c o m p o n e n t s , we obtain a n o t h e r tensor of the same
order,

i//jfc... = r , Y * . . . + S / / * . . . ,
m

called

m

M

t h e sum

of

the original

tensors

( P r o b l e m 1).
Given two tensors of arbitrary order, e.g., r

and s

iJk

products

or

Ujkim = ijk im,
r

their

components.

c a l l e d t h e product

s

These

products

l m 9

define

let u s f o r m t h e
another

tensor

o f t h e o r i g i n a l t e n s o r s . I t s o r d e r is t h e s u m

of the orders of the given tensors ( P r o b l e m 2). Special cases are the p r o d u c t
o f a s c a l a r a n d a t e n s o r (fy = Afy) a n d t h e t e n s o r o b t a i n e d b y m u l t i p l y i n g
t h e c o m p o n e n t s o f s e v e r a l v e c t o r s (t

=

ijk

Let r

u OjW ).
i

k

b e a n a r b i t r a r y t e n s o r o f o r d e r n. P i c k i n g t h e c o m p o n e n t s in

ijkLmmP

w h i c h t w o g i v e n i n d i c e s a r e e q u a l (r

)

ijnmmmP

and applying the summation

c o n v e n t i o n , w e o b t a i n a n o t h e r t e n s o r (///...^ = ///,/...;?) o f o r d e r η - 2 . T h e
p r o c e s s is c a l l e d contraction

w i t h r e s p e c t t o t h e t w o i n d i c e s in q u e s t i o n

( P r o b l e m 3 ) . A s i m p l e e x a m p l e is t h e t r a c e o f a s e c o n d o r d e r

tensor,

t r ί = ία, w h i c h is itself a s c a l a r .
In particular, the process of c o n t r a c t i o n m a y be applied t o a p r o d u c t
w i t h r e s p e c t t o i n d i c e s t a k e n f r o m e a c h o f t h e t w o f a c t o r s (r sij

= t ).

ijk

An

iki

e x a m p l e is t h e s c a l a r p r o d u c t Μ,υ, o f t w o v e c t o r s . If o n e o f t h e t w o f a c t o r s is
a s e c o n d - o r d e r t e n s o r a n d t h e o t h e r t h e K r o n e c k e r t e n s o r (tijdj

= t ),

k

o p e r a t i o n y i e l d s t h e o r i g i n a l t e n s o r . T h u s , δ„ is a l s o c a l l e d t h e unit
o f o r d e r t w o . O t h e r e x a m p l e s a r e t h e powers

the

ik

tensor

of a s e c o n d - o r d e r t e n s o r / ,

symbolically d e n o t e d by / , f , ... a n d defined as t h e second o r d e r tensors
2

3

Up tpj> Up tpq tqj>
It s o m e t i m e s h a p p e n s t h a t , g i v e n a set o f 3 " q u a n t i t i e s / ( / , y . . . , / ) , t h e
q u e s t i o n a r i s e s w h e t h e r t h e y d e f i n e a t e n s o r . It is c l e a r t h a t t h i s q u e s t i o n
c a n b e a n s w e r e d b y c h e c k i n g w h e t h e r t h e t(ij...,

/) t r a n s f o r m a c c o r d i n g t o

( 1 . 1 6 ) . A n e a s i e r m e a n s is t o u s e t h e s o - c a l l e d quotient
o f t h i s l a w s t a t e s t h a t , e . g . , t(i j

k) a r e t h e c o m p o n e n t s o f a t e n s o r t

9 9

t(i j

9 9

k)UiVjW

k

law. A t y p i c a l f o r m
ijk

is a s c a l a r f o r a n y c h o i c e o f v e c t o r s u

i9

v

j9

w.
k

if

I n f a c t , if t h i s is

the case, (1.15) yields
2

T(p, q, r)U

p

Since U ,
p

V W = t(i,j,
q

r

k)u VjW
t

k

= /(/,y, k)c U c
pi

p

Vc

qj

q

rk

W.

(1.21)

r

V , W a r e a r b i t r a r y , it f o l l o w s f r o m ( 1 . 2 1 ) t h a t
Q

r

T(p,

q, r) = CpiCqjCrkt&j,

k).

