Smith ConstitutiveEquationsForAnisotropicAndIsotropicMaterials

CONSTITUTIVE EQUATIONS FORANISOTROPIC AND ISOTROPIC MATERIALS

MECHANICS AND PHYSICSOF DISCRETE SYSTEMS

VOLUME 3

Editor:

GEORGE C. SIHInstitute ofFracture and Solid Mechanics

Lehigh UniversityBethlehem, PA, USA

~~~

NORTH-HOLLANDAMSTERDAM • LONDON • NEW YORK • TOKYO

CONSTITUTIVE EQUATIONSFOR ANISOTROPIC

AND ISOTROPIC MATERIALS

GERALD F. SMITH

Department ofMechanical Engineering and MechanicsLehigh University

Bethlehem, PA, USA

~~~1994

NORTH-HOLLANDAMSTERDAM • LONDON • NEW YORK • TOKYO

PREFACE

Constitutive equations are employed to define the response of

materials which are subjected to applied fields. If the applied fields are

small, the classical linear theories of continuum mechanics and con

tinuum physics are applicable. In these theories, the constitutive equa

tions employed will be linear. If the applied fields are large, the linear

constitutive equations in general will no longer adequately describe the

material response. We thus consider constitutive expressions of the

forms W == "p(E, ... ) and T == <p(E, ... ) where "p(E, ... ) and <p(E, ... ) are

scalar-valued and tensor-valued polynomial functions respectively. The

material considered will generally possess some symmetry properties.

This imposes restrictions on the form of the response functions "p(E, ... )

and <p(E, ... ). Thus, the expressions W == "p(E, ... ) and T == <p(E, ... ) are

required to be invariant under the group A which defines the material

symmetry. We employ results from invariant theory and group repre

sentation theory to determine the form of the functions "p(E, ... ) and

<p(E, ... ). The results obtained are of considerable generality. The poly

nomial functions are assumed to be of degree ~ n in some cases but in

other cases this restriction is removed. The computations leading to

particular results may prove to be tedious. We plan to remedy this de

fect in a subsequent publication where computer-aided procedures will

be discussed which lead to the automated generation of constitutive

expressions.

I would like to express my appreciation to Mrs. Dorothy Radzelo

vage for her careful preparation of the typescript, to my wife Marie for

her assistance in the preparation of this book as well as for her help

with many of the computations involved and to Professor Ronald Rivlin

whose pioneering work in continuum mechanics provided the motiva

tion and inspiration leading to the discussion of constitutive equations

appearing here.

vii

CONTENTS

Introduction to the Series.

Preface .

v

Vll

1.1 Introduction.......

1.2 Transformation Properties of Tensors

1.3 Description of Material Symmetry . .

1.4 Restrictions Due to Material Symmetry .

1.5 Constitutive Equations . . . . . .

Chapter I BASIC CONCEPTS. 1

1

3

7

9

11

Chapter II GROUP REPRESENTATION THEORY

Chapter III ELEMENTS OF INVARIANT THEORY

2.1

2.2

2.3

2.4

2.5

2.6

3.1

3.2

Introduction. . . . . . .

Elements of Group Theory. . .

Group Representations . .

Schur's Lemma and Orthogonality Properties. .

Group Characters .. ....

Continuous Groups. . . . . .

Introduction. . . . . . . . .

Some Fundamental Theorems .

15

15

15

20

24

28

36

43

43

44

Chapter IV INVARIANT TENSORS 53

4.1 Introduction......... 53

4.2 Decomposition of Property Tensors. . . . . .. 56

4.3 Frames, Standard Tableaux and Young Symmetry

Operators. . . . . . . . . . . . . . . . . . 62

ix

x ContentsContents xi

Chapter VI ANISOTROPIC CONSTITUTIVE EQUATIONS

AND SCHUR'S LEMMA. . . . . . 133

7.1 Introduction................. 159

7.2 Reduction to Standard Form. . . . . . . . . . 160

7.3 Integrity Bases for the Triclinic, Monoclinic, Rhombic,

Tetragonal and Hexagonal Crystal Classes . 163

7.3.1 Pedial Class, C1, 1 . . . . . . . . . . . . .. 167

4.4 Physical Tensors of Symmetry Class (n1n2 ... ) . . . 69

4.5 The Inner Product of Property Tensors and Physical

Tensors. . . . . . . . . . . . . . . . . . . . 76

4.6 Symmetry Class of Products of Physical Tensors . . 79

4.7 Symmetry Types of Complete Sets of Property Tensors 88

4.8 Examples.................. 99

4.9 Character Tables for Symmetric Groups 52' ... , 58 103

201

201

GENERATION OF INTEGRITY BASES:

CONTINUOUS GROUPS . . . .

7.3.2 Pinacoidal Class, Ci , I; Domatic Class, Cs, m;

Sphenoidal Class, C2, 2 . 167

7.3.3 Prismatic Class, C2h, 2/m

Rhombic-pyramidal Class, C2v' mm2

Rhombic-disphenoidal Class, D2, 222 170

7.3.4 Rhombic-dipyramidal Class, D2h , mmm . 171

7.3.5 Tetragonal-disphenoidal Class, S4' 4Tetragonal-pyramidal Class, C4' 4 172

7.3.6 Tetragonal-dipyramidal Class, C4h' 4/m . 173

7.3.7 Tetragonal-trapezohedral Class, D4 , 422

Ditetragonal-pyramidal Class, C4v' 4mm

Tetragonal-scalenohedral Class, D2d, 42m 175

7.3.8 Ditetragonal-dipyramidal Class, D4h' 4/mmm 176

7.3.9 Trigonal-pyramidal Class, C3, 3 180

7.3.10 Ditrigonal-pyramidal Class, C3v' 3m

Trigonal-trapezohedral Class, D3, 32 181

7.3.11 Rhombohedral Class, C3i' 3Trigonal-dipyramidal Class, C3h' 6

Hexagonal-pyramidal Class, C6, 6. 182

7.3.12 Ditrigonal-dipyramidal Class, D3h, 6m2

Hexagonal-scalenohedral Class, D3d, 3m

Hexagonal-trapezohedral Class, D6, 622

Dihexagonal-pyramidal Class, C6v' 6mm. 184

7.3.13 Hexagonal-dipyramidal Class, C6h , 6/m . 188

7.3.14 Dihexagonal-dipyramidal Class, D6h , 6/mmm 191

7.4 Invariant Functions of a Symmetric Second-Order

Tensor: C3 195

7.5 Generation of Product Tables 199

8.1 Introduction...........

Chapter VIII

159

133

133

139

144

149

153

109

109

109

114

117

121

128

GENERATION OF INTEGRITY BASES:

THE CRYSTALLOGRAPHIC GROUPS

Introduction. . . . . . . . . .

Application of Schur's Lemma: Finite Groups

The Crystal Class D3 . . . . . . .

Product Tables . . . . . . . . .

The Crystal Class S4' . . . . . . . .

The Transversely Isotropic Groups T1 and T2

Introduction. . . . . . . . . . . . . .

Averaging Procedure for Scalar-Valued Functions.

Decomposition of Physical Tensors . . . . . . .

Averaging Procedures for Tensor-Valued Functions

Examples. . . . . . . . . . . . . .

Generation of Property Tensors . . . .

6.1

6.2

6.3

6.4

6.5

6.6

Chapter VII

Chapter V GROUP AVERAGING METHODS

5.1

5.2

5.3

5.4

5.5

5.6

Chapter IX GENERATION OF INTEGRITY BASES: THE

CUBIC CRYSTALLOGRAPHIC GROUPS 265

10.1

250 10.2

10.3

256

259 10.4

260

262

xii

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

Contents

Identities Relating 3 x 3 Matrices. . . . .

The Rivlin-Spencer Procedure . . . .

Invariants of Symmetry Type (n1 ... np) .

Generation of the Multilinear Elements of an Integrity

Basis. . . . . . . . . . . . . . . . . . . . . . .

Computation of lPn, Pnl ... np ' Qn, Qnl ... np . . . . .

Invariant Functions of Traceless Symmetric' Second-Order

Tensors: R3 . . . . . . . . . . . . . . . . . . . .

An Integrity Basis for Functions of Skew-Symmetric

Second-Order Tensors and Traceless Symmetric Second

Order Tensors: R3 . . . . . . . . . . . . . . . . .

An Integrity Basis for Functions of Vectors and Traceless

Symmetric Second-Order Tensors: 03.Transversely Isotropic Functions .

8.10.1 The Group T1 .

8.10.2 The Group T2 .

202

207

216

223

226

232

Contents

9.4.4 Functions of n Symmetric Second-Order Tensors

Sl'···' Sn: T d', 0. . . . . . . . . . .

9.5 Hexoctahedral Class, 0h' m3m. . . . . . . . . . . .

9.5.1 Functions of Quantities of Type r 9: 0h. . . . .


Sl'···' Sn: 0h . . . . . . . . . . . . . . . .

Chapter X IRREDUCIBLE POLYNOMIAL CONSTITUTIVE

EXPRESSIONS . . . . . . . .

Introduction. . . .

Generating Functions. . .

Irreducible Expressions: The Crystallographic Groups. .

10.3.1 The Group D2d . . . . . . . . . . . . . . .

Irreducible Expressions: The Orthogonal Groups R3' °3 .

10.4.1 Invariant Functions of a Vector x: R3 ...

10.4.2 Invariant Functions of a Vector x: 03 . . .

10.4.3 Scalar-Valued Invariant Functions of Three

Vectors x, y, z: R3 .

10.4.4 Scalar-Valued Invariant Functions of Three

xiii

290

291

294

295

297

297

300

303

304

310

313

316

316

9.1 Introduction.............. 265

9.2 Tetartoidal Class, T, 23. . . . . . . . 269

9.2.1 FunctionsofQuantitiesofTypesf1,f2,f3,f4: T 270

9.2.2 Functions of n Vectors Pl··· Pn: T . . . . .. 275


Sl'''·' Sn: T. . . . . . . . . . . . . . . .. 276

9.3 Diploidal Class, T h' m3. . . . . . . . . . . . . .. 278

9.3.1 Functions of Quantities of Type r 8: T h. . 280

9.3.2 Functions of Quantities of Types r 1,r2,r3,r4: T h 281

9.4 Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m . 282

9.4.1 Functions of Quantities of Types r l' r 3' r 4: T d' ° 283

9.4.2 Functions ofn Vectors P1,···,Pn: T d . . . 287

9.4.3 Functions of Quantities of Type r 5: T d' ° 287

Vectors x, y, z: 03 . . . . . . . . . .10.4.5 Invariant Functions of a Symmetric Second-Order

Tensor S: R3 . . . . . . . . . . . . . . . .

10.4.6 Invariant Functions of a Symmetric Second-Order

Tensor S: °3 . . . . . . . . . . . . . . . .

10.4.7 Invariant Functions of Symmetric Second-Order

Tensors R, S: 03 . . . . . . . . . . . . . .

10.5 Scalar-Valued Invariant Functions of a Traceless

Symmetric Third-Order Tensor F: R3 , 03 . . . . . .10.6 Scalar-Valued Invariant Functions of a Traceless

Symmetric Fourth-Order Tensor V: R3 .

References

Index . . . . . . . . . . . . . . . . . .

317

319

320

320

323

325

327

333

I

BASIC CONCEPTS

1.1 Introduction

Constitutive equations are employed to define the response of a

material which is subjected to a deformation, an electric field, a

magnetic field, ... or to some combination of these fields. Constitutive

equations are of the forms

W == 7P(E, F, ... ), T == 4>(E, F, ... ) (1.1.1)

where 7P(E, F, ... ) denotes a scalar-valued function and 4>(E, F, ... ) a

tensor-valued function of the tensors E, F, ..... The order and sym

metry of the tensors appearing in (1.1.1) would be specified. For

example, the response of an elastic material which is subjected to an

infinitesimal deformation is defined by the stress-strain law

T·· == C··k/lEk/l1J 1J t:. t:.'T·· == T.. Ek/l == E/lk1J J1' t:. t:.

(1.1.2)

where Tij , Eke and Cijke are the components of the stress tensor T, the

strain tensor E and the elastic constant tensor C respectively. As a

further example, we consider the case where the yield function Y for a

material depends on the stress history. We assume that Y is a function

of the stresses T1 ==T(71)' T2 ==T(72)'··· at the instants 71,72'··· .Thus, we have

Y == "/·(T~. Tg ... )0/ 1J' 1J'

(1.1.3)

where 7P is a scalar-valued function of the components Tf. , Tg , ... of the1J 1J

tensors T 1, T2 , ....

2 Basic Concepts [Ch. I Sect. 1.2] Transformation Properties of Tensors 3

where e! . e· is the dot product of the vectors e! and e· and represents1 J 1 J

the cosine of the angle x! ox·. In (1.2.1)1' we employ the usual1 J

summation convention where the repeated subscript j indicates sum-

mation over the values 1,2,3 which j may assume. Thus, A··e· == A·lelIJ J 1

+Ai2e2 +Ai3e3' We shall use this convention throughout the book.

Similarly, the ei may be expressed as linear combinations of the ei. We

see that

The constitutive equations which define the response of a

material are of the form T == 4>(E, F, ... ) where T, E, F, ... are tensors of

specified order and symmetry. It is necessary to discuss the manner in

which the components of a tensor transform when we pass from one

reference frame to another. We restrict consideration to the case where

the reference frames employed are rectangular Cartesian coordinate

systems. Thus, the tensors appearing in the constitutive expressions

will be Cartesian tensors.

Let x denote the reference frame with mutually orthogonal

coordinate axes xI,x2,x3' We denote by eI,e2,e3 the unit base vectors

which lie along the coordinate axes xl' x2' x3 respectively. Let x'

denote the reference frame with the same origin as the reference frame

x and with mutually orthogonal coordinate axes xl' x2' x3' The unit

b t '" l' I th d' t ' , ,ase vec ors e1' e2' e3 Ie a ong e coor Ina e axes Xl' x2' x3

respectively. We define the orientation of the reference frame x' with

respect to the reference frame x by expressing the set of mutually

orthogonal unit base vectors el' e2' e3 as linear combinations of the unit

base vectors e1,e2,e3' We have

(1.2.2)

(1.2.1)e! . e· == A··1 J IJ

e! == A·· e·1 IJ J'

1.2 Transformation Properties of Tensors

The relevant mathematical disciplines required for dealing with

this problem are the theory of invariants and the theory of group

representations. The problem of determining the general form of a

function 4>(E, F, ... ) which is invariant under a group of transformations

constitutes the first main problem of the theory of invariants. The

second main problem of invariant theory is concerned with the deter

mination of the relations existing among the terms appearing in the

general expression for 4>(E, F, ... ). The theory of group representations

is essential if we are to deal with problems of considerable generality.

It provides a systematic procedure for reducing the problem of

determining the form of a constitutive expression to a number of much

simpler problems. The concepts and results from group representation

theory and invariant theory which we shall require will be discussed in

Chapters II and III respectively.

There are restrictions imposed on the forms of the functions

appearing in (1.1.1), ... , (1.1.3) if the material possesses symmetry

properties. The material symmetry may be specified by listing the set

of symmetry transformations, each of which carries the reference

configuration into another configuration which is indistinguishable from

the reference configuration. We may alternatively specify the material

symmetry by listing a set of equivalent reference frames x, A2x, ...

which are obtained by subjecting the reference frame x to the set of

symmetry transformations. Then, the forms which a constitutive

equation assumes when referred to each of the equivalent reference

frames are required to be the same. This, of course, imposes on the

form of the constitutive equation restrictions which are characterized by

saying that the constitutive equation is invariant under the group of

transformations A defining the symmetry properties of the material.

Our main concern in this book will be the determination of the general

form of functions 7P(E, F, ... ) and 4>(E, F, ... ) which are invariant under a

group A.

4 Basic Concepts [Ch. I Sect. 1.2] Transformation Properties of Tensors 5

S· th b t '" d f t f thInce ease vec ors e1' e2' e3 an e1' e2' e3 orm se s 0 ree

mutually orthogonal unit vectors, we have

e! . e! = boo e· . e· = boo1 J IJ' 1 J IJ

where bij is the Kronecker delta which is defined by

Thus, if the base vectors e! and e· associated with the reference frames1 1

x' and x respectively are related by the equation e! = A·· e·, then the1 IJ J

components Xi and Xi of a vector X when referred to the reference

frames x' and x respectively are related by

(1.2.10)X! A.. Ak· == X! b·k == Xk' == Ak· X·1 IJ J 1 1 J J.

With (1.2.6) and (1.2.9)2' we obtain

(1.2.4)

(1.2.3)

b·· == 0 if i f= j.IJb·· == 1 if i == j,IJ

With (1.2.1), ... , (1.2.3), we have X! == A··X·.1 IJ J (1.2.11)

Let A = [Aijl denote a 3 X 3 matrix where the entry in row i and

column j is given by A··. Let AT denote the transpose of A whereIJ

AT = [Aij]T = [Aji]. Then the relations (1.2.6) may be written as

ei . ej == Aikek . Aj£e£ == AikAj£bk£ == AikAjk == bij ,

ei · ej = Akiek · A£j ee = AkiA£j c5k£ = AkiAkj = c5ij .

Thus, the q~antities Aij (i,j == 1,2,3) satisfy

A·kA·k == b.. , Ak·Ak· == 8·· .1 J IJ 1 J IJ

(1.2.5)

(1.2.6)

We refer to the Xi which transform according to (1.2.11) as the com

ponents of an absolute vector or of a polar vector.

Let C1! 1· and C

1· 1· (i1,···,in == 1,2,3) denote the components

1··· n 1··· nof a three-dimensional nth-order tensor C when referred to the reference

frames x' and x respectively. If the base vectors e! and e· associated1 1

with the reference frames x' and x are related bye! == A.. e·, then1 IJ J

(1.2.12)

Thus, the transformation rule for a second-order tensor T is given by

A vector X may be expressed as a linear combination of the base

vectors e· and also as a linear combination of the base vectors e!. Thus,1 1

where E3 = [c5ijl is the 3 X 3 identity matrix. A matrix A which satisfies

(1.2.7) is referred to as an orthogonal matrix.

(1.2.14)

(1.2.13)

T·· == -TooIJ JlS·· = SOOIJ Jl'

The three-dimensional second-order tensors S == [S .. ] and T == [T.. ] are1J 1J

said to be symmetric and skew-symmetric respectively if

(1.2.7)AAT == E3 ,

where X· and X! are the components of the vector X when referred to1 1

the reference frames x and x' respectively. With (1.2.1) and (1.2.8),

and have 6 and 3 independent components respectively. We frequently

associate an axial vector t with a skew-symmetric second-order tensor

T. Thus, let

x = X· e· == X! e!1 1 1 1

X! e! == X! A·· e· == X· e·1 1 1 IJ J J J' X! A·· == X·.

1 IJ J

(1.2.8)

(1.2.9)t· = -2

1c··kT·k

1 IJ J' T·k = C·k· t·J J 1 l'(1.2.15)

6 Basic Concepts [Ch. I Sect. 1.3] Description of Material Symmetry 7

where the t i (i == 1,2,3) are the components of t and where Cijk is the

alternating symbol defined bySets of three quantities which transform according to the rule (1.2.21)

are referred to as the components of an axial vector. The magnetic

field vector H, the magnetic flux density vector B and the cross product

X X Y of two absolute (polar) vectors are examples of axial vectors.

{

I if ijk == 123, 231, 312 ;c··k == -1 if ijk == 132,321,213;

IJ 0 otherwise.(1.2.16)

t! == (det A) A·· t· .1 IJ J (1.2.21)

We observe that

With (1.2.6)1' (1.2.20)2 may be written as

(1.3.1)(Ae). == A.. e· (i,j == 1,2,3)1 IJ J

Symmetry transformations occurring In the description of the

symmetry properties of crystalline materials are denoted by I, C, Ri ,

denote the vectors into which e· IS carried by a symmetry trans-1

formation. The matrix A == [Aij] whose entries appear in (1.3.1) will be

an orthogonal matrix and the unit vectors (Ae)i (i == 1,2,3) will form a

set of unit base vectors for a rectangular Cartesian coordinate system

Ax which is said to be equivalent to the coordinate system x. Each

symmetry transformation associated with the material determines an

equivalent coordinate system Ax and an orthogonal matrix A. The

symmetry properties of the material may be defined by listing the set

of matrices Al = [At] = I, A2 = [AD], ... which correspond to the set of

symmetry transformations. The set of matrices {AI' A2, ... } forms a

three-dimensional matrix group which we refer to as the symmetry

group A.

The symmetry properties of a material may be described by

specifying the set of symmetry transformations which carry the material

from an original configuration to other configurations which are

indistinguishable from the original. Let e1' e2' e3 denote the unit base

vectors of a rectangular Cartesian coordinate system x whose

orientation relative to some preferred directions in the material is

specified. Let (Ae)i defined by

1.3 Description of Material Symmetry

(1.2.20)

(1.2.17)

(1.2.19)

det A == Cijk Ali A2j A3k == Cijk Ail Aj2 Ak3 '

c··k A· A· Ak == C det A ,IJ Ip Jq r pqr

c··k A . A . A k == c det AIJ pI qJ r pqr '

Cijk Cij £ == 2 bk£ ' Cijk == Cjki == Ckij

where det A denotes the determinant of A. With (1.2.18) and (1.2.19),

A· t! == -21 c··k A· A· Ak C t == -21 (det A)c C t,IS 1 IJ IS JP q pqr r pqs pqr r

t! == -21 c· ·kT!k == -21 c· ·kA. Ak T == -21 c· ·kA. Ak ct. (1.2.18)1 IJ J IJ JP q pq IJ JP q pqr r

We note that, in contrast to the alternating symbol c··k defined above,IJ

we employ c··k in Chapter IV to denote the alternating tensor whoseIJcomponents in a right-handed Cartesian coordinate system are given as

in (1.2.16) but whose components in a left-handed Cartesian coordinate

system are given by -1 if ijk == 123, 231, 312; 1 if ijk == 132, 321, 213;

and 0 otherwise. With (1.2.15) and (1.2.16), we have

The components t! of the axial vector t when referred to the reference1

frame x' are given by

8 Basic Concepts [Ch. ISect. 1.4] Restrictions Due to Material Symmetry 9

(1.4.4)

(1.4.3)

(1.4.2)

(1.4.1)

oo1

-1/2 -~/2 0

S2 == ~/2 -1/2 O.

o 0 1

1oo

T!. == A· A· TIJ Ip Jq pq'

-1/2 ~/2 0

Sl == -~/2 -1/2 0

o 0 1

where T!. and E!. are the components of the tensors T and E whenIJ IJ

referred to the x' frame. With (1.2.13), we have

Equations (1.4.1), ... ,(1.4.3) enable us to define the functions 4>i/-..).Thus,

where T·· and E·· are the components of the second-order tensors T andIJ IJ

E when referred to the reference frame x. Let x' be a reference frame

whose base vectors e! are related to the base vectors e· of the reference1 1

frame x bye! == A·· e·. If we employ x' as the reference frame, the1 IJ J

constitutive equation (1.4.1) is given by

Let the constitutive equation defining the material response be

given by

1.4 Restrictions Due to Material Symmetry

(1.3.2)[

a 0 0](a, b, c) == 0 b 0 == diag (a, b, c) .

o 0 c

I == (1, 1, 1), C == (-1, -1, -1),

R1 == (-1, 1, 1), ~ == ( 1, -1, 1), R3 == ( 1, 1, -1),

D1 == (1, -1, -1), D2 == (-1, 1, -1), D3 == (-1, -1, 1),

Di, Ti, Mj and 8j (i == 1,2,3; j == 1,2). I is the identity transformation.

C is the central inversion transformation. Ri is the reflection trans

formation which transforms a rectangular Cartesian coordinate system

into its image in the plane normal to the xi axis. The rotation trans

formation Di transforms a rectangular Cartesian coordinate system into

that obtained by rotating it through 1800 about the x· axis. The trans-1

formation Ti transforms a rectangular Cartesian coordinate system into

its image in the plane passing through the x· axis and bisecting the1

angle between the other two axes. The transformations M1 and M2transform a rectangular Cartesian coordinate system x into the systems

obtained by rotating the system x through 1200 and 2400 respectively

about a line passing through the origin and the point (1,1,1). The

transformations 81 and 82 transform a rectangular Cartesian coordinate

system x into the syste~s obtained by rotation of the system x through

1200

and 2400

respectively about the x3 axis. Corresponding to each of

these transformations is a matrix which relates the base vectors of the

coordinate system x and the coordinate system into which x is trans

formed. We shall employ the notation

The matrices I, C, ... , Sl' S2 corresponding to the symmetry trans

formations I, C, ... , 81, 82 are as follows:

[100] [001] [010]T 1 == 001, T2 == 010, T3 == 100,010 100 001

(1.3.3)If x and x' are equivalent reference frames, i.e., x and x' are related by

a symmetry transformation, the Tij must be the same functions of the

Ek£ as the Tij are of the Ek£· Thus, 4>ij(".) == 4>ij( .. ') and, with (1.4.4),

10 Basic Concepts [Ch. I Sect. 1.5] Constitutive Equations 11

(1.4.5) (1.4.8)

Let A == {AI' A2

, ...} denote the symmetry group defining the symmetry

properties of the material under consideration. The matrices AI' A2, ...

comprising A relate the base vectors associated with the equivalent

reference frames x, A2x,.... Then, the restrictions due to material

symmetry require that the function <Pi/Eke) must satisfy (1.4.5) for all

matrices A = [Ar) belonging to the symmetry group A. A function

<Pi/Eke) which s~tisfies (1.4.5) for all A belonging to A is said to be

invariant under the group A.

More generally, the restrictions due to material symmetry which

are imposed on the function rP· . ( ... ) appearing in the constitutivelI···lm

equation

T· '. == rP· . (Ek k' Fk k' ... ) ,lI···lm lI···lm 1"· n 1··· p

(1.4.6)

where the T ij are the components of the stress tensor and the FkA are

the deformation gradients. The xk = xk(XA) are the coordinates in the

deformed state of a point located at XA in the undeformed state. The

requirement of invariance under rotation of the physical system imposes

the restriction that

must hold for all proper orthogonal matrices Q = [Qij]' Thus, the

function rPij ( ... ) is a second-order tensor-valued function of the three

vectors Fkl' Fk2' Fk3 which is invariant under the three-dimensional

proper orthogonal group R3 . Equation (1.4.9) is then a special case of

(1.4.7).

where T· ., Ek

k and Fk k are the components of tensors T,lI···lm 1··· n 1··· p

E and F, are that

rP· . (Ak

IJ ••• Ak IJ E IJ IJ, Ak IJ ••• Ak IJ F IJ IJ, ••• )lI ... lm It:-I nt:-n t:-I···t:-n It:-I pt:-p t:-I···t:-P

(1.4.7)== A· . ... A· . rP· . (Ek k' Fk k' ... )

lIJI lwm JI···Jm 1··· n 1··· p

must hold for all matrices A = [Aij] belonging to the symmetry group

A defining the symmetry of the material being considered. A function

<p. . (Ek

k' Fk k' ... ) which satisfies (1.4.7) for all AlI···lm 1'" n 1'" p

belonging to A is said to be invariant under the group A.

There may be restrictions of the form (1.4.7) imposed on the

form of a constitutive equation which do not arise from material

symmetry considerations. For example, we may assume that the

response of an elastic material is given by

1.5 Constitutive Equations

The functions "p(E, F, ... ) and 4>(E, F, ... ) appearing In the

constitutive equations (1.1.1) are usually taken to be polynomial

functions. There are various procedures which enable us to generate

polynomial expressions which are invariant under a group A. The

resulting expressions will in general contain redundant terms. With the

aid of results from the theory of group representations, we may readily

compute the number of linearly independent terms of given degrees in

E, F, ... which are invariant under A. This enables us to devise a

systematic procedure for eliminating redundant terms from polynomial

constitutive expressions. As a consequence, the number and degrees of

the basis elements appearing in the general form of a polynomial

constitutive expression is determinate. We may, of course, consider a

constitutive equation 1/J = 1/J(E, F, ... ) where 1/J(E, F, ... ) is a non

polynomial single-valued function of E, F, ... which is invariant under A.

The problem would then be to determine a set of invariants Ij(E,F, ... )

12 Basic Concepts [Ch. I Sect. 1.5] Constitutive Equations 13

(1.5.1)

(j==l, ... ,p) such that any single-valued function 7P(E,F, ... ) which is

invariant under A is expressible as a single-valued function of the

I. (E, F, ... ) which are said to form a function basis. There is an

jxtensive literature devoted to this type of problem. See, for example,

Rivlin and Ericksen [1955], Wang [1969], Smith [1971], Boehler [1977]

for cases where A is a continuous group and von Mises [1928], Smith

[1962a], Boehler [1978] and Bao [1987] for cases where A is one of the

crystallographic groups. The arguments involved in generating function

bases can become quite intricate so that the possibility of errors arising

is a consideration. Further, suppose that it has been established that

the invariants II' ... , Ip form a function basis and that none of the Ij(j == 1,... , p) is expressible as a single-valued function of the remaining

invariants of the set II' ... , Ip . This (see Bao and Smith [1990]) does

not preclude the existence of another set of invariants J l' ... , Jq (q < p)

which also forms a function basis. There seems to be no systematic

procedure for determining the minimal number of basis elements

comprising a function basis for functions 7P(E, F, ... ) which are invariant

under a group A. Consequently, we shall restrict consideration to cases

where the functions 7P(E, F, ... ) and 4>(E, F, ... ) appearing in constitutive

equations are polynomial functions. This is the path followed in the

classical theory of invariants.

We first consider problems where the functions 7P( ... ) and 4>( ... )

are polynomials of total degree:S N. For example, let the constitutive

expression be given by

T·· == CookXk + Cook"XkX"1J 1J 1J ~ ~ ~

where Xk

and T·· are the components of a vector and a second-order1J

tensor respectively. The restrictions imposed on the tensors Cijk and

Cijk

£ by the requirement that (1.5.1) shall be invariant under the group

A are that

COOk == A· A· Ak C ,1J 1p Jq r pqr

must hold for all A == [A.. ] belonging to A. Tensors which satisfy these1J

restrictions are said to be invariant under A. We may proceed by

determining the general form of the tensors Cijk and Cijk£ which are

invariant under A and then substitute into (1.5.1) to determine the

general form of the constitutive equation. We discuss this procedure in

Chapter IV. We may employ other procedures for generating con

stitutive expressions of the form (1.5.1) which are invariant under a

finite group A. Thus, we apply a group - averaging technique in

Chapter V and employ results based on Schur's Lemma in Chapter VI.

The procedures of Chapters V and VI are well adapted to computer

aided generation of constitutive equations. We are in the process of

producing computer programs based on these procedures which will

facilitate the automatic generation of constitutive expressions.

We next remove the restriction that the polynomial constitutive

expressions be truncated at degree N. Thus, let 7P(E, F, ... ) be a scalar

valued function which is invariant under A. We may determine a set of

polynomial functions I.(E, F, ... ) (j == 1,... , p), each of which is invariantJ

under A, such that any polynomial function 7P(E, F, ... ) which is in-

variant under A is expressible as a polynomial in the Ij(E, F, ... ). The

invariants I.(E, F, ... ) are said to form an integrity basis or a polynomialJ

basis. This yields the canonical form for scalar-valued functions which

are invariant under A. Similar results may be obtained for vector

valued and tensor-valued functions which are invariant under A. In

Chapter VII, we obtain results for 27 of the 32 crystallographic groups

which enable us to determine the general form of constitutive

expressions 7P(E, F, ... ) and 4>(E, F, ... ) where there are no limitations as

to the number or order of the tensors appearing as arguments of 7P( ... )

and 4>( ... ). We are able to attain this level of generality because the

inequivalent irreducible representations r l' ... ,rr associated with these

We will find in general that some of the terms appearing in (1.5.3) are

redundant. For example, we may have II 12 = 11. This is referred to

as a syzygy. In Chapter X, we employ generating functions to assist in

the generation of constitutive expressions which contain no redundant

terms and which are referred to as irreducible constitutive expressions.

groups are finite in number and are of dimensions one or two. In

Chapter VIII, we employ a procedure involving Young symmetry

operators to generate constitutive expressions for functions of vectors

and second-order tensors which are invariant under the three

dimensional orthogonal group 03 or one of the continuous subgroups of

°3 . The number of inequivalent irreducible representations associated

with the group 03 is not finite and consequently there is no hope of

attaining generality comparable to that found in Chapter VII. We

again utilize Young symmetry operators in Chapter IX to generate

constitutive expressions for the five remaining (cubic) crystallographic

groups. The number r of inequivalent irreducible representations r l'

... ,r associated with some of these groups is large and some of therrepresentations are three-dimensional. This contributes to the technical

difficulties so that only partial results are given.

The integrity bases II' ... ,Ip generated in Chapters VII, VIII, IX

are irreducible in the sense that no invariant Ik belonging to the

integrity basis is expressible as a polynomial in the remaining elements

of the integrity basis. Suppose that

14 Basic Concepts

.. k1jJ(E, F, ... ) = f(I1, 12, ... , Ip ) = Cij ... k II Id ... Ip .

[Ch. I

(1.5.3)

II

GROUP REPRESENTATION THEORY

2.1 Introduction

In this chapter, we discuss results from group theory and group

representation theory which will be required subsequently. The

constitutive equations which describe the response of a material

possessing symmetry properties are subject to the requirement that

they be invariant under the group A defining the material symmetry.

The determination of the canonical forms of such expressions leads to

the consideration of invariant-theoretic problems. It is frequently

possible and in some cases necessary to reduce the invariant-theoretic

problem to consideration of a number of simpler problems. The theory

of group representations furnishes a systematic procedure for converting

a large and sometimes almost intractable problem into a number of

much more manageable problems. Definitive treatments of group

representation theory may be found in the treatises authored by

Boerner [1963], Littlewood [1950], Lomont [1959], Murnaghan [1938a],

Van der Waerden [1980], Weyl [1946] and Wigner [1959].

2.2 Elements of Group Theory

Suppose that we have a set A of elements {a, b, c, ... } and a

multiplication rule which associates with each pair of elements (a, b)

taken in a given order another element of A. We denote the product of

b by a as abo The set of elements A is said to form a group if

(i) the associative law (ab)c == a(bc) holds for all a, b, c, ... in A;

15

16 Group Representation Theory [Ch. II Sect. 2.2] Elements of Group Theory 17

where 8ij is the Kronecker delta defined by (1.2.4). The set of six 2 x 2

matrices AI' ... ' A6 defined below forms a group where the multiplica

tion rule is that of matrix multiplication.

The products AiAj (i,j == 1, ... ,6) are listed in Table 2.1.

[-1 0] [ 1/2 -{3/2] [ 1/2

A4 = 0 1 ' A5 = _...]3/2 -1/2' A6 = ...]3/2

-{3/2 ]-1/2 '

(2.2.4)

{3/2 ].-1/2

[

-1/2A -

3 - {3/2{3/2 ],-1/2[

-1/2A -2- -{3/2

(ii) there exists a unique identity element e in A such that e a == a e == a

holds for all a in A;

(iii) for each element a in A, there exists a unique inverse a-I such that

aa-1 == a-I a == e.

We observe that in general ab and ba differ. If in addition to (i), (ii)

and (iii), we have ab == ba for all a, b in A, then A is said to be an

abelian group. If the number n of elements comprising A is finite, we

refer to A as a finite group and say that its order is n. We may also

consider groups for which the number of elements comprising the group

is not bounded. For example, consider the set of all non-singular n x n

matrices A, B, C, ... where

All A12 A1n

A==A21 A22 A2n

(2.2.1)

AnI An2 Ann

Aij denotes the entry in row i, column j of the array (2.2.1). We

employ the usual matrix multiplication rule where the entry (AB)ij In

row i, column j of the product AB of B by A is given by

Table 2.1 Product Table

Al A2 A3 A4 A5 A6

Al Al A2 A3 A4 A5 A6A2 A2 A3 Al A6 A4 A5A3 A3 Al A2 A5 A6 A4A4 A4 A5 A6 Al A2 A3A5 A5 A6 A4 A3 Al A2A6 A6 A4 A5 A2 A3 Al

In (2.2.2), the repeated subscript k indicates summation over the range

1 to n. Thus, A·kBk· == A·lB l · +A· 2B2· + ... + A· B .. The set of all1 ] 1 ] 1 ] In nJ

non-singular n x n matrices with the multiplication rule (2.2.2) forms a

group for which the identity element is the n X n identity matrix En

given by

(En)" == 8.. (i,j == 1,... , n)IJ IJ

(2.2.2)

(2.2.3)

In Table 2.1, the product AiAj appears at the intersection of row i and

column j. We observe that all of the products AiAj (i,j == 1, ... ,6) are

elements of the set AI' ... ' A6. The matrix Al is the identity element of

the group. We see from Table 2.1 that each of the A· has an inverse1

which we may denote by Ail. Thus A2l == A3, Ail == A2, A4"l == A4,

.... We denote the matrix group comprised of the matrices AI'···' A6given in (2.2.4) by A == {AI' ... ' A6} == {AK} (K == 1, ... ,6). We shall

frequently suppress (1<: == 1, ... ,6) and denote the group by {AK}.

If every element of a group is expressible as a product of the

18 Group Representation Theory [Ch. II Sect. 2.2] Elements of Group Theory 19

As another example, we consider the group comprised of the n!

permutations of the integers 1,2,... , n. Let

elements comprising a subset of the group, we say that the elements of

this subset are generators of the group. For example, A2 and A4 are

generators of the group A given by (2.2.4) since

(2.2.5)s = ( 1

sl

2 t = ( 1t 1

(2.2.8)

B is a subgroup of a group A if it is comprised of a subset of the

elements of A which themselves form a group. The group A is of course

a subgroup of itself. We refer to a subgroup B of A for which at least

one element of A is not an element of B as a proper subgroup of A. We

see from Table 2.1 that the following are (proper) subgroups of the

matrix group A = {AI' ... ' A6} defined by (2.2.4):

(2.2.10)

( 1 2 3 4 5 6 ), ( 1 2 3 4 5

~),s= 2 3 4 5 6 1 t= 2 1 4 3 6

(2.2.9)

ts = ( ~2 3 4 5

~ ), st = ( ~2 3 4 5 6 ).4 3 6 5 2 5 4 1 6

A permutation which replaces sl by s2' s2 by s3' ... , sm -1 by sm' sm

by sl is said to be a cycle of length m and is denoted by (sl s2 ... sm).

Any permutation of the symbols 1,2, ... , n may be written as the

product of '1' '2' ···"n cycles of lengths 1,2, ... , n respectively where

denote the permutations which replace 1 by sl' 2 by s2' , n by Sn and

1 by t l , 2 by t2, ... , n by tn respectively. Each of the sl' ' sn takes on

one of the values 1, ... ,n and no two of the sl, ... ,sn take on the same

value. The n! permutations of 1, ... , n together with the appropriate

multiplication rule given below constitutes the symmetric group Sn.

We define the product ts of s by t to mean that we first apply s to the

symbols 1,... , n and then apply t. Thus, 1 is replaced by sl and then sl

is replaced by t s1 ; ... ; n is replaced by Sn and Sn is replaced by tsn.

For example, in the case where n = 6, we have

(2.2.6)

Thus, the product of any pair of elements of 0 is an element of D. The

inverse of each element of 0 lies in D. 0 possesses the identity element

Al and the associative law holds in 0 since it holds in A. The set of

elements A4D = A4{AI' A2, A3} = {A4, A5, A6} is referred to as a (left)

coset of 0 in A. We note that A = 0+ A4 D.

We say that an element b of a group A is conjugate to an

element a if there is an element c of A such that cac-1 = b. We may

choose an element a of A and then generate the set of elements cac-1

where a is fixed and c runs through all of the elements of A. We refer

to this set of elements as the class of the group A generated by a. We

may split the elements of A into p subsets which form classes Cl ,... ,Cp .

We see from Table 2.1 that the group A defined by (2.2.4) has 3 classes

given by

The order of a class Ci is the number Ni of elements comprising the

class.

(2.2.11)

The cycle structure of a permutation is denoted by

h t .'j . . d·f 1were a erm J IS omltte 1 'j = 0 and where j is written as j. For

(2.2.7)

20 Group Representation Theory [Ch. II Sect. 2.3] Group Representations 21

example, the permutations (2.2.9) may be written as set of non-singular n x n matrices such that if a b == c, then

The cycle structures of the permutations s, t, ts and st are given by 6,

23, 133 and 133 respectively. Permutations which have the same cycle

structure 1II 2/2 ... nIn belong to the same class I ( == 1'1 2/2 ... nIn

or '1 '2 ... In) of 5n · For example,

We say that the matrices D(e), D(a), D(b), ... form a matrix repre

sentation of dimension n of the group A. From (2.3.1), we see that

D(a) D(e) == D(ae) == D(a) so that

s == (1 2 3 4 5 6),

ts == (1) (3) (5) (246),

t == (1 2) (3 4) (5 6),

st == (1 3 5) (2) (4) (6).(2.2.12)

D(a) D(b) == D(c).

D(e) == En == [<5ij] (i,j == 1, ... , n)

(2.3.1)

(2.3.2)

s == (1 2 3 5 4 6), u == (1 3 5 6 2 4) (2.2.13)where En is the n x n identity matrix. Also, from (2.3.1), D(a) D(a-1)

== D(aa-1) == D(e) == En and hence

A matrix D is said to be orthogonal if its inverse D-1 is the transpose

of D, i.e., if

have the same cycle structure. We observe that u rsr-1 where

r == (1) (2 3 5 6 4) and r-1 == (1) (2 4 6 5 3). The 0 rder h, of a class I

of a symmetric group 5n is the number of permutations comprising the

class I of 5n .

The classes of the symmetric group 52 == {e, (12)} are given by

(2.3.3)

(2.3.4)

2.3 Group Representations

where, for example, the class denoted by 3 consists of permutations

comprised of a single cycle of length 3.

Let e, a, b, c, ... denote the elements of a group A where e is the

identity element of the group. Let D(e), D(a), D(b), D(c), ... denote a

(2.3.5)Dt == D-1

where DJ. == D.. and where D.. denotes the complex conjugate of D...1J J1 1J 1J

We indicate the manner in which we may define a matrix

representation {D(A1), ... ,D(AN)} == {D1,... ,DN} == {DK } of A which

describes the transformation properties under the symmetry group A

== {A1,···,AN} == {AK} of the components of a tensor T. Consider the

case where T is a second-order three-dimensional tensor whose com

ponents when referred to the reference frame x are given by T··1J

(i,j == 1,2,3). Let t denote the column vector whose entries T 1,···, T 9are given by

where DJ = Dji . A matrix D is said to be unitary if its adjoint Dt is

the inverse of D, i.e., if

(2.2.15)

(2.2.14)2: (12)

13 : e == (1) (2) (3);

12: (1) (23), (2) (31), (3) (12);

3: (123), (132)

where the classes 12 and 2 are denoted by the cycle structure of the

permutations comprising these classes. The classes of the symmetric

group 53 are given by

22 Group Representation Theory [Ch. II Sect. 2.3] Group Representations 23

Thus, with (2.3.8) and (2.3.12), we have

The entries in the column vector t when referred to the reference

frames x and x' are given by Ti and Ti where, with (2.3.7),

The C··k, D" k may be obtained from (2.3.6). Let x and x' == Ax denoteIJ IJreference frames whose base vectors ei and ei are related by ei == Aijej

where the matrix A == [Aij ] appears in the group {AK}. The com

ponents of T when referred to the reference frames x and x' are given

by Tij and Tij where, with (1.2.13),

where(2.3.16)

(2.3.15)

(2.3.14)

SDK S-1 (K == 1,... , N)

Dir(A) Drn(B) == CijkAjpAkqDpqrCrstBs£BtmD£mn

== CookA. Ak 8 8 tB IJB t D IJIJ JP q ps q s~ m ~mn

== C··kA. B IJAk B D IJIJ JP p~ q qm ~mn

== Cijk (AB)j£ (AB)km D£mn

== Din(AB).

where S is non-singular also forms a matrix representation of A which is

said to be equivalent to the representation {DK}. The matrices (2.3.15)

define the transformation properties of the column vector u == S t under

A. Thus, if t' == DK t, then

The set of matrices D(AK) == DK (K == 1,... ,N) which describes the

transformation properties of t under the group A == {AK} then forms a

matrix representation of A which we shall denote by {DK} (K == 1, ... ,N)

or by {DK}. If the matrices comprising {DK} are n x n matrices, we

say that {DK} is an n-dimensional matrix representation of A.

The set of matrices

(2.3.8)

(2.3.7)

(2.3.9)

(2.3.11)

(2.3.10)

T'k == D·k·T.J J 1 1

C" k D'klJ == 8'1J Doo k CklJ == 8'1J 8· .IJ J ~ 1~ , IJ ~n 1~ In

T· == C··kT·k1 IJ J'

T!. == A· A· T .IJ Ip Jq pq

T! == C··kT!k == CookA. Ak T1 IJ J IJ JP q pq

== COOk A· Ak D T == D· (A) TIJ JP q pqr r Ir r

T. == CookT·k, T! == C··kT!k'1 IJ J 1 IJ J

Let

where

With (2.3.7), (2.3.9) and (2.3.10),

Corresponding to each matrix A in {AK}, there IS a matrix D(A)

== [Dir(A)] which relates the Ti and Ti by (2.3.11). We observe that if

A, Band AB are elements of {AK}, then

D· (A) == C··kA. Ak D .lr IJ JP q pqr

D(A) D(B) == D(AB).

(2.3.12)

(2.3.13)

If S can be chosen so that

SDKS-1 =l F: ::] (K=l,... ,N), (2.3.17)

we say that the representation {DK} is reducible. If there is no S such

that (2.3.17) holds for all K == 1,... , N, the representation is said to be

irreducible. If S can be chosen so that

24 Group Representation Theory [Ch. IISect. 2.4] Schur's Lemma and Orthogonality Properties 25

holds for K == 1, ... , N, we say that {DK} decomposes into the direct sum

of the representations {FK } and {GK }. It may be shown (see Wigner

[1959], p.74) that a matrix representation {DK}of a finite group A is

equivalent to a representation {RDK R- I } where the RDK R- I are

unitary. If the matrices DK are real, we may determine a matrix V

such that the matrices comprising the representation {VDKV-I} are

orthogonal. This also holds for the continuous groups considered in this

book. We may thus restrict consideration to cases where the DK are

either unitary matrices or orthogonal matrices.

We multiply (2.4.3) on the left and right by DK and note that the DKare unitary to obtain

(2.4.3)

(2.4.4)

dDt< = Dt<Ct (K = 1,... , N).

that

for all DK comprising an n-dimensional irreducible repre

sentation of A, then C == AEn where En is the n x n identity

matrix.

The argument leading to these results may be found in Wigner [1959],

Murnaghan [1938] or any of the references listed at the end of §2.1. We

indicate below the manner in which (iv) may be established.

Since the adjoint of AB is (AB)t = BtAt, we see from (2.4.2)

(2.3.18)

Thus, with (2.4.2), the adjoint ct of C and hence the Hermitian

matrices C +d and i (C - ct) also commute with each of the DK"

Consider then the case where L is Hermitian, i.e., L == Lt or L·. == Loo,1J J1

and satisfies

2.4 Schur's Lemma and Orthogonality Properties

Let {DK } and {RK } denote n-dimensional and m-dimensional

irreducible matrix representations of the finite group A == {A1,... ,AN}.

We may assume that the matrices DK == D(AK) and RK == R(AK)

(K == 1, ... , N) are unitary. We consider the problem of determining the

form of the n x m matrix C which satisfies DK L == LDK (K == 1,... , N). (2.4.5)

DKC==CRK (K==l, ... ,N). (2.4.1) Since L is Hermitian, we may determine a matrix T such that T-ILT

== M is diagonal. With (2.4.5),

Schur's Lemma tells us that

(iii) C is non-singular, i.e., det C :I 0 if n == m and the representations

{DK} and {RK} are equivalent;

(iv) if {DK} == {RK} in (2.4.1) so that C satisfies

DKC==CDK (K==l,oo.,N) (2.4.2)

(2.4.7)

(2.4.8)

(2.4.6)

where

or

Suppose that the DK and hence the FK are 3 x 3 matrices and that M

== diag(Al,A2,A3)· Further, suppose that Al == A2 =I A3· Then, from

C == 0 if n:l m;

C == 0 if the representations {DK} and {RK} are inequivalent;

(i)

(ii)

26 Group Representation Theory [Ch. II Sect. 2.4] Schur's Lemma and Orthogonality Properties 27

(2.4.9)

(2.4.15)

(2.4.13)

(2.4.14)

(2.4.12)

(i,j == 1, ... , n)

(i == 1, ... , n; j == 1, ... , m)

(K == 1,... , N) .

N K-K'" D· R· == 0K~ Ir JS==1

N K-K'" D· D· == Ars 8·· .K~ Ir JS IJ

==1

We set i == j in (2.4.15) in order to determine Ars . Thus,

where the Ck£ are arbitrary.

(2.4.14) to obtain

unitary, Rj(l = Rk or (Rj(\j = R~. We then set Ck£ = 8kr 8£s in

(2.4.11) to obtain

where r is chosen from the set 1, ... , nand s from the set 1,... , m. If we

consider the case where {DK} == {RK}, then (2.4.10) becomes

Since the representation {DK } is irreducible, Schur's Lemma yields the

result that P == AEn. Then, upon setting RL == DL in (2.4.9), we have

N K -K

KL Dik Ck£Dj£ == A8ij==1

Let {DK } and {RK } denote inequivalent irreducible unitary

representations of dimensions nand m respectively of the group A

== {A1,... ,AN}. Let

N -1P == L DLCRL

L==l

where C is an arbitrary n x m matrix. Then,

(2.4.7), we see that the matrices FK = [F~l are such that F~ = 0 for

ij == 13, 31, 23, 32 and K == 1, ... ,N. The representation {FK}

== {T-1DK T} is then the direct sum of two representations which

contradicts the assumption that {DK } is irreducible. We conclude that

(2.4.7) implies that {DK} is reducible unless Al == A2 == A3 == A, i.e.,

unless M == AE3. Since {DK} is assumed to be irreducible, we see that

M = ,\E3 and hence, from (2.4.8)1' L = TMT-1 = ,\E3. Similarly, we

may arrive at the result that in (2.4.5) the n x n Hermitian matrix L is

equal to ,\ En. Thus, the Hermitian matrices C + et and i (C - ct) and

hence C = !(C + ct) -! i· i (C - ct) must be multiples of the identity

matrix.

(2.4.17)

(2.4.16)

(i,j, r, s == 1, ... , n) .N K-K N"D. D· == - 8.. 8~ Ir JS n IJ rs

K==l

N K-K'" D· D· == N8rs == Ars nK~ Ir IS==1

where we have noted that DK is unitary and that 8ii == n. Substituting

this expression for Ars into (2.4.15) gives

(2.4.10)

(2.4.9), we have

N -1== L DMCRM RK == PRK (K == 1, ... ,N)

M==l

where we have set DKDL = DM, RKRL = RM and R~ = RL1Rj(1.

Since {D1<) and {RK } are inequivalent irreducible representations of A,

Schur's Lemma tells us that all entries of P == [P ij ] are zero. With

where the Ck

£ are arbitrary and where we have noted that, since RK is

(i == 1,... , n; j == 1,... , m) (2.4.11)The equations (2.4.12) and (2.4.17) give the orthogonality relations for

irreducible representations of a finite group A == {AI' ... ' AN}. Suppose

that {DK} is two-dimensional (n == 2). We observe from (2.4.17) that

28 Group Representation Theory [Ch. IISect. 2.5] Group Characters 29

where we have noted that tr ABC == tr BCA == tr CAB. Now, let

r == {DK} and r' == {RK} be inequivalent irreducible unitary repre-

may be thought of as a set of four mutually orthogonal vectors of

lengths ~N /2 in an N-dimensional space. Thus ai ai == N/2, bi hi == N/2,

a· b. == 0 a· c· == 0 .... Suppose that {RK} is a one-dimensional... , 1 1 '1 1 '

representation of A. Then,

(2.5.4)

(2.5.3)

NE XKXK == N,

K==l

NE XKXK == O.

K==l

N K-KED.. D..K==l 11 JJ

N K-KED.. R..K==l 11 JJ

where we have used (2.3.13). We denote by Xk the common value of

the XK == tr DK == tr D(AK) for the AK belonging to the class Ck. We

then denote the characters of the r inequivalent irreducible repre-

sentations r1"'" rr by

respectively. With (2.5.3), the orthogonality relations for the char

acters are given by

NE X· X'K == N 8i · (i,j == 1,... , r). (2.5.5)K==l lK J J

Thus, the r characters (2.5.4) may be considered to form a set of r

mutually orthogonal vectors in an N-dimensional space. If AK and ALbelong to the same class of A, i.e., if AK == AllALAM for some group

element AM (see §2.2), then

XK = tr DK = tr D:N{DLDM = tr DMD:N{DL = tr DL = XL (2.5.6)

These relations are referred to as the orthogonality relations for the

characters of irreducible representations. We note that the number of

inequivalent irreducible representations of a finite group A is equal to

the number r of classes C1,... , Cr of A. We denote these irreducible

representations by r 1"'" r r and their characters by

sentations of A. Let (X1"",XN) and (Xl, ... ,XN) denote the characters

of rand r' respectively. We may set i == rand j == s in (2.4.17) and

(2.4.12) to obtain

(2.5.2)

(2.4.18)

(2.4.19)

D!I,···,Dri = aI"" ,aN;

Db,··· ,Dr2 = hI"" ,hN ;

D~I , ... , D~I = cI" .. ,cN;

Dh,· .. ,D~2 = dI ,· .. ,dN

the four sets of N quantities (N == order of A)

2.5 Group Characters

Let r == {DK} denote an n-dimensional matrix representation of

the group A == {AI ,... , AN}' The character of the representation r is

given by (Xl"'" XN) where

XK = trDK = D~ = Dfl + ... + D~n (2.5.1)

and where tr DK is referred to as the trace of the matrix DK = [D~l.

Equivalent representations {DK} and {S DK S-1} have the same

characters since

Ril,· .. ,Rri = fI,· .. ,fN

is seen from (2.4.12) and (2.4.17) to be a vector of length ~ in an N

dimensional space which is orthogonal to the vectors (2.4.18) arising

from {DK}. Thus, fiIi == N; fi ai == 0, ... , fi di == O.

30 Group Representation Theory [Ch. II Sect. 2.5] Group Characters 31

(2.5.13)

(2.5.14)

(2.5.15)

rXK == .E ci XiK (K == 1,... ,N)

1==1

where the summation in (2.5.14)3 4 is over the r classes of A and where,Nk denotes the order of the class Ck of A. We further note with

(2.5.5), (2.5.8) and (2.5.13) that

1 N _ 1 r _ 2 2N E XKXK==N ENk Xk Xk==cl+···+ cr

K==1 k==l

where the c· are non-negative integers. If1

or

where (X 1,... ,XN) and (Xi1 ,... ,XiN) are the characters of the repre

sentations rand r· respectively. The orthogonality relations (2.5.5)1

enable us to determine the number ci of times the irreducible repre-

sentation r i appears in the decomposition (2.5.10) of r. Let r 1 denote

the I-dimensional identity representation of A where rk = 1

(K == 1,... ,N). With (2.5.5) and (2.5.13),

(2.5.7)

(2.5.10)

rE Nk X·k X·k == N<5i· (i,j == 1, ... , r) (2.5.8)

k== 1 1 J J

where the summation is over the r classes of A. With (2.5.8), it is seen

that

A matrix representation r == {DK} of A may be decomposed into

the direct sum of the r inequivalent irreducible representations

r i = {Dk} (i = I, ... ,r) associated with A. Thus we may determine a

matrix S such that

form a set of r mutually orthogonal unit vectors in an r-dimensional

space.

where X.k is the value which the character of r· assumes for the group1 1

elements belonging to the class Ck. Let Nk be the number of group

elements comprising the class Ck. We may then express the ortho

gonalilty relations for the group characters as

where the expression on the right denotes a matrix with cl matrices

Dk, ... , cr matrices Dk lying along the diagonal with zeros elsewhere.

For example,

(2.5.16)

Upon taking the trace of both sides of (2.5.10), we see with (2.5.2) that

(2.5.17)

we must have c· == 1 for some i and c· == 0 (j == 1,... , r; j f:. i). Thus, the1 J

condition (2.5.16) indicates that the representation r whose character is

given by (Xl'.'" XN) is an irreducible representation.

Let us consider the group A == {AI'"'' A6} where the AK are

defined by (2.2.4). With (2.2.7), we see that the AI' ... ' A6 may be split

into the three classes(2.5.12)

(2.5.11)

r .tr SDKS-1 == tr DK == E ci tr Dk (K == 1,... ,N)

i==l


Since the number of inequivalent irreducible representations of A is

equal to the number of classes comprising A, there are three inequiv

alent irreducible representations associated with A which we denote by

r 1 == {FK}, r 2 == {GK} and r 3 == {HK}. These are given by

r1 : F 1,···,F6 == 1, 1, 1, 1, 1, 1·,

r2 : G1,· .. ,G6 == 1, 1, 1, -1, -1, -1; (2.5.18)

r3 : H1,···,H6 == AI' A2, A3, A4, A5, A6

where F 1 == 1 indicates that F 1 is a 1 X 1 matrix with entry 1, i.e.,

F 1 == [1], and where the A1,... ,A6 are defined by (2.2.4). r 1 is the

identity representation. r 2 is obtained by setting GK == det AK( == [det AK ]) which furnishes a representation since det AKAL== det A!{ det AL. r 3 is the representation furnished by the group

elements AI' ... ' A6. With (2.2.4) and (2.5.18), the characters of the

representations r l' r2' r3 are given by

form a set of six mutually orthogonal vectors as IS required by the

orthogonality relations (2.4,.12) and (2.4.17).

We now construct the character table for the group A. We

observe from (2.5.17) that AI' AK (K == 2,3) and AK (K == 4,5,6)

comprise the classes C1, C2 and C3. Thus,

Xik == XiK (k == K == 1), Xik == XiK (k == 2; K == 2,3),

Xik = XiK (k = 3; K = 4, 5, 6) . (2.5.21)

Table 2.2 listed below is the character table for the group A. The entry

appearing in the row headed r· and the column headed C· is the value1 J

X.. which the character of the representation r· assumes for the AK~ 1

belonging to the class C·. The entries in the row headed N· give theJ J

orders of the the classes CJ.. The X.. are determined from (2.5.19) and

IJ(2.5.21 ).

Table 2.2 Character Table: A

Since the (Xil,.",Xi6) (i == 1,2,3) satisfy (2.5.16), the representations

r l' r2 and r3 are irreducible. For example, t (X31 X31 +... +X36X36)

== 1. We observe that

1 6 1, 1, 1, 1,F 11 ,···,F11 1, 1·,

G1 6 1, 1, 1, -1, -1, -1 ;11,···,G11 ==HI 6 1, -1/2, -1/2, -1, 1/2, 1/2 ;11,···,H11 ==HI 6 0, ~/2; -~/2, -~/2, ~/2;

(2.5.20)

12'···' H12 == 0,

HI 6 0, -{3/2, {3/2, 0, -{3/2, {3/2;21'···' H21 ==HI 6 1, -1/2, -1/2, 1, -1/2, -1/222' ... ,H22 ==

(XII'···' X16 ) == (1, 1, 1, 1, 1, 1);

(X21'···' X26) == (1, 1, 1, -1, -1, -1);

(X31 ,... , X36) == (2, -1, -1, 0, 0, 0).

