CONSTITUTIVE EQUATIONS FOR ANISOTROPIC AND ISOTROPIC MATERIALS
MECHANICS AND PHYSICS OF DISCRETE SYSTEMS
CONSTITUTIVE EQUATIONS FOR ANISOTROPIC AND ISOTROPIC MATERIALS
VOLUME 3
GERALD F. SMITH Editor:
GEORGE C. SIH
Department of Mechanical Engineering and Mechanics Lehigh University Bethlehem, PA, USA
Institute ofFracture and Solid Mechanics Lehigh University Bethlehem, PA, USA
~ ~
~
~ ~
~ 1994
NORTH-HOLLAND AMSTERDAM • LONDON • NEW YORK • TOKYO
NORTH-HOLLAND AMSTERDAM • 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 continuum physics are applicable. In these theories, the constitutive equations 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 ==
Thus, the expressions W == "p(E, ...) and T ==
required to be invariant under the group A which defines the material symmetry. We employ results from invariant theory and group representation theory to determine the form of the functions "p(E, ... ) and
~
n in some cases but in
The computations leading to
particular results may prove to be tedious. We plan to remedy this defect 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 Radzelovage 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 motivation and inspiration leading to the discussion of constitutive equations appearing here. vii
CONTENTS Introduction to the Series.
v
Preface . Chapter I
Vll
BASIC CONCEPTS.
1
1.1
Introduction.......
1
1.2
Transformation Properties of Tensors
3
1.3
Description of Material Symmetry . .
7
1.4
Restrictions Due to Material Symmetry .
9
1.5
Constitutive Equations . . . . . .
Chapter II
11
GROUP REPRESENTATION THEORY
15
2.1
Introduction. . . . . . .
15
2.2
Elements of Group Theory. . .
15
2.3
Group Representations . .
20
2.4
Schur's Lemma and Orthogonality Properties. .
24
2.5
Group Characters . .
28
2.6
Continuous Groups. . . . . .
Chapter III
....
36
ELEMENTS OF INVARIANT THEORY
43
3.1
Introduction. . . . . . . . .
43
3.2
Some Fundamental Theorems .
44
INVARIANT TENSORS
53
Chapter IV
4.1
Introduction.........
4.2
Decomposition of Property Tensors. . .
53
4.3
Frames, Standard Tableaux and Young Symmetry
. . ..
Operators. . . . . . . . . . . . . . . . . .
ix
56 62
xi
Contents
Contents
x
4.4
Physical Tensors of Symmetry Class (n1n2 ... ) . . .
4.5
The Inner Product of Property Tensors and Physical
7.3.2
69
7.3.3
Pinacoidal Class, C i , I; Domatic Class, C s, m; Sphenoidal Class, C 2 , 2 . Prismatic Class, C 2h , 2/m Rhombic-pyramidal Class, C 2v' mm2
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
7.3.4
Rhombic-disphenoidal Class, D 2 , 222 Rhombic-dipyramidal Class, D 2h , mmm .
4.9
Character Tables for Symmetric Groups 52' ... , 58
103
7.3.5
Tetragonal-disphenoidal Class, S 4' 4
Chapter V
GROUP AVERAGING METHODS
109
5.1
Introduction. . . . . . . . . . . . . .
109
5.2
Averaging Procedure for Scalar-Valued Functions.
109
5.3
Decomposition of Physical Tensors . . . . . . .
114
5.4
Averaging Procedures for Tensor-Valued Functions
117
5.5
Examples. . . . . . . . . . . . . .
121
5.6
Generation of Property Tensors
128
Chapter VI
. . . .
AND SCHUR'S LEMMA. . . . . .
7.3.6
Tetragonal-dipyramidal Class, C 4h' 4/m .
173
7.3.7
Tetragonal-trapezohedral Class, D4 , 422 Ditetragonal-pyramidal Class, C 4v' 4mm Tetragonal-scalenohedral Class, D 2d , 42m Ditetragonal-dipyramidal Class, D 4h' 4/ mmm
180
Trigonal-trapezohedral Class, D 3 , 32 7.3.11 Rhombohedral Class, C 3i' 3
181
Trigonal-dipyramidal Class, C 3h' 6 Hexagonal-pyramidal Class, C 6 , 6. 7.3.12 Ditrigonal-dipyramidal Class, D 3h , 6m2
133
6.2
Application of Schur's Lemma: Finite Groups
133
6.3
139
6.4
The Crystal Class D 3 . . . . . . . Product Tables . . . . . . . . .
144
Hexagonal-scalenohedral Class, D 3d , 3m Hexagonal-trapezohedral Class, D 6 , 622
6.5
The Crystal Class S 4' . . . . . . . .
149
Dihexagonal-pyramidal Class, C 6v' 6mm.
6.6
The Transversely Isotropic Groups T1 and T2
153
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 . 7.3.1
Pedial Class, C 1, 1 . . . . . . . . . . . . . .
7.3.13 Hexagonal-dipyramidal Class, C 6h , 6/m . 7.3.14 Dihexagonal-dipyramidal Class, D 6h , 6/mmm 7.4
159
176
Trigonal-pyramidal Class, C 3 , 3 7.3.10 Ditrigonal-pyramidal Class, C 3v' 3m
Introduction. . . . . . . . . .
THE CRYSTALLOGRAPHIC GROUPS
175
7.3.9
133
GENERATION OF INTEGRITY BASES:
171 172
6.1
Chapter VII
170
Tetragonal-pyramidal Class, C 4' 4
7.3.8
ANISOTROPIC CONSTITUTIVE EQUATIONS
167
7.5
182
184 188 191
Invariant Functions of a Symmetric Second-Order Tensor: C 3
195
Generation of Product Tables
199
Chapter VIII
GENERATION OF INTEGRITY BASES: CONTINUOUS GROUPS
. . . .
201
163 167
8.1
Introduction...........
201
xii
xiii
Contents
Contents
8.2
Identities Relating 3 x 3 Matrices. . . . .
202
9.4.4
8.3
The Rivlin-Spencer Procedure . . . .
207
290
8.4
Invariants of Symmetry Type (n1 ... n p ) .
216
Sl'···' Sn: T d', 0. . . . . . . . . . . Hexoctahedral Class, 0h' m3m. . . . . . . . . . . .
8.5
Generation of the Multilinear Elements of an Integrity
9.5.1
Functions of Quantities of Type r 9: 0h. . . . .
294
9.5.2
Functions of n Symmetric Second-Order Tensors
8.6 8.7 8.8
Basis. . . . . . . . . . . . . . . . . . . . . . .
223
Computation of lPn, P nl ... n p ' Qn, Qnl ... n p . . . . . Invariant Functions of Traceless Symmetric' Second-Order
226
Tensors: R3 . . . . . . . . . . . . . . . . . . . .
232
An Integrity Basis for Functions of Skew-Symmetric
Sl'···' Sn: 0h . . . . . . . . . . . . . . . . Chapter X
291
295
IRREDUCIBLE POLYNOMIAL CONSTITUTIVE EXPRESSIONS . . .
. . . . .
297
10.1 Introduction. . . . 10.2 Generating Functions. . .
300
10.3 Irreducible Expressions: The Crystallographic Groups. .
303 304
259
10.3.1 The Group D 2d . . . . . . . . . . . . . . . 10.4 Irreducible Expressions: The Orthogonal Groups R3' 3 .
.
260
10.4.1 Invariant Functions of a Vector x: R3 . . .
313
8.10.2 The Group T2 .
262
10.4.2 Invariant Functions of a Vector x: 03 . . .
316
Second-Order Tensors and Traceless Symmetric SecondOrder Tensors: R3 8.9
9.5
Functions of n Symmetric Second-Order Tensors
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
250
An Integrity Basis for Functions of Vectors and Traceless Symmetric Second-Order Tensors: 03.
256
8.10 Transversely Isotropic Functions . 8.10.1 The Group T1
297
°
310
10.4.3 Scalar-Valued Invariant Functions of Three Chapter IX
GENERATION OF INTEGRITY BASES: THE CUBIC CRYSTALLOGRAPHIC GROUPS
265
9.1
Introduction..............
265
9.2
Tetartoidal Class, T, 23. . . . . . . .
269
9.2.1
FunctionsofQuantitiesofTypesf 1,f2 ,f 3,f4 : T
270
9.2.2
Functions of n Vectors Pl··· Pn : T . . . . . . Functions of n Symmetric Second-Order Tensors
275
9.2.3 9.3
9.4
Vectors x, y, z: R3
.
316
10.4.4 Scalar-Valued Invariant Functions of Three Vectors x, y, z: 03
. . . . . . . . . .
317
10.4.5 Invariant Functions of a Symmetric Second-Order Tensor S: R3 . . . . . . . . . . . . . . . .
319
10.4.6 Invariant Functions of a Symmetric Second-Order
°
Tensor S: 3 . . . . . . . . . . . . . . . . 10.4.7 Invariant Functions of Symmetric Second-Order
320
Tensors R, S: 03 . . . . . . . . . . . . . .
320
Sl'''·' Sn: T. . . . . . . . . . . . . . . . . Diploidal Class, T h' m3. . . . . . . . . . . . . . .
276 278
9.3.1
Functions of Quantities of Type r 8: T h. .
280
10.5 Scalar-Valued Invariant Functions of a Traceless
9.3.2
Functions of Quantities of Types
r 1,r2,r3,r4 : T h
281
Symmetric Third-Order Tensor F: R3 , 03
Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m .
282
10.6 Scalar-Valued Invariant Functions of a Traceless
283 287
Symmetric Fourth-Order Tensor V: R3 . References
325
287
Index . . . . . . . . . . . . . . . . . .
333
9.4.1 9.4.2 9.4.3
Functions of Quantities of Types r l' r 3' r 4: T d' Functions ofn Vectors P1,···,Pn: T d . . . Functions of Quantities of Type r 5: T d'
°
°
. . . . . .
323
327
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·· 1J == C··k/lEk/l 1J t:. t:.'
T·· 1J == T.. J1'
Ek/lt:. == E/lt:.k
(1.1.2)
where T , Eke and C ijke are the components of the stress tensor T, the ij 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 T 1 ==T(71)' T 2 ==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 the 1J 1J tensors T 1, T 2 , ....
2
Basic Concepts
[Ch. I
There are restrictions imposed on the forms of the functions
Sect. 1.2]
Transformation Properties of Tensors
3
1.2 Transformation Properties of Tensors
appearing in (1.1.1), ... , (1.1.3) if the material possesses symmetry
The constitutive equations which define the response of a
properties. The material symmetry may be specified by listing the set
material are of the form T == 4>(E, F, ... ) where T, E, F, ... are tensors of
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
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
symmetry by listing a set of equivalent reference frames x, A 2x, ... which are obtained by subjecting the reference frame x to the set of
the reference frames employed are rectangular Cartesian coordinate
symmetry transformations.
will be Cartesian tensors.
Then, the forms which a constitutive
systems.
Thus, the tensors appearing in the constitutive expressions
equation assumes when referred to each of the equivalent reference frames are required to be the same.
Let x denote the reference frame with mutually orthogonal
This, of course, imposes on the
form of the constitutive equation restrictions which are characterized by
coordinate axes xI,x2,x3' We denote by eI,e2,e3 the unit base vectors
saying that the constitutive equation is invariant under the group of
which lie along the coordinate axes xl' x2' x3 respectively.
transformations A defining the symmetry properties of the material.
denote the reference frame with the same origin as the reference frame
Our main concern in this book will be the determination of the general
x and with mutually orthogonal coordinate axes xl' x2' x3'
form of functions 7P(E, F, ... ) and 4>(E, F, ... ) which are invariant under a
'" ' x2' , x3, vec t ors e1' e2' e3 l'Ie aI ong th e coor d'Inat e axes Xl' respectively. We define the orientation of the reference frame x' with
group A.
Let x' The unit
b ase
respect to the reference frame x by expressing the set of mutually The relevant mathematical disciplines required for dealing with this problem are the theory of invariants and the theory of group representations.
orthogonal unit base vectors el' e2' e3 as linear combinations of the unit base vectors e1,e2,e3' We have
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.
e! == A·· e· 1 IJ J'
e! . e· == A·· 1 J IJ
(1.2.1)
The
second main problem of invariant theory is concerned with the deter-
where e! . e· is the dot product of the vectors e! and e· and represents
mination of the relations existing among the terms appearing in the
the cosine of the angle x! ox·.
general expression for 4>(E, F, ... ). The theory of group representations
summation convention where the repeated subscript j indicates sum-
is essential if we are to deal with problems of considerable generality. determining the form of a constitutive expression to a number of much
mation over the values 1,2,3 which j may assume. Thus, A··e· == A·lel IJ J 1 + A i2 e2 + A i3 e3' We shall use this convention throughout the book. Similarly, the ei may be expressed as linear combinations of the ei. We
simpler problems. The concepts and results from group representation
see that
It provides a systematic procedure for reducing the problem of
1
J
1
1
J
J
In (1.2.1)1' we employ the usual
theory and invariant theory which we shall require will be discussed in Chapters II and III respectively.
(1.2.2)
Basic Concepts
4
thease b '" · SInce vec t ors e1' e2' e3 an d e1' e2' e3 f orm se t s
0f
[Ch. I
Sect. 1.2]
th ree
With (1.2.6) and (1.2.9)2' we obtain
mutually orthogonal unit vectors, we have e! . e! 1 J
= booIJ'
e· . e· 1 J
X!1 A.. IJ A kJ· == X!1 b·1k == X k' == A kJ· X·J.
= booIJ
x' and x respectively are related by the equation e! = A·· e·, then the 1 IJ J components Xi and Xi of a vector X when referred to the reference
b·· == 0 if i IJ
f=
j.
(1.2.4)
= Akiek · A£j ee = AkiA£j c5 k £ = AkiA kj = c5ij .
= [Aijl
Ak·A · == 8··IJ . 1 kJ
(1.2.11)
We refer to the Xi which transform according to (1.2.11) as the com(1.2.5)
ponents of an absolute vector or of a polar vector. Let C1!
Thus, the q~antities Aij (i,j == 1,2,3) satisfy
Let A
frames x' and x respectively are related by X!1 == A··X·. IJ J
ei . ej == Aikek . Aj £e£ == AikAj £ bk £ == AikA jk == bij ,
A·kA· 1 Jk == b.. IJ ,
(1.2.10)
Thus, if the base vectors e!1 and e·1 associated with the reference frames
With (1.2.1), ... , (1.2.3), we have
ei · ej
5
(1.2.3)
where bij is the Kronecker delta which is defined by b·· == 1 if i == j, IJ
Transformation Properties of Tensors
1·
1··· n
and C1·
1·
1··· n
(i 1,···,i n == 1,2,3) denote the components
of a three-dimensional nth-order tensor C when referred to the reference (1.2.6)
frames x' and x respectively.
If the base vectors e!1 and e·1 associated
with the reference frames x' and x are related bye! == A.. e·, then 1 IJ J
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 where IJ AT = [Aij]T = [Aji ]. Then the relations (1.2.6) may be written as AAT == E 3 ,
(1.2.12) Thus, the transformation rule for a second-order tensor T is given by
(1.2.7) (1.2.13)
3 = [c5ijl is the 3 X 3 identity matrix. A matrix A which satisfies (1.2.7) is referred to as an orthogonal matrix.
where E
The three-dimensional second-order tensors S == [S ..] and T == [T..] are 1J 1J said to be symmetric and skew-symmetric respectively if
A vector X may be expressed as a linear combination of the base vectors e·1 and also as a linear combination of the base vectors e!. Thus, 1
x = X·1 e·1 ==
S··IJ
= SOOJl'
T··IJ == -Too Jl
(1.2.14)
(1.2.8)
and have 6 and 3 independent components respectively. We frequently
where X· and X! are the components of the vector X when referred to
associate an axial vector t with a skew-symmetric second-order tensor T. Thus, let
1
X!1 e!1
1
the reference frames x and x' respectively. With (1.2.1) and (1.2.8), X!1 e!1 == X!1 A··IJ e·J == X·J e·J'
X!1 A··IJ == X·. J
(1.2.9)
t·1
= -21 c··kT·k IJ J'
T·Jk
= C·k· J 1 t·l'
(1.2.15)
Basic Concepts
6
[Ch. I
Sect. 1.3]
where the t i (i == 1,2,3) are the components of t and where Cijk is the alternating symbol defined by
Description of Material Symmetry
t! == (det A) A·· t· . 1 IJ J
7
(1.2.21)
Sets of three quantities which transform according to the rule (1.2.21) I if ijk == 123, 231, 312 ; c··k == { -1 if ijk == 132,321,213; IJ 0 otherwise.
(1.2.16)
are referred to as the components of an axial vector.
field vector H, the magnetic flux density vector B and the cross product X
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 whose IJ components 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 (1.2.17)
The magnetic
X
Y of two absolute (polar) vectors are examples of axial vectors.
1.3 Description of Material Symmetry
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
The components
t!1 of the
axial vector t when referred to the reference
specified. Let (Ae)i defined by
frame x' are given by 1 1 A . Ak T == -21 c· ·kA. Ak c t . (1.2.18) t!1 == -2 c·IJ·kT!k J == -2 c·IJ·k JP q pq IJ JP q pqr r We observe that
(i,j == 1,2,3)
(1.3.1)
denote the vectors into which e·1 IS carried by a symmetry transformation. The matrix A == [A ij ] 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
det A == Cijk Ali A 2j A 3k == Cijk Ail Aj2 Ak3 ' c··k IJ A·Ip A·Jq Ak r == Cpqr det A ,
set of unit base vectors for a rectangular Cartesian coordinate system (1.2.19)
c··k IJ A pI. A qJ. Ar k == cpqr det A ' Cijk Cij £ == 2 bk £ '
(Ae). == A.. e· 1 IJ J
Ax which is said to be equivalent to the coordinate system x.
symmetry transformation associated with the material determines an equivalent coordinate system Ax and an orthogonal matrix A.
Cijk == Cjki == Ckij
Each The
symmetry properties of the material may be defined by listing the set
where det A denotes the determinant of A. With (1.2.18) and (1.2.19),
of matrices Al
= [At] = I,
symmetry transformations. 1 A·IS t!1 == -21 c··k IJ A·IS A·JP Ak q Cpqr t r == -2 (det A)c pqs Cpqr t, r (1.2.20)
A 2 = [AD], ... which correspond to the set of
The set of matrices {AI' A 2 , ... } forms a
three-dimensional matrix group which we refer to as the symmetry group A. Symmetry transformations occurring In the description of the
With (1.2.6)1' (1.2.20)2 may be written as
symmetry properties of crystalline materials are denoted by I, C, R , i
8
[Ch. I
Basic Concepts
Di , T i , Mj and 8j (i == 1,2,3; j == 1,2). I is the identity transformation. C is the central inversion transformation. R i is the reflection transformation which transforms a rectangular Cartesian coordinate system into its image in the plane normal to the xi axis. The rotation trans-
formation T i transforms a rectangular Cartesian coordinate system into its image in the plane passing through the x·1 axis and bisecting the The transformations M 1 and M 2 transform a rectangular Cartesian coordinate system x into the systems 0
1 ~/2
0
Sl == -~/2 -1/2 o 0
0 1
-1/2
0
angle between the other two axes.
o o
1
o o
formation D i transforms a rectangular Cartesian coordinate system into that obtained by rotating it through 180 about the x·1 axis. The trans-
9
Restrictions Due to Material Symmetry
Sect. 1.4]
-1/2 -~/2 S2 == ~/2
o
-1/2 0
0 O. 1
1.4 Restrictions Due to Material Symmetry
Let the constitutive equation defining the material response be
0
obtained by rotating the system x through 120 and 240 respectively about a line passing through the origin and the point (1,1,1).
given by
The (1.4.1)
transformations 8 1 and 82 transform a rectangular Cartesian coordinate system x into the syste~s obtained by rotation of the system x through 0
0
120 and 240 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 transformed. We shall employ the notation a
0 0]
o
0
== 0 b 0 == diag (a, b, c) .
where T·· and E·· are the components of the second-order tensors T and IJ IJ E when referred to the reference frame x. Let x' be a reference frame whose base vectors e!1 are related to the base vectors e·1 of the reference frame x bye! == A·· e·. If we employ x' as the reference frame, the 1 IJ J constitutive equation (1.4.1) is given by
(1.3.2)
(1.4.2)
The matrices I, C, ... , Sl' S2 corresponding to the symmetry trans-
where T!. and E!. are the components of the tensors T and E when IJ IJ referred to the x' frame. With (1.2.13), we have
(a, b, c)
[
c
formations I, C, ... , 81, 82 are as follows: T!.IJ == A·Ip A·Jq T pq' I == (1, 1, 1),
C == (-1, -1, -1), Equations (1.4.1), ... ,(1.4.3) enable us to define the functions
R 1 == (-1, 1, 1),
~
D 1 == (1, -1, -1),
D 2 == (-1, 1, -1),
== ( 1, -1, 1),
R3 == ( 1, 1, -1),
(1.4.3)
4>i/-..).
Thus,
D 3 == (-1, -1, 1),
(1.4.4) If x and x' are equivalent reference frames, i.e., x and x' are related by
T 1 == 100] 001, [ 010
T 2 == [001] 010, 100
T 3 == [010] 100, 001
(1.3.3)
a symmetry transformation, the Tij must be the same functions of the
k
E £ as the T ij are of the E k £· Thus, 4>ij(".) == 4>ij( .. ') and, with (1.4.4),
[Ch. I
Basic Concepts
10
Sect. 1.5]
Constitutive Equations
11
(1.4.5)
(1.4.8)
Let A == {AI' A , ... } denote the symmetry group defining the symmetry 2 properties of the material under consideration. The matrices AI' A 2 , ...
where the T ij are the components of the stress tensor and the F are kA the deformation gradients. The xk = xk(XA) are the coordinates in the
comprising A relate the base vectors associated with the equivalent
deformed state of a point located at XA in the undeformed state. The
reference frames x, A x,.... Then, the restrictions due to material 2 symmetry require that the function
requirement of invariance under rotation of the physical system imposes
matrices A = [A
r ) belonging
s~tisfies
to the symmetry group A.
the restriction that
A function
(1.4.5) for all A belonging to A is said to be
invariant under the group A.
must hold for all proper orthogonal matrices Q = [Qij]'
More generally, the restrictions due to material symmetry which are imposed on the function rP·
. ( ... ) appearing in the constitutive
lI···lm
equation
Thus, the
function rPij ( ... ) is a second-order tensor-valued function of the three vectors F kl' Fk2' F k3 which is invariant under the three-dimensional proper orthogonal group R3 . Equation (1.4.9) is then a special case of
T· '.
lI···lm
where T·
==
., E
lI···lm
(1.4.6)
rP·
. (E k' F k k' ... ) , lI···lm k 1"· n 1··· p k and F
k 1··· n
k are the components of tensors T, k 1··· p
E and F, are that
rP·
. (A
lI ... lm
(1.4.7).
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
IJ
k It:-I
•••
A
IJ
E IJ
k nt:-n t:-I···t:-n
== A· . ... A· . rP· lIJI
IJ,
Ak
IJ
It:-I
•••
Ak
IJ
pt:-p
F IJ
functions. IJ, ••• )
t:-I···t:-P
(1.4.7)
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
. (E k k' F k k' ... ) 1··· n 1··· p
lwm JI···Jm
aid of results from the theory of group representations, we may readily
must hold for all matrices A = [A ij ] belonging to the symmetry group A defining the symmetry of the material being considered. A function
E, F, ... which are invariant under A.
systematic procedure for eliminating redundant terms from polynomial
. (E
k' F k k' ... ) which satisfies (1.4.7) k 1'" n 1'" p belonging to A is said to be invariant under the group A.
for
all
A
lI···lm
There may be restrictions of the form (1.4.7) imposed on the
compute the number of linearly independent terms of given degrees in This enables us to devise a
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
form of a constitutive equation which do not arise from material
constitutive equation 1/J
symmetry considerations.
polynomial single-valued function of E, F, ... which is invariant under A.
For example, we may assume that the
response of an elastic material is given by
= 1/J(E, F, ... ) where 1/J(E, F, ... ) is a non-
The problem would then be to determine a set of invariants Ij(E,F, ... )
Basic Concepts
12
[Ch. I
(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]
Sect. 1.5]
13
Constitutive Equations
COOk 1J == A·1p A·Jq A k r C pqr , must hold for all A == [A..] belonging to A. Tensors which satisfy these 1J restrictions are said to be invariant under A. We may proceed by
for cases where A is a continuous group and von Mises [1928], Smith
determining the general form of the tensors C ijk and C ijk £ which are invariant under A and then substitute into (1.5.1) to determine the
[1962a], Boehler [1978] and Bao [1987] for cases where A is one of the
general form of the constitutive equation. We discuss this procedure in
crystallographic groups. The arguments involved in generating function
Chapter IV.
bases can become quite intricate so that the possibility of errors arising
stitutive expressions of the form (1.5.1) which are invariant under a
is a consideration.
finite group A.
Further, suppose that it has been established that
We may employ other procedures for generating conThus, we apply a group - averaging technique in
the invariants II' ... , I p form a function basis and that none of the Ij (j == 1,... , p) is expressible as a single-valued function of the remaining
The procedures of Chapters V and VI are well adapted to computer-
invariants of the set II' ... , I p . This (see Bao and Smith [1990]) does
aided generation of constitutive equations.
not preclude the existence of another set of invariants J l' ... , J q (q < p)
producing computer programs based on these procedures which will
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.
For example, let the constitutive
expression be given by T··1J == CookX k + Cook"XkX" 1J 1J ~ ~ ~
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 scalarvalued function which is invariant under A. We may determine a set of polynomial functions I.(E, F, ... ) (j == 1,... , p), each of which is invariant J
under A, such that any polynomial function 7P(E, F, ... ) which is ininvariants I.(E, F, ... ) are said to form an integrity basis or a polynomial J basis. This yields the canonical form for scalar-valued functions which are invariant under A.
(1.5.1)
where X
and T·· are the components of a vector and a second-order k 1J tensor respectively. The restrictions imposed on the tensors C ijk and
£ by the requirement that (1.5.1) shall be invariant under the group ijk A are that
C
We are in the process of
variant under A is expressible as a polynomial in the Ij(E, F, ... ). The
We first consider problems where the functions 7P( ... ) and 4>( ... ) are polynomials of total degree:S N.
Chapter V and employ results based on Schur's Lemma in Chapter VI.
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
14
[Ch. I
Basic Concepts
groups are finite in number and are of dimensions one or two.
In
Chapter VIII, we employ a procedure involving Young symmetry
II
operators to generate constitutive expressions for functions of vectors and
second-order
tensors
which
are
invariant
under
the
three-
GROUP REPRESENTATION THEORY
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.
2.1 Introduction
We In this chapter, we discuss results from group theory and group
again utilize Young symmetry operators in Chapter IX to generate constitutive expressions for the five remaining (cubic) crystallographic groups.
... ,rr
The number r of inequivalent irreducible representations
r l'
associated with some of these groups is large and some of the
representations are three-dimensional. This contributes to the technical
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
difficulties so that only partial results are given.
the consideration of invariant-theoretic problems. The integrity bases II' ... ,I p generated in Chapters VII, VIII, IX are irreducible in the sense that no invariant I
belonging to the
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
k integrity basis is expressible as a polynomial in the remaining elements
of group representations furnishes a systematic procedure for converting
of the integrity basis. Suppose that
a large and sometimes almost intractable problem into a number of
..
1jJ(E, F, ... )
k
= f(I 1, 12 , ... , Ip ) = C ij ... k II Id ... Ip
.
(1.5.3)
We will find in general that some of the terms appearing in (1.5.3) are 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
redundant.
the generation of constitutive expressions which contain no redundant
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
terms and which are referred to as irreducible constitutive expressions.
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
[Ch. II
Group Representation Theory
(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.
Sect. 2.2]
where 8ij is the Kronecker delta defined by (1.2.4). The set of six 2 x 2 matrices AI' ... ' A 6 defined below forms a group where the multiplication rule is that of matrix multiplication. -1/2
A -
2- [ -{3/2
We observe that in general ab and ba differ. If in addition to (i), (ii) 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
{3/2 ],
-1/2
A -
3 - [ {3/2
-1/2
-{3/2 ] -1/2 '
(2.2.4)
and (iii), we have ab == ba for all a, b in A, then A is said to be an abelian group.
17
Elements of Group Theory
A4
=[
-1 0
0]
1 '
A5
[ 1/2
= _ ...]3/2
-{3/2] -1/2'
A6
=
[
1/2 ...]3/2
{3/2 ]. -1/2
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
The products AiAj (i,j == 1,... ,6) are listed in Table 2.1. Table 2.1
All
A 12
A 1n
A 21
A 22
A 2n
Product Table
Al
A2
A3
A4
A5
A6
Al A2
Al A2
A2
A3
A4
A5
A6
A4
A5
A3
A3
A5
A6
A4
A4
A4
Al A5
Al A2
A6
A2
A3
We
A5
A5
A6
A4
Al A3
A2
employ the usual matrix multiplication rule where the entry (AB)ij In
A6
A6
A4
A5
A2
Al A3
A==
AnI
A n2
(2.2.1)
Ann
Aij denotes the entry in row i, column j of the array (2.2.1).
A3
A6
Al
row i, column j of the product AB of B by A is given by (2.2.2)
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
In (2.2.2), the repeated subscript k indicates summation over the range
elements of the set AI' ... ' A 6. The matrix Al is the identity element of
1 to n. Thus, A·kB l ]· + A·12B 2]· + ... + A·In BnJ.. The set of all 1 1 k]· == A·lB non-singular n x n matrices with the multiplication rule (2.2.2) forms a
the group. We see from Table 2.1 that each of the A· has an inverse 1 which we may denote by Ail. Thus A 2l == A 3, Ail == A 2 , A4"l == A , 4 .... We denote the matrix group comprised of the matrices AI'···' A 6
group for which the identity element is the n
X
n identity matrix En
given in (2.2.4) by A == {AI'... ' A6} == {A K } (K == 1,... ,6). We shall frequently suppress (1<: == 1,... ,6) and denote the group by {A }. K
given by (En)" == 8.. IJ IJ
(i,j == 1,... , n)
(2.2.3)
If every element of a group is expressible as a product of the
18
Group Representation Theory
[Ch. II
Sect. 2.2]
As another example, we consider the group comprised of the n!
elements comprising a subset of the group, we say that the elements of this subset are generators of the group. For example, A 2 and A 4 are generators of the group A given by (2.2.4) since
permutations of the integers 1,2,... , n. Let s
=(
(2.2.5) 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
19
Elements of Group Theory
1 sl
2
t= ( 1
(2.2.8)
t1
denote the permutations which replace 1 by sl' 2 by s2'
, n by Sn and
1 by t l , 2 by t 2 , ... , n by t n 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
matrix group A = {AI'... ' A 6 } defined by (2.2.4):
symbols 1,... , n and then apply t. Thus, 1 is replaced by sl and then sl (2.2.6)
is replaced by t s1 ; ... ; n is replaced by Sn and Sn is replaced by t sn . For example, in the case where n = 6, we have
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 A 4 D = A4 {AI' A 2 , A 3} = {A4, A 5, A 6} is referred to as a (left) coset of 0 in A. We note that A = 0+ A 4 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 C l ,... ,C p . We see from Table 2.1 that the group A defined by (2.2.4) has 3 classes
s=
(
1 2
2 3
3 4
4 5
5
6
6 1
),
t=
(
1
2
3
4
5
2
1
4
3
6
~), (2.2.9)
ts = (
~
2 4
3 3
4
6
5 5
~ ),
st = (
~
2
3
4
5
2
5
4
1
6 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 (2.2.10)
given by The cycle structure of a permutation is denoted by
(2.2.7) (2.2.11) The order of a class Ci is the number Ni of elements comprising the class.
. omltte . d·f 1 is written as j. For h were a t erm J.'j IS 1 'j = 0 and where j
[Ch. II
Group Representation Theory
20
example, the permutations (2.2.9) may be written as s
== (1 2 3 4 5 6),
Sect. 2.3]
set of non-singular n x n matrices such that if a b
== (1 2) (3 4) (5 6),
t
D( a) D(b)
(2.2.12) ts
== (1) (3) (5) (246),
st
== (1 3 5) (2) (4) (6).
== (1 2 3 5 4 6),
u
have the same cycle structure.
== (1 3 5 6 2 4) We observe that u
(2.3.1)
==
sentation of dimension n of the group A. D(a) D(e)
D (e)
(2.2.13)
From (2.3.1), we see that
== D(ae) == D(a) so that
1'1 2 /2 ... nIn
or '1 '2 ... In) of 5 n · For example, s
== D( c).
== c, then
We say that the matrices D(e), D( a), D(b), ... form a matrix repre-
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 ... n In belong to the same class I (
21
Group Representations
== En == [<5ij] (i,j == 1,... , n)
(2.3.2)
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
rsr- 1 where (2.3.3)
== (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
A matrix D is said to be orthogonal if its inverse D- 1 is the transpose
class I of 5 n .
of D, i.e., if
r
The classes of the symmetric group 52 2: (12)
== {e, (12)} are given by (2.2.14)
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
where
DJ = Dji
12: (1) (23), (2) (31), (3) (12);
(2.2.15)
3: (123), (132) where, for example, the class denoted by 3 consists of permutations comprised of a single cycle of length 3. 2.3 Group Representations
Let e, a, b, c, ... denote the elements of a group A where e is the Let D(e), D(a), D(b), D(c), ... denote a
. A matrix D is said to be unitary if its adjoint
Dt
is
the inverse of D, i.e., if
Dt == D- 1 where DJ. 1J
13 : e == (1) (2) (3);
identity element of the group.
(2.3.4)
(2.3.5)
== D.. and where D.. denotes the complex conjugate of D... J1
1J
1J
We indicate the manner in which we may define a matrix representation {D(A 1 ), ... ,D(A N )} == {D 1 ,... ,D } == {D } of A which K N describes the transformation properties under the symmetry group A == {A 1 ,···,A N } == {A K } of the components of a tensor T. Consider the
case where T is a second-order three-dimensional tensor whose components 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 ,···, T 1 9 are given by
Group Representation Theory
22
[Ch. II
Group Representations
Sect. 2.3]
23
Thus, with (2.3.8) and (2.3.12), we have Let
Dir(A) Drn(B) == CijkAjpAkqDpqrCrstBs£BtmD£mn T'Jk == D·k·T. J 1 1
T·1 == C··kT· k IJ J'
(2.3.7)
== CookA. IJ IJ JP Ak q 8ps 8q t B s~IJB t m D ~mn
where C"IJ k D'JklJ~ == 8'1J 1~
DooIJ k CklJ~n == 8'1J 8· . 1~ In
,
IJ == C··kA. IJ JP B p~IJA k q B qm D ~mn
(2.3.8)
(2.3.14)
== C ijk (AB)j£ (AB)km D£mn
The C·· k , D" k may be obtained from (2.3.6). Let x and x' == Ax denote IJ IJ reference frames whose base vectors ei and ei are related by ei == Aijej
== Din(AB).
where the matrix A == [A ij ] appears in the group {AK }. The components of T when referred to the reference frames x and x' are given
The set of matrices D(A K ) == D K (K == 1,... ,N) which describes the transformation properties of t under the group A == {A K } then forms a
by T ij and Tij where, with (1.2.13),
matrix representation of A which we shall denote by {D K } (K == 1,... ,N) or by {D K }. If the matrices comprising {D K } are n x n matrices, we
T!.IJ == A·Ip A·Jq T pq .
(2.3.9)
The entries in the column vector t when referred to the reference frames x and x' are given by T i and Ti where, with (2.3.7), T.1 == CookT· IJ Jk ,
T!1 == C··kT!k' IJ J
The set of matrices SDK S-1 (K == 1,... , N)
(2.3.15)
(2.3.10) where S is non-singular also forms a matrix representation of A which is said to be equivalent to the representation {D K }. The matrices (2.3.15) define the transformation properties of the column vector u == S t under
With (2.3.7), (2.3.9) and (2.3.10), T!1 == C··kT!k IJ J == CookA. IJ JP Ak q T pq
say that {D K } is an n-dimensional matrix representation of A.
(2.3.11)
A. Thus, if t' == D K t, then
== COOk IJ A·JP A k q D pqr T r == D·Ir (A) T r
(2.3.16)
where D·lr (A) == C··kA. IJ JP Ak q D pqr .
(2.3.12)
Corresponding to each matrix A in {A K }, there IS a matrix D(A) == [Dir(A)] which relates the Ti and T i by (2.3.11). We observe that if
K =l
SD S-
1
F:
::]
(K=l,... ,N),
(2.3.17)
we say that the representation {D K } is reducible. If there is no S such that (2.3.17) holds for all K == 1,... , N, the representation is said to be
A, Band AB are elements of {AK }, then D(A) D(B) == D(AB).
If S can be chosen so that
(2.3.13)
irreducible. If S can be chosen so that
24
[Ch. II
Group Representation Theory
Sect. 2.4]
25
Schur's Lemma and Orthogonality Properties
for all D K comprising an n-dimensional irreducible representation of A, then C == AEn where En is the n x n identity matrix.
(2.3.18)
The argument leading to these results may be found in Wigner [1959], holds for K == 1,... , N, we say that {D K } decomposes into the direct sum of the representations {FK } and {GK }. It may be shown (see Wigner [1959], p.74) that a matrix representation {D
R- I }
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.
K}of a finite groupI A is
equivalent to a representation {RDK where the RD K R- are unitary. If the matrices D K are real, we may determine a matrix V such that the matrices comprising the representation {VD KV-I} are orthogonal. This also holds for the continuous groups considered in this We may thus restrict consideration to cases where the D K are either unitary matrices or orthogonal matrices. book.
Since the adjoint of AB is (AB)t
= BtAt,
we see from (2.4.2)
that
dDt< = Dt
(K
= 1,... , N).
(2.4.3)
We multiply (2.4.3) on the left and right by D K and note that the D K are unitary to obtain (2.4.4)
2.4 Schur's Lemma and Orthogonality Properties
Thus, with (2.4.2), the adjoint
Let {D K } and {RK } denote n-dimensional and m-dimensional irreducible matrix representations of the finite group A == {A 1,... ,A N }.
matrices C +
We may assume that the matrices DK == D(AK ) and RK == R(A K ) (K == 1,... , N) are unitary. We consider the problem of determining the
and satisfies
d
and i (C -
DK L == LD K (2.4.1)
of C and hence the Hermitian
also commute with each of the DK"
Consider then the case where L is Hermitian, i.e., L ==
form of the n x m matrix C which satisfies
DKC==CRK (K==l, ... ,N).
ct)
ct
(K == 1,... , N).
Lt
or L·. == Loo, 1J
J1
(2.4.5)
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 (2.4.6)
n:l m;
or
(i)
C == 0 if
(ii)
C == 0 if the representations {D K } and {RK } are inequivalent;
(iii)
C is non-singular, i.e., det C
:I 0 if n == m
and the representations
(2.4.7) where (2.4.8)
{D K } and {RK } are equivalent; (iv)
Suppose that the D K and hence the F K are 3 x 3 matrices and that M == diag(Al,A2,A3)· Further, suppose that Al == A2 =I A3· Then, from
if {D K } == {RK } in (2.4.1) so that C satisfies
DKC==CD K (K==l,oo.,N)
(2.4.2)
[Ch. II
Group Representation Theory
26
(2.4.7), we see that the matrices F K = 32 and K == 1, ... ,N.
[F~l are such that F~ = 0 for
ij == 13, 31, 23,
The
representation {FK } == {T- 1D T} is then the direct sum of two representations which K
contradicts the assumption that {D K } is irreducible. We conclude that (2.4.7) implies that {D K } is reducible unless Al == A2 == A3 == A, i.e., unless M == AE . Since {D } is assumed to be irreducible, we see that 3 K M = ,\E3 and hence, from (2.4.8)1' L = TMT- 1 = ,\E3 . Similarly, we
Schur's Lemma and Orthogonality Properties
Sect. 2.4]
unitary, Rj(l = Rk or (Rj(\j (2.4.11) to obtain
N K-K '" ~ D·Ir R·JS == 0 K ==1
= R~.
27
We then set C k £ = 8kr 8£s in
(i == 1, ... , n; j == 1,... , m)
(2.4.12)
where r is chosen from the set 1,... , nand s from the set 1,... , m. If we consider the case where {D K } == {RK }, then (2.4.10) becomes
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 + hence C = !(C + ct) -! i· i (C -
ct)
et and i (C -
ct)
must be multiples of the identity
matrix. Let {D } and {R } denote inequivalent irreducible unitary K K representations of dimensions nand m respectively of the group A == {A 1,... ,A N }. Let
P ==
N
L
L==l
-1
(2.4.9)
DLCRL
(K == 1,... , N) .
and
Since the representation {D K } is irreducible, Schur's Lemma yields the result that P == AEn. Then, upon setting R L == D L in (2.4.9), we have N
K
-K
L Dik Ck £ Dj £ K ==1
(i,j == 1,... , n)
== A8ij
(2.4.14)
where the Ck £ are arbitrary. (2.4.14) to obtain
N K-K '" D·Ir D·JS == Ars 8··IJ . ~
where C is an arbitrary n x m matrix. Then,
(2.4.13)
(2.4.15)
K ==1
We set i == j in (2.4.15) in order to determine Ars . Thus, (2.4.10) ==
N
L
M==l
-1
DMCRM RK == PRK
K-K
K ==1
where we have set DKD L = DM , RKRL = R M and R~ = RL1Rj(1. Since {D <) and {R } are inequivalent irreducible representations of A, K 1 Schur's Lemma tells us that all entries of P == [P ij ] are zero. With (2.4.9), we have
(i == 1,... , n; j == 1,... , m)
N
'" ~ D·Ir D·IS == N 8rs == Ars n
(K == 1, ... ,N)
(2.4.11)
(2.4.16)
where we have noted that D K is unitary and that 8ii == n. Substituting this expression for Ars into (2.4.15) gives N K-K N "D. ~ Ir D·JS == -n 8.. IJ 8rs
(i,j, r, s == 1,... , n) .
(2.4.17)
K==l
The equations (2.4.12) and (2.4.17) give the orthogonality relations for irreducible representations of a finite group A == {AI' ... ' AN}. Suppose
where the C £ are arbitrary and where we have noted that, since R K is k
that {D K } is two-dimensional (n == 2). We observe from (2.4.17) that
28
Group Representation Theory
[Ch. II
= aI""
= hI"" , D~I = cI" ..
,h N ;
D~I , ...
,cN;
Dh,· .. ,D~2
=
29
(2.4.12) to obtain
,aN;
Db,··· ,Dr2
Group Characters
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
the four sets of N quantities (N == order of A)
D!I,···,Dri
Sect. 2.5]
N
N K-K ED.. D..
(2.4.18)
K==l
11
E
JJ
XKXK == N,
K==l
(2.5.3)
d I ,· .. ,d N
N
N K-K ED.. R..
K==l
may be thought of as a set of four mutually orthogonal vectors of
11
E
JJ
XKXK == O.
K==l
lengths ~N /2 in an N-dimensional space. Thus ai ai == N/2, b i hi == N/2, a· c·1 == 0 ' .... Suppose that {RK } is a one-dimensional ... , a·1 b.1 == 0 ' 1
These relations are referred to as the orthogonality relations for the
representation of A. Then,
inequivalent irreducible representations of a finite group A is equal to
Ril,· .. ,Rri
=
(2.4.19)
f I ,· .. ,fN
characters of irreducible representations. We note that the number of
the number r of classes C1,... , Cr of A. representations by
r 1"'"
We denote these irreducible
r r and their characters by
is seen from (2.4.12) and (2.4.17) to be a vector of length ~ in an N(2.5.4)
dimensional space which is orthogonal to the vectors (2.4.18) arising from {D K }. Thus, fiIi == N; fi ai == 0, ... , fi d i == O.
respectively.
With (2.5.3), the orthogonality relations for the char-
acters are given by
2.5 Group Characters
N
Let r == {D K } 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 = trD K = 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 {D K } and {S D K S-1} have the same characters since (2.5.2) where we have noted that tr ABC == tr BCA == tr CAB.
r == {D K }
Now, let
and r' == {RK } be inequivalent irreducible unitary repre-
E X· X'K == N 8iJ· K==l lK J
(i,j == 1,... , r).
(2.5.5)
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 A L belong to the same class of A, i.e., if A K == All ALA M for some group element AM (see §2.2), then X
K
= tr DK = tr D:N{DLDM = tr DMD:N{DL = tr DL = XL
(2.5.6)
where we have used (2.3.13). We denote by Xk the common value of the XK == tr D K == tr D(A K ) for the AK belonging to the class Ck . We then denote the characters of the r inequivalent irreducible representations
r 1"'" r r by
Group Representation Theory
30
[Ch. II
(2.5.7)
where X.k is the value which the character of 1
r·1 assumes
r
(i,j == 1, ... , r)
or
(2.5.8)
where the summation is over the r classes of A. With (2.5.8), it is seen
31
Group Characters
r
XK == .E ci XiK 1==1
for the group
elements belonging to the class C . Let N k be the number of group k elements comprising the class C . We may then express the orthok gonalilty relations for the group characters as E N X·k X·k == N<5 i · k== 1 k 1 J J
Sect. 2.5]
(2.5.13)
(K == 1,... ,N)
where (X ,... ,XN) and (X ,... ,XiN) are the characters of the repre1 i1 sentations rand r·1 respectively. The orthogonality relations (2.5.5) enable us to determine the number ci of times the irreducible representation r i appears in the decomposition (2.5.10) of r. Let r 1 denote the I-dimensional identity representation of A where =1
rk
(K == 1,... ,N). With (2.5.5) and (2.5.13),
that (2.5.14)
form a set of r mutually orthogonal unit vectors in an r-dimensional space. A matrix representation r == {D K } of A may be decomposed into the direct sum of the r inequivalent irreducible representations
r i = {Dk}
where the summation in (2.5.14)3 ,4 is over the r classes of A and where denotes the order of the class C of A. k k (2.5.5), (2.5.8) and (2.5.13) that
N
We further note with
(i = I,... ,r) associated with A. Thus we may determine a 1 N _ 1 r _ 2 2 N E XKXK==N EN k Xk Xk==cl+···+ cr K==1 k==l
matrix S such that
(2.5.15)
(2.5.10)
where the c·1 are non-negative integers. If where the expression on the right denotes a matrix with cl matrices
Dk, ... , cr matrices Dk lying along the diagonal with zeros elsewhere.
(2.5.16)
For example,
(2.5.11)
we must have c· == 1 for some i and c· == 0 (j == 1,... , r; j f:. i). Thus, the 1 J condition (2.5.16) indicates that the representation r whose character is given by (Xl'.'" XN) is an irreducible representation.
Upon taking the trace of both sides of (2.5.10), we see with (2.5.2) that r
.
tr SD K S- 1 == tr DK == E ci tr Dk i==l
(K == 1,... ,N)
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' ... ' A 6 may be split into the three classes
(2.5.12) (2.5.17)
32
[Ch. II
Group Representation Theory
Sect. 2.5]
33
Group Characters
Since the number of inequivalent irreducible representations of A is
form a set of six mutually orthogonal vectors as
equal to the number of classes comprising A, there are three inequiv-
orthogonality relations (2.4,.12) and (2.4.17).
alent irreducible representations associated with A which we denote by
r 1 == {FK }, r 2 == {GK } and r 3 == {HK }.
r 1 : F 1,···,F6 == 1, r 2 : G 1,· .. ,G6 == 1, r 3 : H 1,···,H6 == AI'
These are given by
1,
1,
1,
1,
1, -1, -1, -1;
Xik
r2
( == [det A K ]) which == det A!{ det A L . r 3 is the representation furnished by the group representations
Xik == XiK (k == K == 1),
A 2, A3 , A4 , A 5 , A 6
is obtained by setting G K == det A K furnishes a representation since det AKA L
elements AI' ... ' A 6 .
We now construct the character table for the group A.