(1.22)

T h i s is in f a c t t h e t r a n s f o r m a t i o n (1.16)i f o r η = 3 . A n o t h e r f o r m o f t h e
q u o t i e n t l a w s t a t e s t h a t t h e set t(i j

9 9

k) d e f i n e s a t e n s o r t

ijk

if t(i j

9 9

k)^

is a

7
v e c t o r f o r a n y c h o i c e o f t h e t e n s o r ry ( P r o b l e m 4 ) . O t h e r v e r s i o n s o f t h e
q u o t i e n t l a w a r e easily i n f e r r e d f r o m t h e s e e x a m p l e s .
A t e n s o r is c a l l e d symmetric

w i t h r e s p e c t t o t w o i n d i c e s if t h e e x c h a n g e

o f t h e s e i n d i c e s d o e s n o t a l t e r t h e c o m p o n e n t s . If t h e e x c h a n g e i n v e r t s t h e
s i g n s o f t h e c o m p o n e n t s , t h e t e n s o r is c a l l e d antimetric.
second-order tensor t

ij9

In t h e case of a

t h e o n l y s y m m e t r y r e l a t i o n is (/,· = *(/. T h e m a t r i x

r e p r e s e n t a t i o n ( 1 . 1 8 ) s h o w s t h a t t h e s y m m e t r i c t e n s o r ty h a s o n l y six
independent components. O n the other h a n d , the only antimetry condition
f o r tij is ίβ = -tij.
indicating

cyclic

Since this implies f = ··· = 0 (three d o t s in general
n

permutation),

there

remain

only

three

independent

c o m p o n e n t s . I t is e a s y t o see t h a t t h e s e p r o p e r t i e s a r e i n d e p e n d e n t o f t h e
coordinate system (Problem 6).
By m e a n s of t h e identity
tij^Htij

+ tjd + Uty-tji)

(1.23)

t h e s e c o n d - o r d e r t e n s o r ty a p p e a r s d e c o m p o s e d i n t o its s y m m e t r i c a n d
antimetric parts,
kij) = Wij +
t

m

= Wij-

tji\

(1.24)

r e s p e c t i v e l y . I n t h e c a s e o f t w o t e n s o r s , r a n d $/,, it is e a s y t o see t h a t
(J

It i m m e d i a t e l y f o l l o w s t h a t
iJ U = ( W) + [(/]) ( W) +

r

s

r

r

S

w0 = W) W) + m m ·

s

r

s

r

s

ί

1

·

2 6

)

T h r e e a r b i t r a r y n o n - c o m p l a n a r v e c t o r s w, y, w f o r m a r i g h t - o r lefth a n d e d vector system. Since t h e d e t e r m i n a n t
U

U

M

Όχ

V

t>

Wj

W

W

X

D =

is e q u a l t o t h e t r i p l e p r o d u c t u*(oxw),
block

1

2

2

2

3

(1.27)

3

3

it r e p r e s e n t s t h e v o l u m e Κ o f t h e

f o r m e d b y t h e t h r e e v e c t o r s , p r e c e d e d b y t h e p o s i t i v e sign if t h e

vector system a n d t h e coordinate system are b o t h right- or left-handed a n d
b y t h e n e g a t i v e sign if o n e o f t h e m is r i g h t - h a n d e d , t h e o t h e r o n e lefth a n d e d . F o r g i v e n v e c t o r s , Κ is a s c a l a r , w h e r e a s D c h a n g e s sign i n a
1

A shorter word for 'parallelepiped', suggested by Flugge in [1].


8
transformation from a right-handed to a left-handed coordinate system.
W e t h e r e f o r e call D a pseudo-scalar
tensor).

(the simplest version of a

pseudo-

W e will n o t d i s c u s s t h i s c o n c e p t h e r e , b u t r a t h e r a v o i d it b y

restriction to right-handed coordinate systems.
A n y p e r m u t a t i o n of the three digits 1 , 2 , 3 m a y be o b t a i n e d by successive
interchanges of t w o adjacent digits. A c c o r d i n g as the n u m b e r of necessary
s t e p s is e v e n o r o d d , t h e p e r m u t a t i o n itself is c a l l e d a n e v e n o r a n o d d
p e r m u t a t i o n o f 1, 2 , 3 . L e t u s d e f i n e a set o f 27 s y m b o l s e

by stipulating

ijk

t h a t t h e i r v a l u e s a r e 1, - 1 o r 0 a c c o r d i n g a s t h e s e q u e n c e ij