(2.5.19)

C1 C2 C3

N· 1 2 3J

r 1 1 1 1

f 2 1 1 -1

f 3 2 -1 °We may readily verify that the orthogonality relations (2.5.8) for the

group characters hold.

(2.5.22)


where E3 is the 3 x 3 identity matrix and

0 1 0 0 0 1

M1 = 0 0 1 , M2 = 1 0 0 (2.5.23)

1 0 0 0 1 0

and take the trace of the resulting expression to obtain dy + d~ + d~

= 6. This is a reflection of the general result that

(2.5.27)

where, with (2.5.18), d1== d2 == 1, d3 == 2. We may set K == 1 in (2.5.26)

where XK = tr RK· With (2.5.14) and (2.5.24), the number ci of times

the irreducible representation f i appears in the decomposition of {RK}

is given by

6ci = t K~l XK XiK = XiI (i = 1, 2, 3). (2.5.25)

We note that the value X.1

of the character of the representation f·1 1

corresponding to the identity element Al of A is equal to the dimension

di of the representation fi. Thus f i appears di times in the decom

position of the regular representation, i.e., there is a matrix S such that

The matrices RK (K =1,... ,6) listed in (2.5.22) describe the

manner In which the AI' ... ' A6 defined by (2.2.4) permute among

themselves when multiplied on the left by the matrices AK (K = 1,

... ,6). Thus, AL [A1,···,A6] = [A1,···,A6]RL. Also, AKAL [A1,... ,A6]

= AK [AI'···' A6] RL = [AI'···' A6] RKRL· We observe that if AKAL= AM' then RKRL = RM. For example, we see from Table 2.1 that

A4A5 = A2 and from (2.5.22) that R4R5 =~. The set of matrices

RK = R(AK) (K = 1,... ,6) then forms a matrix representation of

dimension 6 of the group A = {AI' ... ' A6) defined by (2.2.4) which we

denote by {RK } and which is referred to as the regular representation

of the group A. The character of {RK } is seen from (2.5.22) and

(2.5.23) to be given by

(2.5.31 )

(2.5.29)

(2.5.30)

KKKS3kRkj = H11 S3j + H12 S4j ,

S4kR~ = H~l S3j + H~2 S4j .

Let

Consider the third and fourth rows of (2.5.28). We have

where we have noted that if ALAK = AM' then RLRK = RM,

HLHK = HM and Hr:1 =HKHll. Thus, the S3j and S4j (j = 1,... ,6)

given by (2.5.30) satisfy (2.5.29)1. In similar fashion, it may be shown

that they also satisfy (2.5.29)2. Proceeding in this manner, we may

6 -1 L 6 -1 LS3j = L (HL )11 R 1j , S4j = L (HL )21 R 1j ·

L=l L=l

With (2.5.30), the left hand side of (2.5.29)1 gives

K 6 -1 L K 6 -1 MS3kRk· = L (HL )11 R1k Rk· = L (HK HM )11 R1J·J L=l J M=l

K 6 -1 M K 6 -1 M= H 11 L (HM )11 R1· +H12 L (HM )21R1·M=l J M=l J

= H¥l S3j + H¥2 S4j

where N is the order of the group A considered and the d1,... , dr are the

dimensions of the inequivalent irreducible representations f 1' ... ' f r

associated with A.

We now indicate the manner in which the rows of the matrix S

effecting the decomposition (2.5.26) may be obtained. We may express

(2.5.26) as

S RK = (FK +GK +2HK) S (K = 1,... ,6). (2.5.28)

(2.5.26)

(2.5.24)(Xl, ... ,X6) = (6,0,0,0,0, 0)

36 Group Representation Theory [Ch. II Sect. 2.6] Continuous Groups 37

show that the rows of S are given by unit vector

(2.5.32)

(2.6.2)

where ¢1 is the angle which the axis of rotation makes with the x3 axis

and ¢2 is the angle which the projection of the rotation axis onto the

Xl' x2 plane makes with the Xl axis. We may characterize the rotation

by specifying the vector

With (2.5.18), (2.5.22) and (2.5.32), we have (2.6.3)

The symmetry properties of isotropic materials and transversely

isotropic materials are defined by continuous groups. For example, the

symmetry of a material possessing rotational symmetry but which lacks

a center of symmetry is defined by the three-dimensional proper

orthogonal group R3. R3 is also referred to as the three-dimensional

rotation group and is comprised of all 3 x 3 real matrices A which

satisfy

1 1 1 1 1 1

1 1 1 -1 -1 -1

1 -1/2 -1/2 -1 1/2 1/2S== (2.5.33)

0 -~/2 ~/2 0 -~/2 ~/2

0 ~/2 -~/2 0 -~/2 ~/2

1 -1/2 -1/2 1 -1/2 -1/2

2.6 Continuous Groups

(2.6.5)

(2.6.4)

(0 :::; 8 :::; 21r).

oo1

det A == ± 1

cos 8 sin 8

Q(8) == -sin 8 cos 8

o 0

where 0 :::; 8 :::; 1r, 0:::; ¢1 :::; 1r, 0:::; ¢2 :::; 21r. The vectors (2.6.3) form a

sphere of radius 1r. The group R3 of all rotations is referred to as a

three-parameter continuous group and the domain over which the

parameters ~l' ~2' ~3 vary is referred to as the group manifold.

The set of all 3 x 3 real matrices A which satisfy

forms the three-dimensional full orthogonal group 03. 03 is a mixed

continuous group comprised of two parts given by (i) the proper

orthogonal matrices A which belong to R3 and for which det A == 1 and

(ii) the improper orthogonal matrices which are of the form CA where

C == diag (-1, -1, -1) and A belongs to R3 . We observe that R3 is a

subgroup of 03. We shall be concerned with other continuous and

mixed continuous subgroups of 03 which define the symmetry

properties of transversely isotropic materials. One such group T1 is

comprised of the set of matrices

(2.6.1)det A == 1.

Consider a rotation of 8 radians whose axis of rotation is given by the The group T1 is a one-parameter continuous group. Another subgroup


of 03' which is denoted by T2 , is comprised of the matrices representations rand r' are (see Wigner [1959], p. 101)

cos (} sin (} 0 -cos (} -sin (} 0 ID· (A) f). (A) dr = (VIn) 8·· fJ V = Idr,Ir JS IJ rs'

Q((}) == -sin (} cos (} 0 , R1Q((}) == -sin (} cos (} 0 (2.6.6) A A

0 0 1 0 0 1 IDir(A) Rjs(A) dr = O.(2.6.9)

A

We consider integrals of the form

where R1 == diag (-1, 1, 1) and where 0 ~ (} ~ 21r. The group T2 is a

mixed continuous group. It is comprised of two disjoint parts given by

(i) the Q((}) and (ii) the matrices R1Q((}).

(2.6.11)

The analysis leading to the determination of the weight function

w(A) == w(~I' ~2' ~3) associated with the three-dimensional rotation

group R3 may be found in Wigner [1959], §10 and §14. We have

Ix(A) x(A) dr = v, Ix(A) x'(A) dr = 0 (2.6.10)

A Awhere X(A) == tr D(A) and X'(A) == tr R(A) denote the characters of the

representations rand r' respectively.

are

The results corresponding to (2.5.3) which give the orthogonality

relations for the characters of the irreducible representations rand r'

(2.6.7)

(2.6.8)If(SA) dr = If(A) drA A

If(A)dr = If(~1""'~n) w(~1'''''~n) d~1 ..·d~nA A

which are referred to as invariant integrals. The integration in (2.6.7)

is over the gr'oup manifold of A. The weight function w(~I'... '~n) is

chosen so that

1r 1r 21r

= I I I 2Po(1-cosO)O-202sin<P1 d<P2 d<P1 dO = 81r2

pO'

000

where Po is a constant and where A corresponds to a rotation through (}

radians about an axis whose direction is given by (~1' ~2' ~3). With

(2.6.3) and (2.6.11), the volume V of the group R3 is given by

The three-dimensional full orthogonal group 03 is a mixed

continuous group comprised of the two cosets formed by (i) the

matrices A belonging to R3 and (ii) the matrices CA where the A

belong to R3 . The invariant integral over 03 is given by

where S is an element of A. We refer to dT==w(~I'... '~n)d~I ... d~n as

an invariant element of volume. We may determine invariant elements

of volume for the continuous groups considered here, i.e., 03 and its

continuous subgroups. The arguments leading to the orthogonality

relations for the irreducible representations and for the characters of the

irreducible representations of a continuous group A are almost identical

to those given in §2.4 and §2.5 with the summation over the group

elements AI' ... ' AN being replaced by integration over the group

manifold of A. Let r == {D(A)} and r' == {R(A)} denote inequivalent

irreducible representations of A which are of dimensions nand m

respectively. We may assume with no loss of generality that rand r'

are unitary representations. The results corresponding to (2.4.12) and

(2.4.17) which give the orthogonality relations for the irreducible

V = Idr = I W(~1'~2'~3)d~1 d~2d~3R3 R3

(2.6.12)


(2.6.13)Jf(A) dT = Jf(A) dT + Jf(CA) dT.

03 R3 R3

The invariant integral over the one-parameter group T1 == {Q(B)}

(0 ~ B~ 27r) defined by (2.6.5) is given by

We may determine the number ck of times that the irreducible

representation {Dk(A)} appears in the decomposition of {D(A)} upon

multiplying both sides of (2.6.17) by Xk(A) and then integrating over

A. Thus, with (2.6.10) and (2.6.17),

Upon taking the trace of both sides of (2.6.16), we have

where the invariant element of volume dT == dB. The invariant integral

associated with the mixed continuous group T2 comprised of the two

cosets formed by (i) the matrices Q(B) belonging to T1 and (ii) the

matrices R1Q(B) where Q(B) belongs to T1 (see 2.6.6) is given by

We are particularly concerned with cases where the functions

f(A) appearing as integrands of the integrals f f(A) dT are the characters

X(A) == tr D(A) of a representation {D(A)} or the product of characters

X(A) == tr D(A) and X'(A) == tr R(A). Let {D1(A)}, {D2(A)}, ... denote

the inequivalent irreducible representations of the continuous group A.

Let {D(A)} denote a representation of A. We may then determine a

matrix S such that

(2.6.20)

(2.6.18)c1=~fx(A)dT.A

ck = ~ JX(A) Xk(A) dT,A

where X(B) == trD(A), Xk(B) == trDk(A), and A denotes a rotation

through B radians about the xl axis.

27r 27r

03: ck = 4~ JX(O) Xk(O) (1- cosO) dO + l7r JX'(O) Xk(O) (1- cosO) dO,o 0

27r 27rc1 = 4~ JX( 0)(1 - cos 0) dO + 4~ JX' (0) (1 - cos 0) dO

o 0

The coefficient c1 appearing in (2.6.17) and (2.6.18) denotes the number

of times the identity representation {D1(A)} appears in the

decomposition of the representation {D(A)}. This gives the number of

linearly independent functions which are linear in the quantities

forming the carrier space of the representation {D(A)} and which are

invariant under A. In (2.6.18)2' we have noted that Xl (A) == 1 where

Xl (A) is the character of the identity representation of A.

We observe that the characters X(A) and Xk(A) appearing in

(2.6.18) are functions of a single variable B for the cases where A is

given by R3 , 03' T1 or T2 . The expressions (2.6.18) appropriate for

the groups R3 , 03' T1 and T2 are listed below.

27rR3 : ck = i7r f X(O)Xk(O)(l-cosO)dO,

o2

f7r (2.6.19)

c1 =i7r X(O)(l-cosO)dOo

(2.6.14)

(2.6.17)

(2.6.16)

(2.6.15)

27rff(A)dT= fg(O)dO, g(O)=f(A(O))T1 0

f f(A) dT = f f(A) dT + f f(R1A) dT.

T2 T1 T127r 27r

= f g(0) dO + f h(0) dOo 0

where g(B) == f( A(B)), h(B) == f( R1A(B)), R1 == diag (-1, 1, 1), and

where the element of volume dT == dB.

42 Group Representation Theory [Ch. II

where X(B) == trD(A), Xk(8) == trDk(A), X'(8) == trD(CA), Xk(8)== tr Dk(CA), A denotes a rotation through 8 radians about the xl axis,

and C == diag (-1, -1, -1).III

21r

T1 : ck = l7r f x(0) Xk(0) dO,o

21r

c1 = 2~ f X(O) dOo

(2.6.21 )

ELEMENTS OF INVARIANT THEORY

(2.6.22)

3.1 Introduction

We consider the problem of determining the general form of a

scalar-valued polynomial function W(x) which is invariant under the

group A == {A}. Let {D(A)} denote the matrix representation which

defines the transformation properties of the quantity x == [xl' ... ' Xn]T

under A. W(x) is said to be invariant under A if

for all A in A. We wish to determine a set of functions II (x) ,... , Ip(x),

each of which is invariant under A, such that any polynomial function

W(x) which is invariant under A is expressible as a polynomial in the

invariants II (x) ,... , Ip(x). The invariants II (x) ,... , Ip(x) are said to

form an integrity basis for functions W(x) which are invariant under A.

The determination of an integrity basis constitutes the first main

problem of invariant theory. We have

(3.1.1)W(x) == W( D(A) x)

where X(8) == trD(A), Xk(8) == trDk(A), and A denotes a rotation

through 8 radians about the x3 axis.

21r 2~

T2 : ck = 4~ f X(O) Xk(O) dO + 4~ f X/(O) xk(O) dO,o 0

21r 21r

c1 = 4~ f X( 0) dO + l7r f X/( 0) dOo 0

where X(8) == tr D(A), Xk(8) == tr Dk(A), X'(8) == tr D(R1A), Xk(8)

= tr Dk(R1A), A denotes a rotation through 8 radians about the x3

axis, and R1 = diag(-I, 1, 1).

W(x) == c· . I i1... I Jp.I ... J (i, ... ,j == 1,2, ... ). (3.1.2)

It may happen that not all of the terms appearing In (3.1.2) are

independent. For example, we may have a relation of the form

1112 = I§. This is not an identity in the II' 12, 13 but becomes an

identity in x when we replace the II' 12, 13 by II (x), I2(x), I3(x). This

is referred to as a syzygy. The second main problem of invariant theory

is to determine a set of relations fi(I1,... , Ip) == 0 (i == 1, ... , q) such that

every syzygy relating the invariants I1,... ,Ip is a consequence of the

43

44 Elements of Invariant Theory [Ch. III Sect. 3.2] Some Fundamental Theorems 45

Theorem 3.3 An integrity basis for polynomial functions

'Y(xl' ... ,xn) = W(xl, x~, x~, ... , xl' x~, x3) of the vectors Xi = [xi,

X2' X3]T (i == 1, ... , n) which are unaltered under cyclic permutation of

the subscripts 1, 2, 3 or equivalently which satisfy

relations fi(I1,... , Ip ) == o. We usually write the expression (3.1.2) in the

form

(3.1.3)

where Wk(x) is a linear combination of all the monomials Ii ... Ii which

are of degree k in x. We are able to compute the number of linearly

independent functions of degree k in x which are invariant under A.

This enables us to determine whether there are any redundant terms in

the expression Wk(x). If any exist, we may employ relations of the

form fi(I1,... , Ip ) == 0 to remove the redundancies. Thorough discussions

of the theory of invariants are given by Grace and Young [1903], Elliott

[1913], Weitzenboch [1923], Schur and Grunsky [1968] and Weyl [1946].

W(X1'···' xn ) == W(Okx1'···' 0kxn) (k == 1,2,3)

where

°1 =[ ~0 0 l °2 =[ ~

1 0 l °3 =[ ~0 1 ]1 0 0 1 0 0

0 1 0 0 1 0

is given by

1. Lxi (i == 1, ... , n) ;

(3.2.3)

(3.2.4)

In (3.2.5), L xi ~2'" denotes the sum of the three terms obtained by

cyclic permutation of the subscripts on the summand. For example,

3.2 Some Fundamental Theorems

In this section, we list some theorems which are useful for pur

poses of determining integrity bases. We shall employ the notation xf

to designate the jth component of the vector xn ' i.e., xn == [xl,x2' ... ]T.


W(x1,x2' ... ,xn ) which satisfy

(3.2.1)

is given by

2.

3.

L(xi ~2 + x~~l) (i,j = 1,... ,n; i ::;j),

"'( i j i j) ( )L...J Xl x2 - x2 x1 i,j == 1, ... ,n; i <j ;

L xi (x~x! + ~3 x~) (i,j, k = 1,... , nj i ::; j ::; k) ,. . k k . k· . k·· .L {xix-I1(X2 - X3) + x-I1Xl (x2 - x3) + Xl xi (x-I2 - x-I3) }

(i,j,k == 1, ... ,n; i ~j ~ k).

(3.2.5)

(3.2.6)

(j, k == 1,... , n; j ~ k). (3.2.2)In order to prove Theorem 3.3, we set

where w == -1/2 + i ~/2 and w2 == -1/2 - i ~/2 are cube roots of unity.


W(I1,I2,... ,Ir , x1,x2' ... 'xn ) which are invariant under a group A for

which the 11,12, , Ir are invariants is formed by adjoining to the

quantities II' 12, , Ir an integrity basis for polynomial functions

V(x1' x2'···' xn ) which are invariant under the group A.

1 1 1

y. == Kx· (i == 1, ... ,n), K== 1 w2 w1 1

1 w w2(3.2.7)


invariants of degree two appearing in (3.2.11). We have from (3.2.7)Then, W(xl'···' xn ) == W(K-1Yl'···' K-1yn ) == V(Yl'·'" Yn) where

V(Y1 ,... , Yn ) = V(KI\:K-1 Y1 ,... , K I\:K-1 Yn) (k = 1,2,3). (3.2.8)

With (3.2.4) and (3.2.7),(3.2.13)

(3.2.9)The result (3.2.5) follows from (3.2.11) and (3.2.13).

where (a, b, c) == diag (a, b, c) as in (1.3.2). Thus, we have

( iii) V( i i 2 i) _ V( i 2 i i )V Y1' Y2' Y3 == Y1' wY2' w Y3 - Y1' w Y2' wY3 (3.2.10)


W(x1'''''xn ) = W(xl, x~, x! ,... ,xr, x~, x3) which are invariant under

all permutations of the subscripts 1,2,3 or equivalently which satisfy

where i == 1, ... , n. An integrity basis is readily seen to be given by

1. Yl (i == 1, ... ,n) ;

(3.2.14)

where

(yl)i1 (y~)h (y!)k1 '" (yr)in (y~)jn (Y3)kn (3.2.12)

from V(Y1' ... ' Yn) which satisfies (3.2.10). This requires that

)1+ ... +jn w2(k1+ ... +kn ) = 1 or (since w3 = 1) that j1 +... +jn +2k1

+... +2kn == 3r where j1 ,... , kn and r are positive integers. Since the

yi are invariants, we may factor out these quantities from (3.2.12).

S~nce yk ~ y~ and y~ ~3 Y~ are invariants, we may also factor out such

terms. The term (3.2.12) is thus expressible as products of invariants

from (3.2.11) and terms of the form (3.2.12) for which i1 == ... == in == 0,

jl + ... + jn ~ 2, k1 + ... + kn ~ 2, j1 + ... + jn + 2k1 + ... + 2kn == 3 or

6. The quantities satisfying these restrictions are seen to be of the form

yk~3 or yk~3Y~y~, These terms are expressible in terms of the

(3.2.16)

(i ,j ,k == 1, ... , n; i ~ j ~ k) .

(i ,j == 1, ... , n; i ~ j) ;E(xi ~2 + xkxi)

'" xi (xj

xk + xj xk)Li123 32

is as follows, where we employ the notation (3.2.6):

2.

1. Ex! (i == 1, ... ,n);

3.

We may proceed as in the argument leading to Theorem 3.3 and

set Yi == KXi (i == 1, ... ,n) where K is given by (3.2.7). Then

W(x1'···' xn) == V(Y1'···' Yn ) where

(3.2.11)(i ,j == 1, , n; i ~ j) ,

(i ,j == 1, , n; i < j) ;

Thus, consider the monomial term

2. ykr1 + Y~~2i j i j

Y2 Y3 - Y3 Y2

3.


The requirement that (3.2.17) holds for k == 1,2,3 is seen from Theorem

3.3 to imply that V(Yl' ... 'Yn) is expressible as a polynomial V1(... ) in

the quantities

(3.2.23)

(3.2.22)

x2u2v2 + x3u3v3 x2w3 +x3w2 ,Y2z2u3v3 +Y3z3u2v2 Y2z2w2 +Y3z3w3

.. kE xPl x~2 ... xPm (i,j,k == 1,..., n; i::; j ::; ... ::; k).

2(Y2z2u3v3 + Y3z3u2v2) == (Y2u3 + Y3u2)(z2v3 + z3v2)

+ (Y2v3 +Y3v2)(z2u3 +z3u2) - (Y2z3 +Y3z2)(u2v3 +u3v2)·

where the matrices D(A1), ... , D(AN) form an n-dimensional matrix rep-

m.

In (3.2.22), E x~l x-iP2 '" x~m' for example, denotes the summation

over all Pl,P2, ... ,Pm chosen from 1,2,... ,m such that Pl,P2, ... ,Pm are

all different. The proof of this theorem may be found in Weyl [1946],

Chapter 2. Theorem 3.4 is, of course, a special case of Theorem 3.4A.

We may employ this theorem to show that the elements of an integrity

basis for functions W(xl' x2' ... ) which are invariant under a finite group

A == {AI' ... ' AN} of order N are of degree ::; N. Thus, consider a

function W(x) == W(xl' ... ' xn) which is invariant under a finite group A

== {AI' ... ' AN} and hence satisfies

Theorem 3.4A An integrity basis for polynomial functions

W( ) - W( 1 1 1. . n n n) h· h . .xl,.",xn - xl,x2, ... ,xm , ... ,xl,x2, ,xm w Ie are InvarIant

under all permutations of the subscripts 1,2, , m is given by

Thus, an integrity basis for functions V(y1' ... ' Yn) invariant under the

KI\K-1 (k = 1,... ,6) is formed by the quantities (3.2.19). The result

(3.2.16) then follows from (3.2.13) and (3.2.19).

1. EXP1 (i == 1, ... ,n);

"i j ( )2. ~ xPl xP2 i,j == 1,... , n; i ::; j ;

(3.2.18)

(3.2.19)

(3.2.20)

i (. 1 ) i j i j ( )Y1 1 == ,... ,n, y2 y3 + y3 y2 i,j == 1,... , n; i ::; j ,

Y~~Y~ + Y~~3Y~ (i,j,k = 1,... ,n; i S;j S; k)

i j i j (.. - 1 ...)Y2 Y3- Y3 Y2 1,J- ,... ,n,1<J,

i j k i j k (.. k - 1 .. < . < k)Y2Y2Y2-Y3Y3Y3 1,J, - ,... ,n,1_J_ .

x2 x3 z3 u3 x2z3 + x3z2 x2u3 + x3u2

Y2 Y3 z2 u2 Y2z3 + Y3z2 Y2u3 + Y3u2

x2 x3 u3 v3 x2u3 + x3u2 x2v3 + x3v2

Y2z2u2 + Y3z3u3 Y2z2v2 + Y3z3v3,

Y3z3 Y2z2 u2 v2

(3.2.21 )

The matrices KI\K-1 (k=1,2,3) are given by (3.2.9) and

and

There are restrictions imposed on the form of V1(... ) by the require

ment that it be invariant under the KOkK-1 (k = 4,5,6). With

(3.2.18), we see that the quantities (3.2.19) remain unaltered under the

KI\K-1 (k = 4,5,6) while the quantities (3.2.20) change sign. With

Theorems 3.1 and 3.2, we see that an integrity basis for functions

V(Y1'''''Yn) invariant under the KI\K-1 (k = 1,... ,6) is given by the

quantities (3.2.19) and products of the quantities (3.2.20) taken two at

a time. The products of the quantities (3.2.20), however, are

expressible in terms of the invariants (3.2.19). This follows from

identities (involving determinants) of the forms


transformation properties of x under A. Then,resentation {D(AK)} which defines the transformation properties of x

under A. Any polynomial function W(x1 ,... , xn ) which satisfies (3.2.23)

may be written asW(x) == W(D(A)x), c·· x· x· == c·· D· (A) D· (A) x xIJ 1 J IJ lr Js r s (3.2.27)

1 NW(x1"" xn) == N L W(D 1J·(AK ) x· ,... , D .(AK ) x.).

K=l J nJ J

Following the argument given by Weyl [1946], we set

(3.2.24)holds for all D(A) belonging to {D(A)}. This implies that

crs == c· ·D· (A)D. (A)IJ lr Js (3.2.28)

Theorem 3.4B is useful In cases where the group A is of low

order. A more general result is of assistance in determining limits on

the degrees of the elements comprising an integrity basis for functions

W(x, y, ... , z) which are invariant under a group A. The quantities x,

y, ... , z transform in the same manner under A and have n independent

components, i.e., x == [xl"'" X ]T. Suppose that W(x) == c·· x· x· isn IJ 1 Jinvariant under A where we may assume that c·· == c·· Let {D(A)}IJ Jl·denote the n-dimensional matrix representation which defines the

Since (3.2.24) is unaltered under all permutations of the values 1,2, ... , N

which K assumes, the function V( ) in (3.2.26) is unaltered under all

permutations of the subscripts 1,2, , N. Then, with Theorem 3.4A, we

see ~ha:t V( ... ) i~ ~xpressible as a polynomial in the quantities L zi,LZl z~, ... , LZl z~ ... z~ where L("') has the same meaning as in

Theorem 3.4A. This establishes the following result:

Theorem 3.4B The elements of an integrity basis for functions

W(x) which are invariant under a finite group A == {AI"." AN} of order

N are of degrees ~ N in x.

(3.2.32)

(3.2.29)

(3.2.30)V(x,y) == V( D(A) x, D(A) y)

holds for all A in A. Consider the quantity

holds for all A in A, i.e., V(x,y) is invariant under A. We' refer to the

process of applying Yk a~ to W(x) as a polarization process.k

Similarly, we may show that repeated application of the polarization

process to an invariant W(x) produces another invariant. Thus, if

W(x) is an invariant of degree q in x, then

V(x,y) == Yk aa c·· x· x· == c.. (x.y. +y.x.).xk IJ 1 J IJ 1 J 1 J

With (3.2.28) and (3.2.29), we see that

for all D(A) forming an n-dimensional representation {D(A)} of A, arise

a aZi ax. Yj ax. W(x) == U(x, y, z) (3.2.31)

1 J

is an invariant of degree (q - 2, 1, 1) in (x, y, z). The manner in which

the polarization process may be employed in the generation of integrity

bases is indicated by the following theorem which is referred to as

Peano's Theorem. The proof of this theorem is given by Weyl [1946].

Theorem 3.5 The elements of an integrity basis for polynomial

functions W(xl'''''~) of m>n quantities ~ = [xi. ... ,X~]T which are

invariant under a group A, i.e., which satisfy

(3.2.26)

(3.2.25)zl, .... ,Z& = D1/A1)xj , , D1/AN) Xj ,

zr,· ..,z* = Dnj(A1)xj , , Dn/AN) Xj .

Then, W(x1"'" xn) given by (3.2.24) is expressible as

52 Elements of Invariant Theory [Ch. III

upon repeated polarization of invariants comprising an integrity basis

for functions of n quantities xl' ... ' xn which are invariant under A.

Further, if

xl xl xl1 2 n

x2 x2 x2

det(xI,x2'···'xn) ==1 2 n (3.2.33)

xn xn xn1 2 n

IV

INVARIANT TENSORS

4.1 Introduction

Expressions of the form

occur In the constitutive relations employed in the classical linear

theories of crystal physics and also in the non-linear generalizations of

these theories. The tensors C· . are referred to as property tensorsIl··· In

and relate physical tensors such as stress tensors, strain tensors, electric

field vectors, .... The tensors E· . and E· . denote physical11··· In 13 ... In

tensors or the outer products of physical tensors, e.g., E· .13 ... 16

= Ei i Ei i· There are restrictions imposed on the form of the34 56

constitutive relations (4.1.1) by the requirement that they must be

invariant under the group A which defines the material symmetry.

Thus, the expression in (4.1.1)2 must satisfy

is invariant under A, the elements of an integrity basis for functions

W(x1' ... ' xm ) of m ~ n quantities which are invariant under A arise

upon repeated polarization of the invariant (3.2.33) and the invariants

comprising an integrity basis for functions of n -1 quantities xl'···' Xu-Iwhich are invariant under A.

(4.1.1)

(4.1.2)

for all A = [Aij ] belonging to the group A. The equations (4.1.2)

impose restrictions on the form of the property tensor C = C· ..Il··· In

With (4.1.1)2' (4.1.2) and (1.2.6) (the A are orthogonal), we see that

(4.1.3)

must hold for all A [Aij ] belonging to A.

53

A tensor C· . whichIl··· In

54 Invariant Tensors [Ch. IV Sect. 4.1] Introduction 55

(4.1.8)

(4.1.7)

(4.1.10)

CK == C~. (K == 1, ... , P),ll··· ln

Then the property tensors appearing in (4.1.6), i.e.,

then a number of terms in (4.1.9) will be redundant. The redundant

terms may be eliminated by inspection in simple cases. We give results

below which assist in the elimination process in more complicated

problems.

where the tensors C1,... , Cp form a complete set of nth-order tensors.

If the physical tensors T and E possess symmetry properties, e.g., if

W == (a1C}. . + ... + apCr .) E· .ll··· ln ll··· ln ll··· ln'

Expressions of the form (4.1.1) which are invariant under A may

then be written as

(4.1.9)

where the a1"'" ap are constants. A set of P linearly independent nth

order tensors which are invariant under A will be referred to as a

complete set of nth-order tensors. In §4.7, we list complete sets of

invariant tensors of orders 1,2, ... for the groups D2h, 0h' R3 , 03 and

T1 ·

form a set of linearly independent nth-order tensors which are invariant

under A. Any nth-order tensor C == C· 1· which is invariant under A11·" n

is expressible as a linear combination of the tensors (4.1.7). Thus,

(4.1.6)

(4.1.5)

(4.1.4)

p=~J (trA)ndT

A

if A is a continuous group. The matrix defining the transformation

properties of the 3n components x f x~ ... x!1 (i1,... , in == 1,2,3) under A11 12 In

is referred to as the Kronecker nth power of A. The trace of the

Kronecker nth power of A is given by (tr A)n. The number of linearly

independent functions which are multilinear in x1,x2"'" xn and which

are invariant under A is equal to the number of times the identity rep

resentation appears in the decomposition of the matrix representation

of A furnished by the Kronecker nth powers of the matrices A

comprising A. This is seen from (2.5.14) and (2.6.18) to be given by

(4.1.4) or (4.1.5) depending on whether A is a finite or a continuous

group. A discussion of the properties of the Kronecker products of

matrices is given by Boerner [1963].

We may employ theorems from Chapter III to generate the P

linearly independent multilinear functions of x1, .."xn which are

invariant under A. Suppose that these invariants are given by

for cases where A is a finite group {A1,... ,AN} and by expressions such

as

satisfies (4.1.3) for all A belonging to A is said to be invariant under A.

In order to determine the general form of C, we first consider the

problem of determining the general expression for functions which are

multilinear in the n vectors xi = [xi, x~, X~]T (i = 1,... , n) and are also

invariant under A. The number P of linearly independent multilinear

functions of xl"'" xn which are invariant under A is given by

56 Invariant Tensors [Ch. IV Sect.4.2] Decomposition of Property Tensors 57

W 2 == (b1V~ . + ... + bMVM . )E. ..Il···In Il···In Il· .. In

If we denote by Wi and W2 the expressions obtained from (4.1.11)

upon eliminating the redundant terms, then the appropriate expression

for W is given by

Suppose that the nth-order tensors C1,... , Cp which are invariant

under A are comprised of the linearly independent isomers of the

tensors U and V. We note that U·· . is an isomer of U·· . ifIpIq ... Ir 1112 ... In

(p,q, , r) is some permutation of (1,2, ... , n). Let U1,... , UN and

VI' ' VM denote the linearly independent isomers of U and V. We

proceed by eliminating the redundant terms in the expressions

(4.2.1)sU == U·· .Ia I,8 ... I,

the permutation of the integers 1,2, ... , n which carrIes 1,2, ... , n into

(Y, {3, ... , ,. Application of s to the tensor U == U·· . yields an isomer1112... I n

of U defined by

We see that the distinct isomers of U form the carrIer space for a

matrix representation {D(si)} (i == 1, ... , n!) of the symmetric group Sn

which is comprised of the n! permutations of the numbers 1,2, ... , n. We

note that there is an irreducible representation of Sn corresponding to

each partition n1n2... of n, i.e., to each set of positive integers

n1 2 n2 2 ... such that n1 + n2 + ... == n. For example, the partitions

of n == 3 are given by 3, 21 and 111. We denote an irreducible

representation of Sn by (n1n2· .. ) where n1n2 ... is a partition of n. The

representation {D(si)} whose carrier space is formed by the independent

isomers of U may be decomposed into the direct sum of irreducible

representations of Sn. Thus, we may determine a matrix K such that

(4.1.11)

(4.1.12)W == Wi + W2·

The number of linearly independent terms in each of the expreSSIons

Wi and W2 is a useful bit of information. If this information is

lacking, it may prove to be tedious to determine whether all of the

redundant terms have been eliminated from WI and W 2. Given this

information, we may proceed by generating the appropriate number of

linearly independent terms rather than by eliminating the redundant

terms. We consider below the problem of determining the number of

linearly independent terms appearing in expressions such as (4.1.11).

(4.2.2)

where the (Ynln2... are positive integers or zero and where the

summation is over the irreducible representations (n1n2 ... ) of Sn. A set

of property tensors which forms the carrier space for an irreducible

representation (n1n2···) of Sn is referred to as a set of tensors of

symmetry type (n1n2... ). The number of tensors comprising a set of

tensors of symmetry type (n1n2··· np ) is given by fnln2 ... np where

4.2 Decomposition of Property Tensors

Let C1, C2, ... be a complete set of nth-order tensors which are

invariant under the group A. This set of tensors is comprised of tensors

U,V, ... together with the distinct isomers of these tensors. Let s denote

(4.2.3)

£1 == n1 + p - 1, £2 == n2 + p - 2 , ... , £p == np .

The number fnln2 ... np is the dimension of the representation

58 Invariant Tensors [Ch. IV Sect.4.2] Decomposition of Property Tensors 59

( n n) of 5 and may be found in the first column of the charactern1 2 ... P n

table for 5n . The character tables for 52' ... ,58 are given in §4.9.

We shall employ the notation The matrices D(sl), ... ,D(s6) appearing in (4.2.8) are given by

(4.2.9)

where h1i == 1 if i == 1, h1i == 0 if i =1= 1. Let us now consider a third

order property tensor

The isomers of U1 arise upon application of the permutations e, (12),

(13), (23), (123), (132) comprising 53 to the subscripts i1i2i3 in (4.2.5).

With (4.2.1) and (4.2.5), we have for example

0 0 0 0 0 0 1

D(sl),···,D(s3) == 0 1 0 0 0 0 1 0

0 0 0 0 1 1 0 0

(4.2.10)1 0 0 0 0 1 0 1 0

D(s4),···,D(s6) == 0 0 1 0 0 0 0 1

0 1 0 0 1 0 0 0

The D(si) form a matrix representation of 53' With (4.2.10), the

character Xi == X( si) == tr D(si) (i == 1,... ,6) of this representation isgiven by

Xl"'" X6 == 3, 1, 1, 1,0, O. (4.2.11)

(4.2.4)

(4.2.6)

(4.2.5)

e12 3 == hI' h2· ... h3· (i1,... , in == 1,2,3)... 11 12 In

Proceeding in this manner, we obtainWith (2.5.14), the number of times the irreducible representation

(n1n2"') occurs in the decomposition of {D(si)} is given by

(4.2.7)

6a == 1 '" x· x~ln2'" , x· == trD(s.)n1n2'" 6.L.-J 1 1 1 1

1==1(4.2.12)

where the summation is over the 3! permutations of 53 and where

X~ln2'" is the value of the character of (nln2"') corresponding to the1

permutation si' We have noted that the Xf1n2'" (i == 1,... ,6) are real.

If s· and s· belong to the same class I of 53' then1 J

With (4.2.7) and similar expressions obtained upon applying the

permutations e, , (132) to U2 and U3, we see that the tensors skUi

(i == 1,2,3; k == 1, ,6) are expressible asXi == Xj == X,, (4.2.13)

where

skU. == U.D .. (sk)1 J Jl (i,j == 1,2,3; k == 1, ... ,6) (4.2.8)The X, and x~Jn2'" are the values of the characters of the

representations {D(si)} and (n1n2"') for the class I' Then, (4.2.12)

may be rewritten as

60 Invariant Tensors [Ch. IV Sect04.2] Decomposition of Property Tensors 61

(4.2.14)

where the summation is over the three classes e; (12), (13), (23); (123),

(132) of 53. These classes are characterized by their cycle structure.

The class denoted by 1II 2/2 3/3 or alternatively by 11/2/3 has II 1

cycles, 12 2-cycles and 13 3-cycles. The number of permutations

belonging to a class, = '112'3 is denoted by h,: The h" X~ln2··· for

S3 are given in the character table for 53 (Table 4.2 in §4.9). With

(4.2.11), (4.2.14) and Table 4.2, we obtain a3 == 1, a21 == 1, alII == O.

Thus, the set of three property tensors U1,U2,U3 defined by (4.2.7)

may be split into two sets of tensors, one of symmetry type (3) and one

of symmetry type (21). With (4.2.3), the number of tensors comprising

each set is given by f3 = X~ = 1 and f21 = X~1 = 2 respectively.

These numbers appear in the first column of the character table for 53

and give the dimensions of the irreducible representations (3) and (21).

We note that the subscript e is used to denote the class of S3 comprised

of the identity permutation.

More generally, the set of distinct isomers of a property tensor U

== U· . forms the carrier space for a representation {D(s.)} of the11 ... 1n . 1

symmetric group 5n whose elements are the n! permutatIons of 1,2, ... ,n.

We may determine the matrices comprising the representation {D(si)}

and then determine the character Xi == tr D(si) of the representation.

The isomers of U may be split into sets which form the carrier spaces

for the irreducible representations (n1n20 .. ) of Sn and which are referred

to as sets of tensors of symmetry types (n1n2... ). The number of sets of

tensors of symmetry type (n1n2... ) arising from the distinct isomers of

U is given by

where the summation in (4.2015)2 is over the classes of 5n , h, is the

order of the class, and the X" X~ln2··· are defined as in (4.2.13). In

the next section, we discuss a procedure which will enable us to

generate the sets of property tensors of symmetry type (n1n2... ) arising

from the isomers of U.

We observe that the tensor Uo .. == 81. (82. 83

0 +83

. 82

. )111213 11 12 13 12 13

may be considered to be the product of the tensors 81. and11

82 . 83. + 830 82. which are of symmetry types (1) and (2) re-

12 13 12 13spectively. The tensors 81

0 and 820 83

0 + 830 82

0 form the carrIer11 12 13 12 13

spaces for the identity representations of the symmetric groups

51 == {e} and 52 == {e, (23)} respectively. We say that U· . 0 forms111213

the carrier space for the identity representation of the direct product of

the groups 51 and 52. Further, the tensor Uo 0 0 and its distinct111213

isomers form the carrier space for a reducible representation of the

symmetric group 53 which we denote by (2) . (1) or by (1) . (2) and refer

to as the direct product of the irreducible representations (1) and (2).

Murnaghan [1937], [1938a]' [1938b] has considered the problem of

determining the decomposition of the direct products of the irreducible

representations (m1m2· 0.) of 5m and (n1n2.. 0) of 5n into the sum of

irreducible representations of Sm + n. In Murnaghan [1937], [1938b],

tables are given which yield the decomposition of the products of

irreducible representations (m1m2... ) of 5m and (nln2... ) of 5n for

cases where m + n :::; 10. For example, we see from Table 8.3 in §8.6 or

from the tables given by Murnaghan [1937] that (2) . (I) == (3) + (21).

As a further example, we consider the determination of the

symmetry type of the SIX distinct Isomers of the tensor

U == 810 81

0 820 82

0 given by11 12 13 14

(4.2.15) (4.2016)

62 Invariant Tensors [Ch. IV Sect. 4.3] Frames, Standard Tableaux and Young Symmetry Operators 63

If we apply one permutation from each of the classes of S4' e.g., e, (12),

(123), (1234), (12)(34) to the six tensors (4.2.16), we find that the

traces of the matrices defining the transformation of the tensors are

given by 6, 2, 0, 0, 2 respectively. This tells us that the tensors (4.2.16)

form the carrier space for a representation of S4 whose character X, is

given by

1,2,... , n in any manner into the n squares. A standard tableau is one in

which the integers increase from left to right and from top to bottom.

For example, the standard tableaux associated with the frames [31] and

[22] are given by

X, == 6, 2, 0, 0, 2 (4.2.17)

1 2 3,

4

1 2 4,

3

1 3 4 and

2

1 2,

3 4

1 3 .

2 4

(4.3.1)

for the classes ,== 14, 122, 13, 4, 22. With (4.2.15), (4.2.17) and the

character table for S4 (Table 4.3 in §4.9), we see that the set of tensors

(4.2.16) is of symmetry type (4) + (31) + (22). Alternatively, we note

that 81. 81. and 82. 82. are both of symmetry type (2) and that the11 12 13 14

tensors (4.2.16) form the carrier space for the reducible representation

(2) . (2) of S4. With Table 8.3 in §8.6 or the tables given by Murnaghan

[1937], we have (2). (2) == (4) + (31) + (22) so that the tensors (4.2.16)

form a set of symmetry type (4) + (31) + (22).

4.3 Frames, Standard Tableaux and Young Symmetry Operators

A partition n1n2... of the positive integer n is a set of positive

integers n1 ~ n2 ~ ... ~ ° such that n1 +n2 +... == n. The frame

[n1n2···] associated with the partition n1n2"· consists of a row of n1

squares, a row of n2 squares, ... arranged so that their left hand ends

are directly beneath one another. Thus, the frames associated with the

partitions 4, 31, 22 of 4 are

We may denote the standard tableaux associated with a frame

a == [n1n2···] by F1,F2\···· We order the standard tableaux by saying

that F? precedes F~ if, upon reading as in a book, the first location in

the two tableaux for which the entries differ has a smaller entry for F?

than has F~. Thus we would denote the three standard tableaux given

on the left of (4.3.1) by F1, F2\ F3 respectively with a == 31. We

denote by ag the permutation which carries F~ into F? Thus,

af~ = (34) : (34) 124 _ 123 (34) F31 - F31 .3 -4 2 - 1 '

af§ = (243) : (243) 134 _ 123 (243) F 31 - F31 .2 -4 3 - 1 '(4.3.2)

a~l = (34) : (34) 123 _ 124(34) F31 - F31 .4 - 3 1 - 2 '

an = (234) : (234) 123 _ 134 (234) F31 - F314 - 2 1 - 3 .

We note that a~ is the inverse of agoLet F? denote a tableau associated with a frame a == [n1n2 ... ].

Let

where the summation in (4.3.3)1 IS over all permutations of the

numbers in F? which leave each number in its own row and where theA tableau is obtained from a frame [n1n2 ...] by inserting the numbers

(4.3.3)

64 Invariant Tensors [Ch. IV Sect. 4.3] Frames, Standard Tableaux and Young Symmetry Operators 65

The Young symmetry operators associated with a standard tableau F?

are given by

summation In (4.3.3)2 is over all permutations of the entries in F?

which leave each number in its own column. The quantity cq is +1 or

-1 according to whether q is an even permutation or an odd

permutation. We recall that e denotes the identity permutation and

that permutations are to be multiplied from right to left. For example,

(13)(12) == (123). In cases where no confusion should arise, we at times

suppress the parenthesis on the permutations, e.g., we will write. 22 12(e + 12) Instead of (e + (12)). Thus, for the standard tableaux F 1 == 34

22 13 hand F2 == 24 ,we ave

There is no summation over the repeated indices In (4.3.7).

example, with ar~ = a~r = (23) and (4.3.4), we haveFor

(4.3.7)

(4.3.8)

= (23 - 132 - 234 +1342) = Qr2 ar~ .

= (23 + 123 +243 +1243) = pF ar~,

a paQa pO' a Qa pO' Qa a(]'rs s . s == r (]'rs s == r . r (]'rs ,

a yO' yO' a(]'rs s == r (]'rs ,

ar~ p~2 = (23)(e +13)(e +24)

22 22(]'12 Q2 == (23)(e - 12)(e - 34)

We have (see Rutherford [1948], p. 16)

(4.3.4)

Qr2 = (e - 13)(e - 24),

Q~2 = (e -12)(e - 34).p~2 = (e +13)(e +24),

Pr2 = (e + 12)(e +34),

y? == P? Q?, (4.3.5) Then, with (4.3.7) and (4.3.8),

22 12where P? and Q? are defined by (4.3.3). For the tableaux F1 == 34

and F22 - 13 we have2 - 24'

(]'22 y2212 2

(]'22 Q22 p22 _ Q22 (]'22 p2212 ·2 2 - 1 12 2

- p22 Q22 (]'22 _ y22 (]'22 .- 1 1 12 - 1 12'y22 (e + 12) (e + 34) (e - 13) (e - 24),1

-22 (e - 13) (e - 24) (e + 12) (e + 34),Yl(4.3.6)

y22 (e + 13) (e + 24) (e - 12) (e - 34),2

-22 == (e - 12) (e - 34) (e + 13) (e + 24).Y2

_ Q22 p22 (]'22- . 1 1 12

In similar fashion, we find that

Y-22 (]'221 12'

(4.3.9)

Let (]'~ denote the permutation which yields F? when applied to F~. (]'2221 y212 == y222 (]'2221

' (]'22 y22 _ y22 (]'2221 1 - 2 21 . (4.3.10)

66 Invariant Tensors [Ch. IV Sect. 4.3] Frames, Standard Tab/eaux and Young Symmetry Operators 67

Y~ = (e + 12 + 13 + 23 + 123 + 132),

YI1 = (e + 12)(e - 13) = (e + 12 - 13 - 132),

Let X = a1sl +a2s2 +... +an!sn! denote an arbitrary linear com-

bination of the permutation operators sl ( = e), s2' where the si

(i = 1,... ,n!) are the n! permutations of the numbers 1, ,n. The Young

symmetry operators satisfy the relations listed below. The arguments

leading to these results may be found in Chapter 2 of Rutherford [1948].

F~ = 123, F21 _ 12 F21 _ 131-3' 2-2' a 21 - (23)21 - ,

(4.3.14)

(23) YI1 = (23)(e +12)(e - 13) = (23 + 132 123 - 12).

y? X Y? = BCt pY? , p = coefficient of e in Y? X ,

Y? X Y~ = BCt p ag Y~, p = coefficient of e in ag. Y? X , (4.3.11)

Applying the Young symmetry operators given In (4.3.14) to

U1 = e123 +e132 yields, with (4.2.7),

Y?XY~ = 0,

where BCt is a non-zero constant.

(4.3.15)

We may employ Young symmetry operators to generate a set of

property tensors of symmetry type (n1n2".). Suppose that

F1,F2,···,Ff (4.3.12)

We have, from (4.3.15),

2

2

2

1 1

1 -2

-2 1

(4.3.16)

yields a set of tensors of symmetry type (n1n2' .. ) provided that

y Ct1 U· . i= O. For example, we have seen that the three distinct

11'·· Inisomers of the tensor U1 = e123 + e132 defined by (4.2.5) may be split

into sets of tensors of symmetry types (3) and (21). We have

are the f standard tableaux associated with the frame Q. Then

(r=l, ... ,f; Q=n1n2.") (4.3.13)

Application of the sl'.'" s6 = e, (12), ... , (132) to the Vk (k = 1,2,3) is

then defined by

s.Vk=s.U K k=U D (s.)K k=V.K~lD (s.)K k (4.3.17)1 1 q q P pq 1 q J JP pq I q

where we have employed (4.2.8). The set of six matrices K-1 D(si) K

(i = 1,... ,6) forms a matrix representation of the symmetric group 53

which is expressible as the direct sum of irreducible representations of

53. Thus, we have

68 Invariant Tensors [Ch. IV Sect. 4.4] Physical Tensors of Symmetry Class (n1 n2 •• .) 69

where the D(si) are defined in (4.2.10) and where

The complete set of nth-order property tensors associated with a

group A is generally comprised of tensors U, V, ... together with the

distinct isomers of these tensors. We may in principle employ Young

symmetry operators to split the sets of tensors comprised of U and its

isomers, V and its isomers, ... into a number of sets of tensors of

symmetry types (n1n2... )' (m1m2... )' .... Let ,8n1n2... be the number

of sets of property tensors of symmetry type (n1n2 ... ) which occur. We

then say that the complete set of nth-order property tensors associated

with the group A is of symmetry type 2:,8n1n2... (n1n2... ) where the

summation is over all partitions of n. The determination of the

symmetry type of a complete set of nth-order property tensors is not a

difficult matter for n ~ 10. This is primarily due to results given by

Murnaghan [1937], [1938b], [1951].

(4.3.19)

(4.3.18)(i == 1,... ,6)

1] [-1 1] [0 -1].o ' -1 0 ' 1 -1

The transformation properties of the independent components ¢>l, ... ,¢>q

is referred to as a tensor of symmetry class (n1n2... ). The tensor

T· . is a three-dimensional nth-order tensor with 3n independent11··· 1n

components. The transformation properties of the T· . under a11···1n

transformation A == [Aij] are given by

4.4 Physical Tensors of Symmetry Class (nln2...)

Let n1n2... denote a partition of n. Let FI denote the first

standard tableau associated with the frame a == [n1n2...]. Let VI be

the Young symmetry operator defined by (4.3.5)2 which is associated

with the standard tableau Fl. Then the tensor

(4.4.2)

(4.4.1)

We may verify that the matrices r 1(si) and r 2(si) form irreducible

matrix representations of the group 53. The characters are given by

tr r 1(si) == 1, 1, 1, 1, 1, 1 and tr r 2(si) == 2, 0, 0, 0, -1, -1. We note

that the permutations si (i == 1,... ,6) defined by (4.2.9) which comprise

53 may be split into the sets sl == e; s2' s3' s4 == (12), (13), (23); s5' s63== (123), (132). These sets form the classes, of 53 denoted by 1 , 12,

3. We recall that the character of a representation of 53 takes on the

same value for all elements of 53 belonging to the same class. We then

see from Table 4.2 in §4.9 that the representations {r1(si)} and {r2(si)}

are equivalent to the irreducible representations of 53 denoted by (3)

and (21) since the characters of {r1(si)} and {r2(si)} are the same as

the characters of (3) and (21) respectively. The tensors VI and

(V2

, V3) defined by (4.3.15) form the carrier spaces for the

representations {r1(si)} and {r2(si)} and hence form sets of tensors of

symmetry types (3) and (21). We thus see that we may employ Young

symmetry operators to conveniently generate sets of tensors of specified

symmetry type.

70 Invariant Tensors [Ch. IV Sect. 4.4] Physical 7'ensors of Symmetry Class (n1 n2 ...) 71

of <Pi i defined by (4.4.1) under a group A are described by a q-1··· n

dimensional matrix representation {R(A)}. We are interested in

determining the matrices R(A) and the quantities tr R(A) which define

the character of the representation {R(A)}.

Let

(4.4.6)

(4.4.7)

We consider the special case where n = 2, I.e., where T· . IS a1112

second-order tensor. The tensor T·· may be expressed as the sum of1112

its symmetric and skew-symmetric parts. Thus,

Then (4.4.6) may be written as

T' = (AxA) T (4.4.8)

The tensors on the right of (4.4.3) are of symmetry classes (2) and (11)

respectively. Thus, there are two partitions of n = 2 given by

n1n2... = 2 and'll respectively. The standard tableaux associated with

the frames [2] and [11] are given by

F11 _ 11 - 2

(4.4.3)

(4.4.4)

where A x A is referred to as the Kronecker square of A and is defined

by

A11A11 A11A12 A12A11 A12A12

A11A21 A11A22 A12A21 A12A22 = [ AnA A12A ].AxA=A21A11 A21A12 A22A11 A22A12 A21A A22A

A21A21 A21A22 A22A21 A22A22 (4.4.9)

respectively. The tensorsLet

A= KT, (4.4.10)

(4.4.5) where, with (4.4.5),

are then tensors of symmetry classes (2) and (11) respectively.

We now consider the special case where the tensors <p. . , 'l/J..1112 1112

and T1• 1· are two-dimensional. The tensor T·· then has four in-1 2 1112

dependent components given by T 11' T 12' T 21 and T22. The trans-

formation of the T· . under a transformation A is defined by1112

<Pl1 1 0 0 0 1 0 0 0

<P12 0 1/2 1/2 0K-1 -

0 1 0 1A= , K= , -

<P22 0 0 0 1 0 1 0 -1

'l/J12 0 1/2 -1/2 0 0 0 1 0

(4.4.11)

The manner in which A transforms under A is given by

72 Invariant Tensors [Ch. IVSect. 4.4] Physical Tensors of Symmetry Class (n1 n2 •••) 73

A11A11 A11A12 +A12A11 A12A12 0

A11A21 A11A22 +A12A21 A12A22 0

. (4.4.13)A21A21 A21A22 +A22A21 A22A22 0

0 0 0 A11A22 - A12A21

where

A' == KT' == K(A X A)T == K(A X A)K-1A (4.4.12)

where the 4 X 4 matrices L = [ t Kik1 Kko] and M = [ Kii K4j] arek==l J

seen from (4.4.11) to be given by

1 0 0 0 0 0 0 0

0 1/2 1/2 0 0 1/2 -1/2 0(4.4.15)L== M-, -

0 1/2 1/2 0 0 -1/2 1/2 0

0 0 0 1 0 0 0 0

We may express the 4 X 4 matrices L, M and A X A as

where iI' i2, jl' j2 take on values 1,2 and where the rows (columns) 1,

2,3,4 of the matrices are those for which i1i2 (jlj2) take on the values

11,12,21,22 respectively. With (4.4.14) ,... , (4.4.16), we have

The 3 X 3 matrix A(2) appearing in the upper left of (4.4.13) is referred

to as the symmetrized Kronecker square of A and defines the

transformation properties of the independent components <7>11' <7>12' <7>22

of the symmetric part <7>. . == -21(T. . +T· . ) of T· .. The 1 X 1(11) 1112 1112 1211 1112

matrix A appearing in the lower right corner of (4.4.13) describes

the transformation of the one independent component 7P12

= !(T12 - T21 ) of the skew-symmetric part of Ti1i2

o

We observe that

(4.4.16)

(2) -1tr A == Kik Kkj (A X A)ji

== L·· (A X A).. (i,j == 1, ... ,4; k == 1,2,3)1J J1

(11) -1tr A == Ki4 K4j (A X A)ji

== Mij (A X A)ji (i,j == 1, ... ,4)

(4.4.14)

where

S - tr A s - tr A2 s == tr An .1- , 2- , ... , n

We may also express (4.4.1 7) in the form

(4.4.18)

74 Invariant Tensors [Ch. IV Sect. 4.4] Physical Tensors of Symmetry Class (n1 n2 •••) 75

trA(2) == l '" h X2 s'l s'2 - 1 (s2+ s )2! 4y , , 1 2 - 2 1 2'

trA(ll)-l"'h X11s'l s'2 - 1(s2 s)- 2! 4y , , 1 2 - 2 1 - 2

(4.4.19)

An nth-order tensor T == T· . may be split into a sum of11·" In

tensors 4>1' 4>2' ... of symmetry classes (n1n2···)' (m1m2···)' ... where the

n1n2··· , m1m2···' ... are partitions of n. For example, the third-order

tensor T == T· . . is expressible as111213

where x~ and xV are the values of the characters of the irreducible

representations (2) and (11) of the symmetric group 52 (see Table 4.1 in

§4.9) for permutations belonging to the class of permutations,. The

cycle structure of the permutations belonging to , is given by 1'1 2'2

where '1 denotes the number of I-cycles and '2 the number of 2-cycles.

The summation in (4.4.19) is over the classes of 52 and h, gives the

order of the class , (h, == 1 for the classes , == 12 and , == 2). More

generally, if A(n1n2"') is the matrix which defines the transformation

properties of the qn n independent components of an nth-order1 2···

tensor of symmetry class (n1n2... ) under a transformation A, we have

(see Lomont [1959], p. 267)

(4.4.22)

where

(4.4.20) (4.4.23)

where X~ln2··· denotes the value of the character of the irreducible

representation (n1n2···) of the symmetric group 5n corresponding to the

class , of permutations. The summation in (4.4.20) is over the classes

, of 5n . The quantities X~ln2··· and h, may be found in the character

tables for 5n (see §4.9). The number of independent components of a

three-dimensional tensor of symmetry class (n1n2... ) is given by

qn n where1 2···

(4.4.21 )

A thorough discussion of tensors of symmetry class (n1n2... ) may be

found in Boerner [1963].

are tensors of symmetry classes (3), (21), (21) and (111) respectively.

With (4.4.21) and the character table for 53 (Table 4.2 in §4.9), we see

that 4>1' 4>2' 4>3 and 4>4 have 10, 8, 8 and 1 independent components

76 Invariant Tensors [Ch. IV Sect. 4.5] The Inner Product of Property Tensors and Physical Tensors 77

respectively. The tensor T given by (4.4.22) is said to be of symmetry

class (3) + 2(21) + (111). We observe that T = Ti i i has 33 = 27123

independent components and has no symmetry in the sense that no

relations such as T··· == T· .. occur. In order to list the111213 121113 i i

independent components of a tensor of symmetry class (21), we let i1 2

take on values 1, 2 and 3 so that, when entered into the frame [21], tte

numbers do not decrease as we move to the right and increase as we

move downwards. Thus,

standard table~ux associated with the frame CY == [n1n2...]. Then the

set of tensors C! . (i == 1,... , q) may be written as11··· In

(4.5.2)... ,

where the aCYl' ... ' a CY are the permutations which carry F~ intos qsFf,... ,Fq. Let Fe denote one of the standard tableaux associated with

the frame f3 == [m1m 2...]. Then a tensor of symmetry class (m1m2... )

may be considered to be given by(4.4.24)11 11 12 12 13 13 22 23

2' 3' 2' 3' 2' 3' 3' 3·

With (4.4.23) and (4.4.24), we have, for example,(4.5.3)

3<PI23 = T123 + T213

3<PI32 = T 132 + T312

T231 '(4.4.25)

where T· . is non-symmetric, i.e., T· . has 3n independent com-11··· 1n 11··· 1n

ponents.

We now consider the set of q functions obtained by taking the

inner product of the q tensors (4.5.2) and the tensor (4.5.3), i.e.,

Each of the components <P~31' <P~13' <P~12' <P~21 IS expressible as alinear combination of the components (4.4.25).

(r == 1,... , q). (4.5.4)

We first note that

and that the permutation (132) is the inverse of the permutation (123).

More generally, we have

4.5 The Inner Product of Property Tensors and Physical Tensors

Let C! . (i == 1,... , q) denote a set of nth-order property11··· In

tensors of symmetry type (n1n2... ). Let ¢. . be a physical tensor of11···1n

symmetry class (m1m2... ). Then the number of linearly independent

functions in the set

(4.5.5)

is equal to one if (n1n2... ) == (m1m2... ) and is equal to zero otherwise.

We now proceed to verify this statement. Let Fr denote one of the

C! . ¢. . (i == 1,... , q)11··· 1n 11··· 1n

(4.5.1)

(4.5.6)

where a denotes a permutation of 1,2,... , n and a-I is the inverse of a.