(2.5.18)
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 A 1,... ,A6 are defined by (2.2.4). r 1 is the identity representation.
With (2.2.4) and (2.5.18), the characters of the
r l' r 2' r 3 are given by 1,
1,
(X21'···' X26) == (1,
1,
1, -1, -1, -1);
(X31 ,... , X36) == (2, -1, -1,
0,
1,
= XiK
(k
= 3;
K
Xik == XiK (k == 2; K == 2,3),
= 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·1 and
the column headed C· is the value
X.. which the character of the representation
r·1
J
assumes for the A K The entries in the row headed N· give the
~
belonging to the class C·. J
J
orders of the the classes C .. The X.. are determined from (2.5.19) and J IJ (2.5.21 ).
0,
C1
C2
C3
N·
1
2
3
r1
1
1
1
f2 f3
1
1
-1
2
-1
1); (2.5.19)
J
0).
Since the (Xil,.",Xi6) (i == 1,2,3) satisfy (2.5.16), the representations r l' r 2 and r 3 are irreducible. For example, (X31 X31 +... + X36 X36) == 1. We observe that
t
1 6 F 11 ,···,F 11
We
Table 2.2 Character Table: A
(XII'···' X16 ) == (1,
1,
required by the
observe from (2.5.17) that AI' A K (K == 2,3) and A K (K == 4,5,6) comprise the classes C1, C 2 and C3 . Thus,
1·,
1,
IS
1,
1,
1,
1,
1,
1·,
G111,···,G 6 == 1, 11
1,
1,
-1,
-1,
-1 ;
HI11,···,H6 == 1, -1/2, -1/2, -1, 11
1/2,
1/2 ;
HI 6 12'···' H 12 == 0, ~/2; -~/2, 0, -~/2, ~/2; HI 6 21'···' H 21 == 0, -{3/2, {3/2, 0, -{3/2, {3/2; HI22' ... ,H 6 == 1, -1/2, -1/2, 1, -1/2, -1/2 22
°
We may readily verify that the orthogonality relations (2.5.8) for the group characters hold.
(2.5.20) (2.5.22)
[Ch. II
Group Representation Theory
34
M1 =
1
0
0
0
1
1
0
0
, M2 =
0
0
1
1
0
0
0
1
0
= 6.
themselves when multiplied on the left by the matrices A K (K = 1, ... ,6). Thus, A L [A 1,···,A6] = [A 1,···,A6]RL . Also, AK A L [A 1,... ,A6] = A K [AI'···' A6] R L = [AI'···' A6] RKRL · We observe that if AKA L = AM' then RKRL = R M . For example, we see from Table 2.1 that
A4 A5 = A2 and from (2.5.22) that R 4 R 5 =~. The set of matrices R K = R(A K ) (K = 1,... ,6) then forms a matrix representation of
dimension 6 of the group A = {AI'... ' A 6 ) defined by (2.2.4) which we
(2.5.27) where N is the order of the group A considered and the d 1,... , d r 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
= tr R K ·
(6,0,0,0,0, 0)
(2.5.24)
With (2.5.14) and (2.5.24), the number ci of times
6
= t K~l XK XiK = XiI
+GK +2HK ) S
(K
= 1,... ,6).
(2.5.28)
KKK
S3kR kj = H 11 S3j
+ H 12 S4j , (2.5.29)
S4kR~ = H~l S3j + H~2 S4j .
the irreducible representation f i appears in the decomposition of {RK } is given by ci
(FK
Consider the third and fourth rows of (2.5.28). We have
(2.5.23) to be given by
where XK
=
S RK
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
=
This is a reflection of the general result that
(2.5.23)
The matrices R K (K = 1,... ,6) listed in (2.5.22) describe the manner In which the AI' ... ' A6 defined by (2.2.4) permute among
(Xl, ... ,X6)
35
Group Characters
and take the trace of the resulting expression to obtain dy + d~ + d~
where E 3 is the 3 x 3 identity matrix and 0
Sect. 2.5]
(i
= 1, 2, 3).
(2.5.25)
Let S3j
=
L6
of the character of the representation f· 11 1 corresponding to the identity element Al of A is equal to the dimension
d i of the representation fi.
Thus f i appears d i times in the decomposition of the regular representation, i.e., there is a matrix S such that (2.5.26) where, with (2.5.18), d 1== d 2 == 1, d 3 == 2. We may set K == 1 in (2.5.26)
L
S4j
=
L6
-1
L
(HL )21 R 1j ·
(2.5.30)
L=l L=l With (2.5.30), the left hand side of (2.5.29)1 gives
K S3k R k· J
=
L6
-1 L K (H L )11 R 1k R k · L=l J
K
We note that the value X.
-1
(H L )11 R 1j ,
= H 11
=
L6
M=l
-1 M (H K H M )11 R 1J·
6 -1 M K 6 -1 M L (HM )11 R 1· + H 12 L (HM )21 R 1· M=l J M=l J
(2.5.31 )
= H¥l S3j + H¥2 S4j where we have noted that if ALA K = AM' then RLRK = R M , 1 HLHK = H M and Hr: = HK Hll. 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
36
Group Representation Theory
[Ch. II
show that the rows of S are given by
Sect. 2.6]
Continuous Groups
37
unit vector (2.6.2) (2.5.32)
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) where 0 :::; 8 :::;
1
1
1
1
1
1
1
1
1
-1
-1
-1
1
-1/2
-1/2
-1
1/2
1/2
0
-~/2
~/2
0
-~/2
~/2
0
~/2
-~/2
0
-~/2
~/2
1
-1/2
-1/2
1
-1/2
-1/2
sphere of radius
0:::; ¢1 :::;
1r,
0:::; ¢2 :::; 21r.
The vectors (2.6.3) form a
The group R3 of all rotations is referred to as a
1r.
three-parameter continuous group and the domain over which the parameters ~ l'
S==
1r,
(2.5.33)
~2' ~3
vary is referred to as the group manifold.
The set of all 3 x 3 real matrices A which satisfy det A == ± 1
(2.6.4)
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
2.6 Continuous Groups
(ii) the improper orthogonal matrices which are of the form CA where
The symmetry properties of isotropic materials and transversely
C == diag (-1, -1, -1) and A belongs to R3 . We observe that R3 is a
isotropic materials are defined by continuous groups. For example, the
subgroup of 03.
symmetry of a material possessing rotational symmetry but which lacks
mixed continuous subgroups of 03 which define the symmetry
a center of symmetry is defined by the three-dimensional proper
properties of transversely isotropic materials.
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 det A == 1.
(2.6.1)
Consider a rotation of 8 radians whose axis of rotation is given by the
We shall be concerned with other continuous and One such group T1 is
comprised of the set of matrices cos 8
sin 8
Q( 8) == -sin 8
cos 8
o o
0
1
o
(0 :::; 8 :::; 21r).
(2.6.5)
The group T1 is a one-parameter continuous group. Another subgroup
[Ch. II
Group Representation Theory
38
of 03' which is denoted by T2 , is comprised of the matrices cos (}
sin (}
0
Q((}) == -sin (}
cos (}
0
0
1
0
-cos (} -sin (}
, R 1Q((}) == -sin (}
0
0
1
0
representations rand r' are (see Wigner [1959], p. 101)
ID· (A) (A) dr = (VIn) 8·· IDir(A) Rjs(A) dr =
0
cos (}
V
I
= dr, A (2.6.9)
O.
A The results corresponding to (2.5.3) which give the orthogonality relations for the characters of the irreducible representations rand r' are
Ix(A) x(A) dr = v,
Ix(A) x'(A) dr = 0
A
A
We consider integrals of the form (2.6.7)
A
(2.6.10)
where X(A) == tr D(A) and X'(A) == tr R(A) denote the characters of the representations rand r' respectively.
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
If(SA) dr = If(A) dr A
IJ fJ rs'
A
(2.6.6)
(i) the Q((}) and (ii) the matrices R 1Q((}).
= If(~1""'~n) w(~1'''''~n) d~1 .. ·d~n
f).JS
Ir
where R 1 == 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
If(A)dr A
39
Continuous Groups
Sect. 2.6]
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 (2.6.8)
(2.6.11)
A
where S is an element of A. We refer to dT==w(~I'... '~n)d~I ... d~n as
where Po is a constant and where A corresponds to a rotation through (}
an invariant element of volume. We may determine invariant elements
radians about an axis whose direction is given by (~1' ~2' ~3).
With
of volume for the continuous groups considered here, i.e., 03 and its
(2.6.3) and (2.6.11), the volume V of the group R3 is given by
continuous subgroups.
The arguments leading to the orthogonality
relations for the irreducible representations and for the characters of the
V=
R3
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'
I dr = I W(~1'~2'~3)d~1 d~2d~3 R3
I I I 2Po(1-cosO)O-202sin
1r 1r 21r
=
(2.6.12) 2 81r pO'
000 The three-dimensional full orthogonal group 03 is a mixed continuous group comprised of the two cosets formed by (i) the
are unitary representations. The results corresponding to (2.4.12) and
matrices A belonging to R3 and (ii) the matrices CA where the A
(2.4.17) which give the orthogonality relations for the irreducible
belong to R3 . The invariant integral over 03 is given by
40
[Ch. II
Group Representation Theory
Jf(A) dT = Jf(A) dT + Jf(CA) dT. 03
R3
(2.6.13)
R3
Sect. 2.6]
41
Continuous Groups
We may determine the number ck of times that the irreducible representation {Dk(A)} appears in the decomposition of {D(A)} upon
The invariant integral over the one-parameter group T1 == {Q( B)}
multiplying both sides of (2.6.17) by Xk(A) and then integrating over
(0 ~ B ~ 27r) defined by (2.6.5) is given by
A. Thus, with (2.6.10) and (2.6.17),
27r ff(A)dT= fg(O)dO, T1 0
g(O)=f(A(O))
ck =
(2.6.14)
~
JX(A) Xk(A) dT,
c1=~fx(A)dT.
(2.6.18)
A
A
The coefficient c1 appearing in (2.6.17) and (2.6.18) denotes the number
where the invariant element of volume dT == dB. The invariant integral
of
associated with the mixed continuous group T2 comprised of the two
decomposition of the representation {D(A)}. This gives the number of
cosets formed by (i) the matrices Q(B) belonging to T1 and (ii) the
linearly independent functions which are linear in the quantities
matrices R 1Q( B) where Q( B) belongs to T1 (see 2.6.6) is given by
forming the carrier space of the representation {D(A)} and which are
T1
(2.6.15)
27r
= f g( 0) dO + f h( 0) dO o g( B) == f( A( B)),
{D 1(A)}
appears
in
the
We observe that the characters X(A) and Xk(A) appearing in given by R3 , 03' T1 or T2 . The expressions (2.6.18) appropriate for
R 1 == diag (-1, 1, 1),
and
the groups R3 , 03' T1 and T2 are listed below. 27r
We are particularly concerned with cases where the functions
R3 : ck
f(A) appearing as integrands of the integrals f f(A) dT are the characters c1
X(A) == tr D(A) and X'(A) == tr R(A). Let {D 1(A)}, {D 2 (A)}, ... denote the inequivalent irreducible representations of the continuous group A. We may then determine a
f
= i7r X(O)Xk(O)(l-cosO)dO, o 27r
X(A) == tr D(A) of a representation {D(A)} or the product of characters
Let {D(A)} denote a representation of A.
representation
(2.6.18) are functions of a single variable B for the cases where A is
0
h( B) == f( R 1A( B)), where the element of volume dT == dB. where
identity
Xl (A) is the character of the identity representation of A.
T1
27r
the
invariant under A. In (2.6.18)2' we have noted that Xl (A) == 1 where
f f(A) dT = f f(A) dT + f f(R1A) dT. T2
times
(2.6.19)
=i7r f X(O)(l-cosO)dO o
where X(B) == trD(A), Xk(B) == trDk(A), and A denotes a rotation through B radians about the xl axis.
matrix S such that
27r
(2.6.16)
03: ck =
27r
4~ JX(O) Xk(O) (1- cosO) dO + l7r JX'(O) Xk(O) (1- cosO) dO,
o
Upon taking the trace of both sides of (2.6.16), we have (2.6.17)
c1 =
0
27r
27r
o
0
4~ JX( 0)( 1 - cos 0) dO + 4~ JX' (0) (1 - cos 0) dO
(2.6.20)
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). 21r
T1 : ck
= l7r f x( 0) Xk( 0) dO,
ELEMENTS OF INVARIANT THEORY
21r
c1
o
= 2~ f X(O) dO
(2.6.21 )
o
where X(8) == trD(A), X (8) == trDk(A), and A denotes a rotation k through 8 radians about the x3 axis.
T2 : ck
c1
= 4~
21r
2~
o
0
21r
21r
o
0
f
= 4~ X( 0) dO + l7r X/( 0) dO
We consider the problem of determining the general form of a group A == {A}.
Let {D(A)} denote the matrix representation which
defines the transformation properties of the quantity x == [xl' ... ' Xn]T (2.6.22)
X(8) == tr D(A), X (8) == tr Dk(A), X'(8) == tr D(R1A), Xk(8) k = tr D k (R1A), A denotes a rotation through 8 radians about the x3 axis, and R 1 = diag(-I, 1, 1). where
3.1 Introduction
scalar-valued polynomial function W(x) which is invariant under the
f X(O) Xk(O) dO + 4~ f X/(O) xk(O) dO,
f
III
under A. W(x) is said to be invariant under A if W(x) == W( D(A) x)
(3.1.1)
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 W(x) == c·
I ...
. I i ... I Jp. J 1
(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), I 2 (x), I 3 (x). This is referred to as a syzygy. The second main problem of invariant theory is to determine a set of relations fi (I 1,... , Ip ) == 0 (i == 1,... , q) such that every syzygy relating the invariants I 1,... ,I p is a consequence of the 43
44
Elements of Invariant Theory
o.
relations fi (I 1 ,... , I p ) == form
[Ch. III
Sect. 3.2]
We usually write the expression (3.1.2) in the
(3.1.3)
Some Fundamental Theorems
Theorem 3.3
An
integrity
basis
for
45
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
where Wk(x) is a linear combination of all the monomials Ii ... Ii which are of degree k in x.
W(X1'···' x n ) == W(Ok x 1'···' 0kxn)
We are able to compute the number of linearly
independent functions of degree k in x which are invariant under A.
°1 =[ ~
If any exist, we may employ relations of the
form fi (I 1 ,... , I p ) == 0 to remove the redundancies. Thorough discussions of the theory of invariants are given by Grace and Young [1903], Elliott
is given by
[1913], Weitzenboch [1923], Schur and Grunsky [1968] and Weyl [1946].
1.
Lxi
3.2 Some Fundamental Theorems
2.
L(xi ~2
An
integrity
basis
for
0 0 1
l °2 =[ ~ l °3 =[ ~ 1 0 0
3.
+ x~~l)
(i,j
]
(3.2.4)
= 1,... ,n; i ::;j),
(i,j == 1, ... ,n; i <j ) ;
(i,j,k == 1, ... ,n; i ~j ~ k).
L xi ~2'"
denotes the sum of the three terms obtained by
cyclic permutation of the subscripts on the summand. For example, (3.2.6)
is given by
(j, k == 1,... , n; j Theorem 3.2
1 0 0
L xi (x~ x! + ~3 x~) (i,j, k = 1,... , nj i ::; j ::; k) , . . k k . k· . k·· . L {xi x-I1(X2 - X3) + x-I1Xl (x2 - x3) + Xl xi (x-I2 - x-I3) }
In (3.2.5), (3.2.1)
0 0 1
(i == 1,... , n) ;
polynomial functions
W(x1,x2' ... ,xn ) which satisfy
0 1 0
(3.2.5)
poses of determining integrity bases. We shall employ the notation xf
Theorem 3.1
0 1 0
"'( j - x2i xj) L...J Xli x2 1
In this section, we list some theorems which are useful for purto designate the jth component of the vector x n ' i.e., x n == [xl,x2' ... ]T.
(3.2.3)
where
This enables us to determine whether there are any redundant terms in the expression Wk(x).
(k == 1,2,3)
An
integrity
~
(3.2.2)
k).
basis
In order to prove Theorem 3.3, we set for
polynomial functions
W(I 1 ,I 2 ,... ,Ir , x1,x2' ... 'x n ) which are invariant under a group A for which the 11 ,1 2 , , I r are invariants is formed by adjoining to the quantities II' 12 , , I r an integrity basis for polynomial functions V(x1' x2'···' x n ) which are invariant under the group A.
1
y.1 == Kx·1
(i == 1, ... ,n),
K==
1 1
1
1 2 w w w
(3.2.7)
w2
where w == -1/2 + i ~/2 and w 2 == -1/2 - i ~/2 are cube roots of unity.
[Ch. III
Elements of Invariant Theory
46
1 Then, W(xl'···' x n ) == W(K- 1Y l'···' K- y n ) == V(Yl'·'" Yn ) where V(Y1 ,... , Yn )
= V(KI\:K- 1 Y1 ,... , K I\:K- 1 Yn )
(k
= 1,2,3).
Sect. 3.2]
Some Fundamental Theorems
47
invariants of degree two appearing in (3.2.11). We have from (3.2.7) (3.2.8)
(3.2.13) With (3.2.4) and (3.2.7), (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
Theorem 3.4
iii) V( i i 2 i) _ V( i 2 i i) Y1' wY2' w Y3 Y1' w Y2' wY3 V ( Y1' Y2' Y3 ==
(3.2.10)
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 (3.2.14)
1. 2.
Yl
(i == 1,... ,n) ;
ykr1 + Y~~2 i j i j Y2 Y3 - Y3 Y2
where
(i ,j == 1, , n; i ~ j) , (i ,j == 1, , n; i
(3.2.11)
< j) ;
3. Thus, consider the monomial term (yl)i 1
from
(y~)h
(y!)k 1 '" (yr)i n
V(Y1' ... ' Yn )
which
)1 + ... +jn w 2(k 1+ ... +kn ) +... + 2k n == 3r
(y~)jn (Y3)kn
satisfies
= 1 or
(3.2.10).
(3.2.12)
This
(since w 3 = 1) that j1
requires
that
+... + jn + 2k 1
where j1 ,... , k n 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.
1.
Ex!
2.
E(xi ~2 + xkxi)
3.
j k '" L ixi1(x2 x3
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 i 1 == ... == in == 0, jl + ... + jn ~ 2, k 1 + ... + k n ~ 2, j1 + ... + jn + 2k 1 + ... + 2k n == 3 or 6. The quantities satisfying these restrictions are seen to be of the form
yk~3
is as follows, where we employ the notation (3.2.6):
or
yk~3Y~y~,
These terms are expressible in terms of the
(i == 1,...
,n);
+ xj32 xk )
(i ,j == 1, ... , n; i ~ j) ;
(3.2.16)
(i ,j ,k == 1, ... , n; i ~ j ~ k) .
We may proceed as in the argument leading to Theorem 3.3 and set
Yi == KXi (i == 1, ... ,n)
where
W(x1'···' xn) == V(Y1'···' Yn ) where
K
is
given
by
(3.2.7).
Then
[Ch. III
Elements of Invariant Theory
48
Sect. 3.2]
Some Fundamental Theorems
The matrices KI\K- 1 (k=1,2,3) are given by (3.2.9) and
49
x2 u 2v 2 + x3 u 3v 3 x2 w3 + x3 w2 Y2 z2u 3v 3 + Y3 z3u 2v 2 Y2 z2w2 + Y3 z3w3
,
(3.2.18) 2(Y2 z2u 3v 3 + Y3 z3u 2v 2) == (Y2 u 3 + Y3 u 2)(z2 v 3 + z3 v 2)
+ (Y2 v 3 + Y3 v 2)(z2 u 3 + z3 u 2) -
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 V 1(... ) in the quantities i yj3 y2
i (.1 == 1,... , n, ) Y1
Y~~Y~ + Y~~3Y~
+ y3i yj2
(3.2.19)
(i,j,k = 1,... ,n; i S;j S; k)
(3.2.20)
(.. .. < . < k) . 1,J, k -- 1,... ,n,1_J_
There are restrictions imposed on the form of V1(... ) by the requirement that it be invariant under the KO K- 1 (k = 4,5,6). With k (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 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 x2
x3
z3
u3
Y2
Y3
z2
u2
x2 x3 z Y3 3 Y2 z2
u3 v3 u2 v2
Theorem 3.4A An integrity basis for polynomial functions 1., ... .,xl,x2, n n ,x n) wh·Ie h are InvarIant . . W( 1 1 W( xl,.",xn ) xl,x2, ... ,xm m under all permutations of the subscripts 1,2, , m is given by
(.. - 1 ...) 1,J- ,... ,n,1<J,
i j k i j k Y2Y2Y2-Y3Y3Y3
a time.
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).
( i,j == 1,... , n; i ::; j ) ,
and Y2i Yj3- Yi3 Yj2
x2 z3 Y2 z3
+ x3 z2 + Y3 z2
x2 u 3 + x3 u 2 Y2 z2u 2 + Y3 z3u 3
x2 u 3 Y2 u 3
(Y2 z3 + Y3 z2)(u2 v 3 + u3 v 2)·
1.
EXP1
2.
" i xj ~x Pl P2
m.
k E x.Pl .x~2 ... xPm
In (3.2.22),
(i == 1,... ,n); (i,j == 1,... , n; i ::; j ) ;
E x~l x-iP2 '"
(i,j,k == 1,..., n; i::; j ::; ... ::; k).
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' ... ' x n ) which is invariant under a finite group A
+ x3 u 2 + Y3 u 2
x2 v 3 + x3 v 2 Y2 z2v 2 + Y3 z3v 3
(3.2.22)
== {AI'... ' AN} and hence satisfies (3.2.23)
, (3.2.21 )
where the matrices D(A 1), ... , D(A N ) form an n-dimensional matrix rep-
[Ch. III
Elements of Invariant Theory
50
resentation {D(A K )} which defines the transformation properties of x under A. Any polynomial function W(x1 ,... , x n ) which satisfies (3.2.23) may be written as 1 N
L
W(x1"" x n ) == N W(D 1J·(AK ) x· ,... , D .(AK ) x.). K=l J nJ J
(3.2.24)
,Z& = D1/A1)xj ,
zr,·.. ,z* = Dnj(A1)xj ,
51
Some Fundamental Theorems
transformation properties of x under A. Then, W(x) == W(D(A)x),
c··IJ x·1 x·J == c·· IJ D·lr (A) D·Js (A) x r x s
(3.2.27)
holds for all D(A) belonging to {D(A)}. This implies that crs == c·IJ·D·lr (A)D.Js (A)
Following the argument given by Weyl [1946], we set
zl,....
Sect. 3.2]
(3.2.28)
holds for all A in A. Consider the quantity
, D1/AN ) Xj , (3.2.25)
V(x,y) == Yk
, Dn/AN ) Xj .
aaxk
c·· x· x· == c.. (x.y. + y.x.). IJ 1 J IJ 1 J 1 J
(3.2.29)
With (3.2.28) and (3.2.29), we see that Then, W(x1"'" x n ) given by (3.2.24) is expressible as
V(x,y) == V( D(A) x, D(A) y)
(3.2.30)
(3.2.26) holds for all A in A, i.e., V(x,y) is invariant under A. We' refer to the Since (3.2.24) is unaltered under all permutations of the values 1,2,... , N which K assumes, the function V(
permutations of the subscripts 1,2, , N. Then, with Theorem 3.4A, we
zi,
~ha:t V( ... ) i~ ~xpressible as a polynomial in the quantities L LZl z~, ... , LZl z~ ... z~ where L("') has the same meaning as in
see
Theorem 3.4A. This establishes the following result: Theorem 3.4B
The elements of an integrity basis for functions
~
N in x.
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.
to W(x) as a polarization process.
k
process to an invariant W(x) produces another invariant.
Thus, if
W(x) is an invariant of degree q in x, then
a Yj ax. a W(x) == U(x, y, z) ax. 1
(3.2.31)
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
Theorem 3.4B is useful In cases where the group A is of low order.
a~
Similarly, we may show that repeated application of the polarization
Zi
W(x) which are invariant under a finite group A == {AI"." AN} of order N are of degrees
process of applying Yk
) in (3.2.26) is unaltered under all
The quantities x,
y, ... , z transform in the same manner under A and have n independent
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
components, i.e., x == [xl"'" Xn ]T. Suppose that W(x) == c··IJ x·1 x·J is invariant under A where we may assume that c··IJ == c·· Jl· Let {D(A)}
(3.2.32)
denote the n-dimensional matrix representation which defines the
for all D(A) forming an n-dimensional representation {D(A)} of A, arise
[Ch. III
Elements of Invariant Theory
52
upon repeated polarization of invariants comprising an integrity basis for functions of n quantities xl' ... ' x n which are invariant under A.
IV
Further, if
det(xI,x2'···'xn )
==
xl 1 2 x1 x n1
xl 2 x2 2 x n2
xl n x n2
INVARIANT TENSORS
(3.2.33)
x nn
4.1 Introduction Expressions of the form (4.1.1)
is invariant under A, the elements of an integrity basis for functions W(x1' ... ' x m ) of m ~ n quantities which are invariant under A arise
occur In the constitutive relations employed in the classical linear
upon repeated polarization of the invariant (3.2.33) and the invariants
theories of crystal physics and also in the non-linear generalizations of
comprising an integrity basis for functions of n -1 quantities xl'···' Xu-I
these theories. The tensors C· . are referred to as property tensors Il··· In and relate physical tensors such as stress tensors, strain tensors, electric
which are invariant under A.
field vectors, ....
The tensors E·
.
11··· In
and E·
.
13 ... In
denote physical
tensors or the outer products of physical tensors, e.g., E·
.
13 ... 16
= Ei i Ei i· 34 56
There are restrictions imposed on the form of the
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 (4.1.2) for all A = [A ij ] 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
[A ij ] belonging to A. 53
A tensor C·
. which Il··· In
[Ch. IV
Invariant Tensors
54
satisfies (4.1.3) for all A belonging to A is said to be invariant under A.
Sect. 4.1]
Introduction
55
Then the property tensors appearing in (4.1.6), i.e.,
In order to determine the general form of C, we first consider the
CK == C~.l ll··· n
problem of determining the general expression for functions which are
(K == 1,... , P),
(4.1.7)
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
form a set of linearly independent nth-order tensors which are invariant
functions of xl"'" x n which are invariant under A is given by
under A. Any nth-order tensor C == C·
1·
11·" n
which is invariant under A
is expressible as a linear combination of the tensors (4.1.7). Thus, (4.1.4) (4.1.8) for cases where A is a finite group {A 1,... ,AN } and by expressions such as
where the a1"'" ap are constants. A set of P linearly independent nthorder tensors which are invariant under A will be referred to as a
p=~J (trA)ndT
(4.1.5)
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
if A is a continuous group. The matrix defining the transformation properties of the 3n components x f x~ ... x!1 (i 1,... , in == 1,2,3) under A 11 12
In
is referred to as the Kronecker nth power of A.
The trace of the
T1 · Expressions of the form (4.1.1) which are invariant under A may then be written as
Kronecker nth power of A is given by (tr A)n. The number of linearly independent functions which are multilinear in x1,x2"'" x n and which are invariant under A is equal to the number of times the identity rep-
W == (a1C}.
.
ll··· l n
+ ... +
apCr
.) E·
ll··· l n
.
ll··· l n'
(4.1.9)
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.
where the tensors C 1,... , C p form a complete set of nth-order tensors. If the physical tensors T and E possess symmetry properties, e.g., if
A discussion of the properties of the Kronecker products of
matrices is given by Boerner [1963].
(4.1.10)
We may employ theorems from Chapter III to generate the P linearly independent multilinear functions of x1, .. "xn which are
then a number of terms in (4.1.9) will be redundant.
invariant under A. Suppose that these invariants are given by
terms may be eliminated by inspection in simple cases. We give results (4.1.6)
The redundant
below which assist in the elimination process in more complicated problems.
[Ch. IV
Invariant Tensors
56
Sect.4.2]
Decomposition of Property Tensors
57
Suppose that the nth-order tensors C 1 ,... , C p which are invariant under A are comprised of the linearly independent isomers of the
the permutation of the integers 1,2, ... , n which carrIes 1,2, ... , n into
tensors U and V. We note that U·· . is an isomer of U··1 . if IpI q ... Ir 11 2 ... In
of U defined by
Let U 1 ,... , UN and VI' ' V M denote the linearly independent isomers of U and V. We proceed by eliminating the redundant terms in the expressions (p,q,
(Y, {3, ... , ,. Application of s to the tensor U
== U·· . yields an isomer 11 12... I n
sU == U·· . Ia I,8 ... I,
, r) is some permutation of (1,2, ... , n).
(4.2.1)
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 (4.1.11) W2
== (b1V~
. Il···I n
+ ... +
If we denote by Wi and W
bMV M . )E. .. Il···I n Il· .. In
2 the
note that there is an irreducible representation of Sn corresponding to each partition n1 n2... of n, i.e., to each set of positive integers n1
expressions obtained from (4.1.11)
upon eliminating the redundant terms, then the appropriate expression
2
of n
n2
2 ...
such that n1
+ n2 + ...
== n. For example, the partitions
== 3 are given by 3, 21 and 111.
We denote an irreducible
representation of Sn by (n1 n2· .. ) where n1 n2 ... is a partition of n. The representation {D(si)} whose carrier space is formed by the independent
for W is given by
isomers of U may be decomposed into the direct sum of irreducible W
== Wi
+
W
2·
(4.1.12)
representations of Sn. Thus, we may determine a matrix K such that
The number of linearly independent terms in each of the expreSSIons Wi and W
2 is
a useful bit of information.
(4.2.2)
If this information is
lacking, it may prove to be tedious to determine whether all of the
where the (Ynln2... are positive integers or zero and where the
redundant terms have been eliminated from WI and W 2.
Given this
summation is over the irreducible representations (n1 n2 ... ) of Sn. A set
information, we may proceed by generating the appropriate number of
of property tensors which forms the carrier space for an irreducible
linearly independent terms rather than by eliminating the redundant
representation (n1 n2···) of Sn is referred to as a set of tensors of
terms.
symmetry type (n1n2 ... ).
We consider below the problem of determining the number of
linearly independent terms appearing in expressions such as (4.1.11).
The number of tensors comprising a set of
tensors of symmetry type (n1 n2··· n p ) is given by f nl n2 ... n p where
4.2 Decomposition of Property Tensors
(4.2.3) £1 == n1
Let C 1 , C 2 , ... 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
The
+p -
1,
number fnl n2 ... np
£2 == n2 is
+p -
the
2 , ... ,
dimension
£p == n p . of the
representation
Invariant Tensors
58
[Ch. IV
Sect.4.2]
Decomposition of Property Tensors
59
(n1 n 2 ... n) P of 5n and may be found in the first column of the character table for 5n . The character tables for 52' ... ,58 are given in §4.9. The matrices D(sl), ... ,D(s6) appearing in (4.2.8) are given by
We shall employ the notation e12 ... 3 == hI'11 h212 · ... h3In · where h1i == 1 if i == 1, order property tensor
(i 1,... , in == 1,2,3)
h1i == 0 if i
=1=
(4.2.9)
1.
(4.2.4)
Let us now consider a third-
D(sl),···,D(s3) ==
0
0
0
1
0
0
0
1
0
0
0
0
1
0
0
0
1
0
0
0
1
0
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
0
0
1
0
1
0
0
0
0
(4.2.10) (4.2.5) D(s4),···,D(s6) == The isomers of U 1 arise upon application of the permutations e, (12), (13), (23), (123), (132) comprising 53 to the subscripts i 1i 2i 3 in (4.2.5). With (4.2.1) and (4.2.5), we have for example (4.2.6)
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 is given by Xl"'" X6 == 3, 1, 1, 1,0, O.
(4.2.11)
Proceeding in this manner, we obtain With (2.5.14), the number of times the irreducible representation (n1 n 2"') occurs in the decomposition of {D(si)} is given by
6
a n1n2'" == 1 '" 6.L.-J
(4.2.7)
1==1
x·1x~ln2'" , x·1 == trD(s.)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 the 1
With (4.2.7) and similar expressions obtained upon applying the permutations e,
, (132) to U 2 and U 3 , we see that the tensors skUi (i == 1,2,3; k == 1, ,6) are expressible as
n permutation si' We have noted that the Xf1 2'" (i == 1,... ,6) are real. If s·1 and s·J belong to the same class I of 53' then (4.2.13)
Xi == Xj == X,, skU. == U.D .. (sk) 1 J Jl where
(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' may be rewritten as
Then, (4.2.12)
60
Invariant Tensors
[Ch. IV
Sect04.2]
Decomposition of Property Tensors
61
(4.2.14)
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
where the summation is over the three classes e; (12), (13), (23); (123),
the next section, we discuss a procedure which will enable us to
(132) of 53.
generate the sets of property tensors of symmetry type (n1n2... ) arising from the isomers of U.
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 1cycles, 12 2-cycles and 13 3-cycles.
belonging to a class,
= '112'3
The number of permutations
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 U 1,U2 ,U 3 defined by (4.2.7) may be split into two sets of tensors, one of symmetry type (3) and one
We observe that the tensor Uo 1.. == 8 . (8 . 8 0 +8 . 8 . ) 111 212 313 312 213 11 213 may be considered to be the product of the tensors 8 . and 111 82 . 83. + 83 0 82. which are of symmetry types (1) and (2) re12 13 12 13 spectively. The tensors 810 and 82 0 83 0 + 8 0 8 0 form the carrIer 312 213 11 12 13 spaces for the identity representations of the symmetric groups
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
51 == {e} and 52 == {e, (23)} respectively. We say that U· . 0 forms 11 1213 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 distinct 11 1213 isomers form the carrier space for a reducible representation of the
and give the dimensions of the irreducible representations (3) and (21).
symmetric group 53 which we denote by (2) . (1) or by (1) . (2) and refer
We note that the subscript e is used to denote the class of S3 comprised
to as the direct product of the irreducible representations (1) and (2).
of the identity permutation.
Murnaghan [1937], [1938a]' [1938b] has considered the problem of
of symmetry type (21). With (4.2.3), the number of tensors comprising
More generally, the set of distinct isomers of a property tensor U
determining the decomposition of the direct products of the irreducible
== U·11 ... 1n . forms the carrier space for a representation {D(s.)} of the . 1 symmetric group 5 n whose elements are the n! permutatIons of 1,2,... ,n.
representations (m1m2· 0.) of 5m and (n1 n2 .. 0) of 5n into the sum of irreducible representations of Sm + n. In Murnaghan [1937], [1938b],
We may determine the matrices comprising the representation {D(si)}
tables are given which yield the decomposition of the products of
and then determine the character Xi == tr D(si) of the representation.
irreducible representations (m1m2 ... ) of 5 m and (nln2 ... ) of 5 n for
The isomers of U may be split into sets which form the carrier spaces
cases where m
for the irreducible representations (n1n20 .. ) of Sn and which are referred
from the tables given by Murnaghan [1937] that (2) . (I) == (3)
to as sets of tensors of symmetry types (n1n2... ). The number of sets of tensors of symmetry type (n1 n2... ) arising from the distinct isomers of
U is given by
For example, we see from Table 8.3 in §8.6 or
+ (21).
As a further example, we consider the determination of the symmetry type of the SIX U == 810 810 820 820 given by 11
(4.2.15)
+ n :::; 10.
12
13
distinct
Isomers
of
the
tensor
14
(4.2016)
62
[Ch. IV
Invariant Tensors
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),
1,2,... , n in any manner into the n squares. A standard tableau is one in
(123), (1234), (12)(34) to the six tensors (4.2.16), we find that the
which the integers increase from left to right and from top to bottom.
traces of the matrices defining the transformation of the tensors are
For example, the standard tableaux associated with the frames [31] and
given by 6, 2, 0, 0, 2 respectively. This tells us that the tensors (4.2.16)
[22] are given by
form the carrier space for a representation of S4 whose character X, is given by
X, == 6, 2, 0, 0, 2 for the classes
1 2 3,
1 2 4,
1 3 4
4
3
2
and
1 2,
1 3 .
3 4
2 4
(4.3.1)
(4.2.17)
,== 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
We may denote the standard tableaux associated with a frame
a == [n1 n2···] by F
1,F 2\····
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?
that 8 . 8 . and 8 . 8 . are both of symmetry type (2) and that the 111 112 213 214
than has F~. Thus we would denote the three standard tableaux given
tensors (4.2.16) form the carrier space for the reducible representation
on the left of (4.3.1) by F
(2) . (2) of S4. With Table 8.3 in §8.6 or the tables given by Murnaghan
denote by
[1937], we have (2). (2)
1, F 2\ F 3 respectively
with a
ag the permutation which carries F~ into F?
== 31. We
Thus,
== (4) + (31) + (22) so that the tensors (4.2.16)
form a set of symmetry type (4)
+ (31) + (22).
af~
=
(34) :
_ 123 (34) 124 -4 3
- F 31 . (34) F 31 2 1 '
af§
=
_ 123 (243) : (243) 2134 -4
. (243) F 331 -- F 31 1 '
=
(34) :
(34) 4123 -_ 3124
- F 31 . (34) F 31 1 2 '
=
(234) : (234) 4123 -_ 2134
31 (234) F 31 1 -- F 3 .
4.3 Frames, Standard Tableaux and Young Symmetry Operators
(4.3.2)
a~l
A partition n1 n2 ... of the positive integer n is a set of positive integers n1 ~ n2 ~ ... ~
° such
that n1
+ n2 +... == n.
an
The frame
[n1 n2···] associated with the partition n1 n2"· 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
We note that a~ is the inverse of
partitions 4, 31, 22 of 4 are
ago
Let F? denote a tableau associated with a frame a
== [n1 n2 ...].
Let (4.3.3) where the summation in (4.3.3)1 A tableau is obtained from a frame [n1 n2 ...] by inserting the numbers
IS
over all permutations of the
numbers in F? which leave each number in its own row and where the
64
[Ch. IV
Invariant Tensors
summation In (4.3.3)2 is over all permutations of the entries in F?
Sect. 4.3]
Frames, Standard Tableaux and Young Symmetry Operators
65
We have (see Rutherford [1948], p. 16)
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
a paQa a Qa a , (]'rs s . s == pO' r (]'rs s == pO' r Qa . r (]'rs
permutation. We recall that e denotes the identity permutation and that permutations are to be multiplied from right to left. For example,
(4.3.7)
a yO' a , (]'rs s == yO' r (]'rs
(13)(12) == (123). In cases where no confusion should arise, we at times suppress the parenthesis on the permutations, e.g., we will write . 12 (e + 12) Instead of (e + (12)). Thus, for the standard tableaux F 22 1 == 34 13 h and F 22 2 == 24 ,we ave Pr 2 = (e + 12)(e + 34),
There is no summation over the repeated indices In (4.3.7). example, with
ar~ p~2
Qr 2 = (e - 13)(e - 24),
(4.3.4)
p~2 = (e + 13)(e + 24),
Q~2
= (e -12)(e -
= (23 -
y? == P? Q?, where P? and Q? are defined by (4.3.3). and F 222 -- 24' 13 we have
(4.3.5) 12 For the tableaux F 22 1 == 34
y22 1
(e + 12) (e + 34) (e - 13) (e - 24),
-22
(e - 13) (e - 24) (e + 12) (e + 34),
132 - 234 + 1342)
(4.3.8)
= Qr 2 ar~ .
Then, with (4.3.7) and (4.3.8), (]'22 y22 12 2 -
p22 Q22 (]'22 _ y22 (]'22 . 1 1 12 1 12' (4.3.9)
(4.3.6) y22 2
+ 13)(e + 24)
22 22 (]'12 Q 2 == (23)(e - 12)(e - 34)
are given by
Yl
= (23)(e
For
= (23 + 123 + 243 + 1243) = pF ar~,
34).
The Young symmetry operators associated with a standard tableau F?
ar~ = a~r = (23) and (4.3.4), we have
(e + 13) (e + 24) (e - 12) (e - 34),
-22 Y2 == (e - 12) (e - 34) (e + 13) (e + 24). Let (]'~ denote the permutation which yields F? when applied to F~.
(]'22 Q22 p22 _ Q22 (]'22 p22 12 ·2 2 1 12 2 _ Q22 p22 (]'22 - .1 1 12
-22 (]'22 Y 1 12'
In similar fashion, we find that (]'2 2 y2 2 == y222 (]'222 ' 21 1 1
(]'22 y22 _ y22 (]'22 21 1 2 21 .
(4.3.10)
66
[Ch. IV
Invariant Tensors
Let X = a1sl + a2 s2 +... + an!sn! denote an arbitrary linear comwhere the si bination of the permutation operators sl ( = e), s2' (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].
Sect. 4.3]
Frames, Standard Tab/eaux and Young Symmetry Operators
F~ = 123, Y~ = (e
YI
p = coefficient of e in Y? X ,
Applying
12
+
13
+
F21 _ 13 2-2' 23
+
123
+
21 - (23) a 21 , 132), (4.3.14)
1 = (e + 12)(e - 13) = (e
(23) y? X Y? = BCt p Y? ,
+
21 _ 12 F1 -3'
67
+
12 - 13 - 132),
YI 1 = (23)(e + 12)(e - 13) = (23 + 132
123 - 12).
the
In
Young
symmetry operators
given
(4.3.14)
to
U1 = e123 + e132 yields, with (4.2.7), Y? X Y~ = BCt p ag Y~, p = coefficient of e in
Y?XY~ =
ag. Y? X ,
(4.3.11)
(4.3.15) 0,
where BCt is a non-zero constant. We may employ Young symmetry operators to generate a set of
We have, from (4.3.15),
property tensors of symmetry type (n1n2".). Suppose that
1, F2,···,Ff
(4.3.12)
F
are the f standard tableaux associated with the frame Q. Then (r=l, ... ,f; Q=n1n2.")
2
1
1
2
1
-2
2
-2
1
(4.3.16)
Application of the sl'.'" s6 = e, (12), ... , (132) to the V k (k = 1,2,3) is then defined by (4.3.13)
yields a set of tensors of symmetry type (n1 n2' .. ) provided that y Ct U· . i= O. For example, we have seen that the three distinct 1 11'·· In isomers of the tensor U 1 = e123 + e132 defined by (4.2.5) may be split into sets of tensors of symmetry types (3) and (21). We have
s.Vk=s.U K q k=U P Dpq (s.)K k=V.K~lD (s.)K q k 1 1 q 1 q J JP pq I
(4.3.17)
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
[Ch. IV
Invariant Tensors
(i == 1,... ,6)
(4.3.18)
Sect. 4.4]
Physical Tensors of Symmetry Class (n 1 n 2
69
•• .)
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
where the D(si) are defined in (4.2.10) and where
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 (n1 n 2... )' (m1m2 ... )' .... Let ,8n1 n 2... 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 (4.3.19)
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
1]
o '
We may verify that the matrices
r 1(si)
matrix representations of the group 53.
[-1 1] [0 -1]. -1 0 ' 1 -1
and
r 2(si)
difficult matter for n
~
10.
This is primarily due to results given by
Murnaghan [1937], [1938b], [1951].
form irreducible
The characters are given by
4.4 Physical Tensors of Symmetry Class (nln2...)
tr r 1(si) == 1, 1, 1, 1, 1, 1 and tr r 2(si) == 2, 0, 0, 0, -1, -1. We note
Let n1 n2 ... denote a partition of n.
that the permutations si (i == 1,... ,6) defined by (4.2.9) which comprise
standard tableau associated with the frame
53 may be split into the sets sl == e; s2' s3' s4 == (12), (13), (23); s5' s6 == (123), (132). These sets form the classes, of 53 denoted by 13, 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 {r 1(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 Young symmetry operator defined by (4.3.5)2 which is associated
the characters of (3) and (21) respectively.
components.
The tensors VI and
(V , V ) defined by (4.3.15) form the carrier spaces for the 2 3 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
with the standard tableau
Fl.
Let F a
I
denote the first
== [n1 n2...]. Let
VI be
Then the tensor (4.4.1)
is referred to as a tensor of symmetry class (n1n2 ... ). The tensor T· . is a three-dimensional nth-order tensor with 3n independent 1 11··· n
The transformation properties of the T·
transformation A == [A ij ] are given by
.
11··· 1n
under a
(4.4.2)
symmetry operators to conveniently generate sets of tensors of specified symmetry type.
The transformation properties of the independent components ¢>l, ... ,¢>q
of
[Ch. IV
Invariant Tensors
70
i
1··· n
71
Physical 7'ensors of Symmetry Class (n 1 n2 ...)
Sect. 4.4]
(4.4.6)
defined by (4.4.1) under a group A are described by a q-
dimensional matrix representation {R(A)}.
We are interested in
determining the matrices R(A) and the quantities tr R(A) which define
Let
the character of the representation {R(A)}. We consider the special case where n
(4.4.7)
= 2,
I.e., where T· 1. IS a 11 2 second-order tensor. The tensor T··1 may be expressed as the sum of 11 2 its symmetric and skew-symmetric parts. Thus,
Then (4.4.6) may be written as
T'
= (AxA)
T
(4.4.8)
(4.4.3) where A x A is referred to as the Kronecker square of A and is defined by The tensors on the right of (4.4.3) are of symmetry classes (2) and (11) respectively. n1n2...
Thus, there are two partitions of n
= 2 and'll
=2
given by
respectively. The standard tableaux associated with AxA=
the frames [2] and [11] are given by
F 111 -_ 21
(4.4.4)
A 11 A 11
A 11 A 12
A 12 A 11
A 12 A 12
A 11 A 21
A 11 A 22
A 12 A 21
A 12 A 22
= [ AnA
A 21 A 11
A 21 A 12
A 22 A 11
A 22 A 12
A 21 A
A 21 A 21
A 21 A 22
A 22 A 21
A 22 A 22
A 12 A ]. A 22 A
(4.4.9)
respectively. The tensors Let (4.4.10)
A= KT, (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
A=
'l/J 12
, K=
1
0
0
0
0
1/2
1/2
0
0
0
0
1
0
1/2 -1/2
0
,
K- 1 --
1
0
0
0
0
1
0
1
0
1
0 -1
0
0
1
11 2
(4.4.11)
dependent components given by T 11' T 12' T 21 and T 22. The transformation of the T· 1. under a transformation A is defined by 11 2
0
The manner in which A transforms under A is given by
[Ch. IV
Invariant Tensors
72
A' == KT' == K(A X A)T == K(A X A)K- 1A
Physical Tensors of Symmetry Class (n 1 n 2
Sect. 4.4]
(4.4.12)
where the 4 X 4 matrices L
=[
t
1 Kik Kko] and M
k==l
where
73
•••)
= [ Kii
J
K 4j ] are
seen from (4.4.11) to be given by
A 11 A 11
A 11 A 12
+ A 12 A 11
A 12 A 12
1
0
A 11 A 21
A 11 A 22
+ A 12 A21
A 12 A 22
0
A 21 A 21
A 21 A 22
+ A22 A21
A 22 A 22
0
L== . (4.4.13)
0
0
0
A 11 A 22 - A 12 A 21
0
0
0
0
1/2 1/2
0
0
1/2 1/2
0
0
0
0
, M-
0
0
0
0
0
1/2
-1/2
0
0
-1/2
1/2
0
0
0
0
1
0
We may express the 4 X 4 matrices L, M and A
X
(4.4.15)
A as
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 (4.4.16)
transformation properties of the independent components <7>11' <7>12' <7>22 of the symmetric part <7>. 1. == -21(T. 1. + T· 1. ) of T· 1.. The 1 X 1 11 2 11 2 12 1 11 2 (11) matrix A appearing in the lower right corner of (4.4.13) describes the
transformation
of
the
one
independent
component
7P12 where iI' i 2 , jl' j2 take on values 1,2 and where the rows (columns) 1, 2,3,4 of the matrices are those for which i 1i 2 (jlj2) take on the values 11,12,21,22 respectively. With (4.4.14) ,... , (4.4.16), we have
= !(T 12 - T 21 ) of the skew-symmetric part of T i1i2 o We observe that tr A
(2)
-1
== Kik Kkj (A X A)ji == L·· (A X A).. 1J
J1
(i,j == 1, ... ,4; k == 1,2,3) (4.4.14)
tr A
(11)
-1 == K i4 K 4j (A X A)ji
where S 1-
== M ij (A X A)ji
tr A ,
s2- tr A2 ,
... ,
(i,j == 1,... ,4) We may also express (4.4.1 7) in the form
sn == tr An .
