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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. . . . .
9.5.2 Functions of n Symmetric Second-Order Tensors
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
9.2.3 Functions of n Symmetric Second-Order Tensors
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
32 Group Representation Theory [Ch. II Sect. 2.5] Group Characters 33
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)
34 Group Representation Theory [Ch. II Sect. 2.5] Group Characters 35
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
38 Group Representation Theory [Ch. II Sect. 2.6] Continuous Groups 39
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)
40 Group Representation Theory [Ch. II Sect. 2.6] Continuous Groups 41
(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.
Theorem 3.1 An integrity basis for polynomial functions
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.
Theorem 3.2 An integrity basis for polynomial functions
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)
46 Elements of Invariant Theory [Ch. III Sect. 3.2] Some Fundamental Theorems 47
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)
Theorem 3.4 An integrity basis for polynomial functions
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.
48 Elements of Invariant Theory [Ch. III Sect. 3.2] Some Fundamental Theorems 49
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
50 Elements of Invariant Theory [Ch. III Sect. 3.2] Some Fundamental Theorems 51
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 ' .. ,
82 Invariant Tensors [Ch. IV Sect. 4.6] Symmetry Class of Products of Physical Tensors 83
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)
84 Invariant Tensors [Ch. IV Sect. 4.6] Symmetry Class of Products of Physical Tensors 85
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)
86 Invariant Tensors [Ch. IV Sect. 4.6] Symmetry Class of Products of Physical Tensors 87
(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
92 Invariant Tensors [Ch. IV Sect. 4.7] Symmetry Types of Complete Sets of Property Tensors 93
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
94 Invariant Tensors [Ch. IV Sect. 4.7] Symmetry Types of Complete Sets of Property Tensors 95
(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
96 Invariant Tensors [Ch. IV Sect. 4.7] Symmetry Types of Complete Sets of Property Tensors 97
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
Table 4.2 Character Table: 53
I 13 12 3
h, 1 3 2
(3) 1 1 1(21) 2 0 -1(111) 1 -1 1
Table 4.3 Character Table: 54
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
Table 4.4 Character Table: 55
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
Table 4.5 Character Table: 56
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
Table 4.6 Character Table: 57
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
(2.5.14) to be given by
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
122 Group A veraging Methods [Ch. V Sect. 5.5] Examples 123
(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
124 Group A veraging Methods [Ch. V Sect. 5.5] Examples 125
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)
166 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 167
(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
168 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 169
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, ...
174 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 175
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
176 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 177
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
178 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 179
(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)·
180 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 181
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.
The matrices E, A, ... , H appearing in Table 7.11 are defined by
(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.
186 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 187
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
188 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 189
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
190 Generation of Integrity Bases: The Crystallographic Groups [Ch. VII Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes 191
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
The matrices E, A, ... , H appearing in Table 7.13 are defined by
(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
(7.3.22) to be given by
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
208 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.3] The Rivlin-Spencer Procedure 209
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
210 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.3] The Rivlin-Spencer Procedure 211
(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~ ·
212 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.3] The Rivlin-Spencer Procedure 213
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)
214 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.3] The Rivlin-Spencer Procedure 215
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.
2. We compute the number 1?2 of linearly independent invariants
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)
3. We compute the number 1?3 of linearly independent invariants
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)
238 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 239
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)
240 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 241
(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:
242 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 243
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
244 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 245
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
246 Generation of Integrity Bases: Continuous Groups [eh. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 247
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).
248 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.7] Traceless Symmetric Second-Order Tensors: R3 249
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)
260 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.10] Transversely Isotropic Functions 261
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).
262 Generation of Integrity Bases: Continuous Groups [Ch. VIII Sect. 8.10] Transversely Isotropic Functions 263
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)
274 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.2] Tetartoidal Class, T, 23 275
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)
276 Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX Sect. 9.2] Tetartoidal Class, T, 23 277
(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
We see from Table 9.5A that the transformation properties of a
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
We see from Table 9.7A that the transformation properties of a
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-
304 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.3] Irreducible Expressions: The Crystallographic Groups 305
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.
308 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.3] Irreducible Expressions: The Crystallographic Groups 309
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.
316 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4] Irreducible Expressions: The Orthogonal Groups R3' 03 317
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)
318 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4] Irreducible Expressions: The Orthogonal Groups R3' 03 319
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)
320 Irreducible Polynomial Constitutive Expressions [Ch. X Sect. 10.4] Irreducible Expressions: The Orthogonal Groups R3' 03 321
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
We consider the problem of determining an integrity basis for
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
We consider the problem of determining an integrity basis for
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
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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