9

k is e i t h e r a n

e v e n p e r m u t a t i o n o f 1, 2 , 3 , a n o d d o n e , o r n o p e r m u t a t i o n a t a l l . I n o t h e r
words, e

m

= ··· = 1 , e

m

= ··· = - 1 a n d e

2 3 3

= ··· =^223=

= *πι = ··· =

0. By m e a n s of these s y m b o l s , t h e d e t e r m i n a n t (1.27) m a y be written as
D = e UiVjW
uk

(1.28)

k

f o r a n y set o f v e c t o r s a n d a n y c h o i c e o f t h e c o o r d i n a t e s y s t e m . I n f a c t , D is
d e f i n e d in m a n y t e x t s b y ( 1 . 2 8 ) . S i n c e w e h a v e r e s t r i c t e d o u r s e l v e s t o r i g h t handed

coordinate

systems, D

is a s c a l a r .

connection with the quotient law that e

ijk

s o m e t i m e s c a l l e d t h e permutation

tensor

From

( 1 . 2 8 ) it f o l l o w s

in

is a n i s o t r o p i c t h i r d - o r d e r t e n s o r ,
o r t h e alternating

t e n s o r . It c a n b e

s h o w n ( P r o b l e m 8) t h a t
Cpij pkl

= $ik fyl ~ ^ilfyk>

e

tpqiepqj = <5ij>

0-29)

2

Cpqr^pqr ~ 6·

L e t Sj b e a n a r b i t r a r y s e c o n d - o r d e r t e n s o r , a n d let u s a s s o c i a t e w i t h it a
k

vector
ti = \e s ,
ijk

c a l l e d its dual

vector.

(1.30)

jk

T h e c o m p o n e n t s o f /, a r e o b v i o u s l y
t\=HS23-S3 )=S ....
2

(1.31)

[23]f

T h e y are identical with the c o m p o n e n t s of the antimetric part of the tensor
Sj

k

a n d h e n c e d o n o t d e p e n d o n its s y m m e t r i c p a r t . O n a c c o u n t o f ( 1 . 3 0 )

and (1.29)!,
^ijk^k

=

i^ijk^kpq^pq

\^kij^kpq pq

=

s

= \ (Sipdjq ~ diqdjp)Spq

= ! ( % ~ ty) = % ] ·

(1.32)

T h u s , the relation
W]

s

=

Uktk

e

(1.33)


9
m a y b e c o n s i d e r e d a s t h e i n v e r s i o n o f ( 1 . 3 0 ) ; it a s s o c i a t e s a n a n t i m e t r i c
tensor

=

w i t h a n y v e c t o r t , c a l l e d its dual

tensor.

k

T h e d u a l v e c t o r o f ujv

k

is
Wi = je UjO .

uk

(1.34)

k

S i n c e its c o m p o n e n t s a r e w =\(u t>3-w i> ),...,
x

2

3

w e h a v e , in

2

symbolic

notation,
w = ±uxv.

(1.35)

O n t h e o t h e r h a n d , ( 1 . 3 4 ) is e q u i v a l e n t t o
w,- = ie UjV
kij

where U

ij9

= \U v

k

ki

= -\UijVj,

k

(1.36)

a c c o r d i n g t o ( 1 . 3 3 ) , is d u a l t o u . T h u s , t h e v e c t o r p r o d u c t

uxv

k

m a y be written as -

UyVj.

Fig. 1.2. Gyro.

If, e . g . , ω is t h e i n s t a n t a n e o u s a n g u l a r v e l o c i t y o f a g y r o ( F i g . 1.2) w i t h
fixed p o i n t O, t h e v e l o c i t y ν o f t h e p o i n t Ρ w i t h r a d i u s v e c t o r r f r o m Ο is
ν = ωΧΓ.

It m a y a l s o b e e x p r e s s e d b y
= e

iJk

where the x

k

a>jx = - QijXj,

(1.37)

k

a r e t h e c o o r d i n a t e s o f Ρ a n d Ω^ = β ^ω/
ί

is d u a l t o a> .

(

k

Problems
1. S h o w t h a t t h e s u m o f t w o t e n s o r s , t j = r j + s j ,
i k

i k

is a t e n s o r .

i k

2 . S h o w t h a t t h e p r o d u c t o f t w o t e n s o r s , tij

= rjj Si

3. Show that contraction of the tensor r

with respect to k and

klm

yields a t h i r d - o r d e r t e n s o r .