(4.4.18)
74
Invariant Tensors
2 trA(2) == l2! '" 4y h , X, s'l 1 s'2 2 --
[Ch. IV
An nth-order tensor T == T·
12 (s2+ 1 s2) '
.
11·" In
(4.4.19)
'l '2 -- 21(s21 - s)2
11s trA(ll)-l"'h - 2! 4y , X, 1 s2
where x~ and
Physical Tensors of Symmetry Class (n 1 n 2
Sect. 4.4]
•••)
75
may be split into a sum of
tensors 4>1' 4>2' ... of symmetry classes (n1 n 2···)' (m1 m 2···)' ... where the n1 n2··· , m1 m2···' ... are partitions of n. For example, the third-order tensor T == T· 1. 1. is expressible as 11
xV are the values of the characters of the irreducible
23
(4.4.22)
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
where
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 n generally, if A(n1 2"') is the matrix which defines the transformation properties of the qn n
1 2···
independent components of an nth-order
tensor of symmetry class (n1n2 ... ) under a transformation A, we have (see Lomont [1959], p. 267) (4.4.20)
(4.4.23)
where X~ln2··· denotes the value of the character of the irreducible representation (n1 n2···) 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 (n1 n2 ... ) is given by qn n
1 2···
where (4.4.21 ) are tensors of symmetry classes (3), (21), (21) and (111) respectively.
A thorough discussion of tensors of symmetry class (n1 n2... ) may be
With (4.4.21) and the character table for 53 (Table 4.2 in §4.9), we see
found in Boerner [1963].
that 4>1' 4>2' 4>3 and 4>4 have 10, 8, 8 and 1 independent components
[Ch. IV
Invariant Tensors
76
respectively. The tensor T given by (4.4.22) is said to be of symmetry
class (3) + 2(21) + (111). We observe that T = T i i i has 33 = 27 123 independent components and has no symmetry in the sense that no
Sect. 4.5]
The Inner Product of Property Tensors and Physical Tensors
standard table~ux associated with the frame
CY == [n1 n2 ...]. Then the . (i == 1,... , q) may be written as
set of tensors C!
11··· In
relations such as T··· == T· .. occur. In order to list the 11 1213 121113 i i independent components of a tensor of symmetry class (21), we let i 1 2
take on values 1, 2 and 3 so that, when entered into the frame [21], tte
77
... ,
(4.5.2)
numbers do not decrease as we move to the right and increase as we
CY where the aCYl' s ...' a qs are the permutations which carry F~ into Let denote one of the standard tableaux associated with
move downwards. Thus,
the frame f3 == [m1m 2...]. Then a tensor of symmetry class (m1m2... )
11 2'
11 3'
12 2'
12 3'
13 2'
13 3'
22 3'
23 3·
Ff,...,Fq.
Fe
may be considered to be given by (4.4.24) (4.5.3)
With (4.4.23) and (4.4.24), we have, for example, where T·
3
= T 123 +
T 213
. is non-symmetric, i.e., T·
11··· 1n
T 231 '
.
11··· 1n
has 3n independent com-
ponents. (4.4.25)
3
We now consider the set of q functions obtained by taking the
= T 132 + T 312
inner product of the q tensors (4.5.2) and the tensor (4.5.3), i.e.,
Each of the components
IS
expressible as a
(r == 1,... , q).
(4.5.4)
We first note that 4.5 The Inner Product of Property Tensors and Physical Tensors
Let
C!11··· In .
(i == 1,... , q) denote a set of nth-order property
tensors of symmetry type (n1n2 ... ). Let ¢.
(4.5.5)
. be a physical tensor of
11··· 1n
symmetry class (m1m2 ... ).
Then the number of linearly independent and that the permutation (132) is the inverse of the permutation (123).
functions in the set
More generally, we have
C!
. ¢.
11··· 1n
.
11··· 1n
(i == 1,... , q)
(4.5.1) (4.5.6)
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
where a denotes a permutation of 1,2,... , n and a-I is the inverse of a.
[eh. IV
Invariant Tensors
78
Sect. 4.6]
Symmetry Class of Products of Physical Tensors
79
(4.5.13)
We further note that
c·11 ... In . ("""'f3.a.T. . )==(2:f3.a:-1C. . )T. . L.J 1 1 11· .. In 1 1 11··· In 11··· In
(4.5.7)
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 where the summation in (4.5.7) is over all of the permutations ai of
f3.1 are real numbers. Let the Young symmetry operator Y~ == Q~ P~ associated with the tableaux F~ be written as 1,2,... , n and where the
values since a~v a~s == e and the coefficient of e in Y~ is not zero. This says that if the frames [n1 n2 ...] and [m1 m2 ...] are different, the q functions (4.5.4) are all zero and that if the frames [n1 n2...] and [m1m2···] are the same, then there is just one linearly independent
(4.5.8) Then, it may be shown (see Boerner [1963], p. 147) that replacing ai by (Til in (4.5.8) will yield
function contained in the set (4.5.4). The number of linearly independent invariants of the form
Y~, i.e.,
(4.5.14) (4.5.9) associated with a material for which the complete set of nth-order property tensors is of symmetry type
With (4.5.7) ,... , (4.5.9), we have
(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 [nl n 2."] and
[mlm2...]
respectively.
If
the
frames
0;
If the frames
0;
. is of symmetry class
Il··· In
(4.5.16)
== [n1n2...] and
(J == [m1m 2···] are different, we see from (4.3.11)5 that
(r == 1,... , q).
and where the physical tensor
is then given by (4.5.17)
(4.5.11)
and (J are the same, we have with (4.3.7) and (4.3.11)3 4.6 Symmetry Class of Products of Physical Tensors
(4.5.12)
In this section, we consider the problem of determining the symmetry class of the product of tensors T and U where the symmetry
where p is the coefficient of e in the expression
classes of T and U are given. We first indicate the manner in which
80
[Ch. IV
Invariant Tensors
(4.6.1) may be obtained when T··1 and U··1 are of symmetry classes (2) and .
.
11 2
.
11 2
11 2
and U··1
are symmetric and skew-
11 2
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
respectively under A.
T 1,... , T 6
U 1,... , U 3 of T
and
and
tr R(A)
2'1'2 2!1 '" h ,X,sl s2
=
trace of the Kronecker product Q x 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),
tr {Q(A) x R(A)} = tr Q(A) tr R(A) =
i! ~ h, xV sIl si
Y , J-L, sl' 1s2' 2 ... s4'4 ·
representation of 54 assumes for the classes, of 54.
= 21( s21 +s2 ) =
The number of
times the irreducible representation (n1 n2 ... ) appears in the decomposition of this representation is given by
!(st - s2) (4.6.5)
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
(4.6.4)
The quantities Il, in (4.6.4) give the values which the character of a
(4.6.2) 2
i(sf - s~)
-- 1.(6 4! sl4 _ 6s22) -- 1. 4! '" h
We note that Q(A) is the symmetrized
y
Q(A)
dependent components TiU . (i == 1,... ,6; j == 1,... ,3) of T· 1. U··1 is the J 11 2 13 4 Kronecker product Q(A) x R(A) of Q(A) and R(A). We note that the
U
Kronecker square A(2) of A. With (4.4.19), tr
81
Symmetry Class of Products of Physical Tensors
matrix which describes the transformation properties of the in-
the symmetry classes of tensors such as
(11) respectIvely, I.e., T·.1
Sect. 4.6]
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 by
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 h, == 1, 6, 8, 6, 3;
Il,
== 6, 0, 0, 0, -2
(4.6.6)
for the classes, == 14 , 122, 13,4,2 2 of 5 n and that (4.6.3)
where the summation is over the classes of 5n and where X~ln2··· gives the values of the character of the irreducible representation (n1 n2 ... ) of
(4.6.7) Thus, the tensor T· 1. U· 1. is of symmetry class (31) 11 2 13 4
+
(211).
5 n for the class ,. We first determine the symmetry class of T· 1. U··1 where T 11 2 13 4 and U are of symmetry classes (2) and (11) respectively. The 18 X 18
We next consider the determination of the symmetry classes of the tensors T· 1. T·· and T· . T· . T·· where T· 1. is symmetric i e 11 2 1314 111213141516 11 2 ' .. ,
82
[Ch. IV
Invariant Tensors
Sect. 4.6]
of symmetry class (2). The transformation properties of the 21 independent components T·T· (i,j == 1,... ,6; i <j) of T· 1. T··1 and the 56 inde11 2 13 4 1 J pendent components T.T.T k (i,j,k==I, ... ,6; i<j
Symmetry Class of Products of Physical Tensors
h, == 1, 6, 8, 6, 3;
II,
== 3, 1, 0, 1, 3
(4.6.11)
for the classes, == 14 , 12 2, 13,4,2 2 of 54 and that h, == 1, 15, 40, 90, 45, 120, 144, 120, 90, 15, 40;
symmetrized Kronecker cube, Q(3)(A), of Q(A) respectively. We have
, .L ,
1 ""'" h tr Q(2)() A == -2' . LJ , tr Q(3)(A) = 3\
X,2'1'2 t 1 t 2 -_
h, x~ tIl t~2
(4.6.12)
A,== 15,3,0,1,3,0,0,1,1,7,3
1 (t 2 + t ) ' 2 2 1 (4.6.8)
tj3 =
~ (t~ + 3t 1t 2 + 2t 3)
2 of 5 6. lor the cIasses 16 ,4 1 2, 13 3, 12 4, 122 2 , 123, 15, 6, 24, 23 ,3
£
The
II,
decomposition of these representations is seen to be given by (4) and (6)
= ~ (sr + s2)'
t 2 = tr Q2(A) = tr Q(A2 )
= ~ (s~ + s4) , (4.6.9)
The summations in (4.6.8)1 and (4.6.8)2 are over the classes of 52 and S3 respectively.
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
where (see Murnaghan [1951]) t 1 = tr Q(A)
83
The quantities x~, X~ are the characters of the
identity representations of 52 and 53 which are denoted by (2) and (3)
+(42) + (222)
+ (22)
respectively.
Thus, T· 1. T··1 is of symmetry 11 2 13 4 class (4) + (22) and T· 1. T· 1. T·.1 is of symmetry class (6) + (42) 11 2 13 4 15 6 + (222).
. is of symMore generally, we may suppose that T == T· 11· .. 1P metry class al(nln2 ... )+a2(mlm2 ... )+ .... Let Q(A) denote the matrix which defines the transformation properties under A of the independent components T 1,.. , T r of T. Then
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) =
tr Q(3)(A)
=
1, (3sf+6srS2+6s4 +9s~) = 1, ~h,v,sIls~2 ... s14, J! (15sY + 45sf S2 + 90sr S4 + 135sr s~ + 120s6
transformation properties under A of the (r + (4.6.10)
We see from (4.6.10) and the character tables for 54 and 56 (Tables 4.3 and 4.5 in §4.9) that
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
~ -1 )
independent com-
(iI' i2, ... , i m == 1, ... ,r; i 1 ~ i 2 ~ ... ~ im ). ponents T i T i ... T 1· 12m have (see Murnaghan [1951])
We
(4.6.14)
[Ch. IV
Invariant Tensors
84
Sect. 4.6]
Symmetry Class of Products of Physical Tensors
85
The tensor T· . T· . is a tensor of order 2p == m whose sym11.. ·lp 1p +1 .. · 12p metry class is denoted by (P1 P2.") x (2) where
where the quantities t 1,... , t m are given by
(4.6.18) 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
The summation in (4.6.18) is over the irreducible representations
Smp, we may employ (4.6.5) to determine the number ,BPIP2"" a ... of times the irreducible representations (P1 P2···)' (q1 q 2···)'
discussed by Murnaghan [1951]. Most of the results of interest for the applications considered here may be obtained from Murnaghan's papers
... of Smp appear in the decomposition of this representatIOn. We then
(see also Table 8.4, p. 232).
fJql q2··· ,
.
(m1m2"') of Sm.
say that the tensor T· . T· . ... T k k (m terms) is of symmetry 11· .. 1P Jl··· Jp 1'" P class ,BPIP2'" (PIP2"')
+ ,BqlQ2'"
(Ql Q2''')
+ ....
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 11· .. 1P (p 1P 2 ... ). Let U·11· .. 1.q be a tensor of order q and . is a tensor of (q q ... ). Then T· 1. U· 1 2 11· .. P 1p +1 .. · 1p+q whose symmetry class is denoted by (P1 P2''') -(q1 q 2"')
symmetry class symmetry class order p + q == n where
The determination of the decomposition (4.6.18) is
We list below (see Smith [1970]) the symmetry classes of physical tensors which arise from the products of vectors E i , F i , ... , symmetric second-order tensors B.. , C", ... and skew-symmetric second1J 1J order tensors a··, (J.. , .... We note that all components of a three1J
1J
dimensional tensor of symmetry class (P1 P 2P 3P4) with P4>0 are zero. Since we are mainly concerned with three-dimensional tensors, we will not list terms such as (P1 P 2P 3P4) with P4>0 in the description of the symmetry classes of the tensors listed below. For example, the symmetry class of EiFjGkH£ is (4) + 3(31) + 2(22) + 3(211) (1111) has no non-zero components. Symmetry Classes: Products of Vectors
The summation in (4.6.16) is over the irreducible representations 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 class 11... 14 11 of T· . U· is given by 11 ... 14 15 (22) . (1) == (32) + (221).
We
suppress the (1111) since a three-dimensional tensor of symmetry class (4.6.16)
(nl n 2"') of Sn'
+ (1111).
(4.6.17)
2. E.E., (2); E.F., (2) 1
J
1
J
+ (11)
Invariant Tensors
86
[Ch. IV
Sect. 4.6]
Symmetry Class of Products of Physical Tensors
87
+ 2(44) + (431) + 3(422) (4.6.19)
+ 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. a··1J' (11)
2. B ij , (2)
aij (3k£ 'mn' (33)
+ 2(321) +
(222) (4.6.21 )
BijBk£Cmn' (6)
+ (51) + 2(42) + (321) + (222)
BijCk£Dmn' (6)
+ 2(51) + 3(42) +
(411) + (33) + 2(321) + (222) (4.6.20)
BijBk£BmnCpq, (8) + (71)
+ 2(62) + (53) + (521) +
+(44) + (431) + 2(422)
aij ak£(3mn 'pq, (44) + 2(431) + (422) + (332)
[Ch. IV
Invariant Tensors
88
Sect. 4.7]
89
Symmetry Types of Complete Sets of Property Tensors
quantities obtained by cyclic permutation of the subscripts on the
4.7 Symmetry Types of Complete Sets of Property Tensors
summand. 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' ...
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 invariant under A. With (4.1.6) and (4.1.7), the complete set of
and/or the results given by Murnaghan [1937], [1951] to determine the
nth-order invariant property tensors may be immediately listed given
symmetry type of a set of tensors comprised of a property tensor and
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 U l' VI'···; ... ;
US' V S' ... j such that these tensors together with their distinct isomers
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 P n may be computed with (4.1.4) or (4.1.5). We observe that sets of three-dimensional property tensors of symmetry type (n1 n2···n p ) with n p > 0, P 2: 4 will be comprised of tensors whose components are all equal to zero. There may be sets of threedimensional property tensors of symmetry type (n1 n2 np) with np P ~ 3 which are comprised of null tensors.
If (n1 n2
> 0,
) represents the
form complete sets of tensors of orders 1,... ,8. Further, the symmetry
symmetry type of a set of property tensors whose components are all
types of the sets of tensors are given.
zero, we indicate this by underlining the (n1n2 ... )' e.g., (2111).
We follow Smith [1970] and
dimension fn1n2 ... of the irreducible representation (n1n2 ... ) gives the number of tensors comprising a set of tensors of symmetry type
employ the notation
U, V, W;
The
6·,
+
(3)
2(21)
+
(111)
(4.7.1)
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,
(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: D 2h The symmetry group D 2h associated with this crystal class defined by
IS
1 tensors whose symmetry types are (3), (21), (21), (111) respectively. We also employ the notation (4.7.2)
E e l122 == el122 + e2233 + e3311 . In (4.7.2), the notation
E (...)
indicates the sum of the three
where the I,... ,D 3 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 invariant under D 2h is given by
[Ch. IV
Invariant Tensors
90
Sect. 4.7]
elll12233' ~2223311' e33331122;
(4.7.3)
420;
(8)
+ 2(71) +
+ 3(62) + (611) + 2(53) + 2(521) + (44) + (431) + (422).
It follows from (4.7.3) that there are no tensors of odd order which are invariant under D 2h . 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.
91
Symmetry Types of Complete Sets of Property Tensors
Consider the set of 6 fourth-order tensors comprised of the distinct isomers of el122 which are given by
1 ,. (2) .
2.
P 2 = 3.
ell' e22' e33;
4.
P4 = 21.
e1111' ~222' e3333;
1·, (4); = 6 . 6 . 6 . 6 0, 6 . 6 06 06 0, 6 . 6 06 06 0,
el122' el133' e2233; 6.
P6 = 183.
11 1 11 2 21 3 21 4
6·, (4) + (31) + (22).
e111111' e222222' e333333;
11 1 21 2 11 3 21 4
111 212 213 114
(40705)
1·, (6) ; We have observed in §4.2 (see (4.2.16), (402.17)) that these tensors form the carrier space for a reducible representation
e333311' e333322;
15 ;
(6) + (51) + (42) ; (4.7.4)
el12233;
90;
(6)+2(51)+3(42)+(411)+
+ (33) + 2(321) + (222). 8.
P 8 = 1641. e11111111' e22222222' e33333333;
1;
(8);
r
of the group 54 of all
permutations of the subscripts iI' i 2 , i 3 , 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
el1111122' el1111133' e22222211' e22222233' e33333311' e33333322;
28;
(8) + (71)
+ (62);
e11112222' elll13333' e22223333; 70; (8)
+ (71)+
+ (62) + (53) + (44);
where the I, ... ,M2 are defined by (1.3.3). With (4.1.4) and Table 9.1 (p. 268), the number P n of linearly independent nth-order tensors which are invariant under 0h is given by
Invariant Tensors
92
[Ch. IV
Sect. 4.7]
(4.7.6)
possess a center of symmetry is the three-dimensional orthogonal group
Symmetry Types of Complete Sets of Property Tensors
93
03 which is comprised of all three-dimensional matrices A such that We see that P n == 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
AAT == ATA == E 3 , det A == ± 1. The number P n of linearly independent nth-order tensors which are invariant under 03 is equal to
and 8 which are invariant under 0h are listed below where the notation
the number of linearly independent multilinear functions of the n
(4.7.1) and (4.7.2) is used.
vectors Xl"'" x n which are invariant under 03' The matrix representation defining the transformation properties of the 3n quantities
2. 4.
P 2 == 1.
L e 11 == 8ij ; 1 ., (2).
P 4 == 4.
1 ., (4) ; L e 1111 ; L(el122 + e2211);
6.
P 6 == 31.
L e111111 ; L
under 03 is comprised of the Kronecker nth powers
A x A x ... x A of the A belonging to 03'
The number P n 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
3', (4) + (22) .
trCA== _e iB _1_e- iB . Since tr{AxAx ... xA)=={trA)n and P n is obtained from the expression {2.6.20)2' we see that
1 ,. (6) ;
(elll122 + elll133) ; 15;
(6)+(51)+(42); Pn
L (el12233 + el13322); 8.
xf xf ... xk
15;
= 4~
27r
f (eiB + 1 + e-iB)n (1- cos B) dB
o
(6) + (42) + (222) . (4.7.7)
(4.7.8)
27r
+ l1r
P 8 == 274. L e 11111111 ; 1 ,. (8) ;
f(- eiB - 1 - e-iB)n (1 - cos B) dB. o
L (elllll122 + elllll133) ;
28;
(8) + (71) + (62);
L
35;
(8)
(e11112222 + elll13333);
From (4.7.8), we obtain
+ (62) + (44) ;
P n == 0
(n odd) , (4.7.9)
L (e11112233 + ell113322); 210;
(8)
+ (71) + 2(62) +
+ (53) + (521) + (44) + (422). where (iii) Isotropic materials with a center of symmetry: 03
The symmetry group associated with isotropic materials which
(£) ==
m!
(:~m)!
is a binomial coefficient and
(0 ) = 1.
Since
P n == 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
[Ch. IV
Invariant Tensors
94
Sect. 4.7]
95
Symmetry Types of Complete Sets of Property Tensors
invariant under 03 are listed below where we employ the notation
which enables one to list the linearly independent isotropic tensors of
(4.7.1). These tensors are referred to as isotropic tensors.
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 2.
P2
= 1.
fJ·· . IJ '
the subscripts i,j, ... , q. These identities play an important role in gen-
1·, (2).
erating integrity bases for functions which are invariant under the
4.
P4
= 3.
fJ ij fJ k £; 3·, (4)
6.
P6
= 15.
fJ ij fJ k £fJ mn ;
8.
P8
= 91.
fJ ij fJ k £fJ mn fJ pq ;
+ (22) .
15 ;
(6) 105 ;
(4.7.10)
group 03 (see §8.2 and Rivlin and Smith [1975]).
+ (42) + (222). (8)
(iv) Isotropic materials without a center of symmetry: R3
+ (62) + (44) + + (422) + (2222) .
The symmetry group associated with isotropic materials which do not possess a center of symmetry is the three-dimensional rotation
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), fJ ij 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 fJ ij fJ k £, fJ ik fJ j £, b b and may be split into sets of tensors of symmetry types (4) and i2 jk (22) which are comprised of 1 and 2 tensors respectively. We note from
group R3 which is comprised of all three-dimensional matrices A such that AAT = ATA = E 3 , det A = 1.We may proceed as in the case of the group 03 to show that the number P n of linearly independent nthorder tensors which are invariant under R3 is given by
Pn
= l7f
211'"
J(eiB + 1 + e-iB)n (1- cos B) d8
(4.7.12)
o where we have employed (2.6.19)2. We have, upon evaluating (4.7.12),
line 4 of (4.7.10) that there are 105 distinct isomers of fJ ij fJ k £ fJ mn fJ pq
PI = 0,
but that there are only 91 linearly independent eighth-order tensors which are invariant under 03. This is due to the existence of identities
Pn
of the form
(n-l)/2( )() = 1 + k'fl 2k 2: -
(n+l)/2( n-
k'f
2
)(
2k ~ 1 2t
(n odd; n fJ i£
fJ·In
fJ·Iq
fJ kj
fJ k £
8kn
8kq
8 . mJ
8m £
8mn
8mq
fJ pj
8p £
8pn
8pq
fJ· . IJ
=
o.
Pn
(4.7.11)
= 1
We recall that
+}; (2k) (2:)
on =
- } ; (2k
~ 1)(2t =i)
~
=i)
3),
(4.7.13)
(n even).
1. Complete sets of tensors of orders 2, ... ,8 which
are invariant under R3 are listed below where we employ the notation Weare assuming that the tensors are three-dimensional. A procedure
(4.7.1).
The tensors 8ij and Cijk appearing below are the Kronecker
96
Invariant Tensors
[Ch. IV
delta defined by (1.2.4) and the alternating tensor (see p. 6). 2. 3.
P 2 == 1. P 3 == 1.
b... IJ '
Sect. 4.7]
Symmetry Types of Complete Sets of Property Tensors
material for which the x3 axis is an axis of rotational symmetry is comprised of the matrices
1·, (2).
Cijk;
1 ,. (111).
cos ()
sin ()
0
-sin ()
cos ()
0
0
1
o 4.
P 4 == 3.
97
bij bk £;
3·, (4)
b 10; c··k IJ £m ;
(0
~
()
~ 21r).
+ (22) . + (2111).
5.
P 5 == 6.
6.
P 6 == 15.
b·· IJ bk £ bmn ; 15 ;
7.
P 7 == 36.
105; c··k IJ b£ m bnp'.
S.
P s == 91.
bij bk £ bmn bpq ; 105 ;
(311)
(4.7.14)
We see from (2.6.21)2 that the number P n of linearly independent nthorder t~nsors which are invariant under T1 is given by 21r
(6)
+ (42) + (222).
Pn =
l1r J(ei8 + 1 + e-i8 )n d8.
(4.7.16)
o
(511)
(S)
+ (4111) + (331) + + (3211) + (22111).
With (4.7.16), we have
+ (62) + (44) + + (422) + (2222) . (n odd; n ~ 3) ,
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 a IJ ~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] ) b·· + bfJ·~J COk - bmJ. COkfJ == fJ - bkJ· C·fJ IJ ck ~m l~m 1 m 1 ~
o.
(4.7.15)
(v) Transversely isotropic materials: T1 The symmetry group T1 associated with a transversely isotropic
Pn
= 1 +( 2)(
i )+( 4)( ~ ) + '" + ( ~ ) ( n/2)
(4.7.17)
(n even).
Complete sets of tensors of orders 1,... ,5 which are invariant under T1 are listed below where we employ the notation (4.7.1) and (4.7.2).
98
[Ch. IV
Invariant Tensors
Sect. 4.8]
99
Examples
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 e311 + e322;
3;
tensors whose components are all zero.
(3) + (21);
This is a consequence of
identities of the form
e312 - e321;
3 ; (21) + (111) . (4.7.19)
4. P4=19. e3333;
1;
(4.7.18)
(4);
where i, ... ,£ take on values 1,2.
e3311 +e3322;
6;
(4)+(31)+(22);
that there are 4 + 5 + 6
= 15
A further consequence of (4.7.19) is 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.
5. P 5 = 51. e33333;
1;
(5); (i) Determine the form of the scalar-valued function
e33311 +e33322;
10;
(5)+(41)+(32);
e33312 - e33321;
10;
(41) + (311);
W = C·· IJ klJ~mn E·· IJ EklJE ~ mn
(4.8.1)
appropriate for the hexoctahedral crystal class 0h where E
is a ij symmetric second-order tensor. From( 4.6.20), we see that E ij E £ E mn k is of symmetry class
e3(e11 + ~2)(e12 - ~1);
30;
(41) + (32) + (311) +
(6)
+ (311) + (221)+ (2111).
+ (42) + (222).
(4.8.2)
From (4.7.7), we see that the general sixth-order tensor invariant under We observe that
there are
SIX
distinct
e
isomers of the tensor
(en + ~2)(e1T ~1) = b'ij cke (i,j, k, = 1,2; c11 = c22= 0, c12 = -c21 = 1)
the group 0h is expressible as a linear combination of the isomers of the tensors
100
Invariant Tensors
L
(e111122 + eIII133);
15;
(6) + (51) + (42),
L(eI12233+eI13322);
15;
(6)+(42)+(222).
101
[Ch. IV
Sect. 4.8]
(4.8.3)
The general expression for the function (4.8.1) is then given by
Examples
(4.8.6)
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
(ii) Determine the form of the symmetric second-order tensorvalued function
tensors comprised of the tensor and its distinct isomers. The argument and 3 linearly independent invariants contained in the three sets of 1,
15 and 15 invariants given by
(4.8.7)
T··1J == C··1J kfJ~mn EkfJE ~ mn
given in §4.5 together with (4.8.2) and (4.8.3) shows that there are 1, 2
appropriate for the hexoctahedral crystal class 0h where Eij is a symmetric second-order tensor. From (4.6.20), we see that T ij E k £E mn is of symmetry class
(6) + (51) + 2(42) + (321) + (222).
(4.8.8)
(4.8.4) Let us denote the sixth-order property tensors (4.8.3) associated with
0h by (4.8.9)
These invariants are given by
II = :L(eUUU)ij ...nEijEk£Emn =
The symmetry types of these three sets of tensors are given in (4.8.3).
:LE~I'
The numbers of linearly independent invariants contained in the sets
12 = :L (eUU22 + eUU33)ij n Eij Ekl! E mn = 13
=
:L(eUI212 +elU313)ij n E ij Ek£Emn
=
L E tl (E22 + E33 ) , :L E U(Et2+ E t3), (4.8.5)
14
=L
(eU2233 + eU3322)ij n Eij E k £E mn
= 6E U E 22 E 33 '
IS
=L
(eU2323 + eU3232\j n E ij E k £E mn
= 2L
E U E~3 '
u··kfJ .. EkfJE . 1J ~mn T 1J ~ mn'
(r) vookfJ 1J ~mn T.. 1J EkfJE ~ mn
(r == 1,... ,15); (4.8.10)
w~:k) fJ 1J
~mn
T··1J E kfJ~ Emn
( r == 1,... ,15)
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
102
[Ch. IV
Invariant Tensors
u··klJ . IJ ~mn EklJE ~ mn'
v(r) E E ijk£mn k£ mn
(r == 1,... ,15); (4.8.11)
(r) E k£ W ijk£mn
E mn
( r 1 == ,... ,15)
Sect. 4.9]
Character Tables for Symmetric Groups 52 ,... ,58
103
where we have noted that E·· k == E· k ·. We then write (4.8.13) as IJ 1J IJ W == C··IJ klJ~mn A··kA IJ ~mn
+ C··IJ klJ
~mn
IJ B··kB IJ ~mn (4.8.15)
respectively. With (4.8.3), we see that these are given by We may employ the procedure discussed in §4.6 to obtain the
Eb 1i b1j EIl; Eb li b1j Ell (E 22 + E 33 ),
Eb 1i bl/E~2
symmetry classes of the tensors E··kE . We have IJ lJm n ' ... , B··kB IJ ~ IJ ~mn
+ E~3)'
EijkE£mn: (6)
E b1i b1/EI2 + EI3)'
E (b 1i b2j
+ (51) + 3(42) + (411) + 2(321) + (222) + (3111);
+ b2i b1j )(E ll + E 22 )E 12; (4.8.12)
(4.8.16) BijkB£mn: (42)
+ (321) + (222) + (3111);
AijkB£mn + BijkAR,mn: (51) (iii) Determine the form of the scalar-valued function
+ (42) + (411) + (321).
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 (4.8.13)
bij bk£b mn 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
which is invariant under the orthogonal group 03. The tensor E == E ijk has 18 independent components. From the remarks following (4.4.23),
linearly independent isotropic invariants of the form (4.8.13) and that
we see that three-dimensional tensors of symmetry classes (3), (21) and
These are given by
lJ E.IJ·kEIJ~mn' W == C.IJ·k ~mn
(111) have 10, 8 and 1 independent components respectively. would indicate that E is of symmetry class (3)
+
This
(21). We may set
3E··IJ k == A··IJ k + Book' IJ (4.8.14) A··IJ k == E··IJ k + E·Jk1· + E kIJ··,
B·· IJ k == 2E·· IJ k - E·Jk1· - E k IJ··
where A and B are of symmetry classes (3) and (21) respectively and
there are 2, 2 and 1 invariants arising from the three terms in (4.8.15).
A··IJ k A··IJ' A· ; B·· B·· k B··· B· · k A··· 11J J kk IJ k IJ' 11J Jkk '
A··11 k B·· . JJ k
(4.8.17)
4.9 Character Tables for Symmetric Groups 5 2 , ... ,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
Character Tables for Symmetric Groups 52 ,... ,58
Sect. 4.9]
[1950]. The character tables for 511 , 512 and 513 are given by Zia-ud-
Table 4.3
Din [1935], [1937]. In these tables, I == 1'12 /2 ... n In denotes the class
Character Table: 54
of a group where 'I is the number of one cycles, '2 is the number of
I
14
122
13
4
22
two cycles, .... The number of permutations comprising the class I is
h,
1
6
8
6
3
(4) (31) (22) (211) (1111)
1 3 2 3 1
1 1 0 -1 -1
1 0 -1 0 1
1 -1 0 1 -1
1 -1 2 -1 1
given by h,. The characters satisfy the orthogonality relations
where (n1 n2···) and (m1 m2···) are inequivalent irreducible representations of Sn and where X~ln2··· and X~lm2··· give the values of
Table 4.4
the characters of (n1 n 2... ) and (m1m2 ... ) for the class I. The quantity
105
Character Table: 55
X~ln2"· is found in the row corresponding to (nln2".) and the column
I
15
13 2
12 3
14
12 2
23
5
headed by I.
h,
1
10
20
30
15
20
24
1 4 5 6 5 4 1
1 2 1 0 -1 -2 -1
1 1 -1 0 -1 1 1
1 0 -1 0 1 0 -1
1 0 1 -2 1
1 -1 1 0 -1 1 -1
1 -1 0 1 0 -1 1
Table 4.1
Character Table:
I
12
2
h,
1
1
(2) (11)
1 1
1 -1
Table 4.2
(5) (41) (32) (311) (221) (2111) (11111)
52
Table 4.5
Character Table: 53
I
13
12
3
h,
1
3
2
(3) (21) (111)
1 2 1
1 0 -1
1 -1 1
0
1
Character Table: 56
I
16
14 2
13 3
12 4 12 2 2 123
15
6
24
23
32
h,
1
15
40
90
45
120
144
120
90
15
40
1 (6) 5 (51) 9 (42) 10 (411) 5 (33) 16 (321) 5 (222) 10 (3111) 9 (2211) (21111) 5 (111111) 1
1 3 3 2 1 0 -1 -2 -3 -3 -1
1 2 0 1 -1 -2 -1 1 0 2 1
1 1 -1 0 -1 0 1 0 1 -1 -1
1 1 1 -2 1 0 1 -2 1 1 1
1 0 0 -1 1 0 -1 1 0 0 -1
1 0 -1 0 0 1 0 0 -1 0 1
1 -1 0 1 0 0 0 -1 0 1 -1
1 -1 1 0 -1 0 -1 0 1 -1 1
1 -1 3 -2 -3 0 3 2 -3 1 -1
1 -1 0 1 2 -2 2 1 0 -1 1
~
o
0:>
Table 4.6
Character Table: 57
I
17
152
143
13 4
13 2 2 1 223
1 25 .
hi
1
21
70
210
105
420
1 6 14 15 14 35 21 20 21 35 14 15 14 6 1
1 4 6 5 4 5 1 0 -1 -5 -4 -5 -6 -4 -1
1 3 2 3 -1 -1 -3 2 -3 -1 -1 3 2 3 1
1 2 0 1 -2 -1 -1 0 1 1 2 -1 0 -2 -1
1 2 2 -1 2 -1 1 -4 1 -1 2 -1 2 2 1
1 1 0 -1 1 -1 1 0 -1 1 -1 1 0 -1 -1
(7) (61) (52) (511) (43) (421) (331) (4111) (322) (3211) (2221) (31111) (22111) (211111) (1111111)
16
124
12 3
13 2
25
2 23
34
7
504
840
630
105
280
504
210
420
720
1 1 -1 0 -1 0 1 0 1 0 -1 0 -1 1 1
1 0 -1 0 0 1 0 0 0 -1 0 0 1 0 -1
1 0 0 -1 0 1 -1 0 -1 1 0 -1 0 0 1
1 0 2 -3 0 1 -3 0 3 -1 0 3 -2 0 -1
1 0 -1 0 2 -1 0 2 0 -1 2 0 -1 0 1
1 -1 1 0 -1 0 1 0 -1 0 1 0 -1 1 -1
1 -1 2 -1 -1 -1 1 2 1 -1 -1 -1 2 -1 1
1 -1 0 1 1 -1 -1 0 1 1 -1 -1 0 1 -1
1 -1 0 1 0 0 0 -1 0 0 0 1 0 -1 1
~ ~ ~
""S
~'
~ ~ ~
;:l CI.l
<:::l
~
'"0 ?""
<
en C't>
Table 4.7 I
("') M-
Character Table: 58 (Continued on next page) 18
16 2
153
144
14 2 2
13 23
~
3
1 5
1 6
1 224
1 2
1 23 2
420
1120
2
2 3
hi
1
28
112
420
210
1120
1344
3360
2520
(8) (71) (62) (611) (53) (521) (5111) (44) (431) (422) (4211) (332) (3311) (3221) (2222) (41111) (32111) (22211) (311111) (221111) (2111111) (11111111)
1 7 20 21 28 64 35 14 70 56 90 42 56 70 14 35 64 28 21 20 7 1
1 5 10 9 10 16 5 4 10 4 0 0 -4 -10 -4 -5 -16 -10 -9 -10 -5
1 4 5 6 1 4 5 -1 -5 -4 0 -6 -4 -5 -1 5 4 1 6 5
1 3 2 3 -2 0 1 -2 -4 0 0 0 0 4 2 0 2 -3 -2 -3
1 3 4 1 4 0 -5 2 2 0 -6 2 0 2 2 -5 0 4 1 4 3
1 1 0 -1 0 0 -1 0 0 0 2 -2 0 0 0 -1 0 0 -1 0 1
1 1 2 -3 2 0 -3 0 -2 4 0 0 -4 2 0 3 0 -2 3 -2 -1
1 1 -1 0 1 -2 2 2 1 -1 0 0 -1 1 2 2 -2 1 0 -1 1
1
1 2 0 1 -2 -1 0 -1 0 1 0 2 1 0 -1 0 -1 -2 1 0 2 1
1 1 -1 0 -1 0 0 0 1 1 0 0 -1 -1 0 0 0 1 0 1 -1
-1
1 2 1 0 1 -2 -1 1 1 -2 0 0 2 -1 -1 1 2 -1 0 -1 -2 -1
-1
1
-1
1
-1
4 1.
-1
ie
~ ~
""S
~
~
('b
""S
~ ~
~
~
CI.l
~ ""S
~
ce:
S S ('b
:;-
n' ~ ""S
<:;) ~
~ CI.l
(J')
"-> (J')
00
~
o
-1
108
Invariant Tensors
[Ch. IV
v rlrl~M~OM~~oo~~oo~~MO~M~rlrl
I
I
I
I
I
I
I
I
I
I
GROUP AVERAGING METHODS 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
I
I
I
I
I
I
rlrlOrlOOrl~~O~~O~~rlOOrlOrlrl
I 00
o ~ o
l.Q
I
I
I
I
I
I
I
I
rlOrlOOrlOOOOrlOOOOOrlOOrlOrl
I
I
over the group A in order to generate scalar-valued and tensor-valued 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
rlrlOrlOOrlOOOOOOOOrlOOrlOrlrl
I
The procedure employed in this chapter involves summation functions which are invariant under A. We follow the discussion given
rlrl~rl~Orl~OOOOOO~rlO~rl~rlrl
I
5.1 Introduction
computations are well adapted to computer-aided generation.
Com-
puter programs are being developed which will carry out the required computations.
I
rlOrlOrlOrlrlrlOOOOrlrlrlOrlOrlOrl
I
o
00
c.o rl
I
I
I
I
I
rlOrl~rlOrlrlrlOO~OrlrlrlOrl~rlOrl
I
I
I
I
I
I
5.2 Averaging Procedure for Scalar-Valued Functions We consider the problem of determining the form of a scalarvalued function W(E.
rlOOrlOrlO~OrlOOrlOrlOrlOrlOOrl
I
I
I
I
I
.) of an nth-order tensor which is invariant
II···I n
under the finite group A== {A1, ... ,A N }. Thus, W(E.
.) must satisfy
II···I n
,..-......
,..-......
"'-"""rl
,..-......
"'-"""rl "'-""""'-"""rl rl "'-""""'-""""'-"""rl rl rl rl "'-""""'-""""'-"""rlrlrlrlrlrlrl
(5.2.1)
,..-......,..-......rl,..-......rlrl~rlrlrlrlrlrlrl
,..-......,..-......rl,..-......rlrl~rl~rl~rl~~rlrl~rlrlrlrl
,..-......rl~rlM~rl~M~~MM~~rl~~rl~rlrl oo~~~l.Ql.Ql.Q~~~~MMM~~M~M~~rl
'--"''--'"'--''''--''''--''''-'''-'''-'''-'''-'''--''''-'''--''''--''''--''''--''''--'''~'-'''--''''-'''--'''
= [A~]
belonging to A. Let E l ,... , E r denote the independent components of the tensor E· . We then consider the equivalent
for all AI{
II· .. I n
problem of determining the form of W(E) == W(E ) where E denotes i the column vector [E 1,... , Er]T. The restrictions corresponding to 109
110
[Ch. V
Group A veraging Methods
Sect. 5.2]
(5.2.1) are that W(E) must satisfy
A veraging Procedure for Scalar- Valued Functions
Invariants of any given degree n in E may be generated in a
similar fashion.
(K == 1,... ,N)
== W(RK E),
W(E)
111
(5.2.2)
Let
En
denote the column matrix whose (r + ~ - I )
entries are given by
where the set of N r x r matrices {R1,... , R N } forms the r-dimensional matrix representation {R } which defines the transformation properties
(5.2.7)
K
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 T
R 1E
+...+ RNE == [
N K N K E R 1· E· , ... , E R . E· K==1 J J K==l rJ J]
are either invariants or zero.
Thus, replacing E by R
(5.2.3)
where the entries are ordered so that E· E· ... E· 11 12
if i 1 i 2 ... in < jlj2 ···jn· For example, when r
(5.2.4)
N E
K
{R~)}
transformation properties of En under A. R~) is a (r + ~ - I) X + nn - rnat rIX . w h·lC h·IS reJ.erre £ d to as the symmetrized Kronecker nth power of R . Let K
1)
(5.2.9)
1, when multiplied on the right by the column vector E,
Then, each of the (r + ~ - I) entries in the column matrix Rri En is
RK ==
K==l
[Rij ]·
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
either an invariant or zero.
Pn
resentation {RK }. With (2.5.14), we have
~
tr Ri =
~
N
E
K==l
tr R K
We may carry out a row reduction procedure on the matrix R
(5.2.6)
1which
The number of linearly independent
invariants of degree n in E is given by
the identity representation appears in the decomposition of the rep-
PI =
denote the matrix representation which defines the
(5.2.5)
K==1
Each row of R
N E
In
(5.2.8)
(r for any R K (K==l, ... , N). We employ the notation
J1 J2
E~, E~E3' E2E~, E~r
K E in (5.2.3) Let
*
precedes E· E· ... E.
== 3 and n == 3, we have
E 3 = [E~, Er E 2' Er E 3' E I E~, E I E 2E 3, E I E~,
leaves the expression unaltered since
R 1 ==
In
n!
= ~ tr Rri = ~
trR~)
=
N
E
K==1
tr R~) ,
E h,(trRK )'l (trRk)'2 ... (trR~),n
(5.2.10)
I
where (see §4.6) the summation in (5.2.10)2 is over the classes I of the
has rank PI to obtain a PI x r matrix U 1· The PI entries in the column
symmetric group Sn and where h, is the order of the class I. We may
vector U 1E then give the set of PI linearly independent invariants of degree one in E.
carry out a row reduction procedure on Rri which has rank p to obtain r +n n a Pn X ( n matrIx Un· The Pn entrIes in the column vector
-1)
.
.
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. (5.2.14)
We now consider the problem of determining the form of functions W(E. ., 11··· 1n E l· 1· and Fl· 1· 1··· n 1··· m Let E and F denote
F· . ) which are bilinear in the components of 11··· 1m and which are invariant under A == {A 1,···, AN}· the column vectors whose entries are the inde-
Let (5.2.15 )
pendent components of E· . and F· . respectively. Thus, 11··· 1n 11··· 1m (5.2.11) Let {RK } and {SK} be the matrix representations of dimensions rand s respectively which define the transformation properties of E and F
The rank of R 11 is given by the number Pll of linearly independent functions, bilinear in E and F, which are invariant under A. We have 1 * 1 N 1 N Pll == N tr R 11 == N tr(RK x SK) == N tr R K tr SK· (5.2.16) K==l K==l
L
L
under A. Let Row reduction of R 11 yields a Pll X r s matrix U 11 · The Pl1 entries in the column matrix U 11 Ell 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 i 1 - i 2, j 1 - j2 is negative. Let
In order to determine the invariants of degrees m in E and n in F, we let E mn denote the column vector whose ( r+m-1)(s+n-1) m n entries are given by
{RK x SK} denote the matrix representation of A which defines the transformation properties of Ell under A. R K x SK is an r s X r s matrix which is referred to as the Kronecker product of R K and SK.
E· E· ... E·1 F· F· ... F· (i 1,... ,i m == 1,... ,r; jl, ... ,jn == 1,... ,s) 11 12 In m J1 J2
With the ordering (5.2.12), we have (i,k == 1,... , r; j,£ == 1,... , s).
(5.2.13)
(5.2.17)
where i 1 ::; i 2 ::; ... ::; im , j 1 ::; j2 ::; ... ::; jn· The entries in Emn are to be ordered so that E· E. ... E.1 F· F· ... F. precedes E k E k ... 11 12 m J1 J2 In 1 2 E k F £ F £ ... F £ if the first non-zero entry in the list i 1 - k 1,... , m 1 2 n im - km, jl - £1'···' jn - £n is negative. Let
Rows (columns) 1,2, ... , s, s+l, ... , 2s, ... , (r-1)s+1, ... , rs are those
(5.2.18)
for which the indices ij (k£) take on values 11, 12, ... , 1s, 21, ... , 2s, ... , r1, ... , r s. The matrix R K x SK may be written as
where
R~)
and
S~)
denote the symmetrized Kronecker mth power of
114
[Ch. V
Group A veraging Methods
Sect. 5.3]
Decomposition of Physical Tensors
115
R K and the symmetrized Kronecker nth power of SK respectively. The
(5.3.2)
rank of R~n is given by P mn
where p
=.1 trR* N
mn
The transformation properties of T under A
=.1 ~ trR(m) trS(n) NK~ k k
(5.2.19)
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
U mn such that the Pmn entries in the column vector U mn Emn give the p mn linearly independent invariants of degree m, n in E, F. We are developing a computer program
which will carry out the required computations.
(i,j
are defined by
= 1,... ,s; K = 1,... ,N)
(5.3.3)
where the matrices SK = [S~] (K = 1,... , N) form a matrix representation 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
The computations involved in generating the matrices R~n will generally be very tedious.
T!1 = S~T. IJ J
=1
K
of a finite group A is equal to the number r of classes of A. We denote these representations by
This would of course
eliminate the possibility of errors arising during the computation.
r 1 = {rl,···,r&}, ... , rr = {rf,···,rk} .
(5.3.4)
Let P be a non-singular s x s matrix and let 5.3 Decomposition of Physical Tensors
T* = PT. In §4.4, we discussed the decomposition of the set of components T·
.
11"· In
of a physical tensor into sets of quantities which form the
carrier spaces for irreducible representations of the general linear group.
(5.3.5)
The matrix representation defining the manner in which T* transforms under A is given by
In this section, we determine sets of linear combinations of the independent 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·
.
11··· 1n
under a transformation A
K
of A
(5.3.6)
since, if T become SKT, then T* = PT becomes PSKT = PS P- 1PT K = Sk T *. We may choose P so that the matrix representation {Sk} is decomposed into the direct sum of irreducible representations. Thus,
are defined by (5.3.1) Let T denote the column vector whose entries T 1,... ,T s are the Independent 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
[Ch. V
Group A veraging Methods
(5.3.8)
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 representations {SK} and
r i respectively.
Suppose that
r == {f1,... ,fN} is
a k-dimensional irreducible representation of A. We may assume that the matrices f
K
Sect. 5.4]
117
We see from (5.3.10), ... ,(5.3.12) that when T is subjected to the transformation AL , i.e., when T is replaced by SLT, then Uij is replaced by fLU... Thus, the transformation properties of the k-dimensional IJ column vector U·· under A are defined by the irreducible representation
r
IJ
== {f1,... ,fN }. We note that for fixed values of (i,j) in (5.3.11), e.g.,
(i,j) == (1,1), matrix.
are unitary (see remarks following (2.3.18)), i.e.,
A veraging Procedures for Tensor- Valued Functions
VII == [ K=1 f: rfm Sfp]
(m==I, ... ,k; p==I, ... ,s) forms a k X s
The number a of linearly independent matrices in the set
V·· == [ r~ S~ ] which may be generated by allowing i and j to IJ K=1 1m JP take on values 1,... , k and 1,... , s respectively is given by the number of
f:
(5.3.9)
times the irreducible representation
The k s column vectors
r
appears in the decomposition of
{SK}' i.e.,
ij Uij -- [U 1 ,... ,Uij]T k
(5.3.10)
(i==I, ... , k; j==I, ... , s)
(5.3.14)
where U··IJ == V··T IJ'
(5.3.11) ..