78 Invariant Tensors [eh. IV Sect. 4.6] Symmetry Class of Products of Physical Tensors 79

We further note that (4.5.13)

c· . ("""'f3.a.T. . )==(2:f3.a:-1C. . )T.. (4.5.7)11 ... In L.J 1 1 11· .. In 1 1 11··· In 11··· In

where the summation in (4.5.7) is over all of the permutations ai of

1,2,... , n and where the f3. are real numbers. Let the Young symmetry1

operator Y~ == Q~ P~ associated with the tableaux F~ be written as

(4.5.8)

Then, it may be shown (see Boerner [1963], p. 147) that replacing ai by

(Til in (4.5.8) will yield Y~, i.e.,

(4.5.9)

With (4.5.7) ,... , (4.5.9), we have

and where eo; is a positive integer (see Rutherford [1948], p. 19). The

quantity p in (4.5.12) may be zero for some values of r but not for all

values since a~v a~s == e and the coefficient of e in Y~ is not zero.

This says that if the frames [n1n2...] and [m1m2...] are different,

the q functions (4.5.4) are all zero and that if the frames [n1n2...] and

[m1m2···] are the same, then there is just one linearly independent

function contained in the set (4.5.4).

The number of linearly independent invariants of the form

(4.5.14)

associated with a material for which the complete set of nth-order

property tensors is of symmetry type

(4.5.15)

We recall that Y~ and Y~ are Young symmetry operators associated

with standard tableaux Fr and F~ which belong to the frames [nln2."]

and [mlm2...] respectively. If the frames 0; == [n1n2...] and

(J == [m1m2···] are different, we see from (4.3.11)5 that

and where the physical tensor <p. . is of symmetry classIl··· In

(4.5.16)

is then given by

(r == 1,... , q). (4.5.11) (4.5.17)

If the frames 0; and (J are the same, we have with (4.3.7) and (4.3.11)3

(4.5.12)

where p is the coefficient of e in the expression

4.6 Symmetry Class of Products of Physical Tensors

In this section, we consider the problem of determining the

symmetry class of the product of tensors T and U where the symmetry

classes of T and U are given. We first indicate the manner in which

80 Invariant Tensors [Ch. IV Sect. 4.6] Symmetry Class of Products of Physical Tensors 81

may be obtained when T·· and U·· are of symmetry classes (2) and. . 1112 1112

(11) respectIvely, I.e., T·. and U·· are symmetric and skew-. 1112 1112

symmetrIc second-order tensors respectively. We also suppose that the

tensors are three-dimensional. Let Q(A) and R(A) be the 6 X 6 and

3 X 3 matrices which describe the transformation properties of the

independent components T 1,... , T6 and U1,... , U3 of T and U

respectively under A. We note that Q(A) is the symmetrized

Kronecker square A(2) of A. With (4.4.19),

(4.6.4)

tr {Q(A) x R(A)} = tr Q(A) tr R(A) = i(sf - s~)

- 1.(6 4 _ 6 2) - 1. '" h '1'2 '4- 4! sl s2 - 4! Y , J-L, sl s2 ... s4 ·

The quantities Il, in (4.6.4) give the values which the character of a

representation of 54 assumes for the classes, of 54. The number of

times the irreducible representation (n1n2... ) appears in the decom

position of this representation is given by

matrix which describes the transformation properties of the in

dependent components TiUJ. (i == 1,... ,6; j == 1, ... ,3) of T· . U·· is the

1112 1314Kronecker product Q(A) x R(A) of Q(A) and R(A). We note that the

trace of the Kronecker product Qx R of the matrices Q and R is equal

to the product tr Q tr R of the traces of Q and R. We then have, with

(4.6.2),

(4.6.1)

(4.6.2)

t Q(A) 1 '" h 2'1'2 1( 2 )r 2! y ,X,sl s2 = 2 s1 +s2

tr R(A) = i! ~ h, xV sIl si2 = !(st - s2)

the symmetry classes of tensors such as

where we have employed the orthogonality properties of the group

characters. With (4.6.4), (4.6.5) and the character table for 54 (Table

4.3), we see that

where sl == tr A, s2 == tr A2 and the summation is over the classes, of

the symmetric group 52' The values of the characters X~, xV of the

irreducible representations (2), (11) of 52 and the orders h, of the

classes of 52 are given in the character table for 52 (Table 4.1 in §4.9).

More generally, the trace of the r X r matrix S(A) which describes the

transformation properties under A of the r independent components of

an nth-order tensor of symmetry class (n1n2... ) is given byh, == 1, 6, 8, 6, 3; Il, == 6, 0, 0, 0, -2

for the classes, == 14, 122, 13,4,22 of 5n and that

(4.6.5)

(4.6.6)

(4.6.3)(4.6.7)

where the summation is over the classes of 5n and where X~ln2··· gives

the values of the character of the irreducible representation (n1n2... ) of

5n for the class ,.

We first determine the symmetry class of T· . U·· where T1112 1314

and U are of symmetry classes (2) and (11) respectively. The 18 X 18

Thus, the tensor T· . U· . is of symmetry class (31) + (211).1112 1314

We next consider the determination of the symmetry classes of

the tensors T· . T·· and T· . T· . T·· where T· . is symmetric i e1112 1314 111213141516 1112 ' .. ,


where (see Murnaghan [1951])

(2)() 1 ""'" h 2'1'2 _ 1 ( 2 )tr Q A == -2' LJ , X, t 1 t2 - 2 t 1 + t2 '. ,£ I 64 3 2 22 32 5lor the c asses 1 , 1 2, 1 3, 1 4, 1 2 , 123, 15, 6, 24, 2 ,3 of 6.

The II, and A, are the characters of representations of 54 and 56

respectively. With (4.6.5) and the character tables for 54 and 56' the

decomposition of these representations is seen to be given by (4) + (22)

and (6) +(42) + (222) respectively. Thus, T· . T·· is of symmetry1112 1314

class (4) + (22) and T· . T· . T·. is of symmetry class (6) + (42)1112 1314 1516

+ (222).

of symmetry class (2). The transformation properties of the 21 indepen

dent components T·T· (i,j == 1,... ,6; i <j) of T· . T·· and the 56 inde-1 J - 1112 1314

pendent components T.T.Tk (i,j,k==I, ... ,6; i<j<k) ofT1· 1· T1· i Ti i1 J - - 12 34 56

are defined by the symmetrized Kronecker square, Q(2)(A), and the

symmetrized Kronecker cube, Q(3)(A), of Q(A) respectively. We have

(4.6.8)

tr Q(3)(A) = 3\ L h, x~ tIl t~2 tj3 = ~ (t~ + 3t1t2 + 2t3). ,

t 1 = tr Q(A) = ~ (sr + s2)' t2 = tr Q2(A) = tr Q(A2) = ~ (s~ + s4) ,

(4.6.9)

h, == 1, 6, 8, 6, 3; II, == 3, 1, 0, 1, 3

for the classes, == 14, 12 2, 13,4,22 of 54 and that

h, == 1, 15, 40, 90, 45, 120, 144, 120, 90, 15, 40;

A,== 15,3,0,1,3,0,0,1,1,7,3

(4.6.11)

(4.6.12)

The summations in (4.6.8)1 and (4.6.8)2 are over the classes of 52 and

S3 respectively. The quantities x~, X~ are the characters of the

identity representations of 52 and 53 which are denoted by (2) and (3)

respectively. We see from Tables 4.1 and 4.2 that X~ = 1, X~ = 1 for

all,. With (4.6.8) and (4.6.9),

trQ(2)(A) = 1, (3sf+6srS2+6s4 +9s~) = 1, ~h,v,sIls~2 ... s14,

tr Q(3)(A) = J! (15sY + 45sf S2 + 90sr S4 + 135sr s~ + 120s6 (4.6.10)

More generally, we may suppose that T == T· . is of sym-11· .. 1P

metry class al(nln2 ... )+a2(mlm2 ... )+ .... Let Q(A) denote the

matrix which defines the transformation properties under A of the in

dependent components T1,.. , Tr of T. Then

where si = tr Ai and where the summation is over the classes of Sp.

The symmetrized Kronecker mth power Q(m)(A) of Q(A) defines the

transformation properties under A of the (r + ~-1 ) independent com-

ponents Ti Ti ... T1· (iI' i2, ... , im == 1, ... ,r; i1 ~ i2 ~ ... ~ im ). We

12mhave (see Murnaghan [1951])

We see from (4.6.10) and the character tables for 54 and 56 (Tables 4.3

and 4.5 in §4.9) that

(4.6.14)


The problem of determining the symmetry class of the product

of two or more tensors of given symmetry classes has been considered

by Murnaghan [1937], [1938b], [1951] and by Littlewood and Richardson

[1934]. Let T· . denote a tensor of order p and symmetry class11· .. 1P

(p P ... ). Let U· . be a tensor of order q and symmetry class1 2 11· .. 1q

(q q ... ). Then T· . U· . is a tensor of order p + q == n1 2 11· ..1P 1p+1 .. ·1p+q

whose symmetry class is denoted by (P1P2''') -(q1q2"') where

The summations in " a and T are over the classes of the symmetric

groups Sm, Smp and Sp respectively. The Pa appearing in (4.6.14) give

the character of a representation of Smp. GiveR the character table for

Smp, we may employ (4.6.5) to determine the number ,BPIP2""

a ... of times the irreducible representations (P1P2···)' (q1q2···)'fJql q2··· , .... of Smp appear in the decomposition of this representatIOn. We then

say that the tensor T· . T· .... Tk k (m terms) is of symmetry11· ..1P Jl···Jp 1'" P

class ,BPIP2'" (PIP2"') + ,BqlQ2'" (QlQ2''') + ....

(4.6.18)

The tensor T· . T· . is a tensor of order 2p == m whose sym-11.. ·lp 1p+1 .. · 12p

metry class is denoted by (P1P2.") x (2) where

The summation in (4.6.18) is over the irreducible representations

(m1m2"') of Sm. The determination of the decomposition (4.6.18) is

discussed by Murnaghan [1951]. Most of the results of interest for the

applications considered here may be obtained from Murnaghan's papers

(see also Table 8.4, p. 232).

We list below (see Smith [1970]) the symmetry classes of

physical tensors which arise from the products of vectors Ei, Fi, ... ,

symmetric second-order tensors B.. , C", ... and skew-symmetric second-1J 1J

order tensors a··, (J .. , .... We note that all components of a three-1J 1J

dimensional tensor of symmetry class (P1P2P3P4) with P4>0 are zero.

Since we are mainly concerned with three-dimensional tensors, we will

not list terms such as (P1P2P3P4) with P4>0 in the description of the

symmetry classes of the tensors listed below. For example, the sym-

metry class of EiFjGkH£ is (4) + 3(31) + 2(22) + 3(211) + (1111). We

suppress the (1111) since a three-dimensional tensor of symmetry class

(1111) has no non-zero components.(4.6.16)

where the quantities t 1,... , tm are given by

The summation in (4.6.16) is over the irreducible representations

(nln2"') of Sn' Murnaghan [1937], [1938b] lists tables giving the

decomposition (4.6.16) for the cases p + q ~ 10 (see also Table 8.3,

p.231, for special cases). For example, if the symmetry classes of

T. . and U· are (22) and (1) respectively, then the symmetry class11... 14 11

of T· . U· is given by11 ... 14 15

Symmetry Classes: Products of Vectors

2. E.E., (2); E.F., (2) + (11)1 J 1 J

(22) . (1) == (32) + (221). (4.6.17)


(4.6.19)

+ 2(44) + (431) + 3(422)

+ 2(44) + 3(431) + 4(422) + (332)

+ 3(44) + 7(431) + 6(422) + 3(332)

Symmetry Classes: Products of Skew-Symmetric Second-Order Tensors

Symmetry Classes: Products of Symmetric Second-Order Tensors

2. Bij , (2)

BijBk£Cmn' (6) + (51) +2(42) + (321) + (222)

BijCk£Dmn' (6) + 2(51) + 3(42) + (411) + (33) + 2(321) + (222)

2. a·· (11)1J'

aij (3k£ 'mn' (33) + 2(321) + (222)

(4.6.21 )

(4.6.20) aij ak£(3mn 'pq, (44) + 2(431) + (422) + (332)

BijBk£BmnCpq, (8) + (71) +2(62) + (53) + (521) +

+(44) + (431) + 2(422)

88 Invariant Tensors [Ch. IV Sect. 4.7] Symmetry Types of Complete Sets of Property Tensors 89

to indicate that each of the tensors U, V, W has six distinct isomers

and that we may determine six linear combinations of the six isomers of

U, say, which may be split into four sets of tensors comprised of 1, 2, 2,

1 tensors whose symmetry types are (3), (21), (21), (111) respectively.

We also employ the notation

4.7 Symmetry Types of Complete Sets of Property Tensors

Complete sets of property tensors of orders 1, 2, ... which are

invariant under a group A are readily obtained with the aid of theorems

given in Chapter III. These enable us to determine sets of linearly

independent functions which are multilinear in the vectors xl' x2' ...

and invariant under A. With (4.1.6) and (4.1.7), the complete set of

nth-order invariant property tensors may be immediately listed given

the set of linearly independent invariants which are multilinear in

xl' ", '~' The determination of a complete set of tensors which are

invariant under a given crystallographic group has been discussed by

Birss [1964], Mason [1960], Fumi [1952], Fieschi and Fumi [1953],

Billings [1969], Smith [1970],.... In Smith [1970], the sets of invariant

tensors of orders 1, ... ,8 are given for each of the crystallographic

groups. These sets of tensors are specified by tensors Ul' VI'···; ... ;

US' VS' ... j such that these tensors together with their distinct isomers

form complete sets of tensors of orders 1,... ,8. Further, the symmetry

types of the sets of tensors are given. We follow Smith [1970] and

employ the notation

U, V, W; 6·, (3) + 2(21) + (111) (4.7.1)

quantities obtained by cyclic permutation of the subscripts on the

summand.

We list below results for a number of cases of interest.

Corresponding results for all of the crystallographic groups may be

found in Smith [1970]. We may employ the procedure discussed in §4.2

and/or the results given by Murnaghan [1937], [1951] to determine the

symmetry type of a set of tensors comprised of a property tensor and

its distinct isomers. We denote by P n the number of linearly in

dependent nth-order tensors which are invariant under the group A.

The value of Pn may be computed with (4.1.4) or (4.1.5). We observe

that sets of three-dimensional property tensors of symmetry type

(n1n2···np) with np > 0, P 2: 4 will be comprised of tensors whose

components are all equal to zero. There may be sets of three-

dimensional property tensors of symmetry type (n1n2 np) with np > 0,

P ~ 3 which are comprised of null tensors. If (n1n2 ) represents the

symmetry type of a set of property tensors whose components are all

zero, we indicate this by underlining the (n1n2... )' e.g., (2111). The

dimension fn1n2... of the irreducible representation (n1n2... ) gives the

number of tensors comprising a set of tensors of symmetry type

(n1n2... ). The values of the fn1n2... may be found in the first column of

the character tables for 52 ,... ,58 .

(i) Rhombic-dipyramidal crystal class: D2h

The symmetry group D2h associated with this crystal class IS

defined by

In (4.7.2), the notation E (...) indicates the sum of the three

(4.7.2)

E el122 == el122 +e2233 +e3311 .

where the I,... ,D3 are defined by (1.3.3). With (4.1.4) and (1.3.3), the

number P n of linearly independent nth-order tensors which are in

variant under D 2h is given by

90 Invariant Tensors [Ch. IV

(4.7.3)

Sect. 4.7] Symmetry Types of Complete Sets of Property Tensors

elll12233' ~2223311' e33331122; 420; (8) +2(71) +

91

It follows from (4.7.3) that there are no tensors of odd order which are

invariant under D2h. The invariant tensors of orders 2, 4, 6 and 8 are

listed below where the notation (4.7.1) and (4.7.2) is employed.

2. P2 = 3. ell' e22' e33; 1 . (2) .,

4. P4 = 21. e1111' ~222' e3333; 1· (4);,

el122' el133' e2233; 6· (4) + (31) + (22).,

6. P6 = 183. e111111' e222222' e333333; 1· (6) ;,

e333311' e333322; 15 ; (6) + (51) + (42) ;(4.7.4)

el12233; 90; (6)+2(51)+3(42)+(411)+

+ (33) +2(321) + (222).

8. P8 = 1641. e11111111' e22222222' e33333333; 1; (8);

el1111122' el1111133' e22222211' e22222233'

e33333311' e33333322; 28; (8) + (71) + (62);

e11112222' elll13333' e22223333; 70; (8) + (71)+

+ (62) + (53) + (44);

+ 3(62) + (611) + 2(53) + 2(521) + (44) + (431) + (422).

Consider the set of 6 fourth-order tensors comprised of the

distinct isomers of el122 which are given by

= 611. 611. 621. 6210 , 611. 621

0 6110 621

0 , 61. 620 62

0 610 , (40705)

1 2 3 4 1 2 3 4 11 12 13 14

We have observed in §4.2 (see (4.2.16), (402.17)) that these tensors form

the carrier space for a reducible representation r of the group 54 of all

permutations of the subscripts iI' i2, i3, i4 whose decomposition is

given by (4) +(31) +(22) where (4), (31) and (22) denote irreducible

representations of 54. The tensors (4.7.5) then form a set of tensors of

symmetry type (4) + (31) + (22) as indicated on line 3 of (4.7.4).

(ii) Hexoctahedral crystal class: 0h

The symmetry group 0h associated with this crystal class IS

defined by

where the I, ... ,M2 are defined by (1.3.3). With (4.1.4) and Table 9.1

(p. 268), the number Pn of linearly independent nth-order tensors which

are invariant under 0h is given by


From (4.7.8), we obtain

(4.7.8)

(n odd) ,Pn == 0

possess a center of symmetry is the three-dimensional orthogonal group

03 which is comprised of all three-dimensional matrices A such that

AAT == ATA == E3, det A == ± 1. The number Pn of linearly in

dependent nth-order tensors which are invariant under 03 is equal to

the number of linearly independent multilinear functions of the n

vectors Xl"'" xn which are invariant under 03' The matrix rep

resentation defining the transformation properties of the 3n quantities

xf xf ... xk under 03 is comprised of the Kronecker nth powers

A x A x ... x A of the A belonging to 03' The number Pn of linearly

independent invariants is given by the number of times the identity

representation appears in the decomposition of the representation

{A x A x ... x A}. If A denotes a rotation through B radians about some

axis, we note that tr A == eiB + 1 +e-iB, C == diag (-I, -1, -1) and

trCA== _eiB _1_e-iB. Since tr{AxAx ... xA)=={trA)n and Pn is

obtained from the expression {2.6.20)2' we see that

27rPn = 4~ f (eiB + 1 +e-iB)n (1- cos B) dB

o27r

+l1rf(- eiB - 1 - e-iB)n (1 - cos B) dB.

o

(4.7.6)

2. P2 == 1. L e11 == 8ij ; 1 . (2).,

4. P4 == 4. L e1111 ; 1 . (4) ;,

L(el122 + e2211); 3' (4) + (22) .,

6. P6 == 31. L e111111 ; 1 . (6) ;,

L (elll122 + elll133) ; 15; (6)+(51)+(42);

L (el12233 + el13322); 15; (6) + (42) + (222) .

(4.7.7)

8. P8 == 274. L e11111111 ;1 . (8) ;,

L (elllll122 + elllll133) ; 28; (8) + (71) + (62);

L (e11112222 + elll13333); 35; (8) + (62) + (44) ;

We see that Pn == 0 if n is odd so that there are no odd order tensors

which are invariant under 0h' Complete sets of tensors of orders 2, 4, 6

and 8 which are invariant under 0h are listed below where the notation

(4.7.1) and (4.7.2) is used.

L (e11112233 +ell113322); 210; (8) + (71) +2(62) +

+ (53) + (521) + (44) + (422).

(iii) Isotropic materials with a center of symmetry: 03

The symmetry group associated with isotropic materials which

(4.7.9)

where (£) == m! (:~m)! is a binomial coefficient and (0 )= 1. Since

Pn == 0 if n is odd, there are no odd order tensors which are invariant

under 03. Complete sets of tensors of orders 2, 4, 6 and 8 which are


(4.7.12)

invariant under 03 are listed below where we employ the notation

(4.7.1). These tensors are referred to as isotropic tensors.

2. P2 = 1. fJ·· . 1· (2).IJ '

,

4. P4 = 3. fJij fJk£; 3· (4) + (22) .,(4.7.10)

6. P6 = 15. fJij fJk£fJmn ; 15 ; (6) + (42) + (222).

8. P8 = 91. fJij fJk£fJmn fJpq ; 105 ; (8) + (62) + (44) ++ (422) + (2222) .

The notation (2222) in (4.7.10)4 indicates that all 14 tensors comprising

the set of three-dimensional tensors of symmetry type (2222) have all of

their components equal to zero. In (4.7.10), fJij denotes the Kronecker

delta defined by (1.2.4). The second line of (4.7.10) indicates that there

are three distinct isomers of bij bk£ which may be obtained upon

permuting the subscripts i, ... ,£. These are given by fJij fJk£, fJik fJj£,

bi2

bjk

and may be split into sets of tensors of symmetry types (4) and

(22) which are comprised of 1 and 2 tensors respectively. We note from

line 4 of (4.7.10) that there are 105 distinct isomers of fJij fJk£ fJmn fJpq

but that there are only 91 linearly independent eighth-order tensors

which are invariant under 03. This is due to the existence of identities

of the form

fJ· . fJi£ fJ· fJ·IJ In Iq

fJkj fJk£ 8kn 8kq = o. (4.7.11)8 . 8m£ 8mn 8mqmJ

fJpj 8p£ 8pn 8pq

Weare assuming that the tensors are three-dimensional. A procedure

which enables one to list the linearly independent isotropic tensors of

orders 8, 10,... is given by Smith [1968a]. There are 14 distinct

identities of the form (4.7.11) which may be obtained upon permuting

the subscripts i,j, ... , q. These identities play an important role in gen

erating integrity bases for functions which are invariant under the

group 03 (see §8.2 and Rivlin and Smith [1975]).

(iv) Isotropic materials without a center of symmetry: R3

The symmetry group associated with isotropic materials which

do not possess a center of symmetry is the three-dimensional rotation

group R3 which is comprised of all three-dimensional matrices A such

that AAT = ATA = E3, det A = 1.We may proceed as in the case of

the group 03 to show that the number Pn of linearly independent nth

order tensors which are invariant under R3 is given by

211'"Pn = l7f J(eiB + 1 + e-iB)n (1- cos B) d8

owhere we have employed (2.6.19)2. We have, upon evaluating (4.7.12),

PI = 0,

(n-l)/2( )() (n+l)/2( )( )Pn = 1 + k'fl 2k 2: - n - k'f

22k~ 1 2t=i

(n odd; n ~ 3), (4.7.13)

Pn = 1 +}; (2k)(2:) - };(2k~ 1)(2t=i) (n even).

We recall that on = 1. Complete sets of tensors of orders 2, ... ,8 which

are invariant under R3 are listed below where we employ the notation

(4.7.1). The tensors 8ij and Cijk appearing below are the Kronecker


delta defined by (1.2.4) and the alternating tensor (see p. 6). material for which the x3 axis is an axis of rotational symmetry is

comprised of the matrices

(4.7.16)

(0 ~ () ~ 21r).

cos () sin () 0

-sin () cos () 0

o 0 1

We see from (2.6.21)2 that the number Pn of linearly independent nth

order t~nsors which are invariant under T1 is given by

21r

Pn = l1r J(ei8 + 1 + e-i8)n d8.o

With (4.7.16), we have

2. P2 == 1. b... 1· (2).IJ '

,

3. P3 == 1. Cijk; 1 . (111).,

4. P4 == 3. bij bk£; 3· (4) + (22) .,

5. P5 == 6. c··k b£ ; 10; (311) + (2111). (4.7.14)IJ m

6. P6 == 15. b·· bk£b ; 15 ; (6) + (42) + (222).IJ mn

7. P 7 == 36. c··k b£ b . 105; (511) + (4111) + (331) +IJ m np'

+ (3211) + (22111).

S. Ps == 91. bij bk£bmn bpq ; 105 ; (S) + (62) + (44) ++ (422) + (2222) .

Complete sets of tensors of orders 1,... ,5 which are invariant under T1are listed below where we employ the notation (4.7.1) and (4.7.2).

Pn = 1+( 2)(i )+( 4)(~ )+'" + ( ~ ) ( n/2) (n even).

We observe from (4.7.14) that there are ten distinct isomers of the

tensor c··k bfJ which may be split into a set of six tensors comprising aIJ ~m

set of tensors of symmetry type (311) and a set of four tensors

comprising a set of symmetry type (2111). All components of the

three-dimensional tensors forming the set of symmetry type (2111) are

zero. This is due to the existence of identities of the form (see Smith

[196Sa] or Kearsley and Fong [1975] )

(n odd; n ~ 3) , (4.7.17)

b·· ck fJ - bk· C·fJ + bfJ· COk - b . COkfJ == o.IJ ~m J l~m ~J 1 m mJ 1 ~(4.7.15)

(v) Transversely isotropic materials: T1

The symmetry group T1 associated with a transversely isotropic

98 Invariant Tensors [Ch. IV Sect. 4.8] Examples 99

e311 + e322; 3; (3) + (21);

e312 - e321; 3 ; (21) + (111) .

4. P4=19. e3333; 1; (4);

e3311 +e3322; 6; (4)+(31)+(22);

5. P5 = 51. e33333; 1; (5);

e33311 +e33322; 10; (5)+(41)+(32);

(4.7.18)

which form a set of tensors of symmetry type (31) + (211). Sets of

tensors of symmetry types (31) and (211) are comprised of f3 = 3 and

f211 = 3 tensors respectively. The set of tensors of symmetry type

(211) formed from the isomers of (ell + e22)(e12 - ~1) is comprised of

tensors whose components are all zero. This is a consequence of

identities of the form

(4.7.19)

where i, ... ,£ take on values 1,2. A further consequence of (4.7.19) is

that there are 4 + 5 + 6 = 15 tensors comprising sets of tensors of

symmetry types (2111), (221) and (311) formed from the isomers of

e3(e11 + e22)(e12 - ~1) whose components are all zero.

4.8 Examples

We give a number of examples of the application of the concepts

discussed above.

(i) Determine the form of the scalar-valued function

appropriate for the hexoctahedral crystal class 0h where Eij is a

symmetric second-order tensor. From(4.6.20), we see that Eij Ek£Emnis of symmetry class

e33312 - e33321; 10; (41) + (311);

e3(e11 + ~2)(e12 - ~1); 30; (41) + (32) + (311) +

+ (311) + (221)+ (2111).

W = C··klJ E·· EklJEIJ ~mn IJ ~ mn

(6) + (42) + (222).

(4.8.1)

(4.8.2)

We observe that there are SIX distinct isomers of the tensor

(en+~2)(e1T ~1) = b'ij cke (i,j, k, e= 1,2; c11= c22= 0, c12 = -c21 = 1)

From (4.7.7), we see that the general sixth-order tensor invariant under

the group 0h is expressible as a linear combination of the isomers of the

tensors

100 Invariant Tensors [Ch. IV Sect. 4.8] Examples 101

L (e111122 + eIII133); 15; (6) + (51) + (42),

L(eI12233+eI13322); 15; (6)+(42)+(222).

(4.8.3) The general expression for the function (4.8.1) is then given by

(4.8.6)

(ii) Determine the form of the symmetric second-order tensor

valued function

appropriate for the hexoctahedral crystal class 0h where Eij is a

symmetric second-order tensor. From (4.6.20), we see that Tij Ek£Emnis of symmetry class

In (4.8.3), we have listed to the right of each tensor the number of

distinct isomers of the tensor and the symmetry type of the set of

tensors comprised of the tensor and its distinct isomers. The argument

given in §4.5 together with (4.8.2) and (4.8.3) shows that there are 1, 2

and 3 linearly independent invariants contained in the three sets of 1,

15 and 15 invariants given by

T·· == C·· kfJ EkfJE1J 1J ~mn ~ mn (4.8.7)

(4.8.4)(6) + (51) + 2(42) + (321) + (222). (4.8.8)

Let us denote the sixth-order property tensors (4.8.3) associated with

0h by

The symmetry types of these three sets of tensors are given in (4.8.3).

The numbers of linearly independent invariants contained in the sets

are seen from (4.5.15), ... ,(4.5.17), (4.8.3) and (4.8.8) to be given by 1,

4 and 4 respectively. Thus, there will be 1, 4 and 4 linearly in

dependent symmetric second-order tensor-valued functions contained in

the sets

These invariants are given by

II = :L(eUUU)ij ...nEijEk£Emn = :LE~I'

12 = :L (eUU22 + eUU33)ij n Eij Ekl! Emn = L Etl (E22 + E33) ,

13 = :L(eUI212 +elU313)ij n Eij Ek£Emn = :LEU(Et2+ Et3),

(4.8.5)

14 = L (eU2233 + eU3322)ij n Eij Ek£Emn = 6EU E22E33 '

IS = L (eU2323 + eU3232\j n Eij Ek£Emn = 2L EU E~3 '

(r)u··kfJ T.. EkfJE . vookfJ T.. EkfJE1J ~mn 1J ~ mn' 1J ~mn 1J ~ mn

w~:k) fJ T·· EkfJ E ( r == 1,... ,15)1J ~mn 1J ~ mn

(4.8.9)

(r == 1, ... ,15);

(4.8.10)

102 Invariant Tensors [Ch. IV Sect. 4.9] Character Tables for Symmetric Groups 52 ,... ,58 103

W (r) E E (1 15)ijk£mn k£ mn r == ,... ,

u··klJ EklJE .IJ ~mn ~ mn'v(r) E E

ijk£mn k£ mn (r == 1,... ,15);

(4.8.11)where we have noted that E··k == E·k·. We then write (4.8.13) as

IJ 1 J

W == C·· klJ A··kA IJ +C·· klJ B··kB IJIJ ~mn IJ ~mn IJ ~mn IJ ~mn

(4.8.15)respectively. With (4.8.3), we see that these are given by

Eb1i b1j EIl;

Ebli b1j Ell (E22 + E33), Eb1i bl/E~2 + E~3)'

E b1i b1/EI2 + EI3)' E (b1i b2j + b2i b1j )(Ell + E22)E12;

We may employ the procedure discussed in §4.6 to obtain the

symmetry classes of the tensors E··kE lJm ' ... , B··kB IJ . We haveIJ ~ n IJ ~mn

EijkE£mn: (6) + (51) +3(42) + (411) +2(321) + (222) + (3111);

(4.8.12) (4.8.16)

BijkB£mn: (42) + (321) + (222) + (3111);

AijkB£mn +BijkAR,mn: (51) + (42) + (411) + (321).

which is invariant under the orthogonal group 03. The tensor E == Eijkhas 18 independent components. From the remarks following (4.4.23),

we see that three-dimensional tensors of symmetry classes (3), (21) and

(111) have 10, 8 and 1 independent components respectively. This

would indicate that E is of symmetry class (3) + (21). We may set

The set of sixth order property tensors associated with the orthogonal

group are seen from (4.7.10) to be given by the 15 isomers of

bij bk£bmn which form a set of tensors of symmetry type (6) + (42) +(222). With (4.5.14), ... ,(4.5.17) and (4.8.16), we see that there are 5

linearly independent isotropic invariants of the form (4.8.13) and that

there are 2, 2 and 1 invariants arising from the three terms in (4.8.15).

These are given by

(iii) Determine the form of the scalar-valued function

W == C. ·k lJ E. ·kEIJIJ ~mn IJ ~mn'(4.8.13)

A··k A··k A··· A·kk ; B··k B··k B··· B·kk · A··k B··k .IJ IJ' 11J J IJ IJ' 11J J ' 11 JJ(4.8.17)

3E··k == A··k +Book'IJ IJ IJ(4.8.14)

A··k == E··k +E· k· +Ek··, B··k == 2E··k - E·k· - Ek··IJ IJ J 1 IJ IJ IJ J 1 IJ

where A and B are of symmetry classes (3) and (21) respectively and

4.9 Character Tables for Symmetric Groups 52 , ... ,58

We list below the character tables for the symmetric groups

52' ... ,58 which are given by Murnaghan [1938a] and by Littlewood

[1950]. The character tables for 59 and 510 may be found in Littlewood

104 Invariant Tensors [Ch. IV Sect. 4.9] Character Tables for Symmetric Groups 52 ,... ,58 105

[1950]. The character tables for 511 , 512 and 513 are given by Zia-ud

Din [1935], [1937]. In these tables, I == 1'12/2 ... n In denotes the class

of a group where 'I is the number of one cycles, '2 is the number of

two cycles, .... The number of permutations comprising the class I is

given by h,. The characters satisfy the orthogonality relations

where (n1n2···) and (m1m2···) are inequivalent irreducible rep

resentations of Sn and where X~ln2··· and X~lm2··· give the values of

the characters of (n1n2... ) and (m1m2... ) for the class I. The quantity

X~ln2"· is found in the row corresponding to (nln2".) and the column

headed by I.

Table 4.1 Character Table: 52

I 12 2

h, 1 1

(2) 1 1(11) 1 -1


I 13 12 3

h, 1 3 2

(3) 1 1 1(21) 2 0 -1(111) 1 -1 1


I 14 122 13 4 22

h, 1 6 8 6 3

(4) 1 1 1 1 1(31) 3 1 0 -1 -1(22) 2 0 -1 0 2(211) 3 -1 0 1 -1(1111) 1 -1 1 -1 1


I 15 132 123 14 122 23 5

h, 1 10 20 30 15 20 24

(5) 1 1 1 1 1 1 1(41) 4 2 1 0 0 -1 -1(32) 5 1 -1 -1 1 1 0(311) 6 0 0 0 -2 0 1(221) 5 -1 -1 1 1 -1 0(2111) 4 -2 1 0 0 1 -1(11111) 1 -1 1 -1 1 -1 1


I 16 142 133 124 1222 123 15 6 24 23 32

h, 1 15 40 90 45 120 144 120 90 15 40

(6) 1 1 1 1 1 1 1 1 1 1 1(51) 5 3 2 1 1 0 0 -1 -1 -1 -1(42) 9 3 0 -1 1 0 -1 0 1 3 0(411) 10 2 1 0 -2 -1 0 1 0 -2 1(33) 5 1 -1 -1 1 1 0 0 -1 -3 2(321) 16 0 -2 0 0 0 1 0 0 0 -2(222) 5 -1 -1 1 1 -1 0 0 -1 3 2(3111) 10 -2 1 0 -2 1 0 -1 0 2 1(2211) 9 -3 0 1 1 0 -1 0 1 -3 0(21111) 5 -3 2 -1 1 0 0 1 -1 1 -1(111111) 1 -1 1 -1 1 -1 1 -1 1 -1 1


I 17 152 143 134 1322 1223 125 . 16 124 123 132 25 223 34 7

hi 1 21 70 210 105 420 504 840 630 105 280 504 210 420 720

(7) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1(61) 6 4 3 2 2 1 1 0 0 0 0 -1 -1 -1 -1(52) 14 6 2 0 2 0 -1 -1 0 2 -1 1 2 0 0(511) 15 5 3 1 -1 -1 0 0 -1 -3 0 0 -1 1 1(43) 14 4 -1 -2 2 1 -1 0 0 0 2 -1 -1 1 0(421) 35 5 -1 -1 -1 -1 0 1 1 1 -1 0 -1 -1 0(331) 21 1 -3 -1 1 1 1 0 -1 -3 0 1 1 -1 0(4111) 20 0 2 0 -4 0 0 0 0 0 2 0 2 0 -1(322) 21 -1 -3 1 1 -1 1 0 -1 3 0 -1 1 1 0(3211) 35 -5 -1 1 -1 1 0 -1 1 -1 -1 0 -1 1 0(2221) 14 -4 -1 2 2 -1 -1 0 0 0 2 1 -1 -1 0(31111) 15 -5 3 -1 -1 1 0 0 -1 3 0 0 -1 -1 1(22111) 14 -6 2 0 2 0 -1 1 0 -2 -1 -1 2 0 0(211111) 6 -4 3 -2 2 -1 1 0 0 0 0 1 -1 1 -1(1111111) 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1

Table 4.7 Character Table: 58 (Continued on next page)