i j k l m

k

mi

is a t e n s o r .
m


10
4 . P r o v e t h a t t h e set t(ij, k) d e f i n e s a t e n s o r t

ijk

if t(i,j, k)ry is a v e c t o r

f o r a n y c h o i c e o f t h e t e n s o r ry.
5. P r o v e a n o t h e r f o r m o f t h e q u o t i e n t l a w .
6. S h o w t h a t t h e p r o p e r t i e s o f s y m m e t r y a n d a n t i m e t r y o f t h e t e n s o r t

ijklm

with respect t o j a n d / are independent of t h e coordinate system.
7. L e t D b e t h e d e t e r m i n a n t o f a s e c o n d - o r d e r t e n s o r Sy w r i t t e n a s a
9

m a t r i x . Verify t h e i d e n t i t y e D

=

ijk

e SuSj s .
lmn

m kn

8. P r o v e t h e i d e n t i t i e s ( 1 . 2 9 ) .
9. S h o w t h a t t h e m o m e n t o f i n e r t i a o f a b o d y f o r a n a x i s w i t h d i r e c t i o n
c o s i n e s μ,, p a s s i n g t h r o u g h t h e o r i g i n O , h a s t h e f o r m Ι=Ι μ μ
υ

ί

w h e r e Iy is

]

the symmetric tensor defined b y t h e m o m e n t s of inertia 7 , . . . a n d t h e
n

negative p r o d u c t s of inertia - 7 3 , ··· with respect t o t h e c o o r d i n a t e system
2

Xi. A s s u m e t h a t t h e b o d y is a g y r o w i t h fixed p o i n t Ο a n d a n g u l a r v e l o c i t y
ω , , a n d find its a n g u l a r m o m e n t u m Z), a n d its k i n e t i c e n e r g y Γ .

1 . 3 . Principal a x e s
In this section w e restrict ourselves t o symmetric cartesian s e c o n d - o r d e r
t e n s o r s , a n d w e will b e m a i n l y c o n c e r n e d w i t h f i n d i n g a c o o r d i n a t e s y s t e m
in w h i c h t h e c o m p o n e n t s o f ty a r e p a r t i c u l a r l y s i m p l e .
L e t μ, d e n o t e a u n i t v e c t o r o f a r b i t r a r y d i r e c t i o n . B y m e a n s o f t h e
equation
s^

= tiM

(1.38)

t h e t e n s o r ty a s s o c i a t e s a v e c t o r sj^ w i t h t h e d i r e c t i o n μ,. If, i n p a r t i c u l a r , μ,
h a s t h e d i r e c t i o n o f t h e c o o r d i n a t e a x i s Xj, t h e i-th c o m p o n e n t o f t h e v e c t o r
(1.38) b e c o m e s
^

= ty.

(1.39)

T h e c o m p o n e n t ty o f t h e g i v e n t e n s o r m a y t h e r e f o r e b e i n t e r p r e t e d a s t h e
i-th c o m p o n e n t o f t h e v e c t o r

a s s o c i a t e d w i t h t h e c o o r d i n a t e a x i s Xj.

L e t u s a s k f o r a v e c t o r μ, s u c h t h a t t h e c o r r e s p o n d i n g v e c t o r sj^ is
p a r a l l e l t o i t . I f it e x i s t s , it d e f i n e s w h a t w e call a principal

axis,

a n d it

satisfies t h e r e l a t i o n
sj

M)

=t

ijMj

=t

Mh

(1.40)

w h e r e / is a s c a l a r ( p o s i t i v e , n e g a t i v e o r z e r o ) . T h e s e c o n d e q u a t i o n ( 1 . 4 0 ) is
equivalent t o
(ty-toy^j

= 0.

(1.41)


11
S i n c e y is a s u m m a t i o n i n d e x , w h e r e a s / is a r b i t r a r y , ( 1 . 4 1 ) r e p r e s e n t s t h r e e
h o m o g e n e o u s l i n e a r e q u a t i o n s , c a l l e d t h e characteristic

system,

for

the

u n k n o w n s μ,. F u r t h e r m o r e , s i n c e μ is a u n i t v e c t o r ,
7

a n d t h e t r i v i a l s o l u t i o n μ/ = 0 m u s t b e d i s c a r d e d . A n o n t r i v i a l
however,

only

exists if t h e d e t e r m i n a n t

v a n i s h e s , i . e . , if t h e characteristic

of

the coefficients

solution,
in

(1.41)

equation

det(^-^) = 0

(1.43)

is s a t i s f i e d .
B e f o r e p r o c e e d i n g t o s o l v e ( 1 . 4 3 ) , let u s s h o w t h a t a n a p p a r e n t l y q u i t e
different p r o b l e m yields t h e s a m e characteristic system. O n a c c o u n t of
( 1 . 3 8 ) , t h e p r o j e c t i o n o f t h e v e c t o r sj^ o n t o t h e d i r e c t i o n μ, is t h e s c a l a r
p = sf u
{ i)

= t Mj.