UIJm -
N K K L: -1m r· S·JP T p (m-l, ... , k)
K==1
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 SL T , i.e., T p by S~qTq, on the right of (5.3.11)4 to obtain N
N
We denote these linearly independent matrices by V 1,... ,Va . The a column vectors U 1 == VI T , ... , U a == VaT then form a quantities of type r.
_
..
L: r¥ S~ SL T = L: r L rMsMT = r Lmn UIJn K==l 1m JP pq q M==1 mn In Jq q
(m,n
= 1,... ,k) (5.3.12)
where we have set
A tensor-valued function T i i (E k k) 1··· mI··· n invariant under the group A == {A 1,... ,A N} if
IS
said to be
A~.... A~. T· I1J 1
. (E k k) == T· . (A kK n ... AkK n En n) 1mJ m J1...Jm I · .. n 11 ...1m 1~ 1 n ~n ~ 1· .. ~n (5.4.1)
holds for all AK belonging to A. Let - 1 f f- 1 fK == L M' (5.3.13)
1) - (f) (f-l). (fK mi L mn M nl'
5.4 Averaging Procedures for Tensor-Valued Functions
(5.4.2) where T 1,... , T s and E 1,... ,E t denote the independent components of
118
[Ch. V
Group Averaging M eihods
With (5.4.2), we may express the
T· . and E· . respectively. 11··· 1m 11· .. 1n restrictions (5.4.1) as
Sect. 5.4]
A veraging Procedures for Tensor- Valued Functions
119
(5.4.8)
U(E) == Q 11 E satisfies (5.4.5) since, with (5.4.7) and (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 representations which define the transformation properties of T and E
of k t matrices obtained from the Qij given in (5.4.6) upon allowing i
respectively under A.
and j to run through the sets 1,... , k and 1,... , t respectively is equal to
From (5.3.5), ... ,(5.3.7), we see that we may
determine a s X s matrix P such that
*
T == PT == TIl +
+ T 1a + 1
The number PI (f) of linearly independent matrices contained in the set
the number of linearly independent functions U(E) which are linear in
.
+Tr1 + ... + T ra r
E and satisfy (5.4.5). We have
. (5.4.4) (5.4.10)
where the matrix representation defining the transformation properties of the Til'.'" T ia . 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)
Let Q 1,... , Qp denote the set of linearly independent k X t matrices obI tained from the set of k t matrices Qij (i==l, ... ,k; j==l, ... ,t). Then the
which are subject to the restrictions
set of PI (f) quantities of type f which are linear in E is given by the column vectors
r K U(E) == U(RK E)
(5.4.5)
(K==l, ... ,N)
(5.4.11)
where f == {r1"'" r N} is a k-dimensional irreducible representation of
A. We refer to U as a quantity of type f.
We may generate functions U(E) of any given degree n in E which
We first consider the case where U == [U 1,... ,U ]T is a linear k function of E == [E 1,... , Et]T. Let
Q..1J ==
[ N-K K] '" L...J f·1m R·In K==l
Let
En
denote the column vector whose (t + ~ -1)
entries are given by the quantities
(5.4.12) (i,m == 1,... , k; j,n == 1,... , t)
denote a set of k t rectangular k X t matrices.
(5.4.6) arranged
We may again employ
the argument used in (5.3.12) to show that rLQ ij == QijRL ·
satisfy (5.4.5).
(5.4.7)
Then, for a given set of values of (i,j), e.g., (i,j) == (1,1), we see that
in
their
natural
order
so
that
E· E· ... E· precedes 11 12 In ~h ~h·· ~jn i~ t~e first of the non-zero entries in the list i 1 - h, 12 - J2 ,... , In - In IS negative. Let {Rif)} denote the matrix representation of A which defines the transformation properties of En under A where the Rif) are the symmetrized Kronecker nth powers of the R K . Let
120
[Ch. V
Group A veraging Methods
Sect. 5.5]
where the column vector Emn has entries given by (5.2.17) with r, s in
~ r¥ Q (~) IJ = L..J 1m (R(n)). k In
(.I,m -- 1,... , k.· ,J,n -- 1,... , (t+n-1)) n .
'
K~l
(5.4.13)
The Q\mn), ... , Q~mn) are P mn (r) mn linearly independent matrices obtained from the set of matrices (5.2.17) being replaced by t, v.
The QLn) (.1 -1 IJ - ,... , k.· , J -- 1,... , (t+n-1)) n form a set of k (t+n-1) n rectangular k
X
(t
+~ -
121
Examples
Q(~rlll) = IJ
1) matrices. The number P n (r) of linearly in-
dependent matrices contained in this set is equal to the number of
l
Er¥ (R~)
N K~l Ip
x S~)).
(5.4.19)
Jq
upon making an appropriate choice of values of i and j.
linearly independent functions U(E) of degree n in E which satisfy (5.4.5) and is given by Pn(r) =
~
E
5.5 Examples
R~) tr I'K
tr
K~l
(5.4.14)
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
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 ~ {A 1,... ,A 6 } where the Ai are given by (2.2.4), i.e.,
given by
(n)
Al
(n)
Q1 En'·'" QPn En·
(5.4.15)
=[01
=
0 ], A2 [-1/2 f3"/2], A3 1 - f3"/2 -1/2
=[ -1/2 f3"/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 ~ [E 1 ,... , Et]T, F ~ [F 1'... ' F v]T respectively and which are subject to the restrictions that (K~l, ... ,N)
(5.4.16)
where the representations {RK} and {SK} define the transformation properties of E and F.
There are P mn (f) linearly independent
functions of degree m, n in E, F which satisfy (5.4.16) where
f3"/2 ]. -1/2
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 Let x ~ [xl' x2]T. The matrix representation {RK } which defines the transformation properties of x under A is given by
under A.
(5.5.2) (5.4.17) With (5.2.6), (5.5.1) and (5.5.2), the number PI of linearly independent The P mn (f) linearly independent functions of type (mn) (mn) Q 1 E mn , ... , Q p mn E mn
r
are given by (5.4.18)
invariants of degree one in x is given by 1 PI ~ 6
E6
K~1
tr A K ~
o.
(5.5.3)
Group A veraging Methods
122
[Ch. V
In order to generate the invariants of degree two in x, we list the symmetrized Kronecker squares
A~)
which define the transformation
Sect. 5.5]
123
Examples
We next list the symmetrized Kronecker cubes of the A
K the transformation properties under A of the column vector
which define
properties under A of the column vector 2]T 2 x2 = [xl' x1 x 2' x2
The
A~)
A~2) =
(5.5.8)
(5.5.4)
are given by
o o o o o o o o 1 o 1
1
0
0
0
1
0
0
0
1
1/4
A~2) =
-~/2
3/4
-1/2 -~/4 ,
~/4
~/2
3/4
o o o
A~2) =
A~2) =
~/2
3/4
-~/4
-1/2
~/4
3/4
-~/2
1/4
1
0
0
0
-1
0
1/4
0
0
1
-~/2
3/4
1/4
f3/2
3/4
-~/4
1/2
~/4
~/4
1/2
-~/4
3/4
~/2
1/4
3/4
-~/2
1/4
, A~2) =
, A~2) =
3f3
-9
3f3
-f3
5
-f3
-3
-3
~
5
~
-3~
-9
-3~
-1
(3) _ 1 A 2 -8
1
1/4 -1
1/4
-1
(5.5.5)
-9
-3~
-3
~
5
~
-3
-~
5
3f3
-9
1
.
-3{3
A~3) = -81 -~
-~
3~
-3~
5
-1
o
o o
1
(3) , A4 -
o
-1
o o
o o
o
-1
o o
o
1
, (5.5.9)
-3f3
3~
9
3~
-~
-3
5
~
-3
9
3
-~
-5
-~
~
-5
~
-3~
-9
-3~
-1
-9
3~
-1
With (5.2.10)1 and (5.5.5), the number P2 of linearly independent invariants of degree 2 in x is With (5.2.10)1 and (5.5.9), the number P3 of linearly independent (5.5.6) The entries in the column matrix
R2 x2
are either invariants or zero
invariants of degree 3 in x is _ 1 P3 - 6"
L6
(3) _ tr Ak - 1.
(5.5.10) K=1 The entries in the column matrix R x3 are either invariants or zero.
3
and are given by
Thus,
303 000
(5.5.7)
303 which yields the result that xI
+ x~
R3* x3 = is the invariant of degree 2 in x.
6~) X3 = 23 LA..
K=l
0
0 0
0
3 0 -1
0
0 0
0
0 -3 0
1
0
x 31 2 X1 X2 2 x1 x 2 x 32
(5.5.11)
124
[Ch. V
Group A veraging Methods
Sect. 5.5]
Examples
125
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}
The linearly independent matrix obtained from this set is VI = V 11' Hence the quantity of type r 3 is given by
is given by (5.5.15)
(5.5.12) There are three inequivalent irreducible representations associated with
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., V 11' There
the group A which are given by (see (2.5.18))
r 1: r 11,···,r61 -_ r 2: r 21,···,r62_r 3:
1·,
1,
1,
1,
1,
1,
1, -1, -1, -1 ;
1,
1,
(5.5.13)
Ii,..·, ~ = AI' A2, A3, A4, AS' A6
where the A K 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
(5.3.10) and (5.3.11).
== 1,2), we set
6
E
.
tr AK tr
K=1
In the expressions V.. ==
r K == SK == AK
1J
f'k· [
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
K==1
tr
A~) tr rk = 1
(i = 1,2,3).
rJ( S~
]
K=1 1m JP
(i,m,j,p
where the A K are given by (5.5.1).
Upon setting (i,j) == (1,1), (1,2), (2,1) and (2,2) in turn, we have
(5.5.16)
We have from (5.4.13), (5.5.1), (5.5.13), (5.5.5) and (5.5.9),
We now employ
6 _
E
would be no need to determine V 12 , V 21 and V . We would usually 22 refrain from generating any more matrices V·· than are actually 1J required.
ri
appears in the decomposition of the representation {SK} is given by Q'1 == 0, Q'2 == 0, Q'3 == 1 where Q'i == ~
Since we know that Q'3 == 1 and hence that there will be only one
Q1
= [ Kt1 (rk)lm(A K )l P ]
t t
= 3
[~ ~] ,
-~ l
Q(2) = [ (N) 1 K=l K 1m
(A~))2p]
= Q[ 0 2 1
0
Q(3) = [ (N) 1 K=l K 1m
(A~))1p]
= 1! [ 1 4 0
0
1
1
0
2
~J
(5.5.17)
[Ch. V
Group A veraging Methods
126
Sect. 5.5]
With (5.4.11), (5.5.4), (5.5.8) and (5.5.17), the quantities of type r 3 are
Q1 x
= 3 [:1], 2
QF)~ = ~ Xl
X1X [22 ;], -x2
Q~3)X3 = £(xy +x~) [Xl].
~
A3 xA3
=!
127
Examples
-~
1
-~
-3
x2
~
3
-3 ~
1 ~
3 -~ -~
1
1 -~ -~
3
A4 xA4
=
0
0
0
0
-1
0
0
0
0
-1
0
0
0
0
1
1
~
~
~
-1
3 -~
~
3
-1 -~
(5.5.18) Thus, the polynomial expression for .z{x) which is truncated after terms of degree three in x is given by
A5 xA 5
=!
-~
-1
-~
3
-1
~
~
3
3
~
~
A6 xA 6
=!
3
-~ -~
3
1
(5.5.19) (5.5.22)
We now consider the problem of determining the form of a twodimensional 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)
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 A K ) tr r K · We recall that tr (AK x A K )
= (tr AK )2.
With (5.3.11), (5.5.13), (5.5.20) and (5.5.22), we see that
the quantities The matrix representation {SK} which defines the transformation properties of T under A = {AI'.'.' A6} is given by (5.5.21 )
x A denotes the Kronecker square of A K . With (5.2.13), K K (5.2.14) and (5.5.1), we have
where A
Al xA1
=
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
1 A2 xA2
=!
-~ -~
3
~
1
-3 -~
~
-3
1 -~
3
~
~
are quantities of types r l' r 2 and r 3 respectively. The numbers P1(r 2), P 2(r 2), P 3 (r 2) of quantities of type r 2
128
Group A veraging Methods
[Ch. V
which are of degrees 1,2,3 in x are seen from (5.4.14) to be given by
Sect. 5.6]
129
Generation of Property Tensors
may be obtained by determining the set of linearly independent functions multilinear in the n vectors xl' x2' ... , x n which are invariant under A.
The number P n of linearly independent nth-order invariant
tensors is 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
where A K x A K x ... x A K denotes the Kronecker nth power of A K . If the P n multilinear invariants are given by
(5.5.25)
(5.6.2)
The expression for T(x) which is invariant under A and of degree ~
then the set of P n invariant tensors is given by
C·1 . ,... , cP . 1' .
3 in x is then given by
11' .. In
Let
(5.5.26)
(5.6.3)
11· .. n
xl X2... ~ denote the column vector whose 3n entries are xfl xf2··· xin
(i 1 ,i 2 ,... ,i n == 1,2,3) ordered so that x f x 7 ... xP- precedes xJ~ x 7 ... x !1 if 11 12 In 1 J2 Jn the first non-zero element of the set i 1 - j l' i 2 - 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).
The matrix representation of A which defines the transformation properties
of
the
column
vector
Xl x2". xn
is
given
by
{SK}
== {AK x A K x ... x A K } where A K x A K x ... x A K is the Kronecker 5.6 Generation of Property Tensors
We may employ the procedure of §5.2 to generate the set of nthorder property tensors associated with the finite group A == {A1,... ,A N }, 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
nth power of A K . The P n linearly independent invariants which are multilinear in x1" .. ,xn are obtained by determining the P n linearly independent rows of the matrix
(5.6.5)
130
[Ch. V
Group A veraging Methods
The P n
X
3n matrix whose rows are the P n linearly independent rows of
(5.6.5) yields P n linearly independent invariants when multiplied on the right by the column vector xlx2' "xn ' The rows of the P n x 3n matrix obtained from (5.6.5) define the set of P n linearly independent nthorder invariant tensors.
The entries of the third row give the 1, 2, 3 components of the invariant
tensor which we may write as C' 1
JIJ2
=
f: Af Af
K=1
Jl
J2
and will either yield an
We consider as an example the problem of generating the invariant tensors of orders 1 and 2 associated with the group C 3V = {AI"'" A 6} where
A1 =
0
1
0
0
0
1
6
E AKxAK =
K=l
-1/2
0
, A2 =
= t A3~' K=1
~/2
-~/2
-1/2
0
0
-1/2
0 0
, A3 =
-~/2
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
0
0
0
0
0
0
0
0
0
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
6
~/2
-1/2
0
0
0
1
(5.6.6)
A4 =
0
0 0
0 0
0
, A5 =
1/2
-~/2
0
-~/2
-1/2
0
0
0
,A6 =
1/2
~/2
0
~/2
-1/2
0
0
0
1
under the group A defined by (5.6.6) is given by
6
C',
Thus, we have PI = 1, P 2 = 2, P 3 = 5, .... The first-order invariant tensors are defined by the PI = 1 linearly independent rows of the matrix 000 000 006
(5.6.8)
= KE
D.. = 1J
(5.6.7)
The first and last rows of
(5.6.9) give the two linearly independent invariant tensors which are
1J
The number P n of linearly independent nth-order tensors invariant
(5.6.9)
For a given row, columns 1, 2, ... ,9 give the 11, 12, 13, 21, 22, 23, 31, 32, 33 components of the invariant tensor.
-1
The second-order invariant
1
tensors are given by the P 2 linearly independent rows of the matrix
invariant tensor or will have all components equal to zero.
0
131
Generation of Property Tensors
For example, the first row of the matrix
f: AK x AK is given by c, ,
K=1
Sect. 5.6]
=1 6
L
K
=1
Af Af = 3(15 1,15 1, 1 J 1 J
+ 15 21, 15 2J,) ' (5.6.10)
Af. Af. 1
J
= 615 31,153J',
VI ANISOTROPIC CONSTITUTIVE EQUATIONS AND SCHUR'S LEMMA
6.1 Introduction In this chapter, we consider constitutive expressions of the form
== 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, T
we follow Smith and Kiral [1978] in the case where A = {A 1,oo.,A } is N 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 nonlinear constitutive expressions.
Xu, Smith and Smith [1987] use this
type of result to form the basis of a procedure which employs a computer 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
[Ch. VI
Anisotropic Constitutive Equations and Schur's Lemma
which describe the response of anisotropic materials.
The tensor
Sect. 6.2]
135
Application of Schur's Lemma: Finite Groups
Multiplying (6.2.5) on the left by Q and on the right by p- 1, we obtain
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
The restrictions imposed on the matrix D = Q C p- 1 by the invariance
requirement that (6.2.1) shall be invariant under the group of
requirements are then given by
transformations A = {AI '00.' AN} defining the symmetry of the material under consideration. Let T and E denote the column vectors (6.2.2) where T 1,... ,T n and E 1,.. ,Em are the independent components of the tensors T· . and E· . respectively. Let {SK} and {RK } denote J1··· Jq 11··· 1P matrix representations of :4 which are of dimensions nand m re-
QSKCp-1 = QCRKP-I or QSKQ-IQCp-1 = QCp-IpRKP-I.
QSKQ-ID = DPRKP-I
(6.2.8)
(K = I, ... ,N).
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
(K = 1,... ,N),
spectively and which define the transformation properties of T and E (6.2.9)
respectively under A. Then, (6.2.1) may be written as T= CE
(6.2.3)
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
Zl Z=QT=
,
, Z·= J Zr
Z·In·
Xj1
Xl
Zj1 X=PE=
Xr
'
X·= J X·Jm·
(6.2.4)
where f l = {rL... ,r~}, ... , f r = {r~, ... ,r~} denote the r inequivalent irreducible representations associated with A which are of dimensions
We see from (6.2.3) and (6.2.4) that the matrix C is subject to the re-
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 r j . Thus, when T
strictions that
(K = 1,... ,N).
(6.2.5)
and E are repl~ced by SK! and RKE, the quantities Zji and Xji are
Let Z= QT,
X=PE.
(6.2.6)
With (6.2.3) and (6.2.6), we have Z = QT = QCE = QCp-1X
replaced by rkZji and rkXji' The column vectors ZI"'" Zr and X 1,,,·,Xr have P1n1, ... ,Prnr and P1m1, ... ,Prmr entries respectively. We have (6.2.10)
or 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]
137
Application of Schur's Lemma: Finite Groups
With (6.2.9), the equation (6.2.7)1' i.e., Z == nx, may be written as
Z2
nIl n 21
n 12 n 22
n 1r n 2r
Xl X2
Zr
n r1
n r2
n rr
Xr
Zl
(6.2.14) (6.2.11)
r KI
and where the matrix
appears on the diagonal n·1 times and m·1 times respectively. With (6.2.13), we obtain r 2 sets of matrix equations
where the matrices n I] are of the form n I] 11
(i,j == 1,... ,r; K == 1,... ,N)
n I] 1mj
n I] ==
(6.2.12) n I] n·1 1
where the repeated superscripts i,j do not indicate summation.
With
(6.2.12), (6.2.14) and (6.2.15), we have
n In·m· ] 1
(6.2.15)
J
. n I11' ] n I12' ] ... are Pi X Pj rna t rIces . Th e rnat rIces an d nIl , n 12 , ... are PIn1 X PIm1' PI n1 X P2 m 2' ... matrices respectively. nAJ.m.
With (6.2.9) and (6.2.11), the restrictions imposed on the matrix n by (6.2.8) are given by
1
J
(6.2.16)
r]K ] nIn·m· 1
(6.2.13)
pI
K
J
where (6.2.16) must hold for K == 1,... ,N. Equation (6.2.16) yields nimj sets of matrix equations
p2
K
(K
= 1,... ,N).
(0: == 1,... ,ni; (3 == 1,... ,mj; K == 1, ... ,N) (6.2.17)
pr
K
for each given set of values of i and j, e.g., (i,j) == (1,1). Schur's Lemma In (6.2.13), the first and last matrices are block diagonal matrices where
(see §2.4) tells us that if n is an n each of the n
X
n matrices
rK
X
n matrix which commutes with
(K == 1,... ,N) comprising an irreducible
138
Anisotropic Constitutive Equations and Schur's Lemma
[Ch. VI
(6.2.18)
AEn of the n x n identity matrix.
Further, if {rK } and {UK} are inequivalent irreducible representations of dimensions n
then D is a multiple
and m respectively of A and if there is an n
X
The Crystal Class D
139
3
quantities Z·l' ... ' Z· , X· 1,... ,X. are quantities of type r· and may be 1ni 1 1mi 1 1 determined by inspection or by employing the procedure discussed in
representation of the group A == {AI'... ' AN}' i.e., if
(K == 1,... ,N),
Sect. 6.3]
... ' z·In· and X·1 X· · §5.2. The numbers n 1· and mI· of quantities Z·I' 1,... 1 1, m of type r are given by 1 1 i (6.2.22)
m matrix D such that (6.2.19)
(K == 1,... ,N),
Given the numbers n·, 1 m·1 (i == 1,... ,r) and the dimensions p.1 of the irreducible representations r i , we may write down the expression for the matrix D.
Some of the entries in the column matrices Z and X
then D must be the zero matrix. Thus, (6.2.17) yields the result that if
may appear as pairs of complex conjugates.
i ~ j the matrix D IJ (.l is the zero matrix and that if i == j the matrix
minor change in procedure appropriate in such cases.
••
O'.fJ
We discuss in §6.5 the
D IJ (.l is a multiple of the p. dimensional identity matrix. We conclude O'.fJ
1
that the matrix D appearing in (6.2.11) is of the form
6.3 The Crystal Class D 3 We observe from §6.2 that the determination of the form of the (6.2.20)
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
+... + n r
and m1
+... + m r
sets
which form the carrier spaces for the irreducible representations of A appearing
(6.2.21 )
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) ; ... ; X ri (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 deter-
1
mining the quantities Zn,···,Zlnl'···' Zrl,···,Zrnr and Xn,···,X 1m1 ,···, X r1 ,... , X rmr appearing in the expressions (6.2.9)3,4 for Z and X. The
y.1 == C··IJ X·J + C··kX. IJ J Xk which is invariant under the group D 3 == {AI'···' A6} where
(6.3.1)
Anisotropic Constitutive Equations and Schur's Lemma
140
A1
=
0
0
0
1
0
0
0
, A2
=
-1/2
~/2
0
-~/2
-1/2
0
0
0
1
, A3
=
[Ch. VI
-1/2
-~/2
0
~/2
-1/2
0
0
0
1
Sect. 6.3]
The Crystal Class D
141
3
(6.3.4)
(6.3.2) 0 A4
=
0
0 -1 0
0 ,A5 0 -1
=
-1/2
~/2
0
~/2
1/2
0
0
0
-1
,A6
=
-1/2
-~/2
0
-~/2
1/2
0
0
0
-1
There are three inequivalent irreducible representations associated with
1,
1,
1,
r 2 : rt, ... , r~ = 1,
1,
1, -1, -1, -1
=
1,
1,
and [X 2 , -X 1]T which are quantities of types
r 2 and r 3 respectively.
3
Let us consider the first term in (6.3.1), i.e., y. == C·. X·. Since I
1J
J
the Y i transform in the same manner as do the Xi' we see from (6.3.4) that Y3 and [Y2' - Y I]T are quantities of types r 2 and r 3 respectively. In equation (6.2.11), we set
the group D3 which are given by
r 1 : ri, ... , r~
Thus, the set of components Xl' X 2 , X 3 may be split into quantities X
1
(6.3.5) (6.3.3)
r 3 : Ii = [
1
0
0]
'
N2 =[
~3=[ -1/2 ~/2 ~/2], -1/2 ~/2 -~/2], -1/2
-1/2
-
We then see from the discussion of §6.2 that the equation Z == DX which is equivalent to y. == C·· X· may be written as 1 1J J
(6.3.6)
~/2
].
-1/2
The quantities Y i and Xi appearing In (6.3.1) are the components of three-dimensional vectors. The transformation properties of
[y 1,y 2 ,y 3]T and [X 1,X 2 ,X 3]T under the group D 3 are defined by the matrix representations {SK} == {A 1 ,···,A6 } and {RK } == {A 1 ,···,A6 } respectively where the A K are defined by (6.3.2). With (5.3.11), (6.3.2)
DII
Drr
where = cl E 1' = c2~ and where E 1 and E 2 denote the one and two-dimensional identity matrices respectively. With (6.3.5), we then have
c1
o o
0
0
c2
0
0
c2
X3 X2
(6.3.7)
-Xl
and (6.3.3), we have We next consider the second term in (6.3.1), i.e., y. == CookX. X . The 1 1J J k
Anisotropic Constitutive Equations and Schur's Lemma
142
[Ch. VI
Sect. 6.3]
The Crystal Class D 3
143
(6.2.11) and (6.2.12) corresponding to the expression y. == C··kX. X k
transformation properties of the quantity
1
IJ
J
are given by
(6.3.8) under the group D 3 are defined by the matrix representation {A~2),... , A~2)} where the A~) are the symmetrized Kronecker squares of the
(6.3.11)
defined by (6.3.2). We observe that the linear combinations of the K components of the quantity (6.3.8) which form quantities of types r 1, r 2 and r 3 respectively are A
The matrix equation Z == D X which is equivalent to y. == C·· X· X 1 IJ k J k may then be written as
r2 :
(6.3.9)
None
XII
The results (6.3.9) may be obtained from inspection or upon application
Z21
n
23 12
n
X 12
Z31
n31 n3112 n33 n33 11 11 12
X 31
21 11
21 12
n
n
23 11
of the procedure of §5.3. Thus, we have
(6.3.12)
X32 where the
nIt nt1
(i
i= j)
are Pi x Pj zero matrices and
nN = c3E 2'
n1~ == c4 E 2· With (6.3.11), we have ==3
X21 +X 22 X2
(6.3.10) Y3
0
0
0
0
0
0
Y2
0
0
c3
0
c4
0
-Y 1
0
0
0
c3
0
c4
3
X1X3 X2X3 2X 1X2 X2X2 1 2
With (6.3.10), we see that the matrices which appear in the equations
(6.3.13)
Anisotropic Constitutive Equations and Schur's Lemma
144
[Ch. VI
6.4 Product Tables We again consider the group D 3 = {AI'···' A 6 } where the Ai are defined by (6.3.2) and where the irreducible representations r 1, r 2 , r 3 associated with D 3 are given by (6.3.3). Let
r 1:
aI' b 1
r 2:
a2' b 2
r 3:
[ a31 a32
(6.4.1)
1
Sect. 6.4]
In Table 6.1, a1 and b 1 denote quantities of type r 1; a2 and b 2 , quantities of type r 2; [a31' a32]T and [b 31 , b 32]T, quantities of type r 3. We may generate the entries appearing on the right of Table 6.1 upon inspection of the manner in which the quantities a1 b 1, a1 b 2 , ... transform under D 3 . For example, the quantity al is of type r l' i.e., a1 is unaltered under all transformations of D 3 · Hence al b 1, a1 b 2 and [a1 b 31 , a1 b 32 ]T transform under D3 in exactly the same manner as do b 1 , b 2 and [b 31 , b 32]T. Consequently a1 b 1 , a1 b 2 and [al b 31 , a1 b 32]T are quantities of types r l' r 2 and r 3 respectively. Since a2 and b 2 are
31 [b ] b 32
quantities of type
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 b 1, b 2, b , b . These are given by alb 1, alb 2, ... , a32 b 31' a32 b 32· We may 31 32 determine 16 linear combinations of these products which may be split into sets of quantities such that the transformation properties under D 3 of each set is defined by one of the representations
r l' r 2 or r 3·
list these sets of quantities in tabular form in Table 6.1.
We
r1
al
r2
a2
r3
[ a31 ] a32
bl b2
31
[b ] b 32
the matrices
ry,... ,r~ = 1,
1, 1, -1, -1, -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 [a31b 31 , a31 b 32 , a32 b 31' a32 b 32]T. This forms the carrier space for the representation r comprised of the matrices X
rk (K=I, ... ,6) where
rt rt is X
the Kronecker square of
rt·
The character of this representation is given by (Xl' ...' X6) = (4, 1, 1,0,
alb l , a2 b 2' a31b 31 + a32 b 32
0,0) since tr(rkxrk)= (tr rk)2 and, from (6.3.3), (trrf,· .. , trJi)
= (2, a1 b 2' a2 b 1' a31b 32 - a32 b 31
b [ al 31 alb 32
r 2,
define the transformation properties of a2 and b 2 under D 3 . We see immediately that a2b2 is invariant under D 3 and hence of type r 1.
rk Product Table: D 3
Table 6.1
145
Product Tables
1[ 1[ b a31 l a32 b 2
a2b32l -a2 b 31
-1, -1, 0, 0, 0).
irreducible representations
r l' r 2
decomposition of the representation
and
r.
r3
appear once each in the
The linear combinations of the
components of the column vector z = [a31 b 31 , a31 b 32 , a32 b 31' a32 b 32]T
r3
may be obtained with
rt rt in (5.3.11).
Thus, the quantities of
which form quantities of types
(5.3.11) upon setting SK = b [ a32 b 2 ], [ a31b 32 + a32 31 ] -a31b 2 a31b 31 - a32 b 32
With (6.3.3) and (2.5.14), we see that the
types
X
r l' r 2 and r 3 are given by
r l' r 2
and
146
[Ch. VI
Anisotropic Constitutive Equations and Schur's Lemma
Sect. 6.4]
147
Product Tables
(i,j==1,2,3) transform under the group considered.
We list the Basic
Quantities table for the group D 3 below. Table 6.2
(6.4.2)
Basic Quantities: D 3
r1
833 , 8 11 + 822
r2
P3' a3' A 12
r
3 [~:J, [:;J, [~~~ l [Sl~~~22l [-;;3]
The information contained in the product table enables us to readily generate the form of constitutive expressions. respectively, and appear as entries in the rows of Table 6.1 headed by
Consider the
problem of determining the forms of the equations
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
Y·==C··X· 1 IJ J'
y. == C··kX.Xk 1 IJ J '
(6.4.3)
readily generate product tables is outlined in §7.5. We note that one must have available the matrices
rt,...,r N (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. 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 secondIJ
J1
order tensors A·· (== -A .. ) which form the carrier spaces for the 1J
which are invariant under D 3 where the Vi' Xi (i==1,2,3) are components of absolute vectors. We see from Table 6.2 or from (6.3.4) that Y3' X 3 are quantities of type r 2 and that [Y2' - Y l]T, [X 2 , -X 1]T are quantities of type r 3. The argument given in §6.2 shows that each of the quantities of type
ri
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
ri
arising from the decomposition of the
components of E. In the case of (6.4.3)1' we have
J1
irreducible representations of the group D 3- We refer to this table as the Basic Quantities table associated with D 3 . The entries in the table
(6.4.4)
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
which is the result (6.3.7). We next determine the form of (6.4.3)2' i.e., Y i == Cijk Xj X k , which is invariant under D 3 . To do this, we make the identifications
Anisotropic Constitutive Equations and Schur's Lemma
148
[Ch. VI
(6.4.5)
Sect. 6.5]
149
The Crystal Class S 4
arising from the Xi X j X k (i,j,k == 1,2,3) are given by
(6.4.9) We see from (6.4.5) and Table 6.1 that the quantities of types and f
3
r 1, r 2
arising from terms quadratic in X 1"",X 3 ~re given by
r l : X~, r 3:
r 2:
XI+X§;
[~~~~l [~f~;§
None;
(6.4.6)
l
Since Y3 and [Y 2' - Y l]T are quantities of types f 2 and
The form of (6.4.3)3' i.e., Y i == Cijk£Xj XkX£, which is invariant under D 3 is then seen from (6.4.9) to be given by
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 We may determine from (6.4.9) and Table 6.1 the decomposition of the (6.4.7)
quartic terms XiXjXkX£ (i, ... ,£== 1,2,3) into quantities of types and
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·1 which form 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 X 3' [X 2' -XI]T assume the role of hI'
r 3.
r 1, r 2
This iterative process may be continued so as to obtain the
decomposition of the X· X· ... X· 11 12
In
for any reasonable value of n.
A
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.
b 2 , ... in Table 6.1. We have
r 1:
2 3' a1 -X -
r 2:
None;
r 3:
[a31 ] = [X I X 3] a32
b 2 == X 3 ;
X 2X 3 '
[
a~l ] = a32
6.5 The Crystal Class S 4
None;
al =XI+X§;
[ 2X I X2 ]XI -X§
The matrices comprising some of the irreducible representations (6.4.8)
cases, the procedure employed differs slightly from that discussed in the
31 2 ] [hb ] - [X -Xl' 32
With (6.4.8) and Table 6.1, the quantities of types
r l' r 2
associated with the group 54 have complex numbers as entries. In such previous section.
and
r3
The group of matrices defining the symmetry of the
crystal class 54 is given by 54 == {A 1 ,···,A4 } == {I, D 3 , D 1T 3 , D 2T 3 } where
150
A nisotropic Constitutive Equations and Schur's Lemma
-1 0 0
0 0 A1 =
0
0
, A2 =
0 0 1
0
0
0 , A3 = -1 0 0 , A4 = 0 0 0 o -1
o -1
o -1
There are four inequivalent irreducible
0 0
rep~esentations
[Ch. VI
0
o.
(6.5.1)
o -1 associated with
f 1: f 2: f 3: f 4:
Ii, .. ·,Ii Ii, .. ·,Ii
1,
1,
1,
1, -1, -1 (6.5.2)
1, -1,
1,
1, -1,
-1,
151
= Cijk £E k £,
T·· 1J
= C··1J k £mn Ek£Emn
(6.5.3)
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 T ij and E ij which form quantities of types f 1,... ,f4 are given by
1
1,
The Crystal Class S 4
We consider the problem of determining the form of the equations T ij
the group 54 (see §7.3.5) which are given by r 11,···, r14 2 2 r 1,···,r4
Sect. 6.5]
-1 1.
The product table and the basic quantities table for the group 54 are
f 1: T 33' T 11 + T22 ;
E 33 , Ell + E 22 ;
f 2: T 11 -T 22 , T 12 ;
Ell - E 22 , E 12 ;
f 3: T 31 +iT 23 ;
E 31 + i E 23 ;
f 4 : T 31 -iT 23 ;
E 31 -iE 23 ·
(6.5.4)
given below. With (6.5.4), application of the procedures in §6.2 or §6.4 shows that Table 6.3
Product Table: 54
f1
al
b1
a1 b 1' a2 b 2' a3 b 4' a4 b 3
f2
a2
b2
a 1b 2, a2 b l' a3 b 3' a4 b 4
f3
a3
b3
a 1b 3, a3 b l' a2 b 4' a4 b 2
f4
a4
b4
a1 b 4' a4 b 1' a2 b 3' a3 b 2
the form of {6.5.3)1 which is invariant under 54 is given by
(6.5.5)
The last expression in (6.5.5) is the complex conjugate of the preceding Table 6.4
Basic Quantities: 54 a3 , A 12 , 833 , 8 11 + 822
equation and may be omitted. We may equate the real and imaginary parts of {6.5.5)3 to obtain
P3' 812 , 811 - 822
(6.5.6)
PI - i P2' a1 + i a2' A 23 + i A 31 , 8 31 + i 8 23 PI
+ i P2'
a1 -
1
a2' A23 - i A 31 , S31 - i S23
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 T
2
153
We next consider the problem of determining the form of (6.5.3)2' i.e., T ij == Cijk£mnEk£Emn' which is invariant under 3 4 . With (6.5.4) and the product table for 3 4 (Table 6.3), we see that the quantities of types given by
r 1'... ' r 4 arising from
terms quadratic in the E ij are
r 1:
11,... ,1 7 ;
r 3:
K 1 +iK 2, K 3 +iK 4 , K 5 +iK 6 , K 7 + iK S ; (6.5.7)
r 2:
J 1,···, J6 ;
r 4:
K 1 -iK 2, K 3 -iK 4 , K 5 -iK 6 , K 7 -iK S ;
where II' ... ' K S are defined by (6.5.S) and where we have employed the form mentioned in (6.5.6).
6.6 The Transversely Isotropic Groups T1 and T2
Let Tl denote the group comprised of the matrices
where
+ E22 )2, E33 (E n + E22 ), (En - E22 )2, EI2' (En - E22 )E 12 , E51 + E~3 ;
11,... ,1 7 = E53' (En
EI1-E~2' E33 E 12 , (En + E22 )E 12 , E51 - E~3' E31 E23 ;
J1,···,J6=E33(En-E22)'
cos B sin B
Q( B) == -sin B cos B
o (6.5.S)
K 1,···,K S == E 33 E 31 , E 33 E 23 , (Ell +E 22 )E 31 , (Ell +E 22 )E 23 ,
0
o o
(0
~ B < 21r).
(6.6.1)
1
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])
(EII-E22)E31' -(EII-E22)E23' E 12 E 31 , -E 12 E 23 · 10 : 1
With (6.5.4), (6.5.7) and (6.5.S), we see that the form of (6.5.3)2 which -ipB Ip : e
is invariant under 3 4 is given by
[
T33
]
TIl + T 22 -
[Tn -T 22 T 12
f2 f S 9
[£1f
]=[
gl
g7
g2 gs
£7 ]
f 14
g6 ]
II
rp
: e ipB
(p == 1,2,... )
(6.6.2)
(p == 1,2,... ).
17 In (6.6.2), the 1 x 1 matrices correspond to the group element Q (B).
J1 (6.5.9)
g12
J6
We list below the product table and the basic quantities table for the group Tl .
Anisotropic Constitutive Equations and Schur's Lemma
154
[Ch. VI
The Transversely Isotropic Groups T
1
and T 2
155
x3 axis and a plane of symmetry which contains the x2 and x3 axes.
Product Table: T1
Table 6.5
Sect. 6.6]
The irreducible representations associated with the group T2 may be bO
defined by listing the matrices corresponding to the group elements
aOb O apB p , Apb p (p == 1,2,... )
Q( B) and R 1. We denote the irreducible representations by (see Van der Waerden [1980])
IP
aOb p , apb O ambn (m,n==1,2, ... ; m+n==p)
10 : 1 ,
1
amB n ,
f O: 1 ,
-1
Anb m
Ip :
aOB p , Apb O AmB n (m,n == 1,2,... ; m + n == p)
Bp
Amb n , anBm
Table 6.6
(m,n==1,2, ... ; m-n==p)
(m,n==I,2, ... ;m-n==p)
Product Table:
Sll- S22+ 2iS 12 S11- S22- 2iS 12
Let T2 denote the group comprised of the matrices
cosB sin B Q(B) == -sinB cosB 0
0
-cos () -sin ()
0 0 1
,
0
]
R 1Q(B) == -sinB
aO
bO
aOb O' AOB O amIb m2 + a m 2b m1 (m = 1,2,... )
fO
AO
BO
aOB O' AOb O amI b m2 - a m 2b m1 (m = 1,2,... )
0
0
Ip
0
cosB 0
T2
10
PI - i P2' a1 - i a2' A23 - i A 31 , S31 - i S23
'2 f2
1
where the first and second matrices correspond to Q( B) and R 1 respectively. The product table and the basic quantities table for the
Table 6.7
PI + iP2' a1 + i a2' A23 + i A 31 , S31 + i S23
'1 f1
e~pO J [ ~
group T2 are given below.
Basic Quantities: T1
P3' a3' A 12 , Sll + S22' S33
10
[ e ~iPO
(6.6.4)
(6.6.3)
amI bnl] [ a m 2 b n2 (m,n = 1,2,... ; m+n = p)
1
where (0 ~ B<27r) and R 1 == diag(-I, 1, 1). The group T2 defines the symmetry of a material which possesses rotational symmetry about the
amIbn2]' [an2bb m1 ] (m,n=I,2, ... ;m-n=p) [ am 2 b nl anI m2
156
Anisotropic Constitutive Equations and Schur's Lemma
Table 6.8
[Ch. VI
Sect. 6.6]
The Transversely Isotropic Groups T] and T
157
2
belong to 10' 10' 'I' 'I' r 1, r 1, '2' r 2 , '3' r 3 respectively. Each of the quantities in (6.6.6) which belongs to a representation Ip' say, is
Basic Quantities: T2
10
P3' Sll + S22' S33
rO
a3' A 12
1'1
PI + i P2] [a 1 + i a 2] [A 23 + i A 31 ] [ S31 + i S23] [ -PI + i P2 ' al - i a2 ' A - i A 3l ' -8 31 + i 823 23
expressible as a linear combination of the quantities in (6.6.8) which belong to Ip. Thus, we have
T31 + i T 23
= Cs X~(XI
+ i X 2) + c6(Xt +
X~)(XI + i X 2), (6.6.9)
We consider the problem of determining the form of a sym-
T 31 - i T 23
= Cs X~(XI
- i X 2 ) + c6(Xt +
X~)(XI -
i X 2 ),
metric second-order tensor-valued function
T··1J == C··1J klJ~m XkXnX ~ m
(6.6.5)
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 (6.6.6)
TII-T22-2iT12
are quantities of types 10' 10' '1'
r l' '2' r 2 respectively and that (6.6.7)
are quantities of types 10' '1'
r1
The
coefficients cs' c6 and c7 are complex constants, e.g., Cs = d + i eS' and S the third and fifth expressions in (6.6.9) may be omitted. We may, of
TIl +T 22 , T 33 , T 31 +iT 23 , T 31 -iT 23 , TII-T22+2iT12'
where (:5' (:6' (:7 are the complex conjugates of c5' c6' c7.
course, express (6.6.9)2 and (6.6.9)4 as
[~~~]
= [::
~:s][~~~~]+[~6 -::][~~~:~I~~~J (6.6.10)
respectively.
Upon employing the
product table for T] (Table 6.5) twice, we see that
X~, X 3(Xt + X~), X~(XI + i X 2), (Xt + X~)(XI + i X 2 ), X~(XI - i X 2 ), (Xt + X~)(XI - i X 2 ), X 3(Xt - X~ + 2 i Xl X 2 ), X3(Xt-X~-2iXlX2)' (Xl +iX2)3, (Xl -iX2 )3
(6.6.8)
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
= F(B, C, ... ) which
of
a
polynomial
constitutive
equation
T
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
Generation of Integrity Bases: The Crystallographic Groups
160
[eh. VII
of W(B,C, ... ). Thus, we obtain results of complete generality for these
Sect. 7.2]
such that 1 · 2 r (K == 1,... ,N), Q SK Q-1 == n1 r K + n2 r K +· ... +. n r r K
groups. The discussion in this chapter follows closely the work of Kiral and Smith [1974] and Kiral, Smith and Smith [1980].
161
Reduction to Standard Form
The five
remaining crystallographic groups which are associated with the cubic crystal system are considered in Chapter IX.
Xl
We note that the QZ==
procedures discussed in this chapter have been employed by Kiral and Eringen
[1990]
to obtain non-linear
constitutive expressions for
We consider the problem of generating an integrity basis for
(7.2.4)
functions W(B, C .... ) which are invariant under a finite group A == {AK } == {A 1,... ,A N }. Let
where i == 1,... ,n1; j == 1,... ,n2; ... ; k == 1,... ,nr . The restrictions imposed (7.2.1)
the
column
vector
(7.2.3)
where the
7.2 Reduction to Standard Form
denote
== X2
X
magnetic crystals which are subjected to deformation, electric and magnetic fields.
, ... , X
whose
entries
are
the
on V( ... ) by the requirement of invariance under A are then given by
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 requirement that it be invariant under {A K } are given by
where i == 1,... ,n1; j == 1,... ,n2; ... ; k == 1,... ,nr . The problem of concern is
to
determine
the
general form
of the
polynomial function
(7.2.2)
V(
where the n x n matrices Sl' ... ' SN form the n-dimensional matrix
Thus, we must determine an integrity basis for functions of n l' n 2'···' n r quantities of types r 1, r 2 ,... , rr respectively which are invariant under
W(Z) == W(SK Z)
(K == 1,... ,N)
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}, ... ,
fr
= {f:k}.
Thus, we may determine a non-singular n
X
n matrix Q
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' ... , n r 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
function V(
,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
163
We may determine the quantities 4>1'···' 4>n 1' .,pI'···' tPn2' ... which arise from the tensors B, C, ... by inspection or upon application of the procedure discussed in Chapter V. For example, r 1 is the identity representation so that
ri< = I (K = I, ... ,N) and the quantities q,i of type
r1
are invariants. The
T
E R by the column vector [B 1,... , B p ] K-1 K
whose entries are the
Inaependent components of B. The R K (K = 1,... ,N) are the matrices comprising the p-dimensional representation {RK } which defines the
1. J 1(
1 ,4>2)'
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
K 2{.,pl ,.,p2) , K 3{4>1 ,.,pI)' ... ;
(7.2.6)
transformation properties of [B 1,... , Bp]T under A.
3. L1(4)1,4>2,4>3)' L2(4>1 ,
7.3
Integrity Bases for the Triclinic, Monoclinic, Rhombic, Tetragonal and Hexagonal Crystal Classes
such that the integrity basis for functions W( 1 ,... ,
multilinear elements of an integrity basis for each of the crystal classes
, the arguments
of the triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal
,pI'''.' ,pn2 for the typical arguments ,pI' ,p2' ,p3' in all possible combinations with repetitions included. Thus, the elements of the
systems. We identify each crystal class by name and also by listing its
integrity basis of degrees 1,2,3 generated from the typical multilinear
classes, we list a table of the form indicated below.
In this section, we consider the problem of generating the typical
Hermann-Maugin and Schoenflies symbols.
For each of these crystal
elements of the integrity basis given by (7.2.6) would be given by 1. J 1(
2. K I (q,i' q,j) (i,j=I, ,nl)'
K 2(,pi' ,pj) (i,j=I, ... ,n2)'
K3(
'h.)
Q
Al
A2
AN
Basic Quantities (B.Q.)
r1
r 11
r 21
1 rN
¢, ¢/, ...
r2
r 21
r 22
2 rN
a, b, ...
rr
rr 1
rr 2
rr N
A,B, ...
J 2(.,pi) (i=I, ... ,n2)' ... ;
(i,j=I, ,nl; k=I, ,n2)'
(7.2.7)
L3(q,i' ,pj' ,pk) (i=I, ,nl; j,k=I, ,n2)'
The letter
L4(.,pi' 1/Jj , tPk) (i,j ,k= 1,... ,n2)' ... .
class. The matrices AI' A 2, ... , AN are the elements of the matrix group
Q
denotes the Schoenflies symbol which identifies the crystal
164
[Ch. VII
Generation of Integrity Bases: The Crystal/ographic Groups
which defines the symmetry properties of the crystal class.
The
matrices AI' ... ' AN are given in terms ~f th~ mat:ices I, C, R 1, ~, R 3, ... defined in §1.3. The matrices r r r are the matrices
1, 2,..., N
defining the irreducible representation rio
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
(7.3.2) Then W is expressible as a polynomial in the quantities
The irreducible repre-
ai (i
sentations associated with the crystallographic groups considered in this chapter are of dimensions one or two. dimension one, then the
r 1'... ' r N
If the representation r is of
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
r 1,... , r N
= 1,... ,n);
f3 j f3 k (j,k = l, ... ,m; j::; k).
(7.3.3)
Theorem 7.2. Let W be a polynomial function of the real and imaginary parts of the complex quantities al ,... , an' f3 1,... , f3 m which satisfies
comprising r are defined in (7.3.4)
terms of the matrices E, A, ... , L listed below.