I 18 162 153 144 1422 1323 135 126 1224 1223 1232

hi 1 28 112 420 210 1120 1344 3360 2520 420 1120

(8) 1 1 1 1 1 1 1 1 1 1 1

(71) 7 5 4 3 3 2 2 1 1 1 1

(62) 20 10 5 2 4 1 0 -1 0 2 -1

(611) 21 9 6 3 1 0 1 0 -1 -3 0

(53) 28 10 1 -2 4 1 -2 -1 0 2 1

(521) 64 16 4 0 0 -2 -1 0 0 0 -2

(5111) 35 5 5 1 -5 -1 0 0 -1 -3 2

(44) 14 4 -1 -2 2 1 -1 0 0 0 2

(431) 70 10 -5 -4 2 1 0 1 0 -2 1

(422) 56 4 -4 0 0 -2 1 1 0 4 -1

(4211) 90 0 0 0 -6 0 0 0 2 0 0

(332) 42 0 -6 0 2 0 2 0 -2 0 0

(3311) 56 -4 -4 0 0 2 1 -1 0 -4 -1

(3221) 70 -10 -5 4 2 -1 0 -1 0 2 1

(2222) 14 -4 -1 2 2 -1 -1 0 0 0 2

(41111) 35 -5 5 -1 -5 1 0 0 -1 3 2

(32111) 64 -16 4 0 0 2 -1 0 0 0 -2

(22211) 28 -10 1 2 4 -1 -2 1 0 -2 1

(311111) 21 -9 6 -3 1 0 1 0 -1 3 0

(221111) 20 -10 5 -2 4 -1 0 1 0 -2 -1

(2111111) 7 -5 4 -3 3 -2 2 -1 1 -1 1

(11111111) 1 -1 1. -1 1 -1 1 -1 1 -1 1

~

o0:>

~~~

""S~'

~

~~

;:lCI.l<:::l

~

'"0?""

<

enC't>("')M-

~

ie

~~

""S~

~('b

""S

~~

~~

CI.l

~""S

~ce:SS('b

:;-n'~""S<:;)

~~

CI.l

(J')"->

(J')00

~

o-1

108 Invariant Tensors [Ch. IV

v

00o~ol.Q

o00c.orl

rlrl~M~OM~~oo~~oo~~MO~M~rlrl

I I I I I I I I I I

rlrlOrlrlrlO~OrlOrlrlOrlOrlrl~Orlrl

I I I I I I I

rl~rlOrl~~~rlrlOOrlrl~~~rlOrlrlrl

I I I I I I I I I

rlrlrlOrlOOOrlrlOOrlrlOOOrlOrlrlrl

I I I I I I

rlrl~rl~Orl~OOOOOO~rlO~rl~rlrl

I I I I I I I

rlrlOrlOOrl~~O~~O~~rlOOrlOrlrl

I I I I I I

rlrlOrlOOrlOOOOOOOOrlOOrlOrlrl

I I I I

rlOrlOOrlOOOOrlOOOOOrlOOrlOrl

I I I

rlOrlOrlOrlrlrlOOOOrlrlrlOrlOrlOrl

I I I I I I

rlOrl~rlOrlrlrlOO~OrlrlrlOrl~rlOrl

I I I I I I

rlOOrlOrlO~OrlOOrlOrlOrlOrlOOrl

I I I I I

,..-......"'-"""rl

"'-""""'-"""rl rl"'-""""'-""""'-"""rl rl rl rl

,..-...... "'-""""'-""""'-"""rlrlrlrlrlrlrl,..-...... "'-"""rl ,..-......,..-......rl,..-......rlrl~rlrlrlrlrlrlrl

,..-......,..-......rl,..-......rlrl~rl~rl~rl~~rlrl~rlrlrlrl

,..-......rl~rlM~rl~M~~MM~~rl~~rl~rlrloo~~~l.Ql.Ql.Q~~~~MMM~~M~M~~rl

'--"''--'"'--''''--''''--''''-'''-'''-'''-'''-'''--''''-'''--''''--''''--''''--''''--'''~'-'''--''''-'''--'''

GROUP AVERAGING METHODS

5.1 Introduction

The procedure employed in this chapter involves summation

over the group A in order to generate scalar-valued and tensor-valued

functions which are invariant under A. We follow the discussion given

by Smith and Smith [1992]. The computational procedure requires the

generation of the Kronecker products and the symmetrized Kronecker

products of the matrices comprising various matrix representations of

the group A. The rationale for the procedure discussed here is that the

computations are well adapted to computer-aided generation. Com

puter programs are being developed which will carry out the required

computations.

5.2 Averaging Procedure for Scalar-Valued Functions

We consider the problem of determining the form of a scalar-

valued function W(E. .) of an nth-order tensor which is invariantII···In

under the finite group A== {A1, ... ,AN}. Thus, W(E. .) must satisfyII···In

(5.2.1)

for all AI{ = [A~] belonging to A. Let E l ,... , Er denote the independent

components of the tensor E· . We then consider the equivalentII· .. I n

problem of determining the form of W(E) == W(Ei) where E denotes

the column vector [E1,... , Er]T. The restrictions corresponding to

109

110 Group A veraging Methods [Ch. V Sect. 5.2] A veraging Procedure for Scalar- Valued Functions 111

(5.2.1) are that W(E) must satisfy

W(E) == W(RK E), (K == 1, ... ,N) (5.2.2)

Invariants of any given degree n in E may be generated in a

similar fashion. Let En denote the column matrix whose (r + ~ - I )

entries are given by

are either invariants or zero. Thus, replacing E by R K E in (5.2.3)

leaves the expression unaltered since

where the set of N r x r matrices {R1,... , R N} forms the r-dimensional

matrix representation {RK } which defines the transformation properties

of E under A. We first consider the case where W(E) is linear in the

components of E. We observe that the r entries of the column vector

(5.2.7)

(5.2.9)

(5.2.10)

NPn = ~ tr Rri = ~ E tr R~) ,

K==1

n! trR~) = E h,(trRK)'l (trRk)'2 ... (trR~),nI

where the entries are ordered so that E· E· ... E· precedes E· E· ... E.11 12 In J1 J2 In

if i1 i2 ... in < jlj2 ···jn· For example, when r == 3 and n == 3, we have

E3 = [E~, ErE2' ErE3' EIE~, EIE2E3, EI E~, E~, E~E3' E2E~, E~r

(5.2.8)

Let {R~)} denote the matrix representation which defines the

transformation properties of En under A. R~) is a (r +~ - I) X

( r +n - 1) t . h· h· £ dn rna rIX w lC IS reJ.erre to as the symmetrized Kronecker

nth power of RK . Let

Then, each of the (r + ~ - I) entries in the column matrix Rri En is

either an invariant or zero. The number of linearly independent

invariants of degree n in E is given by

(5.2.6)

(5.2.4)

(5.2.3)

(5.2.5)

[ ]

TN K N K

R1E +...+ RNE == E R1· E· , ... , E R . E·K==1 J J K==l rJ J

Each row of R1, when multiplied on the right by the column vector E,

yields either an invariant or zero. The number PI of linearly in

dependent invariants of degree 1 in E is given by the number of times

the identity representation appears in the decomposition of the rep

resentation {RK }. With (2.5.14), we have

NPI = ~ tr Ri = ~ E tr RK

K==l

for any RK (K==l, ... , N). We employ the notation

* N N KR1 == E RK == E [Rij ]·

K==1 K==l

We may carry out a row reduction procedure on the matrix R1which

has rank PI to obtain a PI x r matrix U1· The PI entries in the column

vector U1E then give the set of PI linearly independent invariants of

degree one in E.

where (see §4.6) the summation in (5.2.10)2 is over the classes I of the

symmetric group Sn and where h, is the order of the class I. We may

carry out a row reduction procedure on Rri which has rank p to obtain

( r + n -1) . . na Pn X n matrIx Un· The Pn entrIes in the column vector

112 Group A veraging Methods [Ch. V Sect. 5.2] A veraging Procedure for Scalar- Valued Functions 113

Un En give the set of Pn linearly independent invariants of degree n in

E.

We now consider the problem of determining the form of

functions W(E. ., F· .) which are bilinear in the components of11··· 1n 11··· 1m

El· 1· and Fl· 1· and which are invariant under A == {A1,···, AN}·1··· n 1··· m

Let E and F denote the column vectors whose entries are the inde-

pendent components of E· . and F· . respectively. Thus,11··· 1n 11··· 1m

Let

(5.2.14)

(5.2.15)

(5.2.11)The rank of R11 is given by the number Pll of linearly independent

functions, bilinear in E and F, which are invariant under A. We have

Let {RK} and {SK} be the matrix representations of dimensions rand

s respectively which define the transformation properties of E and F

under A. Let

1 * 1 N 1 NPll == N tr R11 == N L tr(RK x SK) == N L tr RK tr SK· (5.2.16)

K==l K==l

Row reduction of R11 yields a Pll X r s matrix U11 · The Pl1 entries in

the column matrix U11Ell give the set of linearly independent

invariants of degrees 1,1 in E, F.

denote the column vector whose rs entries are the products E·F·1 J

(i == 1,... ,r; j == 1,... ,s) which are ordered so that E· F· precedes E· F·11 J1 12 J2

if the first non-zero entry in the list i1 - i2, j1 - j2 is negative. Let

{RK x SK} denote the matrix representation of A which defines the

transformation properties of Ell under A. RK x SK is an r s X r s

matrix which is referred to as the Kronecker product of RK and SK.

With the ordering (5.2.12), we have

In order to determine the invariants of degrees m in E and n in

(r+m-1)(s+n-1)F, we let Emn denote the column vector whose m n

entries are given by

where i1 ::; i2 ::; ... ::; im , j1 ::; j2 ::; ... ::; jn· The entries in Emn are to

be ordered so that E· E. ... E. F· F· ... F. precedes Ek Ek ...11 12 1m J1 J2 In 1 2

Ek F£ F£ ... F£ if the first non-zero entry in the list i1 - k1,... ,m 1 2 n

im - km, jl - £1'···' jn - £n is negative. Let

(5.2.17)E· E· ... E· F· F· ... F· (i1,... ,im == 1, ... ,r; jl, ... ,jn == 1, ... ,s)11 12 1m J1 J2 In

(5.2.13)(i,k == 1, ... , r; j,£ == 1, ... , s).

Rows (columns) 1,2, ... , s, s+l, ... , 2s, ... , (r-1)s+1, ... , rs are those

for which the indices ij (k£) take on values 11, 12, ... , 1s, 21, ... , 2s, ... ,

r1, ... , r s. The matrix RK x SK may be written as

(5.2.18)

where R~) and S~) denote the symmetrized Kronecker mth power of

114 Group A veraging Methods [Ch. V Sect. 5.3] Decomposition of Physical Tensors 115

RK and the symmetrized Kronecker nth power of SK respectively. The

rank of R~n is given by(5.3.2)

where p is the number of linearly independent invariants of degree

m,n in ~:F. The quantities trRr) and trS~) are given by ex

pressions of the form (5.2.10)2. Row reduction of R~n yields a matrix

Umn such that the Pmn entries in the column vector Umn Emn give

the p linearly independent invariants of degree m, n in E, F.mn

The computations involved in generating the matrices R~n will

generally be very tedious. We are developing a computer program

which will carry out the required computations. This would of course

eliminate the possibility of errors arising during the computation.

T! = S~T. (i,j = 1,... ,s; K = 1,... ,N) (5.3.3)1 IJ J

where the matrices SK = [S~] (K = 1,... , N) form a matrix rep

resentation of A. The SK may be determined from inspection of

(5.3.1). The matrix representation {Sl ,... , SN} = {SK} is in general

reducible and may be decomposed into the direct sum of irreducible

representations. The number of inequivalent irreducible representations

of a finite group A is equal to the number r of classes of A. We denote

these representations by

P =.1 trR* =.1 ~ trR(m) trS(n)mn N mn NK~ k k

=1(5.2.19)

The transformation properties of T under AK are defined by

r 1 = {rl,···,r&}, ... , rr = {rf,···,rk} .

Let P be a non-singular s x s matrix and let

(5.3.4)

The matrix representation defining the manner in which T* transforms

under A is given by

since, if T become SKT, then T* = PT becomes PSKT = PSKP-1PT

= SkT *. We may choose P so that the matrix representation {Sk} is

decomposed into the direct sum of irreducible representations. Thus,

5.3 Decomposition of Physical Tensors

In §4.4, we discussed the decomposition of the set of components

T· . of a physical tensor into sets of quantities which form the11"· In

carrier spaces for irreducible representations of the general linear group.

In this section, we determine sets of linear combinations of the in

dependent components of a tensor which form the carrier spaces for the

irreducible representations of the group A = {AI' ... , AN} which defines

the symmetry of the material under consideration. The transformation

.properties of the components T· . under a transformation AK of A11··· 1n

are defined by

T* = PT. (5.3.5)

(5.3.6)

(5.3.1)

Let T denote the column vector whose entries T 1,... ,Ts are the In-

dependent components of T i i' i.e.,1··· n

The coefficients (Yi in (5.3.7) are positive integers and are seen from


116 Group A veraging Methods [Ch. V Sect. 5.4] A veraging Procedures for Tensor- Valued Functions 117

(5.3.14)

We denote these linearly independent matrices by V1,... ,Va . The a

column vectors U1 == VIT , ... , Ua == VaT then form a quantities of

type r.

We see from (5.3.10), ... ,(5.3.12) that when T is subjected to the trans

formation AL, i.e., when T is replaced by SLT, then Uij is replaced by

fLU... Thus, the transformation properties of the k-dimensionalIJ

column vector U·· under A are defined by the irreducible representationIJ

r == {f1,... ,fN }. We note that for fixed values of (i,j) in (5.3.11), e.g.,

(i,j) == (1,1), VII == [ f: rfm Sfp] (m==I, ... ,k; p==I, ... ,s) forms a k X sK=1

matrix. The number a of linearly independent matrices in the set

V·· == [ f: r~ S~ ] which may be generated by allowing i and j toIJ K=1 1m JP

take on values 1, ... , k and 1, ... , s respectively is given by the number of

times the irreducible representation r appears in the decomposition of

{SK}' i.e.,

(5.3.9)

(5.3.8)

(5.3.11)

(5.3.10)

.. N K KIJ - L: - (- k)U - r· S· T m-l, ... ,m 1m JP pK==1

(i==I, ... , k; j==I, ... , s)

U·· == V··TIJ IJ'

- [Uij Uij]TUij - 1 ,... , k

where

The k s column vectors

where tr rk denotes ~he complex conjugate of tr rk and where tr SK

(K==I, ... ,N) and tr fk. (K==I, ... ,N) give the characters of the rep

resentations {SK} and ri respectively. Suppose that r == {f1,... ,fN} is

a k-dimensional irreducible representation of A. We may assume that

the matrices f K are unitary (see remarks following (2.3.18)), i.e.,

form carrier spaces for the k-dimensional irreducible representation r.In (5.3.10) and (5.3.11), i and j may be any pair of values chosen from

the sets (1, ... ,k) and (1, ... ,s) respectively. We may replace T by SLT ,

i.e., Tp by S~qTq, on the right of (5.3.11)4 to obtain

N N _ ..L: r¥ S~ SL T = L: r L rMsMT = r L UIJ (m,n = 1,... ,k)

K==l 1m JP pq q M==1 mn In Jq q mn n(5.3.12)

5.4 Averaging Procedures for Tensor-Valued Functions

A tensor-valued function T i i (Ek k) IS said to be1··· mI··· n

invariant under the group A == {A1,... ,AN} if

A~.... A~. T· . (Ek k) == T· . (AkK n ... Ak

K n En n)I1J 1 1mJ m J1...JmI· .. n 11 ...1m 1~1 n~n ~1· ..~n

(5.4.1)where we have set

(5.3.13)(5.4.2)

holds for all AK belonging to A. Let

f - 1 f f- 1K == L M'

(f-1) - (f) (f-l).K mi - L mn M nl' where T 1,... , Ts and E1,... ,Et denote the independent components of

118 Group Averaging M eihods [Ch. V Sect. 5.4] A veraging Procedures for Tensor- Valued Functions 119

T· . and E· . respectively. With (5.4.2), we may express the11··· 1m 11· .. 1n

restrictions (5.4.1) as

U(E) == Q11 E

satisfies (5.4.5) since, with (5.4.7) and (5.4.8),

(5.4.8)

SK T(E) == T(RK E) (5.4.3) (5.4.9)

which holds for K == 1,... , N where {SK} and {RK} are the matrix rep

resentations which define the transformation properties of T and E

respectively under A. From (5.3.5), ... ,(5.3.7), we see that we may

determine a s X s matrix P such that

where the matrix representation defining the transformation properties

of the Til'.'" Tia. is fi' Thus, the problem of determining the form of

T(E) which is stbject to (5.4.3) may be replaced by a number of

simpler problems. We seek to determine the form of expressions U(E)

which are subject to the restrictions

* .T == PT == TIl + + T 1a + +Tr1 + ... + Tra

1 r. (5.4.4)

The number PI (f) of linearly independent matrices contained in the set

of k t matrices obtained from the Qij given in (5.4.6) upon allowing i

and j to run through the sets 1,... , k and 1,... , t respectively is equal to

the number of linearly independent functions U(E) which are linear in

E and satisfy (5.4.5). We have

(5.4.10)

Let Q1,... , Qp denote the set of linearly independent k X t matrices ob-I

tained from the set of k t matrices Qij (i==l, ... ,k; j==l, ... ,t). Then the

set of PI (f) quantities of type f which are linear in E is given by the

column vectors

where f == {r1"'" r N} is a k-dimensional irreducible representation of

A. We refer to U as a quantity of type f.

We first consider the case where U == [U1,... ,Uk]T is a linear

function of E == [E1,... , Et]T. Let

r K U(E) == U(RK E) (K==l, ... ,N) (5.4.5)(5.4.11)

We may generate functions U(E) of any given degree n in E which

satisfy (5.4.5). Let En denote the column vector whose (t +~ -1)entries are given by the quantities

denote a set of k t rectangular k X t matrices. We may again employ

the argument used in (5.3.12) to show that

(5.4.12)

arranged in their natural order so that E· E· ... E· precedes11 12 In

~h~h·· ~jn i~ t~e first of the non-zero entries in the list i1 - h,12 - J2 ,... , In - In IS negative. Let {Rif)} denote the matrix rep-

resentation of A which defines the transformation properties of En

under A where the Rif) are the symmetrized Kronecker nth powers of

the RK. Let

(5.4.7)

(5.4.6)(i,m == 1,... , k; j,n == 1,... , t)[N-K K]Q.. == '" f· R·1J L...J 1m InK==l

rLQij == QijRL·

Then, for a given set of values of (i,j), e.g., (i,j) == (1,1), we see that

120 Group A veraging Methods [Ch. V Sect. 5.5] Examples 121

where the column vector Emn has entries given by (5.2.17) with r, s in

(5.2.17) being replaced by t, v. The Q\mn), ... , Q~mn) are Pmn (r)mn

linearly independent matrices obtained from the set of matrices

Q(~) = ~ r¥ (R(n)). (. - 1 k.· - (t+n-1))IJ L..J 1m k In I,m - ,... , ,J,n - 1, ... , n . (5.4.13)K~l '

Th QLn) (. -1 k.· - (t+n-1)) (t+n-1)e IJ 1 - ,... , , J - 1, ... , n form a set of k n

rectangular k X (t +~ - 1) matrices. The number P n(r) of linearly in-

dependent matrices contained in this set is equal to the number of

linearly independent functions U(E) of degree n in E which satisfy

(5.4.5) and is given by

Q(~rlll) = l Er¥ (R~) x S~)).IJ N K~l Ip Jq

upon making an appropriate choice of values of i and j.

(5.4.19)

(5.4.14)

(5.4.15)

Pn(r) = ~ Etr R~) tr I'KK~l

where tr R~) is given by (5.2.10). Let Q\n) ,... , Q~n) denote the set of

linearly independent k X (t + ~ - 1) matrices obtai~ed from the Q~?-).IJ

Then, the set of Pn(f) quantities of type f which are of degree n in E is

given by

Q(n) (n)1 En'·'" QPn En·

5.5 Examples

In this section, we give some examples of the application of the

procedures discussed above to the determination of functions which are

invariant under the group A ~ {A1,... ,A6} where the Ai are given by

(2.2.4), i.e.,

Al =[01

01

], A2=[-1/2 f3"/2], A3=[ -1/2 -f3"/2],- f3"/2 -1/2 f3"/2 -1/2

(5.5.1)

We may similarly generate functions U(E,F) which are of

degrees m,n in the components of E ~ [E1,... , Et]T, F ~ [F1' ... ' Fv]T

respectively and which are subject to the restrictions that

f3"/2 ].-1/2

where the representations {RK} and {SK} define the transformation

properties of E and F. There are Pmn (f) linearly independent

functions of degree m, n in E, F which satisfy (5.4.16) where

(K~l,... ,N) (5.4.16)We may employ the method of §5.2 to generate scalar-valued functions

W(xi) of a two-dimensional vector xi (i ~ 1,2) which are invariant

under A. Let x ~ [xl' x2]T. The matrix representation {RK } which

defines the transformation properties of x under A is given by

The Pmn (f) linearly independent functions of type r are given by

(5.5.3)

(5.5.2)

With (5.2.6), (5.5.1) and (5.5.2), the number PI of linearly independent

invariants of degree one in x is given by

1 6PI ~ 6 E tr AK ~ o.

K~1(5.4.18)

(5.4.17)

Q(mn) (mn)1 Emn , ... , Qpmn Emn


(5.5.4)

In order to generate the invariants of degree two in x, we list the

symmetrized Kronecker squares A~) which define the transformation

properties under A of the column vector

[2 2]T

x2 = xl' x1x2' x2

We next list the symmetrized Kronecker cubes of the AK which define

the transformation properties under A of the column vector

(5.5.8)

(5.5.10)

, (5.5.9)

3~

-3

~

-1

ooo1

9

~

-5

3~

-9 3f3

-f3 -3

5 ~

-3~ -1

oo

-1

o

3~

5

~

-9

o1

oo

-1 3f3

-f3 5

-3 ~

-3~ -9

-1

ooo

A(3) _ 12 -8

A(3) -, 4-

9 -3f3

-~ -3

-5 -~

-3~ -1

ooo1

oo1

o

-3{3 -9 -3~

5 ~ -3

-~ 5 -~

-9 3~ -1

o

oo

-1

~

-3

3f3

1

ooo

0 0 0 0 x31

* 6~) 3 0 3 0 -1 2R3 x3 = LA.. X3 = 2 X1X2

(5.5.11)0 0 0 0 2

K=l x1x20 -3 0 1 x3

2

_ 1 6 (3) _P3 - 6" L tr Ak - 1.

K=1

1 -3~

A~3) =-81 -~ 53 -~

-3~ -9

The entries in the column matrix R3x3 are either invariants or zero.

Thus,

With (5.2.10)1 and (5.5.9), the number P3 of linearly independent

invariants of degree 3 in x is

(5.5.7)

(5.5.6)

303

000

303

which yields the result that xI + x~ is the invariant of degree 2 in x.

The entries in the column matrix R2 x2 are either invariants or zero

and are given by

The A~) are given by

1 0 0 1/4 -~/2 3/4

A~2) = 0 1 0 A~2) = ~/4 -1/2 -~/4 ,

0 0 1 3/4 ~/2 1/4

1/4 ~/2 3/4 1 0 0

A~2) = -~/4 -1/2 ~/4 , A~2) = 0 -1 0 (5.5.5)

3/4 -~/2 1/4 0 0 1

1/4 -~/2 3/4 1/4 f3/2 3/4

A~2) = -~/4 1/2 ~/4 , A~2) = ~/4 1/2 -~/4 .

3/4 ~/2 1/4 3/4 -~/2 1/4

With (5.2.10)1 and (5.5.5), the number P2 of linearly independent

invariants of degree 2 in x is


The linearly independent matrix obtained from this set is VI = V11'

Hence the quantity of type r 3 is given by

which shows that the invariant of degree 3 In x IS given by

(3xy - x~) x2'

We next consider the problem of determining the form of a two

dimensional vector-valued function z (x) which is invariant under A

where z == [zl' z2]T and x == [xl' X2]T. The matrix representation {SK}

which defines the transformation properties of z under A == {AI ,... ,A6}

is given by

(5.5.12)(5.5.15)

We have from (5.4.13), (5.5.1), (5.5.13), (5.5.5) and (5.5.9),

Q1 = [ Kt1 (rk)lm(AK)lP ] = 3 [~ ~] ,

Q(2) = [ t (N) (A~)) ] = Q[ 0 2

-~l1 K=l K 1m 2p 2 1 0(5.5.17)

Q(3) = [ t (N) (A~)) ] = 1! [ 1 0 1

~J1 K=l K 1m 1p 4 0 1 0

(5.5.16)

Since we know that Q'3 == 1 and hence that there will be only one

linearly independent matrix in the set Vij (i,j = 1,2) given by (5.5.14),

we need only generate one non-zero matrix of the set, e.g., V11' There

would be no need to determine V12, V21 and V22

. We would usually

refrain from generating any more matrices V·· than are actually1J

required.

The numbers PI (r3)' ... , P3(r3) of quantities of type r 3 which

are invariant under A and are of degrees 1, 2, 3 in x are seen from

(5.4.14), (5.5.1), (5.5.5) and (5.5.9) to be given by

Pi(r3) = ~ t tr A~) tr rk = 1 (i = 1,2,3).K==1

r1: 1 1 _ 1, 1, 1, 1, 1, 1·r 1,···,r6 - ,

r2: 2 2_ 1, 1, 1, -1, -1, -1 ; (5.5.13)r 1,···,r6 -

r 3: Ii,.. ·,~ = AI' A2, A3, A4, AS' A6

There are three inequivalent irreducible representations associated with

the group A which are given by (see (2.5.18))

where the AK are defined by (5.5.1). From (5.5.12) and (5.5.13), we

see that the transformation properties of z == [zl' z2]T are defined by the

representation r 3 = {rk} and we refer to z as a quantity of type r 3.

We may employ the method of §5.3 to obtain this result. Thus, with

(5.3.8), (5.5.1) and (5.5.13), we see that the number Q'i of times r iappears in the decomposition of the representation {SK} is given by

6 .Q'1 == 0, Q'2 == 0, Q'3 == 1 where Q'i == ~ E tr AK tr f'k· We now employ

K=1 [ 6 _ ](5.3.10) and (5.3.11). In the expressions V.. == E rJ( S~ (i,m,j,p

1J K=1 1m JP

== 1,2), we set rK == SK == AK where the AK are given by (5.5.1).

Upon setting (i,j) == (1,1), (1,2), (2,1) and (2,2) in turn, we have

Thus, the polynomial expression for .z{x) which is truncated after terms

of degree three in x is given by

With (5.4.11), (5.5.4), (5.5.8) and (5.5.17), the quantities of type r 3 are

Q1x = 3 [:1], QF)~= ~ [22X1X

;], Q~3)X3 = £(xy +x~) [Xl].2 Xl -x2 x2

(5.5.18)

126 Group A veraging Methods [Ch. V

(5.5.19)

Sect. 5.5] Examples 127

~ ~ 3 0 0 0

A3 xA3 =! -~ 1 -3 ~ 0 -1 0 0

-~ -3 1 ~A4 xA4 =

0 0 -1 0

3 -~ -~ 1 0 0 0 1

1 -~ -~ 3 1 ~ ~ 3

A5 xA5 =! -~ -1 3 ~A6 xA6 =! ~ -1 3 -~

-~ 3 -1 ~ ~ 3 -1 -~

3 ~ ~ 3 -~ -~ 1

(5.5.22)

We now consider the problem of determining the form of a two

dimensional second-order tensor-valued function Tij{xk) (i,j,k = 1,2)

which is invariant under the group A defined by (5.5.1). Let

(5.5.20)

The matrix representation {SK} which defines the transformation

properties of T under A = {AI'.'.' A6} is given by

(5.5.21 )

where AK x AK denotes the Kronecker square of AK. With (5.2.13),

(5.2.14) and (5.5.1), we have

1 0 0 0 1 -~ -~ 3

0 1 0 0A2 xA2 =! ~ 1 -3 -~

Al xA1 = 0 0 1 0 ~ -3 1 -~

0 0 0 1 3 ~ ~

With (5.3.8), (5.5.13) and (5.5.22), the number ai of times r i appears in

the decomposition of {SK} = {AK x AK} is given by a· = 1 (i = 1 2 3)1 6 2 -i l' ,

where l¥i = 6 K~l (tr AK) tr r K · We recall that tr (AK x AK)

= (tr AK)2. With (5.3.11), (5.5.13), (5.5.20) and (5.5.22), we see that

the quantities

are quantities of types r l' r 2 and r 3 respectively.

The numbers P1(r2), P2(r2), P3(r2) of quantities of type r 2

128 Group A veraging Methods [Ch. V Sect. 5.6] Generation of Property Tensors 129

which are of degrees 1,2,3 in x are seen from (5.4.14) to be given by

With (5.4.13), (5.5.8) and (5.5.9), the quantity of type f 2 which is of

degree 3 in x is

may be obtained by determining the set of linearly independent

functions multilinear in the n vectors xl' x2' ... , xn which are invariant

under A. The number P n of linearly independent nth-order invariant

tensors is given by

where AK x AK x ... x AK denotes the Kronecker nth power of AK. If

the Pn multilinear invariants are given by

(5.5.25) (5.6.2)

The expression for T(x) which is invariant under A and of degree

~ 3 in x is then given by

then the set of P n invariant tensors is given by

1 cPC· . ,... , . '.11' ..In 11· ..1n

(5.6.3)

(5.5.26)

Let xlX2...~ denote the column vector whose 3n entries are xflxf

2··· xi

n(i1,i2,... ,in == 1,2,3) ordered so that x f x7... xP- precedes xJ~ x

J7... x

J!1 if

11 12 In 1 2 n

the first non-zero element of the set i1 - j l' i2 - j2 ,... , in - jn is

negative. For example,

(5.6.4)

where we have employed (5.5.7), (5.5.11), (5.5.19), (5.5.23) and (5.5.25).

5.6 Generation of Property Tensors

We may employ the procedure of §5.2 to generate the set of nth

order property tensors associated with the finite group A == {A1,... ,AN},

i.e., the set of nth-order tensors which are invariant under A. We recall

that the set of linearly independent nth-order tensors invariant under A

The matrix representation of A which defines the transformation

properties of the column vector Xlx2".xn is given by {SK}

== {AK x AK x ... x AK} where AK x AK x ... x AK is the Kronecker

nth power of AK. The Pn linearly independent invariants which are

multilinear in x1" .. ,xn are obtained by determining the Pn linearly in

dependent rows of the matrix

(5.6.5)

130 Group A veraging Methods [Ch. V Sect. 5.6] Generation of Property Tensors 131

The Pn X 3n matrix whose rows are the Pn linearly independent rows of

(5.6.5) yields Pn linearly independent invariants when multiplied on the

right by the column vector xlx2' "xn ' The rows of the Pn x 3n matrix

obtained from (5.6.5) define the set of Pn linearly independent nth

order invariant tensors. For example, the first row of the matrix

f: AK x AK is given by c, , = f: Af Af and will either yield anK=1 JIJ2 K=1 Jl J2

invariant tensor or will have all components equal to zero.

We consider as an example the problem of generating the

invariant tensors of orders 1 and 2 associated with the group C3V

= {AI"'" A6} where

0 0 -1/2 ~/2 0 -1/2 -~/2 0

A1 = 0 1 0 , A2 = -~/2 -1/2 0 , A3 = ~/2 -1/2 0

0 0 1 0 0 0 0 1

(5.6.6)

-1 0 0 1/2 -~/2 0 1/2 ~/2 0

A4 = 0 0 , A5 = -~/2 -1/2 0 ,A6 = ~/2 -1/2 0

0 0 0 0 0 0 1

The number Pn of linearly independent nth-order tensors invariant

under the group A defined by (5.6.6) is given by

The entries of the third row give the 1, 2, 3 components of the invariant

tensor which we may write as C' = t A3~' The second-order invariant1 K=1 1

tensors are given by the P 2 linearly independent rows of the matrix

3 0 0 0 3 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

60 0 0 0 0 0 0 0 0

E AKxAK = 3 0 0 0 3 0 0 0 0 (5.6.9)K=l 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 6

For a given row, columns 1, 2, ... ,9 give the 11, 12, 13, 21, 22, 23, 31,

32, 33 components of the invariant tensor. The first and last rows of

(5.6.9) give the two linearly independent invariant tensors which are

6C', = E Af Af = 3(151,151, + 152, 152,)

1J K 1 J 1 J 1 J'=1 (5.6.10)6

D.. = L Af. Af. = 6153,153,1J K 1 J 1 J'

=1

(5.6.7)

Thus, we have PI = 1, P 2 = 2, P3 = 5, .... The first-order invariant

tensors are defined by the PI = 1 linearly independent rows of the

matrix

000

000

006

(5.6.8)

VI

ANISOTROPIC CONSTITUTIVE EQUATIONS

AND SCHUR'S LEMMA

6.1 Introduction

In this chapter, we consider constitutive expressions of the form

T == CE where T and E are column vectors, C is a matrix and where

the expression T == CE is invariant under a group A. In §6.2 and §6.3,

we follow Smith and Kiral [1978] in the case where A = {A1,oo.,AN

} is

finite to show that application of Schur's Lemma (see §2.4) enables us

to essentially reduce the problem of determining the form of C to that

of determining the decomposition of the sets of components of T and E

into sets of quantities of types r 1,... ,rr (see §5.2, §5.4) where r 1,... ,rr

denote the irreducible representations of A. We introduce in §6.4 the

notion of product tables which enables us to conveniently generate non

linear constitutive expressions. Xu, Smith and Smith [1987] use this

type of result to form the basis of a procedure which employs a com

puter program to automatically generate constitutive expressions which

are invariant under a given crystallographic group A. In §6.6, we follow

Smith and Bao [1988] to indicate the manner in which these procedures

may be extended to the case where the group A is continuous.

6.2 Application of Schur's Lemma: Finite Groups

We consider constitutive expressions of the form

(6.2.1)

133'

134 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.2] Application of Schur's Lemma: Finite Groups 135

where T 1,... ,Tn and E1,.. ,Em are the independent components of the

tensors T· . and E· . respectively. Let {SK} and {RK} denote11··· 1P J1··· Jq

matrix representations of :4 which are of dimensions nand m re-

spectively and which define the transformation properties of T and E

respectively under A. Then, (6.2.1) may be written as

which describe the response of anisotropic materials. The tensor

E· . may denote the outer product of a number of tensors, e.g.,J1···Jq

E· . = F. F· G· . , so that (6.2.1) may be a non-linear expression.J1···J4 J1 J2 J3J4

There are restrictions imposed on the form of (6.2.1) by the

requirement that (6.2.1) shall be invariant under the group of

transformations A = {AI '00.' AN} defining the symmetry of the material

under consideration. Let T and E denote the column vectors

(6.2.9)

(K = 1,... ,N),

Multiplying (6.2.5) on the left by Q and on the right by p-1, we obtain

QSKCp-1 = QCRKP-I or QSKQ-IQCp-1 = QCp-IpRKP-I.

The restrictions imposed on the matrix D = QC p-1 by the invariance

requirements are then given by

QSKQ-ID = DPRKP-I (K = I, ... ,N). (6.2.8)

The sets of matrices QSKQ-I (K=I, ... ,N) and PRKP-I

(K = 1,oo.,N) form matrix representations of the group A which are

equivalent to the representations {SK} and {RK} respectively. We

may determine Q and P so that

(6.2.2)

where C is an n x m matrix. The restrictions on (6.2.3) imposed by the

requirement of invariance under A are given by

T= CE (6.2.3) Zl Zj1 Xl Xj1Z=QT= Z·= , X=PE= X·=, J ' J

Zr Z· Xr X·In· Jm·

We see from (6.2.3) and (6.2.4) that the matrix C is subject to the re

strictions that

or (6.2.10)

where f l = {rL...,r~}, ... , f r = {r~, ...,r~} denote the r inequivalent

irreducible representations associated with A which are of dimensions

PI'"'' Pr respectively, i.e., the rk,···, r~ are PI X PI matrices, ... ,

Pr x Pr matrices. The column vectors Zji (i = 1,oo.,nj)' Xji (i = 1,... ,mj)

have Pj components each and are quantities of type rj . Thus, when T

and E are repl~ced by SK! and RKE, the quantities Zji and Xji are

replaced by rkZji and rkXji' The column vectors ZI"'" Zr and

X1,,,·,Xr have P1n1, ... ,Prnr and P1m1, ... ,Prmr entries respectively.

We have

(6.2.6)

(6.2.4)

(6.2.5)(K = 1,... ,N).

X=PE.Z= QT,

Let

With (6.2.3) and (6.2.6), we have Z = QT = QCE = QCp-1X

Z=DX, (6.2.7)where nand m are the number of components of T and E respectively.

136 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.2] Application of Schur's Lemma: Finite Groups 137

With (6.2.9), the equation (6.2.7)1' i.e., Z == nx, may be written as

where the repeated superscripts i,j do not indicate summation. With

(6.2.12), (6.2.14) and (6.2.15), we have

and where the matrix r KI appears on the diagonal n· times and m·

1 1

times respectively. With (6.2.13), we obtain r2 sets of matrix equations

Zl nIl n 12 n 1rXl

Z2 n 21 n 22 n 2r X2

Zr n r1 n r2 n rr Xr

where the matrices n I] are of the form

n I] n I]11 1mj

n I] ==

n I] n I]n·1 n·m·1 1 J

(6.2.11)

(6.2.12)

(i,j == 1,... ,r; K == 1,... ,N)

(6.2.14)

(6.2.15)

Th t . n I] n I] t . d nIl n 12e rna rIces 11' 12' ... are Pi X Pj rna rIces an , , ... are

PIn1 X PIm1' PIn1 X P2m 2' ... matrices respectively.

With (6.2.9) and (6.2.11), the restrictions imposed on the matrix

n by (6.2.8) are given by

nAJ.m.1 J

(6.2.16)

r]K

nI ]n·m·1 J

(6.2.13)

pIK

p2K

prK

(K = 1,... ,N).

where (6.2.16) must hold for K == 1, ... ,N. Equation (6.2.16) yields nimj

sets of matrix equations

(0: == 1, ... ,ni; (3 == 1, ... ,mj; K == 1, ... ,N) (6.2.17)

In (6.2.13), the first and last matrices are block diagonal matrices where

for each given set of values of i and j, e.g., (i,j) == (1,1). Schur's Lemma

(see §2.4) tells us that if n is an n X n matrix which commutes with

each of the n X n matrices r K (K == 1, ... ,N) comprising an irreducible

138 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.3] The Crystal Class D 3 139

then D is a multiple AEn of the n x n identity matrix. Further, if {rK}

and {UK} are inequivalent irreducible representations of dimensions n

and m respectively of A and if there is an n X m matrix D such that

quantities Z·l' ... ' Z· , X· 1,... ,X. are quantities of type r· and may be1 1ni 1 1mi 1

determined by inspection or by employing the procedure discussed in

§5.2. The numbers n1· and mI· of quantities Z·I' ... ' z· and X· 1,... , X·1 In· 11m·

of type r i are given by 1 1

(6.2.22)

(6.2.18)(K == 1,... ,N),

representation of the group A == {AI' ... ' AN}' i.e., if

then D must be the zero matrix. Thus, (6.2.17) yields the result that if

i ~ j the matrix D IJ (.l is the zero matrix and that if i == j the matrix•• O'.fJ

D IJ (.l is a multiple of the p. dimensional identity matrix. We concludeO'.fJ 1

that the matrix D appearing in (6.2.11) is of the form

(K == 1,... ,N), (6.2.19)Given the numbers n·, m· (i == 1,... ,r) and the dimensions p. of the1 1 1irreducible representations ri, we may write down the expression for

the matrix D. Some of the entries in the column matrices Z and X

may appear as pairs of complex conjugates. We discuss in §6.5 the

minor change in procedure appropriate in such cases.

6.3 The Crystal Class D3

(6.2.20)

(6.2.21 )

We observe from §6.2 that the determination of the form of the

constitutive equation T == CE which is invariant under the finite group

A is trivial once we have decomposed the set of n components of T and

the set of m components of E into n1 +... +nr and m1 +... +mr sets

which form the carrier spaces for the irreducible representations of A

appearing in the decompositions (6.2.9)1 and (6.2.9)2 of the

representations {SK} and {RK}. We must give a procedure for

generating the quantities Zli (i == 1,... , n1) ; ... ; Xri (i == 1, ... , mr)

appearing in the expressions (6.2.9) for Z and X. We consider as an

example the problem of determining the form of the constitutive

equation

and where Ep. is the Pi x Pi identity matrix. We proceed by deter1

mining the quantities Zn,···,Zlnl'···' Zrl,···,Zrnr and Xn,···,X1m1,···,

Xr1 ,... , Xrmr appearing in the expressions (6.2.9)3,4 for Z and X. The

y. == C·· X· +C··kX. Xk1 IJ J IJ J

which is invariant under the group D 3 == {AI'···' A6} where

(6.3.1)

Thus, the set of components Xl' X2, X3 may be split into quantities X3

and [X2, -X1]T which are quantities of types r2 and r3 respectively.

Let us consider the first term in (6.3.1), i.e., y. == C·. X·. SinceI 1J J

the Yi transform in the same manner as do the Xi' we see from (6.3.4)

that Y3 and [Y2' - YI]T are quantities of types r 2 and r 3 respectively.

In equation (6.2.11), we set

140 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI

0 0 -1/2 ~/2 0 -1/2 -~/2 0

A1 = 0 1 0 , A2 = -~/2 -1/2 0 , A3 = ~/2 -1/2 0

0 0 0 0 1 0 0 1

(6.3.2)

0 0 -1/2 ~/2 0 -1/2 -~/2 0

A4 = 0 -1 0 ,A5 = ~/2 1/2 0 ,A6 = -~/2 1/2 0

0 0 -1 0 0 -1 0 0 -1

There are three inequivalent irreducible representations associated with

the group D3 which are given by

r 1 : ri, ... , r~ = 1, 1, 1, 1, 1, 1

Sect. 6.3] The Crystal Class D3 141

(6.3.4)

(6.3.3)

[1 0] N

2=[ -1/2 ~/2], ~3=[ -1/2 -~/2],

r 3 : Ii = 0 ' - ~/2 -1/2 ~/2 -1/2

r 2 : rt, ... , r~ = 1, 1, 1, -1, -1, -1 (6.3.5)

We then see from the discussion of §6.2 that the equation Z == DX

which is equivalent to y. == C·· X· may be written as1 1J J

The quantities Yi and Xi appearing In (6.3.1) are the components

of three-dimensional vectors. The transformation properties of

[y1,y2,y3]T and [X1,X2,X3]T under the group D3 are defined by the

matrix representations {SK} == {A1,···,A6} and {RK} == {A1,···,A6}

respectively where the AK are defined by (6.3.2). With (5.3.11), (6.3.2)

and (6.3.3), we have

We next consider the second term in (6.3.1), i.e., y. == CookX. Xk

. The1 1J J

(6.3.6)

(6.3.7)

c1 0 0 X3o c2 0 X2o 0 c2 -Xl

where DII = clE1' Drr = c2~ and where E1 and E2 denote the one

and two-dimensional identity matrices respectively. With (6.3.5), wethen have

~/2 ].-1/2

142 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.3] The Crystal Class D 3 143

transformation properties of the quantity

(6.3.8)

(6.2.11) and (6.2.12) corresponding to the expression y. == C··kX. Xk1 IJ Jare given by

under the group D3 are defined by the matrix representation {A~2),... ,

A~2)} where the A~) are the symmetrized Kronecker squares of the

AK

defined by (6.3.2). We observe that the linear combinations of the

components of the quantity (6.3.8) which form quantities of types r 1,

r2 and r3 respectively are

(6.3.11)

r2

: None (6.3.9)

The results (6.3.9) may be obtained from inspection or upon application

of the procedure of §5.3. Thus, we have

The matrix equation Z == D X which is equivalent to y. == C··k X· Xk1 IJ Jmay then be written as

XII

Z21 n21 n21 n23 n23X1211 12 11 12

(6.3.12)

Z31 n31 n31 n33 n33 X3111 12 11 12

X32

where the nIt nt1 (i i= j) are Pi x Pj zero matrices and nN = c3E2'

n1~ == c4E2· With (6.3.11), we have

==3 (6.3.10) X2 +X21 2X2

Y3 0 0 0 0 0 03

Y2 0 0 0 0X1X3

(6.3.13)c3 c4

-Y1 0 0 0 0X2X3

c3 c42X1X2X2X2

1 2

With (6.3.10), we see that the matrices which appear in the equations

144 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.4] Product Tables 145

6.4 Product Tables

We again consider the group D3 = {AI'···' A6} where the Ai are

defined by (6.3.2) and where the irreducible representations r 1, r 2, r 3

associated with D3 are given by (6.3.3). Let

r 1: aI' b1

r2: a2' b2(6.4.1)

r3: [ a311[b31

]a32 b32

denote quantities whose transformation properties under D 3 are defined

by the representations indicated. There are 16 distinct products which

may be obtained upon taking the product of one of the elements from

the set aI' a2' a31' a32 with one of the elements from the set b1, b2,

b31

, b32. These are given by alb1, alb2, ... , a32b31' a32b32· We may

determine 16 linear combinations of these products which may be split

into sets of quantities such that the transformation properties under D3

of each set is defined by one of the representations r l' r 2 or r 3· We

list these sets of quantities in tabular form in Table 6.1.

Table 6.1 Product Table: D3

r1 al b l alb l , a2b2' a31b31 + a32b32

r2 a2 b2 a1b2' a2b1' a31b32 - a32b31

r3[ a31 ] [ b

31] [ al

b311[a31

bl1[a2b32l

a32 b32 alb32 a32b2 -a2b31

[ a32b2 ], [ a31b32 + a32b

31 ]-a31b2 a31b31 - a32b32

In Table 6.1, a1 and b1 denote quantities of type r1; a2 and b2,

quantities of type r 2; [a31' a32]T and [b31 , b32]T, quantities of type

r3. We may generate the entries appearing on the right of Table 6.1

upon inspection of the manner in which the quantities a1b1, a1b2, ...

transform under D 3. For example, the quantity al is of type r l' i.e.,

a1 is unaltered under all transformations of D3· Hence alb1, a1b2 and

[a1b31 , a1 b32]T transform under D3 in exactly the same manner as do

b1, b2 and [b31 , b32]T. Consequently a1b1, a1b2 and [alb31 , a1b32]T

are quantities of types rl' r2 and r3 respectively. Since a2 and b2 are

quantities of type r 2, the matrices ry, ...,r~ = 1, 1, 1, -1, -1, -1

define the transformation properties of a2 and b2 under D 3. We see

immediately that a2b2 is invariant under D3 and hence of type r 1.

The remaining entries in Table 6.1 may also be found from inspection

although more of an effort is required than is the case for the obvious

results mentioned above.

We may also generate the entries on the right of Table 6.1 upon

application of the procedures outlined in §5.2 and §5.3. Consider for

example the quantity [a31b31 , a31b32, a32b31' a32b32]T. This forms

the carrier space for the representation r comprised of the matrices

rk X rk (K=I, ... ,6) where rt X rt is the Kronecker square of rt·The character of this representation is given by (Xl' ... ' X6) = (4, 1, 1,0,

0,0) since tr(rkxrk)= (tr rk)2 and, from (6.3.3), (trrf,· .. , trJi)

= (2, -1, -1, 0, 0, 0). With (6.3.3) and (2.5.14), we see that the

irreducible representations r l' r 2 and r 3 appear once each in the

decomposition of the representation r. The linear combinations of the

components of the column vector z = [a31b31 , a31b32, a32b31' a32b32]T

which form quantities of types r l' r2 and r3 may be obtained with

(5.3.11) upon setting SK = rt X rt in (5.3.11). Thus, the quantities of

types r l' r2 and r3 are given by

146 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.4] Product Tables 147

(i,j==1,2,3) transform under the group considered. We list the Basic

Quantities table for the group D 3 below.

(6.4.4)

(6.4.3)y. == C··kX.Xk1 IJ J 'Y·==C··X·1 IJ J'

which is the result (6.3.7). We next determine the form of (6.4.3)2'

i.e., Yi == Cijk Xj Xk, which is invariant under D 3. To do this, we make

the identifications

The information contained in the product table enables us to

readily generate the form of constitutive expressions. Consider the

problem of determining the forms of the equations

r 1 833 , 811 +822

r 2 P3' a3' A12

r3 [~:J, [:;J, [~~~ l [Sl~~~22l [-;;3]

Table 6.2 Basic Quantities: D3

which are invariant under D3 where the Vi' Xi (i==1,2,3) are com

ponents of absolute vectors. We see from Table 6.2 or from (6.3.4) that

Y3' X3 are quantities of type r 2 and that [Y2' - Yl]T, [X2, -X1]T are

quantities of type r 3. The argument given in §6.2 shows that each of

the quantities of type r i arising from decomposition of the components

of T in the expression T == C E is expressible as a linear combination of

each of the quantities of type r i arising from the decomposition of the

components of E. In the case of (6.4.3)1' we have

(6.4.2)

It is convenient to also list in tabular form the linear combinations

of the components of polar (absolute) vectors Pi' axial vectors ai'

symmetric second-order tensors S·· ( == 8.. ) and skew-symmetric second-IJ J1

order tensors A·· (== -A.. ) which form the carrier spaces for the1J J1

irreducible representations of the group D3- We refer to this table as

the Basic Quantities table associated with D 3. The entries in the table

may be determined upon application of the procedures of §5.2 and §5.3.

Examples of this procedure are given by (6.3.4) and (6.3.10). The

entries in the Basic Quantities tables may more readily be determined

from inspection of the manner in which the components Pi' ai' Sij' Aij

respectively, and appear as entries in the rows of Table 6.1 headed by

r l' r 2 and r 3. Product tables for all of the 32 crystallographic groups

have been given by Xu [1985]. Another procedure which enables us to

readily generate product tables is outlined in §7.5. We note that one

must have available the matrices rt, ... ,rN(i == 1, ... , r) defining the

irreducible representations of a group A in order to construct the

product table for A. The irreducible representations for all of the

crystallographic groups may be found in Chapters VII and IX.

148 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.5] The Crystal Class S 4 149

(6.4.5)arising from the Xi Xj Xk (i,j,k == 1,2,3) are given by

We see from (6.4.5) and Table 6.1 that the quantities of types r1, r2and f 3 arising from terms quadratic in X1"",X3 ~re given by

(6.4.9)

Since Y3 and [Y2' - Yl]T are quantities of types f 2 and r 3 respectively,

we see from (6.4.6) that the forms of (6.4.3)2 consistent with the

invariance requirement is given by

r l : X~, XI+X§; r 2: None;

r 3: [~~~~l [~f~;§l(6.4.6) The form of (6.4.3)3' i.e., Yi == Cijk£Xj XkX£, which is invariant under

D3 is then seen from (6.4.9) to be given by

(6.4.7)

This result is equivalent to that given by (6.3.13). We next employ

Table 6.1 to determine the terms of degree three in the X· which form1

quantities of types r 1, r 2 and r 3. We let the terms in (6.4.6) assume

the role of the aI' a2'''. and X3' [X2' -XI]T assume the role of hI'

b2, ... in Table 6.1. We have

r 1: -X2 al =XI+X§; None;a1 - 3'

r 2: None; b2 == X3;(6.4.8)

r 3: [a31] = [XIX3] [a~l] = [ 2XIX2 ]- [h31

] [X2

]a32 X2X3 ' a32 XI -X§b32 - -Xl'

With (6.4.8) and Table 6.1, the quantities of types r l' r 2 and r 3

We may determine from (6.4.9) and Table 6.1 the decomposition of the

quartic terms XiXjXkX£ (i, ... ,£== 1,2,3) into quantities of types r 1, r 2and r 3. This iterative process may be continued so as to obtain the

decomposition of the X· X· ... X· for any reasonable value of n. A11 12 In

computer program has been written which enables us to carry out this

iterative procedure for most of the crystallographic groups. This should

preclude the introduction of errors.

6.5 The Crystal Class S4

The matrices comprising some of the irreducible representations

associated with the group 54 have complex numbers as entries. In such

cases, the procedure employed differs slightly from that discussed in the

previous section. The group of matrices defining the symmetry of the

crystal class 54 is given by 54 == {A1,···,A4} == {I, D3, D1T3, D2T 3}

where

150 A nisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.5] The Crystal Class S 4 151

-1 0 0 0 0 o -1 0We consider the problem of determining the form of the0 0

A1 = 0 0 , A2 = o -1 0 , A3 = -1 0 0 , A4 = 0 o . (6.5.1) equations

0 0 1 0 0 0 o-1 0 o -1 Tij = Cijk£Ek£, T·· = C··k£ Ek£E (6.5.3)1J 1J mn mn

There are four inequivalent irreducible rep~esentations associated with

the group 54 (see §7.3.5) which are given by

f 1: 1 r1 1, 1, 1, 1r 1,···, 4

f 2: 2 2 1, 1, -1, -1r 1,···,r4(6.5.2)

f 3: Ii, .. ·,Ii 1, -1, 1, -1

f 4: Ii, .. ·,Ii 1, -1, -1, 1 .

The product table and the basic quantities table for the group 54 are

given below.

Table 6.3 Product Table: 54

f 1 al b1 a1b1' a2b2' a3b4' a4b3

f 2 a2 b2 a1b2, a2bl' a3b3' a4b4

f 3 a3 b3 a1b3, a3b l' a2b4' a4b2

f 4 a4 b4 a1b4' a4b1' a2b3' a3b2

which are invariant under 54. In (6.5.3), T·· and E·· are three-. ~ ~

dimensional symmetric second-order tensors. From Table 6.4, we see

that the linear combinations of the Tij and Eij which form quantities of

types f 1,... ,f4 are given by

f 1: T33' T 11 + T22 ; E33, Ell + E22 ;

f 2: T 11 -T22, T 12 ; Ell - E22 , E12 ;(6.5.4)

f 3: T31 +iT23 ; E31 + i E23 ;

f 4: T31 -iT23 ; E31 -iE23 ·

With (6.5.4), application of the procedures in §6.2 or §6.4 shows that

the form of {6.5.3)1 which is invariant under 54 is given by

(6.5.5)

Table 6.4 Basic Quantities: 54

a3 A12, 833 , 811 + 822,

P3' 812, 811 - 822

PI - i P2' a1 + i a2' A23 + i A31 ,

PI + i P2' a1 - 1 a2' A23 - i A31 ,

831 + i 823

S31 - i S23

The last expression in (6.5.5) is the complex conjugate of the preceding

equation and may be omitted. We may equate the real and imaginary

parts of {6.5.5)3 to obtain

(6.5.6)

We prefer to write (6.5.5)3 in the form (6.5.6).

152 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.6] The Transversely Isotropic Groups T l and T2 153

6.6 The Transversely Isotropic Groups T1 and T2

Let Tl denote the group comprised of the matrices

where II' ... ' KS are defined by (6.5.S) and where we have employed the

form mentioned in (6.5.6).r 1: 11,... ,17 ; r 3: K1 +iK2, K3 +iK4, K5 +iK6, K7 + iKS ;

(6.5.7)

r 2: J1,···, J6 ; r 4: K1 -iK2, K3 -iK4, K5 -iK6, K7 -iKS ;

where

We next consider the problem of determining the form of

(6.5.3)2' i.e., Tij == Cijk£mnEk£Emn' which is invariant under 34.

With (6.5.4) and the product table for 34 (Table 6.3), we see that the

quantities of types r1' ... ' r4 arising from terms quadratic in the Eij are

given by

11,... ,17 = E53' (En + E22)2, E33(En + E22), (En - E22)2,

EI2' (En - E22)E12, E51 + E~3 ;

cos B sin B

Q(B) == -sin B cos B

o 0

oo1

(0 ~ B< 21r). (6.6.1)

K1,···,KS == E33 E31 , E33 E23 , (Ell +E22)E31 , (Ell +E22)E23 ,

(EII-E22)E31' -(EII-E22)E23' E12 E31 , -E12 E23 ·

J1,···,J6=E33(En-E22)' EI1-E~2' E33 E12,

(En + E22)E12, E51 - E~3' E31E23 ;(6.5.S) The group Tl defines the symmetry of a material which possesses

rotational symmetry about the x3 axis. The irreducible representations

associated with the group Tl are all one-dimensional and are given by

(see Van der Waerden [19S0])

With (6.5.4), (6.5.7) and (6.5.S), we see that the form of (6.5.3)2 which

is invariant under 34 is given by

[ T33 ] [£1 f2 £7 ]II

TIl + T22 - fS f9 f14 17

[Tn -T22]=[ gl g2 g6 ]J1

(6.5.9)T 12 g7 gs g12J6

10 : 1

Ip :-ipB (p == 1,2, ... ) (6.6.2)e

rp : e ipB (p == 1,2, ... ).

In (6.6.2), the 1 x 1 matrices correspond to the group element Q(B).

We list below the product table and the basic quantities table for thegroup Tl .

154 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.6] The Transversely Isotropic Groups T1 and T2 155

Table 6.5

IP

Table 6.6

Product Table: T1

bO aObOapBp, Apbp (p == 1,2,... )

aObp, apbOambn (m,n==1,2, ... ; m+n==p)

amBn, Anbm (m,n==1,2, ... ; m-n==p)

Bp aOBp, ApbOAmBn (m,n == 1,2,... ; m + n == p)

Ambn, anBm (m,n==I,2, ... ;m-n==p)

Basic Quantities: T1

x3 axis and a plane of symmetry which contains the x2 and x3 axes.

The irreducible representations associated with the group T2 may be

defined by listing the matrices corresponding to the group elements

Q(B) and R1. We denote the irreducible representations by (see Van

der Waerden [1980])

10 : 1 , 1

f O: 1 , -1

[ e~iPO e~pOJ [ ~ ](6.6.4)

1Ip :

0

where the first and second matrices correspond to Q(B) and R1respectively. The product table and the basic quantities table for the

group T2 are given below.

10 P3' a3' A12, Sll + S22' S33 Table 6.7 Product Table: T2

'1 PI + iP2' a1 + i a2' A23 + i A31 , S31 + i S23 10 aO bO aObO' AOBOf 1 PI - i P2' a1 - i a2' A23 - i A31 , S31 - i S23 amIbm2 + am2bm1 (m = 1,2, ... )

'2 Sll- S22+ 2iS12 f O AO BO aOBO' AObOf 2 S11- S22- 2iS12 amIbm2 - am2bm1 (m = 1,2, ... )

[amIbn2] [an2bm1]b' b (m,n=I,2, ... ;m-n=p)am2 nl anI m2

[amIbnl]b (m,n = 1,2,... ; m+n = p)am2 n2

Let T2 denote the group comprised of the matrices

cosB sin B 0 -cos () -sin () 0

Q(B) == -sinB cosB 0 , R1Q(B) == -sinB cosB 0 (6.6.3)

0 0 1 0 0 1

where (0 ~ B<27r) and R1 == diag(-I, 1, 1). The group T2 defines the

symmetry of a material which possesses rotational symmetry about the

Ip

156 Anisotropic Constitutive Equations and Schur's Lemma [Ch. VI Sect. 6.6] The Transversely Isotropic Groups T] and T2 157

We consider the problem of determining the form of a sym

metric second-order tensor-valued function

10 P3' Sll + S22' S33

r O a3' A12

[PI + i P2] [a1 + i a2] [A23 + i A31] [ S31 + i S23]

1'1 -PI + i P2 ' al - i a2 ' A23 - i A3l ' -831 + i 823

T31 + i T23 = Cs X~(XI + i X2) + c6(Xt + X~)(XI + i X2),

(6.6.9)

T31 - i T23 = Cs X~(XI - i X2) + c6(Xt + X~)(XI - i X2),

belong to 10' 10' 'I' 'I' r 1, r 1, '2' r 2, '3' r 3 respectively. Each ofthe quantities in (6.6.6) which belongs to a representation Ip' say, is

expressible as a linear combination of the quantities in (6.6.8) which

belong to Ip. Thus, we have

(6.6.5)T·· == C·· klJ XkXnX1J 1J ~m ~ m

Table 6.8 Basic Quantities: T2

which is of degree three in the components of a polar (absolute) vector

and which is invariant under the group T]. From Table 6.6, we see

that

TIl +T22, T33, T 31 +iT23, T 31 -iT23,

TII-T22+2iT12' TII-T22-2iT12(6.6.6)

where (:5' (:6' (:7 are the complex conjugates of c5' c6' c7. The

coefficients cs' c6 and c7 are complex constants, e.g., Cs = dS + i eS' and

the third and fifth expressions in (6.6.9) may be omitted. We may, of

course, express (6.6.9)2 and (6.6.9)4 as

are quantities of types 10' 10' '1' r l' '2' r 2 respectively and that

(6.6.7)

are quantities of types 10' '1' r 1 respectively. Upon employing the

product table for T] (Table 6.5) twice, we see that

X~, X3(Xt + X~), X~(XI + i X2), (Xt + X~)(XI + i X2),

X~(XI - i X2), (Xt + X~)(XI - i X2), X3(Xt - X~ + 2 i XlX2),

X3(Xt-X~-2iXlX2)' (Xl +iX2)3, (Xl -iX2)3 (6.6.8)

[~~~] = [:: ~:s][~~~~]+[~6 -::][~~~:~I~~~J(6.6.10)

There are three other transversely isotropic groups which are denoted

by T3 , T4 and T5 (see §8.10). Results for these groups similar to those

given above may be found in Smith and Bao [1988].

VII

GENERATION OF INTEGRITY BASES: THE

CRYSTALLOGRAPHIC GROUPS

7.1 Introduction

The procedures employed in Chapters IV, V and VI enable us to

determine the form of a polynomial constitutive equation T

= F(B, C, ... ) which is invariant under a group A provided that F( ... ) is

of specified degrees nl' n2' ... in B, C, .... In this chapter, we remove

this restriction and consider the problem of generating an integrity

basis for scalar-valued polynomial functions W(B, C, ... ) which are

invariant under a crystallographic group A. We recall that an integrity

basis is formed by polynomial functions 11' 12' ... , each of which is

invariant under A, such that any scalar-valued polynomial function of

the tensors B, C, ... which is invariant under A is expressible as a

polynomial in the elements 11' 12, ... of the integrity basis. Pipkin and

Rivlin [1959] have shown that the problem of determining the general

form of a tensor-valued polynomial function T = F(B, C, ... ) which is

invariant under A may be reduced to that of determining an integrity

basis for scalar-valued functions W(B, C, ... , T). We consequently

concentrate on the generation of integrity bases for scalar-valued

functions invariant under a group A. We give examples in §7.3 and §7.4

of the manner in which we may generate the form of tensor-valued

invariant functions. In this chapter, we consider the 27 crystallographic

groups associated with the triclinic, monoclinic, rhombic, tetragonal

and hexagonal crystal systems. For these cases, we make no

restrictions as to the number or kinds of tensors appearing as arguments

159

160 Generation of Integrity Bases: The Crystallographic Groups [eh. VII Sect. 7.2] Reduction to Standard Form 161