(1.44)

ij

O b v i o u s l y ρ is a f u n c t i o n o f t h e d i r e c t i o n μ,, a n d w e m a y c o n s e q u e n t l y a s k
f o r t h e d i r e c t i o n s f o r w h i c h ρ is s t a t i o n a r y . T h i s q u e s t i o n s t i p u l a t e s a n
e x t r e m u m p r o b l e m , subject t o the side c o n d i t i o n (1.42) a n d solved

by

setting
θ ,
— ( t

u

m

θ
- /μ,μ,) = — [(t - td )Mj]
u

u

= 0,

(1.45)

where M s a Lagrangean multiplier. Carrying out the differentiation

and

m a k i n g u s e o f t h e s y m m e t r y o f ty, w e o b t a i n t h e e q u a t i o n

2(*Ρΐ-* ρΜ
δ

= >

0·

0

4 6

)

w h i c h is in fact e q u i v a l e n t t o ( 1 . 4 1 ) . M u l t i p l i c a t i o n o f b o t h s i d e s o f ( 1 . 4 1 )
b y μ, y i e l d s
^ - 1 0 ^

=0

o r , o n a c c o u n t o f ( 1 . 4 2 ) a n d ( 1 . 4 4 ) , t=p.

(1.47)

It f o l l o w s t h a t t h e L a g r a n g e a n

multiplier belonging t o a solution of (1.41) represents the c o r r e s p o n d i n g
stationary value of the projection (1.44).
Proceeding n o w to the solution of the characteristic equation (1.43), we
w r i t e it in t h e f o r m
I

hi

h -t

hi

hi

2

t

23

hz~t

I =0.

(1.48)


12
Developing t h e left-hand side a n d o r d e r i n g with respect t o p o w e r s of ' , w e
have
-'

+ ' ( ' + -)-'[('22'33 -*23)+ - ] + d e t ^= 0

3

(1.49)

2

π

or
' -7
3

' -7
2

( 1 )

( 2 )

'-7

= 0,

(1.50)

+ ^23 + — ,

(1.51)

( 3 )

where t h e coefficients a r e

J(i)

' n + '">

=

/ ( 2 ) = -^22^33

J(3) = Ί 1 '22'33 - Ί ι /23

+ 2'

2 3

'

3 1

^12 = d e t '//.

A m o r e c o n c i s e f o r m o f ( 1 . 5 1 ) ( P r o b l e m 1) is

Ai)

'//>

=

^(3) \fitijtjktki ~ %jtjihk + '//'//'*:*)·
=

T h e c h a r a c t e r i s t i c e q u a t i o n ( 1 . 5 0 ) is o f t h e t h i r d d e g r e e i n Λ I t h a s t h r e e
r o o t s , c a l l e d t h e principal

values o f t h e t e n s o r ty. A s s t a t i o n a r y v a l u e s o f /?,

they are independent of the coordinate system. According t o the lemma of
Vi6ta, t h e coefficients 7
the

principal

values

( 1 )

,7

( 2 )

and

and 7

( 3 )

hence

in ( 1 . 5 0 ) m a y b e e x p r e s s e d in t e r m s o f

are themselves

independent

of

the

c o o r d i n a t e system. T h e y c a n b e expressed, a c c o r d i n g t o (1.52), in t e r m s of
the traces of /, t

2

and t .
3

O n e o f t h e r o o t s o f ( 1 . 5 0 ) is a l w a y s r e a l . L e t u s d e n o t e it a s t h e first
principal value t . F o r t = t t h e characteristic system (1.41) h a s a t least o n e
x

Y

r e a l s o l u t i o n μ) s a t i s f y i n g ( 1 . 4 2 ) . T h i s s o l u t i o n d e f i n e s t h e first p r i n c i p a l
a x i s o f ty. L e t u s i n t r o d u c e a n e w c o o r d i n a t e s y s t e m x[ t h e first a x i s o f
w h i c h c o i n c i d e s w i t h t h e p r i n c i p a l a x i s μ).
e q u a t i o n (1.40) takes t h e f o r m

=

In this system the second

. W e t h u s h a v e t' = t
n

u

/

2 1

= /

3 1

=0,

a n d t h e c h a r a c t e r i s t i c e q u a t i o n ( 1 . 4 8 ) , w r i t t e n in t h e s y s t e m Λ / , r e d u c e s t o
ti-t

0

0

0

'22->

'23

0

= 0

(1.53)

' 3 3 - '