E=[~ ],A =[-1/2 -~/2 0
~/2l B=[-1/2 -~/2l F=[ 0 -1/2 {3/2 -1/2
Then W is expressible as a polynomial in the real and imaginary parts
-:l
of the quantities
(7.3.1)
G=
[-1/2
{3/2
~/2 1/2
] _[1/2 ,H-
-{3/2
_[ l L=[-: ~
-~/2] ,K1/2
165
0
1
1
0
ai (i
l
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
= 1,... ,n);
13j 13k , 13j ,8k (j ,k
= I,... ,m;
j::; k).
(7.3.5)
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'·.·' A and where the;e are r i~equiv~ent irreducible representations r 1 = {rK }, ... , rr = {rK } assocIated wIth A. The results generated
r
upon application of Theorems 7.1 and 7.2 will generally contain a number of redundant terms which should be eliminated. In cases where
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
included in the list of typical basic invariants, we may employ the
symmetric second-order tensor S respectively which form carrier spaces
following systematic procedure.
for the irreducible representations r l' r 2 , ... associated with the various
dimensional identity representation comprised of matrices
crystallographic groups.
(K
The integrity bases given in this chapter may be obtained upon repeated application of the following theorems.
there is a question regarding whether redundant terms have been
= 1,... ,N).
We note that r 1 is the one-
rk = 1
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 (i = 2, ... ,r).
In order to determine the typical multilinear basis elements of degree Theorem 7.1.
Let W be a polynomial function of the real
quantities al ,... , an' 13 1 ,... , 13 m which satisfies
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
166
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
= 2, ... , r;
upon setting i,j
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
167
i:S; j. The number P ij of linearly independent invariants which are bilinear in a quantity of type f i and a quantity of
that the degrees of the elements of an integrity basis are not greater
type f j is given by
elements of the integrity bases for the triclinic, ... , hexagonal crystal
P ij
=
N
1 N
E
K=l
.
than the order N of the group A .
tr rk tr r~\.'
(7.3.6)
These invariants may be obtained upon application of Theorem 7.1
results may be employed. 7.3.1
and/or Theorem 7.2. The typical multilinear basis elements of degree two may then be obtained upon inspection of these (~) sets of N
.
.
k
invariants. Similarly we may generate Pijk = ~ ~ tr r k tr rk tr r K linearly independent invariants which are multili~ea~ in quantities of types
r i , rj
and
r k for
the (rt 1) cases where i, j, k
= 2, ... ,rj
i ~j :::; k.
Pedial Class, C 1, 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. 7.3.2
The typical multilinear basis elements of degree three are obtained upon inspection of these (rt 1) sets of invariants.
We also give examples of the manner in which these
classes below.
.
We list the typical multilinear
Pinacoidal Class, C i' I Domatic Class, C s, m
We may generate
Sphenoidal Class, C 2' 2
P ijk £ linearly independent invariants which are multilinear in quantities of types f i , f j , f k and f£ where 1 P·· k £ = N 1J
N . . k £ E trrk trrk trrK trrK · K=l
Table 7.1 (7.3.7)
Irreducible Representations: C i , C s , C 2
I I
B. Q.
There are Qijk£ linearly independent invariants multilinear in quan-
I
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 P ijk £ - Qijk£
1
1
a, a', ...
invariants which, together with the Qijk£ invariants which are products
1
-1
b, b' , ...
of invariants of degree two, form a set of P ijk £ linearly independent invariants multilinear in the four quantities of types f i , r j , r and r £. k Inspection of the (r!2) sets of Pijk€ - Qijk£ invariants obtained by
Table 7.1A
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
Ci
aI' a2' a3' A 23 , A31 , A 12 , S11' S22' S33' S23' S31' S12
integrity basis of degrees 2, 3, 4, 5, .... It is necessary for each crystal class considered to determine when this iterative procedure may be
Cs
P2' P3' aI' A 23 , S11' S22' S33' S23
terminated.
C2
PI' aI' A 23 , SII' S22' S33' S23
For example, we may employ Theorem 3.4B which says
168
Generation of Integrity Bases: The Crystallographic Groups
[Ch. VII
Application of Theorem 7.1 immediately yields the result that the typical multilinear elements of the integrity basis for the groups C i , C s , C 2 are given by
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
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 p
p
x2' x3 1. a;
(7.3.8)
169
xl xi.
)
(p == 1, , n , (p,q
= 1,
(7.3.12)
, llj P ~ q)
2. bb'.
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
which forms an integrity basis for functions W (xl ,... , x n ) invariant under C s . We next generate the canonical form symmetric tensor-valued function
identification (7.3.9) 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
T(S)
of a second-order
of a single second-order
symmetric tensor S == [Sij] which is invariant under the group C s. With Table 7.1A, we see that Sll' S22' 833 , 823 are quantities of type r 1 (i.e., invariants) and that 831 , 812 are quantities of type r 2. With (7.3.8), we then see that an integrity basis for functions W(S) which are invariant under C s is formed by the invariants
(7.3.9) with repetitions allowed will yield x 1f xJ' f xi2 Xj2 ,
xP1 X~J
... ,
x 1P x!1 J (i 'J.
== 1"2 3·, i _< J.),
(7.3.13) (7.3.10)
(i,j == 1,2,3; p,q == 1,... , n; p
V(S) == a S31
This forms an integrity basis for functions W(x1' ... 'xn ) of the n polar vectors xl' ... ' X n which are invariant under the group C i . In order to generate an integrity basis for functions W(x1 ,... , x n ) of n polar vectors which are invariant under C s , we employ Tables 7.1 and 7.1A and make the identifications a, a ,
_ l I n n. - x2' x3 ,... , x2' x3 '
h, h',
= xl,·..,xr .
,
A function V(S) of type
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
readily seen to be expressible in the form
where a and b are polynomial functions of the
11,... ,1 7. Since TIl' T 22 , T 33 , T 23 and T 31 , T 12 are quantities of types r 1 and r 2 respectively, we see that the general expression for a symmetric second-order tensor-valued function T(S) which is invariant under C s is given by
T == (7.3.11)
+ b S12
r 2 is
a1
a2 S31 + a3 S 12
a4 S 31 + a5 S 12
+ a3 S 12 a4 S31 + a5 S 12
a6
a7
a7
a8
a2 S31
(7.3.14)
where the ai == ai (1 1,... ,1 7) are polynomials In the invariants 11,... ,1 7 defined by (7.3.13).
r1 C 2h aI' A23 , 5 11 ,
r2
r3
Table 7.3A PI
P2' P3
a2' a3' A31 , A 12 , 5 12 , 5 31
P2' a3' A 12 , 5 12
P3' a2' A31 , 5 31
P2' a2' A31 , 5 31
P3' a3' A 12 , 5 12
5 22 , 5 33 , 5 23
C 2v PI' 5 11 , 5 22 , 5 33 D2
5 11 , 5 22 , 5 33
r4
aI' A 23 , 5 23 PI' aI' A23 , 5 23
r1 D 2h
Basic Quantities: D 2h
5 11 , 5 22 , 5 33
r2
r3
r4
aI' A23 , 5 23
a2' A31 , 5 31
a3' A 12 , 5 12
r6 r 7 rS PI
P2
P3
Repeated application of Theorem 7.1 yields the result that the Application of Theorem 7.1 twice will yield the result that the
typical multilinear elements of an integrity basis for D 2h are
typical multilinear elements of an integrity basis for the groups C 2h' 1. a;
C 2v ' D 2 are
2. bb', ee', dd', AA', BB', CC', DD'; (7.3.16)
1. a; 2. b b', cc', dd'; 3. bed.
(7.3.15)
3. bcd, bAB, bCD, cAC, eBD, dAD, dBC; 4. beBC, beAD, bdBD, bdAC, edCD, edAB, ABCD.
172
7.3.5
Generation of Integrity Bases: The Crystallographic Groups
[Ch. VII
Sect. 7.3] Integrity Bases for the Triclinic, . . . , IIexagonal Crystal Classes
Tetragonal-disphenoidal Class, S 4' 4
the transformations in Table 7.4. We see from Table 7.4A that, for the
Tetragonal-pyramidal Class, C 4' 4
group S 4' 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
Table 7.4
Irreducible Representations: S4' C 4
integrity basis for functions of n polar vectors which are invariant under
S 4 essentially gives the general result.
This integrity basis has been
obtained by Smith and Rivlin [1964].
Inspection of this result shows
S4
I
C4
I
f1
1
1
1
1
¢, ¢', ...
f2
1
1
-1
-1
'ljJ, 'ljJ', ...
1. ¢;
f3
1
-1
-1
a, b, ...
2. 'ljJ 'ljJ',
f4
1
-1
a, b, ...
3. 'ljJab + 'ljJab,
D 3 D 1T 3 D 3 R 1T 3
D 2T 3
B. Q.
It.2 T 3
that the typical multilinear elements of the integrity basis for S 4 and C 4 are
ab
+ a b,
ab ;
'ljJab-'ljJab; abed-abed.
'Basic Quantities: S 4' C 4
Table 7.4A
f1
f2
f3
a3
P3
PI - i P2' a1 + i a2
PI + i P2' a1 - i a2
A 23 + iA 31
A 23 - iA 31 S31- iS 23
A 12 , S33' Sll + S22
S12' Sll- S22
S31+ iS 23
f4
7.3.6 Tetragonal-dipyramidal Class, C 4h ,4/m Table 7.S
Irreducible Representations: C 4h
It.2T 3
C
1
1
1
1
1
1
¢, ¢', ...
-1
-1
1
1
-1
-1
'ljJ, 'ljJ', ...
-1
1
-1
-1
a, b, ...
1
-1
-1
1
-1
-1
-1
-1
-1
-1
-1
1
1
-1
-1
1
-1
-1
1
C 4h
I
f1
1
1
f2
1
1
f3
1
-1
f4
1
-1
-1
fs
1
1
1
Since quantities of type f 1 are invariants, the general problem is
f6
1
1
-1
the determination of an integrity basis for functions of arbitrary
f7
1
-1
fS
1
-1
C4
ab-
(7.3.17) -1
4. abcd+abcd,
S4
173
P3' a3 A 12 , 5 33 , 5 11 +5 22
5 12 , 5 11 - 5 22
P1+iP2' a1+ ia2
P1-iP2' a1- ia 2
A 23 + iA 31 5 31 + i 5 23
A 23 - iA 31 5 31 - i 5 23
In Table 7.4, the quantities ¢, ¢',
, 'ljJ, 'ljJ', ... are real quantities.
The quantities a == al + i a2' b == b 1 + i b 2 , are complex quantities and a, b, ... denote the complex conjugates of a, b, ... respectively.
numbers of quantities of types f 2' f 3 and f 4 which are invariant under
D 3 R 1T 3
-1
R 3 D 1T 3
D 2T 3
B. Q.
a, b, ... ~, ~',
...
7], 7]', ...
A, B, ... -1
A, B, ...
174
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
Basic Quantities: C 4h
Table 7.5A
r1 C 4h
r2
a3 A 12 , S33' SII + S22
S12' SII- S22
r5 C 4h
r6
7.3.7
r3 al + i a2
al- ia2 A 23 - iA 31
S31 + i S23
S31- i S23
r7
rS
b == b 1 + i b 2 , , A == Al + i A 2 , B == B 1 + i B 2 , ... are complex quantities and a, b, , A, B, ... denote the complex conjugates of a, b, ... , A, B, ... respectively.
real
quantities.
The
quantities
a == al
Table 7.6
Irreducible Representations: D 4' C 4v' D 2d
D4
Dl
D2
D3
I
RIT 3
~T3
R 3T 3
D3
CT3 D 3T 3
RIT3
T3
D2
D3
D 3T 3
D 2T 3
~T3 DIT 3
1
1
1
1
1
1
1
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
1
1
1
1
-1
-1
-1
-1
E
F
-F
-E
-K
-L
L
K
C 4v
I
~
Rl
D 2d
I
Dl
rl r2 r3 r4
1
r5
Pl- i P2
In Table 7.5, the quantities ¢,¢', ... , 'l/J,'l/J', ... , ~,~', ... , "l,"l', ... are
+ i a2'
175
Tetragonal-trapezohedral Class, D 4 , 422 Ditetragonal-pyramidal Class, C 4v' 4mm Tetragonal-scalenohedral Class, D 2d , 42m
r4
A 23 + iA 31
PI + i P2
P3
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
B. Q.
T3
¢,¢', ... 1/;, 1/;', ...
v,v', ... r,r', ...
[:~l[~~l· .
It is found upon repeated application of Theorems 7.1 and 7.2 that the typical multilinear elements of the integrity basis for C 4h are
Table 7.6A
r1 1. ¢; -
-
, , ,
D4
2. ab, AB, 'l/J'l/J , ~~ , TJ"l ; 3. 'l/Jab, 'l/JAB, ~aA, "laA, 'l/J~TJ;
Basic Quantities: D 4' C 4v' D 2d
r2
r3
r4
S12
SII - S22
[PI] [al ] [An ] [823 ] P2 ' a2 ' A 31 ' -S31
SII - S22
[PI] [a2 ] [A31 ] [831 ] P2 ' -al ' -A 23 ' S23
SII - S22
[PI] [al ] [A 23 ] [823 ] P2 ' -a2 ' -A 31 ' S31
P3' a3 S33' SII + S22
A 12
P3
a3
S33' SII + S22
A 12
S12
a3
P3
S33' SII + S22
A 12
S12
(7.3.18)
4. abed, abAB, abAB, ABCD, 'l/J~aA, 'l/JTJaA, ~TJab, ~"lAB ;
C 4v
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.,
D 2d
r5
a b ± a b, ... , "l A abc ± TJ A a b c) are typical multilinear elements of the integrity basis.
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
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
176
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
177
typical multilinear elements of an integrity basis for D 4' C4v and D 2d are 1.
Table 7.7
Irreducible Representations: D 4h (Continued)
2. al b1 + a2b 2 , 1/; 1/;', v v', 7 7';
D 4h
C
R1
f1 f2
1
3. 1/;( al b 2 - a2 b l)' v( al b 2 + a2 b l)' 7(alb 1 - a2 b 2)' 1/; v 7;
(7.3.19)
4. al b 1cl d 1 + a2 b 2 c2 d 2 , 1/; v( al b 1 - a2 b 2),
1/; 7( al b 2 + a2 b 1),
1/
7(al b 2 - a2 b l) ;
f3 f4
5. 1/;( al b 1cl d 2 + al b 1d 1c2 + al cl d 1b 2 + b 1cl d 1a2
-a2b2c2dl-a2b2d2cl-a2c2d2bl-b2c2d2al)'
Ditetragonal-dipyramidal Class, D 4h , 4/mmm
7.3.8
Table 7.7
Irreducible Representations: D 4h
D 4h
I
D1
D2
D3
f1
1
1
1
1
r2
f3
1 1
-1 -1
-1
1
-1
-1
r4
f5
CT3
-1 E
F
-F
-E
~T3
R3T 3
1
-1
1/;, 1/;', ...
-1
-1
1
v,v', ...
-1
-1
-1
7,7', ...
1
-L
L
K
1
-1
-1
-1
1
fg
1
-1
-1
1
-1
-1
1
fg
1
1
1
-1
-1
-1
-1
-K
-L
L
K
E
F
-F
-E
1
1
-1
-1
1
-1
-1
-1
1
1
1
1
1
D1T 3
D2T 3
-1
1/;, 1/;', ...
-1
-1
1
v,v', ...
-1
-1
-1
-1
7,7', ...
F
-F
-E
-K
-L
L
f6
-1
-1
-1
-1
-1
-1
-1
1
-1
-1
-1
-1 -1
-1
-1
-1
flO
-E
-F
F
E
-1
1
E
-1
B. Q.
1
f5
r7 rg rg
D3T 3
-1
K
K
[:~l[~~l . · e,e', ...
-1
'f/, 'f/', ... (}, (}',
-1
L
-L
-K
...
",', ...
[~~l[:~l . ·
[:~l[~~l· .
Table 7.7A
Basic Quantities: D 4h
f1 D4h
f2
f3
f4
5 12
5 11 - 522
f5
a3 5 33 , 5 11 + 522
[a
A12
1] [A23] [523 ]
a2 ' A31 ' -5 31
f 7 flO P3
[~;]
e, e',·..
1
r7
flO
~
B. Q.
1
1
1
f6
-K
R 1T 3
1
T3
~
-1
'f/, 'f/', ... (}, (}',
",',
... The matrices E, F, K, L appearing In Table 7.7 are defined by ...
[~~l[:;l . ·
(Continued on next page)
(7.3.1). Repeated application of Theorem 7.1 yields the result that the typical multilinear elements of the integrity basis for D 4h are given by
Generation of Integrity Bases: The Crystallographic Groups
178
[Ch. VII
Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes
1.
,(alblcIA I -a2 b 2c2A 2)' ,(AIBIClal -A 2B 2 C 2a2)'
2. al b l + a2 b 2' Al B l + A 2B 2 , 'ljJ'ljJ', vv', TT', ~~', 17 r/, f)f)', ,,';
('ljJ~f), 'ljJ17" v~17, vf),) ( al b l - a2b2),
3. 'ljJ(alb 2 - a2 b l)' 'ljJ(A l B 2 - A 2B l ), v(al b 2 + a2 b l)' v(A l B 2 + A 2B l ),
(al b 2 + a2bl)('ljJ~" 'ljJ17f), T~TJ, Tf),),
T(alb l -a2 b 2)' T(AlB l -A 2B 2 ), ~(alAl +a2 A 2)' 17(a l A 2 - a 2A l)'
(v~" v17f), T~f), T17,) ( al b 2 - a2bl)'
f)(a l A 2 +a2 A l)' ,(a l A l - a 2A 2)' 'ljJvT, 'ljJf)" 'ljJ~17, v~f), V17" T~" T17f);
(AI B I - A 2B 2 )( 1/J~O, 'ljJ17" v~TJ, vO,),
4. al b l cl d l + a2 b 2c2d 2' Al B l C l D l + A 2B 2C 2D 2 ,
(AI B 2 - A2BI)(v~" v1J O, T~O, T1J,),
(alb l -a2 b 2)(A l B l -A 2B 2 ),
('ljJv" 'ljJTO, VTTJ, "10,) (alAI +a2A2)'
('ljJv, ~" 17f))( al b l - a2 b 2)' ('ljJT, ~f), 1J,)( al b 2 + a2 b l)'
(al Al - a2 A 2)( 'ljJ17 T, 'ljJv~, VTO, ~170),
(VT, ~17, f),)(a l b 2 - a 2b l)' ('ljJv, ~" 17f))(A l B l -A 2B 2 ),
('ljJVTJ, 'ljJT~, VT" ~17,)(al A 2 + a2AI)'
('ljJT, ~f), 17,)(A l B 2 + A 2B I ), (VT, ~17, f),)(A l B 2 - A 2B I ),
(a I A 2 - a 2A I)('ljJvO, 'ljJT" VT~, ~O,);
('ljJ" v~, T17)(a I A 2 + a2 A I)' ('ljJ~, v" Tf))(a I A 2 - a 2A I)' 'ljJv~" 'ljJV17f), 'ljJT~f), 'ljJT17" VT~17, VTf)" ~17f),;
5. 'ljJ( al b l cl d 2 + al b l d l c2 + al ci d l b 2 + b i cl d l a2
- a2 b 2c2d I - a2 b 2d 2c l - a2 c2d 2b l - b 2c2d 2a l)' 'ljJ(A I B I C I D 2 + Al B I D I C 2 + Al C I D I B 2 + B I C l D I A 2 - A 2B 2 C2D l - A 2B 2D2C I - A 2C 2D 2B I - B 2C 2D 2A I ),
(7.3.20)
('ljJ~" 'ljJ1J O, T~17, TO,)(A I B 2 +A 2B I ),
(alb 2 + a2 b l)(A I B 2 + A 2B l ), (alb 2 - a2 b l)(A l B 2 - A 2B I ),
('ljJ1J, vf), T,)(aIA I +a2 A 2)' ('ljJf), V17, T~)(aIAI-a2A2)'
179
6. ~77(al hI - ~b2)(AI B2 + A2B I ), B')'(al hI - ~h2)(AI B 2 + A2B ), I b c 'ljJ~(al b i ci A 2 - a2 2 2A I)' 'ljJ~(AlB I C I a2 - A 2B 2C 2a l)' 1/J17(a l b l c I A I +a2 b 2c2A 2)' 'ljJTJ(AIBIClal +A 2B 2C 2a2)' 1/JO(alb l ci Al - a2 b 2c2A 2)' 1/JO(A I B I C I al - A2B2C2a2)' TP')'(alhI cI A2 + ~b2c2AI)' TP')'(A I B I C I ~ + A2B2C2al)'
(~1J, B')')(alhI cId 2 + al hI d l c2 + al cId l b 2 + hI cI d l a2 - a2 b 2c2 d l - a2 b 2d 2c I - a2 c2d 2b l - b 2c2d 2a l)'
'ljJ(aIb2+a2bl)(AIBI-A2B2)' 'ljJ(aIbl-a2b2)(AIB2+A2BI)'
(~77, B')')(A I B I C I D2 + Al B I D I C2 + Al C I D I B + B C D A 2 I I I 2
v(alb2-a2bl)(AIBI-A2B2)' v(albl-a2b2)(AIB2-A2BI)'
- A 2B 2 C 2D I - A 2B 2D 2 C I - A 2C 2D B - B C D A ), 2 I 2 2 2 I b (al b 2 - a2 l)(A I B I C I D2 + Al B I D I C2 + Al C I D I B 2 + B I C D A I I 2
T(al b 2 - a2 b l)(A I B 2 + A2B I ), T(al b 2 + a2 b l)(A I B 2 - A2B I ), ~(al b l ci Al + a2 b 2c2A 2)' ~(AI B I C I al + A 2B 2C 2a 2)'
17(alb l c l A 2 - a2 b 2c2A I)' 17(A I B I C I a2 - A 2B 2C 2a l)' f)(al b i ci A 2 + a2 b 2c2A I)' f)(A I B I C I a2 + A 2B 2C 2a l)'
- A 2B 2C 2D I - A 2B 2D 2C I - A 2 C 2D B - B C D A ), 2 I 2 2 2 I
(AI B2 - A2B I )(alhI cI d2 + alhI d l c2 + al cI d l h 2 + hI cI d l a2 - a2 b 2c2d I - a2 b 2d 2c I - a2c2d2bI - b2c2d2al)·
180
7.3.9
Generation of Integrity Bases: The Crystallographic Groups
[Ch. VII
Trigonal-pyramidal Class, C 3' 3
Sect. 7.3] Integrity Bases for the Triclinic, ... , Hexagonal Crystal Classes
7.3.10
181
Ditrigonal-pyramidal Class, C 3v' 3m Trigonal-trapezohedral Class, D 3 , 32
Irreducible Representations: C 3
Table 7.8
r1 r2 r3
1
1
1
w
w2
a, b, ...
1
w2
w
a, b, ...
C3
r3
r1
r2
P3' a3
PI - i P2' a1 - i a2
PI + i P2' al + i a2
A 23 -iA 31 , S31- iS 23
A 23 +iA 31 , S3I +iS 23
A 12 , S33' Sll + S22
SII- S22+ 2iS 12
r l' r 2' r3
respectively.
S2
R1
R 1S 1
R 1S2
D3
I
Sl
S2
D1
D 1S I
D 1S2
1
1
1
1
1
1
1
1
1
-1
-1
-1
1jJ, 1jJ', ...
E
A
B
-F
-G
-H
[:;l[~;l . ·
Table 7.9A
Basic Quantities:
r1
We see from Table 7.8A that P3' PI - i P2' PI + i P2 are
quantities of types
SI
Sll- S22- 2iS 12
complex quantities and a, b, ... denote the complex conjugates of a, b, ... Thus, generation of an
C 3v ' D 3
I
r3
In Table 7.8, w==-1/2+i~/2 and w2==-1/2-i~/2. We note that w 3 == 1. The quantities a == al + i a2' b == b 1 + i b 2 ,... are respectively.
Irreducible Representations:
C 3v
r1 r2
C3
Basic Quantities:
Table 7.8A
Table 7.9
B. Q.
I
C 3v
r2
B. Q.
C 3v' D 3
r3
P3
a3
S33' Sll + S22
A 12
[ PI ] [ a2 ] [ A 31 ] [8 31 ] [ 2 8 12 ] P2 '-a1 ' -A 23 ' S23 ' S11 - S22
S33' Sll + S22
P3' a3 A 12
[ P2 ] [ a2 ] [ A 31 ] [8 31 ] [ 28 12 ] -PI ' -a1 ' -A 23 ' S23 ' Sll - S22
D3
integrity basis for functions of the n polar vectors Pl, ... ,Pn which are invariant under C 3 will yield the desired result.
This integrity basis
has been obtained by Smith and Rivlin [1964]. Upon inspection of the
The matrices E, A, ... , H appearing in Table 7.9 are defined by
result given in this paper, we see that the typical multilinear elements
(7.3.1). We see from Table 7.9A that the transformation properties of
of an integrity basis for C 3 are given by
P3 and [P2' - Pl]T under the group D 3 are defined by the matrices comprising the representations r 2 and r 3 respectively. Since quantities
1.
ab-ab;
3. abc+abc,
(7.3.21 )
abc-abc.
of type
r 1 are
invariants, we see that knowledge of an integrity basis
for functions of n polar vectors PI' ... ' Pn which are invariant under D 3 will suffice to enable us to determine the result required. This integrity
We observe that an argument essentially identical to that employed in
basis has been generated by Smith and Rivlin [1964]. With the aid of
Theorem 3.3 to obtain (3.2.11) will also yield the result (7.3.21).
this result, we readily see that the typical multilinear elements of an
182
.Generation of Integrity Bases: The Crystallographic Groups
[Ch. VII
integrity basis for C 3v and D 3 are given by
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
Table 7.10A
Basic Quantities:
r1
1. ¢;
C 3h
2. al b 1 + a2 b 2' 1/; 1/;' ;
(7.3.22)
r2
a3
3. a2 b 2c2 - al b 1c2 - b 1cl a2 - cl al b 2 , 1/;(al b 2 - a2 b l) ;
r3
PI - i P2
A 12 , 5 33 5 11 + 5 22 5 11 - 5 22 + 2i5 12
C 3i , C 3h , C 6
PI + i P2
5 11 - 5 22 - 2i5 12
(Continued)
r4
P3
183
rS al- i a2
r6 al + ia2
A 23 -iA 31 A +iA 23 31 5 31 -i5 23 5 + i 5 31 23
4. 1/;(alblcl-a2b2cl-b2c2al-c2a2bl).
C6
Rhombohedral Class, C 3i' 3 Trigonal-dipyramidal Class, C 3h' ()
7.3.11
C 3i
I
C 3h
I
C6
I
S2
C
CS 1
CS 2
Sl
S2
R3
~Sl
R 3S2
Sl
S2
D3
D 3S 1
D 3S 2
1
1
1
w
1 w2
1
w2
w
1
1
-1
-1
Sl
PI + iP 2
a3
al- i a2
al + ia2
5 11 + 5 22 5 11 - 5 22 + 2i5 12
C 3i , C 3h , C 6
Irreducible Representations:
PI - iP2
A 12 ,5 33
Hexagonal-pyramidal Class, C 6' 6 Table 7.10
P3
In Table 7.10, w == -1/2 + i ~/2 and w 2 == -1/2 - i ~/2. quantities ¢J and ~ are real.
+ i b 2, ... ,
r5
r6
1
w
1 w2
1
w2
w
The quantities a == al
+ i a2' b == b 1 +
B == B 1 + i B 2 , ... are complex quantities and a, b, ... ,A, 13, ... denote the complex conjugates of a, b, ... ,A, B, ... A == Al
+ i A2,
The
B. Q.
respectively.
r1 r2 r3 r4
5 11 - 5 22 - 2i5 12
A 23 -iA 31 A +iA 23 31 5 31 -i5 23 S31+ i5 23
Let W be a polynomial function of the quantities
1
1
1
¢J, ¢J', ...
¢, ... ,a, a, b, b, ... ,~, ... ,A, A, B, 13, ... which is invariant under the first
1
w
w2
a, b, ...
three transformations of Table 7.10. It is seen from the results (7.3.21)
1
w2
w
for the group C 3 that W is expressible as a polynomial in the quantities
-1
-1
-1
a, b, ... ~,~', ...
-w _w 2
_w 2
A, B, ...
-w
A, B, ...
-
obtained from the typical multilinear quantities
-
¢, ab, abc, A13, aAB
(7.3.23)
~, aA, abA, ABC.
(7.3.24)
and Table 7.10A
r1 C 3i
Basic Quantities:
r2
C 3i' C 3h' C 6
r3
a3
a1 - i a2
al + ia2
A 12
A 23 -iA 31
A 23 + iA 31 5 31 + i 5 23
5 31 -i5 23 5 11 + 5 22 Sll- 5 22 + 2iS 12 5 33
Sll- 5 22 - 2i5 12
r4
rS
r6
P3
Pl- i P2
PI + i P2
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 C 3i' C 3h and C 6 are
184
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
1.
2. a b, A 13, ~~'; 3. abc, aAB, ~aA;
(7.3.25)
4. abA13, ~abA, ~ABC; 5. aABCD; 6. ABCDEF. 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 D EF of these invariants are typical multilinear elements of the integrity basis.
7.3.12
Ditrigonal- dipyramidal Class, D 3h , 6m2 Hexagonal- scalenohedral Class, D 3d , 3m
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
Table 7.11
Irreducible Representations:
D 3h
I
SI
S2
R3
D 3d
I
SI
S2
D6
I
SI
C 6v
I
SI
r1 r2 r3 r4 r5
D 3h , D 3d , D 6 , C 6v
C
R 3S1 CS 1
R 3S2 CS 2
S2
D3
D3S1
D 3S2
S2
D3
D3S1
D 3S2
B. Q.
'l/J,'l/J', ...
1
1
1
1
-1
-1
-1
~,~',
1
1
1
-1
-1
-1
7], 7]', ...
E
A
B
-E
-A
--B
[1;}[:;}..·
r6
E
A
B
E
A
B
[:;}[~;} ...
D 3h
R1
D2
D 3d
D2S1 R 1S1
D 2S2 R 1S2 D 2S2 R 1S2
B. Q.
1
...
Hexagonal- trapezohedral Class, D 6 , 622 Dihexagonal- pyramidal Class, C 6v' 6mm
The matrices E, A, ... , H appearing in Table 7.11 are defined by (7.3.1). We observe that the quantities
+ i A2,
+ i a2'
A == Al - i A2 ,
a == al - i a2'
B == B I
+ i B 2,
13 == B I - i B 2 , ... ,
_ (7.3.26) b == b l + i b 2 , b == b l - i b 2 , ....
D1
R 1S1 D 1S1
R 1S2 D 1S2
R1
D6
D1
D 1S1
D 1S2
D2
C 6v
~
~SI
~S2
R1
D 2S1 R 1S1
1
1
1
1
1
1
-1
-1
-1
-1
-1
-1
'l/J, 'l/J', ...
1
1
1
-1
-1
-1
~,~', ...
-1
-1
-1
1
1
1
7], 7]', ...
r1 r2 r3 r4 r5 r6
With (7.3.22) and (7.3.26), we see that the typical multilinear elements of an integrity basis for polynomial functions of
,A ,B ,B , ... , a1,a2,b 1,b 2, ... which are invariant under the 1 2 1 2
F
G
H
-F
-G
-H
-F
-G
-H
-F
-G
-H
[1;]'[:;]'·. [:;}[~;} ...
185
186
Generation of Integrity Bases: The Crystallographic Groups
[Ch. VII
Table 7.11A
r1 D 3h
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
187
and
r2 a3
r 3 r4
r5
r6
[:~]
[a
P3
1 23 ].[A ] a2 A 31
A 12 S33
23 [5 -S31
J
S11 + S22
1], l/J~, aA+aA, ~(ab-ab), l/J(aA-aA), ~(AB-AB),
[
abA-ahA, ABC-ABC, e(abc+ahc),
(7.3.28)
l/J(a bA + ahA), ~(aAB + aAB), l/J(ABC + ABC).
25 12 ] S11- S22
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.
D3d
a3 A 12
P3
[:~]
A 31 2 -a1 ' -A 23
[a J[
S33
P3' a3
S33
A 12
S11 + 5 22
C 6v P3
a3
S33 A 12 S11 + S22
redundant terms, that the typical multilinear elements of an integrity basis for D 3h ,D 3d , D 6 and C 6v are
12 31 [ 5 ] [ 25 ] S23 ' S11- S22
S11 +S22
D6
l
Application of Theorem 7.1 then yields the result, after elimination of
[:~].[:~ ] 23
[A A
J
[
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),
4.
25 12 ] S11- S22
31 [-A , [5S23 ] 23
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),
[:~l~~J
l
¢;
1](aA + aA ), l/J e 1] ;
23 ] '-5 [5 31 31
A 31
1.
(7.3.29)
~1](aA-aA), e1](ah-ab), e1](AB-AB);
5. [
25 12 ] S11- S22
(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);
group of transformations 1, 2, 3, 10, 11, 12 of Table 7.11 are
¢, 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)
6.
ABCDEF + ABCDEF, l/J(aABCD + aABCD);
7.
l/J(ABCDEF - ABCDEF).
We note that the quantities A, A, ... , F, F, a, a, ... , c, c appearing In (7.3.29) are defined by (7.3.26).
Generation of Integrity Bases: The Crystallographic Groups
188
7.3.13
[Ch. VII
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
Table 7.12A
Hexagonal-dipyramidal Class, C 6h ' 6/m
Basic Quantities:
Table 7.12
Irreducible Representations:
C 6h
SI
I
S2
D3
D 3S 1
C 6h
D 3S2
C 6h
C 6h
f2
f1
f
f3
f
S
a1- i a2
a3
6
al + i a2
A 23 - iA 31 A 23 +iA 31
A 12 , S33
B. Q.
189
S11- S22+2 i S12
S11 + S22
S11- S22-2 i S12
S31- i S23
S31 + i S23
1
1
1
1
1
1
¢,q/, ...
1
w
w2
1
w
w2
a; b, ...
f3 f4
1
w2
w
1
w2
w
a, b, ...
1
1
1
-1
-1
-1
~, ~', ...
f5 f6
1
w
w2
-1
_w 2
A, B, ...
1
w2
w
-1
-w _w 2
-w
A,]3, ...
f7 f8
1
1
1
1
1
1
7r, 7r', ...
quantities ¢,
1
w
w2
1
w
w2
X, Y, ...
fg
1
w2
w
1
w2
w
flO
1
1
1
-1
-1
-1
X, Y, ... 8,8', ...
A == Al + i A2 , X == Xl + i X 2 , x == Xl + i x2 are complex and a, A, X, x denote the complex conjugates of a, A, X, x respectively. The
f 11 f 12
1
w
w2
-1
_w 2
x, y, ...
first three transformations of Table 7.12 in the same manner as do the
1
w2
w
-1
-w _w 2
-w
x, y, ...
quantities ¢ and a and
C 6h
C
f 1 f2
CS I
CS 2
1
R 3S 1
R 3S2
f 12
P3
PI - i P2
PI + i P2
In Table 7.12, w == -1/2 + i ~/2 and w 2 == -1/2 - i ~/2.
The
The quantities a == a1
+ i a2'
~, 1r,
quantities ¢, ~,
1r,
8 are real quantities.
8 and a, A, X, x and a,
A, X, x transform under the
a associated with Table 7.8 (crystal class C 3)
¢, ~, 7r, 8, ab, aA, aX, ax, AB, AX, Ax,
-1
-1
~, ~', ...
_w 2
XV, XX, xy, abc, abA, abX, abx, ABC,
A, B, ...
-w
A,]3, ...
ABa, ABX, ABx, XYa, XYA, XYZ, XYx, xya, xyA, xyX, xyz, aAX, aAx, aXx, AXx.
1
w
1
w2
w
1
w2
w
1
w
1
w2
-1
-1
(7.3.30)
w
-1
-1
-1
-1
-1
7r, 7r', ...
-w _w 2
_w 2
X, Y, ...
-w
X, Y, ...
Under any of the remaining nine transformations of Table 7.12, the
1
1
1
w
1
w2
w
8,8', ... x, y, ... x, y, ...
quantities (7.3.30) either remain invariant or change sign.
w2
_w 2
-1
f9
-1
-w _w 2 -w
flO
-1
-1
-1
invariant under the first three transformations of Table 7.12 are
-w _w 2
r8
-1
-1
8, a, A, X, x, a, A, X, x which are
a, b, ...
w
w2
1r,
a, b, ...
1
1
functions of the quantities ¢, ~,
1
w2
1
typical multilinear elements of the integrity basis for polynomial
¢,¢', ...
w2
1
B. Q.
1
1
f3 f4
f ll f 12
R3
1
1
f6 f7
f 11
under the transformations of Table 7.8. We see from (7.3.21) that the
f1 f2
r5
C 6h
f7
-1
_w 2
-w _w 2 -w
-1
-1 -1 1
-
-
Repeated
application of Theorem 7.2 will then yield, upon elimination of redundant terms, the result that the typical multilinear elements of an
Generation of Integrity Bases: The Crystallographic Groups
190
[Ch. VII
Sect. 7.3] Integrity Bases for the Triclinic, . . . , Hexagonal Crystal Classes
191
1rABCDEx, 1rABCxyz, 1rAxyzuv,
integrity basis for C 6h are
~XYZUVx, ~XYZxyz, ~Xxyzuv,
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,
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.
1rabX, 1rABX, 1rXYZ, 1rxyX, 1raAx,
7.3.14
~abA, ~ABC, ~XYA, ~xyA, ~aXx,
The matrices E, A, ... , H appearing in Table 7.13 are defined by
8abx, 8ABx, 8XYx, 8xyz, 8aAX,
(7.3.1).
1r~ax, 1r~AX, 1r8aA, 1r8xX, ~8aX, ~8Ax;
5.
Dihexagonal-dipyramidal Class, D 6h , 6/mmm
We shall restrict consideration to functions of quantities of
aABCD, aABXY, aABxy, aXYZU, aXYxy, axyzu,
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
ABCXx, ABCXx, XYZAx, XYZAx, abxAX,
vectors, skew-symmetric second-order tensors and symmetric secondorder tensors.
abxAX, abxAX, xyzAX, xyzAX,
functions of quantities of all twelve types f 1' ... f 12 is given by Kiral,
1rabAx, 1rABaX, 1rxyaX, 1rXYAx, ~abXx, ~ABXx, ~XYAa, ~xyAa,
The general result yielding an integrity basis for
Smith and Smith [1980].1 The number of basis elements in the general (7.3.31)
8abAX, 8ABax, 8XYax, 8xyAX,
case is very large leading us to present only partial results here. shall employ the notation
... ,
1r~abx, 1r~ABx, 1r~XYx, 1r~xyz, 1r~aAX,
1r8abA, 1r8ABC, 1r8XYA, 1r8xyA, 1r8aXx,
We
a
== al + i a2'
a
== al - 1 a2' ... ;
(7.3.32)
~8abX, ~8ABX, ~8XYZ, ~8xyX, ~8aAx;
6.
ABCDEF, ABCDXY, ABXYZU, XYZUVW, ABCDxy, ABxyzu,
We also use the notation R( ABXY... ) and I( ABXY ... ) to denote the
xyzuvw, XYZUxy, XYxyzu, aAxXVZ, aAXxyz, aXxABC,
real and imaginary parts of ABXY.... Expressions such as I(AB, XV,
1raABCx, 1raxyzA, 1rABCDX, 1rABxyX, 1rxyzuX,
~aXYZx, ~axyzX, ~XYZUA, ~XYxyA, ~xyzuA,
7.
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
8aABCX, 8aXYZA, 8XYZUx, 8XYABx, 8ABCDx;
in (2.10) in Kiral, Smith and Smith [1980] should be replaced by I(AB)I(XY),
ABCDEXx, XYZUVAx, xyzuvAX,
I(AB)I(abXY), I(XY)I(abxy) respectively.
Table 7.13
Irreducible Representations:
D6h
I
SI
[1
1
S2
D1
1
111 -1
[2
1
1
1
[3
1
1
1
1
r4
1
1
1
-1
r5
E
DIS I
A
B
F
-1 G
[6
E
A
B
[7
1
1
111
[8
1
1
1
1
[9 flO
E
[11
f I2
E
1
-F
-1
-1
-1
111
1
1
-1
A
B
F
A
-G
B
-F
-1
G
-G
D 1S2
D2
-1
H
-ll
~
D 2S 1
1
-H
3
D 3S 1
D 3S 2
B.Q.
1
1
1
¢>, ¢>', .
-1
-1
-1
1
1
1
'ljJ,'ljJ', ..
-1
-1
-1
-1
-1
-1
e, e', .
1
1
1
-1
-1
-1
ry, ry',
-B
[1~l[:~l . ·
-F -F
-1 -1
-G -G
-H -ll
-F -F
-E E
1
-1 -1
-1 -1
-G -G
1 -1 -1
1
-1
H
D
1
1
-1
D 2S2
1
1
-1
~
co
D 6h
-H -ll
-E E
-A
..
A
B
[:~l[~~l···
1
1
1r,
1
1
p,p', .
-1
-1
f), f)', ..
-1
-A A
-1
-B B
1r', .
",', .
[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
en ('b
(Continued)
~
;+~
D 6h
C
CS 1 CS 2
R1
R 1S 1
R 1S 2
IL.2
IL.2 S1
IL.2 S2
R3
R 3S 1
R 3S 2
B.Q.
~
~ ~
~
:1.
[1
1
1
1
1
1
[2
1
1
1
-1
-1
r3 r4 r5 [6
1
I I
E
A
-1
1
1
1
-1
-1
-1
B
F
G
n
1
1
-1
-1
-1
-F
-1
-G
1
-1
1
1
1
¢>, ¢>', ..
1
'ljJ,'ljJ', ..
-1
e, e', .
?"i
1
-1
-1
-1
ry, ry',
S. C'b
-H
-E
-A
-B
B
-F
-G
-II
-F
-G
-H
-1
-1
-1
-1
-1
-1
-1
-1
-1
1
B
-1
-1
-1
1r,
1
-1
-1
-1
p,p', .
1
1
1
1
-1
1
1
-1
-1
-1
-1
-1
-1
rIO
-1
-1
-1
1
1
1
-1
-1
-1
1
1
1
r 11
I -E
-A
-B
-F
-G
-ll
F
G
II
E
A
B
G
II
F
[1~l[:~l· . [:;l[~;l· .
A
-1
F
..
E
-1
-B
G
~
-1
A
-A
~
-1
E
I -E
to C/)
-1
r7 r8 r9
r 12
~
II
-E
-A
-B
1r', .
(), ()',
..
",', .
[i~l[~~l· . [~~l[~;l· .
C/)
~ ""1
~ ;;. -.~.
~ C'b ~
~
~
c
~
::.. ~
c.e:::
~
::.. Q ~
C/) C/) ~
\I)
~
co w
194
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
Table 7.13A
f1
Basic Quantities: D 6h f2 f5
D 6h
a3 S33' S11 + S22
lPI(ab).
A 12
[ a 1] [A 23 ] [5 23 ] a2 ' A 31 ' -S31
Sect. 7.4] Invariant Functions of a Symmetric Second-Order Tensor: C
195
3
7.4 Invariant Functions of a Symmetric Second-Order Tensor: C 3 f6 [ 25 12 ] S11- S22
f S f 11
[~~]
P3
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 C 3 = {AI' A 2 , A 3 } = {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
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
associated with the group C 3 which are seen from Table 7.S to be given by
which are invariant under the group D 6h .
1.
¢;
2.
11'11", pp', R(AB, XV, ab) ;
f 2:
r 1 r1 1 l' 2' r 3 2 2 r 1 , r2 2, r 3
1, w, w2
f 3:
ri,~,ri
1, w2, w
f 1:
3.
lPI(AB, XV, ab), pl(AX), I(ABa, aXY, abc);
4.
lPR(ABa, aXY, abc), pR(AaX), lPpR(AX),
1, 1, 1 (7.4.1 )
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
R(ABXY, ABa:b, abXY), I(AB) I(XY), I(AB) I(ab), I(ab) I(XY);
5.
quantity of type
r 1, i.e., an invariant.
We may then set
lPI(ABXY, ABa:b, abXY), pI(Aa:bX), pl(ab) R(AX), (7.4.2) 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); (7.3.33)
6.
where W is a polynomial function of the elements II' 12 , ... of an integrity basis for functions of S which are invariant under C 3.
lPR(ABCDa:, a:XYZU, ABa:XY), We observe from Table 7.SA that y
iY 2' where Y1 and Y2 are components of the polar vector y, is a quantity of type r 2. Suppose
R(ABCDEF, XYZUVW, ABCDXY, ABXYZU),
that the general polynomial form of y is given by
I(AB) I(abXY), I(XY) I(ABa:b);
7.
~I(ABCDEF, XYZUVW,
y = cl VI (S)
+ ... + cmVm(S)
(7.4.3)
ABCDXY, ABXYZU),
pI(ABCXYZ, ABCDEX, AXYZUV), 1/JpI(ABCa:X, Aa:XYZ), I(AB) R(a:XYZU), I(XY) R(ABCDa:);
S.
= Y1 -
pR(ABCa:X, Aa:XYZ, AabcX), lPpR(Aa:bX),
~pR(ABCXYZ, ABCDEX, AXYZUV),
where the c1' ... ' cm are polynomial functions of the elements 11'.'" In of an integrity basis for functions of S which are invariant under C 3. Let
x = Xl
+ ix2
r 3.
We see from (7.3.21) that
the typical multilinear elements of an integrity basis for functions W( ¢, a, b, c, ... , a:,
I(AB) I(XYZUVW), I(XY) I(ABCDEF).
denote a quantity of type
b, c, ... ) are given by
196
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
Sect. 7.4] Invariant Functions of a Symmetric Second-Order Tensor: C 3
197
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, xS 2 , xST and xT 2. The expression
1. ¢; 2. ab;
(7.4.4)
(7.4.3) for y = Y1 -iY2 is then given by
3. abc where ¢, ¢', ... ; a, b, c, ... and a, b, c, ... denote quantities of types
r 2 and r 3 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
r2
(7.4.8)
r l'
and a quantity of type
r3
yields an invariant.
quantities V1(S), ... ,Vm(S) in (7.4.3) are quantities of type
where the ci = ci(1 1,... ,1 14 ). parts of (7.4.8), we obtain
Upon equating the real and imaginary
Since the
r 2, it
is seen
that xV 1(S), ... ,xV m (S) are also invariants. The quantities V 1(S), ... ,Vm(S) are obtained upon eliminating x from those elements of an
(7.4.9)
integrity basis for functions of x and S which are linear in x. We make the identifications
a, b, c
x,S,T;
a, b, c
x,8,T
(7.4.5)
where the c1, ... ,c5 are polynomial functions of the 11,... ,1 14 given in (7.4.7). The expression (7.4.2) for Y3' i.e., Y3 = W(1 1,···, 114 ), together with the expression (7.4.8) or (7.4.9) gives the general expression for y = F(S) which is invariant under C 3 .
where we have employed Table 7.8A and where
Consider next the problem of determining the general expression x=x1- ix 2'
S=S31- iS 23'
T=SII- S22+ 2iS I2'
x=xl+ ix 2'
8=S31+ iS 23'
T=SII- S22- 2iS 12'
for a symmetric second-order tensor-valued polynomial function U (7.4.6)
= R(S) of the symmetric second-order tensor S which is invariant
We see from (7.4.4) and (7.4.5) that the elements of an integrity basis
under the group C 3 . We see from Table 7.8A or from (7.4.5) and (7.4.6) that the quantities U33 , U 11 + U22 are of type r 1 and that U 31 -iU 23 , U 11 - U22 +2iU 12 are of type r 2. With (7.4.2) and
11' 12, ... for functions of S are given by
(7.4.8), we have immediately
11,... ,114=S33,SII+S22' S8, ST+8T, ST-8T, TT, S3+8 3 , S3-5 3 , S2 T + 5 2 T , S2 T -5 2 T, S T 2 + 8 1'2, S T 2 - 8 1'2, T 3 + T3 , T 3 -
(7.4.7)
T3.