where i == 1, ... ,n1; j == 1, ... ,n2; ... ; k == 1, ... ,nr . The restrictions imposed

on V( ... ) by the requirement of invariance under A are then given by

(7.2.4)

(7.2.3)

(K == 1, ... ,N),

Xl

X2, ... , X ==

X

QZ==

QS Q-1 r 1 · r 2 · . r rK == n1 K + n2 K + ... + nr K

where the <Pi' tPj' ... , Xk are quantities of types r l' r 2' ... , r r respectively.

The transformation properties under A of the quantities A...../.. X. 0/1' o/J' ... , k

are then defIned by the sets of matrices r K1 r 2 rr (K == 1 N), K'···' K ,... ,

respectively. We set

such that

(7.2.1)

7.2 Reduction to Standard Form

We consider the problem of generating an integrity basis for

functions W(B, C .... ) which are invariant under a finite group A

== {AK} == {A1,... ,AN}. Let

of W(B,C, ... ). Thus, we obtain results of complete generality for these

groups. The discussion in this chapter follows closely the work of Kiral

and Smith [1974] and Kiral, Smith and Smith [1980]. The five

remaining crystallographic groups which are associated with the cubic

crystal system are considered in Chapter IX. We note that the

procedures discussed in this chapter have been employed by Kiral and

Eringen [1990] to obtain non-linear constitutive expressions for

magnetic crystals which are subjected to deformation, electric and

magnetic fields.

denote the column vector whose entries are the independent

components of the tensors B, C, .... We set W(B, C, ... ) == W(Z). The

restrictions imposed on the polynomial function W(Z) by the re

quirement that it be invariant under {AK} are given by

where the n x n matrices Sl' ... ' SN form the n-dimensional matrix

representation {SK} of A which defines the transformation properties of

Z under A. The representation {SK} may be decomposed into the

direct sum of the r inequivalent irreducible representations associated

with the group A. We denote these representations by f 1 = {fk}, ... ,f r = {f:k}. Thus, we may determine a non-singular n X n matrix Q

W(Z) == W(SK Z) (K == 1,... ,N) (7.2.2)

where i == 1, ... ,n1; j == 1, ... ,n2; ... ; k == 1,... ,nr . The problem of concern

is to determine the general form of the polynomial function

V(<P1, ... ,<Pn1' tP1,···,tPn2'···' X1,···,Xnr) which is consistent with the re

strictions (7.2.5) for the case where the n1' n2' ... , nr are arbitrary.

Thus, we must determine an integrity basis for functions of n n nl' 2'···' rquantities of types r1, r2 ,... , rr respectively which are invariant under

A. The problems of determining the forms of various scalar-valued

functions W(B, C, ... ), W*(D, E, ... ) which are invariant under A are all

special cases of the problem of determining the form of (7.2.4) which is

consistent with the restrictions (7.2.5). The difference in the problems

arises in that the numbers n1' n2' ... , nr used in (7.2.4) depend upon the

particular case considered. Since we produce the general form of the

162 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 163

Q Al A2 AN Basic Quantities (B.Q.)

r 1 r 1 r 1 r 1 ¢, ¢/, ...1 2 N

r 2 r 2 r 2 r 2 a, b, ...1 2 N

(7.2.7)

rr rr rr rr A,B, ...1 2 N

function V(<PI' ... ' <Pnl' .,pI'···' tPn2' ... , Xl'''·' Xnr) for arbitrary values of

nl' n2'···' nr, the results obtained will be all inclusive.

We proceed by determining the typical elements of the integrity

basis for polynomial functions of the basic quantities <PI'···' <Pnl'

,pI'... ' tPn2' ... of types r l' r 2' ... which are invariant under the

crystallographic group A. Suppose that the typical elements of the

integrity basis of degrees 1,2,3, ... are given by

1. J 1(<PI) , J2(.,pI)' ... ;

2. K1(4>1 ,4>2)' K2{.,pl ,.,p2) , K3{4>1 ,.,pI)' ... ; (7.2.6)

3. L1(4)1,4>2,4>3)' L2(4>1 ,<P2,.,pl) , L3(<PI ,.,pI ,.,p2) , L4(.,pl ,.,p2,.,p3) , ....

These invariants are multilinear in the arguments indicated and are

such that the integrity basis for functions W( <PI ,... ,<Pnl' tPl ,... ,,pn2' ... )depending on the nl+n2 +... quantities 4>1 ,... ,<Pnl' .,pI ,oo.,,pn2' ... is

obtained by substituting in the invariants (7.2.6) the arguments

<PI' ... ' <Pnl for the typical arguments <PI' <P2' <P3' , the arguments

,pI'''.' ,pn2 for the typical arguments ,pI' ,p2' ,p3' in all possible

combinations with repetitions included. Thus, the elements of the

integrity basis of degrees 1,2,3 generated from the typical multilinear

elements of the integrity basis given by (7.2.6) would be given by

1. J1(<Pi) (i=I, ... ,nl)' J2(.,pi) (i=I, ... ,n2)' ... ;

2. KI (q,i' q,j) (i,j=I, ,nl)' K2(,pi' ,pj) (i,j=I, ... ,n2)'

K3(<Pi' tPj) (i= 1, ,n1; j= 1,... ,n2)' ... ;

3. L1(<Pi' <Pj' 4\) (i,j,k=l, ... ,nl)'

L2(q,i' q,j' 'h.) (i,j=I, ,nl; k=I, ,n2)'

L3(q,i' ,pj' ,pk) (i=I, ,nl; j,k=I, ,n2)'

L4(.,pi' 1/Jj , tPk) (i ,j ,k= 1,... ,n2)' ... .

We may determine the quantities 4>1'···' 4>n1' .,pI'···' tPn2' ... whicharise from the tensors B, C, ... by inspection or upon application of the

procedure discussed in Chapter V. For example, r 1 is the identity rep

resentation so that ri< = I (K = I, ... ,N) and the quantities q,i of type

r1 are invariants. The <Pi arising from the tensor B are given by

multiplying (see §5.2) the linearly independent rows of the matrixN TE RK by the column vector [B1,... , Bp] whose entries are the

K-1Inaependent components of B. The RK (K = 1, ... ,N) are the matrices

comprising the p-dimensional representation {RK} which defines the

transformation properties of [B 1,... , Bp]T under A.

7.3 Integrity Bases for the Triclinic, Monoclinic, Rhombic, Tetragonal

and Hexagonal Crystal Classes

In this section, we consider the problem of generating the typical

multilinear elements of an integrity basis for each of the crystal classes

of the triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal

systems. We identify each crystal class by name and also by listing its

Hermann-Maugin and Schoenflies symbols. For each of these crystal

classes, we list a table of the form indicated below.

The letter Q denotes the Schoenflies symbol which identifies the crystal

class. The matrices AI' A2, ... , AN are the elements of the matrix group

164 Generation of Integrity Bases: The Crystal/ographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 165

Theorem 7.2. Let W be a polynomial function of the real and

imaginary parts of the complex quantities al ,... , an' f3 1,... , f3m which

satisfies

Then W is expressible as a polynomial in the quantities

(7.3.2)

(7.3.3)

(7.3.4)

f3j f3k (j,k = l, ... ,m; j::; k).ai (i = 1,... ,n);

which defines the symmetry properties of the crystal class. The

matrices AI' ... ' AN are given in terms ~f th~ mat:ices I, C, R1, ~,

R3, ... defined in §1.3. The matrices r1, r2,...,rNare the matrices

defining the irreducible representation rio The irreducible repre

sentations associated with the crystallographic groups considered in this

chapter are of dimensions one or two. If the representation r is of

dimension one, then the r 1'... ' rN are 1 x 1 matrices whose entries

consist of either a real or a complex number. If the representation r is

of dimension two, the matrices r1,... , rN comprising r are defined in

terms of the matrices E, A, ... , L listed below.

Then W is expressible as a polynomial in the real and imaginary parts

of the quantities

We consider the case indicated in the table at the beginning of

this section where the group A = {AK} is comprised of the matrices

AI'·.·' Ar and where the;e are r i~equiv~ent irreducible representations

r 1 = {rK}, ... , rr = {rK} assocIated wIth A. The results generated

upon application of Theorems 7.1 and 7.2 will generally contain a

number of redundant terms which should be eliminated. In cases where

there is a question regarding whether redundant terms have been

included in the list of typical basic invariants, we may employ the

following systematic procedure. We note that r 1 is the one

dimensional identity representation comprised of matrices rk = 1

(K = 1,... ,N). The quantities cP, cP', ... of type r 1 are invariants. The

typical element of the integrity basis of degree one is given by cP. There

are no invariants of degree one in the quantities of type r. (i = 2, ... ,r).I

In order to determine the typical multilinear basis elements of degree

two, we proceed by generating the invariants which are bilinear in a

quantity of type r i and a quantity of type r j for the (~) cases obtained

E=[~0 ] [-1/2 ~/2l B=[-1/2 -~/2l F=[ -: l,A =-~/2 -1/2 {3/2 -1/2 0

(7.3.1)

[-1/2 ~/2 ] _[1/2 -~/2] _[0 1

l L=[ -: ~lG= ,H- ,K-{3/2 1/2 -{3/2 1/2 1 0

The entries cP, cP', .. · ; a, b, ... ; ... appearing in the rows headed r l' r 2' ...

indicate the notation employed to denote quantities of type r l'

quantities of type r 2, ... which we also refer to as basic quantities. We

also list Basic Quantity tables which give the linear combinations of the

components Pi' ai' Aij and Sij (i,j =1,2,3) of an absolute (polar) vector

p, an axial vector a, a skew-symmetric second-order tensor A and a

symmetric second-order tensor S respectively which form carrier spaces

for the irreducible representations r l' r 2 , ... associated with the various

crystallographic groups.

The integrity bases given in this chapter may be obtained upon

repeated application of the following theorems.

Theorem 7.1. Let W be a polynomial function of the real

quantities al ,... , an' 131,... , 13m which satisfies

ai (i = 1,... ,n); 13j 13k, 13j ,8k (j ,k = I,... ,m; j::; k). (7.3.5)


(7.3.6)

upon setting i,j = 2, ... , r; i:S; j. The number Pij of linearly independent

invariants which are bilinear in a quantity of type f i and a quantity of

type f j is given by

1 N . .Pij = N E tr rk tr r~\.'

K=l

that the degrees of the elements of an integrity basis are not greater

than the order N of the group A . We list the typical multilinear

elements of the integrity bases for the triclinic, ... , hexagonal crystal

classes below. We also give examples of the manner in which these

results may be employed.

B. Q.

b, b' , ...

a, a', ...1

-11

1

I

I

I

C i aI' a2' a3' A23, A31 , A12, S11' S22'

S33' S23' S31' S12

Table 7.1 Irreducible Representations: C i , Cs , C2

Cs P2' P3' aI' A23, S11' S22' S33' S23

C2 PI' aI' A23, SII' S22' S33' S23

Table 7.1A

7.3.2 Pinacoidal Class, C i' IDomatic Class, Cs, m

Sphenoidal Class, C2' 2

7.3.1 Pedial Class, C1, 1

Since materials belonging to this crystal class possess no

symmetry properties, there are no restrictions imposed on the form of

constitutive relations defining the material response.

(7.3.7)

These invariants may be obtained upon application of Theorem 7.1

and/or Theorem 7.2. The typical multilinear basis elements of degree

two may then be obtained upon inspection of these (~) sets ofN . . k

invariants. Similarly we may generate Pijk = ~ ~ tr rk tr rk tr rK

linearly independent invariants which are multili~ea~ in quantities of

types ri, rj and rk for the (rt1) cases where i, j, k = 2, ... ,rj i ~j :::; k.

The typical multilinear basis elements of degree three are obtained

upon inspection of these (rt1) sets of invariants. We may generate

Pijk£ linearly independent invariants which are multilinear in quantities

of types f i, f j , f k and f£ where

1 N . . k £P··k£ = N E trrk trrk trrK trrK ·

1J K=l

There are Qijk£ linearly independent invariants multilinear in quan

tities of types r i, r j , r k and r£ which arise as products of elements of

the integrity basis of degree two. We then determine Pijk£ - Qijk£

invariants which, together with the Qijk£ invariants which are products

of invariants of degree two, form a set of Pijk£ linearly independent

invariants multilinear in the four quantities of types f i, r j , r k and r £.

Inspection of the (r!2) sets of Pijk€ - Qijk£ invariants obtained by

choosing i, j, k, £ so that i, j, k, £ = 2, ... , r; i:S; j :s; k :s; £ will then yield

the typical multilinear basis elements of degree four. We proceed in

this fashion to determine the typical multilinear elements of the

integrity basis of degrees 2, 3, 4, 5, .... It is necessary for each crystal

class considered to determine when this iterative procedure may be

terminated. For example, we may employ Theorem 3.4B which says


Application of Theorem 7.1 immediately yields the result that

the typical multilinear elements of the integrity basis for the groups Ci,

Cs , C2 are given by

b, b' in b b' by all possible combinations of two quantities from (7.3.11)2

with repetitions allowed gives the set of invariants

1. a;

2. bb'.(7.3.8)

p p )x2' x3 (p == 1, , n ,

xl xi. (p,q = 1, , llj P ~ q)(7.3.12)

We consider the problem of determining an integrity basis for

functions W(x1' ... 'xn) of n polar vectors x1, ... ,xn which are invariant

under the group C i . With Tables 7.1 and 7.1A, we make the

identification

The typical multilinear element of the integrity basis involving

quantities b, b~, ... is seen from (7.3.8) to be given by b b'. Replacing

b, b' by all possible combinations of two quantities chosen from the list

(7.3.9) with repetitions allowed will yield

(7.3.14)

(7.3.13)

a1 a2S31 + a3S12 a4S31 + a5S12

T == a2S31 + a3S12 a6 a7

a4S31 + a5S12 a7 a8

which forms an integrity basis for functions W (xl ,... , xn) invariant

under C s .

We next generate the canonical form of a second-order

symmetric tensor-valued function T(S) of a single second-order

symmetric tensor S == [Sij] which is invariant under the group Cs.

With Table 7.1A, we see that Sll' S22' 833, 823 are quantities of type

r1 (i.e., invariants) and that 831 , 812 are quantities of type r2. With

(7.3.8), we then see that an integrity basis for functions W(S) which are

invariant under Cs is formed by the invariants

A function V(S) of type r 2 is readily seen to be expressible in the form

V(S) == a S31 +b S12 where a and b are polynomial functions of the

11,... ,17. Since TIl' T22, T33, T23 and T31 , T 12 are quantities of types

r 1 and r 2 respectively, we see that the general expression for a sym

metric second-order tensor-valued function T(S) which is invariant

under Cs is given by

(7.3.9)

(7.3.11)

(7.3.10)

, _ lIn n.a, a , - x2' x3 ,... , x2' x3 '

h, h', = xl,· ..,xr .

x f x f 2 2 xP x!1 (i J. == 1 2 3· i _< J.),1 J' xi Xj , ... , 1 J ' " ,

xP X~ (i,j == 1,2,3; p,q == 1,... , n; p<q).1 J

This forms an integrity basis for functions W(x1' ... 'xn) of the n polar

vectors xl' ... ' X n which are invariant under the group Ci .

In order to generate an integrity basis for functions W(x1 ,... , xn)

of n polar vectors which are invariant under Cs , we employ Tables 7.1

and 7.1A and make the identifications

With (7.3.8), the typical multilinear elements of the integrity basis are

given by a and b b'. Replacing a by each of the entries in (7.3.11)1 and

where the ai == ai (11,... ,17) are polynomials In the invariants 11,... ,17defined by (7.3.13).

r1 r2 r3 r4 r6 r7 rS

D 2h 511 , 522, 533 aI' A23, 523 a2' A31, 531 a3' A12, 512 PI P2 P3

r1 r2 r3 r4

C2h aI' A23, 511 , PI P2' P3 a2' a3' A31, A12,

522, 533, 523 512, 531

C2v PI' 511, 522, 533 aI' A23, 523 P2' a3' A12, 512 P3' a2' A31, 531

D2 511 , 522, 533 PI' aI' A23, 523 P2' a2' A31, 531 P3' a3' A12, 512

Table 7.3A Basic Quantities: D2h

Application of Theorem 7.1 twice will yield the result that the

typical multilinear elements of an integrity basis for the groups C2h'

C2v' D 2 are

Repeated application of Theorem 7.1 yields the result that the

typical multilinear elements of an integrity basis for D2h are

1. a;

2. bb', ee', dd', AA', BB', CC', DD';1. a;

2. b b', cc', dd';

3. bed.

(7.3.15)

(7.3.16)3. bcd, bAB, bCD, cAC, eBD, dAD, dBC;

4. beBC, beAD, bdBD, bdAC, edCD, edAB, ABCD.

172 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , IIexagonal Crystal Classes 173

the transformations in Table 7.4. We see from Table 7.4A that, for the

group S4' the quantities P3' PI - i P2' PI + i P2 belong to the

representations f 2' f 3' f 4 respectively. Thus, the determination of an

integrity basis for functions of n polar vectors which are invariant under

S4 essentially gives the general result. This integrity basis has been

obtained by Smith and Rivlin [1964]. Inspection of this result shows

that the typical multilinear elements of the integrity basis for S4 and

C4 are

7.3.5 Tetragonal-disphenoidal Class, S4' 4Tetragonal-pyramidal Class, C4' 4

Table 7.4 Irreducible Representations: S4' C4

S4 I D3 D1T3 D2T3 B. Q.

C4 I D3 R1T3 It.2T3

f 1 1 1 1 1 ¢, ¢', ...

f 2 1 1 -1 -1 'ljJ, 'ljJ', ...

f 3 1 -1 -1 a, b, ...

f 4 1 -1 -1 a, b, ...

1. ¢;

2. 'ljJ 'ljJ', a b + a b, a b - a b ;

3. 'ljJab + 'ljJab, 'ljJab-'ljJab;

4. abcd+abcd, abed-abed.

(7.3.17)

Table 7.4A 'Basic Quantities: S4' C4

f 1 f 2 f 3 f 4

S4 a3 P3 PI - i P2' a1 + i a2 PI+ i P2' a1 - i a2

A12, S33' Sll+ S22 S12' Sll- S22 A23+ iA31 A23- iA31

S31+ iS23 S31- iS23

C4 P3' a3 P1+iP2' a1+ ia2 P1-iP2' a1- ia2

A12, 533, 511+522 512, 511- 522 A23+ iA31 A23- iA31

531+ i 523 531 - i 523

In Table 7.4, the quantities ¢, ¢', , 'ljJ, 'ljJ', ... are real quantities.

The quantities a == al + i a2' b == b1 + i b2, are complex quantities and

a, b, ... denote the complex conjugates of a, b, ... respectively.

Since quantities of type f 1 are invariants, the general problem is

the determination of an integrity basis for functions of arbitrary

numbers of quantities of types f 2' f 3 and f 4 which are invariant under

7.3.6 Tetragonal-dipyramidal Class, C4h,4/m

Table 7.S Irreducible Representations: C4h

C4h I D3 R1T3 It.2T3 C R3 D1T3 D2T 3 B. Q.

f 1 1 1 1 1 1 1 1 1 ¢, ¢', ...

f 2 1 1 -1 -1 1 1 -1 -1 'ljJ, 'ljJ', ...

f 3 1 -1 -1 1 -1 -1 a, b, ...

f 4 1 -1 -1 1 -1 -1 a, b, ...

f s 1 1 1 1 -1 -1 -1 -1 ~, ~', ...

f 6 1 1 -1 -1 -1 -1 1 1 7], 7]', ...

f 7 1 -1 -1 -1 1 -1 A, B, ...

f S 1 -1 -1 -1 1 -1 A, B, ...


In Table 7.5, the quantities ¢,¢', ... , 'l/J,'l/J', ... , ~,~', ... , "l,"l', ... are

real quantities. The quantities a == al + i a2' b == b1 + i b2, ,

A == Al + i A2, B == B1 + i B2, ... are complex quantities and a, b, ,A, B, ... denote the complex conjugates of a, b, ... , A, B, ... respectively.

It is found upon repeated application of Theorems 7.1 and 7.2 that the

typical multilinear elements of the integrity basis for C4h are

r1 r2 r3 r4

C4h a3 al+ i a2 al- ia2

A12, S33' SII+ S22 S12' SII- S22 A23+ iA31 A23- iA31

S31+ i S23 S31- i S23

r5 r6 r7 rS

C4h P3 PI+ i P2 Pl- i P2

Table 7.6 Irreducible Representations: D4' C4v' D2d

7.3.7 Tetragonal-trapezohedral Class, D4, 422

Ditetragonal-pyramidal Class, C4v' 4mm

Tetragonal-scalenohedral Class, D2d, 42m

Basic Quantities: D4' C4v' D 2dTable 7.6A

D4 I Dl D2 D3 CT3 RIT 3 ~T3 R3T 3C4v I ~ R l D3 D3T 3 RIT 3 ~T3 T 3 B. Q.

D 2d I Dl D2 D3 D3T 3 D2T3 DIT 3 T 3

r l 1 1 1 1 1 1 1 1 ¢,¢', ...r 2 1 -1 -1 1 -1 1 1 -1 1/;, 1/;', ...r 3 1 -1 -1 1 1 -1 -1 1 v,v', ...r 4 1 1 1 1 -1 -1 -1 -1 r,r', ...

r 5 E F -F -E -K -L L K [:~l[~~l· ..

Basic Quantities: C4hTable 7.5A

1. ¢;- - , , ,

2. ab, AB, 'l/J'l/J , ~~ , TJ"l ;

3. 'l/Jab, 'l/JAB, ~aA, "laA, 'l/J~TJ; (7.3.18)

4. abed, abAB, abAB, ABCD, 'l/J~aA, 'l/JTJaA, ~TJab, ~"lAB ;

5. ~aABC, ~Aabc, TJaABC, TJAabc.

The presence of the complex invariants a b, ... , TJ A abc In (7.3.18)

indicates that both the real and imaginary parts of a b, ... , TJ A abc (i.e.,

a b ± a b, ... , "l A abc ± TJ A a bc) are typical multilinear elements of the

integrity basis.

r 1 r 2 r 3 r4 r 5

D4 P3' a3

S33' SII+ S22 A12 S12 SII - S22 [PI] [al ] [An ] [823 ]P2 ' a2 ' A31 ' -S31

C4v P3 a3

S33' SII+ S22 A12 S12 SII - S22 [PI] [a2 ] [A31 ] [831 ]P2 ' -al ' -A23 ' S23

D2d a3 P3

S33' SII+ S22 A12 S12 SII - S22 [PI] [al ] [A23 ] [823 ]P2 ' -a2 ' -A31 ' S31

The matrices E, F, K, L appearing in Table 7.6 are defined by

(7.3.1). Repeated application of Theorem 7.1 yields the result that the


typical multilinear elements of an integrity basis for D4' C4v and D2d

are

1. <P;Table 7.7 Irreducible Representations: D4h (Continued)

2. alb1 +a2b2, 1/; 1/;', v v', 7 7'; D4h C R1 ~ ~ T3 D1T3 D2T3 D3T3 B. Q.

3. 1/;(alb2 - a2bl)' v(alb2 +a2bl)' 7(alb1 - a2b2)' 1/; v 7;(7.3.19) f 1 1 1 1 1 <P, <p', ...

4. alb1cld1 +a2 b2 c2 d2, 1/; v(alb1 - a2 b2),f 2 1 -1 -1 1 -1 1 -1 1/;, 1/;', ...

1/; 7(alb2 +a2 b1), 1/ 7(alb2 - a2bl) ; f 3 -1 -1 1 1 -1 -1 1 v,v', ...

5. 1/;(alb1cld2 +alb1d1c2 +alcld1b2 +b1cld1a2 f 4 1 1 1 -1 -1 -1 -1 7,7', ...

-a2b2c2dl-a2b2d2cl-a2c2d2bl-b2c2d2al)' f 5 E F -F -E -K -L L K [:~l[~~l ..·f 6 -1 -1 -1 -1 -1 -1 -1 -1 e,e', ...r7 -1 1 -1 -1 -1 'f/, 'f/', ...

7.3.8 Ditetragonal-dipyramidal Class, D4h, 4/mmm rg -1 -1 -1 -1 (}, (}', ...rg -1 -1 -1 -1 ",', ...

Table 7.7 Irreducible Representations: D4h flO -E -F F E K L -L -K [~~l[:~l ..·D4h I D1 D2 D3 CT3 R1T3 ~T3 R3T3 B. Q.

The matrices E, F, K, L appearing In Table 7.7 are defined by

(7.3.1). Repeated application of Theorem 7.1 yields the result that the

typical multilinear elements of the integrity basis for D4h are given by

f 1 f 2 f 3 f 4 f 5 f 7 flO

D4h a3[a1] [A23] [523 ] [~;]

533, 511+ 522 A12 512 511 - 522a2 ' A31 ' -531

P3

f 1 1 1 1 1 1 <p, <p', ...

r2 1 -1 -1 1 -1 1 1 -1 1/;, 1/;', ...

f 3 1 -1 -1 -1 -1 1 v,v', ...

r4-1 -1 -1 -1 7,7', ...

f 5 E F -F -E -K -L L K [:~l[~~l· ..f 6 1 1 1 e, e',· ..r7 1 -1 -1 -1 1 -1 'f/, 'f/', ...

f g 1 -1 -1 1 -1 -1 1 (}, (}', ...

f g 1 1 1 -1 -1 -1 -1 ",', ...

flO E F -F -E -K -L L K [~~l[:;l ..·(Continued on next page)

Table 7.7A Basic Quantities: D4h


(7.3.20)

1. <P;

2. alb l + a2b2' AlBl + A2B2, 'ljJ'ljJ', vv', TT', ~~', 17r/, f)f)', ,,';

3. 'ljJ(alb2 - a2bl)' 'ljJ(Al B2 - A2Bl ), v(alb2 + a2bl)' v(Al B2 + A2Bl ),

T(alb l -a2b2)' T(AlB l -A2B2), ~(alAl +a2A2)' 17(al A2 - a2Al)'

f)(al A2 +a2Al)' ,(al Al - a2A2)' 'ljJvT, 'ljJf)" 'ljJ~17, v~f), V17" T~" T17f);

4. alb l cld l + a2b2c2d2' AlBl Cl Dl + A2B2C2D2,

(alb2 + a2bl)(AIB2 + A2Bl ), (alb2 - a2bl)(Al B2 - A2BI ),

(alb l -a2b2)(Al Bl -A2B2),

('ljJv, ~" 17f))(alb l - a2b2)' ('ljJT, ~f), 1J,)(alb2 + a2bl)'

(VT, ~17, f),)(a l b2 - a2bl)' ('ljJv, ~" 17f))(Al Bl -A2B2),

('ljJT, ~f), 17,)(Al B2 + A2BI ), (VT, ~17, f),)(A l B2 - A2BI ),

('ljJ1J, vf), T,)(aIAI +a2A2)' ('ljJf), V17, T~)(aIAI-a2A2)'

('ljJ" v~, T17)(aI A2 + a2AI)' ('ljJ~, v" Tf))(aI A2 - a2AI)'

'ljJv~" 'ljJV17f) , 'ljJT~f), 'ljJT17" VT~17, VTf)" ~17f),;

5. 'ljJ(alb l cl d2 + alb l d l c2 + alcidl b2 + b i cld l a2

- a2b2c2dI - a2b2d2cl - a2c2d2bl - b2c2d2al)'

'ljJ(AI BI CI D2 + AlBI DI C2 + AlCI DI B2 + BI Cl DI A2

- A2B2C2Dl - A2B2D2CI - A2C2D2BI - B2C2D2AI ),

'ljJ(aIb2+a2bl)(AIBI-A2B2)' 'ljJ(aIbl-a2b2)(AIB2+A2BI)'

v(alb2-a2bl)(AIBI-A2B2)' v(albl-a2b2)(AIB2-A2BI)'

T(alb2 - a2bl)(AI B2 + A2BI ), T(alb2 + a2bl)(AI B2 - A2BI ),

~(alb l ciAl + a2b2c2A2)' ~(AIBI CIal + A2B2C2a2)'

17(alb l cl A2 - a2b2c2AI)' 17(AIBI CI a2 - A2B2C2al)'

f)(al b i ciA2 + a2b2c2AI)' f)(A IBI CI a2 + A2B2C2al)'

,(alblcIAI -a2b2c2A2)' ,(AIBIClal -A2B2C2a2)'

('ljJ~f), 'ljJ17" v~17, vf),) (albl - a2b2),

(alb2 + a2bl)('ljJ~" 'ljJ17f), T~TJ, Tf),),

(v~" v17f), T~f), T17,) (alb2 - a2bl)'

(AIBI - A2B2)( 1/J~O, 'ljJ17" v~TJ, vO,),

('ljJ~" 'ljJ1JO, T~17, TO,)(AI B2 +A2BI ),

(AI B2 - A2BI)(v~" v1JO, T~O, T1J,),

('ljJv" 'ljJTO, VTTJ, "10,) (alAI +a2A2)'

(alAl - a2A2)( 'ljJ17T, 'ljJv~, VTO, ~170),

('ljJVTJ, 'ljJT~, VT" ~17,)(alA2 + a2AI)'

(aI A2 - a2AI)('ljJvO, 'ljJT" VT~, ~O,);

6. ~77(al hI - ~b2)(AIB2 + A2BI ), B')'(alhI - ~h2)(AIB2 + A2BI ),

'ljJ~(al bi ciA2 - a2b2c2AI)' 'ljJ~(AlBI CI a2 - A2B2C2al)'

1/J17(al b l cI AI +a2b2c2A2)' 'ljJTJ(AIBIClal +A2B2C2a2)'

1/JO(albl ci Al - a2b2c2A2)' 1/JO(AI BI CI al - A2B2C2a2)'

TP')'(alhIcIA2 + ~b2c2AI)' TP')'(AIBICI~ +A2B2C2al)'

(~1J, B')')(alhIcId2 + alhIdl c2 +alcIdl b2 + hIcIdl a2

- a2b2c2dl - a2b2d2cI - a2c2d2bl - b2c2d2al)'

(~77, B')')(AIBICID2 + AlBIDIC2 + Al CIDIB2 + BI CIDIA2

- A2B2C2DI - A2B2D2CI - A2C2D2BI - B2C2D

2A

I),

(alb2 - a2bl)(AIBICID2 + AlBIDIC2 + AlCIDIB2 + BICIDIA2

- A2B2C2DI - A2B2D2CI - A2C2D2BI - B2C2D

2A

I),

(AIB2 - A2BI )(alhIcId2 + alhIdl c2 + alcIdl h2 + hIcIdl a2

- a2b2c2dI - a2b2d2cI - a2c2d2bI - b2c2d2al)·


Table 7.8 Irreducible Representations: C3

7.3.9 Trigonal-pyramidal Class, C3' 3

We observe that an argument essentially identical to that employed in

Theorem 3.3 to obtain (3.2.11) will also yield the result (7.3.21).

r1 r2 r3

C3 P3' a3 PI - i P2' a1 - i a2 PI + i P2' al + i a2

A12, S33' Sll+ S22 A23 -iA31 , S31- iS23 A23 +iA31 , S3I +iS23

SII- S22+ 2iS12 Sll- S22- 2iS12

Basic Quantities: C3v' D 3

7.3.10 Ditrigonal-pyramidal Class, C3v' 3m

Trigonal-trapezohedral Class, D3, 32

Table 7.9 Irreducible Representations: C3v' D 3

C3v I SI S2 R1 R1S1 R1S2D3 I Sl S2 D1 D1SI D1S2 B. Q.

r1 1 1 1 1 1 1 <p, <p', ...

r2 1 1 1 -1 -1 -1 1jJ, 1jJ', ...

r3 E A B -F -G -H [:;l[~;l ..·Table 7.9A

The matrices E, A, ... , H appearing in Table 7.9 are defined by

(7.3.1). We see from Table 7.9A that the transformation properties of

P3 and [P2' - Pl]T under the group D3 are defined by the matrices

comprising the representations r 2 and r 3 respectively. Since quantities

of type r1 are invariants, we see that knowledge of an integrity basis

for functions of n polar vectors PI' ... ' Pn which are invariant under D3will suffice to enable us to determine the result required. This integrity

basis has been generated by Smith and Rivlin [1964]. With the aid of

this result, we readily see that the typical multilinear elements of an

r1 r2 r3

C3v P3 a3[ PI ] [ a2 ] [ A31 ] [831 ] [ 2 812 ]

S33' Sll+ S22 A12 P2 '-a1 ' -A23 ' S23 ' S11 - S22

D3 P3' a3[ P2 ] [ a2 ] [ A31 ] [831 ] [ 2812 ]

S33' Sll+ S22 A12 -PI ' -a1 ' -A23 ' S23 ' Sll - S22

(7.3.21 )

B. Q.I

1. <p;

2. ab+ab, ab-ab;

3. abc+abc, abc-abc.

In Table 7.8, w==-1/2+i~/2 and w2==-1/2-i~/2. We

note that w3 == 1. The quantities a == al + i a2' b == b1 + i b2,... are

complex quantities and a, b, ... denote the complex conjugates of a, b, ...

respectively. We see from Table 7.8A that P3' PI - i P2' PI + i P2 are

quantities of types rl' r2' r3 respectively. Thus, generation of an

integrity basis for functions of the n polar vectors Pl, ... ,Pn which are

invariant under C3 will yield the desired result. This integrity basis

has been obtained by Smith and Rivlin [1964]. Upon inspection of the

result given in this paper, we see that the typical multilinear elements

of an integrity basis for C3 are given by

r1 1 1 <p, <P', ...

r2 1 w w2 a, b, ...

r3 1 w2 w a, b, ...

Table 7.8A Basic Quantities: C3

182 .Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 183

integrity basis for C3v and D3 are given by Table 7.10A Basic Quantities: C3i , C3h , C6 (Continued)

1. ¢;

2. alb1 +a2b2' 1/; 1/;' ;

3. a2b2c2 - alb1c2 - b1cla2 - clalb2, 1/;(alb2 - a2b l) ;

4. 1/;(alblcl-a2b2cl-b2c2al-c2a2bl).

7.3.11 Rhombohedral Class, C3i' 3Trigonal-dipyramidal Class, C3h' ()

Hexagonal-pyramidal Class, C6' 6

(7.3.22)

r1 r2 r3 r4 rS r6

C3h a3 PI - i P2 PI + i P2 P3 al- i a2 al + ia2

A12, 533 A23-iA31 A23+iA31511+ 522 511 - 522+ 2i512 511 - 522- 2i512 531-i523 531+ i 523

C6 P3 PI - iP2 PI + iP2

a3 al- i a2 al + ia2

A12,533 A23-iA31 A23+iA31511 + 522 511- 522+ 2i512 511 - 522- 2i512 531-i523 S31+ i523

In Table 7.10, w == -1/2 + i~/2 and w2 == -1/2 - i ~/2. The

quantities ¢J and ~ are real. The quantities a == al + i a2' b == b1 ++ i b2, ... , A == Al + i A2, B == B1 + i B2, ... are complex quantities and

a, b, ... ,A, 13, ... denote the complex conjugates of a, b, ... ,A, B, ...

respectively. Let W be a polynomial function of the quantities

¢, ... ,a, a, b, b, ... ,~, ... ,A, A, B, 13, ... which is invariant under the first

three transformations of Table 7.10. It is seen from the results (7.3.21)

for the group C3 that W is expressible as a polynomial in the quantities

obtained from the typical multilinear quantities

Table 7.10 Irreducible Representations: C3i , C3h , C6

C3i I Sl S2 C CS1 CS2

C3h I Sl S2 R3 ~Sl R3S2 B. Q.

C6 I Sl S2 D3 D3S1 D3S2

r1 1 1 1 1 1 1 ¢J, ¢J', ...

r2 1 w w2 1 w w2 a, b, ...

r3 1 w2 w 1 w2 w a, b, ...r4 1 1 1 -1 -1 -1 ~,~', ...r5 1 w w2 -1 -w _w2 A, B, ...

r6 w2 _w2 - -1 w -1 -w A, B, ...

and

¢, ab, abc, A13, aAB (7.3.23)

Table 7.10A Basic Quantities: C3i' C3h' C6 ~, aA, abA, ABC. (7.3.24)

r1 r2 r3 r4 rS r6

C3i a3 a1 - i a2 al + ia2 P3 Pl- i P2 PI + i P2

A12 A23 -iA31 A23 + iA31533 531 -i523 531 + i 523

511+ 522 Sll- 522+ 2iS12 Sll- 522- 2i512

The quantities (7.3.23) remain invariant and the quantities (7.3.24) all

change sign under any of the last three transformations of Table 7.10.

With Theorem 7.2, we then see that the typical multilinear elements of

an integrity basis for C3i' C3h and C6 are

184 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII

With (7.3.22) and (7.3.26), we see that the typical multilinear elements

of an integrity basis for polynomial functions of <p, <p', ... , 'l/J, 'l/J', ... , ~, ~', ... ,7],7]', ... ,A1,A2,B1,B2, ... , a1,a2,b1,b2, ... which are invariant under the

A == Al + i A2, A == Al - i A2, B == BI + i B2, 13 == BI - i B2, ... ,

_ (7.3.26)a == al + i a2' a == al - i a2' b == b l + i b2, b == b l - i b2, ....

7.3.12 Ditrigonal- dipyramidal Class, D3h, 6m2

Hexagonal- scalenohedral Class, D 3d, 3m

Hexagonal- trapezohedral Class, D6, 622

Dihexagonal- pyramidal Class, C6v' 6mm

Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 185

Table 7.11 Irreducible Representations: D 3h, D 3d , D6, C6v

D3h I SI S2 R3 R3S1 R3S2

D3d I SI S2 C CS1 CS2

D6 I SI S2 D3 D3S1 D3S2 B. Q.

C6v I SI S2 D3 D3S1 D3S2

r1 <P, <p', ...r2 1 1 1 'l/J,'l/J', ...r3 1 1 -1 -1 -1 ~,~', ...r4 1 1 1 -1 -1 -1 7], 7]', ...

r5 E A B -E -A --B [1;}[:;} ..·r6 E A B E A B [:;}[~;} ...

D3h R1 R1S1 R1S2 D2 D2S1 D2S2

D3d D1 D1S1 D1S2 R1 R1S1 R1S2

D6 D1 D1S1 D1S2 D2 D2S1 D2S2 B. Q.

C6v ~ ~SI ~S2 R1 R1S1 R1S2

r1 1 1 1 1 1 1 <p, <p', ...r2 -1 -1 -1 -1 -1 -1 'l/J, 'l/J', ...r3 1 1 1 -1 -1 -1 ~,~', ...

r4 -1 -1 -1 1 1 1 7], 7]', ...

r5 F G H -F -G -H [1;]'[:;]'·..r6 -F -G -H -F -G -H [:;}[~;} ...

(7.3.25)

1. <P;

2. a b, A 13, ~~';

3. abc, aAB, ~aA;

4. abA13, ~abA, ~ABC;

5. aABCD;

6. ABCDEF.


(7.3.1). We observe that the quantities <p, 'l/J, ~, 7], [AI' A2]T, [aI' a2]T

associated with Table 7.11 transform under transformations 1, 2, 3, 10,

11, 12 of Table 7.11 in the same manner as do the quantities <p, 'l/J, 'l/J, <p,

[aI' a2]T, [aI' a2]T under the transformations of Table 7.9 associated

with the crystal classes C3v and D3. Let us employ the notation

The presence of the complex invariants a b, ... ,ABC D E F in (7.3.25)

indicates that both the real a.nd imaginary parts a b ± a b, ... ,

ABC D E F ± ABC DEF of these invariants are typical multilinear

elements of the integrity basis.


Table 7.11A and

1. ¢;

We note that the quantities A, A, ... ,F, F, a, a, ... , c, c appearing In

(7.3.29) are defined by (7.3.26).

The quantities (7.3.27) remain invariant while the quantities (7.3.28)

change sign under all of the remaining transformations of Table 7.11.

Application of Theorem 7.1 then yields the result, after elimination of

redundant terms, that the typical multilinear elements of an integrity

basis for D3h ,D3d, D6 and C6v are

2. ah+ab, AB+AB, l/Jl/J', ~~', 1]1]';

3. abc-ahc, aAB-aAB, l/J(ah-ab), l/J(AB-AB), e(aA-aA),

1](aA + aA), l/J e 1] ;

4. abAB+ahAB, (ah-ab)(AB-AB), l/J(abc+ahc),

~(aAB+ aAB), e(a bA + ahA), e(ABC + ABC),

1](abA-ahA), 1](ABC-ABC), l/Je(aA+aA), (7.3.29)

~1](aA-aA), e1](ah-ab), e1](AB-AB);

5. (abc+ahc)(AB-AB), aABCD-aABCD, l/J(abAB-ahAB),

l/Je(abA-ahA), l/Je(ABC-ABC), l/J1](ABC+ABC),

l/J1](a bA + ahA), ~ 1](a be + ahc), ~ 1](aAB + aAB);

6. ABCDEF + ABCDEF, l/J(aABCD + aABCD);

7. l/J(ABCDEF - ABCDEF).

(7.3.28)

1], l/J~, aA+aA, ~(ab-ab), l/J(aA-aA), ~(AB-AB),

abA-ahA, ABC-ABC, e(abc+ahc),

l/J(a bA + ahA), ~(aAB + aAB), l/J(ABC + ABC).

¢, l/Jl/J', ee', AB+AB, ab+ab, l/J(AB-AB), l/J(ab-ab),

e(aA-aA), abc-abc, aAB-aAB, (7.3.27)

l/J(abc+abc), l/J(aAB+aAB), ~(abA+abA),~(ABC+ABC)

group of transformations 1, 2, 3, 10, 11, 12 of Table 7.11 are

r1 r2 r3 r4 r5 r6

D3h a3 P3 [a1].[A

23] [:~]A12

a2 A31

S33

[523J [ 2512 ]

S11+ S22 -S31 S11- S22

D3d a3 P3

[:~] [a2J[A31lA12

-a1 ' -A23

S33[ 531 ] [ 25

12]

S11 +S22 S23 ' S11- S22

D6 P3' a3 [:~].[:~]S33

[A23

] [523J [ 2512 ]A12 A31 '-531 S11- S22S11+ 522

C6v P3 a3 [:~l~~JS33

[

A31l[531

][ 2512 ]A12 -A23 , S23 S11- S22S11+ S22


ABa, ABX, ABx, XYa, XYA, XYZ, XYx,(7.3.30)

xya, xyA, xyX, xyz, aAX, aAx, aXx, AXx.

f 7 f 11 f 12

C6h P3 PI - i P2 PI + i P2

Basic Quantities: C6hTable 7.12A

¢, ~, 7r, 8, ab, aA, aX, ax, AB, AX, Ax,

XV, XX, xy, abc, abA, abX, abx, ABC,

Under any of the remaining nine transformations of Table 7.12, the

quantities (7.3.30) either remain invariant or change sign. Repeated

application of Theorem 7.2 will then yield, upon elimination of re

dundant terms, the result that the typical multilinear elements of an

f 1 f 2 f 3 f S f 6

C6h a3 a1- i a2 al + i a2

A12, S33 A23- iA31 A23+iA31

S11+ S22 S11- S22+2 i S12 S11- S22-2 i S12 S31- i S23 S31+ i S23

In Table 7.12, w == -1/2 + i~/2 and w2 == -1/2 - i ~/2. The

quantities ¢, ~, 1r, 8 are real quantities. The quantities a == a1 + i a2'

A == Al + i A2, X == Xl + i X2, x == Xl + i x2 are complex and a, A, X, x

denote the complex conjugates of a, A, X, x respectively. The

quantities ¢, ~, 1r, 8 and a, A, X, x and a, A, X, x transform under the

first three transformations of Table 7.12 in the same manner as do the

quantities ¢ and a and a associated with Table 7.8 (crystal class C3)under the transformations of Table 7.8. We see from (7.3.21) that the

typical multilinear elements of the integrity basis for polynomial

functions of the quantities ¢, ~, 1r, 8, a, A, X, x, a, A, X, x which are

invariant under the first three transformations of Table 7.12 are

C6h I SI S2 D3 D3S1 D3S2 B. Q.

f 1 1 1 1 1 1 1 ¢,q/, ...

f 2 1 w w2 1 w w2 a; b, ...

f 3 1 w2 w 1 w2 w a, b, ...

f 4 1 1 1 -1 -1 -1 ~, ~', ...f 5 1 w w2 -1 -w _w2 A, B, ...

f 6 1 w2 w -1 _w2 -w A,]3, ...

f 7 1 1 1 1 1 1 7r, 7r', ...

f 8 1 w w2 1 w w2 X, Y, ...

f g 1 w2 w 1 w2 w X, Y, ...

flO 1 1 1 -1 -1 -1 8,8', ...

f 11 1 w w2 -1 -w _w2 x, y, ...

f 12 1 w2 w -1 _w2 -w x, y, ...

C6h C CSI CS2 R3 R3S1 R3S2 B. Q.

f 1 1 1 1 1 1 1 ¢,¢', ...

f 2 1 w w2 1 w w2 a, b, ...

f 3 1 w2 w 1 w2 w a, b, ...

f 4 1 1 1 -1 -1 -1 ~, ~', ...

r5 1 w w2 -1 -w _w2 A, B, ...

f 6 1 w2 w -1 _w2 -w A,]3, ...

f 7 -1 -1 -1 -1 -1 -1 7r, 7r', ...

r8 -1 -w _w2 -1 -w _w2 X, Y, ...

f 9_w2 _w2 - -

-1 -w -1 -w X, Y, ...

flO -1 -1 -1 1 1 1 8,8', ...

f ll -1 -w _w2 1 w w2 x, y, ...

f 12 -1 _w2 1 w2 - -

-w w x, y, ...

Table 7.12 Irreducible Representations: C6h

7.3.13 Hexagonal-dipyramidal Class, C6h' 6/m


integrity basis for C6h are

1. ¢;

2. ab, AB, XV, xy, 1r1r', ~~', 88';

3. abc, ABa, XYa, xya, AXx, ~aA, ~xX, 8ax, 8AX, 1raX, 1rAx, 1r~8;

4. abAB, abXV, abxy, ABXV, ABxy, XYxy, aAXx, aAXx, aAXx,

1rabX, 1rABX, 1rXYZ, 1rxyX, 1raAx,

1rABCDEx, 1rABCxyz, 1rAxyzuv,

~XYZUVx, ~XYZxyz, ~Xxyzuv,

8ABCDEX, 8ABCXYZ, 8AXYZUV.

The presence of the complex invariants ab, AB, ... , 8AXYZUV in

(7.3.31) indicates that both the real and imaginary parts of these

invariants are typical multilinear elements of the integrity basis.

7.3.14 Dihexagonal-dipyramidal Class, D6h, 6/mmm


(7.3.1). We shall restrict consideration to functions of quantities of

types f 1, f 2, f 5, f 6, f 8 and f 11 only. We see from Table 7.13A that

this will furnish an integrity basis for functions of polar vectors, axial

vectors, skew-symmetric second-order tensors and symmetric second

order tensors. The general result yielding an integrity basis for

functions of quantities of all twelve types f 1' ... f 12 is given by Kiral,

Smith and Smith [1980].1 The number of basis elements in the general

case is very large leading us to present only partial results here. We

shall employ the notation

~abA, ~ABC, ~XYA, ~xyA, ~aXx,

8abx, 8ABx, 8XYx, 8xyz, 8aAX,

1r~ax, 1r~AX, 1r8aA, 1r8xX, ~8aX, ~8Ax;

5. aABCD, aABXY, aABxy, aXYZU, aXYxy, axyzu,

ABCXx, ABCXx, XYZAx, XYZAx, abxAX,

abxAX, abxAX, xyzAX, xyzAX,

1rabAx, 1rABaX, 1rxyaX, 1rXYAx,

~abXx, ~ABXx, ~XYAa, ~xyAa,

8abAX, 8ABax, 8XYax, 8xyAX,

1r~abx, 1r~ABx, 1r~XYx, 1r~xyz, 1r~aAX,

1r8abA, 1r8ABC, 1r8XYA, 1r8xyA, 1r8aXx,

(7.3.31)

a == al + i a2'

... ,

a == al - 1 a2' ... ; (7.3.32)

~8abX, ~8ABX, ~8XYZ, ~8xyX, ~8aAx;

6. ABCDEF, ABCDXY, ABXYZU, XYZUVW, ABCDxy, ABxyzu,

xyzuvw, XYZUxy, XYxyzu, aAxXVZ, aAXxyz, aXxABC,

1raABCx, 1raxyzA, 1rABCDX, 1rABxyX, 1rxyzuX,

~aXYZx, ~axyzX, ~XYZUA, ~XYxyA, ~xyzuA,

8aABCX, 8aXYZA, 8XYZUx, 8XYABx, 8ABCDx;

7. ABCDEXx, XYZUVAx, xyzuvAX,

We also use the notation R(ABXY... ) and I(ABXY... ) to denote the

real and imaginary parts of ABXY.... Expressions such as I(AB, XV,

ab) denote the set of quantities I(AB), I(XV), I(ab). Expressions such

as ~I(AB, XV, ab) denote the set of invariants ~I(AB), ~I(XV),

1We note that the terms I(AB)I(XY), I(AB)I(abXY), I(Xy)I(abxy) appearing

in (2.10) in Kiral, Smith and Smith [1980] should be replaced by I(AB)I(XY),

I(AB)I(abXY), I(XY)I(abxy) respectively.

Table 7.13 Irreducible Representations: D6h~

co~

D6h I SI S2 D1 DISI D1S2 D2 D2S1 D2S2 D3 D3S1 D3S2 B.Q.

[1

[2

[3

r 4

r 5

[6

[7

[8

[9

flO

[11

f I2

1

1

1

1

E

E

1

1

1

E

E

1

1

1

1

A

A

1

1

1

1

A

A

111

1 -1 -1

1 1

1 -1 -1

B F G

B -F -G

111

1 -1 -1

111

1 -1 -1

B F G

B -F -G

1

-1

-1

H

-ll

-1

1

-1

H

-H

-1

-1

1

-F

-F

1

-1

-1

-F

-F

1

-1

-1

1

-G

-G

-1

-1

1

-G

-G

1

-1

-1

1

-H

-ll

1

-1

-1

-H

-ll

1

1

-1

-1

-E

E

1

-1

-1

-E

E

1

1

-1

-1

-A

A

1

1

-1

-1

-A

A

1

1

-1

-1

-B

B

1

1

-1

-1

-B

B

¢>, ¢>', .

'ljJ,'ljJ', ..

e, e', .ry, ry', ..

[1~l[:~l ..·[:~l[~~l···1r, 1r', .p,p', .

f), f)', ..

",', .

[i~l[~~l· ..[~~l[~~l· ..

G1~

~~

"i

~o'~

~

a~

~

:1.~

to~\I)C'b

~

~~

C1""1

c.e:::C/)

~0-~"'i~

~;::roo~.

G1""1<:l~~\I)

Q~

<~~

Table 7.13 (Continued) en('b~

;+-

[6 I E A B -F -G -II

r 5 I E A B F G n

r12 I -E -A -B F G II

r 7 -1 -1 -1 -1 -1 -1

r 8 -1 -1 -1 1 1

r 9 -1 -1 -1 -1 -1 -1

rIO -1 -1 -1 1 1 1

r11 I -E -A -B -F -G -ll

~

cow

~C'b~~

~c~

::..

Q~C/)C/)

~\I)

~""1

~;;.-.~.

~~

~~

~

:1.~

~c.e:::~::..

to~C/)~C/)

?"i

S.C'b

B.Q.

¢>, ¢>', ..

'ljJ,'ljJ', ..

e, e', .ry, ry', ..

[1~l[:~l· ..[:;l[~;l· ..1r, 1r', .p,p', .

(), ()', ..

",', .

[i~l[~~l· ..[~~l[~;l· ..

1

1

-1

-1

B

B

-1

-1

1

1

-B

-B

R3S2

A

-1

-1

1

A

-1

-1

1

1

-A

-A

R3S1R3

E

1

1

-1

-1

E

-1

-1

1

1

-E

-EII

II

-1

-1

1

-1

-H

-1

-H

IL.2S2

G

G

1

-1

-1

-1

1

1

-1

-G

-G

IL.2S1

F

F

IL.2

1

-1

-1

-1

1

-1

-F

-F

R 1S2

-1

-1

R1S1

1

-1

-1

R1

1

-1

1

-1

1 1

1 1

1

1

CS1 CS2C

1

1

1

D6h

[1

[2

r 3r 4

194 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.4] Invariant Functions of a Symmetric Second-Order Tensor: C3 195

lPI(ab). We list below the typical multilinear elements of an integrity

basis for functions of quantities of types f l' f 2' f 5' f 6' f Sand f 11

which are invariant under the group D6h.

f 1 f 2 f 5 f 6 f S f 11

D6h a3[a1] [A23] [523 ] [ 2512 ] [~~]a2 ' A31 ' -S31 S11- S22 P3

S33' S11+ S22 A12

Table 7.13A Basic Quantities: D6h 7.4 Invariant Functions of a Symmetric Second-Order Tensor: C3

We consider the problem of determining the general form of a

vector-valued polynomial function y = F(S) of a second-order sym

metric tensor S = [Sij] which is invariant under the group C3 = {AI'

A2, A3} = {I, Sl' S2}· The matrices I, Sl and S2 are defined in (1.3.3).

There are three inequivalent irreducible representations f l' f 2' f 3

associated with the group C3 which are seen from Table 7.S to be given

by

1. ¢;

(7.3.33)

where w=-1/2+i~/2 and w2=-1/2-i~/2. We see from Table

7.SA that the component Y3 of the polar (absolute) vector y is a

quantity of type r1, i.e., an invariant. We may then set

(7.4.2)

(7.4.3)

f 1: r 1 r1 1 1, 1, 1l' 2' r 3

f 2: 222 1, w w2 (7.4.1 )r 1, r 2, r 3 ,

f 3: ri,~,ri 1, w2 w,

y = clVI (S) + ... +cmVm(S)

where the c1' ... ' cm are polynomial functions of the elements 11'.'" In of

an integrity basis for functions of S which are invariant under C3. Let

x = Xl + ix2 denote a quantity of type r 3. We see from (7.3.21) that

the typical multilinear elements of an integrity basis for functions W(¢,

a, b, c, ... , a:, b, c, ... ) are given by

where W is a polynomial function of the elements II' 12, ... of an

integrity basis for functions of S which are invariant under C3.

We observe from Table 7.SA that y = Y1 - iY2' where Y1 and Y2

are components of the polar vector y, is a quantity of type r 2. Suppose

that the general polynomial form of y is given by

lPR(ABCDa:, a:XYZU, ABa:XY),

pR(ABCa:X, Aa:XYZ, AabcX), lPpR(Aa:bX),

R(ABCDEF, XYZUVW, ABCDXY, ABXYZU),

I(AB) I(abXY), I(XY) I(ABa:b);

~I(ABCDEF,XYZUVW, ABCDXY, ABXYZU),

pI(ABCXYZ, ABCDEX, AXYZUV), 1/JpI(ABCa:X, Aa:XYZ),

I(AB) R(a:XYZU), I(XY) R(ABCDa:);

~pR(ABCXYZ,ABCDEX, AXYZUV),

I(AB) I(XYZUVW), I(XY) I(ABCDEF).

11'11", pp', R(AB, XV, ab) ;

lPI(AB, XV, ab), pl(AX), I(ABa, aXY, abc);

lPR(ABa, aXY, abc), pR(AaX), lPpR(AX),

R(ABXY, ABa:b, abXY), I(AB) I(XY), I(AB) I(ab), I(ab) I(XY);

lPI(ABXY, ABa:b, abXY), pI(Aa:bX), pl(ab) R(AX),

1/JpI(AaX), I(ABCDa:, a:XYZU, ABa:XY), I(AB) R(aXY),

I(XY) R(ABa), I(AB) R(abc), I(XY) R(abc);

7.

6.

4.

S.

5.

2.

3.

196 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.4] Invariant Functions of a Symmetric Second-Order Tensor: C3 197

1. ¢;

2. ab;

3. abc

(7.4.4)

functions of x and S which are linear in x are seen from (7.4.4) and

(7.4.5) to be given by xS, xT, xS2, xST and xT2. The expression

(7.4.3) for y = Y1 -iY2 is then given by

where the ci = ci(11,... ,114). Upon equating the real and imaginary

parts of (7.4.8), we obtain

where ¢, ¢', ... ; a, b, c, ... and a, b, c, ... denote quantities of types r l'

r2 and r3 respectively and where both the real and imaginary parts of

ab and abc are typical multilinear elements of the integrity basis. We

see from (7.4.4) that yx is an invariant, i.e., the product of a quantity

of type r 2 and a quantity of type r 3 yields an invariant. Since the

quantities V1(S), ... ,Vm(S) in (7.4.3) are quantities of type r 2, it is seen

that xV1(S), ... ,xVm (S) are also invariants. The quantities V1(S),

... ,Vm(S) are obtained upon eliminating x from those elements of an

integrity basis for functions of x and S which are linear in x. We make

the identifications

(7.4.8)

(7.4.9)

We see from (7.4.4) and (7.4.5) that the elements of an integrity basis

11' 12, ... for functions of S are given by

where we have employed Table 7.8A and where

where the c1, ... ,c5 are polynomial functions of the 11,... ,114 given in

(7.4.7). The expression (7.4.2) for Y3' i.e., Y3 = W(11,···, 114), together

with the expression (7.4.8) or (7.4.9) gives the general expression for

y = F(S) which is invariant under C3.

Consider next the problem of determining the general expression

for a symmetric second-order tensor-valued polynomial function U

= R(S) of the symmetric second-order tensor S which is invariant

under the group C3. We see from Table 7.8A or from (7.4.5) and

(7.4.6) that the quantities U33, U11 + U22 are of type r1 and that

U31 -iU23, U11 - U22 +2iU12 are of type r2. With (7.4.2) and

(7.4.8), we have immediately

(7.4.6)

(7.4.5)

x,8,T

x,S,T;a, b, c

a, b, c

x=x1- ix2' S=S31- iS23' T=SII- S22+ 2iS I2'

x=xl+ ix2' 8=S31+ iS23' T=SII- S22- 2iS12'

11, ... ,114=S33,SII+S22' S8, ST+8T, ST-8T, TT,

S3+83, S3-53, S2T + 52 T , S2 T -52 T, (7.4.7)

S T2 + 8 1'2, S T2 - 8 1'2, T3 + T3, T3 - T3. (7.4.10)

The invariants x V 1(8), xV2(8), ... appearing in an integrity basis for

198 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.5] Generation of Product Tables 199

where d1,... , eS are polynomial functions of the invariants 11,... ,114given by (7.4.7) and where Sand T are defined by (7.4.6). Similarly,

the problem of determining the general form of a symmetric third-order

tensor-valued polynomial function D = G(S) is also readily solved. We

note that the components D··k of D satisfy the relationsIJ

(7.4.11 )

and that there are 10 independent components of D. It has been shown

by Kiral, Smith and Smith [1980] that the linear combinations of the

components of D which form quantities of types r1 and r2 are given by

(7.4.12)

The general expression for the function D = G(S) which IS invariant

under C3 is then given by

7.5 Generation of Product Tables

In §6.4, we have constructed product tables which list the

quantities forming carrier spaces for the irreducible representations

which arise from the decomposition of the product of two irreducible

representations. These tables play a central role in the application of

Schur's Lemma to the generation of constitutive equations. The con

struction of the product table associated with a group A is facilitated if

the typical multilinear elements of an integrity basis for A are given.

We consider as an example the group D3. We list below the product

table for D3 (see §6.4) where ¢, <Ii and ¢, ¢' and [aI' a2]T, [b1, b2]T

denote quantities of types r 1 and r 2 and r 3 respectively.

Table 7.14 Product Table: D3

r 1 ¢ ¢' ¢¢', ¢¢', alb i + a2b2

r 2 ¢ ¢' ¢¢', ¢¢', alb2- a2b I

r 3 [:~] [:~] [¢bl] [al¢'j [~b2] [a2~'J [alb2+ a2bl]¢ b2 ' a2¢' , -¢ b l ' -al'ljJ" alb l - a2b2

There are 16 quantities which arise as products of each of the four

entries ¢, 'ljJ, aI' a2 in the first column of Table 7.14 with each of the

The typical elements of the integrity basis for functions of ¢, <Ii, ... ,'ljJ, ¢', ... , aI' a2' b1, b2, ... which are invariant under D3 are seen from


1. ¢;

DIll - 3D122 = W3(11,... ,114), D222 - 3D211 = W4(11,... ,114),

D113 + D223 = WS(11,···, 114), D333 = W6(11,... ,114),

D133 - i D233 = £1 S +£2T +£352 +£45 l' +£51'2, (7.4.13)

. -2DIll +DI22-1(D222+D211) = glS + g2T + g3S

+ g45 l' + g51'2,

. -2 - - -2D311 -D322+2ID312 = h1S+ h2T+h3S +h4ST+hST

where the f1,... , hS are polynomial functions of the invariants 11'···' 114given by (7.4.7) and where Sand T are defined by (7.4.6).

2.

3.

4.

a1b1 + a2b2' 'ljJ 'ljJ' ;

a2b2c2 - a1b1c2 - b1cla2 - clalb2, 'ljJ( alb2 - a2b1) ;

'ljJ(alb1cl - a2b2cl - b2c2al - c2a2bl)·

(7.S.1)

200 Generation of Integrity Bases: The Crystallographic Groups [eh. VII

four entries <Ii, 'l/J', bl , b2 in the second column. Those products which

are invariants are quantities of type fl. We see from (7.5.1) thatVIII

(7.5.2)

IS a quantity of type f 2. Similarly, we see from (7.5.2) that

I == a1b1 +a2b2 is an invariant and that

IS a quantity of type f 3. We observe from (7.5.1) that

J == 'l/J(alb2 - a2b1) and K == a2b2c2 - alb l c2 - b l cla2 - clalb2 areinvariants. Hence, the quantities

GENERATION OF INTEGRITY BASES: CONTINUOUS GROUPS

8.1 Introduction

In this chapter, we consider the problem of determining an

integrity basis for polynomial functions of vectors and/or second-order

tensors which are invariant under a group A which is the three

dimensional orthogonal group or one of its continuous subgroups. In

the previous chapter, we obtained results of complete generality for the

crystallographic groups considered. This was possible because a crystal

lographic group A is a finite group and hence has only a finite number r

of inequivalent irreducible representations fl, ... ,fr . We then deter

mined the form of polynomial functions of n1 quantities of type f l' ... ,

nr quantities of type f r which are invariant under A where n1 ,... , nr are

arbitrary. This constitutes the general result. The numbers of

inequivalent irreducible representations associated with the continuous

groups considered here are not finite. There is consequently no hope of

obtaining results of generality comparable to those given in Chapter

VII. We thus restrict consideration to the determination of the form of

polynomial functions of vectors and/or second-order tensors which are

invariant under a continuous group A.

This problem has been discussed by Rivlin and Spencer for the

groups R3 and 03. Their procedure makes extensive use of matrix

identities which are generalizations of the Cayley-Hamilton identity.

We discuss the generation of these identities in §8.2. We outline in §8.3

the Rivlin-Spencer procedure as applied to the generation of the

canonical forms of scalar-valued and tensor-valued polynomial functions

(7.5.3)

(7.5.4)

(7.5.5)

are of type f 3. The quantities (7.5.5) then appear as entries in row 3,

column 3 of Table 7.14. Thus, from inspection of the list of typical

multilinear elements of an integrity basis for the group D3, we may

immediately determine most of the entries in the product table for D 3.

The remaining entries in the product table for D3 may be readily

determined by inspection.

are invariants, i.e., of type f l , and hence will appear as entries in row 1,

column 3 of the product table. With (7.5.2), we see that the product

'l/J'l/J' of two quantities of type f 2 is an invariant. Then, from (7.5.1),

'l/J(alb2 - a2b1) is an invariant and hence

201

202 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.2] Identities Relating 3 X 3 Matrices 203

(8.2.1)

(8.2.2)

(8.2.3)

M3 - (tr M)M2+! [(tr M)2 - tr M2JM

- i [(tr M)3 - 3 tr M tr M2 +2 tr M3JE3 = o.

Hamilton identity

Rivlin and Spencer have employed (8.2.1) and other identities which

may be referred to as generalized Cayley-Hamilton identities to

generate the canonical forms of scalar-valued and second-order tensor

valued polynomial functions of three-dimensional skew-symmetric

second-order tensors AI' A2, ... and three-dimensional symmetric

second-order tensors 81, 82, ... which are invariant under the proper

orthogonal group R3 . We briefly discuss their procedure in §8.3.

We have observed in §4.7 (iii) that the 105 distinct isomers of

the tensor

are invariant under the group R3 (also the group 03). The number of

linearly independent three-dimensional eighth-order tensors which are

invariant under R3 is given by the number P 8 of linearly independent

multilinear functions of the eight three-dimensional vectors Xl'···' X8which are invariant under R3 . We have (see §4.7 (iv))

27r1 f iB _. B 8P8 == 27r (e +1 - e 1 ) (1 - cos B) dB == 91.

oThere are then 105 - 91 == 14 linearly independent linear combinations

of the isomers of 8· . 8. . 8· . 8.. which have all of their components1112 1314 1516 1718

equal to zero. The isomers of 8· . 8· . 8· . 8· . form the carrier space. . 1112 1314 1516 1718

for a reducIble representatIon of the symmetric group S8 whose de-

composition is seen from (4.7.10) to be given by (8) + (62) + (44) ++ (422) + (2222). The 14 tensors forming the carrier space for the

irreducible representation (2222) are those which have all components

equal to zero. It has been shown by Smith [1968] that there is a

8.2 Identities Relating 3 x 3 Matrices

In this section, we derive identities which relate 3 x 3 matrices.

A well-known example of such an identity is furnished by the Cayley-

of two symmetric second-order tensors 81, S2 which are invariant under

R3 . The generalization of this problem to the case of functions of n

symmetric second-order tensors and m skew-symmetric second-order

tensors has been thoroughly discussed by Rivlin and Spencer in a

sequence of papers. A lucid outline of their work is given by Spencer

[1971]. We next follow the discussion of Smith [1968b] and consider the

problem of determining the multilinear elements of the bases for

functions of n traceless symmetric second-order tensors B1,... , Bn and m

skew-symmetric second-order tensors AI' ... ' Am which are invariant

under R3 . This leads us to consider in §8.4 the notion of sets of

functions of symmetry type (n1 ... np). We discuss in §8.5 and §8.6 the

use of Young symmetry operators to generate the sets of functions of

given symmetry types (n1." np ) which comprise the multilinear

elements of the bases required. Given these sets of functions, we may

readily generate the remaining (non-multilinear) basis elements. This

procedure is applied in §8.7 to generate the multilinear basis elements

for functions of n traceless symmetric second-order tensors B1,···, Bnwhich are invariant under R3 . In §8.8, we generate the multilinear

basis elements for scalar-valued functions of m skew-symmetric second

order tensors AI' ... ' Am and n traceless symmetric second-order tensors

B1,... , Bn which are invariant under R3 . In §8.9, we generate the

multilinear basis elements for scalar-valued functions of vectors and

traceless symmetric second-order tensors which are invariant under the

full orthogonal group 03. In §8.10.1 and §8.10.2, we consider the

generation of the multilinear basis elements for scalar-valued functions

of vectors and second-order tensors which are invariant under the

transverse isotropy groups T] and T2 respectively.

204 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.2] Identities Relating 3 X 3 Matrices 205

correspondence between these tensors and the standard tableaux

associated with the frame [222 2] which are given by

The tensor associated with the first standard tableau of (8.2.4) is given

by

8i1i3i5i7 811131516 811131417 811131416 8i1i3i4i5 8i1i2i5i7 811121516.... , .... , .... , .... , .... , .... , .... ,12141618 12141718 12151618 12151718 12161718 13141618 13141718

(8.2.6)

8i1i2i4i7 811121416 8i1i2i4i5 8i1i2i3i7 8i1i2i3i6 8i1i2i3i5 8~1~2~3~4.... , .... , .... , .... , . . . . , .... ,1315161S 1315171S 13161718 14151618 1415171S 14161718 15161718

If the tensor (8.2.5) is three-dimensional, it is a null tensor. For any of

the 38 possible choices of values which i1,... , is may assume, at least two

rows (and at least two columns) of the determinant will be the same

and the component will be zero. If the tensor (8.2.5) is four

dimensional, it is not a null tensor, e.g., b i ; ; : = 1. The 14 three

dimensional null tensors associated with the standard tableaux (8.2.4)

are given by

(8.2.7)

(8.2.9)

8i1i3i5i7M1 M2 M3 ==0i2i4i6i8 i3i4 i5i6 i7i8

8~1~3~4~7 M~. M~. M~. == 012151618 1314 1516 1718

where the notation (8.2.5) is employed. With (8.2.5), we have

M1M2M3 +M2M3M1 +M3M1M2 +M1M3M2 +M3M2M1

+M2M1M3 - (M1M2 +M2M1) tr M3 - (M2M3 +M3M2) tr M1

- (M3M1 + M1M3) tr M2 - M1 (tr M2M3 - tr M2 tr M3)(8.2.8)

- M2 (tr M3M1 - tr M3 tr M1) - M3 (tr M1M2 - tr M1 tr M2)

- E3 (tr M1 tr M2 tr M3 - tr M1 tr M2M3 - tr M2 tr M3M1

- tr M3 tr M1M2 + tr M1M2M3 + tr M3M2M1) == o.

This is the generalized Cayley-Hamilton identity which was obtained in

this manner by Rivlin [1955]. If we set M 1 == M2 == M 3 == M in (8.2.8),

we recover the Cayley-Hamilton identity (8.2.1). Further identities