&

or
( Ί - ' ) [ ' - ('22 + ' 3 3 ) ' + '22'33 - '23 ] = 0 .
2

The remaining
expression

principal values

between

(1.54)

2

' , '
n

m

are obtained

by equating the

square brackets to zero. T h e discriminant

of the


13
c o r r e s p o n d i n g q u a d r a t i c e q u a t i o n is
('22 + ' 3 3 ) - 4 ( & * 3 3 " Φ
S i n c e it is n o n - n e g a t i v e , i

n

and t

(1.55)

= (*22 " *33Ϋ + 4 $ ·

2

a r e r e a l , a n d it f o l l o w s t h a t a s y m m e t r i c

m

tensor of order t w o only a d m i t s real principal values.

Let us characterize t h e principal axes corresponding t o t

and t

u

v e c t o r s μ]

1

a n d μ]

11

by the

m

respectively. T h e y a r e also real, a n d since they a r e

solutions of the characteristic system, we have
(*y ~ hAj )μ} = 0 ,

(tij - ίηιδ^μ}

1

11

= 0.

(1.56)

M u l t i p l y i n g t h e first o f t h e s e e q u a t i o n s b y μ / , t h e s e c o n d o n e b y μ} , a n d
11

1

substracting t h e results, w e obtain
(ήι-ήιι)Λ

? Π

=σ.

(1.57)

It f o l l o w s t h a t t h e p r i n c i p a l a x e s c o r r e s p o n d i n g t o d i f f e r e n t

principal

values a r e o r t h o g o n a l . In consequence, t h e tensor h a s a u n i q u e system of
principal axes p r o v i d e d t h e t h r e e principal values a r e different. If t = t ,
u

t h e d i s c r i m i n a n t ( 1 . 5 5 ) m u s t v a n i s h ; h e n c e t' = 0 a n d t' = t
13

21

33

m

= t = tm. I t
u

f o l l o w s t h a t t h e c o o r d i n a t e s y s t e m xi a n d i n c o n s e q u e n c e a n y c o o r d i n a t e
s y s t e m c o n t a i n i n g t h e a x i s x[ d e f i n e s a p r i n c i p a l s y s t e m . A s l o n g a s t\ is
d i f f e r e n t f r o m t = t , t h e p r i n c i p a l a x i s x[ is u n i q u e ; o t h e r w i s e , i . e . , if
u

h = hi = hn>

a n

m

Y coordinate system defines a system of principal axes.

I n p r i n c i p a l a x e s t h e t e n s o r ty is r e p r e s e n t e d b y a d i a g o n a l m a t r i x ,
0

0

hi

0

0

'in J

It is o b v i o u s t h a t a l s o t h e p o w e r s o f / , d e f i n e d in S e c t i o n 1.2 a s
hptpqtqj*

-

(1.58)

,, a r e represented b y d i a g o n a l matrices
0

0

0

if.

0

0

0

tt

•*P PJ>
ip

g

(1.59)

T h e i r p r i n c i p a l a x e s a r e t h o s e o f t, a n d t h e i r p r i n c i p a l v a l u e s a r e t h e p o w e r s
of
S i n c e t h e p r i n c i p a l v a l u e s t\,... satisfy t h e c h a r a c t e r i s t i c e q u a t i o n ( 1 . 5 0 ) ,
we have
<2)'l + A3)'

(1.60)


14
T h e three e q u a t i o n s (1.60) m a y b e c o n d e n s e d i n t o a single o n e , t h e soc a l l e d Hamilton-Cayley

equation
ΐ=Ι

( 1 )

ί

+Ι ί

2

+ Ι δ,

{2)

(1.61)

0)

w h e r e δ d e n o t e s t h e u n i t t e n s o r . I n p r i n c i p a l a x e s ( 1 . 6 1 ) is i n f a c t e q u i v a l e n t
t o (1.60). A s a t e n s o r e q u a t i o n , (1.61) r e m a i n s valid in t h e f o r m
Uptpqtqj — h^iptpj

+ h^)Uj +

(1.62)

in a n y c a r t e s i a n c o o r d i n a t e s y s t e m .
B y m e a n s o f ( 1 . 6 1 ) it is p o s s i b l e t o e x p r e s s a n y p o w e r o f / in t e r m s o f t , t
2

a n d δ. T h e f o u r t h p o w e r , e . g . , is g i v e n b y
t

= / ( ΐ ) ( / ( ΐ ) / + 7(2)/ + /(3)<*) + I(2)* + / < 3 ) '

4

2

=

2

+ hi))*

2

+ (Ai) (2) + / ( 3 ) ) ' + W ( 3 ) A

(1-63)

7

H e r e t h e coefficients of t , t a n d δ a r e p o l y n o m i a l s in t h e invariants 7
2

a n d 7(3), o f d e g r e e 2 t o 4 i n /,·,· s i n c e , a c c o r d i n g t o ( 1 . 5 2 ) , 7
of

t h e first,

second

a n d third

degree respectively.