The invariants x V 1(8), xV 2(8), ... appearing in an integrity basis for
(7.4.10)
198
[Ch. VII
Generation of Integrity Bases: The Crystallographic Groups
where d 1,... , eS are polynomial functions of the invariants 11,... ,1 14 given 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 relations IJ
Sect. 7.5]
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.
(7.4.11 )
199
Generation of Product Tables
These tables play a central role in the application of
Schur's Lemma to the generation of constitutive equations. The construction of the product table associated with a group A is facilitated if
and that there are 10 independent components of D. It has been shown
the typical multilinear elements of an integrity basis for A are given.
by Kiral, Smith and Smith [1980] that the linear combinations of the
We consider as an example the group D 3 . We list below the product
components of D which form quantities of types
r 1 and r 2 are given by
(7.4.12)
The general expression for the function D
= G(S)
which IS invariant
under C 3 is then given by DIll - 3D 122 = W 3(1 1,... ,1 14 ), D 113 + D 223
= W S(1 1,···, 114 ),
D 222 - 3D 211 = W4(1 1,... ,1 14 ), D 333
= W 6(1 1,... ,1 14 ),
D 133 - i D 233 = £1 S + £2T + £3 52 + £45 l' + £51'2,
.
DIll +DI22-1(D222+D211)
table for D 3 (see §6.4) where ¢,
Product Table: D 3
r1
¢
¢'
¢¢', ¢¢', al b i + a2 b 2
r2
¢
¢'
¢¢', ¢¢', al b 2 - a2 b I
r3
[:~] [:~]
l] [al¢'j [~b2] [a2~' J [alb2+ a2bl] [¢b ¢ b ' a2¢' , -¢ b -al'ljJ" alb a2b2 2
l '
l -
The typical elements of the integrity basis for functions of ¢,
(7.4.13)
(7.3.22) to be given by
-2
= gl S + g2 T + g3 S
+ g45 l' + g51'2, . -2 - -2 D 311 -D322+2ID312 = h 1S+ h 2T+h 3S +h4ST+hST where the f 1,... , h S are polynomial functions of the invariants 11'···' 114 given by (7.4.7) and where Sand T are defined by (7.4.6).
1.
¢;
2.
a1 b 1 + a2 b 2' 'ljJ 'ljJ' ;
3.
a2 b 2c2 - a1 b 1c2 - b 1cl a2 - clalb 2 , 'ljJ( al b 2 - a2 b 1) ;
4.
'ljJ(al b 1cl - a2 b 2c l - b 2c2a l - c2 a 2b l)·
(7.S.1)
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
200
Generation of Integrity Bases: The Crystallographic Groups
four entries
[eh. VII
are invariants are quantities of type fl. We see from (7.5.1) that
VIII (7.5.2)
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. 'l/J(al b 2 - a2 b 1) is an invariant and hence
Then, from (7.5.1),
GENERATION OF INTEGRITY BASES: CONTINUOUS GROUPS
8.1 Introduction In this chapter, we consider the problem of determining an
(7.5.3) a quantity of type f 2 . Similarly, we see from (7.5.2) that I == a1 b 1 + a2 b 2 is an invariant and that
IS
integrity basis for polynomial functions of vectors and/or second-order tensors which are invariant under a group A which is the threedimensional 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-
(7.5.4)
lographic group A is a finite group and hence has only a finite number r of inequivalent irreducible representations fl, ... ,fr .
We then deter-
that
mined the form of polynomial functions of n1 quantities of type f l' ... ,
J == 'l/J(al b 2 - a2 b 1) and K == a2 b 2c2 - al b l c2 - b l cl a2 - cl al b 2 are invariants. Hence, the quantities
n r quantities of type f r which are invariant under A where n1 ,... , n r are
IS
a
quantity
of
type
f 3.
We
observe
from
(7.5.1)
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 (7.5.5)
obtaining results of generality comparable to those given in Chapter VII. We thus restrict consideration to the determination of the form of
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 D 3 may be readily determined by inspection.
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
201
Generation of Integrity Bases: Continuous Groups
202
[Ch. VIII
of two symmetric second-order tensors 8 1, 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
Sect. 8.2]
203
Hamilton identity
M3 - (tr M)M2
+! [(tr M)2 - tr M2JM
(8.2.1)
-i [(tr M)3 - 3 tr M tr M2 + 2 tr M3JE3 = o.
tensors has been thoroughly discussed by Rivlin and Spencer in a sequence of papers.
Identities Relating 3 X 3 Matrices
A lucid outline of their work is given by Spencer
[1971]. We next follow the discussion of Smith [1968b] and consider the
Rivlin and Spencer have employed (8.2.1) and other identities which
problem of determining the multilinear elements of the bases for
may be referred to as generalized Cayley-Hamilton identities to
functions of n traceless symmetric second-order tensors B 1,... , B n and m skew-symmetric second-order tensors AI' ... ' Am which are invariant
generate the canonical forms of scalar-valued and second-order tensor-
under R3 .
second-order
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." n p ) which comprise the multilinear
elements of the bases required.
Given these sets of functions, we may
readily generate the remaining (non-multilinear) basis elements.
valued polynomial functions tensors
of three-dimensional skew-symmetric
AI' A 2 , ...
and
three-dimensional
symmetric
second-order tensors 8 1, 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
This
(8.2.2)
procedure is applied in §8.7 to generate the multilinear basis elements for functions of n traceless symmetric second-order tensors B 1 ,···, B n which are invariant under R3 . In §8.8, we generate the multilinear
are invariant under the group R3 (also the group 03). The number of linearly independent three-dimensional eighth-order tensors which are
basis elements for scalar-valued functions of m skew-symmetric second-
invariant under R3 is given by the number P 8 of linearly independent
order tensors AI' ... ' Am and n traceless symmetric second-order tensors
multilinear functions of the eight three-dimensional vectors Xl'···' X8
B 1,... , B n 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.
which are invariant under R3 . We have (see §4.7 (iv)) 27r
1 P 8 == 27r
f (eiB + 1 - e _. B 8(1 - cos B) dB == 91. 1 )
(8.2.3)
o
In §8.10.1 and §8.10.2, we consider the
generation of the multilinear basis elements for scalar-valued functions
There are then 105 - 91 == 14 linearly independent linear combinations
of vectors and second-order tensors which are invariant under the
of the isomers of 8· 1. 8. 1. 8· 1. 8.. which have all of their components
transverse isotropy groups T] and T2 respectively.
equal to zero. The isomers of 8· 1. 8· 1. 8· 1. 8· 1. form the carrier space
11
.
2 13 4 15 6 1718 .
11
2 13 4 15 6 17 8
for a reducIble representatIon of the symmetric group S8 whose decomposition is seen from (4.7.10) to be given by (8)
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-
+ (422) + (2222).
+ (62) + (44) +
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
[eh. VIII
Generation of Integrity Bases: Continuous Groups
204
correspondence between these tensors and the standard tableaux
Sect. 8.2]
Identities Relating 3 X 3 Matrices
205
where the notation (8.2.5) is employed. With (8.2.5), we have
associated with the frame [222 2] which are given by
12
12
12
12
12
13
13
34, 56 78
34,
35, 46 78
35, 47 68
36, 47 58
24, 56 78
57 68
13
13 25, 4 7 68
13
14
14
14
15
26, 4 7 58
25, 36
25, 37 68
26, 37 58
26. 37 48
57 68
8i1i3i5i7M1 M 2 M 3 ==0 i 2i 4i 6i 8 i 3i 4 i 5i 6 i 7i 8
24, (8.2.4)
25, 46 78
78
(8.2.7)
where the M i = [Mjkl are 3 x 3 matrices.
Upon expanding (8.2.5), we
obtain
M 1M 2 M 3 + M 2 M 3 M 1 + M 3 M 1 M 2 + M 1 M 3 M 2 + M 3 M 2 M 1
The tensor associated with the first standard tableau of (8.2.4) is given
+ M 2M 1M 3 -
by
- (M 3 M 1 + M 1M 3 ) tr M 2 - M 1 (tr M 2 M 3 - tr M 2 tr M 3 )
8· 1.
8· 1.
8· 1.
8· 1. 11 8
8· 1.
8· 1.
8· 1.
8· 1.
8· 1. 15 2 8· 1. 17 2
8· 1.
11 2
8~1~3~5~7 == 12141618
13 2
11 4
13 4
15 4 8· 1. 17 4
11 6
13 6 8· 1. 15 6
8· 1.
17 6
(M 1M 2 + M 2 M 1 ) tr M 3 - (M2 M 3 + M 3 M 2 ) tr M 1 (8.2.8)
- M 2 (tr M 3 M 1 - tr M 3 tr M 1 ) - M 3 (tr M 1M 2 - tr M 1 tr M ) 2
13 8 8· 1. 15 8 8· 1. 17 8
- E 3 (tr M 1 tr M 2 tr M 3 - tr M 1 tr M 2 M 3 - tr M 2 tr M 3M 1
(8.2.5)
- tr M 3 tr M 1M 2
+ tr M 1M 2M 3 + tr M 3M 2M 1 ) == o.
This is the generalized Cayley-Hamilton identity which was obtained in
If the tensor (8.2.5) is three-dimensional, it is a null tensor. For any of
this manner by Rivlin [1955]. If we set M 1 == M 2 == M 3 == M in (8.2.8), we recover the Cayley-Hamilton identity (8.2.1). Further identities
the 38 possible choices of values which i 1,... , is may assume, at least two
may be obtained upon applying the other null tensors in the set (8.2.6)
rows (and at least two columns) of the determinant will be the same
to
and the component will be zero.
M~. M~. M~. . For example, the identity 1314
1516
1718
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
12151618
1314
1516
(8.2.9)
1718
is equivalent to
8i1i3i5i7 811131516 811131417 811131416 8i1i 3i 4i 5, 8i1i2i5i7 .... , 811121516 .... , , .... .... , .... , .... , .... 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 13141718 13141618 12 5 7 8 12 6 7 8 12 5 6 8 12 4 6 8 12 4 7 8 (8.2.6) 8i1i 2i 4i 7 811121416 8i1i2i4i5 8i1i2i3i7 8i1i2i3i6 8i1i2i3i5 , , . 1. 1. 1. , .... , .... .... , .... , .... 14161718 14151618 14 5 7 S 13161718 1315161S 1315171S
8~1~3~4~7 M~. M~. M~. == 0
8~1~2~3~4
15161718
T
T
T
(M 2 - M 2 )(M 1 - M 1 ) M 3 T
+ (M 2 T
+ M T3 (M 2 T
T
T
M 2 )(M 1 - M 1 ) T
T
M 2 ) M 3 (M 1 - M 1 ) - (M 2 - M 2 )(M 1 - M 1 ) tr M 3 {
T }
T
T
-M3 tr M 1(M 2 -M2 ) -E3~r{(M1-M1)(M2-M2)M3} -trM3 tr{M 1(M 2 -M;)}]
(8.2.10)
= O.
206
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
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 M i (i == 1,2,3) as the sum of a skew-symmetric matrix Ai and a symmetric
Sect. 8.3]
207
tPl (AI ,A2,Aa ) = 2(A1A2Aa + AaA2A1) - Aa tr Al A2 - Al tr A A 2 a
= 0,
tP2( Al ,A2,Aa ) = Al A2Aa - AaA2A1 + A2Aa A1 - Al Aa A2 + AaA A 1 2
matrix Si. Thus,
- A 2A 1A 3 - 2E3 tr Al A 2 A 3 == 0,
1 T Ai == 2 (Mi - M i ), M·1 == A·1
The Rivlin-Spencer Procedure
+ 8·1
1( T) Si == 2 M i + M i '
(8.2.11)
tPa(A1,A2,Sa)
(i == 1,2,3)
= Al A2Sa - Sa A2Al + Al Sa A2 - A2Sa A1 + SaAl A2 - A2 A 18 3 - (AI A2 - A 2 A 1 ) tr 8 == 0, 3
where the Ai are skew-symmetric 3 x 3 matrices and the 8 i are symmetric 3 x 3 matrices. With (8.2.11), the identity (8.2.10) may be
tP4( Al ,A2,Sa)
= Al A2Sa + Sa A2Al + Al Sa A2 + A2Sa A1 + SaAlA2
+ A 2 A 18 3 -
written as
(AI A2 + A 2 A 1 ) tr 8 3 - 8 3 tr Al A 2
- E 3 (2 tr Al A 28 3 - tr 8 3 tr Al A 2) == 0, (8.2.12)
The identity (8.2.12) was obtained by Spencer and Rivlin [1962].
may apply each of the 14 null tensors in (8.2.6) to
We
Mf3i4Mr5i6MP7i8 to
tP5(Sl,S2,Aa)
- (8 1A3 + A38 1) tr 8 2 - A 3 (tr 8 182 - tr 8 tr 8 ) == 0, 1 2
tP6(Sl,S2,Sa) = SlS2 Sa + Sa S2S1 + S2 SaSl
+ SlSa S2 + SaSlS2 (8 28 3 + 8 3 8 2 ) tr 8 1 - (8 18 3 + S381) tr 8 2
obtain other identities. It has been shown by Rivlin and Smith [1975]
+ 8 2 S 18 3 -
that the resulting identities may be written as
- (8 18 2 + 8 28 1) tr 8 3 - 8 1(tr 8 28 - tr 8 tr 8 ) 3 2 3
- S2( tr SaSl - tr Sa tr Sl) - Sa( tr SlS2 - tr Sl tr S2) - E 3 (tr 8 1 tr 8 2 tr 8 3 - tr 8 1 tr 8 8 - tr 8 tr 8 8 2 3 2 3 1 (8.2.13)
(8.2.14)
= SlS2 Aa + AaS2S1 + SlAa S2 + S2AaSl + 8281A 3 + A38182 - (8 2A 3 + A 382) tr 81
- tr 8 3 tr 8 18 2 + 2 tr 8 18 28 ) == 0. 3
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
where the 4>1 (... ) , ... , 4>6 (... ) are defined by
these identities to generate the canonical forms of scalar-valued and
208
Generation of Integrity Bases: Continuous Groups
second-order
tensor-valued
functions
of
three-dimensional
[Ch. VIII
Sect. 8.3]
The Rivlin-Spencer Procedure
209
skew-
Upon introducing expressions of the form (8.3.3) into (8.3.1), we see
symmetric second-order tensors AI' A 2 , ... and symmetric second-order
that P is expressible as a polynomial in the traces of products formed
tensors Sl' S2' ... which are invariant under the proper orthogonal group
from the matrices Sl and S2. Similarly, we see that P(Sl' S2) may be
R3 . We note that the canonical forms obtained are also invariant under the full orthogonal group 03. We briefly outline their procedure as it
expressed as the sum of a number of products formed from the matrices
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-
8 1 and 82 together with E3 , with coefficients which are polynomials in traces of products formed from the matrices Sl and 8 . 2 We set M equal to Sl in (8.2.1) and multiply the resulting expression on the left by S2 to obtain
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
(8.3.4)
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
= [Sn J is expressible as
f3 p q r P == C + C· . 1.. .. S· . S· .... S· . (p,q, ... ,r == 1 or 2). n==l I1J 1 2J2 ... InJn I1J 1 12J2 InJn
L
We say that S2S~ is reducible, i.e., S2S~ is expressible as a polynomial in matrix products of degrees (p, q) in (8 1,8 2) where p:::; 3, q:::; 1, p + q<4 with coefficients which are polynomials in the traces of matrix
(8.3.1)
products.
We denote this by writing S2S~ ~ O. We take the trace of
(8.3.4) to obtain
A matrix-valued polynomial function P(Sl' S2)' P == [P ij ] is expressible as (3
p
q
(8.3.5)
r
p .. == C·· + "'" C··· . 1.. .. S· . S· .... S· . (p,q, ... ,r == 1 or 2). IJ IJ LJ IJ I1J 1 2J2 ... InJn I1J 1 12J2 InJn n==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 C ij , C ... inin and i1h C·· .. must be invariant under R3 . We see from §4.7 that these IJ ... InJn tensors must be expressible in terms of the outer products of Kronecker deltas. For example,
We say that tr S2S~ is reducible, i.e., tr S2S~ is expressible as a polynomial in traces of matrix products of 8 1 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
(8.3.3)
(8.3.6)
i
We say that the symmetric matrix-valued function SyS2 + S2 S equivalent to -8 182S1 and denote this by
is
210
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
(8.3.7) We first determine the canonical form for symmetric matrix-
Sect. 8.3]
4. Symmetric: Sf' Sf S2 + S2 s f, SyS2S1 + Sl S2Sy,
S~Sl S2 + S2S1s~, S~Sl + Sl s~, S~ ; Skew-Symmetric:
S~S2 - S2S~,
Sr S2S1 - Sl S2S1,
Sl S2 S1S2 - S2 S1S2 S1'
S~Sl S2 -
(8.3.14)
S1S~ - S~S1,
S2 S1S~,
S~Sl -
Sl S~.
(8.3.8)
We see immediately from (8.3.11) that Sf ~ 0, SfS2 ± S2 s f ~ 0,
2. 2. Symmetric: 821, 8 182 + 828 1, 82'
(8.3.9)
Skew-Symmetric: 8 182 - 828 1 . 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 8 1 and 82 are given by
3. Symmetric:
Sys~ + S~sy,
Sl S~Sl ' S2SrS2'
Sl S2 S1S2 + S2 S1S2 S1'
matrices 8 1 and 82 . The symmetric and skew-symmetric matrix-valued functions of total degree 1,2, ... in 8 1 and 82 are linear combinations of
1. Symmetric: 8 1 , 82 .
211
4 The 2 = 16 matrix products of total degree 4 in Sl and S2 are given by
valued and skew-symmetric matrix-valued functions of the symmetric
the matrix products of 8 1 and 82 listed below. We note that there are 2n distinct monomial matrix products of total degree n.
The Rivlin-Spencer Procedure
S~, Sr S2 + S2 Sr, Sl S2 S1' S2 S1S2' S~Sl + Sl S~, S~;
Skew-Symmetric: S1S2 - S2 Sr,
S~Sl - Sl S~ ·
(8.3.10)
S~Sl ± Sl S~ ~ 0, S~ ~ O.
We replace S2 by
S~ in
(8.3.6) and (8.3.7) to
obtain (8.3.15)
In similar fashion, we see that S2Sr S2 ~ - (Sr S~ + S~ Sr). We replace S3 by Sr in {8.2.14)6 to obtain Sr S2S1 + Sl S2 Sr + 2 S~ S2 + 2 S2 Sf
~0
(8.3.16)
Similarly, S~Sl S2 + S2 S1S~ S ~ O. Upon setting A3 = Sl S2 - S2 1 and S3 = Sl S2 + S2 S1 in (8.2.14)5 and {8.2.14)6 respectively, we find that and conclude that Sr S2S1 + Sl S2 Sr
We see from (8.2.1) upon replacing M by 8 1 and 82 in turn that
~ O.
(8.3.11)
Sl S2 S1S2 - S2 S1S2 S1 ~ -(Sr S~ -
We have, with (8.3.7),
Sl S2 S1 ~ -(S1 S2 + S2 S1),
S2 S1S2
~ -(S~Sl + Sl S~).
Sl S2 S1S2 + S2 S1S2 S1 ~ Sr S~ + (8.3.12)
With (8.3.10) ,... , (8.3.12), the basis elements of degree 3 are given by
3. Symmetric: SIS2+S2SI,
S~Sl +Sl S~;
Skew-Symmetric: SIS2 - S2 SI,
S~Sl - Sl S~ .
S~ S1), (8.3.17)
S~ Sr.
We see from (8.3.14) ,.. , (8.3.17) that the basis elements of degree 4 are given by
4. Symmetric: SIS~ + S~Sr ; (8.3.13)
(8.3.18)
Skew-Symmetric:
sys~ - S~Sr,
Sr S2S1- Sl S2Sr,
S~Sl S2 -
S2 S1S~.
212
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
Sect. 8.3]
213
The Rivlin-Spencer Procedure
The 25 == 32 matrix products of total degree 5 in 81 and 8 2 are given by
where ... indicates the 32 terms obtained from the preceding 32 terms
5. Symmetric: Sf, S1 S2 + s2 s 1, Sf S2S1 + SI s2 s f ' sI s 2s I,
sf, Sf, S~ are obviously reducible. Since all symmetric matrix-valued products of degree 5 are reducible, we see immediately that
sfs§ + s§sf, Sy S2S1S2 + S2 S1S2 Sy, SyS§SI + SI s§sy,
upon interchanging the subscripts 1 and 2. All terms containing s~, Sf,
SI S2 SyS2 + S2 SyS2S1' SI S2 S1S2 S1' S2 Sf S2""; (8.3.19) Skew-Symmetric: S1 S2 - S2 S1, Sf S2S1 - SIS2 Sf, SfS§ - S§Sf, Sy S2S1S2 - S2 S1S2 Sy, SyS§SI - SIs§sy, Proceeding in this fashion, we may readily show that all of the matrix
SIS2 SyS2 - S2 SyS2S1' ...
products in (8.3.21) are reducible.
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
We conclude that every symmetric matrix-valued polynomial in the matrix products of 81 and 8 2 which is of degree 6 or less is expressible in the form
the skew-symmetric terms are either reducible or equivalent to either
SyS~SI - SI S~SI or S§SyS2 - S2SyS~,
Thus, the basis elements of
degree 5 are given by
T(SI' S2)
= a OE 3 + alSI + a2 S2 + a3 Sy + a4(SI S2 + S2 S1)
(8.3.23)
+ a5S~ + a6(Sy S2 + S2 Sy) + a7(SI S~ + S~SI) + a8(SIS~ + SIS~)
5. Symmetric: None; (8.3.20)
where aO, ... ,a8 are polynomials in the traces of matrix products. Also every skew-symmetric matrix-valued polynomial in the matrix products
We consider next the 26 == 64 monomial matrix products of total degree six. These are listed below.
6.
A(SI' S2)
S~, Sf S2' S1 S2S1' S~S2Sy, SyS2S~, S1S~, Sy S2S1S2' SfS~SI' Sy S2SyS2'
Sy S2S1S2S1'
SIS§S~, S§S1, SyS~,
S2SyS2S1' S2SyS2Sy, S2S1S2Sf,
Sy S2S1S§,
S§Sy)
+ b6(SyS~SI -
sys~sy,
S2 S1 S2'
SIS2S1S2S1S2' 3 2 8 18 228 218 2 , 81 8 2281 8 28 1, 8 18 28 l' ...
= b O(SI S2 -
+ b3(SyS§ -
SI S2S1, S2 Sf,
SIS2SyS2' SIS2SyS2S1' SI S2S1S2Sy,
SyS~SIS2' SIS~SI' SIS2SIS~,
"of S1 and S2 which is of degree 6 or less is expressible in the form
(8.3.21 )
SIS2S1S~SI'
S2 S1) + b l (Sy S2 - S2 Sy)
+ b 4(Sy S2S1 -
SI S§SI)
SI S2 Sy)
+ b 7(S§Sy S2 -
+ b 2(SI S~ -
+ b5(S~SI S2 -
S§SI)
S2 S1S~)
S2SyS§)
(8.3.24)
where b O,... ,b 7 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
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
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
Sect. 8.3]
The Rivlin-Spencer Procedure
215
(8.3.5) together with relations such as
+ p T S1 is
expressible as a polynomial in symmetric matrix-valued matrix
tr Sl S2 S3 == tr S2 S3S1 == tr S3 S1S2'
products of degree 6 or less.
tr Sl S2S3S4 == tr S4 S1S2S3 == tr S3S4S1S2 == tr S2S3S4S1'
Sl p
+p
T
Applying (8.3.23) again, we see that
S1 is expressible in the form (8.3.23). Similarly SI P - p T S1 is
expressible in the form (8.3.24).
nomial in matrix products of degree 7 is expressible in the forms
(8.3.27)
tr Sl S2 S3S4 == tr S4S3S2S1
We may argue in this fashion to
establish that every symmetric and skew-symmetric matrix-valued poly-
tr SI S2 S3 == tr S3 S2S1'
in order to eliminate redundant terms. The basic invariants are seen to be given by
(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
1. tr SI' tr S2;
8, 9, ....
2. tr sy, tr SI S2' tr S2. 2' 3. tr S~, tr S~ S tr Sl S~, tr S3. 2' 2' 4. tr Sy S~.
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, ... ,b 7 in (8.3.23) and (8.3.24) are polynomials
(8.3.28)
in the traces of matrix products. We now determine a basis for these
We indicate the manner in which terms tr(P + P T ) may be excluded
quantities. Consider the invariants tr SiP (i == 1,2) where P is a matrix
from the basis (8.3.28) for the case where the P
product of total degree 5 in SI and S2. We have
and are listed in (8.3.14). We see from (8.3.5) that tr S2S~ ~ 0. Upon
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
tr S2 S1S2 S1 = tr Sl S2 S1S2
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 + p T) for the
cases where P is of degree 1,... ,5.
The terms P
+pT
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).
tr Sl S~Sl = tr S2SyS2 = tr
Thus, the
trace of any matrix product is expressible as a polynomial in traces of
We employ matrix identities as in (8.3.4) and
~ tr sYs~.
(8.3.29)
sYs~.
(8.3.30)
We also have, from (8.3.27),
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,....
are of degree 4
setting S2 = Sl in (8.3.5), we have tr Sf ~ 0. We note that tr Sy S2S1 =tr Sl S2 Sy = 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
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
+p T
From the discussion above, we see that
tr Sf ~ 0, tr
tr(S~S2 + S2S~) ~ 0,
tr(SyS2S1 + Sl S2Sy)
~ 0,
S~ ~ 0, tr(S~Sl + Sl S~) ~ 0, tr(S~Sl S2 + S2S1 S~) ~ 0,
tr(Sl S2 S1S2 + S2 S1S2 S1)
(8.3.31)
~ tr SyS~, tr Sl S~Sl = tr S2SyS2= tr sysy.
216
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
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.
Invariants of Symmetry Type (n 1 ... npJ
Sect. 8.4]
where P
= [P ji]
is non-singular.
217
Then, the matrices p- 1 D(s) P which
describe the transformation properties of the invariants J 1"'" J r under the n! permutations s of 5 n also form a r-dimensional matrix representation of the group 5 n 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
8.4 Invariants of Symmetry Type (uI." up) Let I 1,... ,I r be a set of linearly independent scalar-valued functions which are multilinear in the tensors B 1,... , B n and which are invariant under the group A.
Let s denote the permutation of the
of 5 n , the representation is said to be reducible. If not, the representation is irreducible. The number of inequivalent irreducible representations associated with 5n is equal to the number of partitions of n, i.e., to the number of solutions in positive integers of
numbers 1,... , n which carries 1 into iI' ... , n into in. Let s Ij (B 1 ,···, Bn ) be defined by (8.4.1)
n1
+ n2 + ... + n p == n,
(8.4.5)
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
We assume that, for each of the n! elements s of the group 5 n of permutations of the numbers 1,... , n, the invariants s Ij (B 1,... , Bn ) are
denote by (4), (31), (22), (211), (1111). The irreducible representation
(8.4.2)
of 5 n associated with the partition n1'" np of n is denoted by (n1'" n p ). The components of the character of the irreducible representation nl···np( s ) . (n1'" n p ) are denoted by X.
The n! matrices D(s) == [Dkj(s)] which describe the behavior of the
Let I ,... ,Ir be a set of invariants which are multilinear in 1 B and which form the carrier space for a r-dimensional reducible
expressible as linear combinations of the II"'" I r . Thus,
(j,k == 1,... ,r).
s Ij == IkD kj (s)
invariants I 1 ,... ,I r under the permutations of 5 n form a matrix representation of dimension r of the group 5 n , i.e., to every element s of 5 n there 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 representation {D(s)}. The character of the representation {D(s)} is denoted by
B
1"'" n representation {D(s)} of 5 n . We may determine a matrix P == [P ji ] so that the representation {P- 1 D(s) P} decomposes into the direct sum of
irreducible representations of 5 n . The invariants J i == Ij P ji which form the carrier space for the representation {P- 1 D(s) P} may thus be split 1·nto
sets J 1,... , Jp ., ... ., J q + I ,... , J r such that each set forms the carrier
space for an irreducible representation (n1 ... np) of 5n .
A set of
invariants which forms the carrier space for an irreducible repre-
x(e),···,x(s n.,),
x(s) == tr D(s)
(8.4.3)
where e denotes the identity permutation. Let
J. == I· p .. 1
J JI
(i ,j == 1,... , r )
(8.4.4)
sentation (n1'" n p ) of 5 n is referred to as a set of invariants of symmetry type (nl'.' np). The number anI'" n p of sets of invariants of symmetry type (nl ... np) arising from the II, ... ,Ir is seen from (2.5.14)
to be given by
218
Generation of Integrity Bases: Continuous Groups
X( s) == tr D( s)
[Ch. VIII
(8.4.6)
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 5 n , i.e., if sl and s2 have the same cycle structure (see §2.2), then
Sect. 8.4]
Invariants of Symmetry Type (n 1 ... npJ
Class 14 :
s == e;
Class 122:
s
Class 13:
s == (123);
s II' s 12 , s 13 == 13 , II' 12 .
Class 4:
s == (1234);
sI 1, sI 2, sI 3 == 13 ,1 2 , II·
Class 22:
s
== (12);
219
s 11' s 12 , s 13 == 11,12 , 13 . s II' s 12 , s 13
== 11,13 , 12. (8.4.11)
== (12) (34); s II' s 12 , s 13 == 11' 12 , 13 .
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
With (8.4.6) and (8.4.7), we have (8.4.8)
invariants left unaltered by the permutation s. Thus,·the quantities X, associated with the classes ;-yI == 14 , 12 2 , 1 3 " 4 22 are given by 3 , 1, 0 , 1, 3 respectively. Then, with (8.4.8) and the character table for 54 (Table
where h, is the order of the class I and where the summation is over the classes of Sn. The quantities X~l··· n p and h, may be found in the character tables for 5 n (n == 2, ... ,8) given in §4.9. Thus, in order to determine the number of sets of invariants of symmetry type (n1 ... n p ) contained in the set of invariants II'··' Ir which form the carrier space for a representation {D(s)} or, equivalently, the number of times the irreducible representation (n1 ... n p )
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··· n p appears in the first column of the character table for 5n and gives the dimension of the irreducible representation (n1 ... n p ), i.e., the number of invariants comprising a set of invariants of symmetry type (n1 ... n p ).
occurs in the decomposition of {D(s)}, we need only determine tr D(s)
In Chapter IV, we introduced Young symmetry operators which
for one permutation from each class I of 5n and then apply (8.4.8). For
were employed to generate sets of property tensors of symmetry type
example, consider the invariants
(n1··· n p ).
We may employ the same procedure to generate sets of
invariants of symmetry type (n1 ... n p ). Let n1 ... np denote a partition 11 == tr B 1B 2 tr B 3B 4 ,
12 == tr B 1B 3 tr B 2B 4 ,
(8.4.9)
13 == tr B 1B 4 tr B 2B 3
squares respectively arranged so that their left hand ends are directly
where B 1,... , B 4 are symmetric second-order tensors. We note that tr B·1 B·J == tr B·J B·.1
of the integer n. Associated with each partition n1 ... n p is a frame [n1··· n p] which consists of p rows of squares containing n1, ... ,n p
(8.4.10)
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
We list below, one element s of each class of S4 and the invariants
top to bottom.
sI 1,... ,sI 3 into which 11,... ,1 3 are carried by the permutation s.
frame corresponding to the partition n1 ... n p of n is given by the
The number of standard tableaux associated with the
220
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
dimension X~l··· np of the irreducible representation (n1'" n p ). 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 3 X21 == 2 and XlII Xe == 1' e e == 1 respectively. ·These standard tableaux are given by 123,12, 3
1 3, 2
1.
2
(8.4.12)
3
Let p(q) be a permutation which interchanges only the numbers in each
Sect. 8.4]
Invariants of Symmetry Type (n 1 ... npJ
221
where e denotes the identity permutation which leaves all integers unaltered. Let Y be the Young symmetry operator associated with a standard tableau corresponding to the partition n1". n p of n. The set of n! invariants (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 ... n p) provided that YI(B 1,···,Bn ) is not identically zero. We may choose X~l···np permutations e, s2' s3' ... such that the X~l'" n p invariants
row (column) among themselves. Let P =
LP, p
Q=
LC qq q
(8.4.13)
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 where P and
(8.4.14)
Q are defined by (8.4.13).
linearly
+ 12 + 13 + 23 + 123 + 132 (8.4.15)
Y(
~)
=
and
all
invariants
standard tableaux associated with the frame [n1'" n p ] and then determining the X~l··· n p permutations which send the first standard tableau into the remaining tableaux. invariant
For example, consider the
(8.4.18) where the Bi are symmetric second-order tensors. We note that tr BiB j == tr B·B·. In order to generate a set of invariants of symmetry type J 1 (21), we observe that
y(~ 2) = (e+12)(e-13) = e + 12-13-132,
Y( ~ 3 ) =
independent
For example, the Young
symmetry operators associated with the tableaux (8.4.12) are given by Y (1 2 3) == e
s Y I(B ,... , B ) are 1 n expressible as linear combinations of the invariants (8.4.17). The permutations s2' s3' ... are obtained (see §4.3) by listing the X~l··· n p are
(e + 13)(e -12) = e + 13 -12 -123, e - 12 - 13 - 23 + 123 + 132
II = Y (
12 ) I = (e + 12 - 13 - 132) tr B 1 tr B2B3
== tr HI tr B2B 3 + tr B 2 tr B 1B 3 - 2 tr B 3 tr B 1B 2 I:
(8.4.19) O.
222
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
Generation of the Multilinear Elements of an Integrity Basis
223
With the character table for 53 (Table 4.2), we see that the character
The standard tableaux associated with the frame [2 1] are 1 2, 3
Sect. 8.5]
(8.4.23) is that associated with the irreducible representation (21) of 53.
1 3 2
(8.4.20)
Application of the permutation (23) to the first tableau In (8.4.20) 8.5 Generation of the Multilinear Elements of an Integrity Basis
yields the second tableau. We have 12 = (2 3)Y(
In this section, we follow Smith [1968b] and outline a procedure
§2 ) I = (23 + 132 -123 -12) tr'B 1 tr B 2B 3
similar to those employed by Young [1977] and Littlewood [1944] (8.4.21)
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-
functions I(B 1,... ,B n ) invariant under a group A. Let I?n denote the number of linearly independent multilinear functions of B ,... , B which 1 n are invariant under A. These invariants form the carrier space for a reducible representation of the group 5n comprised of the n! pern denote the number of times the mutations of 1,2, ... , n. Let P n 1··· p irreducible representation (n1 ... n p ) appears in the decomposition of this
binations of II' 12 and are listed below. Table 8.1
enabling us to generate the multilinear elements of an integrity basis for
s II' s 12: S3
s
e
(12)
(13)
(23)
(123)
(132)
s 11
II
11
-1 1-1 2
12
-1 1-1 2
12
s 12
12
-1 1-1 2
representation or, equivalently, the number of sets of invariants of symmetry type (n1 ... np) arising from the set of I?n invariants. Let Qn
12
11
-1 1-1 2
11
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
D(e)
D(23)
=[ =[
~ ~
o ], 1
D(12) == [1 -1 ] , D(13) == [-1 0 -1 -1
1 ] , D(123) == [-1 o -1
1] , D(132) == [ 0 0 1
~l -1] .
be the. nun:ber of invariants multilinear in B 1,... , B n which are of the form 1~1 ... I~q where the 11'... ' Iq are elements of the irreducible integrity basis and i 1,... , iq are positive integers or zero. These Qn invariants also form the carrier space for a reducible representation of Sn. Let Qn1'" n p denote the number of times the irreducible representation (n1··· n p ) appears in the decomposition of this representation, i.e., the number of sets of invariants of symmetry type (n1 ... n p ) arising from the set of Qn invariants. We note that
(8.4.22)
-1
The character of the representation {D(s)} defined by (8.4.22) is given by
(8.5.1) p h were Xenl··· n is t h e d imension of the irreducible representation (n1'" np) and where the summation is over the set of all partitions We give methods below for determining I?n P n n , 1'·' p' Qn, Qnl." n p ' Let us assume for the moment that we are able to determine these quantities. We proceed as follows. nl'" np of n.
x(e), X(12), X(13), X(23), X(123), X(132) == 2, 0, 0, 0, -1, -1. (8.4.23)
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
3. We compute the number 1?3 of linearly independent invariants
which are linear in B 1. Suppose that I? 1 == p and that the p invariants are given by 11 (B 1 ) ,... , I p (B 1). These invariants are of symmetry type
which are multilinear in B 1,B 2,B 3 and the numbers P3,P21,P111 of sets of invariants of symmetry types (3), (21), (111) respectively where
(1) and give the set of integrity basis elements of degree one in B 1. 2. We compute the number 1?2 of linearly independent invariants which are multilinear in B 1 and B 2 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 B 1 and B 2 and which arise as products of elements of the
1P3=X~P3+X~IP21+X~11P111 =P3+ 2P 21 +P 111 ·
The Q3 invariants which are multilinear in B 1, B 2 , B 3 and which are products of integrity basis elements of degrees 1 and 2 are given by (i)
the p3 invariants (8.5.6)
integrity basis of degree one. These are given by which may be split into ( Pj21 sets of Ii (B 1) Ij (B 2 )
(i,j == 1,... , p).
(8.5.2)
1 invariants of symmetry
type (3), ip(p2 -1) sets of X~ = 2 invariants of symmetry type (21)
(ii) (8.5.3)
the 3pq invariants Ii (B 1) J j (B 2 , B 3 ), Ii (B 2 ) J j (B 3 , B 1), Ii (B 3 ) J j (B 1, B 2 )
and the ( ~ ) invariants
We note that p2 = ( p!1 ) + (
X~ =
and ( ~ ) sets of X~11 = 1 invariants of symmetry type (111);
1 These invariants may be replaced by the ( P! ) invariants
(i == 1,... , p; j == 1,... , q)
Ii (B 1) Ij (B 2 ) - Ii (B 2 ) Ij (B 1),
~).
(i,j == 1,... , p; i<j).
(8.5.5)
(8.5.4)
(8.5.7)
which may split into pq sets of invariants of symmetry type (3) and pq sets of symmetry type (21);
The invariants (8.5.3) are unaltered
under interchange of B 1 and B 2 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 B 1 and B 2 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 - Q 2 and P 11 - Q 11 sets of invariants of symmetry 2 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 - Q 11
== rand
that the elements of the integrity basis which are bilinear in B 1, B 2 and of symmetry types (2) and (11) are given by J 1(B 1, B 2 ),···, J q (B 1 , B 2 ) and K 1(B 1, B 2), .. ·, K r (B 1, B 2 ) respectively.
(iii)
the 3pr invariants I i (B 1) Kj (B 2 , B 3 ), Ii (B 2) K j (B 3 , B 1), Ii (B 3 ) K j (B 1, B 2 )
(i == 1,... , p; j == 1,... , r)
(8.5.8)
which may be split into pr sets of invariants of symmetry type (21) and pr sets of invariants of symmetry type (Ill). Thus, we have Q 3 == ( p+2 3 ) +pq,
{ 2 -l)+pq+pr, Q 21 ==31 PP
Q 111 = (~)+pr.
(8.5.9)
The integrity basis then will have P 3 - Q3' P 21 - Q 21 and PIll - Q 111
226
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
Sect. 8.6]
Computation of lPn' P n
1"·
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
n' Qn' Q P n1 ··· np
227
In (8.6.1), 'I' '2' "·"n gives the cycle structure of a class, of permutations 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·" n p
determine the stage at which the iterative procedure may be termi-
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"· n p are listed in the
nated. The above procedure is formal in the sense that we assume that
character tables for Sn (n = 2,3, ... ) given in §4.9.
particular problem, we must give an argument which will enable us to
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
case where n1". n p = n, we have (8.6.1), we then have n! ¢n
which relate the invariants so that the number of linearly independent multilinear invariants of degree n in B 1,... , B n which are products of integrity basis elements is less than Qn. The existence of syzygies does
=
X!y = 1
For the particular
for all classes of Sn.
E, h, sll s~2 ... sn'n.
With
(8.6.3) ,
It has been shown by Schur [1927] that the quantities
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
¢n1 ¢n1.. · n p
=
considered.
¢n1+1 4>n 2
4> n 2-1
¢nl+p-1 4>n 2+p-2
(8.6.4)
4>n p
where
8.. 6 Computation of Pn, Pnl.... np' Qn, Qnl"" np Let b l ,... , b s denote the independent components of a tensor B chosen from the set of tensors B 1,... , B n , each of which transforms in the same manner under the group A. Let the transformation properties
=
4>4
¢5
4>2
o
1
(8.6.5)
of the column vector [b 1,... , bs]T under the group A be defined by the sdimensional matrix representation {S(A) }. Corresponding to each
The number IPn of linearly independent functions which are multilinear in B1,... , Bn and which are invariant under the group A is obtained by
partition n1 ... np of n, we define the quantity
taking the average over the group A of the quantity sr. We denote this by (8.6.1)
(8.6.6)
where (8.6.2)
where M.V. stands for mean value.
The number P n1··· n p of sets of
228
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
invariants of symmetry type (n1 ... n p ) arising from the I?n invariants multilinear in B 1,... , B n is obtained by taking the average over the group A of the quantity
Sect. 8.6]
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.
(8.6.7)
229
n Computation of lPn' P n n' Qn' Qn 1·" p l · · · P
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
If A is a continuous group, the averaging process indicated in (8.6.6)
characters of irreducible representations to determine the decomposition
and (8.6.7) is accomplished by integrating over the group manifold.
of the reducible representation S, the value of whose character for the
The number Qn of functions which are multilinear in B 1,···, B n , 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(B 1, B 2 ) and J(B 1, B 2 ) where
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 . S·IS gIven . b y 4' 1 '""" nl·" n p where the sumth e representatIon L..J h ,
X, X,
mation is over the five classes of 5~. ' We collect the results in tabular form below. Decomposition of Representations S, T, U
Table 8.2
(8.6.8)
Class
14
122
13
4
22
From these invariants, we obtain the following three sets of invariants
Class member
e
(12)
(123)
(1234)
(12)(34)
which are multilinear in B 1,... ,B 4 :
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
3
1
0
-1
-1
2
0
-1
0
2
X~l X~2
We then have Q 4 == 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) (4)
+ (22)
and (4)
+ (22)
Q31 = 1 and Q22 == 3.
respectively.
+ (31) + (22),
Thus, we find that Q4
= 3,
230
Generation of Integrity Bases: Continuous Groups
[eh. VIII
The procedure indicated above can become tedious if the
Sect. 8.6]
Table 8.3
Computation of lPn' P n
1"·
n' Qn' Q n
n
pl·" P
231
Decomposition of (mI ... m p ) . (n1 ... nq)
number of invariants comprising the carrier space of a reducible representation is large. In practice, it is usually preferable to employ results due to Murnaghan [1937]. We note that the invariants I(B 1, B 2) and J(B 1, B 2) given by (8.6.8) form carrier spaces for irreducible representations (2) and (2) respectively. The set of invariants (8.6.9) forms
2.
(1) . (1) == (2)+(11)
3.
(2) · (1) == (3)+(21),
4.
(3)· (1) == (4)+(31), (21)· (1) == (31)+(22)+(211), (111) . (1) == (211 )+(1111), (2)· (2) == (4)+(31 )+(22),
the carrier space for a reducible representation which we refer to as the
(11)· (1) == (21)+(111)
(2) . (11) == (31)+(211),
(11)· (11) == (22)+(211)+(1111)
product of the representations (2) and (2) and denote, by (2) . (2). The decomposition of these product representations is discussed in §4.6.
5.
+ 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
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
(211)· (1) == (311)+(221)+(2111),
(1111)· (1) == (2111)+(11111), (3)· (2) = (5)+(41)+(32), (21) · (2) = (41)+(32)+(311)+(221), (111)· (2) = (311)+(2111),
decompositions of (mI ... mp) . (n1 ... nq) for all cases such that m1 + ...
sentations (2) and (2) and denote by (2) x (2). The decomposition of
(31)· (1) == (41)+(32)+(311),
(22) . (1) == (32)+(221),
This problem has been considered by Murnaghan [1937] who lists the
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-
(4)· (1) == (5)+(41),
(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),
procedure leading to Table 8.2 or may be found in Murnaghan [1951].
(31) . (2) = (51 )+(42)+(411 )+(33)+(321),
We note that some caution is required when determining the decom-
(22) . (2)
position of a symmetrized product of the representations (n1". n p ) and
= (42)+(321 )+(222), (211) . (2) = (411)+(321)+(3111)+(2211),
(n1 .. · n p ). For example, the quantity a1 b 1 forms the carrier space for a representation (2) of S2 since a1 b 1 is unaltered under interchange of a and b. The carrier space for the symmetrized product (2) x (2) of this
(1111). (2) == (3111)+(21111),
representation is formed by the single quantity a1 b 1c1 d 1 which is of symmetry type (4). In this case, we have (2)x(2) ==(4) rather than
(211)· (11) = (321)+(222)+(3111)+(2211)+(21111),
(4) + (22) as listed in Table 8.4.
(3) . (3) = (6)+(51 )+(42)+(33),
We note that the dimension of a
(3)· (3) = (6)+(51)+(42)+(33),
= (51)+(411), (31)· (11) = (42)+(411)+(321)+(3111), (22)· (11) = (32)+(321)+(2211),
(4)· (11)
(1111)· (11) = (2211)+(21111)+(111111),
reducible representation must be equal to the sum of the dimensions of
(3)· (21) = (51)+(42)+(411)+(321),
the irreducible representations into which it is decomposed. This serves
(21) . (21) = (42)+(411 )+(33)+2(321 )+(3111 )+(222)+(2211), (21) . (111) == (321)+(3111)+(2211)+(21111), (111)· (111) == (222)+(2211)+(21111)+(111111)
as a check and should enable us to avoid errors in degenerate cases such as that mentioned above.
(3)· (111) = (411)+(3111),
232
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
Table 8.4
Sect. 8.7]
Traceless Symmetric Second- Order Tensors: R3
as isotropic functions.
233
This problem differs from that considered by
Spencer and Rivlin [1959 a, b] and Spencer [1961] only in that we
4.
(2) x (2) == (4)+(22),
(11) x (2) == (22)+(1111)
6.
(3) x (2) == (6)+(42),
(2) x (3) == (6)+(42)+(222),
impose the restriction that tr Bi == 0 (i == 1,... , n). We borrow from the discussions of Spencer and Rivlin the results that
(11) x (3) == (33)+(2211)+(111111) (i)
8.
(2) x (4) == (8)+(62)+(44)+(422)+(2222),
less and are of the form tr BiBj ... B k ;
(11) x (4) == (44) + (3311 )+ (2222)+ (221111 )+ (11111111 ) 10.
(2) x (5) == (10)+(82)+(64)+(622)+(442)+(4222)+(22222),
(ii)
the multilinear terms appearing in the general expressions for A(B 1,... , B n ) and S(B 1,... , B n ) are of the forms
(11)x(5) == (55)+(4411)+(3322)+(331111)+(222211) + (2 2 111111 )+ (1111111111 ) 12.
the multilinear elements of an integrity basis are of degree six or
(2) x (6) == (12)+(10,2)+(84)+(822)+(66)+(642)+(6222) +(444)+(4422)+(42222)+(222222)
(BiBj
B k - B k ··· BjBi ) tr B£
Bm ,
(BiBj
B k + B k .. · BjBi ) tr B£
Bm
(8.7.1)
respectively where the BiBj ... B k ± B k ... BjBi are of degree five or less; 8.7
Invariant Functions of Traceless Symmetric Second-Order Tensors: R3
In this section-, we employ the procedure of §8.5 to generate the
(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,
multilinear elements of an integrity basis for functions of an arbitrary number of three-dimensional symmetric second-order traceless tensors
tr B 1B 2B 3 == tr B 2B 3B 1 == tr B 3B 1B 2 ,
B 1,... , B n which are invariant under the three-dimensional proper ortho-
tr B 1B 2B 3 == tr B 3B 2B 1.
gonal group R3 . We then generate the multilinear elements appearing
(8.7.2)
in the general expressions for skew-symmetric second-order tensor-
The results (i) and (ii) above are critical in that they indicate when the
valued functions A(B 1,... , B n ) and for traceless symmetric second-order tensor-valued functions S(B 1,... ,B n ) which are invariant under R3 .
iterative procedures to be employed may be terminated.