may be obtained upon applying the other null tensors in the set (8.2.6)

to M~. M~. M~. . For example, the identity1314 1516 1718

where the M i = [Mjkl are 3 x 3 matrices. Upon expanding (8.2.5), we

obtain

is equivalent to

T T T T T T(M2 - M2 )(M1 - M1) M3 +M3 (M2 - M2 )(M1 - M 1)

T T T T+ (M2 - M2 ) M3 (M1 - M1) - (M2 - M2 )(M1 - M1 ) tr M3T { T } T T (8.2.10)

-M3 tr M1(M2 -M2) -E3~r{(M1-M1)(M2-M2)M3}

-trM3 tr{M1(M2 -M;)}] = O.

(8.2.5)

(8.2.4)

1 324,5 768

1 526.3 748

8· .1118

8· .1318

8· .1518

8· .1718

1 426,3 75 8

1 324,567 8

8· .1116

8· .1316

8· .1516

8· .1716

1 425,3 76 8

1 236,4758

1 425,3 67 8

1 235,476 8

8· .1114

8· .1314

8· .1514

8· .1714

1 235,467 8

1 326,4 75 8

8· .1112

8· .1312

8· .1512

8· .1712

1 234,5 768

1 325,4 76 8

8~1~3~5~7 ==12141618

1 234,5 67 8

1 325,467 8

206 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.3] The Rivlin-Spencer Procedure 207

(8.2.11)

(8.2.14)

A matrix S is said to be symmetric if S == ST. A matrix A is said to be

skew-symmetric if A == -AT. We may express each of the matrices Mi(i == 1,2,3) as the sum of a skew-symmetric matrix Ai and a symmetric

matrix Si. Thus,

1 T 1 ( T)Ai == 2 (Mi - Mi ), Si == 2 Mi +Mi '

M· == A· + 8· (i == 1,2,3)1 1 1

where the Ai are skew-symmetric 3 x 3 matrices and the 8i are

symmetric 3 x 3 matrices. With (8.2.11), the identity (8.2.10) may be

written as

(8.2.12)

The identity (8.2.12) was obtained by Spencer and Rivlin [1962]. We

may apply each of the 14 null tensors in (8.2.6) to Mf i Mr i MP i to3 4 5 6 7 8

obtain other identities. It has been shown by Rivlin and Smith [1975]

that the resulting identities may be written as

<PI (A1,A2,A3) == 0, <PI (A2,A3,A1) == 0, <PI (A3,A1,A2) == 0,

<P2(Al ,A2,A3) == 0, <P3(A1,A2,83) == 0, <P3(A2,A3, 81) == 0,

(8.2.13)

<P3(A3,A1, 82) == 0, <P4(A1,A2, 83) == 0, <P4(A2,A3, 81) == 0,

<P4(A3,A1,82) == 0, <P5( 81, 82,A3) == 0, <P5( 82, 83,A1) == 0,

where the 4>1 (... ) , ... , 4>6 (... ) are defined by

tPl (AI ,A2,Aa) = 2(A1A2Aa + AaA2A1) - Aa tr AlA2 - Al tr A2Aa = 0,

tP2(Al ,A2,Aa) = AlA2Aa - AaA2A1+A2AaA1 - AlAaA2 +AaA1A2- A2A1A3 - 2E3 tr AlA2A3 == 0,

tPa(A1,A2,Sa) = AlA2Sa - SaA2Al +AlSaA2 - A2SaA1 +SaAlA2- A2A183 - (AI A2 - A2A1) tr 83 == 0,

tP4( Al ,A2,Sa) = AlA2Sa +SaA2Al +AlSaA2 +A2SaA1 +SaAlA2

+A2A183 - (AI A2 +A2A1) tr 83 - 83 tr Al A2

- E3(2 tr Al A283 - tr 83 tr AlA2) == 0,

tP5(Sl,S2,Aa) = SlS2Aa +AaS2S1+SlAaS2 +S2AaSl

+ 8281A3 + A38182 - (82A3 + A382) tr 81

- (81A3 +A381) tr 82 - A3(tr 8182 - tr 81tr 82

) == 0,

tP6(Sl,S2,Sa) = SlS2Sa +SaS2S1+S2SaSl +SlSaS2 +SaSlS2

+82S183 - (8283 +8382) tr 81 - (8183 +S381) tr 82

- (8182 + 8281) tr 83 - 81(tr 8283 - tr 82 tr 83

)

- S2(tr SaSl - tr Sa tr Sl) - Sa(tr SlS2 - tr Sl tr S2)

- E3(tr 81 tr 82 tr 83 - tr 81 tr 8283 - tr 82

tr 83

81

- tr 83 tr 8182 +2 tr 818283) == 0.

8.3 The Rivlin-8pencer Procedure

In a series of papers (see Rivlin [1955], Spencer and Rivlin [1959a,

b; 1960; 1962], Spencer [1961; 1965]), Rivlin and Spencer have employed

the matrix identities given in §8.2 as well as identities which arise from

these identities to generate the canonical forms of scalar-valued and


second-order tensor-valued functions of three-dimensional skew

symmetric second-order tensors AI' A2, ... and symmetric second-order

tensors Sl' S2' ... which are invariant under the proper orthogonal group

R3 . We note that the canonical forms obtained are also invariant under

the full orthogonal group 03. We briefly outline their procedure as it

applies to the special case of generating the form of functions of two

symmetric second-order tensors Sl and S2 which are invariant under

R3 . We follow Rivlin and Spencer and refer to these functions as scalar

valued and matrix-valued isotropic functions of the symmetric matrices

Sl' S2· A complete discussion of their method is given by Spencer

[1971]. An outline of the computations yielding the canonical forms for

isotropic functions of symmetric matrices is given by Rivlin and Smith

[1970].

A scalar-valued polynomial function P(Sl' S2) of the symmetric

matrices Sl = [stJ, S2 = [SnJ is expressible as

f3 p q rP == C + L C· . .. .. S· . S· .... S· . (p,q, ... ,r == 1 or 2). (8.3.1)

n==l I1J 112J2 ... InJn I1J 1 12J2 InJn

A matrix-valued polynomial function P(Sl' S2)' P == [Pij ] is expressible

as

(3 p q rp .. == C·· + "'" C··· . .. .. S· . S· .... S· . (p,q, ... ,r == 1 or 2).

IJ IJ LJ IJ I1J 112J2 ... InJn I1J 1 12J2 InJnn==l (8.3.2)

The requirement that the functions (8.3.1) and (8.3.2) be invariant

under R3 imposes the restrictions that the tensors Cij , Ci1h

... inin and

C·· .. must be invariant under R3 . We see from §4.7 that theseIJ ... InJn

tensors must be expressible in terms of the outer products of Kronecker

deltas. For example,

(8.3.3)

Upon introducing expressions of the form (8.3.3) into (8.3.1), we see

that P is expressible as a polynomial in the traces of products formed

from the matrices Sl and S2. Similarly, we see that P(Sl' S2) may be

expressed as the sum of a number of products formed from the matrices

81 and 82 together with E3, with coefficients which are polynomials in

traces of products formed from the matrices Sl and 82.

We set M equal to Sl in (8.2.1) and multiply the resulting ex

pression on the left by S2 to obtain

(8.3.4)

We say that S2S~ is reducible, i.e., S2S~ is expressible as a polynomial

in matrix products of degrees (p, q) in (81,82) where p:::; 3, q:::; 1,

p +q<4 with coefficients which are polynomials in the traces of matrix

products. We denote this by writing S2S~ ~ O. We take the trace of

(8.3.4) to obtain

(8.3.5)

We say that tr S2S~ is reducible, i.e., tr S2S~ is expressible as a poly

nomial in traces of matrix products of 81 and 82 which are of lower

total degree in Sl and S2 than is S2S~ . We denote this by tr S2S~ ~ O.

If we replace S3 by Sl in the expression <P6( ... ) defined in (8.2.14), we

have

SyS2 + S2Sy + SlS2S1 ~ O. (8.3.6)

We say that the symmetric matrix-valued function SyS2 + S2Si is

equivalent to -8182S1 and denote this by


(8.3.7) The 24 = 16 matrix products of total degree 4 in Sl and S2 are given by

(8.3.14)

Skew-Symmetric: S~S2 - S2S~, SrS2S1 - SlS2S1, S1S~ - S~S1,

SlS2S1S2 - S2S1S2S1' S~SlS2 - S2S1S~, S~Sl - Sl S~.

4. Symmetric: Sf' SfS2 + S2s f, SyS2S1 + SlS2Sy, Sys~ + S~sy,

Sl S2S1S2 + S2S1S2S1' SlS~Sl ' S2SrS2'

S~SlS2 + S2S1s~, S~Sl + Sl s~, S~ ;

We first determine the canonical form for symmetric matrix

valued and skew-symmetric matrix-valued functions of the symmetric

matrices 81 and 82. The symmetric and skew-symmetric matrix-valued

functions of total degree 1,2, ... in 81 and 82 are linear combinations of

the matrix products of 81 and 82 listed below. We note that there are

2n distinct monomial matrix products of total degree n.

1. Symmetric: 81, 82 . (8.3.8)

2. Symmetric: 2 8182 + 8281, 82 .81, 2' (8.3.9)Skew-Symmetric: 8182 - 8281 .

We see immediately from (8.3.11) that Sf ~ 0, SfS2 ± S2s f ~ 0,

S~Sl ± SlS~ ~ 0, S~ ~ O. We replace S2 by S~ in (8.3.6) and (8.3.7) toobtain

(8.3.13)

(8.3.17)

(8.3.15)

(8.3.18)

Skew-Symmetric: sys~ - S~Sr, SrS2S1- Sl S2Sr, S~SlS2 - S2S1S~.

In similar fashion, we see that S2Sr S2 ~ - (Sr S~ + S~ Sr). We replaceS3 by Sr in {8.2.14)6 to obtain

SrS2S1 + SlS2Sr + 2 S~ S2 + 2 S2Sf ~ 0 (8.3.16)

and conclude that SrS2S1 + SlS2Sr ~ O. Similarly, S~SlS2 + S2S1S~~ O. Upon setting A3 = Sl S2 - S2S1 and S3 = SlS2 + S2S1 in

(8.2.14)5 and {8.2.14)6 respectively, we find that

SlS2S1S2 - S2S1S2S1 ~ -(Sr S~ - S~ S1),

Sl S2S1S2 + S2S1S2S1 ~ Sr S~ + S~ Sr.

We see from (8.3.14) ,.. , (8.3.17) that the basis elements of degree 4 aregiven by

4. Symmetric: SIS~ + S~Sr ;

(8.3.11)

With (8.3.10) ,... , (8.3.12), the basis elements of degree 3 are given by

3. Symmetric: SIS2+S2SI, S~Sl+SlS~;

Skew-Symmetric: SIS2 - S2SI, S~Sl - SlS~ .

We have, with (8.3.7),

SlS2S1 ~ -(S1S2 + S2S1), S2S1S2 ~ -(S~Sl + Sl S~). (8.3.12)

We see from (8.2.1) upon replacing M by 81 and 82 in turn that

No one of the terms in (8.3.8) and (8.3.9) is reducible. We refer to

these matrix products as basis elements. The 23 == 8 matrix products of

total degree 3 in 81 and 82 are given by

3. Symmetric: S~, SrS2 + S2Sr, SlS2S1' S2S1S2' S~Sl + Sl S~, S~;(8.3.10)

Skew-Symmetric: S1S2 - S2Sr, S~Sl - SlS~ ·


The 25 == 32 matrix products of total degree 5 in 81 and 82 are given by

5. Symmetric: Sf, S1S2 + s2s1, SfS2S1 + SIs2s f ' sIs 2s I,

sfs§ +s§sf, SyS2S1S2 +S2S1S2Sy, SyS§SI +SIs§sy,

SIS2SyS2 + S2SyS2S1' SIS2S1S2S1' S2Sf S2"";(8.3.19)

Skew-Symmetric: S1S2 - S2S1, SfS2S1 - SIS2Sf, SfS§ - S§Sf,

SyS2S1S2 - S2S1S2Sy, SyS§SI - SIs§sy,

SIS2SyS2 - S2SyS2S1' ...

where ... above indicates the terms obtained upon interchange of 81 and

S2 in the preceding terms. We may show in the manner employed

above that all of the symmetric terms in (8.3.19) are reducible and that

the skew-symmetric terms are either reducible or equivalent to either

SyS~SI - SIS~SI or S§SyS2 - S2SyS~, Thus, the basis elements of

degree 5 are given by

5. Symmetric: None;

(8.3.20)

We consider next the 26 == 64 monomial matrix products of total degree

six. These are listed below.

6. S~, SfS2' S1S2S1' S~S2Sy, SyS2S~, SIS2S1, S2Sf,

S1S~, SyS2S1S2' SfS~SI' SyS2SyS2' SyS2S1S2S1' sys~sy,

SIS2SyS2' SIS2SyS2S1' SIS2S1S2Sy, SIS§S~, S2S1S2'(8.3.21 )

S2SyS2S1' S2SyS2Sy, S2S1S2Sf, S§S1, SyS~, SyS2S1S§,

SyS~SIS2' SIS~SI' SIS2SIS~, SIS2S1S2S1S2' SIS2S1S~SI'2 2 2 8 8382

81828182, 8182818281, 1 2 l' ...

where ... indicates the 32 terms obtained from the preceding 32 terms

upon interchanging the subscripts 1 and 2. All terms containing s~, Sf,

sf, Sf, S~ are obviously reducible. Since all symmetric matrix-valued

products of degree 5 are reducible, we see immediately that

Proceeding in this fashion, we may readily show that all of the matrix

products in (8.3.21) are reducible.

We conclude that every symmetric matrix-valued polynomial in

the matrix products of 81 and 82 which is of degree 6 or less is

expressible in the form

T(SI' S2) = aOE3 +alSI +a2S2 +a3Sy +a4(SIS2 +S2S1)(8.3.23)

+a5S~ +a6(SyS2 +S2Sy) +a7(SIS~ +S~SI) +a8(SIS~ +SIS~)

where aO, ... ,a8 are polynomials in the traces of matrix products. Also

every skew-symmetric matrix-valued polynomial in the matrix products

"of S1 and S2 which is of degree 6 or less is expressible in the form

A(SI' S2) = bO(SIS2 - S2S1) +bl (SyS2 - S2Sy) +b2(SIS~ - S§SI)

+b3(SyS§ - S§Sy) +b4(SyS2S1 - SIS2Sy) +b5(S~SIS2 - S2S1S~)

+ b6(SyS~SI - SIS§SI) + b7(S§SyS2 - S2SyS§) (8.3.24)

where bO,... ,b7 are polynomials in the traces of matrix products.

Consider a matrix product S1P(S1,S2) where P(S1,S2) IS a

matrix product of degree 6. We may write this as

(8.3.25)


in order to eliminate redundant terms. The basic invariants are seen tobe given by

tr SlS2S3 == tr S2S3S1 == tr S3S1S2' tr SIS2S3 == tr S3S2S1'

tr SlS2S3S4 == tr S4S1S2S3 == tr S3S4S1S2 == tr S2S3S4S1' (8.3.27)

tr Sl S2S3S4 == tr S4S3S2S1

tr Sf ~ 0, tr(S~S2 + S2S~) ~ 0, tr(SyS2S1 + Sl S2Sy) ~ 0,

tr S~ ~ 0, tr(S~Sl + Sl S~) ~ 0, tr(S~Sl S2 + S2S1 S~) ~ 0, (8.3.31)

tr(SlS2S1S2 + S2S1S2S1) ~ tr SyS~, tr Sl S~Sl= tr S2SyS2= tr sysy.

(8.3.29)

(8.3.30)

(8.3.5) together with relations such as

1. tr SI' tr S2;

2. tr sy, tr SIS2' tr S2.2'

(8.3.28)3. tr S~, tr S~ S tr SlS~, tr S3.

2' 2'

4. tr Sy S~.

We indicate the manner in which terms tr(P +P T ) may be excluded

from the basis (8.3.28) for the case where the P +p T are of degree 4

and are listed in (8.3.14). We see from (8.3.5) that tr S2S~ ~ 0. Upon

setting S2 = Sl in (8.3.5), we have tr Sf ~ 0. We note that tr SyS2S1

=tr SlS2Sy = tr S~S2 = tr S2S~ ~ 0. We may set S3 = Sl in (8.2.14)6'

multiply on the left by S2 and take the trace of the resulting expression

to obtain

tr S2S1S2S1 = tr Sl S2S1S2 ~ tr sYs~.

We also have, from (8.3.27),

tr SlS~Sl = tr S2SyS2 = tr sYs~.

From the discussion above, we see that

With (8.3.23) and (8.3.24), we see that SlP is expressible as a matrix

polynomial in matrix products of degree 6 or less. Then SlP +p T S1 is

expressible as a polynomial in symmetric matrix-valued matrix

products of degree 6 or less. Applying (8.3.23) again, we see that

Slp +p T S1 is expressible in the form (8.3.23). Similarly SIP - p T S1 is

expressible in the form (8.3.24). We may argue in this fashion to

establish that every symmetric and skew-symmetric matrix-valued poly

nomial in matrix products of degree 7 is expressible in the forms

(8.3.23) and (8.3.24) respectively. An identical argument enables us to

reach the same conclusion for polynomials in matrix products of degrees

8, 9, .... Thus, symmetric and skew-symmetric matrix-valued poly

nomials of arbitrary degree are expressible in the forms (8.3.23) and

(8.3.24) respectively.

The coefficients aO, ... ,b7 in (8.3.23) and (8.3.24) are polynomials

in the traces of matrix products. We now determine a basis for these

quantities. Consider the invariants tr SiP (i == 1,2) where P is a matrix

product of total degree 5 in SI and S2. We have

since tr Si{P - p T) == O. For any matrix product P of degree 5, we have

seen that P + pT is expressible in the form (8.3.23) with coefficients

which are polynomials in the traces of matrix products of degree 4 or

less. Then, tr SiP (i == 1,2) is expressible as a polynomial in the traces

of matrix products of degrees 5 or less. The same argument holds for

the cases where P is a matrix product of degree 6,7,.... Thus, the

trace of any matrix product is expressible as a polynomial in traces of

matrix products of degree 5 or less. Let P denote a matrix product.

Since tr P = ~ tr(P + PT), we consider the terms tr(P + pT) for the

cases where P is of degree 1,... ,5. The terms P + p T where P is a

matrix product of degree 1, ... ,5 are listed in (8.3.8), (8.3.9), (8.3.10),

(8.3.14) and (8.3.19). We employ matrix identities as in (8.3.4) and

216 Generation of Integrity Bases: Continuous Groups [Ch. VIIISect. 8.4] Invariants of Symmetry Type (n1 ... npJ 217

where P = [Pji] is non-singular. Then, the matrices p-1 D(s) P which

describe the transformation properties of the invariants J 1"'" Jr under

the n! permutations s of 5n also form a r-dimensional matrix rep

resentation of the group 5n which is said to be equivalent to the

representation {D(s)}. If there is a proper subspace of the carrier space

for the representation {D(s)} which is invariant under all permutations

of 5n , the representation is said to be reducible. If not, the repre

sentation is irreducible. The number of inequivalent irreducible repre

sentations associated with 5n is equal to the number of partitions of n,

i.e., to the number of solutions in positive integers of

Thus, the only term of degree 4 which need be included in the basis is

tr SIS~. In similar fashion, we may verify that all terms tr(P +P T) ~ 0

where P is a matrix product of degree 5.

8.4 Invariants of Symmetry Type (uI." up)

Let I1,... ,Ir be a set of linearly independent scalar-valued func

tions which are multilinear in the tensors B1,... , Bn and which are

invariant under the group A. Let s denote the permutation of the

numbers 1, ... , n which carries 1 into iI' ... , n into in. Let s Ij (B1,···, Bn)

be defined by n1 + n2 + ... + np == n, (8.4.5)

We assume that, for each of the n! elements s of the group 5n of

permutations of the numbers 1, ... , n, the invariants s Ij (B1,... , Bn ) are

expressible as linear combinations of the II"'" Ir . Thus,

Thus, there are five partitions of n == 4 given by 4, 31, 22, 211, 1111 and

hence five inequivalent irreducible representations of 54 which we

denote by (4), (31), (22), (211), (1111). The irreducible representation

of 5n associated with the partition n1'" np of n is denoted by (n1'" np).

The components of the character of the irreducible representationnl···np( )(n1'" np) are denoted by X. s .

Let I1,... ,Ir be a set of invariants which are multilinear in

B B and which form the carrier space for a r-dimensional reducible1"'" n

representation {D(s)} of 5n . We may determine a matrix P == [Pji] so

that the representation {P-1 D(s) P} decomposes into the direct sum of

irreducible representations of 5n . The invariants J i == Ij Pji which form

the carrier space for the representation {P-1 D(s) P} may thus be split

1·nto sets J 1 J. . J +I J such that each set forms the carrier,... , p , ... , q ,... , r

space for an irreducible representation (n1 ... np) of 5n . A set of

invariants which forms the carrier space for an irreducible repre

sentation (n1'" np) of 5n is referred to as a set of invariants of

symmetry type (nl'.' np). The number anI'" np of sets of invariants of

symmetry type (nl ... np) arising from the II, ... ,Ir is seen from (2.5.14)

to be given by(8.4.4)

(8.4.3)

(8.4.2)

(8.4.1)

x(s) == tr D(s)

(j,k == 1, ... ,r).

(i ,j == 1, ... , r )

x(e),···,x(s ,),n.

J. == I· p ..1 J JI

where e denotes the identity permutation. Let

s Ij == IkDkj (s)

The n! matrices D(s) == [Dkj(s)] which describe the behavior of the

invariants I1,... ,Ir under the permutations of 5n form a matrix rep

resentation of dimension r of the group 5n, i.e., to every element s of 5nthere corresponds a r X r matrix D(s) such that to the product u == t s of

two permutations corresponds the matrix D(u) == D(t) D(s). The

invariants II"'" Ir are said to form the carrier space for the repre

sentation {D(s)}. The character of the representation {D(s)} is denoted

by

218 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.4] Invariants of Symmetry Type (n1 ... npJ 219

With (8.4.6) and (8.4.7), we have

(8.4.11)

s 11' s 12, s 13 == 11,12, 13.

s II' s 12, s 13 == 11,13, 12.

s II' s 12, s 13 == 13, II' 12.

sI1, sI2, sI3 == 13,12, II·

s II' s 12, s 13 == 11' 12, 13.

s == e;

s == (12);

s == (123);

s == (1234);

s == (12) (34);

Class 14:

Class 122:

Class 13:

Class 4:

Class 22:

With (8.4.2) and (8.4.11), we see that the trace of the matrix D(s)

associated with a transformation s of (8.4.11) is given by the number of

invariants left unaltered by the permutation s. Thus,·the quantities X,

associated with the classes ;-y == 14 12 2 1 3 4 22 are given by 3 1 0 1I , , " , , , ,

3 respectively. Then, with (8.4.8) and the character table for 54 (Table

4.3) in §4.9, we see that 04 == 022 == 1, 031 == 0211 == 01111 == o. Hence,

the set of invariants (8.4.9) may be split into two sets of symmetry

types (4) and (22) which are comprised of X~ = 1 and X~2 = 2

invariants respectively. The quantity X~l··· np appears in the first

column of the character table for 5n and gives the dimension of the

irreducible representation (n1 ... np), i.e., the number of invariants

comprising a set of invariants of symmetry type (n1 ... np ).

In Chapter IV, we introduced Young symmetry operators which

were employed to generate sets of property tensors of symmetry type

(n1··· np). We may employ the same procedure to generate sets of

invariants of symmetry type (n1 ... np). Let n1 ... np denote a partition

of the integer n. Associated with each partition n1 ... np is a frame

[n1··· np] which consists of p rows of squares containing n1, ... ,npsquares respectively arranged so that their left hand ends are directly

beneath one another. A tableau is obtained from a frame by inserting

the numbers 1,2,... , n in any order into the n squares. A standard

tableau is one in which the integers increase from left to right and from

top to bottom. The number of standard tableaux associated with the

frame corresponding to the partition n1 ... np of n is given by the

(8.4.9)

(8.4.6)

(8.4.8)

(8.4.10)

X( s) == tr D(s)

tr B· B· == tr B· B·.1 J J 1

11 == tr B1B2 tr B3B4, 12 == tr B1B3 tr B2B4,

13 == tr B1B4 tr B2B3

where B1,... , B4 are symmetric second-order tensors. We note that

We list below, one element s of each class of S4 and the invariants

sI1,... ,sI3 into which 11,... ,13 are carried by the permutation s.

where h, is the order of the class I and where the summation is over

the classes of Sn. The quantities X~l··· np and h, may be found in the

character tables for 5n (n == 2, ... ,8) given in §4.9.

where the summation is over the elements s of 5n . If the permutations

sl and s2 belong to the same class I of permutations of 5n , i.e., if sl

and s2 have the same cycle structure (see §2.2), then

Thus, in order to determine the number of sets of invariants of

symmetry type (n1 ... np) contained in the set of invariants II'··' Irwhich form the carrier space for a representation {D(s)} or, equi

valently, the number of times the irreducible representation (n1 ... np)

occurs in the decomposition of {D(s)}, we need only determine tr D(s)

for one permutation from each class I of 5n and then apply (8.4.8). For

example, consider the invariants

220 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.4] Invariants of Symmetry Type (n1 ... npJ 221

Let p(q) be a permutation which interchanges only the numbers in each

row (column) among themselves. Let

dimension X~l··· np of the irreducible representation (n1'" np). For

example, the partitions of n == 3 are given by 3, 21, 111. The frames

corresponding to these partitions are I I I I, EfJ ' §.From the character table for 53 (Table 4.2), we see that the numbers of

standard tableaux associated with the frames 3, 21 and 111 are given by

X3 == 1 X21 == 2 and XlII == 1 respectively. ·These standard tableauxe 'e e

are given by

where e denotes the identity permutation which leaves all integers

unaltered.

(8.4.16)

obtained when s runs through the group 5n will be spanned by one set

of X~l···np invariants of symmetry type (nl ... np) provided that

YI(B1,···,Bn) is not identically zero. We may choose X~l···np permu

tations e, s2' s3' ... such that the X~l'" np invariants

Let Y be the Young symmetry operator associated with a

standard tableau corresponding to the partition n1". np of n. The set of

n! invariants

(8.4.12)1 .23

1 3,2

123,12,3

P = LP, Q = LCq q (8.4.13)p q

where cq is plus or minus one according to whether q is an even or an

odd permutation. The sums in (8.4.13) are taken over all row per

mutations p and column permutations q respectively. The Young

symmetry operator Y associated with a tableau is given by

Y == PQ (8.4.14)

are linearly independent and all invariants s Y I(B1,... , B

n) are

expressible as linear combinations of the invariants (8.4.17). The

permutations s2' s3' ... are obtained (see §4.3) by listing the X~l··· np

standard tableaux associated with the frame [n1'" np] and then

determining the X~l··· np permutations which send the first standard

tableau into the remaining tableaux. For example, consider the

invariant

where P and Q are defined by (8.4.13). For example, the Young

symmetry operators associated with the tableaux (8.4.12) are given by

Y (1 2 3) == e + 12 + 13 + 23 + 123 + 132

y(~ 2) = (e+12)(e-13) = e + 12-13-132,

(8.4.15)

Y( ~ 3 ) = (e + 13)(e -12) = e + 13 -12 -123,

Y ( ~) = e - 12 - 13 - 23 + 123 + 132

(8.4.18)

where the Bi are symmetric second-order tensors. We note that tr BiBj

== tr B·B·. In order to generate a set of invariants of symmetry typeJ 1

(21), we observe that

II = Y ( 12 ) I = (e + 12 - 13 - 132) tr B1 tr B2B3(8.4.19)

== tr HI tr B2B3 + tr B2 tr B1B3 - 2 tr B3 tr B1B2 I: O.

222 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.5] Generation of the Multilinear Elements of an Integrity Basis 223

The standard tableaux associated with the frame [2 1] are

1 2, 1 33 2 (8.4.20)

With the character table for 53 (Table 4.2), we see that the character

(8.4.23) is that associated with the irreducible representation (21) of 53.

(8.4.21)

x(e), X(12), X(13), X(23), X(123), X(132) == 2, 0, 0, 0, -1, -1. (8.4.23)

(8.5.1)

8.5 Generation of the Multilinear Elements of an Integrity Basis

In this section, we follow Smith [1968b] and outline a procedure

similar to those employed by Young [1977] and Littlewood [1944]

enabling us to generate the multilinear elements of an integrity basis for

functions I(B1,... ,Bn) invariant under a group A. Let I?n denote the

number of linearly independent multilinear functions of B1,... , Bn

which

are invariant under A. These invariants form the carrier space for a

reducible representation of the group 5n comprised of the n! per

mutations of 1,2, ... , n. Let Pn n denote the number of times the1··· pirreducible representation (n1 ... np) appears in the decomposition of this

representation or, equivalently, the number of sets of invariants of

symmetry type (n1 ... np) arising from the set of I?n invariants. Let Qn

be the. nun:ber of invariants multilinear in B1,... , Bn which are of the

form 1~1 ... I~q where the 11'... ' Iq are elements of the irreducible integrity

basis and i1,... , iq are positive integers or zero. These Qn invariants

also form the carrier space for a reducible representation of Sn. Let

Qn1'" np denote the number of times the irreducible representation

(n1··· np) appears in the decomposition of this representation, i.e., the

number of sets of invariants of symmetry type (n1 ... np) arising from

the set of Qn invariants. We note that

h nl··· np h dwere Xe is t e imension of the irreducible representation

(n1'" np) and where the summation is over the set of all partitions

nl'" np of n. We give methods below for determining I?n Pn n, 1'·' p'

Qn, Qnl." np ' Let us assume for the moment that we are able to

determine these quantities. We proceed as follows.

(8.4.22)~l-1] .-1

o ] [1 -1 ] [-1, D(12) == , D(13) ==1 0 -1 -1

1 ] [-1 1] [ 0, D(123) == , D(132) ==o -1 0 1

Table 8.1 s II' s 12: S3

s e (12) (13) (23) (123) (132)

s 11 II 11 -11-12 12 -11-12 12

s 12 12 -11-12 12 11 11 -11-12

12 = (2 3)Y( §2 ) I = (23 + 132 -123 -12) tr'B1 tr B2B3

D(e) = [ ~

D(23) = [ ~

The character of the representation {D(s)} defined by (8.4.22) is given

by

Application of the permutation (23) to the first tableau In (8.4.20)

yields the second tableau. We have

The invariants 11' 12 then form a set of invariants of symmetry type

(21). The invariants sll' sl2 for s belonging to S3 are linear com

binations of II' 12 and are listed below.

The invariants 11' 12 form the carrier space for a representation {D(s)}

of the group S3 where, with (8.4.2), the matrices D(s) are given by

224 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.5] Generation of the Multilinear Elements of an Integrity Basis 225

1. We compute the number 1?1 of linearly independent invariants

which are linear in B1. Suppose that I?1 == p and that the p invariants

are given by 11 (B1) ,... , Ip (B1). These invariants are of symmetry type

(1) and give the set of integrity basis elements of degree one in B1.


which are multilinear in B1 and B2 and the numbers P 2 and P 11 of

sets of invariants of symmetry types (2) and (11) respectively which

arise from the IP2 invariants. There are Q2 = p2 invariants which are

bilinear in B1 and B2 and which arise as products of elements of the

integrity basis of degree one. These are given by

Ii (B1) Ij (B2) (i,j == 1, ... , p).

These invariants may be replaced by the ( P!1

) invariants

(8.5.2)


which are multilinear in B1,B2,B3 and the numbers P3,P21,P111 of

sets of invariants of symmetry types (3), (21), (111) respectively where

1P3=X~P3+X~IP21+X~11P111=P3+ 2P21 +P111· (8.5.5)

The Q3 invariants which are multilinear in B1, B2, B3 and which are

products of integrity basis elements of degrees 1 and 2 are given by

(i) the p3 invariants

(8.5.6)

which may be split into ( Pj21 sets of X~ = 1 invariants of symmetry

type (3), ip(p2 -1) sets of X~ = 2 invariants of symmetry type (21)

and ( ~ ) sets of X~11 = 1 invariants of symmetry type (111);

(ii) the 3pq invariants

and the ( ~ ) invariants

(8.5.3)Ii(B1) Jj (B2, B3), Ii (B2) Jj (B3, B1), Ii(B3) Jj (B1, B2)

(i == 1, ... , p; j == 1, ... , q) (8.5.7)

(iii) the 3pr invariants

The integrity basis then will have P3 - Q3' P 21 - Q21 and PIll - Q111

which may split into pq sets of invariants of symmetry type (3) and pq

sets of symmetry type (21);

which may be split into pr sets of invariants of symmetry type (21) and

pr sets of invariants of symmetry type (Ill).

(8.5.9)( p+2 ) 1 { 2Q3 == 3 +pq, Q21 ==3PP -l)+pq+pr,

Q111 = (~)+pr.

Thus, we have

Ii(B1) Kj (B2, B3), Ii(B2) Kj (B3, B1), Ii(B3) Kj (B1, B2)

(i == 1, ... , p; j == 1, ... , r) (8.5.8)

Ii(B1) Ij (B2) - Ii(B2) Ij (B1), (i,j == 1,... , p; i<j). (8.5.4)

We note that p2 = ( p!1 ) + ( ~). The invariants (8.5.3) are unaltered

under interchange of B1 and B2 and each of the invariants (8.5.3)

constitutes a set of invariants of symmetry type (2). The invariants

(8.5.4) change sign under interchange of B1 and B2 and each of the

invariants (8.5.4) forms a set of invariants of symmetry type (11).

Thus, we have Q2 = ( p!1 ) and Q11 = ( ~). The integrity basis must

then contain P 2 - Q2 and P 11 - Q11 sets of invariants of symmetry

types (2) and (11) respectively. These may be generated with aid of

the methods of §8.4. We suppose that P2 - Q2 == q, P 11 - Q11 == rand

that the elements of the integrity basis which are bilinear in B1, B2 and

of symmetry types (2) and (11) are given by J 1(B1, B2),···, Jq(B1, B2)

and K1(B1, B2), .. ·, Kr(B1, B2) respectively.

226 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.6] Computation of lPn' Pn n' Qn' Q1"· P n1 ··· np

227

sets of invariants of symmetry types (3), (21) and (111) respectively

which may be generated with the aid of the methods of §8.4.

We continue this iterative procedure so as to determine the

multilinear elements of the integrity basis of degrees 4,5, .... For each

particular problem, we must give an argument which will enable us to

determine the stage at which the iterative procedure may be termi

nated. The above procedure is formal in the sense that we assume that

the Qn (n = 2,3, ... ) multilinear invariants of degree n which arise as

products of elements of the integrity basis are linearly independent.

This is usually the case if n is small. However, there may be syzygies

which relate the invariants so that the number of linearly independent

multilinear invariants of degree n in B1,... , Bn which are products of

integrity basis elements is less than Qn. The existence of syzygies does

not cause any problems in the cases considered below. The number of

invariants in the integrity bases obtained coincides with the number

obtained upon employing different procedures which indicates that the

formal procedure does not develop any problems for the cases

considered.

In (8.6.1), 'I' '2' "·"n gives the cycle structure of a class, of per

mutations of the symmetric group Sn, i.e., 'I gives the number of one

cycles, '2 the number of two cycles, .... The summation is over the

classes of Sn and hI' gives the order of the class. The quantity X~l·" np

is the value of the character of the irreducible representation (n1 ... np)

of Sn for the class I' of Sn. The quantities hI" X~l"· np are listed in the

character tables for Sn (n = 2,3, ... ) given in §4.9. For the particular

case where n1". np = n, we have X!y = 1 for all classes of Sn. With

(8.6.1), we then have

n! ¢n = E h, sll s~2 ... sn'n. (8.6.3) ,,It has been shown by Schur [1927] that the quantities <Pnl". np are

expressible in terms of the quantities ¢1'···' ¢n. Thus,

¢n1 ¢n1+1 ¢nl+p-1

¢n1.. · np = 4>n2-1 4>n2 4>n2+p-2 (8.6.4)

<Pnp- p+l ¢np-p+2 4>np

where <Po = 1 and any <P with a negative subscript is zero. For example,

where

8..6 Computation of Pn, Pnl.... np' Qn, Qnl"" np

Let b l ,... , bs denote the independent components of a tensor B

chosen from the set of tensors B1,... , Bn, each of which transforms in

the same manner under the group A. Let the transformation properties

of the column vector [b1,... , bs]T under the group A be defined by the s

dimensional matrix representation {S(A)}. Corresponding to each

partition n1 ... np of n, we define the quantity

(8.6.5)

(8.6.6)

The number P of sets ofn1··· np

<P3 4>4 ¢5

<P321 = <PI 4>2 <P3

o 1 <PI

The number IPn of linearly independent functions which are multilinear

in B1,... , Bn and which are invariant under the group A is obtained by

taking the average over the group A of the quantity sr. We denote this

by

where M.V. stands for mean value.

(8.6.1)

(8.6.2)

228 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.6] Computation of lPn' Pn n' Qn' Qn n1·" pl··· P

229

Table 8.2 Decomposition of Representations S, T, U

Class 14 122 13 4 22

Class member e (12) (123) (1234) (12)(34)

Class order h, 1 6 8 6 3

X,: S 6 2 0 0 2

X,: T 3 1 0 1 3

X,: U 3 1 0 1 3

X~ 1 1 1 1 1

X~l 3 1 0 -1 -1

X~2 2 0 -1 0 2

by considering the manner in which the invariants (8.6.9) transform

under one element of each of the classes 14, 122 13 4 22 of, , ,permutations of 54 and then determining the trace of the associated

transformation matrix. This is given by the number (= X,) of

invariants which remain unaltered under a permutation belonging to

the class,. We then employ the orthogonality properties of the

characters of irreducible representations to determine the decomposition

of the reducible representation S, the value of whose character for the

class, of 54 is given by X,. Thus, with (8.4.8), the number of times

the irreducible representation (n1 ... np) appears in the decomposition of

th . S· . b 1 '""" h nl·" npe representatIon IS gIven y 4' L..J , X, X, where the sum-

mation is over the five classes of 5~. ' We collect the results in tabular

form below.

(8.6.8)

(8.6.7)

From these invariants, we obtain the following three sets of invariants

which are multilinear in B1,... ,B4 :

The number Qn of functions which are multilinear in B1,···, Bn ,

which are invariant under A and which arise as products of elements of

the integrity basis of degree less than n is determined by inspection.

For example, suppose that n == 4 and that the typical multilinear

elements of the integrity basis of degree less than 4 are I(B1, B2) and

J(B1, B2) where

If A is a continuous group, the averaging process indicated in (8.6.6)

and (8.6.7) is accomplished by integrating over the group manifold.

invariants of symmetry type (n1 ... np) arising from the I?n invariants

multilinear in B1,... , Bn is obtained by taking the average over the

group A of the quantity <Pnl ... np. Thus,

We then have Q4 == 12. The three sets of invariants (8.6.9), (8.6.10)

and (8.6.11) form the carrier spaces for reducible representations S, T

and U of dimensions 6, 3 and 3 respectively of the symmetric group 54·We wish to determine the decomposition of these representations. We

may determine the decomposition of the representation S, for example,

With (8.4.8) and Table 8.2, we see that the decompositions of the

representations S, T and U whose carrier spaces are formed by the

invariants (8.6.9), (8.6.10) and (8.6.11) are given by (4) + (31) + (22),

(4) + (22) and (4) + (22) respectively. Thus, we find that Q4 = 3,

Q31 = 1 and Q22 == 3.

230 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.6] Computation of lPn' P n n' Qn' Qn n1"· pl·" P

231

2. (1) . (1) == (2)+(11)

3. (2) · (1) == (3)+(21), (11)· (1) == (21)+(111)

4. (3)· (1) == (4)+(31), (21)· (1) == (31)+(22)+(211),

(111) . (1) == (211 )+(1111), (2)· (2) == (4)+(31 )+(22),

(2) . (11) == (31)+(211), (11)· (11) == (22)+(211)+(1111)

5. (4)· (1) == (5)+(41), (31)· (1) == (41)+(32)+(311),

(22) . (1) == (32)+(221), (211)· (1) == (311)+(221)+(2111),

(1111)· (1) == (2111)+(11111), (3)· (2) = (5)+(41)+(32),

(21) · (2) = (41)+(32)+(311)+(221), (111)· (2) = (311)+(2111),

(3)· (11) = (41)+(311), (21)· (11) = (32)+(311)+(221)+(2111),

(111) . (11) = (221)+(2111)+(11111)

6. (5)· (1) == (6)+(51), (41)· (1) = (51)+(42)+(411),

(32)· (1) == (42)+(33)+(321), (311)· (1) == (411)+(321)+(3111),

(221) . (1) = (321)+(222)+(2211),

(2111) . (1) = (3111)+(2211)+(21111),

(11111). (1) = (21111)+(111111), (4). (2) = (6)+(51)+(42),

(31) . (2) = (51 )+(42)+(411 )+(33)+(321),

(22) . (2) = (42)+(321 )+(222),

(211) . (2) = (411)+(321)+(3111)+(2211),

(1111). (2) == (3111)+(21111), (3)· (3) = (6)+(51)+(42)+(33),

(4)· (11) = (51)+(411), (31)· (11) = (42)+(411)+(321)+(3111),

(22)· (11) = (32)+(321)+(2211),

(211)· (11) = (321)+(222)+(3111)+(2211)+(21111),

(1111)· (11) = (2211)+(21111)+(111111),

(3) . (3) = (6)+(51 )+(42)+(33),

(3)· (21) = (51)+(42)+(411)+(321), (3)· (111) = (411)+(3111),

(21) . (21) = (42)+(411 )+(33)+2(321 )+(3111 )+(222)+(2211),

(21) . (111) == (321)+(3111)+(2211)+(21111),

(111)· (111) == (222)+(2211)+(21111)+(111111)

The procedure indicated above can become tedious if the

number of invariants comprising the carrier space of a reducible repre

sentation is large. In practice, it is usually preferable to employ results

due to Murnaghan [1937]. We note that the invariants I(B1, B2) and

J(B1, B2) given by (8.6.8) form carrier spaces for irreducible repre

sentations (2) and (2) respectively. The set of invariants (8.6.9) forms

the carrier space for a reducible representation which we refer to as the

product of the representations (2) and (2) and denote, by (2) . (2). The

decomposition of these product representations is discussed in §4.6.

This problem has been considered by Murnaghan [1937] who lists the

decompositions of (mI ... mp) . (n1 ... nq) for all cases such that m1 + ...

+ mp + n1 + ... + nq ~ 9. We record in Table 8.3 the results of

Murnaghan [1937], pp. 483-487, which are required below. The set of

invariants (8.6.10) forms the carrier space for a reducible representatio~

of S4 which we refer to as the symmetrized product of the repre

sentations (2) and (2) and denote by (2) x (2). The decomposition of

such representations (see §4.6) has been considered by Murnaghan

[1951]. We list in Table 8.4 the decompositions of the symmetrized

products required below. These results may be obtained by the

procedure leading to Table 8.2 or may be found in Murnaghan [1951].

We note that some caution is required when determining the decom

position of a symmetrized product of the representations (n1". np) and

(n1 .. · np). For example, the quantity a1 b1 forms the carrier space for a

representation (2) of S2 since a1b1 is unaltered under interchange of

a and b. The carrier space for the symmetrized product (2) x (2) of this

representation is formed by the single quantity a1 b1c1d1 which is of

symmetry type (4). In this case, we have (2)x(2) ==(4) rather than

(4) + (22) as listed in Table 8.4. We note that the dimension of a

reducible representation must be equal to the sum of the dimensions of

the irreducible representations into which it is decomposed. This serves

as a check and should enable us to avoid errors in degenerate cases such

as that mentioned above.

Table 8.3 Decomposition of (mI ... mp) . (n1 ... nq)

232 Generation of Integrity Bases: Continuous Groups [Ch. VIIISect. 8.7] Traceless Symmetric Second- Order Tensors: R3 233

respectively where the BiBj ... Bk ± Bk... BjBi are of degree five

or less;

(ii) the multilinear terms appearing in the general expressions for

A(B1,... , Bn) and S(B1,... , Bn) are of the forms

(i) the multilinear elements of an integrity basis are of degree six or

less and are of the form tr BiBj ... Bk;

(8.7.1)(BiBj Bk - Bk··· BjBi) tr B£ Bm ,

(BiBj Bk +Bk.. · BjBi) tr B£ Bm

as isotropic functions. This problem differs from that considered by

Spencer and Rivlin [1959 a, b] and Spencer [1961] only in that we

impose the restriction that tr Bi == 0 (i == 1,... , n). We borrow from the

discussions of Spencer and Rivlin the results that

Table 8.4

4. (2) x (2) == (4)+(22), (11) x (2) == (22)+(1111)

6. (3) x (2) == (6)+(42), (2) x (3) == (6)+(42)+(222),

(11) x (3) == (33)+(2211)+(111111)

8. (2) x (4) == (8)+(62)+(44)+(422)+(2222),

(11) x (4) == (44) + (3311)+ (2222)+ (221111 )+ (11111111)

10. (2) x (5) == (10)+(82)+(64)+(622)+(442)+(4222)+(22222),

(11)x(5) == (55)+(4411)+(3322)+(331111)+(222211)

+ (2 2 111111)+ (1111111111)

12. (2) x (6) == (12)+(10,2)+(84)+(822)+(66)+(642)+(6222)

+(444)+(4422)+(42222)+(222222)

(iii) the trace of a matrix product of symmetric matrices is unaltered

by cyclic permutation of the factors in the product 'and is also

unaltered if the order of the factors is reversed. Thus,

The results (i) and (ii) above are critical in that they indicate when the

iterative procedures to be employed may be terminated.

We first consider the problem of generating the multilinear

elements of an integrity basis for functions of the traceless symmetric

second-order tensors B1,... , Bn which are invariant under R3 . The

matrix which defines the transformation properties under A of the

column vector [B11 , B12, B13, B22, B23, B33 ]T whose entries are the

six independent components of a three-dimensional symmetric second

order tensor B' is the symmetrized Kronecker square A(2) of A.

8.7 Invariant Functions of Traceless Symmetric Second-Order

Tensors: R3

In this section-, we employ the procedure of §8.5 to generate the

multilinear elements of an integrity basis for functions of an arbitrary

number of three-dimensional symmetric second-order traceless tensors

B1 B which are invariant under the three-dimensional proper ortho-,... , n

gonal group R3 . We then generate the multilinear elements appearing

in the general expressions for skew-symmetric second-order tensor

valued functions A(B1,... , Bn) and for traceless symmetric second-order

tensor-valued functions S(B1,... ,Bn) which are invariant under R3 .

The non-linear terms in these general expressions may be readily

generated from the multilinear terms. Since the restrictions imposed on

functions of second-order tensors by the requirements of invariance

under the proper and full orthogonal groups are identical, the results

obtained here also apply for the full orthogonal group 03. We refer to

functions which are invariant under the proper or full orthogonal groups

tr B1B2B3 == tr B2B3B1 == tr B3B1B2,

tr B1B2B3 == tr B3B2B1.(8.7.2)

234 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 235

Suppose that A is the matrix corresponding to a rotation through B

radians about the x3 axis, i.e.,

We see from (4.4.17)1' (4.4.18) (or (5.2.10)) and (8.7.3) that

tr A(2) = ~(tr A)2 +~tr A2

= ! (eiB + 1 + e-iB)2 +! (e2iB + 1 + e-2iB)

= e2iB +eiB +2 +e-iB +e-2iB.

cos B sin B

A = -sin B cos B

o 0

oo1

(8.7.3)

(8.7.4)

The number Pn1 ... np of sets of invariants of symmetry type (nl ... np)

arising from these IPn invariants is given by

271"

P Ul .. · Up = l1r J<PUl .. ·up(l-cosB)dB. (8.7.8)o

The <Pul ... up are defiued by (8.6.3) aud (8.6.4) iu terms/>f the Sr where

Sr = e2irB + eirB + 1 + e- irB + e- 2irB . (8.7.9)

The quantities 1P1,... , 1P6 may be computed from (8.7.7) and are given

by

1P1 = 0, 1P2 = 1, 1P3 = 1, IP4 = 5, 1P5 = 16, 1P6 = 65. (8.7.10)

The number IPn of linearly independent scalar-valued functions which

are multilinear in B1,... , Bn and which are invariant under R3 is seen

with (2.6.19)2' (8.6.6) and (8.7.6) to be given by

271"Pu = 2~ Jsf(l-cosB)dB. (8.7.7)

o

where B is a symmetric traceless second-order tensor, i.e., tr B = O. We

observe that tr B' = Bii is invariant under the group R3 . The six

independent components of B' may be split into two sets comprised of

tr B' and the five independent components of the traceless tensor B

respectively. The quantity tr B' forms the carrier space for the identity

representation of R3 . The five independent components of the traceless

tensor B form the carrier space for an irreducible representation of R3 ,

the value of whose character for the class of R3 comprised of rotations

through (} radians about some axis is seen from (8.7.4) to be given by

We now generate the typical multilinear elements of the integ

rity basis. We list in Table 8.6 below the quantities Pn1 ... np' Qnl". npand X~l···Up for those u1."up for which PUl ... UP f:. o. The PUl ... uP

We list in Table 8.5 the mean values over the group R3 of the

quantities <pp ... <Pq, i.e., l1r J<pp ... <Pq (1 - cos B) dB, for all positive

values of p, ... , q such that p + ... + q ~ 6.

Table 8.5 Mean Values over R3 of </>p ... </>q

</>p ... </>q </>1 </>2 </>1 </>3 </>2</>1 </>t </>4M.V. (</>p ... </>q) 0 1 1 1 1 1 1

</>p ... </>q </>3</>1 </>~ </>2</>r </>1 </>5 </>4</>1 </>3 </>2M.V. (</>p ... </>q) 1 3 3 5 1 2 3

</>p ... </>q </>3</>1 </>~</>1 </>2</>t </>1 </>6 </>5</>1 </>4</>2M.V. (</>p ... </>q) 4 6 9 16 2 2 5

</>p ... </>q </>4</>1 </>~ </>3</>2</>1 </>3</>Y </>~ </>2</>1 </>rM.V. (</>p ...</>q) 5 5 9 14 21 36 65

(8.7.5)

(8.7.6)

We may set

B' = B + 1(tr B')E3, B = B' -1 (tr B')E3,

236 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 237

may be readily computed with the aid of Table 8.5, (8.6.4) and (8.7.8).

The quantity X~I··· np gives the number of invariants comprising a set

of invariants of symmetry type (n1 ... np). The values of X~I"· np are

found in the first column of the character tables for 5n (n = 2,3, ... )

given in §4.9. The computations yielding the QnI... np are indicated

below. We observe that 1P1 = 0 so that there are no invariants of

degree one. We have 1P2 = 1; P2 = 1, P 11 = o. Since there are no

invariants of degree one, there are no invariants of degree two which

arise as products of integrity basis elements of degree one. Hence,

Q2 = 0; Q2 = Q11 = o. We then have P2 - Q2 = 1 set of invariants of

symmetry type (2) appearing in the integrity basis. This set is com

prised of X~ = 1 invariant which is given by

We see as in §8.4 or §8.6 that the invariants (8.7.13) form a set of

invariants of symmetry type (4) + (22). We also note (see §8.6) that

the invariants (8.7.13) form the carrier space for a representation which

is referred to as the symmetrized product of the irreducible

representations (2) and (2). This is denoted by (2) x (2) and, from

Table 8.4, we have that (2) x (2) = (4) + (22). Thus, Q4 = Q22 = 1,

Q31 = Q211 = Q1111 = o. The integrity basis will then contain

P22 - Q22 = 2 - 1 = 1 set of invariants of symmetry type (22) which is

comprised of X~2 = 2 invariants. These are given by (see §8.4)

(8.7.14)

Y(123) tr B1B2B3 = (e +12 +13 +23 + 123 +132) tr B1B2B3

=6trB1B2B3; (3) (8.7.12)

where we have employed (8.7.2). We next see that IP4 = 5; P4 = 1,

P22 = 2, P31 = P211 = P 1111 =0. There are three linearly indepen

dent multilinear invariants which arise as products of invariants of the

form (8.7.11). These are given by

tr B1B2 tr B3B4, tr B1B3 tr B2B4, tr B1B4 tr B2B3. (8.7.13)

where we have noted that tr B1B2 = tr B2B1. The designation (2) in

(8.7.11) indicates that tr B1B2 forms a set of invariants of symmetry

type (2). We next observe that 1P3 = 1; P3 = 1, P21 = PIll = O.

There are no invariants of degree three arising as products of invariants

of lower degree. Hence, Q3 = Q3 = Q21 = Q111 = O. The integrity

bases will then contain P3 - Q3 = 1 set of invariants of symmetry type

(3) which consists of a single invariant since X~ = 1. This is given by

5. tr B1B2 tr B3B4B5, (~) = 10, (2)· (3) = (5) + (41) + (32);

6. tr B1B2 tr B3B4 tr BSB6, IS, (2) x (3) = (6) + (42) + (222);

(8.7.15)

We further observe that 1P5 = 16; Ps = P41 = P32 = P221 = P11111

= 1; 1P6 = 65; P6 = P222 = 2, P42 = 3 and P321 = P3111 = 1. The

multilinear invariants of degree 1,1,1,1,1 in B1,... , BS and of degree

1,1,1,1,1,1 in B1,... , B6 which arise as products of elements of the

integrity basis of lower degree may be divided into sets of invariants

which form carrier spaces for reducible representations of the symmetric

groups 55 and 56. We list below a typical invariant from each of these

sets the number of invariants in the set and the representation for,which these invariants form the carrier space. The irreducible represen

tations into which these representations may be decomposed are given

in Tables 8.3 and 8.4 and are also listed. The quantities QnI ... np for

n1 +... +np = 5,6 appearing in Table 8.6 may then be immediately

determined.

(8.7.11)Y(12) tr B1B2 = (e + 12) tr B1B2 = tr B1B2 + tr B2B1

= 2tr B1B2; (2)


Table 8.6 Scalar-Valued Invariant Functions of B1,···, Bn: R3 6.

n1 .. · np 2 3 4 22 S 41 32 221 11111 6 42 321 3111 222

P 1 1 1 2 1 1 1 1 1 2 3 1 1 2nl .. ·np

Qnl···np 0 0 1 1 1 1 1 0 0 2 3 1 0 2

nl .. ·np 1 1 1 2 1 4 S S 1 1 9 16 10 SXe

Ij(B1,B2,B3,B4) tr BSB6, 30, (2)· (22) = (42) + (321) + (222);

tr B1B2B3 tr B4BSB6, 10, (3) X (2) = (6) + (42).

We see from (8.7.1S) that QS = Q41 = Q32 = 1, Q6 = 2, Q42 = 3,

Q321 = 1, Q222 = 2. The remaining Qnl ... np (n1 +... +np = S or 6)

are zero. We list the results in Table 8.6.

12

S. JO(B1,B2,B3,B4,BS)=Y 3 trB1B2B3B4BS' (11111);4S (8.7.17)

J 1(B1, B2, B3, B4, BS)'···' JS(B1, B2, B3, B4, BS)

= [e, (45), (23), (23)(45), (2453)J V( i ~) tr B1B2B3B4B5' (221)j

K 1(B1,···, B6),···, K 10(B1,.. ·, B6)

= [ e, (34), (354), (3654), (234), (2354), (23654), (24)(35),

(24)(365), (25364)J V(i2 3 ) tr B1B2B3B4B5B6' (3111).

We see with Table 8.6 that Pn1 ... np - Qnl ... np = 1 if n1." np = 2,3,

22,221, 11111,3111 and is zero otherwise. The typical multilinear

elements of an integrity basis are then comprised of one set of

invariants of each of the symmetry types

We may then apply the procedure of §8.4 to generate the typical multi

linear elements of an integrity basis for functions of traceless symmetric

second-order tensors B1, B2, ... which are invariant under R3 . These are

comprised of the sets of invariants listed below.

(2), (3), (22), (221), (11111), (3111). (8.7.16)

We next indicate the manner in which one may generate the

non-linear elements of an integrity basis given the typical multilinear

elements (8.7.17). We list only the typical non-linear elements. For

example, the n(n - 1) invariants tr B[Bj (i,j = 1,... ,nj i t= j) are

elements of the integrity basis. We list only the typical invariant

tr BrB2' We obtain the non-linear elements of the integrity basis upon

identifying certain of the tensors B1,... , Bn in the multilinear basis

elements. Thus, all of the non-linear basis elements of degree six may

be obtained upon identifying tensors in the invariants K 1(B1,... , B6), ... ,

K10(B1,···, B6)·

Consider the reducible representation of the group

defined by the matrices D(e), ... , D(132) which describe the manner in

which the invariants Ki (B1, B2, B3, B4, BS' B6) (i = 1,... ,10) transform

under the permutations (8.7.18). We may form two sets of invariants,

the elements of which are linear combinations of the K1,... , K10. The

2. tr B1B2, (2) ;

3. tr B1B2B3, (3) ;

4. I1(B1,B2,B3,B4)' I2(B1,B2, B3, B4)

= [e, (23) ] V ( ~ ~ ) tr B1B2B3B4, (22);

53 = { e, (12), (13), (23), (123), (132) } (8.7.18)


(8.7.22)

(8.7.23)1 '"" 3111() 1 (4! ~ X s == 24 10 - 6·2 + 8 ·1 + 6·0 - 3 ·2) == 0

Similarly, the number of linearly independent linear combinations of

the K1,... , K10 which are symmetric in B1, B2, B3, B4 is given by the

number of times the identity representation appears in the decom

position of the representation of the group 54 of permutations of 1,2,3,

4 whose carrier space is formed by the set K 1,... , K 10 of invariants of

symmetry type (3111). This number is

where the summation is over the permutations of 54 which are divided

into five classes denoted by their cycle structures 14, 122, 13, 4, 22 and

comprised of 1, 6, 8, 6, 3 permutations respectively. Since B5 and B6are not affected by permutations of the subscripts 1, ... ,4, the values of

the characters of the representation considered corresponding to the

classes 14, 122, 13,4,22 of 54 may be read off from the values 10, -2,

1, 0, -2 of the character of the irreducible representation (3111) of 56

(see Table 4.5) corresponding to the classes 16, 142, 133, 124, 1222 of

56. In similar fashion, we may show that there are no basis elements of

degrees 6 in BI ; (5, 1), (4,2), (3,3) in BI , B2; (3,2, I), (2,2,2) in BI , B2,

B3; there are 4 basis elements of degrees (2,1,1,1,1) in B1, B2, B3, B4,

B5 and 1 basis element of degrees (2,2,1,1) in B1, B2, B3, B4 which

arise from the K 1,... K10. The number of typical non-linear basis

elements arising from the sets of invariants of symmetry types

(2), (3), ... , (221) may be obtained in the same manner. We list below

the typical multilinear elements of the integrity basis together with the

typical non-linear basis elements obtained from them by the

identification process.(8.7.21)

(8.7.20)

(8.7.19)

( ~23)_ (~23)sY 5 - Y 5

6 6

sInce

where the summation is over the s belonging to the group (8.7.18). We

note that e and (12), (13), (23) and (123), (132) belong to the classes 16,

142 and 133 of 56 respectively. The values X3111 (s) of the character of

the irreducible representation (3111) are found in the character table for

S6 (Table 4.5). The invariant which is symmetric in B1, B2, B3 is given

by

elements of one set are symmetric in B1, B2, B3, i.e., they are invariant

under the group (8.7.18). Each of these invariants forms a carrier space

for the identity representation of 53. The invariants comprising the

second set form the carrier space for a reducible representation of 53

which does not contain the identity representation. These invariants

will vanish identically when we set B1 == B2 == B3 in them. The basis

elements of degrees 3,1,1,1 in B1, B4, B5, B6 are obtained upon setting

B1 == B2 == B3 in the first set of invariants which are symmetric in

B1, B2, B3. The number of linearly independent linear combinations of

the K 1,... , K 10 which are symmetric in B1, B2, B3 is equal to the

number of times the identity representation occurs in the decom

position of the representation of the group (8.7.18) whose carrier space

is formed by the set K1,... ,K10 of invariants of symmetry type (3111).

This number, is given by

for all s belonging to the group (8.7.18). The element of the integrity

basis of degree 3,1,1,1 in B1, B4, B5, B6 is then given by

2. Two tensors:

One tensor:


5. Five tensors: Ji(B1, B2, B3, B4, B5) (i = 0, 1,2,3,4,5);

Four tensors: Ji(B1, B1, B2, B3, B4) (i = 1,2);

Three tensors: J1(B1, B1, B2, B2, B3); upon inserting 1,1,2,3,4,5 in the frame and the standard tableaux

left to right in each of the rows. The number of linearly independent

invariants of degree 2,1,1,1,1 in B1,B2,B3,B4,B5' of degree 3,1,1,1

in B1, B2, B3, B4 and of degree 2,2,1,1 in B1, B2, B3, B4 which may be

obtained from the set of invariants Ki(Bl ,... , B6) (i = 1,... ,10) of

symmetry type (3111) appearing in (8.7.17) upon appropriately iden

tifying tensors is given by the number of standard tableaux obtained

upon inserting the integers (1,1,2,3,4,5), (1,1,1,2,3,4) and (1,1,2,