( 1 )

,7

( 2 )

( 1 )

and 7

,7

( 2 )

are

( 3 )

In a similar w a y

f , . . . m a y b e r e d u c e d . I t f o l l o w s t h a t a n y p o w e r series i n f,
5

s = AO + Bt + Ct

+ Dt +

2

3

(1.64)

2

(1.65)

can be reduced to three terms

s=fo + gt + ht

w i t h c o e f f i c i e n t s / , g, h t h a t a r e p o w e r series i n t h e i n v a r i a n t s 7
A deviator

( 1 )

,7

( 2 )

,7

( 3 )

.

is a s y m m e t r i c t e n s o r / ' t h e t r a c e o f w h i c h is z e r o . If t j is a n
t

arbitrary symmetric tensor,
t ^ t i j - ^ i j
is a d e v i a t o r s i n c e t' =0.
h

(1.66)

O n the other h a n d , by the inversion of (1.66),
tu = t' + \t d ,
u

kk

u

(1.67)

t h e t e n s o r ty a p p e a r s d e c o m p o s e d i n t o a d e v i a t o r a n d a n i s o t r o p i c t e n s o r .
O n a c c o u n t o f ( 1 . 6 6 ) , t h e p r i n c i p a l a x e s o f t[j c o i n c i d e w i t h t h e o n e s o f t

ij9

a n d t h e p r i n c i p a l v a l u e s o f ty a r e t \ - \ t

k k

, . . . . T h e principal axes of the

i s o t r o p i c p a r t o f ty a r e a r b i t r a r y , a n d t h e t h r e e p r i n c i p a l v a l u e s a r e \ t
Problems
1. V e r i f y t h e e x p r e s s i o n s ( 1 . 5 2 ) .

k k

.

15
2. S h o w t h a t the d e c o m p o s i t i o n (1.67) of a symmetric tensor into a
d e v i a t o r a n d a n i s o t r o p i c t e n s o r is u n i q u e .

1.4. Invariants a n d i s o t r o p i c t e n s o r f u n c t i o n s
Let us consider functions of o n e or m o r e tensors. Provided such a
f u n c t i o n is itself a s c a l a r , w e c a l l it a scalar-valued
t e n s o r s . I n a s i m i l a r m a n n e r , w e d e f i n e vectorfunctions.

function

of the given

o r tensor-valued

tensor

For example, the expansion
X = C+C t j
iJ i

+ C t jt +
iJkl i

.··

kl

(1.68)

w h e r e C , Q , , C ^ / . - . a r e c o n s t a n t t e n s o r s o f o r d e r s 0 , 2 , 4 , . . . , is a s c a l a r valued function of the second-order tensor t

ij9

U

S

+ Gjkltkl

=

whereas

+ Qjklmn hltmn+

(1 · 6 9 )

is a t e n s o r - v a l u e d f u n c t i o n o f o r d e r 2 .
A s a scalar,

λ

in ( 1 . 6 8 ) is i n d e p e n d e n t

H o w e v e r , if t h e n u m e r i c a l v a l u e s o f t h e

of the coordinate

system.

are prescribed independent of

t h e c o o r d i n a t e s y s t e m , t h e t r a n s i t i o n f r o m o n e s y s t e m t o t h e o t h e r r e s u l t s in
d i f f e r e n t v a l u e s f o r λ a n d a l s o f o r t h e c o m p o n e n t s s^ in ( 1 . 6 9 ) s i n c e , w i t h
the components C

ij9

0 , * / , . . . , t h e f u n c t i o n a l f o r m s o f t h e r i g h t - h a n d s i d e s in

( 1 . 6 8 ) a n d ( 1 . 6 9 ) d e p e n d o n t h e c h o i c e o f t h e c o o r d i n a t e s y s t e m . If t h i s is
n o t t h e c a s e , t h e t e n s o r f u n c t i o n is c a l l e d isotropic.