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
We first consider the problem of generating the multilinear elements of an integrity basis for functions of the traceless symmetric second-order tensors B 1,... , B n which are invariant under R3 . The matrix which defines the transformation properties under A of the column vector [B 11 , B12 , B 13 , B 22 , B 23 , B 33 ]T whose entries are the six independent components of a three-dimensional symmetric secondorder tensor B' is the symmetrized Kronecker square A(2) of A.
234
Generation of Integrity Bases: Continuous Groups
[eh. VIII
Suppose that A is the matrix corresponding to a rotation through B radians about the x3 axis, i.e.,
A =
cos B sin B
o
-sin B cos B
o
o
0
~(tr
(8.7.3)
1
(8.7.4)
l1r J
(8.7.8)
o
B
= B' -1 (tr B')E3,
is invariant under the group R3 .
= e2irB + eirB + 1 + e- irB + e- 2irB .
(8. 7.9)
The quantities 1P 1,... , 1P6 may be computed from (8.7.7) and are given by
1P2 = 1, 1P3 = 1,
1P5 = 16,
IP4 = 5,
1P6 = 65.
(8.7.10)
We list in Table 8.5 the mean values over the group R3 of the (8.7.5)
where B is a symmetric traceless second-order tensor, i.e., tr B = O. We
= Bi i
P Ul .. · Up =
1P 1 = 0,
We may set
observe that tr B'
The number Pn1 ... np of sets of invariants of symmetry type (nl ... np) arising from these IP n invariants is given by
Sr
A)2 +~tr A 2
1(tr B')E3,
235
3
The f the Sr where
= ! (eiB + 1 + e-iB )2 +! (e2iB + 1 + e- 2iB ) = e 2iB + eiB + 2 + e- iB + e- 2iB . B' = B +
Traceless Symmetric Second-Order Tensors: R
271"
We see from (4.4.17)1' (4.4.18) (or (5.2.10)) and (8.7.3) that tr A(2) =
Sect. 8.7]
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
l1r J
quantities
values of p, ... , q such that p + ... Table 8.5
cos B) dB, for all positive
+ q ~ 6.
Mean Values over R3 of p ... q
p ... q M.V. (p... q)
1 0
p ... q M.V. (p ... q)
31 1
p ... q 31 4 M.V. (p ... q)
2 1
~ 3
~1 6
1 1
3 1
21 1
t 1
4 1
2r 3
1 5
5 1
41 2
32 3
2t 9
1 16
6 2
51 2
4 2 5
321 9
3Y 14
21 36
r 65
(8.7.6) The number IPn of linearly independent scalar-valued functions which are multilinear in B 1,... , B n and which are invariant under R3 is seen with (2.6.19)2' (8.6.6) and (8.7.6) to be given by
Pu
= 2~
o
41 5
~ 5
~ 21
We now generate the typical multilinear elements of the integ-
271"
Jsf(l-cosB)dB.
p ... q M.V. (p ...q)
(8.7.7)
rity basis. We list in Table 8.6 below the quantities Pn1 ... np' Qnl". n p and X~l···Up for those u1."up for which P Ul ... UP f:. o. The P Ul ... uP
Generation of Integrity Bases: Continuous Groups
236
[Ch. VIII
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 ... n p ).
The values of X~I"· np are
Sect. 8.7]
Traceless Symmetric Second-Order Tensors: R 3
We see as in §8.4 or §8.6 that the invariants (8.7.13) form a set of invariants of symmetry type (4) is
given in §4.9. The computations yielding the QnI... n p are indicated below. We observe that 1P 1 = 0 so that there are no invariants of
representations (2) and (2).
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 = Q 11 = o. We then have P 2 - Q 2 = 1 set of invariants of symmetry type (2) appearing in the integrity basis. This set is com-
+ (22).
We also note (see §8.6) that
the invariants (8.7.13) form the carrier space for a representation which
found in the first column of the character tables for 5n (n = 2,3, ... )
degree one.
referred
to
as
the symmetrized product
of the irreducible
This is denoted by (2) x (2) and, from
Table 8.4, we have that (2) x (2) = (4)
+ (22).
Thus, Q4 = Q 22 = 1, Q31 = Q 211 = Q 1111 = o. The integrity basis will then contain P 22 - 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)
prised of X~ = 1 invariant which is given by
(8.7.14)
Y(12) tr B 1B 2 = (e + 12) tr B 1B 2 = tr B 1B 2 + tr B 2B 1
(8.7.11)
= 2tr B 1B 2; (2) where we have noted that tr B 1B 2 = tr B 2B 1. The designation (2) in (8.7.11) indicates that tr B 1B 2 forms a set of invariants of symmetry type (2). We next observe that 1P 3 = 1; P 3 = 1, P 21 = PIll = O. There are no invariants of degree three arising as products of invariants Hence, Q3 = Q3 = Q 21 = Q111 = O. The integrity bases will then contain P 3 - Q3 = 1 set of invariants of symmetry type
of lower degree.
(3) which consists of a single invariant since X~ Y(123) tr B 1B 2B 3 = (e + 12 + 13
= 1.
+ 23 +
This is given by
123 + 132) tr B 1B 2B 3
=6trB 1B 2B 3; (3) where we have employed (8.7.2).
237
We next see that IP4 = 5;
(8.7.12) P4 = 1,
We further observe that 1P 5 = 16; P s = P 41 = P 32 = P 221 = P11111 = 1; 1P 6 = 65; P 6 = P 222 = 2, P42 = 3 and P 321 = P 3111 = 1. The multilinear invariants of degree 1,1,1,1,1 in B 1,... , BS and of degree 1,1,1,1,1,1 in B 1,... , B 6 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 representations into which these representations may be decomposed are given in Tables 8.3 and 8.4 and are also listed. The quantities QnI ... n p for
n1 +... + np = 5,6 appearing in Table 8.6 may then be immediately determined.
P 22 = 2, P 31 = P 211 = P 1111 =0. There are three linearly independent multilinear invariants which arise as products of invariants of the
5.
tr B 1B 2 tr B3B4 B5 , (~) = 10,
(2)· (3) = (5) + (41)
form (8.7.11). These are given by
6.
tr B 1B 2 tr B 3B 4 tr BSB6 ,
(2) x (3) = (6)
tr B 1B 2 tr B 3B 4 , tr B 1B 3 tr B 2B4 , tr B 1B 4 tr B 2B 3. (8.7.13)
IS,
+ (32);
+ (42) + (222); (8.7.15)
238
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
Ij(B1,B2,B3,B4) tr BSB 6 , 30,
(2)· (22) = (42)
tr B 1B 2B 3 tr B4B SB 6 ,
X
10,
(3)
(2) = (6)
Sect. 8.7]
+ (321) + (222);
+ (42).
S.
JO(B1,B2,B3,B4,BS)=Y
1 2 3
trB1B2B3B4BS'
4
(11111);
S We see from (8.7.1S) that QS = Q41 = Q32 = 1, Q 6 = 2, Q42 = 3, Q321 = 1, Q222 = 2. The remaining Qnl ... np (n1 + ... + n p = S or 6) are zero. We list the results in Table 8.6. Table 8.6
Scalar-Valued Invariant Functions of B 1,···, B n : S 41
32 221 11111 6
R3
2
3
4
22
P nl .. ·np
1
1
1
2
1
1
1
1
1
2
3
1
1
2
Qnl···np nl .. ·np Xe
0
0
1
1
1
1
1
0
0
2
3
1
0
2
1
1
1
2
1
4
S
S
1
1
9
16
10
S
42 321 3111 222
(8.7.17)
J 1(B 1, B 2, B 3 , B 4 , BS)'···' J S(B 1, B 2, B 3 , B 4 , BS) = [e, (45), (23), (23)(45), (2453)J
6.
n1 .. · n p
239
Traceless Symmetric Second-Order Tensors: R3
V( i ~) tr B1B2B3B4B5'
(221)j
K 1(B 1,···, B 6),···, K 10 (B 1,.. ·, B 6) = [ e, (34), (354), (3654), (234), (2354), (23654), (24)(35),
(24)(365), (25364)J
V(
i
2 3 ) tr B1B2B3B4B5B6' (3111).
We next indicate the manner in which one may generate the We see with Table 8.6 that P n1 ... 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 (2), (3), (22), (221), (11111), (3111).
(8.7.16)
We may then apply the procedure of §8.4 to generate the typical multilinear elements of an integrity basis for functions of traceless symmetric second-order tensors B 1, B 2 , ... which are invariant under R3 . These are comprised of the sets of invariants listed below. tr B 1B 2,
(2) ;
3.
tr B 1B 2B 3,
4.
I1(B1,B2,B3,B4)'
(3) ;
(23) ] V (
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 B 1,... , 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(B 1,... , B 6 ), ... ,
K10 (B 1,···, B 6)· Consider the reducible representation of the group
53 = { e, (12), (13), (23), (123), (132) }
2.
= [e,
non-linear elements of an integrity basis given the typical multilinear
(8.7.18)
defined by the matrices D(e), ... , D(132) which describe the manner in I2(B 1,B 2, B 3, B 4 )
~ ~)
tr B1B2B3B4, (22);
which the invariants Ki (B 1, B 2, B 3, B 4, B S' B 6) (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 K 1,... , K . The 10
240
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
Sect. 8.7]
241
Traceless Symmetric Second-Order Tensors: R3
(8.7.22)
elements of one set are symmetric in B 1, B 2 , B 3 , i.e., they are invariant under the group (8.7.18). Each of these invariants forms a carrier space The invariants comprising the
Similarly, the number of linearly independent linear combinations of
second set form the carrier space for a reducible representation of 53
the K 1,... , K 10 which are symmetric in B 1 , B 2 , B 3 , B 4 is given by the number of times the identity representation appears in the decom-
for the identity representation of 53.
which does not contain the identity representation.
These invariants
will vanish identically when we set B 1 == B 2 == B 3 in them. The basis elements of degrees 3,1,1,1 in B 1, B 4 , B 5 , B 6 are obtained upon setting B 1 == B 2 == B 3 in the first set of invariants which are symmetric in B 1 , B 2 , B 3 . The number of linearly independent linear combinations of the K 1 ,... , K 10 which are symmetric in B 1 , B 2 , B 3 is equal to the number of times the identity representation occurs 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 1 '"" 1 ( 10 - 6·2 + 8 ·1 4! ~ X3111() s == 24
+ 6·0 -
3 ·2) == 0
(8.7.23)
position of the representation of the group (8.7.18) whose carrier space
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
is formed by the set K 1,... ,K 10 of invariants of symmetry type (3111). This number, is given by
comprised of 1, 6, 8, 6, 3 permutations respectively. Since B 5 and B 6 are not affected by permutations of the subscripts 1, ... ,4, the values of
(8.7.19)
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,
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 ,
1, 0, -2 of the character of the irreducible representation (3111) of 56 (see Table 4.5) corresponding to the classes 16 , 14 2, 133, 124, 1222 of
142 and 133 of 56 respectively. The values X3111 (s) of the character of
56. In similar fashion, we may show that there are no basis elements of
the irreducible representation (3111) are found in the character table for
degrees 6 in B I ; (5, 1), (4,2), (3,3) in B I , B2; (3,2, I), (2,2,2) in B I , B2, B3 ; there are 4 basis elements of degrees (2,1,1,1,1) in B 1 , B 2 , B3 , B4 ,
S6 (Table 4.5). The invariant which is symmetric in B 1, B 2 , B 3 is given by (8.7.20)
B5 and 1 basis element of degrees (2,2,1,1) in B 1 , B 2 , B3 , B4 which arise from the K 1,... K 10 . 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.
sInce
We list below
the typical multilinear elements of the integrity basis together with the sY
(
~23)_ (~23) 5 - Y 5 6
typical (8.7.21)
non-linear
identification process.
6
for all s belonging to the group (8.7.18). The element of the integrity basis of degree 3,1,1,1 in B 1, B 4 , B 5 , B 6 is then given by
basis
2.
Two tensors: One tensor:
elements
obtained
from
them
by
the
242
3.
4.
5.
6.
Generation of Integrity Bases: Continuous Groups
[eh. VIII
Sect. 8.7]
Traceless Symmetric Second-Order Tensors: R3
243
left to right in each of the rows. The number of linearly independent
Three tensors:
tr B 1B 2B 3;
Two tensors:
tr ByB2 ;
One tensor:
tr B 3. l'
Four tensors:
2tr B 1B 2B 3B4 -tr B 1B 3B 2B 4 -tr B 1B 2B4B 3,
tifying tensors is given by the number of standard tableaux obtained
2 tr B 1B 3B 2B4 - tr B 1B 2B 3B4 - tr B 1B 3B4B 2;
upon inserting the integers (1,1,2,3,4,5), (1,1,1,2,3,4) and (1,1,2,
invariants of degree 2,1,1,1,1 in B1,B2,B3,B4,B5' of degree 3,1,1,1 in B 1, B 2, B3, B4 and of degree 2,2,1,1 in B 1, B 2, B 3, B4 which may be obtained from the set of invariants K i (B l ,... , B 6) (i = 1,... ,10) of symmetry type (3111) appearing in (8.7.17) upon appropriately iden-
Three tensors:
tr ByB 2B 3 - tr B I B 2B I B 3 ;
Two tensors:
tr ByB~ - tr B I B 2B I B 2 ;
2,3,4) respectively into the boxes of the frame [3111] associated with the partition 3111. Thus, we obtain the four standard tableaux
(8.7.24)
Five tensors:
Ji (B 1, B 2, B 3, B 4 , B 5)
(i = 0, 1,2,3,4,5);
Four tensors:
Ji (B 1, B 1, B 2, B 3, B4 )
(i = 1,2);
Three tensors:
J1(B 1, B 1, B 2, B 2, B 3);
Six tensors:
Ki(B1,B2,B3,B4,B5,B6)
(i
Five tensors:
Ki(B1,B1,B2,B3,B4,B5)
(i = 1,2,3,4);
Four tensors:
K1(B1,B1,B1,B2,B3,B4)'
1 1 2 3 4 5
,
1 1 3 , 2 4 5
1 1 4 , 2 3 5
115 2 3 4
upon inserting 1,1,2,3,4,5 in the frame and the standard tableaux
= 1, ... ,10);
K (B 1, B 1, B 2, B 2, B3, B4) + K 2(B 1, B 1, B 2, B 2, B3, B4 )· 1
111 2 3 4
and
112 2 3 4
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 (B 1,... ,B6) by Bl,B1,B2,B3,B4,B5'''.' and a single linearly independent invariant of degree 2,2, 1, 1 In of basis elements of various degrees in B 1, B 2 ,... which may be obtained from a set of invariants of symmetry type (n1 ... np) upon identifying
B 1,B 2 ,B3,B4 upon replacing B 1,B 2,... ,B6 in the K i (B 1,... ,B6) by B 1, B 1, B 2, B 2, B 3 , 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
certain of the tensors. Consider, for example, the frame associated with
the frame [3111].
the partition 3111 of 6.
The irreducible representations of the group R3 are of dimensions 1,3,5,7, .... The independent components of a vector or a skew-symmetric second-order tensor, a traceless symmetric second-order tensor, a traceless symmetric third-order tensor,... form the carrier spaces for
We now outline a graphical method for determining the number
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
Generation of Integrity Bases: Continuous Groups
244
[Ch. VIII
irreducible representations of R3 of dimensions 3,5,7, ... respectively. The values of the characters of these representations corresponding to
Sect. 8.7]
245
Traceless Symmetric Second-Order Tensors: R3
R n1 ... np
= 2~
the class of rotations through fJ radians are given by
27r
JcPnl ... np(eiO + 1 +e-iO)(l-cosO)dO
(8.7.28)
o
where the
Let
Snl". np denote the number of sets of skew-symmetric second-order (8.7.25)
tensor-valued functions of symmetry type (nl". np) which are invariant under R3 and which arise from the products of skew-symmetric second-
We next generate the general expression for a skew-
order tensor-valued functions of degree m < n in the B·1 and invariants
symmetric second-order tensor-valued function which is invariant under
of degree n - m in the B i . We list below in Table 8.7 the quantities Rn1 np ' Snl... np and X~l ..·np for those nl ... n p for which
respectively.
R3 and multilinear in the traceless symmetric second-order tensors B l , B2 , ... , Bn . The 5n independent components of the tensor B~ · ... B!1. form the carrier space for a 5n _dimensional reducible IlJl InJn representation of R3 whose character corresponding to a rotation through 0 radians is given by sf = (e2iO + eiO + 1 + e-iO + e-2iO The
functions of symmetry type (nl." n p ) is found in the first column of the character table for the symmetric group Sn. The computations yielding
three independent components of a skew-symmetric second-order tensor
the Snl." np are given below.
t.
R nl np (8.7.28).
and nl +... + n p ~ 5. The R nl ... n p are computed from The number X~l·" np of functions comprising a set of
f::. 0
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 IRn
= l1r
27r
J
sf(eiO + 1 + e-iO)(l- cos 0) dO,
o
(8.7.26)
The quantities 1R 1,... ,1R 5 may be computed from (8.7.26) and are given by
Table 8.7
Skew-Symmetric Tensor-Valued Functions of B 1,... ,Bn : 11
21
111
31
211
41
32
311
221
Rnl· .. n p
1
1
1
2
2
2
2
3
1
1
Snl···np nl···np Xe
0
0
0
1
1
2
1
3
1
1
1
2
1
3
3
4
5
6
5
4
n1 .. · n p
R3
2111
From Table 8.7, we see that Rnl ... np -Snl ... n p = 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
1R 1 = 0,
1R 2 = 1,
1R 3 = 3,
1R4 = 12,
1R5 = 45.
(8.7.27)
one set of functions of each of the symmetry types (11), (21), (111), (31), (211) and (32). These are given by
The number R n 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
246
Generation of Integrity Bases: Continuous Groups
[eh. VIII
[e, (34), (23)J Y(
~
(8.7.29)
2 4 ) (B 1B2B3B4 -B4B3B2B 1), (31),
[e, (23), (243)J Y(! 2 ) (B 1B2B3B4 -B4B3B2B 1), (211);
s.
Traceless Symmetric Second-Order Tensors: R3
247
[ e, ..., (23)(45)]Y( ~ ~ 3 ) to BIB2B3B4B5 - B5B4B3B2Bl will yield a set of null matrices. Application of the symmetry operators [e, ...,
3.
4.
Sect. 8.7]
[e, (23), (45), (345), (23)(45)J Y(
~ ~
(23)(45)JY(
1~ 4) to BIB2B3B4B5 -BSB4B3B2Bl will yield a set of
matrices 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 as basis elements. Some care is clearly required in choosing the symmetry
5 ) (BIB2B3B4B5
operators which generate the set of basic skew-symmetric matrices of - BSB4B3B2B1)' (32).
symmetry type (32). We consider next the generation of the expression for a traceless
The multilinear elements of degrees 1,1,1,1 in B 1 , B 2 , B 3 , B 4 and 1,1,1,1,1 in B1,B2,B3,B4,BS which arise as products of the terms in
symmetric second-order tensor-valued function which is invariant under
(8.7.29) with the invariants (8.7.17) may be divided into sets of
R3 and multilinear in the traceless symmetric second-order tensors
functions which form carrier spaces for reducible representations of the
B 1, B 2 , ... B n .
symmetric groups 54 and 55. We list below a typical term from each of
metric second-order tensor form the carrier space for an irreducible
these sets, the number of terms in the set and the representation for
representation of R3 whose character corresponding to a rotation through f) radians is given by e 2iO + e iO + 1 + e- iO + e- 2iO . The number
which these functions form the carrier space.
The decomposition of
The five independent components of a traceless sym-
of times this representation appears in the decomposition of the Sn-
these representations may be found in Table 8.3 (p. 231).
dimensional representation whose carrier space is formed by the Sn independent components of the tensor
Ifn
(iii) Y(
~
(8.7.30)
2 )B 1B2B3 tr B4B5, 20,
~ ) B1B2B3 tr B4B5,
10,
I1J1
InJn
is given by
21r
Jsl(e2i8 + ei8 + 1 + e-i8 + e-2i8 )(1 - cos 8) d8
(8.7.31)
o .."1r"1r h were sl == e 2if) + e if) + 1 + e -if) + e -2if) . The quantItIes u 1'···' Us may
(21)· (2) == (41)
(iv) Y(
= 2~
Bf . ... B~.
+ (32) + (311) + (221);
(111)· (2)
=
(311)
be computed from(8.7.31) and are given by (8.7.32)
+ (2111).
The number Tn1 ... np of sets of traceless symmetric second-order The Sn1 ... np appearing In Table 8.7 are determined from (8.7.30).
We observe that application of the symmetry operators
tensor-valued functions of symmetry type (nl ... np) arising from the If n functions is given by
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
248
Sect. 8.7]
21r T nl .. ·np
==...L 21r
J nl· .. np (e2i8 + ei8 + 1 + e-i8 + e-2iO)(1 - cos 8) d8
[e, (23), (243)J
o
(8.7.33)
where the
v( ~
Let
Unl ... np be the number of sets of traceless symmetric second-order tensor-valued functions of symmetry type (nl". n p ) which arise from
i
the products of functions such as B 1, B 1B 2 + B 2B 1 - E3 tr B 1B 2 , ... with the invariants (8.7.17). We list the quantities T nl n p ' U nl ... n p and X~l· .. np in Table 8.8 for the n1 ... np where Tnl np::JO and
T nl U nl
3
21
4
31
np 1 1
1
2
2
2
22 211 1111
5
41 32 311 221 2111
249
3
2 )(B 1B2B3B4 +B4B3B2B1),
}B1B2B3B4 +B 4B3B2B1),
(34),
(211),
(1111);
3 }B1B2B3B4BS + BSB4B3B2B1)' (2111),
[ e, (34), (354), (234), (2354), (24)(3S)J
Table 8.8 Traceless Symmetric Tensor-Valued Functions of B 1,... ,Bn :R3 1 2
v(!
s. ~,(23), (354~ v( ~
nl + ... + np ~ 5. The T nl n p are computed from (8.7.33). The computations yielding the U nl n p are given below.
nl". n p
Traceless Symmetric Second-Order Tensors: R
v( i
2 3 ) (B1B2B3B4BS + BSB4B3B2B1)'
(311).
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
np 0 0
X~l··· n p 1 1
1 1
1 2
2 1
2 3
2 1 2
1
1234231
0
0234130
symmetric groups 53' 54 and 55.
1
from each of these sets, the number of terms in the set and the
3
1
4
5
6
5'
4
which form the carrier spaces for reducible representations of the We indicate below a typical term
representation for which these functions form the carrier space.
We
employ Table 8.3 to obtain the decompositions. With Table 8.8, we have T nl ... np - Unl ... np == 1 if nl ... np == 1,2,21, 22, 211, 1111, 311, 2111 and equals zero otherwise. The typical comprised of one set of functions of each of the symmetry types (1),
B 1 tr B 2B 3 , 3,
4.
(B 1B2 + B2B1 - i E3 tr B1B2) tr B3B4, 6,
(2), (21), (22), (211), (1111), (311), (2111). These are given by 1.
B 1,
2.
B1B2+B2B1-iE3trB1B2'
3.
[e,(23)JVn3)(B1B2B3+B3B2B1)'
4.
[e, (23)J
== (3) +
3.
multilinear traceless symmetric second-order tensor basis elements are
(1)· (2)
(21);
(2)· (2)
== (4) + (31) + (22),
(1);
B 1 tr B 2B 3B 4 ,
4,
(1)· (3) == (4) + (31);
(2); S. (21);
vO D(B1B2B3B4 + B4B3B2B1),
B1 V( ~ ~ )tr B2B3B4B5, 10, B1trB2B3trB4B5'
(22),
(8.7.34)
15,
(1). (22)
= (32) + (221),
(1)·(4)+(1)·(22)
== (5) + (41) + (32)+(221),
(8.7.35)
250
Generation of Integrity Bases: Continuous Groups
(B 1B 2 + B2B 1 -
i E3 tr B1B2) tr B3B4BS'
10,
[eh. VIII
(2)· (3)
Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd-Order Tensors: R
3
251
representations are denoted by (ml". m q ) and (nl." n p ) where ml". m q and nl ... np are partitions of m and n respectively. The characters of
= (5) + (41) + (32),
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£
y(~ 3)(B1B2B3+B3B2B1)trB4BS' 20, (21)'(2)
inequivalent
= (41) + (32) + (311) + (221).
irreducible representations
of 5 = SmSn denoted by
(ml ... m q , nl ... n p ) whose characters are Xm1 ··· m q(s') Xnl .. ·np(s").
The value of the U nl ... np appearing in Table 8.8 follow Immediately from (8.7.35).
8.8 An Integrity Basis for Functions of Skew-Symmetric Second-Order Tensors and Traceless Symmetric Second-Order Tensors: R3
Consider the set of invariants I.(A J I ,... , A m' B l ,... , B) n (J. -- 1,... , r) 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'.'" I . This set of r 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 B 1,...,Bn into
B . 1,... , B .n. We define the invariant sI.(A J l ,... , A m' B l ,... , B) n by J J
In this section, we generate the typical multilinear basis elements for scalar-valued functions W(A l ,.·., Am' B l ,... , B n ) of the skew-symmetric second-order tensors AI'''.' Am and the traceless sym-
(8.8.1 ) where s = s's". We may then determine a r x r matrix D(s) such that
metric second-order tensors B l ,... , B n which are invariant under the proper orthogonal group R3 . We note that the restrictions imposed on
(8.8.2)
W(A l ,···, Am' B l ,···, B m ) by the requirements of invariance under R3 and by the requirement of invariance under 03 are the same. Thus, the
which describes the transformation properties of the II' ... ' I r under a
integrity basis generated here will also form an integrity basis for
permutation s = s's" of SmSn.
functions of AI' ... ' Am' B l ,... , B n 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, ... ,Am and 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
The m!n! matrices D(s) = [Dk·(s)]
furnish a r-dimensional representation of SmSn' The set of
invari~ts
Il, ... ,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 representation (ml". m q , nl"· n q ) will be referred to as a set of invariants of symmetry type (mI." m q , nl ... n p ). The number of invariants comprising a set of invariants of symmetry type (mI ... mq, nl ... n p ) is given b y Xml···mq Xnl .. ·nq were h Xml ... m q an d Xnl ··· n p are the values of e e e e the characters of the representations (mI ... m q ) and (nl ... np) corresponding to the identity permutation e.
Generation of Integrity Bases: Continuous Groups
252
[Ch. VIII
The number IPm ,n of linearly independent functions which are multilinear in AI"'" Am' BI,oo., B n and which are invariant under R3 is given by I?m,n
= 2~
253
Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd- Order Tensors: R3
+ + np :s; 6. mI +... + mq == 0 mI
00'
We also exclude from Table 8.9 the cases where and nI
+... + n p == 0
since these refer to invariants
involving only the BI, ... ,Bn or only the AI, ... ,Am .
27r
J(s])m (sq)n (1 - cos 0) dO,
o iO s1 = e + 1 + e- iO ,
Table 8.9 (8.8.3)
sq
= e 2iO + eiO + 1 + e- iO + e- 2iO
Invariant Functions of AI"'" Am' B I ,···, B n : R3
mI·oomq, nI·oonp
some axis of the three independent components of a three-dimensional
respectively. The number Pm1 ... m q, n1 ... n p of sets of invariants of symmetry type (mloo, mq, nI'" np) arising from these IPm, n invariants is given by
27r
P m1 ... m q, n1°o· n p l - 27r
J
o
(8 .. 84)
1,211
1,41
1,32
1,311 1,221
1
1
1
2
2
2
2
3
1
Qm1°o·mq, n1· oon p n m Xem1··· q Xen1··· p
0
0
0
1
1
2
1
3
1
1
2
1
3
3
4
5
6
5
skew-symmetric second-order tensor and the five independent components of a three-dimensional traceless symmetric second-order tensor
1,31
P m1°o· m q, n1··· n p where s1. and s1 are the traces of the matrices which describe the transformation properties under a rotation through 0 radians about
1,11 1,21 1,111
mI'" mq, nloo, np
1,2111 11,11 11,21
11,111
11,31
11,211
2,1
2,2
2,3
P m1··· m q, n1··· n p
1
1
1
1
2
2
1
2
2
Qm1" .mq, n1" .np n m Xem1··· q Xen1··· p
1
0
0
0
1
2
0
1
2
4
1
2
1
3
3
1
1
1
2,1111
mI·oomq, nI·oonp
.p
2,21 2,4
2,31
2,22
2,211
111,2 111,3 21,1
where the 4>~1'" mq' 4>~1'" np are defined by (8.6.3) and (8.6.4) with
m1. oom q, n1°o .np
2
3
2
4
1
1
1
1
1
the Sr appearing in (8.6.3) being replaced by s~ and s~ respectively
Qm1°o·mq,n1°o·np oon m Xem1.·· q Xen1· p
1
3
2
4
0
1
1
1
0
2
1
3
2
3
1
1
1
2
21,3
21,21
21,111
3,11 3,3
3,21
where
s~ =
eirO + 1 + e-irO ,
s~ =
e2irO + eirO + 1 + e- irO + e- 2irO .
(8.8.5) mI'" m q, nI'" np
Let Qm1
m n n be the number of sets of invariants of symmetry ... q, 1'" P type (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 quantities p Q and mI'" mq nI'" np for those m1°o.mq, n1 np' m1.oomq, n1.oonp Xe Xe m1 ... m q, n1 np for which P m1 ... mq , nl... np is not zero. Spencer and
21,2 21,11
3,111
P mI" .mq, n1" .np
1
1
1
3
1
2
1
2
2
Qml···mq,n1···np m1· oom q n1··· n p Xe Xe
0
1
1
3
2
1
0
2
2
2
2
2
4
2
1
1
2
1
Rivlin [1962] have shown that the integrity basis elements are of degree
The typical multilinear elements of an integrity basis for
six or less which enables us to consider only cases for which
functions W(A I ,... , Am' B I ,···, B n ) which are invariant under R3 may
254
Generation of Integrity Bases: Continuous Groups
[Ch. VIII
Sect. 8.8] Skew-Symmetric and Traceless Symmetric 2nd-Order Tensors: R
be split into three sets:
Y"( (i)
Invariants
which
involve
traceless
255
3
i)
tr Al (B 1B 2B3 - B3B 2B 1), (1,111),
symmetric second-order tr (AI A 2 - A 2 A 1)(B 1B 2 - B 2B 1), (11,11),
tensors B 1,... ,Bn only. These are given by (8.7.17).
tr (AI A 2 + A 2 A 1)(B 1B 2 + B 2B 1), (2,2), (ii)
Invariants which involve skew-symmetric second-order tensors
[e, (23)J
AI'·'"Am only. These are readily seen to·be given by tr A 1A 2 and tr Al A 2A 3 which are of symmetry types (2,0) and (111,0)
y'n
2) tr B 1(AI A2A3 - A3A2A 1), (21,1); (8.8.6)
1
respectively.
2
5. (iii)
Y"
Invariants involving both the A 1,... ,Am and the B 1,... ,Bn . With Table 8.9, we see that the typical multilinear elements of
3 4
tr B 1B 2B 3B 4B 5 , (0,11111),
5
~,
the integrity basis involving both the AI'···' Am and B 1,· .. , B n are comprised of one set of invariants of each of the symmetry types
, (1~ 2)
(45), (23), (23)(45), (2453EY"
[e, (34), (23E Y"(
~
4
tr BIB2B3B4B5' (0,221),
2 4 ) tr Al (B 1B 2B 3B4 - B4 B 3B B ), (1,31), 2 1
(1,11), (1,21), (1,111), (1,31), (1,211), (1,32), (11,11), (11,21), (11,111), (11,31), (2,1), (2,2),
[e, (23), (243E Y"(! 2) tr Al (B 1B2B3B4 - B4B 3B2B ), (1,211), 1
(2,21), (2,211), (21,1), (21,2), (3,11), (3,3).
~
3 ) tr{A 1A2 + A2Al){BIB2B3 + B 3B 2B 1), (2,21),
~, (23E y,,( ~
2 )tr(A1A2 - A2Al)(BIB2B3 - B 3B 2B 1), (11,21),
[e, (23E Y"(
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 B 1,· .. , B n respectively. tr B 1B 2 , (0,2),
3.
tr B 1B 2B 3 , (0,3),
[e, (23)J y,,(~ [e, (23)J y"
[e, (23E Y'(
tr Al A 2A 3 , (111,0),
tr Al (B 1B 2 - B 2B 1), (1,11),
4.
1 ) tr(AIA2-A2Al)(BIB2B3-B3B2BI)' (11,111), Y"( §
tr Al A2 , (2,0);
2.
tr (AI A 2+A 2A 1)B 1, (2,1);
Dtr B1B2B3B4, (0,22),
(1 2 ) tr Al (B1B2B3 - B3B2B1),
2) tr{B 1B2 + B2Bl)(AIA2A3 - A3A2A 1), (21,2),
Y'(123) tr (AI A2B 1A 3B2 - AIA2B2A3Bl)' (3,11); 6.
(1,21),
j
[e, (34), (354), (3654), (234), (2354), (23654), (24)(35), (24)(365), (25364E Y"(
i
2 3 ) tr B 1B2B3B4B5B6' (0,3111),
256
[Ch. VIII
Generation of Integrity Bases: Continuous Groups
[e, (23), (243~Y"(l2) tr (AI A2 + A2A 1)(B1B 2B3B 4
3
257
linear basis elements for scalar-valued functions of Xl'''.' X , B ,... , B m 1 n which are invariant under 03. The Young symmetry operators Y( ... )
(2,211),
are applied to the subscripts on the B 1,... , B n . The notation (mI ... m q , lll"· n p ) employed below indicates the symmetry type of the sets of
A2A1)(B1B2B3B4
invariants. The mI ... mq refers to the vectors X1'~'." and the n1··· np refers to the tensors B 1,···, B n .
- B4B3B2B1)' (11,31), [e, (23), (45), (23)(45),
Vectors and Traceless Symmetric 2nd- Order Tensors: 0
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-
+ B4B 3B2B 1),
[e, (34), (23~y"G 2 4) tr(A1A2 -
Sect. 8.9]
(345~ y"G ~ 5) tr Al (B 1B 2B 3B 4B 5
2. - B5B4B3B2B1)' (1,32), 3.
4. 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{X 1,... , X m ,B 1,... "Bn ) of the ,absolute vectors Xl'.··' X m and the traceless symmetric second-order tensors B 1,···, B n which are invariant under the full orthogonal group 03. It is readily
1
shown by the method adopted by Pipkin and Rivlin [1959], §5, that any
2 Y 3 tr B 1B 2B 3B 4B 5 , (0,11111), -4 5
polynomial function W{X1,... , Xm , B 1,... , B n ) which is invariant under 03 is expressible as a polynomial in the quantities
[e, (45), (23), (23)(45), (2453~Y( ~
5.
i)
~, (23~ Y (~ .
.
.
T
1 2 3] .
(8.9.2)
(1) ~
2) tr (Xl T
tr B1B2B3B4B5' (0,221),
X; - ~X~) B 1B 2B3,
(11,21),
T
tr(X1~ -~X1)B1B2B3' (11,111),
where i,j=l, ... ,m and Xi denotes the column vector [X ,X ,X
Y
The quantities tr IlK' IlL and lIM are scalar-valued, symmetric matrixvalued and skew-symmetric matrix-valued polynomial functions of B I , ... , B n which are invariant under the group R3 . The general form of the
~, (23~ Y (~
3 ) tr (Xl
X; + ~X~) (B1B 2B 3 + B 3B 2B 1),
(2,21);
Generation of Integrity Bases: Continuous Groups
258
6.
[Ch. VIII
[e, (34), (354), (3654), (234), (2354), (23654), (24)(35), (24)(365),
(25364~ Y(
[e, (34), (23~Y(~
2 4
i
Sect. 8.10]
Transversely Isotropic Functions
259
The problem discussed above has been considered by Smith [196S]. Results of greater generality are available. Thus, an integrity basis for
2 3 ) tr BlB2B3B4B5B6' (0,3111),
functions of m absolute vectors, n symmetric second-order tensors and p skew-symmetric second-order tensors which are invariant under 03 has been given by Smith [1965] and by Spencer [1971].
)tr(XlX;-~Xr)(BlB2B3B4 -;- B 4B 3 B 2B 1), (11,31),
[e, (23), (243~Y: (12) tr (Xl ~ - ~Xl )(Bl B2B3B4 T
8.10 Transversely Isotropic Functions
T
There are five groups T1 ,... , T5 which define the symmetry
- B 4B 3B 2B 1), (11,211),
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), R 1 = diag (-1,1,1)
T3 : Q(O),
Ra = diag (1, 1, -1)
T4 :
R1
Q(O),
= diag (-1,1,1),
(S.10.1)
Ra = diag (1,1, -1)
T5 : Q(O), D 2 = diag (-1, 1, -1)
7.
~,
(23), (45), (23)(45),
(345~
yG ~
In (S.10.1), Q(O) denotes the matrix
5) tr (XIX;
T
- X2Xl)(BIB2B3B4B5 - B5B4B3B2Bl)'
(11,32),
[e, (23), (34), (354~ y(~ 3)tr(XlX;+~Xr)(BlB2B3B4B5 + B5B4B3B2Bl)'
(2,2111),
Q(O)
=
cos 0
sin 0
0
-sinO
cosO
0
0
1
o
which corresponds to a rotation about the x3 axis.
[e, (34), (354), (234), (2354), (24)(35~ Y ( +
~
R 1 and
Ra
cor-
respond to reflections in planes perpendicular to the xl axis and the x3 axis respectively.
123)
(S.10.2)
tr (Xl X2 T
~Xr)(Bl B2BiB4B5 + B5B4B3B2Bl)' (2,311).
D 2 corresponds to a rotation through ISO degrees
about the x2 axis. We restrict consideration here to the groups T1 and T2 . We list
Generation of Integrity Bases: Continuous Groups
260
[Ch. VIII
Sect. 8.10]
261
Transversely Isotropic Functions
the irreducible representations associated with these groups (see §6.6)
The typical multilinear elements of an integrity basis for functions of
and the linear combinations of the components Pi of an absolute vector
¢, ... , a, f3, ..., A, B, ... which are invariant under T1 are given by
p, .the components ai of an axial vector a, the components A ij of a skew-symmetric second-order tensor A and the components Sij of a
1.
¢;
2.
ap, A13; Aap.
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].
We list in Table S.12 the first few irreducible representations /0'
r l'
/2'
r2
The presence of the complex invariants a~, ...
typical multilinear elements of an integrity basis for functions W(p, q, ... , S, T, U, ... ) of the vectors p, q, ... and the symmetric secondorder tensors S, T, U, ... which are invariant under T1 . These are given by the real and imaginary parts of
2.
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
Table S.12 Irreducible Representations and Basic Quantities: T1 1
(PI + iP 2)(q1- iq2)' (S13+iS23)(T13-iT23)' (PI + iP2)(S13 - iS 23 ), (Sll - S22 + 2iS 12 )(T11 - T 22 - 2iT12);
matrices corresponding to the group element Q( 8) which define the
/0
,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
associated with T1 · The second column gives the 1 x 1
representations.
ap, ..., Aap in (S.10.3) indicates af3 ± ap, ... , Aap ± Aaf3 of
that both the real and imaginary parts
1.
8.10.1 The Group T1 /1'
3.
(S.10.3)
¢, ¢', ...
f3 ,
P3' a3' A 12 , S11 + S22' S33
(S.10.4)
3.
(Sll - S22 + 2iS 12 )(Pl - iP2)(ql - iq2)' (Sll-S22+2iS12)(Pl-iP2)(T13-iT23)' (S11 - S22 + 2iS I2 )(T13 - iT 23 )(U I3 - iU 23 )·
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 and 1
e -i8
a,
e ilJ
a, p, .
..
PI
+ iP2' a1 + ia2' A 13 + iA 23 , S13 + iS 23
P1 - iP2' a1 - ia2' A 13 - iA 23 , S13 - iS 23
e- 2i8 A, B, .
S11 - S22 + 2iS 12
e 2i8
Sll - S22 - 2iS 12
A, 13, .
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
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 R 1. 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. Table 8.13 Irreducible Representations and Basic Quantities:
1
/0
1,
rO
1, -1
"p, "p', ...
T2
P3' S33' Sll + S22 a3' A 12
Sect. 8.10]
Transversely Isotropic Functions
263
The quantities
2.
"p"p',
3.
"p(°1 f3 2 - a2 f3 1)' "p(A 1B 2 - A 2B 1),
4.
"p(A 1a2,82 - A 20 1f3 1 ),
a1f32+a2f31'
A 1B 2 +A 2B 1; A 10 2f3 2 + A 20 1f3 1;
(8.10.6)
(a1,82 - a2 f3 1)(A 1B 2 - A 2B 1)·
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,
2iO
12
lfe-0
0
C, C,
, S, T, U, V, ... ) of the skew-symmetric second-order tensors A, B, and the symmetric second-order tensors S, T, U, V, ... which are invariant under T2 . These are given by
1J' [0l O1]A[A]: ' [BB:],..
e 2iO
The group T2 is generated by the matrices Q( 8) and R 1. With the results of §8.10.1, we observe that the typical multilinear elements of an integrity basis for functions of the quantities
2.
S13 T 13+ S23 T 23'
f3 1, f3 2 , ... , AI' A2, B 1, B 2, ... which are invariant under the subgroup T1 of T2 are given by 3.
1.
2.
a1 f3 2+ a 2f3 1'
3.
Al a2{j2 + A 2a l{jl'
"p; a1,82- a 2f3 1'
A 1B 2 +A 2B 1, A 1B 2 -A 2B 1;
Al a2{j2 - A 2a l{jl·
(8.10.5)
A 12B 12 , A 13B 13 + A23 B 23 , A13 S13 + A23 S23 , (Sll-S22)(T11-T22)+4S12T12;
A12(B13C23 - B 23 C 13 ),
A12(B13S23 - B 23 S13 ),
A12 (S13 T 23 - S23 T 13)' A12 (S11 - S22)T 12 - A 12 (T 11 - T 22 )S12' (Sll - S22)(A 13 B 13 - A23 B 23 ) + 2S12(A13B23 + A23 B 13 ),
264
Generation of Integrity Bases: Continuous Groups
[eh. VIII
(Sll - S22)(A 13 T 13 - A23 T 23 ) + 2S12(A13T23 + A 23 T 13 ), (Sll - S22)(T13 U 13 - T 23 U23 ) + 2S 12 (T13U 23 + T 23 U 13 );
IX
(8.10.7) 4.
GENERATION OF INTEGRITY BASES: THE CUBIC
A 12 (Sll - S22)(B 13 C23 + B 23 C 13 ) - 2A12S12(B13C13 - B 23 C 23 ),
CRYSTALLOGRAPHIC GROUPS
A 12 (Sll - S22)(B 13 T 23 + B 23 T 13) - 2A12S12(B13T 13 - B 23 T 23)' A 12 (S11 - S22)(T13U23 + T 23 U13) - 2A 12 S12 (T13U13 - T 23 U23 ),
9.1 Introduction
((S11 - S22)T12 - (T11 - T22)S12)(A13B23 - A23 B 13 ),
In this chapter, we consider the problem of generating integrity
((S11 - S22)T12 - (T11 - T22)S12)(A13U23 - A 23 U13)'
bases for functions which are invariant under a given cubic crystal-
((S11-S22)T12-(T11-T22)S12)(U13V23- U23 V 13 )·
The elements of. an integrity basis for functions of n symmetric secondorder 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].
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 skewsymmetric second-order tensor A and a symmetric second-order tensor
S respectively which form carrier spaces for the irreducible representations r a' r b'·" 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
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 a,b,... denote the number of linearly n,m, ... independent functions which are multilinear in n quantities of type fa'
I?
m quantities of type f b , ... and which are invariant under the group A = {AI'''·' AN}· We have
I?::~',::. = & f
(tr rK)n (tr
K=l
r~)m...
(9.1.1)
where rj(, r~, ... (K = 1,... ,N) are the matrices comprIsmg the irreducible representations fa' f b , .... The number of sets of invariants of .. f rom th e IP a,b,... In· symm e t ry t ype (n1 ... np, mI ... m q , ... ) arIsIng n,m, ... variants is given by
N pa,b,... =1. ""A.. (ra)A.. (r b ) (912) nl·" n p , ml". mq,... N L...J ""nl"· np K ""ml"· m q K··· ..
K=l
where
Sect. 9.1]
267
Introduction
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
3! 4>3(r)
= (tr r)3 + 3 tr r
tr r 2 + 2 tr
:r3.
(9.1.5)
The expression for
1
(9.1.6)
o The values of the quantities tr r, tr r 2, ... and 4>1 (r),
(9.1.3)
which arise as products of elements of the integrity basis. The Qa, b, ... are to be determined from inspection with the nl." np,ml··· m q , ... aid of Tables 8.3 and 8.4.
In (9.1.3), the non-diagonal entries in the determinant are obtained by increasing (decreasing) by one the subscript on
,
The number of sets of basic invariants of . . b P a, b, ... symmetry type (n1··· np, mI··· m q , ... ) IS qIven y n n m m 1"· p, 1··· q, ... - Qa, b, ... provided that the invariants comprising the nl n p , ml"· mq, ...
Qnl a, b, sets of invariants are linearly independent. The ... n p , 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
(9.1.4)
E, A, B, F, G, H and I, C, R 1, ... , M 2 which appear in the sets of matrices defining the two-dimensional and three-dimensional irreducible
where the summation is over the classes, of the symmetric group Sn
representations associated with the cubic groups are defined by (7.3.1)
and where h, gives the order and 1'1 2'2 ... n,n the cycle structure of
and (1.3.3) respectively." We employ the notation 2:( ... ) to indicate the
n!
= Lh,{tr r)'l {tr r 2 )'I ... {tr rn),n
268
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
sum of the three quantities obtained by cyclic permutation of the sub-
Table 9.2
scripts on the summand. For example, EX1Y1
= x1 Y1 + x2 Y2 + x3 Y3'
(9.1.7)
f
trr
E A,B
2
3
-1
F,G,H I
4
5
6
7
0
1
-1
0
1
0
1
0
1
0
1
3
6
10
15
21
-3
6
-10
15
1
2
2
-1
2
1
D 2T 1, D 3T 1, D 1T 2 D 3T 2, D 1T 3, D 2T 3 (I, D 1, D 2, D 3) . M 1 (I, D 1, D 2, D 3) · M2
E
A,B
Values of tr f, tr f2, ... : The Cubic Groups
C
tr r 2 trr3 tr r4 tr r5 tr r 6 trr7 trrB tr r 9
2
2
2
2
2
2
2
2
2
Rl'~' R 3 T 1, T 2, T 3 D 1T 1, D 2T 2, D 3T 3
-1
-1
2
-1
-1
2
-1
-1
2
F,G,H
0
2
0
2
0
2
0
2
0
D 1, D 2, D 3
1
3
3
3
3
3
3
3
3
3
CT1' CT2' CT3
C
-3
3
-3
3
-3
3
-3
3
-3
1
3
1
3
1
3
1
3
1
Rl'~'~
D 1, D 2, D 3 CT1' CT2 ' CT3
-1
3
-1
3
-1
3
-1
3
-1
R 1T 1, ~T2' ~T3 ~Tl' R 3T 1, R 1T 2
~T2' R 1T 3, ~T3
1
-1
1
3
1
-1
1
3
1
-1
-1
-1
3
-1
-1
-1
3
-1
(I, D 1, D 2, D 3) . M 1 (I, D 1, D 2, D 3) . M2
0
0
3
0
0
3
0
0
3
(C, R 1,
~,
R 3)· M 1
(C, Rl'~'~) ·M2
~T2' R 1T 3, ~T3
(C, R 1,
~, ~).