2,3,4) respectively into the boxes of the frame [3111] associated with

the partition 3111. Thus, we obtain the four standard tableaux

3.

4.

Three tensors:

Two tensors:

One tensor:

Four tensors:

Three tensors:

Two tensors:

tr B1B2B3;

tr ByB2;

tr B3.l'

2tr B1B2B3B4 -tr B1B3B2B4 -tr B1B2B4B3,

2 tr B1B3B2B4 - tr B1B2B3B4 - tr B1B3B4B2;

tr ByB2B3 - tr BIB2BIB3;

tr ByB~ - tr BI B2BIB2; (8.7.24)1 1 2 ,345

1 1 3 ,245

1 1 4 ,235

115234

The irreducible representations of the group R3 are of dimensions

1,3,5,7, .... The independent components of a vector or a skew-sym

metric second-order tensor, a traceless symmetric second-order tensor, a

traceless symmetric third-order tensor,... form the carrier spaces for

upon inserting 1, 1, 1,2,3,4 and 1, 1,2,2,3,4 respectively in the frame

[3111]. This tells us that we may obtain four linearly independent

invariants of degree 2,1,1,1 in B1,B2,B3,B4,B5 upon replacing

Bl,B2,B3,B4,B5,B6 in the Ki(B1,... ,B6) by Bl,B1,B2,B3,B4,B5'''.'

and a single linearly independent invariant of degree 2,2, 1, 1 In

B1,B2,B3,B4 upon replacing B1,B2,... ,B6 in the Ki(B1,... ,B6) by

B1, B1, B2, B2, B3, B4. We note that we are unable to obtain any

standard tableaux upon inserting (1,1,1,1,1,1), (1,1,1,1,2,3), ... into

the frame [3111].

6. Six tensors: Ki(B1,B2,B3,B4,B5,B6) (i = 1, ... ,10);

Five tensors: Ki(B1,B1,B2,B3,B4,B5) (i = 1,2,3,4);

Four tensors: K1(B1,B1,B1,B2,B3,B4)'

K1(B1, B1, B2, B2, B3, B4) +K2(B1, B1, B2, B2, B3, B4)·

We now outline a graphical method for determining the number

of basis elements of various degrees in B1, B2 ,... which may be obtained

from a set of invariants of symmetry type (n1 ... np) upon identifying

certain of the tensors. Consider, for example, the frame associated with

the partition 3111 of 6. We obtain a tableau upon inserting integers

1,2, ... into the boxes of the frame. If two or more of the integers are

the same, we say that the tableau is standard if the integers increase

from top to bottom in each of the columns and are non-decreasing from

111234

and 112234


irreducible representations of R3 of dimensions 3,5,7, ... respectively.

The values of the characters of these representations corresponding to

the class of rotations through fJ radians are given by

27rRn1 ... np = 2~ JcPnl ... np(eiO + 1 +e-iO)(l-cosO)dO

o(8.7.28)

Table 8.7 Skew-Symmetric Tensor-Valued Functions of B1,... ,Bn: R3

n1 .. · np 11 21 111 31 211 41 32 311 221 2111

R 1 1 1 2 2 2 2 3 1 1nl· .. np

S 0 0 0 1 1 2 1 3 1 1nl···npnl···np 1 2 1 3 3 4 5 6 5 4Xe

where the <Pnl". np are defined by (8.6.3), (8.6.4) and (8.7.9). Let

Snl". np denote the number of sets of skew-symmetric second-order

tensor-valued functions of symmetry type (nl". np) which are invariant

under R3 and which arise from the products of skew-symmetric second

order tensor-valued functions of degree m < n in the B· and invariants1

of degree n - m in the Bi. We list below in Table 8.7 the quantities

Rn1 np ' Snl... np and X~l.. ·np for those nl ... np for which

Rnl np f::. 0 and nl +... +np ~ 5. The Rnl ... np are computed from

(8.7.28). The number X~l·" np of functions comprising a set of

functions of symmetry type (nl." np) is found in the first column of the

character table for the symmetric group Sn. The computations yielding

the Snl." np are given below.

(8.7.26)

(8.7.25)

respectively. We next generate the general expression for a skew

symmetric second-order tensor-valued function which is invariant under

R3 and multilinear in the traceless symmetric second-order tensors

Bl , B2, ... , Bn . The 5n independent components of the tensor

B~ · ... B!1. form the carrier space for a 5n_dimensional reducibleIlJl InJn

representation of R3 whose character corresponding to a rotation

through 0 radians is given by sf = (e2iO + eiO + 1 + e-iO + e-2iOt. The

three independent components of a skew-symmetric second-order tensor

form the carrier space for an irreducible representation of R3 whose

character corresponding to a rotation through fJ radians is given by

eifJ +1 +e-iO. The number of times this representation appears in the

decomposition of the 5n_dimensional representation is given by

27rIRn = l1r Jsf(eiO + 1 + e-iO)(l- cos 0) dO,

o

The quantities 1R1,... ,1R5 may be computed from (8.7.26) and are given

by

The number Rn n of sets of skew-symmetric second-order tensor1"· p

valued functions of symmetry type (nl". np) arising from these IRn

functions is given by

1R1 = 0, 1R2 = 1, 1R3 = 3, 1R4 = 12, 1R5 = 45. (8.7.27)

From Table 8.7, we see that Rnl ... np -Snl... np = 1 ifnl ... np = 11,21,

111, 31, 211, 32 and is zero otherwise. Thus, the typical multilinear

skew-symmetric second-order tensor basis elements are comprised of

one set of functions of each of the symmetry types (11), (21), (111),

(31), (211) and (32). These are given by


3.

4.

s.

(8.7.29)

[e, (34), (23)J Y( ~ 2 4 ) (B1B2B3B4 -B4B3B2B1), (31),

[e, (23), (243)J Y(! 2 ) (B1B2B3B4 -B4B3B2B1), (211);

[e, (23), (45), (345), (23)(45)J Y( ~ ~ 5 ) (BIB2B3B4B5

- BSB4B3B2B1)' (32).

[ e, ... , (23)(45)]Y( ~ ~ 3 ) to BIB2B3B4B5 - B5B4B3B2Bl will yield aset of null matrices. Application of the symmetry operators [e, ... ,

(23)(45)JY( 1~ 4) to BIB2B3B4B5 -BSB4B3B2Bl will yield a set ofmatrices which are equivalent to the set of skew-symmetric matrices

comprising the set of symmetry type (32) which arises from matrices of

the form Y( j 2 )B1B2B3 tr B4B5. Consequently, the set of matrices

fe, ... , (23)(45)] Y(1 ~ 4) (BIB2B3B4B5 - B5B4B3B2Bl) cannot serve asbasis elements. Some care is clearly required in choosing the symmetry

operators which generate the set of basic skew-symmetric matrices of

symmetry type (32).

The multilinear elements of degrees 1,1,1,1 in B1, B2, B3, B4 and

1,1,1,1,1 in B1,B2,B3,B4,BS which arise as products of the terms in

(8.7.29) with the invariants (8.7.17) may be divided into sets of

functions which form carrier spaces for reducible representations of the

symmetric groups 54 and 55. We list below a typical term from each of

these sets, the number of terms in the set and the representation for

which these functions form the carrier space. The decomposition of

these representations may be found in Table 8.3 (p. 231).

We consider next the generation of the expression for a traceless

symmetric second-order tensor-valued function which is invariant under

R3 and multilinear in the traceless symmetric second-order tensors

B1, B2, ... Bn. The five independent components of a traceless sym

metric second-order tensor form the carrier space for an irreducible

representation of R3 whose character corresponding to a rotation

through f) radians is given by e2iO + eiO + 1 + e-iO + e-2iO. The number

of times this representation appears in the decomposition of the Sn

dimensional representation whose carrier space is formed by the Sn

independent components of the tensor Bf . ... B~. is given byI1J1 InJn

The Sn1 ... np appearing In Table 8.7 are determined from

(8.7.30). We observe that application of the symmetry operators

(8.7.31)

(8.7.32)

21rIfn = 2~ Jsl(e2i8 + ei8 + 1 +e-i8 + e-2i8)(1 - cos 8) d8

o

h 2if) + if) + 1 + -if) + -2if) Th .."1r"1rwere sl == e e e e . e quantItIes u1'···' Us may

be computed from(8.7.31) and are given by

The number Tn1 ... np of sets of traceless symmetric second-order

tensor-valued functions of symmetry type (nl ... np) arising from the Ifn

functions is given by

(8.7.30)

(iii) Y( ~ 2 )B1B2B3 tr B4B5, 20,

(21)· (2) == (41) + (32) + (311) + (221);

(iv) Y( ~ ) B1B2B3 tr B4B5, 10, (111)· (2) = (311) + (2111).


21rT ==...L J<P (e2i8 + ei8 + 1 + e-i8 + e-2iO)(1 - cos 8) d8nl .. ·np 21r nl· .. np

o (8.7.33)

where the <Pnl ... np are given by (8.6.3), (8.6.4) and (8.7.9). Let

Unl ... np be the number of sets of traceless symmetric second-order

tensor-valued functions of symmetry type (nl". np) which arise from

the products of functions such as B1, B1B2 + B2B1 - i E3 tr B1B2, ...

with the invariants (8.7.17). We list the quantities Tnl np ' Unl ... npand X~l·.. np in Table 8.8 for the n1 ... np where Tnl np::JO and

nl + ... + np ~ 5. The Tnl np are computed from (8.7.33). The com-

putations yielding the Unl np are given below.

[e, (23), (243)J v(! 2 )(B1B2B3B4 +B4B3B2B1), (211),

v( ~ }B1B2B3B4 +B4B3B2B1), (1111);

s. ~,(23), (34), (354~ v( ~ 3 }B1B2B3B4BS + BSB4B3B2B1)' (2111),

[ e, (34), (354), (234), (2354),

(24)(3S)J v( i 2 3 ) (B1B2B3B4BS + BSB4B3B2B1)' (311).

nl". np 1 2 3 21 4 31 22 211 1111 5 41 32 311 221 2111

Table 8.8 Traceless Symmetric Tensor-Valued Functions of B1,... ,Bn:R3

With Table 8.8, we have Tnl ... np - Unl ... np == 1 if nl ... np == 1,2,21,

22, 211, 1111, 311, 2111 and equals zero otherwise. The typical

multilinear traceless symmetric second-order tensor basis elements are

comprised of one set of functions of each of the symmetry types (1),

(2), (21), (22), (211), (1111), (311), (2111). These are given by

1234231

0234130

1 1 4 5 6 5' 4

3. B1 tr B2B3, 3, (1)· (2) == (3) + (21);

4. (B1B2 + B2B1 - i E3 tr B1B2) tr B3B4, 6, (2)· (2)

== (4) + (31) + (22),

B1 tr B2B3B4, 4, (1)· (3) == (4) + (31);

S. B1V( ~ ~ )tr B2B3B4B5, 10, (1). (22) = (32) + (221), (8.7.35)

B1trB2B3trB4B5' 15, (1)·(4)+(1)·(22)

== (5) + (41) + (32)+(221),

The multilinear traceless symmetric second-order tensor-valued

functions of total degree five or less which arise as products of the basis

elements (8.7.34) and the invariants (8.7.17) may be split into sets

which form the carrier spaces for reducible representations of the

symmetric groups 53' 54 and 55. We indicate below a typical term

from each of these sets, the number of terms in the set and the

representation for which these functions form the carrier space. We

employ Table 8.3 to obtain the decompositions.

(8.7.34)

1. B1, (1);

2. B1B2+B2B1-iE3trB1B2' (2);

3. [e,(23)JVn3)(B1B2B3+B3B2B1)' (21);

4. [e, (23)J vO D(B1B2B3B4 + B4B3B2B1), (22),

Tnl np 1 1 1 2 2 2 2 1

Unl np 0 0 1 1 2 2 1 0

X~l··· np 1 1 1 2 1 3 2 3

250 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd-Order Tensors: R3 251

(B1B2 + B2B1 - i E3 tr B1B2) tr B3B4BS' 10, (2)· (3)

= (5) + (41) + (32),

y(~ 3)(B1B2B3+B3B2B1)trB4BS' 20, (21)'(2)

= (41) + (32) + (311) + (221).

The value of the Unl ... np appearing in Table 8.8 follow Im

mediately from (8.7.35).

8.8 An Integrity Basis for Functions of Skew-Symmetric Second-Order

Tensors and Traceless Symmetric Second-Order Tensors: R3

representations are denoted by (ml". m q) and (nl." np ) where ml". m qand nl ... np are partitions of m and n respectively. The characters of

these representations are denoted by Xm1·" mq(s') and Xn1"· np(s")

where s' and s" are elements of Sm and Sn respectively. There are k£

inequivalent irreducible representations of 5 = SmSn denoted by

(ml ... m q, nl ... np ) whose characters are Xm1··· m q(s') Xnl .. ·np(s").

Consider the set of invariants I.(AI A Bl B) (J. - 1 r)J ,... , m' ,... , n - ,... ,

which are such that application of any permutation s = s's" of SmSn

will send each of the Ij into a linear combination of II'.'" Ir . This set of

invariants will form the carrier space for a r-dimensional representation

of SmSn. Let s' be the permutation which carries AI'''.' Am into

Ai1,· .. ,Aim and s" the permutation which carries B1,... ,Bn into

BJ. ,... , B

J.. We define the invariant sI.(Al A Bl B) by1 n J ,... , m' ,... , n

where s = s's". We may then determine a r x r matrix D(s) such that

which describes the transformation properties of the II' ... ' Ir under a

permutation s = s's" of SmSn. The m!n! matrices D(s) = [Dk·(s)]

furnish a r-dimensional representation of SmSn' The set of invari~tsIl, ... ,Ir may be split into sets of invariants where each set of invariants

forms the carrier space for an irreducible representation of SmSn. A set

of invariants which forms the carrier space for an irreducible repre

sentation (ml". mq, nl"· nq) will be referred to as a set of invariants of

symmetry type (mI." m q, nl ... np ). The number of invariants com

prising a set of invariants of symmetry type (mI ... mq, nl ... np) is givenb ml···mq nl .. ·nq h ml mq d nl ny Xe Xe were Xe ... an Xe ··· p are the values of

the characters of the representations (mI ... mq) and (nl ... np) corre

sponding to the identity permutation e.

In this section, we generate the typical multilinear basis

elements for scalar-valued functions W(Al ,.·., Am' B l ,... , Bn) of the

skew-symmetric second-order tensors AI'''.' Am and the traceless sym

metric second-order tensors Bl ,... , Bn which are invariant under the

proper orthogonal group R3 . We note that the restrictions imposed on

W(Al ,···, Am' Bl ,···, Bm ) by the requirements of invariance under R3

and by the requirement of invariance under 03 are the same. Thus, the

integrity basis generated here will also form an integrity basis for

functions of AI' ... ' Am' Bl ,... , Bn which are invariant under the full

orthogonal group 03. This problem has been considered by Spencer

and Rivlin [1962] and by Spencer [1965]. The procedure employed here

differs from that adopted by Spencer and Rivlin. Let 5 = SmSn denote

the direct product of the symmetric groups Sm and Sn. The group 5 is

comprised of elements of the form s's" where s' is an element of the

group Sm of all permutations of the subscripts l, ... ,m on the Al, ... ,Amand s" is an element of the group Sn of all permutations of the

subscripts l, ... ,n on the Bl, ... ,Bn . Let k and £ denote the number of

inequivalent irreducible representations of Sm and Sn respectively. The

(8.8.1 )

(8.8.2)

252 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd- Order Tensors: R3 253

The number IPm n of linearly independent functions which are,multilinear in AI"'" Am' BI,oo., Bn and which are invariant under R3 is

given by

mI + 00' +np :s; 6. We also exclude from Table 8.9 the cases where

mI +... +mq == 0 and nI +... +np == 0 since these refer to invariants

involving only the BI, ... ,Bn or only the AI, ... ,Am .

where the 4>~1'" mq' 4>~1'" np are defined by (8.6.3) and (8.6.4) with

the Sr appearing in (8.6.3) being replaced by s~ and s~ respectively

where

s~ = eirO +1 +e-irO, s~ = e2irO+ eirO+ 1 +e-irO+ e-2irO. (8.8.5)

Let Qm1 m n n be the number of sets of invariants of symmetry... q, 1'" Ptype (mI'" mq, nI'" np) arising as products of elements of the integrity

basis. This number may be determined from inspection with the aid of

results given in Tables 8.3 and 8.4. We list in Table 8.9 the quantitiesp Q and mI'" mq nI'" np for those

m1°o.mq, n1 np' m1.oomq, n1.oonp Xe Xem1... mq, n1 np for which P m1... mq, nl... np is not zero. Spencer and

Rivlin [1962] have shown that the integrity basis elements are of degree

six or less which enables us to consider only cases for which

where s1. and s1 are the traces of the matrices which describe the

transformation properties under a rotation through 0 radians about

some axis of the three independent components of a three-dimensional

skew-symmetric second-order tensor and the five independent com

ponents of a three-dimensional traceless symmetric second-order tensor

respectively. The number Pm1 m n1 n of sets of invariants of... q, ... psymmetry type (mloo, mq, nI'" np) arising from these IPm, n invariants

is given by27r

P -l J<p' <p" (I- cosO) dB (8 84)m1 ... m q, n1°o· np - 27r m1°o· m q n1··· np ..o

The typical multilinear elements of an integrity basis for

functions W(AI ,... , Am' B I ,···, Bn ) which are invariant under R3 may

Table 8.9 Invariant Functions of AI"'" Am' B I ,···, Bn : R3

mI·oomq, nI·oonp 1,11 1,21 1,111 1,31 1,211 1,41 1,32 1,311 1,221

P 1 1 1 2 2 2 2 3 1m1°o·m q, n1···np

Qm1°o·mq, n1· oonp 0 0 0 1 1 2 1 3 1

m1···m q n1···np 1 2 1 3 3 4 5 6 5Xe Xe

mI'" mq, nloo, np 1,2111 11,11 11,21 11,111 11,31 11,211 2,1 2,2 2,3

P 1 1 1 1 2 2 1 2 2m1···m q, n1···np

Qm1" .mq, n1" .np 1 0 0 0 1 2 0 1 2

m1···m q n1···np 4 1 2 1 3 3 1 1 1Xe Xe

mI·oomq, nI·oonp 2,21 2,4 2,31 2,22 2,211 2,1111 111,2 111,3 21,1

.p 2 3 2 4 1 1 1 1 1m1. oomq, n1°o .np

Qm1°o·mq,n1°o·np 1 3 2 4 0 1 1 1 0

m1.··m q n1· oonp 2 1 3 2 3 1 1 1 2Xe Xe

mI'" m q, nI'" np 21,2 21,11 21,3 21,21 21,111 3,11 3,3 3,21 3,111

P 1 1 1 3 1 2 1 2 2mI" .mq, n1" .np

Qml···mq,n1···np 0 1 1 3 2 1 0 2 2

m1· oomq n1···np 2 2 2 4 2 1 1 2 1Xe Xe

(8.8.3)

27rI?m,n = 2~ J(s])m (sq)n (1 - cos 0) dO,

os1 = eiO +1 +e-iO, sq = e2iO +eiO +1 +e-

iO +e-2iO

254 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd-Order Tensors: R3 255

(8.8.6)

Y"( i)tr Al (B1B2B3 - B3B2B1), (1,111),

tr (AI A2 - A2A1)(B1B2 - B2B1), (11,11),

tr (AI A2 + A2A1)(B1B2 + B2B1), (2,2),

[e, (23)J y'n 2) tr B1(AIA2A3 - A3A2A1), (21,1);

12

Y" 3 tr B1B2B3B4B5, (0,11111),45

, (1 2)~, (45), (23), (23)(45), (2453EY" ~ 4 tr BIB2B3B4B5' (0,221),

[e, (34), (23E Y"( ~ 2 4 ) tr Al (B1B2B3B4 - B4B3B2B1), (1,31),

[e, (23), (243E Y"(! 2) tr Al (B1B2B3B4 - B4B3B2B1), (1,211),

[e, (23E Y"( ~ 3 ) tr{A1A2 + A2Al){BIB2B3 + B3B2B1), (2,21),

~, (23E y,,( ~ 2 )tr(A1A2 - A2Al)(BIB2B3 - B3B2B1), (11,21),

"( 1 )Y § tr(AIA2-A2Al)(BIB2B3-B3B2BI)' (11,111),

[e, (23E Y'( j 2) tr{B1B2 + B2Bl)(AIA2A3 - A3A2A1), (21,2),

Y'(123) tr (AIA2B1A3B2 - AIA2B2A3Bl)' (3,11);

5.

(1,11), (1,21), (1,111), (1,31), (1,211), (1,32),

(11,11), (11,21), (11,111), (11,31), (2,1), (2,2),

(2,21), (2,211), (21,1), (21,2), (3,11), (3,3).

(i) Invariants which involve traceless symmetric second-order

tensors B1,... ,Bn only. These are given by (8.7.17).

(ii) Invariants which involve skew-symmetric second-order tensors

AI'·'"Am only. These are readily seen to·be given by tr A1A2and tr AlA2A3 which are of symmetry types (2,0) and (111,0)

respectively.

(iii) Invariants involving both the A1,... ,Am and the B1,... ,Bn.

With Table 8.9, we see that the typical multilinear elements of

the integrity basis involving both the AI'···' Am and B1,· .. , Bnare comprised of one set of invariants of each of the symmetry

types

be split into three sets:

2. tr B1B2, (0,2), tr AlA2, (2,0);

3. tr B1B2B3, (0,3), tr Al A2A3, (111,0),

tr Al (B1B2 - B2B1), (1,11), tr (AIA2+A2A1)B1, (2,1);

The typical multilinear elements of the integrity basis are listed below.

The Young symmetry operators Y'( ... ) and Y"( ... ) are to be applied to

the subscripts on the AI' ... ' Am and the B1,· .. , Bn respectively.

4. [e, (23)J y,,(~Dtr B1B2B3B4, (0,22),

[e, (23)J y" (1 2) tr Al (B1B2B3 - B3B2B1), (1,21),

6. [e, (34), (354), (3654), (234), (2354), (23654), (24)(35),

(24)(365), (25364E Y"(i2 3 ) tr B1B2B3B4B5B6' (0,3111),

256 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.9] Vectors and Traceless Symmetric 2nd- Order Tensors: 0 3 257

[e, (23), (243~Y"(l2) tr (AIA2 +A2A1)(B1B2B3B4

+B4B3B2B1), (2,211),

[e, (34), (23~y"G 2 4) tr(A1A2 - A2A1)(B1B2B3B4

- B4B3B2B1)' (11,31),

[e, (23), (45), (23)(45), (345~ y"G ~ 5) tr Al (B1B2B3B4B5

- B5B4B3B2B1)' (1,32),

expressions for tr IlK' IlL and ITM may be read off from the results

(8.7.17), (8.7.34) and (8.7.29) respectively. We list the typical multi

linear basis elements for scalar-valued functions of Xl'''.' Xm, B1,... , Bnwhich are invariant under 03. The Young symmetry operators Y( ... )

are applied to the subscripts on the B1,... , Bn. The notation (mI ... mq,

lll"· np) employed below indicates the symmetry type of the sets of

invariants. The mI ... mq refers to the vectors X1'~'." and the

n1··· np refers to the tensors B1,···, Bn.

2.

(8.9.2)

8.9 An Integrity Basis for Functions of Vectors and Traceless

Symmetric Second-Order Tensors: 03

In this section, we generate the multilinear basis elements for

scalar-valued functions W{X1,... , X ,B1,... "B ) of the absolute vectorsm n ,Xl'.··' Xm and the traceless symmetric second-order tensors B1,···, Bnwhich are invariant under the full orthogonal group 03. It is readily

shown by the method adopted by Pipkin and Rivlin [1959], §5, that any

polynomial function W{X1,... , Xm , B1,... , Bn) which is invariant under

03 is expressible as a polynomial in the quantities

. . . T

where i,j=l, ... ,m and Xi denotes the column vector [X1,X2,X3] .The quantities tr IlK' IlL and lIM are scalar-valued, symmetric matrix

valued and skew-symmetric matrix-valued polynomial functions of BI ,

... , Bn which are invariant under the group R3 . The general form of the

3.

4.

5.

12

Y 3 tr B1B2B3B4B5, (0,11111),-4

5

[e, (45), (23), (23)(45), (2453~Y(i ~)tr B1B2B3B4B5' (0,221),

~, (23~ Y (~ 2) tr (XlX; - ~X~) B1B2B3, (11,21),

(1) T TY ~ tr(X1~ -~X1)B1B2B3' (11,111),

~, (23~ Y (~ 3 ) tr (XlX; +~X~) (B1B2B3 + B3B2B1), (2,21);

258 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.10] Transversely Isotropic Functions 259

6. [e, (34), (354), (3654), (234), (2354), (23654), (24)(35),

(24)(365), (25364~ Y(i2 3 ) tr BlB2B3B4B5B6' (0,3111),

[e, (34), (23~Y(~ 2 4 )tr(XlX;-~Xr)(BlB2B3B4

-;- B4B3B2B1), (11,31),

(12) T T[e, (23), (243~Y: tr (Xl~ - ~Xl )(Bl B2B3B4

- B4B3B2B1), (11,211),

The problem discussed above has been considered by Smith [196S].

Results of greater generality are available. Thus, an integrity basis for

functions of m absolute vectors, n symmetric second-order tensors and

p skew-symmetric second-order tensors which are invariant under 03has been given by Smith [1965] and by Spencer [1971].

8.10 Transversely Isotropic Functions

There are five groups T1 , ... , T5 which define the symmetry

properties of materials which are referred to as being transversely

isotropic. We define these groups by listing the matrices which

generate the groups.

T1 : Q(O)

T2 : Q(O), R1 = diag (-1,1,1)

T3 : Q(O), Ra = diag (1, 1, -1) (S.10.1)

T4 : Q(O), R1 = diag (-1,1,1), Ra = diag (1,1, -1)

T5 : Q(O), D2 = diag (-1, 1, -1)

which corresponds to a rotation about the x3 axis. R 1 and Ra cor

respond to reflections in planes perpendicular to the xl axis and the x3

axis respectively. D2 corresponds to a rotation through ISO degrees

about the x2 axis.

We restrict consideration here to the groups T1 and T2 . We list

7. ~, (23), (45), (23)(45), (345~ yG ~ 5) tr (XIX;

T- X2Xl)(BIB2B3B4B5 - B5B4B3B2Bl)' (11,32),

[e, (23), (34), (354~ y(~ 3)tr(XlX;+~Xr)(BlB2B3B4B5+ B5B4B3B2Bl)' (2,2111),

(123) T[e, (34), (354), (234), (2354), (24)(35~ Y ~ tr (XlX2

+ ~Xr)(BlB2BiB4B5 + B5B4B3B2Bl)' (2,311).

In (S.10.1), Q(O) denotes the matrix

cos 0 sin 0 0

Q(O) = -sinO cosO 0

o 0 1

(S.10.2)


The typical multilinear elements of an integrity basis for functions of

¢, ... , a, f3, ... , A, B, ... which are invariant under T1 are given by

The presence of the complex invariants ap, ... , Aap in (S.10.3) indicates

that both the real and imaginary parts af3 ± ap, . .. , Aap ± Aaf3 of

a~, ... ,Aap are typical multilinear elements of the integrity basis.

With (S.10.3) and the right hand column of Table S.12, we may list the

typical multilinear elements of an integrity basis for functions

W(p, q, ... , S, T, U, ... ) of the vectors p, q, ... and the symmetric second

order tensors S, T, U, ... which are invariant under T1 . These are given

by the real and imaginary parts of

the irreducible representations associated with these groups (see §6.6)

and the linear combinations of the components Pi of an absolute vector

p, .the components ai of an axial vector a, the components Aij of a

skew-symmetric second-order tensor A and the components Sij of a

symmetric second-order tensor S which form the carrier spaces for these

representations. We then give the typical multilinear elements of an

integrity basis for functions of quantities associated with the first few

irreducible representations. Spencer [1971] gives a lucid account of the

procedures employed by Rivlin [1955], Smith and Rivlin [1957], Pipkin

and Rivlin [1959] and Adkins [1960 a, b] to obtain integrity bases for

functions invariant under T1 and T2 . See also Ericksen and Rivlin

[1954]. Integrity bases for functions of vectors and second-order tensors

which are invariant under any of the groups T1 , ... , T5 have been

obtained by Smith [19S2].

1. ¢;

2. ap, A13;3. Aap.

(S.10.3)

(Sll - S22 + 2iS12)(Pl - iP2)(ql - iq2)'

(Sll-S22+2iS12)(Pl-iP2)(T13-iT23)'

(S11 - S22 + 2iS I2)(T13 - iT23)(UI3 - iU23)·

8.10.1 The Group T1

We list in Table S.12 the first few irreducible representations /0'

/1' r l' /2' r 2 associated with T1 · The second column gives the 1 x 1

matrices corresponding to the group element Q(8) which define the

representations. The third column gives the notation identifying

quantities which transform according to /0' /1' r l' .... In the last

column, we give the linear combinations of the components Pi' ai' Aij ,

S·· which form carrier spaces for the irreducible representations.1J

1.

2.

3.

(PI + iP2)(q1- iq2)' (S13+iS23)(T13-iT23)'

(PI + iP2)(S13 - iS23), (Sll - S22 + 2iS12)(T11 - T 22 - 2iT12);

(S.10.4)

Table S.12 Irreducible Representations and Basic Quantities: T1

/0 1 ¢, ¢', ...

-i8 f3e a, , ..

e ilJ a, p, .e-2i8 A, B, .

e 2i8 A, 13, .

P3' a3' A12, S11 + S22' S33

PI + iP2' a1 + ia2' A13 + iA23, S13 + iS23

P1 - iP2' a1 - ia2' A13 - iA23, S13 - iS23

S11 - S22 +2iS12

Sll - S22 - 2iS12

We note that the quantities [Pl,P2,P3]T, [a1,a2,a3]T, [A23,A31,A12]T

where the p., a·, A·· are the components of a vector, an axial vector and1 1 1J

a skew-symmetric second-order tensor, respectively, transform in the

same manner under the group T1 . Thus, an integrity basis for

functions of arbitrary numbers of vectors, axial vectors, symmetric and

skew-symmetric second-order tensors may be read off from the result

(S.10.4).


8.10.2. The Group T2

We list in Table 8.13 the first few irreducible representations /0'

r 1, /1' /2 associated with the group T2 · In the second column, we list

the matrices associated with these representations which correspond to

the group elements Q(9) and R1. The third column gives the notation

identifying quantities which transform according to the representations

/0' r 0' /1' /2· In the last column, we give the linear combinations of

the components Pi' ai' Aij , Sij which form carrier spaces for the

irreducible representations.

The quantities <p, 01 f32 + 02,81' AlB2 + A2B1 and A 10 2f32 + A20 1,81

remain invariant under R1 while the quantities "p, 01 f32 - 02f3 1'

AlB2 - A2B1 and Ala2 f32 - A2a l,81 change sign under RI . The

typical elements of an integrity basis are given by the invariants <p,

01,82 + a2,81' together with the products of the quantities "p,

a1f32-02,81' taken two at a time (see Theorems 3.1 and 3.2, p.44).

Some of these products are redundant. Upon eliminating the redundant

terms, we see that the typical multilinear elements of an integrity basis

for functions of <p, <p', ... , AI' A2, B1, B2, ... which are invariant under T2

are given by

1. <p;

With (8.10.6) and the right hand column of Table 8.13, we may list the

typical multilinear elements of an integrity basis for functions W(A, B,

C, , S, T, U, V, ... ) of the skew-symmetric second-order tensors A, B,

C, and the symmetric second-order tensors S, T, U, V, ... which are

invariant under T2 . These are given by

"p"p', a1f32+a2f31' A1B2 +A2B1;(8.10.6)

"p(°1 f32 - a2f3 1)' "p(A1B2 - A2B1), A 10 2f32 +A20 1f3 1;

"p(A1a2,82 - A20 1f31), (a1,82 - a2 f3 1)(A1B2 - A2B1)·

2.

3.

4.

fe-2iO0 1[0 1] [A] [B ]

12 l 0 e2iOJ' lOA:' B: ,...

Table 8.13 Irreducible Representations and Basic Quantities: T2

/0 1, 1 <P, <p', ... P3' S33' Sll + S22

rO 1, -1 "p, "p', ... a3' A12

The group T2 is generated by the matrices Q(8) and R1. With the

results of §8.10.1, we observe that the typical multilinear elements of an

integrity basis for functions of the quantities <p, <p', ... , "p, "p', ... , aI' a2'

f3 1, f32, ... , AI' A2, B1, B2, ... which are invariant under the subgroup T1

of T2 are given by

1. <p, "p;

2. a1 f32+ a2f3 1' a1,82- a2f3 1' A1B2 +A2B1, A1B2 -A2B1;

3. Ala2{j2 +A2a l{jl' Ala2{j2 - A2a l{jl· (8.10.5)

2. A12B12, A13B13 + A23B23, A13S13 +A23S23,

S13T13+ S23T23' (Sll-S22)(T11-T22)+4S12T12;

3. A12(B13C23 - B23C13), A12(B13S23 - B23S13),

A12(S13T23 - S23T13)' A12(S11 - S22)T12 - A12(T11 - T22)S12'

(Sll - S22)(A13B13 - A23B23) +2S12(A13B23 +A23B13),

264 Generation of Integrity Bases: Continuous Groups

(Sll - S22)(A13T13 - A23T23) +2S12(A13T23 +A23T13),

(Sll - S22)(T13U13 - T23U23) +2S12(T13U23 + T23U13);

[eh. VIII

IX

(8.10.7)

4. A12(Sll - S22)(B13C23 +B23C13) - 2A12S12(B13C13 - B23C23),

A12(Sll - S22)(B13T23 +B23T13) - 2A12S12(B13T13 - B23T 23)'

A12(S11 - S22)(T13U23 +T23U13) - 2A12S12(T13U13 - T23U23),

((S11 - S22)T12 - (T11 - T22)S12)(A13B23 - A23B13),

((S11 - S22)T12 - (T11 - T22)S12)(A13U23 - A23U13)'

((S11-S22)T12-(T11-T22)S12)(U13V23- U23V13)·

The elements of. an integrity basis for functions of n symmetric second

order tensors and m vectors which are invariant under T2 are given by

Adkins [1960b]. An integrity basis for functions of n symmetric

second-order tensors, m vectors and p skew-symmetric second-order

tensors which are invariant under T2 has been obtained by Smith [1982].

GENERATION OF INTEGRITY BASES: THE CUBIC

CRYSTALLOGRAPHIC GROUPS

9.1 Introduction

In this chapter, we consider the problem of generating integrity

bases for functions which are invariant under a given cubic crystal

lographic group. For each of the cubic crystal classes, we list the

transformations defining the material symmetry and the matrices

defining the irreducible representations r a' r b' ... associated with the

crystallographic group. We also list the linear combinations of the

components (P1,P2,P3)' (a1,a2,a3)' (A23,A31,A12) and (Sll,S22,S33'

S23' S31' S12) of an absolute vector p, an axial vector a, a skew

symmetric second-order tensor A and a symmetric second-order tensor

S respectively which form carrier spaces for the irreducible rep

resentations r a' rb'·" and are referred to as quantities of types r a'

r b' .... General results comparable to those obtained in Chapter VII

are given only in the case of the crystallographic group T. Results of

complete generality for the crystal classes T d and 0 are given by Kiral

[1972]. These results are very lengthy. We consequently give only

partial results for these groups and for the remaining crystallographic

groups T h and 0h. In all cases, we may use the results to generate

integrity bases for functions of n vectors and for functions of n

symmetric second-order tensors. These integrity bases are equivalent to

those obtained by Smith and Rivlin [1964] and by Smith and Kiral

[1969] respectively.

We employ the procedure involving Young symmetry operators

265

266 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.1] Introduction 267

where

Npa,b,... =1. ""A.. (ra)A.. (r b ) (912)

nl·" np, ml". mq,... N L...J ""nl"· np K ""ml"· mq K··· ..K=l

(9.1.5)

(9.1.6)

3! 4>3(r) = (tr r)3 +3tr r tr r 2 +2 tr :r3.

<P3{r)

<P311 (r) = 1o

The expression for <P311 (r) is seen from (9.1.3) to be given by

the class,. For example, h, = 1, 3, 2 and '1'2'3 = 300, 110, 001 for

the three classes of S3 (see Table 4.2) so that (9.1.4) yields

The values of the quantities tr r, tr r 2, ... and 4>1 (r), <P2(r), ... for the

matrices r comprising the two-dimensional and three-dimensional irre

ducible representations associated with the cubic groups (see Smith

[1968 b]) are listed below in Tables 9.1 and 9.2. The quantities

IP a,b,... and P a, b, ... may be calculated with (9.1.1), ... ,n,m,... nl." np, mI." mq, ...

(9.1.3) and Tables 9.1 and 9.2. Let Qna, b, ·n·· m m denote the1··· p, 1·" q, ...

number of sets of invariants of symmetry type (n1 ... np,m1 ... mq, ... )

which arise as products of elements of the integrity basis. The

Qa, b, ... are to be determined from inspection with thenl." np,ml··· m q, ...

aid of Tables 8.3 and 8.4. The number of sets of basic invariants of

( ) . . b P a, b, ...symmetry type n1··· np, mI··· mq, ... IS qIven y n n m m1"· p, 1··· q, ...

- Qa, b, ... provided that the invariants comprising thenl np , ml"· mq, ...

Qa, b, sets of invariants are linearly independent. Thenl ... np , mI· .. m q, ...

sets of basic invariants of symmetry types (n1 ... np, mI ... m q, ... ) may

be generated with the aid of Young symmetry operators. The matrices

E, A, B, F, G, H and I, C, R1, ... , M2 which appear in the sets of

matrices defining the two-dimensional and three-dimensional irreducible

representations associated with the cubic groups are defined by (7.3.1)

and (1.3.3) respectively." We employ the notation 2:( ... ) to indicate the

(9.1.3)

(9.1.4)

which was introduced in Chapter VIII to generate the multilinear

elements of an integrity basis for functions of n quantities of type fa' m

quantities of type f b,.... Let I? a,b,... denote the number of linearlyn,m, ...independent functions which are multilinear in n quantities of type fa'

m quantities of type f b , ... and which are invariant under the group A

= {AI'''·' AN}· We have

I?::~',::. =&f (tr rK)n (tr r~)m... (9.1.1)K=l

where rj(, r~, ... (K = 1,... ,N) are the matrices comprIsmg the irre

ducible representations fa' f b, .... The number of sets of invariants of

m t t ( ) .. f th IP a,b,... ·sym e ry ype n1 ... np, mI ... m q, ... arIsIng rom e In-n,m, ...variants is given by

In (9.1.3), the non-diagonal entries in the determinant are obtained by

increasing (decreasing) by one the subscript on <P as we move from a

column to its neighbor on the right (left). Also <PO = 1 and any <P with

a negative subscript is zero. The quantities <pn{r) appearing in (9.1.3)

are defined by

n! <pn{r) = Lh,{tr r)'l {tr r 2)'I ... {tr rn),n,where the summation is over the classes, of the symmetric group Sn

and where h, gives the order and 1'1 2'2 ... n,n the cycle structure of

268 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.2] Tetartoidal Class, T, 23 269

sum of the three quantities obtained by cyclic permutation of the sub

scripts on the summand. For example,

f trr tr r 2 trr3 tr r4 tr r5 tr r 6 trr7 trrB tr r 9

E 2 2 2 2 2 2 2 2 2

A,B -1 -1 2 -1 -1 2 -1 -1 2

F,G,H 0 2 0 2 0 2 0 2 0

1 3 3 3 3 3 3 3 3 3

C -3 3 -3 3 -3 3 -3 3 -3

Rl'~'~T 1, T 2, T 3 1 3 1 3 1 3 1 3 1

D1T 1, D2T 2, D3T3

D1, D2, D3

CT1' CT2' CT3 -1 3 -1 3 -1 3 -1 3 -1

R1T 1, ~T2' ~T3

~Tl' R3T 1, R1T 2

~T2' R1T 3, ~T31 -1 1 3 1 -1 1 3 1

D2T 1, D3T 1, D1T 2D3T 2,D1T 3, D2T3

-1 -1 -1 3 -1 -1 -1 3 -1

(I, D1, D2, D3) . M10 3

(I, D1, D2, D3) . M20 0 3 0 0 3 0

(C, R1, ~, R3)· M13 0 0 -3

(C, Rl'~'~) ·M20 0 -3 0 0

EX1Y1 = x1Y1 +x2Y2 +x3Y3'

EX1Y2Z3 = x1Y2z3 +x2Y3z1 +x3Y1z2'

Table 9.1 Values of tr f, tr f2, ... : The Cubic Groups

(9.1.7)

Table 9.2 Values of <PI (f), <p2(f), ... : The Cubic Groups

r <PI <P2 <P3 <P4 <P5 <P6 <P7 </>8 <P9

E 2 3 4 5 6 7 8 9 10

A,B -1 0 1 -1 0 1 -1 0 1

F,G,H 0 1 0 1 0 1 0 1 0

I 3 6 10 15 21 28 36 45 55

C -3 6 -10 15 -21 28 -36 45 -55

Rl'~' R3T 1, T 2, T3 1 2 2 3 3 4 4 5 5

D1T 1, D2T 2, D3T 3

D1, D2, D3

CT1' CT2' CT3 -1 2 -2 3 -3 4 -4 5 -5

R1T 1, ~T2' R3T 3

~Tl' ~Tl' R1T 21

~T2' R1T3, ~T31 0 0 1 1 0 0 1

D2T 1, D3T 1, D1T 21

D3T 2, D1T3, D2T 3-1 0 0 1 -1 0 0 -1

(I, D1, D2, D3) . M1(I, D1, D2, D3) · M2

0 0 1 0 0 1 0 0 1

(C, R1, ~, ~). M1

(C, R1, ~, ~). M20 0 -1 0 0 1 0 0 -1

9.2 Tetartoidal Class, T, 23

In Table 9.3 below, the matrices I, D1, ... are defined by (1.3.3),

w = -1/2 + i.J3/2, w2 = -1/2 - i.J3/2 and xi = [xL x~, x~{ Quantities

of types r l' r2' ... are denoted by <P, <P', ... , a, b, ... respectively and are

listed in the last column of the table. The quantities a = a1 + ia2'

270 Generation of Integrity Bases: The Cubic Crystallographic Groups [eh. IX Sect. 9.2] Tetartoidal Class, T, 23 271

b = b1 + ib2,... are complex and a = a1 - ia2' b = b1 - ib2,... denote

the complex conjugates of a, b, ... respectively.

Table 9.3 Irreducible Representations: T

variant under the group T. We see from §7.3.3 that multilinear

functions of the quantities <p, <P', ... , a, b, ... , a, b, ... , Xl' x2' x3'''. which

are invariant under the subgroup D2 = {I, D1, D2, D3} of Tare

expressible in terms of functions of the form

T I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 B.Q.

r1 1 1 1 <P, ¢i, ...r2 1 w w2 a, b, ...

r3 1 w2 w a, b, ...

r4 I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 x1'~'''·

(9.2.1 )

where we have set

(9.2.2)

9.2.1 Functions of Quantities of Types rl' r2' r3' r4: T

We consider the problem of generating the typical multilinear

elements of an integrity basis for functions W(<p, <Ii, ... , a, b, ... , a, b, ... ,Xl' x2' x3' ... ) of quantities of types r l' r2' r3 and r4 which are in-

r 1 511 +522 +S33

r 2 511 +w2S22 +wS33

r 3 811 + w822 + w2833

r 4 [PI' P2' P3]T, [aI' a2' a3]T, [A23, A31 , A12]T, [823,831, S12]T

(9.2.3)

The functions (9.2.1) may be replaced by the equivalent set of functions

a, b, ... , xIYl +w2x2Y2 +wX3Y3'

2x1Y3z2+ w x2Y1z3+wx3Y2z1' ...

a, b, ... , xIYl +wX2Y2 +w2x3Y3'

2x1Y3z2 +wX2Y1z3 +W x3Y2z1' ...

where 2: (... ) is defined by (9.1.7) and w, w2 are cube roots of unity.

The functions (9.2.3) are grouped according to the manner in which

they transform under T. Thus, the functions designated by r1, r2, r3

are quantities of types rl' r2' r3 respectively. The determination of an

integrity basis for functions of quantities of types rl' r2' r3 which are

invariant under T has been considered in §7.3.9. We see from (7.3.21)

that the elements of the integrity basis are given by the quantities of

type rl' the product of each quantity of type r2 with each quantity of

type r 3 and the products taken three at a time of the quantities of type

r 2· We observe that each product of two quantities of type r 2 of the

form

(xIYl +w2x2Y2 + wX3Y3)(zlu2v3 + w2z2u3vl +wZ3ulv2) (9.2.4)

Basic Quantities: TTable 9.3A

In Table 9.3, we employ (I, D1, D2, D3) . M1 to denote M1,

D1M1, D2M1, D3M1. Entries in the row headed by r 2 indicate that

the 1 dimensional matrices comprising r 2 which correspond to I, D1,222 2D2, D3, ... , M2, D1M2, D2M2, D3M2 are 1,1,1,1, ... , W , W , W , W

respectively. In the row headed by r 4' the entries indicate that the

3 x 3 matrices comprising r 4 which correspond to I, D1, .. · , D3M2 are

given by I, D1, ... , D3M2 respectively.

272 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX

ab +ab, ab - ab ; L>lx~, (0,2);

3. abc, (3,0); a (xlxr +wx~x~ +w2x1x~), (1,2);

The typical multilinear elements of an integrity basis for T are listed

below.

Sect. 9.2] Tetartoidal Class, T, 23 273

Table 9.4 Invariant Functions of a,b, ... ; x1,x2' ... : T

nl··· np 2 3 111 4 31 22 5 41 32

p4 1 1 2 1 1 2 1n1··· np

Q!l· .. np 0 0 0 1 0 1 1 1 1

n1··· np 1 1 1 1 3 2 1 4 5Xe

nl··· np 311 6 51 42 411 33 222

p4 4 2 3 1n1 .. · np

Q!l'" np 3 2 3

n1··· np 6 5 9 10 5 5Xe

nl· .. np, ml .. · mq 1,2 1,21 1,4 1,31 1,22 2,2 2,21

p2,4 1 1 2 1 1 1 1n1 ...np , mI··· mqQ2,4 0 0 1 1 1 0 0. n1 ... np, mI ... mq

n1 .. · np m1· .. mq 1 2 1 3 2 1 2Xe Xe

2.

1. ¢;

(9.2.7)

(9.2.5)(xIY1 + wX2Y2 + w2x3Y3) L: z1u2v3' ... ,

(xIY2z3 +wX2Y3z1 +w2x3Y1z2) L:ul v1' ....

(i) invariants which are functions of xl' x2' ... and are comprised of

one set each of invariants of symmetry types (2), (3), (111), (4),

(31), (41), (6).

With the aid of the above observations, we may conclude that the basis

elements involving the xl' x2' ... only are of degree 6 or less, the basis

elements involving a, b, ... , a, b, ... only are of degree 2 or 3 and the

basis elements involving (a, b, ... ; xl' x2' ... ) are of degrees (1,2), (1,3),

(1,4), (2,2) and (2,3) in quantities of type (r2,r4) respectively. We

list in Table 9.4 the values of P!l'" np"'" Q~l~.. np, mI'" mq for the

nl'" np and n1'" np' mI'" mq of interest. The P!l'" np"" are

determined from (9.1.2) and Table 9.2. The Q!l'" np, ... are obtained

by inspection. With (7.3.21) and Table 9.4, we see that the typical

multilinear elements of an integrity basis for functions of ¢,... a, b, ... ,

a, b, ... , xI'~' ... which are invariant under T are given by ¢, ab, abc

and

Also, each product of two quantities of type r 2 of the form

is expressible as a linear combination of the eighty functions of the form

is expressible as a linear combination of the forty functions of the forms

(ii) invariants which are functions of a, b, ... ; xl' x2' ... and are com

prised of one set each of invariants of symmetry types (1,2),

(1,21), (1,4), (2,2), (2,21).

4. Exlxrx~xf' (0,4);

[e, (34), (234)J Y(l2 3)L>lxI(x~x~-x~xj), (0,31);

(Continued on next page) (9.2.8)


Some of the invariants appearing in (9.2.8) are complex functions. For

example,

[e, (23)] Y(j 2) a {xl(x~x~ + x~x~)

+ wx~(x~x~ +xIx~) +w2x!(xIx~ +x~x~)}, (1,21);

ab (xlxI +w2x~x~ +wx§x~), (2,2);

(9.2.10)

9.2.2 Functions ofn Vectors Pl'''.'Pn: T

We see from Table 9.3A that the transformation properties of a

vector P = [PI' P2' P3]T under the group T are defined by the repre

sentation r 4. The typical basic invariants which are multilinear in the

qua~titi~s :r1'''.'~ of type r 4 are given in (9.2.8). We set x.=p._ [I I I]T (.. . I I- PI' P2' P3 I = 1,...n) In (9.2.8) to obtaIn the typical multilinear

elements of an integrity basis for functions of n vectors p p1'·'" n

invariant under T. These are given by

3. LPhp~p~ +p~p~), (3);

4. LPlpIPfpf, (4);

[e, (34), (234)JY(12 3)LPlpI(p~p~-p~p§), (31);

5. [e, (45), (345), (2345)J Y (1 2 3 4 ) L PIPfpf(p~p~ - p§p~), (41);

6. Y(123456)LPlpIP~Pf(p~p~-P~Pg), (6).(9.2.9)

abc = (a1b1c1 - a1b2c2 - b1c2a2 - c1a2b2)

+ i(a1b1c2 +b1c1a2 +c1a1b2 - a2b2c2)·

5. [e, (45), (345), (2345)J YO 2 34) L>Ix~xf(x~x~- x§x~), (0,41);

a (xlxIx~xf +wx~x~x~x~+w2x§x~x~x§), (1, 4);

[e, (23)J YO 2) ab {xhx~x~+x~x~)

+ w2x~(x~x~ +xIx~) +wx§(xIx~ +x~xV}, (2,21);

Both the real and imaginary parts of the invariant (9.2.9) are basic

invariants. We have indicated the symmetry types of most of the sets

of invariants appearing in (9.2.8). For example, ab (xlxI + w2x~x~

+ wX§x§) is of symmetry type (2,2). The first entry in (2,2) indicates

that the invariant is of symmetry type (2) under permutation of a and

b. The second entry in (2, 2) indicates that the invariant is of

symmetry type (2) under permutation of the superscripts on the x's.

The Young symmetry operators in (9.2.8) are applied to the

superscripts on the x's.

The s!m~e~ry operators Y( ... ) are to be applied to the superscripts on

the PI' P2' P3· Results equivalent to (9.2.10) are given by Smith and

Rivlin [1964]. We observe that an integrity basis for functions of a

single vector P = [PI' P2' P3]T is obtained upon setting PI = P2 =... = P6 = P in (9.2.10). The terms of symmetry type (111), (31) and

(41) will vanish in this case and only terms arising from the sets of

invariants of symmetry types (2), (3), (4) and (6) will yield integrity

basis elements. These are given by

(9.2.11)


(9.2.14)

3. LShSIlS~I' (3,0);

E[ShSII (S~2 - S~3) + SiiS~1(Sh - Sh)

+S~ISh(S~2-S~3)J, (3,0);

L SIIxIx~, (1,2); E Sh(x~x~ - x§x~), (1,2);

Exl(x~x~ + x§x~), (0,3); Lxl(x~x~ - x~x~), (0,111);

in (9.2.8) to obtain an integrity basis for functions of the symmetric

second-order tensors Sl ,... , Sn' It is convenient to consider the

invariants obtained from (9.2.8) and (9.2.14) to be functions of the two

kinds of quantities (Sil' S12' S~3) and (S13' S~I' Sb) = (xi. x~, x~)(i = 1, ... ,n). The symmetry types of the sets of invariants appearing

below are indicated by (mI'" mq, nl"" np) where the mI'" mq and

n1'" np pertain to the symmetry properties o~ the. set. of invariants

under perrr:ut~tio:ns of the superscripts on the (8 11 , 822, 833) (i = 1, ... ,n)

and the (xl' x2' x3) respectively. Th~ s~mrr:etry operators Y{ ... ) below

apply to the superscripts on the (xl,x2,x3)' We recall that E{ ... ). d' t . h b' ~81 2 3 81 2 3In lca es summatIon on t e su scrIpts, e.g., L..J 1lXIX1 = 11X1X1

+ S~2x~x~ + Shx§xl The typical multilinear elements of an integrity

basis for functions of Sl'"'' Sn which are invariant under T are given by

(9.2.13)

(9.2.12)

Sl1 + S22 + S33' Sl1 + w2S22 + wS33'T

Sl1 + wS22 + w2S33' [ S23' S31' S12J

9.2.3 Functions of n Symmetric Second-Order Tensors Sl ,... , Sn: T

With Table 9.3A, we see that

5. Ep~(p2q3 - P3q2)' Ephl(P2q3 - P3q2)'

E PIqI(P2q3 - P3q2)' E q~(P2q3 - P3q2);

6. Epf(p~ - p~), Epf(P2q2 - P3q3) +2 Ephl (p~ - p~),

Epf(q~ - q~) +8 Ephl(P2q2 - P3q3) +6 EphI(p~ - p~),

Ephl(q~ - q~) + 3 EpIqI(P2q2 - P3Q3) + L PIQ~(p~ - p~),

E Qt(p~ - p§) + 8 E Q~Pl (P2Q2 - P3Q3) + 6 E QIPI(Q~ - Q§),

LQf(P2Q2 - P3Q3) +2 LQ~Pl(Q~ - Q~), E Qt(Q~ - Q~).

4. EPf, Ephl' EphI, Eplq~, Eqf, EpI(P2q2- P3q3)'

EpI(q~ - q~), EqI(P2q2 - P3q3);

An integrity basis for functions of two vectors p and q which are

invariant under the group T is seen from (9.2.10) to be given by

are quantities of types f 1, f 2, f 3, f 4 respectively. We may set (Continued on next page)

278 Generation of Integrity Bases: The Cubic Crystallographic Groups [eh. IX Sect. 9.3] Diploidal Class, T h' m3 279

In (9.2.15), the xi, xk, x~ denote S~3' S~l' S12' Results equivalent to

(9.2.15) are given by Smith and Kiral [1969].

[e, (45)J Vn 4) ESbSI1x~(x~x3+ x~x~), (2,21);

[ e, (45)J V (~ 4) E SbSI1x~(x~x3 - x~x~), (2,21);

[ e, (45), (345), (2345)J V(1 2 3 4)ExIx~xf(x~x3 - x§x~), (0,41);

4. [ e, (34)J V(~ 3) ESbxt(x~x~+ x~x~), (1,21);

[ e, (34)J V(~ 3) ESbxt(x~x~- x~x~), (1,21);

ESbSt1x~xf' (2,2); ESbSI1(x~4-x~x~),

[e, (34), (234)JV(12 3) Exlxt(x~x~-x~x~),

Exlxtx~xf' (0,4);

Irreducible Representations: T hTable 9.5

T h C, R1,~,R3 (C, R1'~' R3)· M1 (C, R1'~' R3)· M2 B.Q.

r1 1 1 1 </>,</>', ...

r2 1 w w2 a, b, ...

r3 1 w2 w a, b, ...

r4 I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 x1'~'''·

r5 -1 -1 -1 II, II', ...

r6 -1 -w _w2 A, B, ...

r7 -1 _w2 -w A,13, ...

r8 C, R1'~'~ (C, R1'~'~) . M1 (C, R1'~'~)· M2 X1,X2,.. ·

Th I, D1, D2, D3 (I, D1, D2, D3)· M1 (I, D1, D2, D3) . M2 B.Q.

r1 1 1 1 </>, </>', ...

r2 1 w w2 a, b, ...

r3 1 w2 w a, b, ...

r4 I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) . M2 x1'~'·"

r5 1 1 1 II, II', ...

r6 1 w w2 A,B, ...

r7 1 w2 w A,13, ...

r8 I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) . M2 X1,X2,.. ·

(9.2.15)

(2,2);

(0,31 );

r 1 Sll +S22 +S33

r 2 Sll + w2S22 +wS33

r3 Sll + wS22 + w2S33

r 4 [aI' a2' a3]T, [A23, A31 , A12]T, [S23' S31' S12]T

9.3 Diploidal Class, T h' m3

In Table 9.5, the matrices I, D1, ... are defined by (1.3.3),

w.= -.1!i+i{3/2, w2 = -1/2-i{3/2, xi = [xLxk,x~lT and Xi = [XLX2,Xa] . The quantities </> and II are real quantities; the quantities

a = al + ia2 and A = Al + iA2 are complex. The complex conjugates of

a and A are denoted by a = al - ia2 and A = Al - iA2 respectively.

The format of Table 9.5 is the same as that of Table 9.3.

Table 9.5A Basic Quantities: T h

280 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.3] Diploidal Class, T h' m3 281

The restrictions imposed on a function w* (Xixi, x~x~, X~X1) by the

requirement of invariance under T h are that W*( ... ) must be unaltered

under cyclic permutations of the subscripts 1,2,3. Thus, W*( ... ) must

satisfy

The general form of functions W*{ ... ) which are consistent with the

restrictions (9.3.2) may be determined upon application of Theorem 3.3.

With (3.2.5), it is seen that the elements of an integrity basis for

functions of X1,... ,Xn which are invariant under T h are of degrees 2, 4

and 6. We list the values of PnS n, QnS n for the n1 ... np of

1'" PI'" Pinterest in Table 9.6.

9.3.1 Functions of Quantities of Type r8: T h


vector p = [PI' P2' P3]T under the group T h are defined by the repre

sentation f S. Functions W(Xl, ... ,Xn) of quantities Xl,,,.,Xn of type

f S which are invariant under the subgroup D2h = {I, C, R1, ~, R3,

D1, D2, D3} of T h are seen from §7.3.4 to be expressible as functions of

the quantities

(9.3.3)

9.3.2 Functions of Quantities of Types rl' r2' r3' r4: T h

We observe from Table 9.3 and Table 9.5 that the restrictions

imposed on scalar-valued functions of quantities of types f l' f 2' f 3' f 4

which are invariant under T h are identical with those imposed by the

requirement of invariance under the group T (see §9.2). The typical

multilinear elements of an integrity basis for functions of quantities of

types f l , f 2, f 3, f 4 which are invariant under T h are thus given by

(9.2.S). We note that any tensor of even order may be decomposed into

a sum of quantities of types f l , f 2, r3, f 4. The procedure leading to

this decomposition is discussed in §5.3. The results (9.2.S) enable us to

The Young symmetry operators appearing in (9.3.3) are applied to the

superscripts on the xj. Substituting Pi for ~ in (9.3.3) will give the

typical multilinear elements of an integrity basis for functions of the

vectors PI' P2' ... which are invariant under T h. These results are

equivalent to those given by Smith and Rivlin [1964].

6. LXIXIX~XfXfXY, (6);

Y(1 2 3 4 5 6) LXlXIX~Xf(X~X~ - x~xg), (6).

2. LXIXI, (2);

4. LXIXIX~Xf, (4);

[ e, (34), (234)J Y(~ 2 3) LXIXI(X~X~- X~X§), (31);

in (9.1.2) would be over the 24 matrices I, D1, , R3M2 comprising the

representation f S of T h. The matrices I, D1, , R3M2 are listed in row

S of Table 9.5. With Table 9.6, we see that the typical multilinear

elements of an integrity basis for functions of Xl' ~,... which are

invariant under T h are comprised of 1, 1, 1, 2 sets of invariants of

symmetry types (2), (4), (31), (6) respectively. These are given by

(9.3.1 )

(9.3.2)

(i,j = 1,... ,n).

w* (XiXi, X~X~, X~X1) = W* (X~X~, X~X1, XiXi)= W* (X~X1, XiXj1' X~X~).

Table 9.6 Invariant Functions of Xl' X2, ... : T h

n1··· np 2 4 31 22 6 51 42 411 33 222

pS 1 2 1 1 4 2 3 1 1 1nl···np

Q~l· .. np 0 1 0 1 2 2 3 1 1 1

nl···np 1 1 3 2 1 5 9 10 5 5Xe

The P~l'" np are obtained from (9.1.2) and Table 9.2. The summation

282 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX 8ect. 9.4] Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m 283

9.4 Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m

determine integrity bases for functions of arbitrary numbers of even

order tensors which are invariant under T h. In particular, we observe

that the restrictions imposed on the scalar-valued function W{Sl ,... , Sn)

of the symmetric second-order tensors Sl' ... ' Sn by the requirement of

invariance under the group T h are identical with those imposed by the

requirement of invariance under the group T. Thus, the typical

multilinear elements of an integrity basis for functions of Sl ,... , Sn

which are invariant under T h are identical with those given by (9.2.15)

for functions of Sl'.'" Sn invariant under T.

(9.4.1 )

(9.4.3)

(9.4.2)

Basic Quantities: 0, Td

T

a= [aI' a2]' a=a1 + ia2' a=a1- ia2'

[ iii]T _ [i i i ]Txi = Xl' x2' x3 ' Yi - Y1' Y2' Y3 .

<P,<P', ... , a,b, ... , x1,x2'·"

¢, ab+ab, abc + abc, Exlx~,

In Table 9.7, the matrices I, D1, ... and E, B, ... are defined by (1.3.3)

and (7.3.1) respectively. In this section, we employ the notation

Table 9.7A

and

9.4.1 Functions of Quantities of Types r 1, r 3, r4: T d' 0

It is readily seen with (9.2.8) that the multilinear functions of

the quantities

of types f l' f 3' r4 which are invariant under the subgroup T = { I, D1,

D2, D3, M1, D1M1, D2M1, D3M1, M2, D1M2, D2M2, D3M2 } of T d

are expressible in terms of functions of the forms

r1 r2 f 3 r4 r5..- - r- - ..... -..- -

[2511- 522- 533]823 PI al A23

0 811+822+833 831 P2 ' a2 ' A31~ (833- 822)

812 P3 a3 A12..... - ..... - ..... - ..... -

- - - - - - ..... -

[2511- 522- 533]PI 823 al A23

Td 811+ 822+ 833 P2 ' 831 a2 ' A31~ (833- 822)

P3 812 a3 A12.... - .... - - - .... -

Irreducible Representations: 0, T dTable 9.7

0 I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) · M2B.Q.

T d I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2

f 1 1 1 1 <P, <p', ...f 2 1 1 1 "p, "p', ...

f 3 E B A a, b, ...

f 4 I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 x1'~'·"

r 5 I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) . M2 Y1'Y2' ...

0 (C, R1'~'~)·T 1 (C, R1'~'~) . T 2 (C, Rl'~'~) . T 3B.Q.

T d (I, D1, D2, D3) . T 1 (I, D1, D2, D3) . T 2 (I, D1, D2, D3) . T 3

f 1 1 1 1 <p, <p', ...r 2 -1 -1 -1 "p, "p', ...

f 3 F G H a, b, ...

f 4 (I, D1, D2, D3) · T 1 (I, Dl , D2, D3) . T 2 (I, Dl , D2, D3) . T 3 xl'~'''·

f 5 (C, Rl'~'~)·T l (C, Rl'~'~) . T 2 (C, Rl'~'~) . T 3 Y1' Y2'''·

284 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.4] Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m 285

(9.4.4)

functions (9.4.5) together with the products of all pairs of the functions

which may be chosen from the set comprised of the functions (9.4.4)

and the imaginary parts of the functions (9.4.5). The integrity basis

obtained in this manner contains a number of redundant terms.

and the real and imaginary parts of the functions

We observe from Table 9.7A that the problem of determining an

integrity basis for functions of n symmetric second-order tensors 81, ... ,

Sn which are invariant under the group T d is equivalent to the problem

of determining an integrity basis for functions of n quantities of type

rl' n quantities of type r3 and n quantities of type r4. An integrity

basis for functions of S1' ... , Sn which are invariant under Td has been

obtained by Smith and Kiral [1969]. With the results given in their

paper, we may immediately list the degrees in a, b, ; x1'~'''. of the

elements of the integrity basis for functions of 4>, , a,b, ... , x1,x2'''.

which are invariant under T d.

We list In Table 9.8 the symmetry types (m1m2' n1 ... np) In the

quantities a, b and xl' x2'." of the sets of invariants which are can

didates for inclusion in the integrity basis, the number P~;m2' nl ... np

of linearly independent sets of invariants of symmetry type

Invariant Functions of a,b, ... ; x1,x2' ... : T d' 0

212

2 1

o 0 0

111

1 0 0

111

2,21 11,111 3,111

0, 22 1, 2 1, 21

1

2

1

1

1

2,3

0,4

1 1

1 1

o 0

2 1

1 1

1 2

1,22 2,2

0,2 0,3

3

1

1

o

1

1,31

3,0

2

1

1

1

o

1

1,4

2,0

Table 9.8(9.4.5)

[ ] ( 1 2) {I 2 3 2 3) 2 1{ 2 3 2 3)e,(23) Y 3 ab xl{x2x3 +x3x2 +W x2 x3x1 +xlx3

+wx~(xIx~ + x~xV}

where E{...) is defined as in (9.1.7) and where w=-1/2+i~/2,

w2 = -1/2 - i~/2. The last twelve transformations of Table 9.7 leave

the functions (9.4.3) and the real parts of the functions (9.4.5) un

altered. They also change the signs of all of the functions (9.4.4) and

all of the imaginary parts of the functions (9.4.5). With Theorems 3.1

and 3.2, we see that the multilinear elements of an integrity basis for

functions of 4>, ... , a, b, ... , xl' x2' ... which are invariant under Tdare

given in terms of the functions (9.4.3) and the real parts of the

286 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.4] Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m 287