If, e . g . , a s c a l a r - v a l u e d

f u n c t i o n / o f t h e v e c t o r s u„ vv , . . . a n d t h e t e n s o r s t u >
y

kh

...satisfies the

mn

equation
/(V

h

Wj, ...,T U ,...)
kh

mn

= / ( ! * , wj, . . . J

k b

u

m n 9

...\

(1.70)

its f u n c t i o n a l f o r m t h u s b e i n g i n d e p e n d e n t o f t h e c o o r d i n a t e s y s t e m , t h e
f u n c t i o n is i s o t r o p i c . S c a l a r - v a l u e d i s o t r o p i c t e n s o r f u n c t i o n s , a s ( 1 . 7 0 ) ,
a r e a l s o c a l l e d invariants.

T h e properties of invariants a n d of m o r e general

isotropic tensor functions have been studied by Rivlin a n d collaborators.
W e o n l y need a few of their results; for t h e o m i t t e d p r o o f s a n d for m o r e
i n f o r m a t i o n , see t h e c o m p r e h e n s i v e a r t i c l e b y S p e n c e r [ 2 ] .
A n y scalar-valued function

o f a n i n v a r i a n t is itself a n i n v a r i a n t .

It

f o l l o w s t h a t a n y set o f t e n s o r s h a v i n g a t least o n e i n v a r i a n t p o s s e s s e s a n
i n f i n i t y o f t h e m . H o w e v e r , it is a l w a y s p o s s i b l e t o e x p r e s s t h e m in t e r m s o f
a s m a l l n u m b e r o f i n d e p e n d e n t i n v a r i a n t s , r e f e r r e d t o a s basic invariants
a s a n integrity

basis.

or

T h e s i m p l e s t i n t e g r i t y b a s i s o f a s c a l a r , e . g . , is t h e


16
s c a l a r itself. A n y f u n c t i o n o f it a l s o is a n i n v a r i a n t . S i m i l a r l y , t w o o r m o r e
scalars represent their o w n integrity basis. In the case of a vector v

h

it is

o b v i o u s t h a t its m a g n i t u d e o r , p r e f e r a b l y , its s q u a r e
V(i = v v
)

i

(1.71)

i

m a y s e r v e a s a n i n t e g r i t y b a s i s . I n t h e c a s e o f t w o v e c t o r s t>, a n d w,-, t h e
simplest basic invariants are their squares a n d their scalar p r o d u c t ,
»(\) = i>i»i>

w

( 1 )

= W/W/,

l(i = v w .
)

i

i

(1.72)

I n S e c t i o n 1.3 w e h a v e e n c o u n t e r e d t h r e e i n d e p e n d e n t i n v a r i a n t s ( 1 . 5 2 )
o f a s y m m e t r i c t e n s o r o f s e c o n d o r d e r . It is o b v i o u s t h a t t h e y c a n b e
reduced to the simpler invariants
'(1) = '/'/>
i.e., to the traces of /, t

'(2) = ' ( / ( / / >

' ( 3 ) Uj tjktki >

=

(1-73)

a n d / . In fact, a c c o r d i n g t o (1.73) a n d (1.52), we

2

3

have
/ ( l ) = '(1),

hi) = W{2)

/(3) =£(2'(3)

-

-

3'(2)'(1) + '(I))-

(1-74)

In principal axes the invariants (1.73) b e c o m e
Ό) = Ί + - >

'(2) =

tf+

'(3) = * ? +

(1-75)

I n t h e c a s e o f a d e v i a t o r t[j, t h e first o f t h e m is z e r o , a n d t h e o t h e r o n e s m a y
a l s o b e w r i t t e n ( P r o b l e m 1) in t h e f o r m s
'<2) = - 2 ( ί ί ϊ / ί „ + · - ) ,

f<3) = 3 f i / i i / i n .
,

(1.76)

In t e r m s of the invariants (1.73), the H a m i l t o n - C a y l e y e q u a t i o n (1.62)
assumes the form
Uptpqtqj = t \)tipt
{

pj

+ »(2'

+ \(t

~

{1)

( 3 )

-3/

( 2 )

/

( 1 )

t )tij
2

X)

+ '

3
(

1 )

)^.

(1.77)

W i t h its h e l p , a n y p o w e r o f t h e t e n s o r ty a n d h e n c e a l s o t h e p o w e r series
s = Ad
u

u

+ Bt

u

+ Ct t j+Dt t t +
ip P

ip pq qj

-

(1.78)

can be reduced to three terms
Sij=fSij

+ gtij + ht t ,
ik kJ

(1.79)

w h e r e t h e c o e f f i c i e n t s a r e p o l y n o m i a l s o r , i n t h e l a s t c a s e , p o w e r series i n
the invariants (1.73).