(m1m2' n1'" np) and the number Q~;m2' nt ... np of sets of invariants

of symmetry type (m1m2' n1 ... np) which arise as products of integrity

basis elements of lower degree. We employ (9.1.2) and Table 9.2 to

calculate P~;m2' nt... np' With (9.4.3), ... , (9.4.5) and Table 9.8, we

see that the typical multilinear elements of an integrity basis for

functions of 4>, ... , a,b, ... , x1,x2'." which are invariant under Tdare

given by

1. 4>;

2. ab +lib, (2,0); L>lxy, (0,2);

4. [ e, (23)J yn 2) Re[a{xhx§x~+x§x~)

+ wx~(x§x~ +xyx~) +w2x~(xyx~ +x§x~)}J, (1,21);

The Young symmetry operators appearing in (9.4.6) are applied to the

superscripts on the xj. We have used the notation a = a1 +ia2'

a = a1 - ia2' b = b1 + ib2, ... ·

9.4.2 Functions of n Vectors PI'.'" Pn: T d


vector P = [PI' P2' P3]T under the group T d are defined by the

irreducible representation r 4. The typical multilinear elements of an

integrity basis for functions of the quantities xl' ... ' Xn of type r 4 are

given in (9.4.6). We replace xi by Pi=[pi,p~,p~lT in the terms in

(9.4.6) involving only the xi to obtain the typical multilinear elements

of an integrity basis for functions of n vectors PI'·'" Pn which are

invariant under T d. These are given by

2. Eplpy, (2);

3. E 1 ( 2 3 2 3) (3); (9.4.7)PI P2P3 +P3P2 '

4. E 1234 ()P1P1P1P1' 4.

[ e, (23)J y(12 ) Re[ ab{xhx§x~+x§x~)

+ w2x~(x§x~ +xyx~) +wx~(xyx~ +x~x~)}J, (2,21);

(9.4.6) Results equivalent to (9.4.7) are given by Smith and Rivlin [1964].

9.4.3 Functions of Quantities of Type r5: T d' 0

We see from Table 9.7A that the problem of determining an

integrity basis for functions of the quantities Y1 ,... , Yn of type r 5 which

are invariant under 0 is equivalent to that of determining an integrity

basis for functions of the n vectors PI'''.' Pn which are invariant under

O. This problem has been considered by Smith [1967]. It may be

readily seen from the result given by Smith [1967] that the typical

multilinear elements of an integrity basis for functions of the quantities

Y1' ... 'Yn of type r 5 which are invariant under the group 0 are

comprised of one set of invariants of each of the symmetry types (2),

288 Generation of Integrity Bases: The Cubic Crystallographic Groups [eh. IX Sect. 9.4] Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m 289

(9.4.11 )

(111), (4), (41), (6), (61) and (9). These are given by

2. LylYI, (2);

3. Lyhy~y~ - Y§Y~), (111);

4. LylYIY~Yf' (4); (9.4.8)

5. [ e, (45), (345), (2345)J ya 2 34) LYIY~Yf(y~yg - Y!Y~), (41);

6. LylYIY~YfY~Y~' (6);

7. [ e, (67), (567), (4567), (34567),

(23456 7)J Y(i 2 3 4 5 6) LYIY~YfY~Y~(Y~Y~ - Y!Y~), (61);

We now establish the result that none of the basis elements in

(9.4.8) are redundant. Let Pn1 ... np denote the number of linearly inde

pendent sets of invariants of symmetry type (n1 ...np). With (9.1.2), ... ,

(9.1.4), we have

P2 = 2\L<P2(rf<),

PIll = 214L {<p~(rf<) - 2<P1 (rf<) <P2(rf<) + <P3(rf<)},

(9.4.9)

P61 = 2\ L {<PI (rf<) <P6(rf<) - <P7(rf<)},

P9 = l4L<P9(rf<)·

The summation in (9.4.9) is over the set of 24 matrices I, D1, ...,~T3

comprising the representation r 5 (see Table 9.7). We see from (9.4.9)

and Table 9.2 that

(9.4.10)

Let Qnl ... np denote the number of sets of invariants of

symmetry type (n1". np) which arise as products of integrity basis

elements of degree lower than n1 + ... +np. We list below in (9.4.11)

the symmetry types of the sets of invariants which arise as products of

the integrity basis elements given in (9.4.8). For example, we list

(2) X (2) to denote that the set of three invariants EylYI EY~Yf,

Eyly~ EYIYf, EylYf EYIY~ is of symmetry type (2) X (2)= (4) + (22).

(2) x (2), (2)· (111), (2) x (3), (2)· (4), (111) x (2),

(4)· (111), {(2) x (2)} . (111), (2)· (41), (2)· (61),

{(2) x (3)}· (111), {(2) x (2)}· (41), (2)· (4)· (111),

(111) x (3), (6)· (111), (4)· (41).

We may employ results such as those given in Tables 8.3 and 8.4

or those given by Murnaghan [1937, 1951] to obtain the decomposition

of these sets into sets of invariants of symmetry types (n1". np) (see

Smith [1968b]). The Qnl ... np may then be read off. We see in this

manner that

(9.4.12)

Since Pn1 ... np - Qnl ... np = 1 for n1". np = 2, 111, ... ,9, we may not

eliminate any of the sets of invariants in (9.4.8) from the set of typical

multilinear elements of the integrity basis.

We see from Table 9.7A that the typical multilinear elements of

an integrity basis for functions of n vectors PI' ... ' Pn which are in

variant under the group 0 are given upon replacing Yi by Pi (i = 1,2,... )

in (9.4.8). We also see from Table 9.7A that the typical multilinear

elements of an integrity basis for functions of n axial vectors a1 ,... , an

which are invariant under the group T d (also the group 0) are given

upon replacing Yi by ai (i = 1,2,... ) in (9.4.8). We also observe from

290 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.5] Hexoctahedral Class, 0 h' m3m 291

Table 9.7A that the typical multilinear elements of an integrity basis

for functions of n skew-symmetric second-order tensors AI' ... ' An which

are invariant under T d (or 0) are obtained upon replacing the yi, y~,

y~ by A~3' A~l' Ab (i = 1,2,... ) in (9.4.8).

9.4.4 Functions ofn Symmetric Second-Order Tensors Sl,.",Sn: T d , 0

With Table 9.7A, we see that

linear elements of the integrity basis for functions of n symmetric

second-order tensors Sl ,... , Sn which are invariant under the group T d

(or the group 0) are listed below where the notation {9.4.15)2 is

employed.

1. ESh, (1,0);

2. ESI1S~1' (2,0); ExIx~, (0,2);

3. EShS~lSf1' (3,0); EShx~xf, (1,2);

Exl(x~x~+ x§x~), (0,3);

T

811 +822 +833, [823,831,812]'

T[2S11 - S22 - S33' ~(S33 - S22)]

are quantities of types r l' r4 and r3 respectively. We may set

c/>= ESu' a= 2Sh -S~2-S~3+~i(Sh-S~2)'

b = 2S~1 - S~2 - S§3 + ~ i(S§3 - S~2)'

(9.4.13)

(9.4.14)

4.

5.

(9.4.16)

In (9.4.6) to obtain an integrity basis for functions of the symmetric

second-order tensors Sl'''.' Sn. It is preferable to proceed as in §9.2.3

and consider the invariants to be functions of the quantities

· · · T[811,822,833] ,

· . · T · · · T[xl,x2,x3] = [S23,S31,S12] (i = 1,... ,n).

(9.4.15)

[ e, (45)J y(~ 4)EShS~lxf(x~x~+ xix~), (2,21);

E(ShS~2-S~2S~1)·Exf(x~x~-xix~), (11,111);

6. E[SbS~l(S~2 - S~3) + S~lSf1(S~2 - S~3)

+ Sf1Sll(S~2 - S§3)} L>i(x~xg- x~x~), (3,111).

We indicate the symmetry type of the sets of invariants by (mI ... mq,

n1." np) where m1"· mq and n1". np pertain to the behavior of the set

of invariants under permutations of the superscripts on the (Sit, S~2'

S~3)' ... , (Sf1' S22' S33) and the superscripts on the (xl, x~, x§), ... ,

(xl' x2' x3) respectively. The symmet~y ~per~tors Y{... ) appearing

below apply to the superscripts on the xl' X2' xJ. The typical multi-

9.5 Hexoctahedral Class, 0h' m3m

In Table 9.9, the matrices I, D1, ... , and E, B, ... are defined by

(1.3.3) and (7.3.1) respectively and

[ ]T _ [i i i ]T . . · T

a = aI' a2' Xi - xl' x2' x3' Yi = [YI' Y2' Ya] ,T ... T ... T (9.5.1)

C = [C1' C2] , Xi = [Xl' X2,Xa] , Yi = [YI' Y2,Ya] ·

292 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.S] Hexoctahedral Class, 0h' m3m 293

°h I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D 1, D2, D3) · M2 B.Q.

f 1 1 1 1 </>,</>', ...

f 2 1 1 1 "p, "p', ...f 3 E B A a, b, ...

f 4 I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 x1'~'''·

f S I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 Y1' Y2' ...

f 6 1 1 1 a,a', ...f 7 1 1 1 13, 13', ...f S E B A C,D, ...

f 9 I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) . M2 X1,X2,.. ·

flO I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) · M2 Y1, Y2,.. ·

°h C, R1'~'~ (C, R1'~'~)'M1 (C, R1'~'~)'M2 B.Q.

f 1 1 1 1 </>, </>', ...f 2 1 1 1 "p,,,p', ...f 3 E B A a, b, ...

f 4 I, D1, D2, D3 (I, D1, D2, D3) · M1 (I, D1, D2, D3) · M2 x1'~''''

f S I, D1, D2, D3 (I, D1, D2, D3) . M1 (I, D1, D2, D3) . M2 Y1'Y2' ...

f 6 -1 -1 -1 a,a', ...f 7 -1 -1 -1 {3, {3', ...f S -E -B -A C, D, ...

f 9 C, R1'~'~ (C, R1'~'~)·M1 (C, Rl'~'~) . M2 X1,X2,.. ·

flO C, R1'~'~ (C, Rl'~'~)'M1 (C, R1'~'~) . M2 Y1,Y2,· ..

Table 9.9 Irreducible Representations: 0h Table 9.9 Irreducible Representations: °h (Continued)

°h (I, D1, D2, D3) . T 1 (I, D1, D2, D3) · T 2 (I, D1, D2, D3) · T 3 B. Q.

f 1 1 1 1 </>, </>', ...

f 2 -1 -1 -1 "p,,,p', ...

f 3 F G H a, b, ...

f 4 (I, D1, D2, D3)· T 1 (I, D1, D2, D3) . T 2 (I, D1, D2, D3)· T 3 x1,x2' ...

f S (C, Rl'~'~)'T 1 (C, R1'~'~)·T 2 (C, R1'~'~) · T 3 Y1'Y2' ...

f 6 1 1 1 a,a', ...f 7 -1 -1 -1 13,13', ...f S F G H C,D, ...

f 9 (I, D1, D2, D3) . T 1 (I, D1, D2, D3) . T 2 (I, D1, D2, D3) . T 3 X1,X2,.. ·

flO (C, R1'~'~)'T 1 (C, R1'~'~) . T 2 (C, R1'~'~)'T 3 Y1, Y2,.. ·

(Continued on next page)

°h (C, R1'~'~)'T 1 (C, R1'~'~) . T 2 (C, R1'~'~)'T 3 B. Q.

r1 1 1 1 </>, </>', ...r2 -1 -1 -1 "p, "p', ...r3 F G H a, b, ...

r4 (I, D1, D2, D3) . T 1 (I, D1, D2, D3) · T 2 (I, D1, D2, D3) · T 3 xl'~'·"

r5 (C, R1'~' R3)· T 1 (C, Rl'~'~)·T 2 (C, R1'~'~)'T 3 Y1' Y2' ...

f 6 -1 -1 -1 a,a', ...r7 1 1 1 {3, 13', ...f S -F -G -H C,D, ...

f 9 (C, R1'~'~)'T 1 (C, R1'~'~) . T 2 (C, Rl'~'~)·T 3 X1,X2,.. ·

flO (I, D1, D2, D3) . T 1 (I, D1, D2, D3) . T 2 (I, D1, D2, D3) . T3 Y 1, Y 2,· ..

294 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.5] Hexoctahedral Class, 0h' m3m 295

The further restrictions imposed on W*( ... ) by the requirement of

invariance under 0h are given by

where i,j = 1,... , n; i ~ j. With Theorem 3.4 of §3.2, we see immediately

that the integrity basis elements for functions of Xl' ~, ... which are

invariant under 0h are of degrees 2, 4 or 6. We list in Table 9.10 the

number Pn9 n of linearly independent sets of invariants of symmetry1'" p

type (nlu, np) and the number Q&l'" np of sets of invariants of

9.5.1 Functions of Quantities of Type r 9: 0h

Functions W(X1,... ,Xn) of the quantities X1,.",Xn of type f 9which are invariant under the subgroup D2h = { I, D1, D2, D3, C, R1,

~, R3 } of 0h are seen from §7.3.4 to be expressible as

W(Xl"u,~) = W*(XiXl, X~X~, X~X1) (i,i = l,u.,nj i ~i). (9.5.2)

f 9 [PI' P2' P3]T

flO

(9.5.4)

Table 9.10 Invariant Functions of Xl' X2, .. ·: 0h

n1··· np 2 4 22 6 51 42 222

p 9 1 2 1 3 1 2 1nl· .. np

Q&l..·np 0 1 1 2 1 2 1

nl .. ·np 1 1 2 1 5 9 5Xe

The restrictions imposed on functions of the n symmetric

second-order tensors Sl'"'' Sn by the requirement of invariance under

the group 0h are identical with the restrictions imposed by the

requirement of invariance under the group T d' Thus, the typical

multilinear elements of an integrity basis for functions of Sl ,... , Sn

which are invariant under 0h are given by the invariants (9.4.16).

9.5.2. Functions of n Symmetric Second-Order Tensors: 0h

We may set Pi = Xi (i = 1,2,... ) in (9.5.4) to obtain the typical multi

linear basis elements for functions of n vectors PI"'" Pn which are

invariant under 0h'

2. EX}Xy, (2);

4. EX}XrX~Xf, (4);

6. EXlXrx~xtxfx~, (6).

symmetry type (n1". np) arising as products of integrity basis elements

of lower degree for cases where nl + ·u +np = 2, 4, 6 and P&lu. np # O.

With Table 9.10, we see that the typical multilinear basis elements for

functions of Xl' X2, ... which are invariant under 0h are comprised of

one set of invariants of each of the symmetry types (2), (4) and (6).

These are given by

(9.5.3)

Basic Quantities: 0h

f i SII +S22 +S33

f 2

f 3 [2Sll - S22 - S33' ~ (S33 - S22)]T

f 4 [S23' S3l' SI2]T

f 5 [aI' a2' a3]T, [A23, A3l, A12]T

Table 9.9A

W*(Xixil' X~X~, X~X1) = W*(X\Xl, X~X1, X~X~)

=W*(X~X1, X~X~, XiXl) =W*(X~X~, Xixil' X~X1)

= W*(X~X~, X~X1, XiXil) = W*(X~X1, XiXl, X~X~)

(10.1.1)

296 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX

Similarly, the restrictions imposed on functions of quantities of types

f 1, f 2, f 3, f 4, f 5 by the requirements of invariance under 0h and

under T d are identical. Hence, the integrity bases for functions of

quantities of types f l' f 3' f 4 and for functions of quantities of type f 5

which are invariant under the group 0h may be obtained from the

results (9.4.6) and (9.4.8) respectively.

x

IRREDUCIBLE POLYNOMIAL CONSTITUTIVE EXPRESSIONS

10.1 Introduction

A scalar-valued polynomial function W(E) of a tensor E which is

invariant under a group A is expressible as a polynomial in the elements

11,... ,ln of an integrity basis. We say that the integrity basis is

irreducible if none of the Ij (j = 1,... ,n) is expressible as a polynomial in

the remaining elements of the integrity basis. We may write the

general expression for W(E) as

W(E)=c. · Ii11... 11

n·n (i1,... ,in =0,1,2, ... ).11··· 1n

Determination of the elements of an integrity basis constitutes the first

main problem of invariant theory. In general, the elements of an

integrity basis are not functionally independent. For example, we may

have 1112 - I~ = O. Such a relation is referred to as a syzygy. A syzygy

is a relation K(11,... , In) = 0 which is not an identity in the 11'"'' In but

which becomes an identity when the Ij are written as functions of E.

The second main problem of invariant theory requires the deter

mination of a set of syzygies Ki(I1,... ,ln) (i = 1,... ,p) such that every

syzygy K(11,... , In) = 0 relating the invariants 11"'" In is expressible in

the form

K(11,... , In) = d1K 1(11 ,... , In) +... +dpK p (11,... , In) (10.1.2)

where the di are polynomials in the 11"'" In' The existence of syzygies

means that in general there will be redundant terms appearing in the

297

298 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.1] Introduction 299

holds for all AK in A. We may follow the procedure outlined by Pipkin

and Rivlin [1959, 1960] to show that P(S) and T(S) are expressible as

Vector-valued functions P(S) and symmetric second-order

tensor-valued functions T(S) of the symmetric second-order tensor S are

said to be invariant under the group A = {AI' A2, ...} if

(10.1.7)

(10.1.8)P(S) = a1J 1(S) + +arJr(S),

T(S) = b1N1(S) + +bsNs(S)(10.1.3)

expression (10.1.1). We may employ the relations Ki (11'.'" In) = 0 to

remove the redundant terms from (10.1.1). The objective is to produce

a general expression for W(E) which does not contain any redundant

terms. Such an expression is referred to as being irreducible.

For example, let W(S) be a scalar-valued polynomial function of

a symmetric second-order tensor S which is invariant under the

orthogonal group 03. An integrity basis for functions of S invariant

under 03 is given by

We then have

(10.1.4)

where the ai' bi are scalar-valued functions of S which are invariant

under A and the Ji(S) and Ni(S) satisfy (10.1.7)1 and (10.1.7)2

respectively. It is assumed that no term Jp(S) in (10.1.8)1 is ex

pressible as

The expression (10.1.4) may be written as

where W(n)(S) denotes a linear combination of the invariants of degree

n in S appearing in (10.1.4). For example,

(10.1.9)

(10.1.11 )

(10.1.10)

Li(I1,... , In; J 1,... , J r) = 0 (i = 1,2, ),

M i(I1,· .. , In; N1,.. ·, Ns) = 0 (i = 1,2, )

Jp(S) = c1J 1(S) + ... + cp_1Jp_1(S) +

+ cp+1Jp+1(S) + . · · + crJr(S)

We wish to determine vector-valued and symmetric second-order

tensor-valued relations

where the ci are scalar-valued polynomial functions of S which are in

variant under A. If this be the case, we say that none of the terms

J1(S), ... , Jr(S) appearing in (10.1.8)1 are redundant. Similarly, we shall

assume that none of the terms N1(S), ... , Ns(S) in (10.1.8)2 are

redundant. In general however, there will be redundant terms

appearing in the expressions (10.1.8). For example, we may have

(10.1.6)

(10.1.5)

We may compute the number ( = 4) of linearly independent invariants

of degree 4 in S. This indicates that there is no linear relation

connecting the four invariants in (10.1.6). In order to show that there

are no redundant terms in the expression (10.1.4), we must show that

the terms of arbitrary degree n appearing in (10.1.4) are linearly

independent. This requires the introduction of the notion of generating

functions which will be discussed in §10.2.

300 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.2] Generating Functions 301

Suppose that we have determined an expression for Z(S) consistent with

(10.2.1). We proceed by writing

where Z(i)(S) is a linear combination of the terms of degree i in S

appearing in Z(S). We may compute the number ni of linearly

independent terms of type r v which are of degree i in the components

of S. If there are mi terms appearing in Z(i)(S), then mi - ni of these

terms are redundant. We eliminate the redundant terms and thus

replace Z(i)(S) by Z(dS) where all terms appearing in Z(i)(S) are

linearly independent. This is to be accomplished for all values of i.

The resulting expression Z(1)(S) + Z(2)(S) +...+ Z(n)(S) + ... would be

the irreducible expression required.

such that all redundant terms appearing in the expressions (10.1.8) may

be eliminated upon application of the relations (10.1.11). The reduced

expression arising from (10.1.8)1' say, is then to be such that the

number of vector-valued terms of degree n in S appearing will be equal

to the number of linearly independent vector-valued terms of degree n

in S which are invariant under A. This is to hold for all values of n.

Such expressions are said to be irreducible.

We observe that, if A is one of the 32 crystallographic groups, we

may employ the Basic Quantities tables appearing in Chapters VII and

IX to read off the decomposition of vectors and second-order tensors

into the sums of quantities of types r l' r2'.... The problem of deter

mining irreducible expressions for P(S) and T(S) consistent with the

restrictions (10.1.7) may then be reduced to a number of simpler

problems which require the determination of irreducible expressions for

functions Z(S) of type r v (v = 1,2,... ) which are subject to the re

strictions that

(10.2.1 )

(10.2.2)

(10.1.12)Let s denote a column vector whose entries are the six inde

pendent components of the symmetric second-order tensor S. Thus

· r Vmust hold for all AK belonging to the group A. The matrIx K

== r v(AK) is the element of the set of matrices comprising the

irreducible representation r v of A which corresponds to the element AKof the group A = {A1,... ,AN}. We note that the arguments employed

in this chapter have been discussed by Smith and Bao [in press].

10.2 Generating Functions

We consider the problem of generating the general form of a

quantity Z(S) of type rv which is invariant under the group A. We

restrict consideration to the case where S is a symmetric second-order

tensor. The restrictions imposed on Z(S) by the requirement of in

variance under A = {AI' A2, ... } are given by

(10.2.3)

Let R(AK) == RK (K = 1,2,... ) denote the matrices comprIsIng

the matrix representation R = {RK} which defines the transformation

properties of s under the group A = {AI' A2, ... }. Let {R~)} denote the

matrix representation which defines the transformation properties of

the monomials

(10.2.4)

of total degree f in the components of s under the group A. The matrix

R~) == R (f)(AK) is referred to as the symmetrized Kronecker fth power

of RK. The numbers of linearly independent functions of type rv

which are of degrees 1 and f respectively in the components of sand

302 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.3] Irreducible Expressions: The Crystallographic Groups 303

We observe that these quantities are the coefficients of x, x2, x3,

respectively in the expansion of

The 3 x 3 and 4 x 4 matrices appearing in (10.2.9) are the symmetrized

Kronecker square RiP and the symmetrized Kronecker cube Rft)respectively of the matrix RK = diag(£I' £2). We see from (10.2.8) and

(10.2.9) that

(10.2.10)

[( ')2 " (')2]T d· (2 2) [ 2 2]Tsl ,sls2' s2 = lag £1' £1£2' £2 sl' sls2' s2 '

[(s~i, (s1)2s2, sl(s2)2, (s2)3]T (10.2.9)

_ d· (3 2 2 3) [3 2 2 3]T- lag 6'1' 6'1£2' £1£2' £2 sl' sls2' sls2' s2 '

(10.2.5)

1 _ 2 (2) f (f)det(E _ x R ) - 1 + x tr RK + x tr RK + ... + x tr RK + ....

6 K (10.2.6)

which are invariant under the finite group A = {AI'''.' AN} are given

by

With (10.2.5) and (10.2.6), we see that the number of linearly in

dependent quantities of type r v which are of degree n In the

components of s and which are invariant under A is given by the

coefficient of xn in the expansion of the quantity

where the ri are the matrices comprising the matrix representation

defining the transformation properties of a quantity of type r v under

the group A. We note that tr R~) is given by the coefficient of xf in

the expansion of the quantity l/det (E6 - x RK ) where E6 is the 6 X 6

identity matrix. Thus, we have

(10.2.7)

Gv(x) is referred to as the generating function for the number of

linearly independent quantities of type r v .

We give an example to indicate how one arrives at the result

(10.2.6). Suppose that RK = diag(£I' £2) is the 2 x 2 matrix which

defines the transformation properties of the column vector [sl' S2]T

under AK. We have

1 _ 1det(E2 - x RK ) - (1 - x£I)(1 - x£2)

_ (1 2 2 3 3)( 2 2 3 3- +x£l+ x £1+ x £1+··· l+x£2+ x £2+ x £2+···)

(10.2.11 )

where RK = diag(£I' £2). This is the result (10.2.6) for the special case

where RK is a two-dimensional diagonal matrix.

(10.2.8)

The transformation properties under AK of the 3 monomials sr, s1s2' s~

of degree 2, the 4 monomials sq, srs2' s1s~, s~ of degree 3, ... are given

by

10.3 Irreducible Expressions: The Crystallographic Groups

We consider the problem of determining irreducible expressions

for scalar-valued functions W(S), vector-valued functions P(S) and sym-


metric second-order tensor-valued functions T(S) of the symmetric

second-order tensor S which are invariant under a given crystal

lographic group A. We list in Table 10.1 (see Smith [1962b]) the

quantities det(E6 - x RK) appearing in the generating function (10.2.7)

for each of the AK appearing in the various crystallographic groups.

The matrices I, C, ... are defined by (1.3.3). The matrix RK is the sym

metrized Kronecker square of AK and defines the transformation of s

(see (10.2.3)) under AK. The procedure described in this section follows

closely that given by Bao [1987].

Table 10.1 Det (E6 - x RK)

I, C (1-x)6

R1,~,R3,D1,D2,D3'(I,C,R1,D1)·T1, (I,C,~,D2)·T2 (1_x)2(1_x2)2

(I,C,~,D3) .T3, (R1,~,D1,D2)· (81,82) (1-x)2(1_x2)2

(I, C, R1,~, R3, D1, D2, D3) . (M1, M2), (I, C) . (81,82) (1_x3)2

(~,R3,D2,D3)· T 1, (R1,~,D1,D3)· T2, (R1,~,D1,D2)· T 3 (1-x2)(1-x4)

(~, D3) . (81,82) (1-x)(1-x+x2)(1-x3)

rf, ... ,~ = E, F, -F, -E, K, L, -L, -K

where

(10.3.2)

The characters of the irreducible representations r1'... ' rS are seen from

(10.3.1) and (10.3.2) to be given by

tr rl, ... ,tr r~ = 1, 1, 1, 1, 1, 1, 1, 1·,

tr ry, ... ,tr r§ = 1, -1, -1, 1, -1, 1, 1, -1;

tr Ii, ···,tr Ii = 1, -1, -1, 1, 1, -1, -1, 1· (10.3.3),

tr rf' ···,tr :r§ = 1, 1, 1, 1, -1, -1, -1, -1;

tr rf, ···,tr ~ = 2, 0, 0, -2, 0, 0, 0, o.

We list below, the linear combinations of the components Pi and Tij of

a vector P and a symmetric second-order tensor T whose trans

formation properties under D2d are defined by the irreducible repre

sentations r 1,...,rS (see Table 7.6A, p.17S).

10.3.1 The Group D2d

There are five inequivalent irreducible representations associated

with the group D2d = {AI'···' AS} = {I, DI , D2, D3, T3, DIT3, D2T3,

D3T3}. These are seen from §7.3.7 to be given by

(10.3.4)

r 1 rl -1· 1 1 1 1 1 1 1 .1'···' 8 -., , , ., , , , ,

ry, ,r§ = 1, -1, -1, 1, -1, 1, 1, -1;

Ii, ,Ii = 1, -1, -1, 1, 1, -1, -1, 1 j

rf, ,r§ = 1, 1, 1, 1, -1, -1, -1, -1;

(10.3.1)

We refer to the quantities listed in (10.3.4) as quantities of types

r 1,...,rS· With (10.2.7), (10.3.3) and Table 10.1, we see that the

generating functions Gv(x) for the number of linearly independent

quantities of type r v which are invariant under the group D2d are given

by

306 Irreducible Polynomial Constitutive Expressions [eh. X Sect. 10.3] Irreducible Expressions: The Crystallographic Groups 307

(10.3.5)

the expression aO(K1,... , K6). The distinct monomial terms appearing

in the polynomial aO(K1,... , K6) are given by the monomial terms

appearing in the expression

(1 +KI +Ky +...)(1 +K2 +K~ +...)... (1 +K6+K~ +...). (10.3.9)

The number of distinct monomials appearing in aO(K1,... , K6) which are

of degree n in 8 is given by the number of terms of degree n in x in the

function obtained from (10.3.9) upon replacing Ki by ~ where j denotes

the degree in 8 of the invariant K·. Thus, the number of distinct1

monomial terms of degree n in 8 appearing in aO(K1,... , K6) is given by

the coefficient of xn in the expression

We see from §7.3.7 that a polynomial function WI (8) of 8 which

IS invariant under the group D2d , i.e., a function of type r l' is

expressible as a polynomial in the quantities

(1 + x + x2 + x3 + ... )2 (1 + x2 + x4 + x6 + ... )3 (1 + x4 + x8 + x12 + ... )(10.3.10)

where we have noted that the degree in 8 of K 1,... , K6 is given by 1,2,

1, 2, 4, 2. Alternatively, we may say that the number of terms of

degree n in 8 appearing in aO(K1,... , K6) is given by the coefficient of

xn in the formal expansion of

We observe that

(10.3.7)

1 (10.3.11 )

With (10.3.6) and (10.3.7), it is seen that the general expression for a

polynomial function of type r 1 is given by

(10.3.8)

where the aO, ... ,a3 are polynomial functions of the invariants K1,···,K6defined by (10.3.6). The invariants K 1,... , K6 are functionally inde

pendent. Thus, there are no polynomial relations K(K1,.. ·, K6) = 0

other than identities such as Ky = Ky. We now determine the number

of monomials of degree n in S which appear in (10.3.8). First, consider

Similarly, the number of monomials of degree n in 8 appearing in the

expressions alL1, a2L2' a3Ll L2 is given by the coefficient of xn in the

expressions

x3 x3

(l-x)2(1-x2)3(1-x4)' (l-x)2(I-x2)3(1-x4)'

(10.3.12)

where we see from (10.3.6) that L1 and L2 are each of degree three in S.


With (10.3.11) and (10.3.12), we have the result that the number of

monomial terms of degree n in S appearing in (10.3.8) is given by the

coefficient of xn in the expansion of

(10.3.15)

(10.3.17)

x + 2x2 + 2x3 + 2x4 + x5

(1 - x)2 (1 - x2)3 (1 - x4)

(10.3.16)

We observe that the number of distinct monomial terms of

degree n in S appearing in the expressions W2(S), W3(S), W4(S) and

V(S) are given by the coefficient of xn in the expansions of

We may now list the general irreducible expressions for vector

valued functions P(S) and symmetric second-order tensor-valued

functions T(S) which are invariant under D2d. With (10.3.4), we see

that [P1' P21T and P3 are quantities of types f S and f 3 respectively.

With (10.3.14), the irreducible expression for P(S) is given by

respectively. The argument leading to (10.3.16) is identical with that

employed to establish (10.3.13). Since the expressions (10.3.16) coincide

with the generating functions (10.3.5) for the number of linearly inde

pendent functions of types r 2,...,r5 respectively, we conclude that the

expressions (10.3.14) are irreducible.

(10.3.13)

r 2: W2(8) = b1812(8U - 822) +b2812(8~1 - 8~3) ++ b3(8 U - 822)823831 + b4823831(8~1 - 8~3)

r 3: W3(8) =(c1 +c2L2)812 + (c3 +c4L2)823831 (10.3.14)

r 4: W4(8) = (d1 +d2L1)(8U - 822) + (d3 +d4L1)(8~3 - 8~1)5

r 5: V(S) = E eiVi + e6L1V1 + e7L1V2 + e8L2V1i=1

where the b1

, b2, ... , eS are polynomial functions of the invariants

K1

,... , K6

given by (10.3.6), where L1 and L2 are invariants given by

(10.3.6) and where V1,... ,VS are defined by

This coincides with the expression for G1(x) given in (10.3.5). The

coefficient of xn in the expansion of G1(x) gives the number of linearly

independent functions of type r l' i.e. invariants, which are of degree n

in S. We see that the number of terms of degree n in S appearing in

(10.3.8) is equal to the number of linearly independent invariants of

degree n in S. We conclude that the expression W1(S) given by

(10.3.8) is irreducible.

We may employ the results of §7.3.7 to show that the general

expressions for polynomial functions of S which are of types r2'·'" r5

are given by

where a1, ... ,a8' b1,· .. ,b4 are polynomials in the invariants K1,... ,K6.

Similarly, with (10.3.4), (10.3.8) and (10.3.14), we see that the general

irreducible expression for T(S) is given by

310 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4] Irreducible Expressions: The Orthogonal Groups R3

, 03 311

TIl +T22 =cO+c1L1 +c2L2+ c3L1L2'

T33 = dO + d1L1 + d2L2 + d3L1L2,

T 12 = (e1 + e2L2)S12 + (e3 + e4L2)S23S31' (10.3.18)

T11 - T22 = (£1 +£2L1)(811 - 822) + (£3 + £4Ll)(8~3 - 8§1)'

T 5[T23, T31] = EgiVi+g6L1V1 +g7L1V2+g8L2V1

i=1

where the cO, ,g8 are polynomials in the invariants K1,... ,K6. The

quantities K1, ,K6, L1, L2 and V1,... ,V5 are defined in (10.3.6) and

(10.3.15) respectively.

The general expression for an nth-order tensor-valued function

T· . (8) which is invariant under D2d is readily generated. We may11·" In . . ..

use the procedure outlined in §5.3 to determIne the hnear combInatIons

of the 3n components of T· . which form quantities of types r 1,... ,11··· 1n

r 5. Xu, Smith and Smith [1987] have produced a computer program

which will automatically generate such results for any of the crystal

lographic groups. We may then employ the results (10.3.8) and

(10.3.14) to immediately list the general irreducible expression for

T· · (8). Results of the form given above have been obtained for11··· 1n .

almost all of the crystallographic groups by Bao [1987].

10.4 Irreducible Expressions: The Orthogonal Groups R3 , 03

syzygies exist, this is reflected in the form of the generating function.

In some cases, the syzygies are known or may be determined. With the

aid of the syzygies, we may establish an irreducible expression. We

observe that the form of the generating function for scalar-valued

functions invariant under R3 , say, would indicate the number and

degrees of the integrity basis elements. In more complicated cases, this

would be a critical piece of information. The generating function would

also indicate the presence (or absence) and degrees of the syzygies

relating the integrity basis elements.

The generating functions GO(.··; R3), G1(... ; R3), G2

( ... ; R3

),

G3(···; R3) for the number of linearly independent scalar-valued, vector

valued, symmetric second-order tensor-valued and skew-symmetric

second-order tensor-valued functions of the vectors xl'.'" xm ' the skew

symmetric second-order tensors AI' ... ' An and the symmetric second

order tensors 81,... , 8p which are invariant under R3 are given by

where i = 0, ... ,3; IXjl, lak l, Is£1 < 1 and

XO(O) = 1, Xl (0) = X3(0) = eiO + 1 +e-iO,

The matrix L(2)(0) in (1004.1) denotes the symmetrized Krollecker

The general expressions for functions of vectors xl' x2' ... , skew

symmetric second-order tensors AI' A2, ... and symmetric second-order

tensors 81, 82, ... which are invariant under an orthogonal group may

be found in Chapter VIII. In most of the simpler cases, these

expressions contain no redundant terms. We may determine generating

functions for' the number of linearly independent functions of given

degree which are invariant under the groups R3 and 03 respectively. If

L(8) =

1

oo

o 0

cos 8 -sin 8

sin 8 cos 8

(10.4.2)

312 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4] Irreducible Expressions: The Orthogonal Groups R3' 03 313

square of L(B). The quantities XO(B), ... , X3(B) are the characters of the

representations of R3 which define the transformation properties of

scalars, vectors, symmetric and skew-symmetric second-order tensors

respectively. The factor (1 - cos B)dB in (10.4.1) is the volume element

associated with the group R3 . With (10.4.2), we have

(10.4.3)

det(E6 - sL(2)(9)) = (1- se2i9)(1_ sei9)(1- s)2(1- se-i9)(1- se-2i9).

The generating functions GO(."; 03)' G1(... ; 03)' G2(···; 03)' G3(.. ·; 03)for the number of linearly independent scalar-valued, vector-valued,

symmetric second-order tensor-valued and skew-symmetric second-order

tensor-valued functions of the vectors xl'.'" xm ' the skew-symmetric

second-order tensors AI'.'" An and the symmetric second-order tensors

Sl'''.' Sp which are invariant under 03 are given (see Spencer [1970]) by

traceless tensors which are invariant under R3 may be readily deter

mined if we are given the generating functions for the number of

linearly independent functions of two-dimensional symmetric 2nth

order tensors which are invariant under the two-dimensional uni

modular group. Functions which are invariant under the two-dimen

sional unimodular group are studied in the classical theory of

invariants. The use of generating functions in classical invariant theory

has been treated by Sylvester [1879a,b; 1882] and Franklin [1880]. It is

possible to follow Spencer [1970] and employ the results on generating

functions in the classical theory to determine the generating functions

of interest here. We have, in fact, obtained the generating functions

given below by employing residue theory. In more complicated cases,

the effort involved in evaluating the integrals becomes inordinate. In

such cases, we might expect that the corresponding results in classical

theory are also unavailable.

Generating functions. The generating functions for the number

(10.4.5)

(10.4.6)

T(x) = Ti/x) = ~15ij + a3xixj' A(x) = Aij(x) = a4cijkxk

Irreducible expressions. The irreducible scalar, vector, sym

metric second-order tensor and skew-symmetric second-order tensor

valued functions of x which are invariant under R3 are given by

where the coefficients aO'.'" a4 are polynomial functions of the invariant

I = x· x. The €ijk in (10.4.6) denotes the alternating tensor (see

remarks following (1.2.16)).

10.4.1 Invariant Functions of a Vector x: R3

Integrity basis:

l=x·x:=xTx;

Gk(X1'''·'Xm , a1, .. ·,an , sl'·"'sp; 03)

= ~Gk(Xl""'Xm' al,· .. ,an, sl""'sp; R3) +

+~Gk(-Xl'''''-Xm' al, .. ·,an, sl""'sP; R3) (k = 0,2,3);(10.4.4)

Gl(xl""'xm, al, .. ·,an, sl'''''sP; 03) =~Gl(Xl'''''Xm' al,· .. ,an,

sl""'sP; R3) -~Gl(-Xl'''''-Xm' al, .. ·,an, sl""'sP; R3)·

The integrals (10.4.1) may be evaluated upon converting the

integrals into contour integrals in the complex plane by setting z = eiB

and then employing residue theory. In some cases, the evaluation of

these integrals may prove to be very difficult. Spencer [1970] has

shown that the generating functions for the number of linearly

independent functions of three-dimensional symmetric nth-order

314 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4J Irreducible Expressions: The Orthogonal Groups R3

• 03 315

of linearly independent scalar, vector, symmetric second-order and

skew-symmetric second-order tensor-valued functions of x which are

invariant under R3 are seen from (10.4.1) and (10.4.3) to be given by

where the cO' t'.... are constants. If we set. ck = 1 (k = 0, 1,2, ... ) and

replace I by x III (10.4.12) where d ( = 2) IS the degree of I = x· x in

the components of x, we obtain

where k = 0, ... ,3 and where XO(O), ... , X3(0) are given by (10.4.2). Inorder to evaluate the integral GO(x; R3), for example, we set

1 2

J1I" Xk(O) (l-cosO)dO

Gk(x; R3) = 2'0 '0 ' Ix 1< 1 (10.4.7)11" 0 (1 - xel )(1 - x)(l - xe-I )

We see that the coefficient of xn in (10.4.13) gives the number of

monomial terms of degree n in x which appear in aO(I). The expression

HO(x; R3) is the same as the generating function GO(x; R3) given by

(10.4.11). Thus, the coefficient of xn in HO(x; R3) is also equal to the

number of linearly independent scalar-valued functions of degree n in x

which are invariant under R3 . Hence, the monomial terms of arbitrary

degree n in x appearing in (10.4.12) are linearly independent. There are

no redundant terms and thus (10.4.12) is irreducible. We may refer to

HO(x; R3) as the generating function for the number of monomial terms

appearing in (10.4.12).

We may also determine the generating functions HI (x; R3),

H2(x; R3), H3(x; R3) for the number of monomial terms in P(x), T(x),

A(x). With (10.4.6), we have

(10.4.8)

(10.4.9)

(10.4.10)

2 cosO = z +z-ldO = dz/iz,z - e iO- ,

in (10.4.7) so as to obtain

-1 J (l-z)2 dzGO(x;R3)=411"i(1_x) z(l-xz)(z-x)' Ixl<1

C

where the contour C is Iz I = I traversed in the counterclockwise

direction. The residues at the simple poles inside C at z = 0 and z = x

are given respectively by

The value of the integral in (10.4.9) is given by 211"i times the sum of

the residues (10.4.10). With (10.4.9) and (10.4.10), we may then

determine the expression for GO(x; R3) and, in similar fashion, the

expressions for G I (x; R3), ... , G3(x; R3). We obtain

(10.4.14)

(10.4.16)

HI (x; R3) = (I +x2 +x4 +...)x = x (1- x2)-1 = GI (x; R3).

(10.4.15)

We set dk=1 (k=0,1,2, ... ) and replace I=x·x by x2, x by x in

(10.4.14) to obtain

Similarly, we see that(10.4.11)

Go(x; R3) =~ ,I-x

I +x2G2(x; R3) =--2'

I-x

The polynomial function ao(I) appearing in (10.4.6)1 is given by

(10.4.12)

We conclude as above that, since Hk(x; R3) = Gk(x; R3) for k = 1,2,3,

the expressions P(x), T(x) and A(x) given in (10.4.6) contain no re

dundant terms and are irreducible.


10.4.2 Invariant Functions of a Vector x: 03

The expressions (10.4.6)1 2 a for W(x), P(x) and T(x) also give, ,the general irreducible polynomial scalar, vector and symmetric second-

order tensor-valued functions of x which are invariant under 03· There

are no skew-symmetric second-order tensor-valued functions of x which

are ~nvariant under 03. This is reflected in the result that the

generating functions Gk(Xj 03) = Gk(Xj R3) for k = 0, 1, 2 and G3(Xj

03) is O. Thus, with (10.4.4) and (10.4.11), we have

GO(Xj 03)=~GO(Xj R3)+!GO(-Xj R3)=(1-x2

)-1,

G1(Xj 03) = ~G1 (Xj R3) - ~G1 (-Xj R3) = x (1 - x2)-1, (10.4.17)

G2

(Xj 03) = ~G2(Xj R3) +!G2(-Xj R3) = (1 +x2)(1-x

2)-1,

G3(Xj 03) = !G3(Xj R3) +~G3(-Xj R3) = o.

. (10.4.20)

F(x, fJ) = det(E3 - xL(fJ)) = (1- xe1fJ)(1- x)(l- xe-ifJ),

G ( . R ) 1 +XYZox, y, z, 3 =--n2-------..:-~-~------(1 - x )(1 - xy)(l - xz)(l - y2)(1 - yz)(1 - z2) ·

The matrix L(9) in (10.4.20)2 is defined in (10.4.2).

Syzygy:

xl x2 xa xl Y1 z1 x·x x·y x·z12 -7- Y1 Y2 Ya x2 Y2 z2 y·x y.y y·z (10.4.21 )

zl z2 za xa Ya za z·x z·y z·z

Any polynomial function W(x, y, z) which is invariant under R3

is expressible as a polynomial in the elements 11'... ' 17 of the integrity

basis (10.4.18). W(x, y, z) may then be written as

Irreducible expression:

10.4.3 Scalar-Valued Invariant Functions of Three Vectors x, y, z: R3

Integrity basis:

(10.4.23)

Integrity basis:

11,... ,I6 =x.x, x·y, x·z, y.y, y·z, Z·Z.

The syzygy (10.4.21) shows immediately that the terms involving Ii

(i ~ 2) are redundant so that (10.4.22) reduces to (10.4.19). 7

10.4.4 .Scalar-Valued Invariant Functions of Three Vectors x, y, z: 03

(10.4.19)

(10.4.18)11,... ,16 =x.x, x·y, x·z, y.y, y·z, z·z,

17 = det (x, y, z) = Cijk xi Yj zk

Twhere x· x = xTx, X = [xl' x2' x3] , and Cijk denotes the alternating

symbol.

Generating function: Irreducible expression:

211"1 J (1 - cos 8) d8

GO(x, y, Zj R3) = 211" F(x, fJ) F(y, fJ) F(z, fJ) , Ix I, 1y I, 1ZI< 1,o

(10.4.24)


Generating function. With (10.4.4) and (10.4.20)3' we have 10.4.5 Invariant Functions of a Symmetric Second-Order Tensor S: R3

Integrity basis:

(10.4.25)1 (10.4.29)

The monomial terms contained In the expression (10.4.24) given

by

Irreducible expressions. The irreducible scalar-valued and sym

metric second-order tensor-valued functions of S which are invariant

under R3 are given by

(10.4.26)(10.4.30)

are identical with the monomial terms contained in

Generating functions:

where the aO"'" a3 are polynomial functions of the invariants 11,1

2,1

3defined by (10.4.29). There are no vector-valued functions P(S) or

skew-symmetric second-order tensor-valued functions A(S) which areinvariant under R3 .

( . _...L 2I1rXk(B)(I-COSB)dB

Gk s, R3) - 211" 0 F(s,8) (k = 0,1,2,3) (10.4.31)

where Is 1< 1; XO(B), ... , X3(B) are given by (10.4.2) and

F(s,8) = (1- se2i8)(1_ sei8)(1-s)2(1-se-iB)(1_se-2iB). (10.4.32)

With (10.4.2), (10.4.31) and (10.4.32), we have

(10.4.27)

_ 1

- (1 - x2)(1 - xy)(1 - xz)(l - y2)(1 - yz)(1 - z2) .

HO(x, y, z; 03) =2 4 ( 22) ( 2 4 )= (1 +x +x + ) 1 +xy +x y + 1 +z +z + .

(10.4.28)

The number of monomial terms of degree m, n, p in x, y, z appearing in

(10.4.27) is given by the coefficient of xm yn zP in the generating

function HO(x, y, z; 03) for the number of monomial terms in W(x, y, z).

This is obtained by replacing 11,12, ... , 16 in (10.4.27) by x2, xy, xz, y2,

yz, z2 respectively. Thus,

With (10.4.25) and (10.4.28), we see that HO(x, y, z; 03) = GO(x, y, z;

03). Hence, the expression (10.4.24) for W(x, y, z) is irreducible. In

similar fashion, we may verify that the expression (10.4.19) for scalar

valued functions of x, y, z which are invariant under R3 is also

irreducible.

Go(s; R3) = 1 2 ,G1(s· R3) - 0(1 - s)(1 - s )(1 - s3) ,- ,

(10.4.33)


10.4.7 Invariant Functions of Symmetric Second-Order Tensors R, S: 03

With (10.4.33), we may readily establish the irreducibility of the

expressions (10.4.30).

Irreducible expressions. The irreducible scalar-valued, sym-

metric second-order tensor-valued and skew-symmetric second-order

tensor-valued functions W(R,S), T(R,S) and A(R,S) which are in

variant under 03 are given by

10.4.6 Invariant Functions of a Symmetric Second-Order Tensor S : 03

The expressions (10.4.30) for W(S) and T(S) also give the

general irreducible polynomial scalar-valued and symmetric second

order tensor-valued functions of S which are invariant under 03. There

are no vector-valued functions P(S) or skew-symmetric second-order

tensor-valued functions A(S) which are invariant under 03. We

observe that the generating functions Gk(s; 03) = Gk(s; R3) for k = 0,

1, 2, 3.

(10.4.37)

Go(r, Sj 03) = GO(r, Sj R3) = (1 + r2s2 + r4s4)/ K(r, s),

G1(r, Sj 03) = ~GI (r, Sj R3) - ~GI (r, Sj R3) = 0,

A(R, S) = (cO + cll lO)(RS - SR) + C2(R2S - SR2) + C3(RS2 - S2R)

+ C4(R2S2 - S2R2) + cs(R2SR - RSR2) + C6(S2RS - SRS2)

+ C7(R2S2R - RS2R2) + cs(S2R2S - SR2S2).

Generating functions. Let

G2(r, s; 03) = G2(r, s; R3)

= (1 +r +s +r2 +rs +s2 +r2s +rs2 +2r2s2 +r3s2

+r2s3 +r4s2 +r3s3 +r2s4 +r4s3 +r3s4 +r4s4)f K(r, s),

The coefficients aO' aI' a2' bO'·'" b17, cO,·'" c8 are polynomial functions

of the invariants 11,... ,19 defined by (10.4.34). There are no vector

valued functions P(R, S) which are invariant under 03.

. _ I 2

J'Tr Xk(8) (1- cos 8) d8

Gk(r, s, R3) - 211" 0 F(r, 8) F(s, 8) (10.4.36)

where 1r I, 1s 1< 1 and k = 0, ... ,3. The quantities XO(8), ... , X3(8) and

F(s,8) are given by (10.4.2) and (10.4.32) respectively. With (10.4.4)

and (10.4.36), we have

(10.4.34)

Integrity basis:

11, ,IS=trR, trS, trR2, trRS, trS2,

16, ,110 = tr R3, tr R2S, tr RS2, tr S3, tr R2S2.

Since the invariants 11,... ,110 defined by (10.4.34) form an

integrity basis, any scalar-valued polynomial function W(R, S) which is

invariant under 03 is expressible as

T(R, S) = (bO+ bl l lO + b21Io)E3 + (b3 + b411O)R+ (bS + b611O)S

+ (b7 + bSl lO)R2 + (bg + blOl lO)(RS + SR)

+ (bn + b1211O)S2 + (b13 + b1411O)(R2S + SR2)

+ (biS + b1611O)(RS2 + S2R) + b17(R2S2 + S2R2),

(10.4.35)

where

K(r, s) = (1 - r)(1 - r2)(I- r3)(1 - s)(1 - s2)

· (1 - s3)(1 - rs)(1 - r2s)(1 - rs2).(10.4.38)

322 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.5] Traceless Symmetric Third-Order Tensor: R3, 03 323

where the cO' c1' c2'." are polynomials in the invariants 11,... ,19.

Smith [1973] has shown that

where the aO' aI' a2 are polynomials in 11,... ,19. The monomial terms

contained in the polynomial expression aO(I1,... ,I9), say, are identical

with the monomial terms contained in

(10.4.45)

With (10.4.37)1 and (10.4.45), we see that HO(r, s; 03) = GO(r, s; 03).

Hence the expression W(R, S) = aO +al l lO +a2110 in (10.4.35) contains

no redundant terms and is irreducible.

Similarly, the number of monomial terms of degree m, n in R, S

appearing in the expressions al (11'·'" 19)110 and a2(11,... , 19)110 are

given by the coefficient of rm sn in the expressions obtained by

multiplying (10.4.44) by r2 s2 and by r4 s4 respectively. The sum of

these two expressions and the expression (10.4.44) give the generating

function HO(r, s; 03) for the number of monomial terms appearing in

aO +all lO +a2110 where ai = ai(11,... , 19). We have

The details of the argument leading to the result that any

symmetric second-order tensor-valued polynomial function of R, S

which is invariant under 03 is expressible in the form T(R, S) given by

(10.4.35)2 may be found in Smith [1973]. It is also shown there that

the generating function H2(r, s; 03) for the number of monomial terms

appearing in the expression (10.4.35)2 is equal to G2(r, s; 03). Hence

the expression T(R, S) contains no redundant terms.

(10.4.42)

(10.4.41 )

(10.4.40)

(10.4.39)i 1 i10W(R, S) = W(I1,.. ·, 110) = c· . 11 ... 110 .11"· 110

where the f30' f3 1, f32 are polynomials in the invariants 11,... ,19. With

(10.4.41), we see immediately that the terms cklfo (k ~ 3) in (10.4.40)

are redundant. Upon eliminating these redundant terms, we obtain

(10.4.35)1' i.e.,

2W(R, S) = aO +a1I10 +a2I10

We may also write (10.4.39) as

222(1 + II +11 + ... )(1 +12 + 12 + ...) ... (1 +19 +19 +... ). (10.4.43)

The number of monomial terms of degree ill, n in R, S appearing in

(10.4.43) is given by the coefficient of rm sn in the expression obtained. 223223from (10.4.43) by replacIng 11,12, ... , 19 by r, s, r , rs, s ,r , r s, rs , s .

This is given by

(10.4.44)

10.5 Scalar-Valued Invariant Functions of a Traceless Symmetric

Third-Order Tensor F: R3 , 03


functions of a traceless symmetric third-order tensor F which are in

variant under R3 . The components Fijk of F satisfy the relations

1

=l/K(r,s).F··· =0 F··· =0 F··· =0.

IJJ ' JIJ ' JJI

(10.5.1)

324 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.6] Traceless Symmetric Fourth-Order Tensor: R3 325

There are seven independent components of F which are given by are invariant under 03 is given by (see Spencer [1970])

(10.5.2)

Let L(0) denote the 7 X 7 matrix which defines the transformation

properties of the seven independent components under a rotation of °radians about the x3 axis, say. We may show that

det(E7 -£L(O))= (10.5.3)

= (1 - fe3iO)(1 - fe2iO)(1 - feiO)(1 - f)(l - fe- iO)(l - fe-2iO)(1 - fe-3iO).

(10.5.6)

This indicates that an integrity basis for functions of a third-order

traceless symmetric tensor F which are invariant under 03 is comprised

of four invariants 11, 12, 13, 14 of degrees 2, 4, 6, 10 respectively and

further that there are no syzygies relating these invariants.

The number of linearly independent polynomial functions of degree n in

F which are invariant under R3 is given by the coefficient of fn in the

expansion of the generating function GO(f; R3) where

G £. R - .l2J'lr (1 - cos 0) dO _ 1 + f15

0(' 3) - 211" 0 det(E7 - £L(O)) - (1- £2)(1 - £4)(1 - £6)(1- £10) ,

(10.5.4)

10.6 Scalar-Valued Invariant Functions of a Traceless Symmetric

Fourth-Order Tensor V: R3


functions of a traceless symmetric fourth-order tensor V which are in

variant under R3. The components V· · .. of V satisfy the 4! relations11121314

(10.6.1 )

(10.6.2)

where (0, (3, " b) is any of the 4! permutations of (1, 2, 3, 4) together

with the relations

where det(E7 -fL(0)) is given by (10.5.3) and where we assume

If I < 1. We have employed residue theory to evaluate the integral.

Spencer [1970] has obtained the same result upon employing a pro

cedure based on results from classical invariant theory. The form of the

generating function (10.5.4) indicates that there are five elements

11,... ,15 of degrees 2, 4, 6, 10, 15 comprising the integrity basis, and

further, that the irreducible expression for a scalar-valued polynomial

function W(F), invariant under R3, is given by

(10.5.5)There are nine independent components of V which are given by

where the coefficients aO and a1 are polynomial functions of the in

variants 11, 12, 13, 14 of degrees 2, 4, 6, 10 respectively. The generating

function for the number of linearly independent functions of F which

V1111' V1112' V1113' V1122' V1123'

V1222' V1223' V2222' V2223·(10.6.3)

326 Irreducible Polynomial Constitutive Expressions [Ch. X

Let L(O) denote the 9 x 9 matrix which defines the transformation

properties of the nine independent components (10.6.3) under a rotation

about the x3 axis. We may show thatREFERENCES

where the aO' ... , a4 are polynomial functions of the invariants 12, 13, 14,

IS' 16, 17. We emphasize that the above statements are conjectures.

We note from (10.6.S) that the number of linearly independent

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respectively.

where we take Iv I< 1. The form of the generating function indicates

that the elements of an integrity basis would consist of invariants 12, 13,

14, IS' 16, 17, IS' 19, 110 of degrees 2,. 3, 4, S, 6, 7, S, 9, 10 in V. We

may further conjecture that the irreducible expression for W(V) is

given by

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(10.6.S)

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The number of linearly independent polynomial functions of degree n in

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327

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Abelian group 16Adjoint matrix 21Alternating symbol 6Alternating tensor 6Axial vector 5, 7- transformation rules 7

Basic quantities 164Basic quantity tables 146, 164- C i , C s' C 2 167- C2h' C2V' D2 170- D 2h 171- 84' C4 172- C4h 174- D 4 , C 4V' D 2d 175- D4h 177- C3 180- C 3V' D 3 181- C 3i' C 3h' C6 182, 183- D3h' D3d' D6 , C 6V 186- C6h 189- D6h 194- T 270- Th 279- 0, T d 283- 0h 294- T1 154,260- T2 156,262Basis, scalar-valued functions- Ci' C s , C 2 168,169- C2h' C2V' D2 170- D2h 171- 84' C4 173- C4h 174- D4 , C 4V' D 2d 176- D4h 178- C 3 180- C3V' D3 182- C 3i' C 3h' C6 184- D3h' D3d' D6 , C 6V 187- C 6h 190

INDEX

- D6h 194- T 273, 275, 277- Th 281-° 286, 288, 291- Td 286,287,288,291- 0h 295- T1 261- T2 263- R3 238, 242, 254- 03 257Basis, tensor-valued functions- Cs 169- C3 195- R3 245,248Binomial coefficient 93

Cayley-Hamilton identity 203- generalized 205Character- of a representation 28- tables 33-- Sn(n = 2, ... ,8) 104-108Characters, orthogonality properties

29Class- of a group 18- order of 18Complete set of tensors 55- D 2h 90- 0h 92- 03 94- R3 96- T1 97Constitutive equations 1, 11- non-polynomial 11Conjugate elements 18Coset 18

Decomposition- of matrix representations 24- of physical tensors 114

333

334 Index Index 335

- of sets of property tensors 56, 88- of representations (m1". mp) .

(n1." nq) 231- of representations (n1". np) x (m)

232Determinant of a matrix 6Dimension of a representation 21Direct- product of groups 61- product of representations 61- sum of representations 24

Equivalent- coordinate systems 7- matrix-valued functions 209- reference frames 2- representations 23

Frame 62Function- invariant under a group 10- basis 12

Generating functions 300Group- characters 28- class of 18- continuous 36- defining material symmetry 7- definition of 15- generators of 18- manifold 37- order of 16- representations 20Group averaging methods 109- scalar-valued funcitons 109- tensor-valued functions 117- generation of property tensors 128

Hermitian matrix 25

Identities- Cayley-Hamilton 203- - generalized 205- relating tensors 94, 96, 99- relating 3 x 3 matrices 202, 207Identity matrix 4

Inner product- property and physical tensors 76Integrity basis 13, 43, 159- irreducible 14, 297see basis, scalar-valued functionsInvariant 10, 43- element of volume 38- integral 38-- over 03 40-- over R3 39-- over T} 40-- over T2 40Invariants- sets of symmetry type (n1." np) 217Irreducible- integrity basis 14, 297- representation 23Irreducible constitutive expressions

14, 298- D 2d 304- functions of vectors: R3 , 0 3

313-318- functions of symmetric tensors: R3 ,

03 319-326Irreducible representation tables- Ci' Cs, C2 167- C2h' C2V' D2 170-D2h 171- S4' C4 172- C4h 173- D 4 , C 4V' D 2d 175- D 4h 176- C3 180- C 3V' D 3 181- C3i' C3 h' C6 182- D 3h, D 3d' D 6 , C 6V 185- C 6h 188- D 6h 192- T 270- T h 279- 0, T d 282- 0h 292- T} 260- T2 262Isomer of a tensor 56Isotropic- functions 233

- - of two symmetric matrices 208- tensors 94

Kronecker- delta 4- product 112- square 71- nth power 54- - symmetrized 111

Material symmetry 7Matrix- adjoint 21- determinant of 6- Hermitian 25- identity 4- identities 203, 205- Kronecker nth power of 54- multiplication 16- orthogonal 4, 21- skew-symmetric 206- symmetric 206- transpose 21- trace of 28- unitary 21

Order- of a group 16- of a class 18Orthogonal- matrix 4, 21- groups 36, 37Orthogonality relations- for irreducible representations 27- for characters 29, 30

Partition 62Peano's theorem 51Permutations 19- cycle 19- - structure 19- class of 20- products of 19Physical tensors 53- decomposition of 114- outer products of 53- of symmetry class (n1n2 ... ) 69

Polarization process 51Polynomial basis 13Product tables 144- for D3 144, 199Proper orthogonal group 36Property tensors 53- complete set of 55- decomposition of sets of 56- sets of symmetry type (n1n2 ... ) 57,

88

Reducible- matrix product 209- trace of matrix product 209Reference frame 3- equivalent 2, 7Representations- dimension of 21- direct products of 61- equivalent 23- group 20- irreducible 23- matrix 21- reducible 23- regular 34Rivlin-Spencer procedure 207

Schur's Lemma 24- application of 133Skew-symmetric tensor 5Subgroup 18Summation convention 3Symmetric group 19- character tables 104- order of a class of 20Symmetric tensor 5Symmetry- group 7, 10- transformations 7Symmetry class- physical tensor 69- products of physical tensors 79Symmetrized Kronecker- nth power 111- square 72Syzygy 14, 43, 297

336

Tableau 62- standard 63- - ordering 63Tensors- alternating 6- Cartesian 3- invariant 13, 53- - complete sets of 55- isomers of 56- Kronecker delta 4- physical 53- property 53- sets of symmetry type (nln2 ... ) 57- skew-symmetric 5- symmetric 5- transformation properties of 3

Index

Trace of a matrix 28Transpose of a matrix 21Transverse isotropy groups 37, 38,153, 259Typical basis elements 162

Unitary matrix 21

Vector- absolute 5- axial 5, 7- polar 5

Weight function 38, 39

Young symmetry operators 64, 220- properties of 66

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