M1
(C, R 1, ~, ~). M2
D 2T 1, D 3T 1, D 1T 2 D 3T 2,D 1T 3, D 2T 3
0
0
-3
0
0
3
0
0
-3
8
8
9
10
-1
0
1
0
1
0
28
36
45
55
-21
28
-36
45
-55
3
3
4
4
5
5
-2
3
-3
4
-4
5
-5
0
0
1
1
0
0
1
1
-1
0
0
1
-1
0
0
1
-1
0
0
1
0
0
1
0
0
1
0
0
-1
0
0
1
0
0
-1
R 1T 1, ~T2' R 3T 3
~Tl' ~Tl' R 1T 2
T 1, T 2, T 3 D 1T 1, D 2T 2, D 3T 3
269
Values of
r
EX1Y2Z3 = x1 Y2z3 + x2 Y3z 1 + x3Y1 z2'
Table 9.1
Tetartoidal Class, T, 23
Sect. 9.2]
9.2 Tetartoidal Class, T, 23 In Table 9.3 below, the matrices I, D 1, ... are defined by (1.3.3), w = -1/2 + i.J3/2, w 2 = -1/2 - i.J3/2 and xi = [xL x~, x~{ Quantities of types r l' r 2' ... are denoted by
270
Generation of Integrity Bases: The Cubic Crystallographic Groups [eh. IX
b = b 1 + ib 2 ,... are complex and a = a1 - ia2' b = b 1 - ib 2 ,... denote the complex conjugates of a, b, ... respectively. Irreducible Representations: T
Table 9.3 T
r1 r2 r3 r4
I, D 1, D 2, D 3
(I, D 1, D2, D3) . M1
(I, D 1, D 2, D 3) . M2
B.Q.
1
1
1
1
w
w2
a, b, ...
1
w2
w
a, b, ...
(I, D 1, D 2, D3) . M1
(I, D 1, D 2, D3) . M2
I, D 1, D 2, D 3
Sect. 9.2]
functions of the quantities
(9.2.1 ) where we have set (9.2.2) The functions (9.2.1) may be replaced by the equivalent set of functions
r2
a, b, ... , xIYl
2
r4
Basic Quantities: T
+ w2x2Y2 + wX3Y3'
x1 Y 3z 2+ w x2Y1z3+wx3Y2z1' ...
which correspond to I, D 1, .. · , D 3M 2 are given by I, D 1, ... , D 3M 2 respectively. Table 9.3A
271
variant under the group T. We see from §7.3.3 that multilinear
which correspond to I, D1, 22 2 D 2, D 3, ... , M 2 , D 1M 2 , D 2M 2 , D 3M 2 are 1,1,1,1, ... , W 2 , W , W , W respectively. In the row headed by r 4' the entries indicate that the 3 x 3 matrices comprising
23
x1'~'''·
In Table 9.3, we employ (I, D 1, D 2 , D 3 ) . M 1 to denote M 1, D 1M 1, D 2M 1, D 3M 1. Entries in the row headed by r 2 indicate that the 1 dimensional matrices comprising
Tetartoidal Class, T,
(9.2.3)
a, b, ..., xIYl + wX2Y2 + w2x3Y3' x1 Y3z2 + wX2Y1z3 where
2: (...)
+ W 2x3Y2z1'
...
is defined by (9.1.7) and w, w 2 are cube roots of unity.
r1
511 + 522 + S33
The functions (9.2.3) are grouped according to the manner in which
r2 r3
5 11 + w2S22 + wS33
they transform under T. Thus, the functions designated by
811 + w8 22 + w2833
are quantities of types
r4
[PI' P2' P3]T, [aI' a2' a3]T, [A 23 , A 31 , A 12]T, [8 23 ,8 31 , S12]T
r l' r 2' r 3 respectively.
r 1, r 2 , r 3
The determination of an
integrity basis for functions of quantities of types
r l' r 2' r 3 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
9.2.1 Functions of Quantities of Types r l' r 2' r 3' r 4: T
type r 3 and the products taken three at a time of the quantities of type
We consider the problem of generating the typical multilinear elements of an integrity basis for functions W(
r l' r 2' r 3
and
r4
r l' the product of each quantity of type r 2 with each quantity of
which are in-
r 2· We observe that each product of two quantities of type r 2 of the form 2 (xIYl + w2x2Y2 + wX3Y3)(zlu2v3 + w z 2u 3v l + wZ3ulv2) (9.2.4)
272
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
is expressible as a linear combination of the forty functions of the forms (xIY1
+ wX2Y2 + w2x3Y3)
(xIY2 z3 + wX2Y3z1
L: z 1u 2v 3' ... ,
+ w2x3Y1z2) L:u l v1'
(9.2.5) ....
Also, each product of two quantities of type r 2 of the form
Sect. 9.2]
Table 9.4
Tetartoidal Class, T,
23
273
Invariant Functions of a,b, ... ; x1,x2' ... : T 111
4
31
1
2
1
0
0
1
0
1
1
1
1
311
6
51
nl··· n p
2
p4 n1··· n p
1
Q!l·.. np
0
np Xn1··· e nl··· n p
3
22
5
41
1
2
1
1
1
1
1
3
2
1
4
5
42
411
33
222
1
5
5
is expressible as a linear combination of the eighty functions of the form
p4 n1 .. · n p
4
2
3
(9.2.7)
Q!l'" np
3
2
3
5
9
1,21
1,4
1,31
1,22
2,2
1
1
2
1
1
1
1
0
0
1
1
1
0
0
1
2
1
3
2
1
2
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'" n p "'" Q~l~.. n p , mI'" mq for the nl'" n p and n1'" np' mI'" mq of interest. The P!l'" n p "" 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 (i)
invariants which are functions of xl' x2' ... and are comprised of one set each of invariants of symmetry types (2), (3), (111), (4),
np Xn1··· e
6
nl· .. n p, ml .. · m q 1,2 p2,4 n1 ...n p , mI··· m q Q2,4 . n1 ... np, mI ... mq n1 .. · n p m1· .. m q Xe Xe
32
10
2,21
The typical multilinear elements of an integrity basis for T are listed below.
1.
¢;
2.
ab + ab,
3.
abc, (3,0);
4.
Exlxrx~xf' (0,4);
ab - ab ;
L>lx~, (0,2);
a (xlxr + wx~x~ + w2x1x~), (1,2);
(31), (41), (6). (ii) invariants which are functions of a, b, ... ; xl' x2' ... and are comprised of one set each of invariants of symmetry types (1,2), (1,21), (1,4), (2,2), (2,21).
[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
[e, (23)] Y
275
We see from Table 9.3A that the transformation properties of a
(1,21);
ab (xlxI + w2x~x~ + wx§x~), (2,2);
vector P = [PI' P2' P3]T under the group T are defined by the representation r 4. The typical basic invariants which are multilinear in the
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 p 1'·'" n invariant under T. These are given by qua~titi~s :r1'''.'~ of type
[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
Tetartoidal Class, T, 23
9.2.2 Functions ofn Vectors Pl'''.'Pn : T
(j 2) a { xl(x~x~ + x~x~) + wx~(x~x~ + xIx~) + w2x!(xIx~ + x~x~)},
5.
Sect. 9.2]
{xhx~x~ + x~x~)
+ w2x~(x~x~ + xIx~) + wx§(xIx~ + x~xV},
(2,21);
3.
LPhp~p~ + p~p~), (3);
4.
LPlpIPfpf, (4); [e, (34), (234)JY(12 3)LPlpI(p~p~-p~p§), (31);
Some of the invariants appearing in (9.2.8) are complex functions. For example, abc = (a1b 1c1 - a1b 2c2 - b 1c2 a 2 - c1 a2 b 2)
+ i( a1b 1c2 + b 1c1a2 + c1 a1 b 2 -
(9.2.9)
a2 b 2c2)·
(9.2.10)
(1 2 3 4 ) L PIPfpf(p~p~ - p§p~),
5.
[e, (45), (345), (2345)J Y
6.
Y(123456)LPlpIP~Pf(p~p~-P~Pg),
(41);
(6).
Both the real and imaginary parts of the invariant (9.2.9) are basic
The s!m~e~ry operators Y( ... ) are to be applied to the superscripts on
invariants. We have indicated the symmetry types of most of the sets
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
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) superscripts on the x's.
are applied to the
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
Tetartoidal Class, T, 23
Sect. 9.2]
277
An integrity basis for functions of two vectors p and q which are (9.2.14)
invariant under the group T is seen from (9.2.10) to be given by
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
4.
EPf, Ephl' EphI,
EpI(q~ - q~), 5.
Ep~(p2q3
E
q~(P2q3 -
Ephl (q~ -
n1'" np pertain to the symmetry properties o~ the. set. of invariants (9.2.12)
P3 q2);
Epf(P2 q2 - P3 q3) + 2 Ephl (p~ -
Epf(q~ - q~) + 8 Ephl (P2 q2 E Qt(p~ - p§) + 8 E
Q~Pl (P2 Q2 -
P3 Q3) +
under perrr:ut~tio:ns of the superscripts on the (8 11 , 8 22 , 8 33 ) (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{ ... ) 2 3 . d'lcat es summatIon . ' I X31 = 8 111X1X1 In on t he sub scrIpts, e.g., ~81 L..J 1l X2
p~),
P3 q3) + 6 EphI(p~ -
q~) + 3 EpIqI(P2 q2 -
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'" m q and
Ephl (P2q3 - P3q 2)'
- P3 q2)'
Epf(p~ - p~),
Eqf, EpI(P2q2- P3q3)'
EqI(P2 q2 - P3 q3);
E PIqI(P2 q3 - P3 q2)' 6.
Eplq~,
+ S~2x~x~ + Shx§xl The typical multilinear elements of an integrity
p~),
basis for functions of Sl'"'' Sn which are invariant under T are given by
L PI Q~(p~ - p~),
P3 Q3) + 6 E QIPI( Q~ - Q§),
LQf(P2 Q2 - P3 Q3) + 2 LQ~Pl (Q~ -
Q~),
E
Qt(Q~ - Q~). 3.
9.2.3 Functions of n Symmetric Second-Order Tensors Sl ,... , Sn: T
LShSIlS~I' E[Sh SII (S~2 -
(3,0);
S~3) + Sii S~1 (Sh -
+S~ISh(S~2-S~3)J, (3,0);
With Table 9.3A, we see that
Sl1 + S22 + S33'
L SII xIx~,
Sl1 + w2S22 + wS33' T
Sl1 + wS22 + w2S33'
Sh)
[ S23' S31' S12J
are quantities of types f 1, f 2 , f 3 , f 4 respectively. We may set
(9.2.13)
(1,2);
E Sh (x~x~ - x§x~), (1,2);
Exl(x~x~ + x§x~), (0,3);
Lxl(x~x~ - x~x~), (0,111); (Continued on next page)
Generation of Integrity Bases: The Cubic Crystallographic Groups [eh. IX
278
4.
Th
[ e, (34)J V(~ 3) ESbxt(x~x~- x~x~), (1,21); (2,2);
ESbSI1(x~4-x~x~),
(9.2.15) (2,2);
[e, (34), (234)JV(12 3) Exlxt(x~x~-x~x~), (0,31 );
Exlxtx~xf' (0,4);
[e, (45)J Vn 4) ESb SI1x~(x~x3 + [ e, (45)J V (~ 4) E SbSI1 x~(x~x3 -
x~x~), (2,21);
In (9.2.15), the xi, xk, x~ denote S~3' S~l' S12' (9.2.15) are given by Smith and Kiral [1969].
Results equivalent to
9.3 Diploidal Class, T h' m3 In Table 9.5, the matrices I, D 1, ... are defined by (1.3.3), w.= -.1!i+i{3/2, w2 = -1/2-i{3/2, xi = [xLxk,x~lT and Xi = [XL X X The quantities and II are real quantities; the quantities a = al
+ ia2
and A = Al
+ iA 2 are
Irreducible Representations: T h (I, D 1, D 2 , D 3) . M 2
B.Q.
1
1
, ', ...
1
w
w2
a, b, ...
1
w2
w
a, b, ...
(I, D 1, D 2 , D 3) · M 1
(I, D 1, D 2 , D ) . M 3 2
x1'~'·"
1
1
II, II', ...
1
w
w2
A,B, ...
1
w2
w
A,13, ...
I, D 1, D 2 , D 3
(I, D 1, D 2 , D 3)· M 1
r1 r2 r3 r4 r5 r6 r7 r8
I, D 1, D 2 , D 3
(I, D 1, D 2, D 3) · M 1
(I, D 1, D 2 , D ) . M 2 3
Th
C, R1,~,R3
(C, R1'~' R3)· M1
(C, R1'~' R3)· M2
B.Q.
1
1
,', ...
1
w
w2
a, b, ...
1
w2
w
a, b, ...
(I, D 1, D 2 , D 3) . M 1
(I, D 1, D 2 , D 3) . M 2
x1'~'''·
II, II', ... A, B, ...
A,13, ...
1
I, D 1, D 2 , D 3
1
X 1,X2 ,..·
x~x~), (2,21);
[ e, (45), (345), (2345)J V(1 2 3 4 )ExIx~xf(x~x3 - x§x~), (0,41);
2, a] .
279
Diploidal Class, T h' m3
Table 9.5
[ e, (34)J V(~ 3) ESbxt(x~x~+ x~x~), (1,21);
ESbSt1x~xf'
Sect. 9.3]
complex. The complex conjugates of
a and A are denoted by a = al - ia2 and A = Al - iA 2 respectively. The format of Table 9.5 is the same as that of Table 9.3.
r1 r2 r3 r4 r5 r6 r7 r8
1
I, D 1, D 2 , D 3
-1
-1
-1
-w
-1 _w 2
-1
_w 2
-w (C, R1'~'~)· M 2
C, R1'~'~
(C, R1'~'~) . M1
Table 9.5A
Basic Quantities: T h
r1 r2
+ S22 + S33 Sll + w2S22 + wS33
r3
Sll
r4
[aI' a2' a3]T, [A 23 , A 31 , A 12]T, [S23' S31' S12]T
Sll
+ wS22 + w2S33
X 1,X2 ,..·
280
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
9.3.1 Functions of Quantities of Type r 8: 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 representation f S. Functions W(Xl, ... ,Xn ) of quantities Xl,,,.,Xn of type f S which are invariant under the subgroup D 2h = {I, C, R 1, ~, R 3 , D 1, D 2 , D 3} of T h are seen from §7.3.4 to be expressible as functions of
the quantities (i,j
= 1,... ,n).
Sect. 9.3]
in (9.1.2) would be over the 24 matrices I, D 1, , R 3M2 comprising the representation f S of T h . The matrices I, D 1, , R 3M2 are listed in row S of Table 9.5. With Table 9.6, we see that the typical multilinear
symmetry types (2), (4), (31), (6) respectively. These are given by
2.
LXIXI,
4.
LXIXIX~Xf,
6. = W* (X~X~,
X~X1, XiXi)
= W* (X~X1, Xi Xj1' X~X~).
(9.3.2)
(2); (4);
[ e, (34), (234)J Y(~ 2 3)
under cyclic permutations of the subscripts 1,2,3. Thus, W*( ... ) must
X~X~, X~X1)
which are
invariant under T h are comprised of 1, 1, 1, 2 sets of invariants of
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
w* (XiXi,
~,...
elements of an integrity basis for functions of Xl'
(9.3.1 )
satisfy
281
Diploidal Class, T h' m3
LXIXIX~XfXfXY,
LXIXI(X~X~- X~X§), (31);
(9.3.3)
(6);
Y(1 2 3 4 5 6) LXlXIX~Xf(X~X~
- x~xg),
(6).
The Young symmetry operators appearing in (9.3.3) are applied to the
xj.
Substituting Pi for ~ in (9.3.3) will give the
The general form of functions W*{ ... ) which are consistent with the
superscripts on the
restrictions (9.3.2) may be determined upon application of Theorem 3.3.
typical multilinear elements of an integrity basis for functions of the
With (3.2.5), it is seen that the elements of an integrity basis for
vectors PI' P2' ... which are invariant under T h.
functions of X 1,... ,Xn which are invariant under T h are of degrees 2, 4 S n for the n1 ... np of and 6. We list the values of P nS n, Qn 1'" P I ' " P interest in Table 9.6.
equivalent to those given by Smith and Rivlin [1964].
2
4
31
22
6
51
42
411
33
222
pS nl···np
1
2
1
1
4
2
3
1
1
1
nl···np Xe
0 1
1 1
0 3
1 2
2 1
2 5
Th
imposed on scalar-valued functions of quantities of types f l' f 2' f 3' f 4
n1··· n p
Q~l· .. np
r l' r 2' r 3' r 4:
We observe from Table 9.3 and Table 9.5 that the restrictions
Invariant Functions of Xl' X 2 , ... : T h
Table 9.6
9.3.2 Functions of Quantities of Types
These results are
3
1
1
1
9
10
5
5
The P~l'" n p are obtained from (9.1.2) and Table 9.2. The summation
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, r 3, f 4 . The procedure leading to this decomposition is discussed in §5.3. The results (9.2.S) enable us to
282
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
determine integrity bases for functions of arbitrary numbers of evenorder tensors which are invariant under T h . In particular, we observe that the restrictions imposed on the scalar-valued function W{Sl ,... , Sn)
8ect. 9.4]
Gyroidal Class, 0, 432; H extetrahedral Class, T d' 43m
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 T
of the symmetric second-order tensors Sl'... ' Sn by the requirement of
a= [aI' a2]'
invariance under the group T h are identical with those imposed by the
iii]T xi = [Xl' x2' x3 '
requirement of invariance under the group T.
283
Thus, the typical
a=a1
+ ia2'
a=a1- ia2' (9.4.1 )
_ [i i i ]T Yi - Y1' Y2' Y3 .
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)
Table 9.7A
Basic Quantities: 0, T d
r1
for functions of Sl'.'" Sn invariant under T.
r2
r4
f3
-
..-
0
9.4 Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m
Td
(I, D 1, D 2, D 3) . M 1
(I, D 1, D 2 , D 3 ) · M 2
I, D 1, D 2, D 3
(I, D 1, D 2, D 3) . M 1
(I, D 1, D 2 , D 3) . M 2
PI
B.Q.
f1 f2 f3 f4
I, D 1, D 2, D 3
(I, D 1, D 2, D 3) . M 1
(I, D 1, D 2, D 3) . M 2 x1'~'·"
r5
I, D 1, D 2, D 3
(I, D 1, D 2, D 3) · M 1
(I, D 1, D 2 , D 3 ) . M2 Y1'Y2' ...
1
1
1
1
1
1
"p, "p', ...
E
B
A
a, b, ...
Td
8 11 + 8 22 + 8 33
[25 11 - 522 - 533] ~ (8 33 - 8 22 )
(C, R1'~'~)· T 1
(C, R1'~'~) . T 2
(C, Rl'~'~) . T 3
T d (I, D 1, D 2, D 3) . T 1
(I, D 1, D 2, D 3) . T 2
(I, D 1, D 2, D 3) . T 3
B.Q.
f1
1
1
1
r2
-1
-1
-1
"p, "p', ...
f3 f4
F
G
H
(I, D 1, D 2, D 3) · T 1
(I, D l , D 2 , D 3) . T 2
(I, D l , D 2, D 3) . T 3
f5
(C, Rl'~'~)· T l
(C, Rl'~'~) . T 2
(C, Rl'~'~) . T 3 Y1' Y2'''·
-
-
.....
-
-..-
A 23
al
P2 ' a2 ' A31 A a3 P3 ..... - ..... - ..... 12 -
-
8 23
P2 ' 831 8 P3 .... - .... 12 -
-
al
-
.....
A 23
a2 ' A 31 A a3 - - .... 12-
9.4.1 Functions of Quantities of Types r 1, r 3 , r 4 : T d' 0 It is readily seen with (9.2.8) that the multilinear functions of the quantities (9.4.2)
0
.....
8 12
-
PI
8 31
~ (8 33 - 8 22 )
-
I, D 1, D 2, D 3
r-
8 23
[25 11 - 522 - 533]
Irreducible Representations: 0, T d
Table 9.7
0
811+822+833
r5
of types f l' f 3' r 4 which are invariant under the subgroup T = { I, D1, D 2, D3, M 1, D 1M 1, D2M 1, D3M 1, M2, D 1M2 , D 2M 2, D3M2 } of T d are expressible in terms of functions of the forms
¢, ab+ab, abc + abc,
Exlx~, (9.4.3)
a, b, ... xl'~'''·
and
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
284
Sect. 9.4]
Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m
285
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). (9.4.4)
The integrity basis
obtained in this manner contains a number of redundant terms. 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
r l'
and the real and imaginary parts of the functions
n quantities of type
r 3 and n
quantities of type
r 4.
An integrity
basis for functions of S1' ... , Sn which are invariant under T d has been obtained by Smith and Kiral [1969]. With the results given in their paper, we may immediately list the degrees in a, b, elements of the integrity basis for functions of 4>, which are invariant under T d. (9.4.5)
{I
2 3 2 3) 1 2) [ e,(23) ] Y ( 3 ab xl {x2x 3 + x3 x 2
2 3) + W 2 x21{ x32x31 + xlx3
+ wx~(xIx~ + x~xV} where
E{...) is
w 2 = -1/2 -
Table 9.8
0,2
0,3
0,4
1
1
1
1
2
111
o
o
o
0
1
1
1
1
1
1
1
212
1,4
1,31
1,22
2,2
2,3
1
2
1
111
1
o
0
2
1
1
1
1
1
1
3
2
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 T dare given in terms of the functions (9.4.3) and the real parts of the
, a,b, ... , x1,x2'''.
3,0
i~/2. The last twelve transformations of Table 9.7 leave
all of the imaginary parts of the functions (9.4.5). With Theorems 3.1
of the
2,0
2
altered. They also change the signs of all of the functions (9.4.4) and
x1'~'''.
Invariant Functions of a,b, ... ; x1,x2' ... : T d' 0
defined as in (9.1.7) and where w=-1/2+i~/2,
the functions (9.4.3) and the real parts of the functions (9.4.5) un-
;
0, 22
2,21
1, 2
0
11,111
1, 21
0
3,111
0
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 candidates for inclusion in the integrity basis, the number P~;m2' nl ... np of linearly independent sets of invariants of symmetry type
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
286
(m1m2' n1'" n p ) and the number Q~;m2' nt ... n p of sets of invariants of symmetry type (m1m2' n1 ... n p ) 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... n p ' With (9.4.3), ... , (9.4.5) and Table 9.8, we see that the typical multilinear elements of an integrity basis for
Sect. 9.4]
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 = b 1 + ib 2, ... · 9.4.2 Functions of n Vectors PI'.'" Pn: T d
functions of 4>,..., a,b, ... , x1,x2'." which are invariant under T dare given by
287
Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m
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
1.
4>;
2.
ab + lib, (2,0);
integrity basis for functions of the quantities xl' ... ' Xn of type
r4
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
L>lxy, (0,2);
of an integrity basis for functions of n vectors PI'·'" Pn which are invariant under T d. These are given by
4.
[ e, (23)J
yn
2) Re[ a{ xhx§x~ + x§x~)
+ wx~(x§x~ + xyx~) + w2 x~(xyx~ + x§x~)} J,
(1,21); (9.4.6)
2.
Eplpy, (2);
3.
E
4.
1234 E P1 P 1P 1P 1' () 4.
1( 2 3 2 3) PI P2 P3 + P3 P2 ' (3);
(9.4.7)
Results equivalent to (9.4.7) are given by Smith and Rivlin [1964].
9.4.3 Functions of Quantities of Type r 5: 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
[ e, (23)J
are invariant under 0 is equivalent to that of determining an integrity
y( 12 ) Re[ ab{ xhx§x~ + x§x~)
+ w2 x~(x§x~ + xyx~) + wx~(xyx~ + x~x~)} J,
(2,21);
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
LylYI, (2);
3.
Lyhy~y~ - Y§Y~),
4.
LylYIY~Yf' (4);
5.
[ e, (45), (345), (2345)J
6.
LylYIY~YfY~Y~'
7.
[ e, (67), (567), (4567), (34567), (23456 7)J
(111);
the integrity basis elements given in (9.4.8). (9.4.8)
ya
Gyroidal Class, 0, 432; Hextetrahedral Class, T d' 43m
2 34) LYIY~Yf(y~yg - Y!Y~), (41);
(6);
Y( i 2 3 4 5 6) LYIY~YfY~Y~(Y~Y~ -
289
Let Qnl ... n p 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
(111), (4), (41), (6), (61) and (9). These are given by 2.
Sect. 9.4]
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) x (2)} . (111),
(4)· (111),
(2)· (4), (2)· (41),
(111) x (2), (2)· (61), (9.4.11 )
Y!Y~),
(61);
{(2) x (3)}· (111), (111) x (3),
{(2) x (2)}· (41),
(6)· (111),
(2)· (4)· (111),
(4)· (41).
We may employ results such as those given in Tables 8.3 and 8.4 We now establish the result that none of the basis elements in
or those given by Murnaghan [1937, 1951] to obtain the decomposition
(9.4.8) are redundant. Let P n1 ... np denote the number of linearly independent sets of invariants of symmetry type (n1 ...np). With (9.1.2), ... , (9.1.4), we have
of these sets into sets of invariants of symmetry types (n1". np) (see
P2 =
PIll
The Qnl ... n p may then be read off.
We see in this
(9.4.12)
2\L
P 61
Smith [1968b]). manner that
= 2\ L {
Since P n1 ... n p - Qnl ... n p = 1 for n1". n p = 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
P9 =
l4L
an integrity basis for functions of n vectors PI' ... ' Pn which are in-
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)
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 9.7 A that the typical multilinear elements of an integrity basis
linear elements of the integrity basis for functions of n symmetric
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~,
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.
9.4.4 Functions ofn Symmetric Second-Order Tensors Sl,.",Sn: T d , 0
1.
ESh, (1,0);
2.
ESI1S~1' (2,0); ExIx~, (0,2);
3.
EShS~lSf1'
y~ by A~3' A~l' Ab (i = 1,2,... ) in (9.4.8).
With Table 9.7 A, we see that T
8 11 + 8 22 + 8 33 ,
[823,831,812]'
[2S 11 - S22 - S33' ~(S33 - S22)] are quantities of types
c/>= b=
ESu'
(9.4.13)
T
2S~1 - S~2 -
S§3 +
EShx~xf,
We may set
4.
-S~2-S~3+~i(Sh-S~2)' ~ i(S§3 - S~2)'
(9.4.16) (9.4.14)
5. [ e, (45)J y(~ 4 )ESh S~lxf(x~x~ +
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
·
.
· T
·
·
·
= [S23,S31,S12]
T
(i
= 1,... ,n).
6.
E[ Sb S~l(S~2 -
(2,21); (11,111);
S~3) + S~l Sf1 (S~2 - S~3)
-
L>i(x~xg x~x~),
(3,111).
(9.4.15)
We indicate the symmetry type of the sets of invariants by (mI ... m q , n1." n p ) 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 The typical multibelow apply to the superscripts on the
xl' X2' xJ.
xix~),
E(ShS~2-S~2S~1)· Exf(x~x~-xix~),
+ Sf1 Sll (S~2 - S§3)}
[811,822,833] , [xl,x2,x3]
(1,2);
Exl(x~x~ + x§x~), (0,3);
r l' r 4 and r 3 respectively.
a= 2S h
(3,0);
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
] a = [aI' a2' T
C = [C 1' C 2] ,
_ [i i i ]T Xi - xl' x2' x3' ... T
.
.
· T
Yi = [YI' Y2' Ya] , ... T
2, a] , Y i = [YI' Y2,Ya] ·
Xi = [Xl' X X
(9.5.1)
292
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
Irreducible Representations: 0h
Table 9.9
°h f 1 f 2
I, D 1, D 2 , D 3
(I, D 1, D 2 , D 3 ) . M 1
(I, D 1, D 2 , D 3 ) · M 2
B.Q.
1
1
1
,', ...
1
1
1
"p, "p', ...
E
B
A
a, b, ...
°h f 1 f 2
B.Q.
1
1
1
, ', ...
1
1
1
"p,,,p', ... a, b, ...
x1'~'''·
I, D 1, D 2 , D 3
(I, D 1, D 2 , D 3 ) · M 1
(I, D 1, D 2 , D 3 ) · M 2
x1'~''''
(I, D 1, D 2, D 3 ) . M 2
Y1' Y2' ...
fS
I, D 1, D 2 , D 3
(I, D 1, D 2, D 3 ) . M 1
(I, D 1, D 2 , D 3 ) . M 2
Y1'Y2' ...
f6 f7
I, D 1, D 2 , D 3
(I, D 1, D 2, D 3 ) . M 1
f 6 f7
1
1
1
a,a', ...
1
1
1
13, 13', ...
f S f9
E
B
A
C,D, ...
I, D 1 , D 2 , D 3
(I, D 1, D 2 , D 3 ) · M 1
(I, D 1, D 2 , D 3 ) . M 2
flO
I, D 1, D 2 , D 3
(I, D 1 , D 2 , D 3 ) · M 1
(I, D 1 , D 2 , D 3 ) · M 2
(I, D 1, D 2 , D 3 ) · T 2
(I, D 1, D 2 , D 3 ) · T 3
B. Q.
-1
-1
-1
a,a', ...
-1
-1
-1
{3, {3', ...
-E
-B
-A
C, D, ...
X 1,X2 ,.. ·
fS f9
C, R1'~'~
(C, R1'~'~)· M 1
(C, Rl'~'~) . M 2
X 1,X2 ,.. ·
Y1, Y2 ,.. ·
flO
C, R1'~'~
(C, Rl'~'~)' M 1
(C, R1'~'~) . M 2
Y 1,Y2 ,· ..
(C, R1'~'~) . T 2
(C, R1'~'~)' T 3
B. Q.
°h
(I, D 1, D 2 , D 3)· T 1
(I, D 1, D , D ) . T 2 2 3
(I, D 1, D 2 , D 3 )· T 3
x1,x2' ...
f S (C, Rl'~'~)' T 1
(C, R1'~'~)· T 2
(C, R1'~' ~) · T 3
Y1'Y2' ...
r1 r2 r3 r4 r5 f6
f S f9
(C, R1'~'~)' M 2
A
fS
f6 f7
(C, R1'~'~)' M 1
h (Continued)
B
(I, D 1, D 2 , D 3 ) . M 2
f3 f4
C, R1'~'~
°
E
(I, D 1, D 2 , D 3 ) . M 1
f 1 f 2
Irreducible Representations:
293
f 3 f4
I, D 1, D 2 , D 3
(I, D 1, D 2 , D 3 ) . T 1
Hexoctahedral Class, 0h' m3m
Table 9.9
f3 f4
°h
Sect. 9.S]
1
1
1
, ', ...
-1
-1
-1
"p,,,p', ...
F
G
H
a, b, ...
(C, R1'~'~)' T 1 1
1
1
, ', ...
-1
-1
-1
"p, "p', ...
F
G
H
a, b, ...
(I, D 1, D 2 , D ) . T 3 1
(I, D 1, D 2 , D 3 ) · T 2
(I, D 1, D 2 , D 3 ) · T 3
xl'~'·"
(C, R1'~' R 3 )· T 1
(C, Rl'~'~)· T 2
(C, R1'~'~)' T 3
Y1' Y2' ...
1
1
1
a,a', ...
-1
-1
-1
13,13', ...
r7
F
G
H
C,D, ...
(C, R1'~'~) . T 2
(C, Rl'~'~)· T 3
X 1,X2 ,.. ·
(I, D 1, D 2, D 3 ) . T 2
(I, D 1, D 2, D 3 ) . T 3
Y 1 , Y 2 ,· ..
-1
-1
-1
a,a', ...
1
1
1
{3, 13', ...
-G
-H
C,D, ...
(I, D 1, D 2 , D 3 ) . T 1
(I, D 1, D 2 , D 3) . T 2
(I, D 1, D 2 , D 3 ) . T 3
X 1,X2 ,.. ·
fS -F f 9 (C, R1'~'~)' T 1
flO (C, R1'~'~)' T 1
(C, R1'~'~) . T 2
(C, R1'~'~)' T 3
Y1, Y2 ,..·
flO (I, D 1 , D 2 , D 3 ) . T 1
(Continued on next page)
Generation of Integrity Bases: The Cubic Crystallographic Groups [Ch. IX
294
Table 9.9A
fi
Basic Quantities: 0h
Sect. 9.5]
symmetry type (n1". n p ) arising as products of integrity basis elements
of lower degree for cases where nl + ·u
SII + S22 + S33
f2
~ (S33 - S22)]T
[2S ll - S22 - S33'
f4
[S23' S3l' SI2]T
f5
[aI' a2' a3]T, [A 23 , A 3l , A 12]T
f9
[PI' P2' P3]T
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) The further restrictions imposed on W*( ... ) by the requirement of invariance under 0h are given by
22
6
51
42
222
1
2
1
3
1
2
1
Q&l..· n p
0
1
1
2
1
2
1
nl .. ·np
1
1
2
1
5
9
5
These are given by
2.
EX}Xy,
(2);
4.
EX}XrX~Xf,
6.
EXlXrx~xtxfx~,
(9.5.4)
(4); (6).
We may set Pi = Xi (i = 1,2,... ) in (9.5.4) to obtain the typical multi-
X~X~, X~X1) = W*(X\Xl, X~X1, X~X~)
= W*(X~X1, X~X~, XiXl) = W*(X~X~, Xixil' X~X1) =
4
functions of Xl' X 2 , ... which are invariant under 0h are comprised of one set of invariants of each of the symmetry types (2), (4) and (6).
Functions W(X1 ,... ,Xn ) of the quantities X 1,.",Xn of type f 9 which are invariant under the subgroup D2h = { I, D 1, D 2 , D 3 , C, R 1,
W*(Xixil'
2
With Table 9.10, we see that the typical multilinear basis elements for
9.5.1 Functions of Quantities of Type r 9: 0h
~,
n1··· n p p9 nl· .. n p
Xe
flO
+ np = 2, 4, 6 and P&l u. np # O.
Invariant Functions of Xl' X 2 , .. ·: 0h
Table 9.10
f3
295
Hexoctahedral Class, 0h' m3m
linear basis elements for functions of n vectors PI"'" Pn which are invariant under 0h' (9.5.3)
W*(X~X~, X~X1, XiXil) = W*(X~X1, XiXl, X~X~)
9.5.2. Functions of n Symmetric Second-Order Tensors: 0h The restrictions imposed on functions of the n symmetric
where i,j = 1,... , n; i ~ j. With Theorem 3.4 of §3.2, we see immediately
second-order tensors Sl'"'' Sn by the requirement of invariance under
that the integrity basis elements for functions of Xl' ~, ... which are
the group 0h are identical with the restrictions imposed by the
invariant under 0h are of degrees 2, 4 or 6. We list in Table 9.10 the 9 number Pn n of linearly independent sets of invariants of symmetry
requirement of invariance under the group T d'
multilinear elements of an integrity basis for functions of Sl ,... , Sn
type (nlu, np) and the number Q&l'" np of sets of invariants of
which are invariant under 0h are given by the invariants (9.4.16).
1'" p
Thus, the typical
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
x
quantities of types f l' f 3' f 4 and for functions of quantities of type f 5
IRREDUCIBLE POLYNOMIAL CONSTITUTIVE EXPRESSIONS
which are invariant under the group 0h may be obtained from the results (9.4.6) and (9.4.8) respectively. 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
i
1
· I 1... 1n·n (i 1,... ,i n =0,1,2, ... ). 11··· 1n 1
W(E)=c.
(10.1.1)
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(1 1,... , 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 determination of a set of syzygies K i (I 1,... ,ln ) (i = 1,... ,p) such that every syzygy K(1 1,... , In) = 0 relating the invariants 11"'" In is expressible in the form K(1 1,... , In) = d 1K 1(11 ,... , In)
+... + d p K p (1 1,... , In)
(10.1.2)
where the d i 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
Vector-valued functions
P(S)
299
and
symmetric second-order
expression (10.1.1). We may employ the relations K i (11'.'" In) = 0 to remove the redundant terms from (10.1.1). The objective is to produce
tensor-valued functions T(S) of the symmetric second-order tensor S are
a general expression for W(E) which does not contain any redundant
said to be invariant under the group A = {AI' A2, ... } if
terms. Such an expression is referred to as being irreducible. (10.1.7) 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
holds for all A K 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
under 03 is given by
+ + arJr(S), b 1N 1(S) + + bsNs(S)
P(S) = a1 J 1(S) (10.1.3) We then have
T(S) =
(10.1.8)
where the ai' b i 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 (10.1.4)
respectively.
It is assumed that no term Jp(S) in (10.1.8)1 is ex-
pressible as Jp(S) = c1 J 1(S)
The expression (10.1.4) may be written as
+ ... + cp _1 J p _ 1(S) +
(10.1.9)
+ cp +1 J p + 1(S) + .·· + crJr(S) (10.1.5) where the ci are scalar-valued polynomial functions of S which are in-
where W(n)(S) denotes a linear combination of the invariants of degree n in S appearing in (10.1.4). For example, (10.1.6) We may compute the number ( = 4) of linearly independent invariants of degree 4 in S.
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 N 1(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
This indicates that there is no linear relation (10.1.10)
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
We wish to determine vector-valued and symmetric second-order
the terms of arbitrary degree n appearing in (10.1.4) are linearly
tensor-valued relations
independent. This requires the introduction of the notion of generating functions which will be discussed in §10.2.
L i (I 1,... , In; J 1,... , J r )
=0
M i (I 1,· .. , In; N 1,.. ·, Ns ) = 0
(i
= 1,2,
), (10.1.11 )
(i = 1,2, )
[Ch. X
Irreducible Polynomial Constitutive Expressions
300
Sect. 10.2]
Generating Functions
301
such that all redundant terms appearing in the expressions (10.1.8) may (10.2.1 )
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
Suppose that we have determined an expression for Z(S) consistent with (10.2.1). We proceed by writing
in S which are invariant under A. This is to hold for all values of n.
(10.2.2)
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' r 2'....
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
rv
(v
= 1,2,... )
which are subject to the re-
strictions that
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
rv
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) the irreducible expression required.
+...+ Z(n)(S) + ... would be
Let s denote a column vector whose entries are the six inde(10.1.12)
pendent components of the symmetric second-order tensor S. Thus
V
· rK must hold for all A K belonging to the group A. The matrIx == r v( A K ) is the element of the set of matrices comprising the
(10.2.3)
of A which corresponds to the element A K of the group A = {A1,... ,AN }. We note that the arguments employed
Let R(A K ) == R K (K = 1,2,... ) denote the matrices comprIsIng the matrix representation R = {RK } which defines the transformation
irreducible representation
rv
in this chapter have been discussed by Smith and Bao [in press].
properties of s under the group A = {AI' A 2 , ... }. Let {R~)} denote the matrix representation which defines the transformation properties of the monomials
10.2 Generating Functions (10.2.4)
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
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 R K . The numbers of linearly independent functions of type r v which are of degrees 1 and f respectively in the components of sand
302
[Ch. X
Irreducible Polynomial Constitutive Expressions
which are invariant under the finite group A = {AI'''.' AN} are given by (10.2.5)
where the
ri are the matrices comprising the matrix representation
defining the transformation properties of a quantity of type the group A. We note that tr
R~)
the expansion of the quantity l/det (E6 - x RK ) where E 6 is the 6 X 6 identity matrix. Thus, we have 1
_
2
(2)
Irreducible Expressions: The Crystallographic Groups
" (')2]T 2 sl s2' s22]T ' [( sl')2 ,sls2' s2 = d·lag (2 £1' £1 £2' £22) [ sl'
[(s~i, (s1)2 s2 , sl(s2)2, (s2)3]T
rv
(10.2.9)
2 3) [3 2 2 3]T _ d· (3 2 - lag 6'1' 6'1£2' £1 £2' £2 sl' sl s2' sls2' s2 '
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
+ ... + xf tr R K(f) + .... (10.2.6) (10.2.10)
With (10.2.5) and (10.2.6), we see that the number of linearly independent quantities of type
303
rv
under is given by the coefficient of xf in
det(E _ x R ) - 1 + x tr R K + x tr R K 6 K
Sect. 10.3]
which are of degree n In the
components of s and which are invariant under A is given by the coefficient of x n in the expansion of the quantity
(10.2.7)
Gv(x) is referred to as the generating function for the number of
We observe that these quantities are the coefficients of x, x 2, x 3 , respectively in the expansion of 1
_
1
det(E2 - x RK ) - (1 - x£I)(1 - x£2) 2 x 3£1+··· 3 ) ( l+x£2+ x 2£2+ 2 x 3£2+···) 3 -_ (1 +x£l+ x 2£1+
linearly independent quantities of type r v .
(10.2.11 )
We give an example to indicate how one arrives at the result Suppose that RK = diag(£I' £2) is the 2 x 2 matrix which defines the transformation properties of the column vector [sl' S2]T (10.2.6).
under A K. We have
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 A K 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]
metric second-order tensor-valued functions T(S) of the symmetric
rf, ... ,~ = E,
second-order tensor S which are invariant under a given crystallographic group A.
We list in Table 10.1 (see Smith [1962b]) the
Irreducible Expressions: The Crystallographic Groups
F, -F, -E, K, L, -L, -K
where
quantities det(E6 - x RK ) appearing in the generating function (10.2.7) for each of the A K appearing in the various crystallographic groups.
(10.3.2)
The matrices I, C, ... are defined by (1.3.3). The matrix R K is the symmetrized Kronecker square of A K and defines the transformation of s (see (10.2.3)) under A K . The procedure described in this section follows closely that given by Bao [1987]. Table 10.1
The characters of the irreducible representations (10.3.1) and (10.3.2) to be given by
rl, ... ,tr r~ = tr ry, ... ,tr r§ = tr
Det (E6 - x R K )
Ii,···,tr Ii = tr rf' ···,tr :r§ = tr rf, ···,tr ~ = tr
I, C
(1-x)6
R1,~,R3,D1,D2,D3'(I,C,R1,D 1)·T1, (I,C,~,D2)·T2
(1_x)2(1_x2)2
(I,C,~,D3) .T3, (R1,~,D1,D2)· (8 1,82)
(1-x)2(1_x 2)2
(I, C, R 1, ~, R 3, D 1, D 2, D 3) . (M 1, M2), (I, C) . (8 1,8 2)
(1_x 3)2
305
1,
1,
1,
1,
r 1'... ' r S are seen from 1·,
1,
1,
1,
1, -1, -1,
1, -1,
1,
1, -1;
1, -1, -1,
1,
1,
1,
1,
1, -1, -1, -1, -1;
2,
0,
0, -2,
1, -1, -1,
0,
0,
0,
1·,
(10.3.3)
o.
We list below, the linear combinations of the components Pi and T ij of a vector P and a symmetric second-order tensor T whose trans-
(~,R3,D2,D3)· T 1, (R1,~,D1,D3)· T 2, (R1,~,D1,D2)· T 3 (1-x2)(1-x4 ) (1-x)(1-x+x2)(1-x3)
formation properties under D 2d are defined by the irreducible representations r 1,... ,rS (see Table 7.6A, p.17S).
There are five inequivalent irreducible representations associated
(10.3.4)
(~, D3) . (8 1,82)
10.3.1 The Group D 2d with the group D 2d = {AI'···' AS} = {I, D I , D2 , D3 , T 3, DI T 3 , D2T 3 , D3T 3}. These are seen from §7.3.7 to be given by 1 rl8 -., -1· r 1'···'
1,
1, 1.,
ry, ,r§ = 1, -1, -1,
Ii, ,Ii = 1, -1, -1, rf, ,r§ = 1, 1, 1,
1, -1, 1,
We refer to the quantities listed in (10.3.4) as quantities of types
1, 1, 1, 1 ., 1,
r 1,... ,rS·
generating functions Gv(x) for the number of linearly independent
1, -1;
1, -1, -1,
1j
1, -1, -1, -1, -1;
With (10.2.7), (10.3.3) and Table 10.1, we see that the
(10.3.1)
quantities of type by
rv
which are invariant under the group D 2d are given
Irreducible Polynomial Constitutive Expressions
306
[eh. X
Sect. 10.3]
Irreducible Expressions: The Crystallographic Groups
307
the expression aO(K 1,... , K 6 ). The distinct monomial terms appearing in the polynomial aO(K 1,... , K 6 ) are given by the monomial terms appearing in the expression
(1 + KI
+ Ky +...) (1 + K2 + K~ +...)... (1 + K6 + K~ +...).
(10.3.9)
(10.3.5) The number of distinct monomials appearing in aO(K 1,... , K 6 ) 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·.1 Thus, the number of distinct
monomial terms of degree n in 8 appearing in aO(K 1,... , K 6 ) is given by the coefficient of x n in the expression We see from §7.3.7 that a polynomial function WI (8) of 8 which invariant under the group D 2d , i.e., a function of type expressible as a polynomial in the quantities
IS
r l'
is
(1
+ x + x 2 + x3 + ... )2 (1 + x 2 + x4 + x6 + ... )3 (1 + x4 + x 8 + x 12 + ...) (10.3.10)
where we have noted that the degree in 8 of K 1,... , K 6 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(K 1,... , K 6 ) is given by the coefficient of xn in the formal expansion of 1
We observe that
(10.3.11 )
(10.3.7) 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 K 1,···,K6 defined by (10.3.6). The invariants K 1,... , K 6 are functionally indeThus, there are no polynomial relations K(K 1,.. ·, K 6 ) = 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 al L1, a2L2' a3Ll L2 is given by the coefficient of x n in the expressions
x3
x3 2 4 (l-x)2(1-x )3(1-x )' (l-x)2(I-x2)3(1-x4 )' (10.3.12)
pendent.
where we see from (10.3.6) that L1 and L2 are each of degree three in S.
[Ch. X
Irreducible Polynomial Constitutive Expressions
308
Sect. 10.3]
Irreducible Expressions: The Crystallographic Groups
309
(10.3.15)
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 x n in the expansion of (10.3.13) This coincides with the expression for G 1(x) given in (10.3.5). The coefficient of x n in the expansion of G 1(x) gives the number of linearly
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 x n in the expansions of
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
(10.3.16)
(10.3.8) is equal to the number of linearly independent invariants of degree n in S.
x + 2x 2 + 2x 3 + 2x4 + x 5 (1 - x)2 (1 - x 2)3 (1 - x4)
We conclude that the expression W 1(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
r 2'·'" r 5
are given by
r 2:
W 2(8)
= b 1812(8 U + b 3(8 U -
8 22 ) + b2812(8~1 -
8~3) + 822)823831 + b4823831 (8~1 - 8~3)
r 3:
W3(8)
= (c1 + c2 L2)8 12 + (c3 + c4L2)823831
r 4:
W4(8)
= (d 1 + d2L1)(8 U -
r 5:
V(S) =
8 22 ) + (d3 + d4L1)(8~3 -
(10.3.14)
8~1)
5
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 independent functions of types r 2,... ,r5 respectively, we conclude that the expressions (10.3.14) are irreducible. We may now list the general irreducible expressions for vectorvalued functions P(S) and symmetric second-order tensor-valued functions T(S) which are invariant under D 2d . With (10.3.4), we see that [P1' P21T and P 3 are quantities of types f S and f 3 respectively. With (10.3.14), the irreducible expression for P(S) is given by
E eiVi + e6 L1V1 + e7L1V2 + e8 L2V1 i=1
where the b , b , ... , eS are polynomial functions of the invariants 1 2 K ,... , K given by (10.3.6), where L1 and L2 are invariants given by 6 1 (10.3.6) and where V 1,... ,VS are defined by
(10.3.17)
where a1, ... ,a8' b 1,· .. ,b4 are polynomials in the invariants K 1,...,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 R , 03 3
311
TIl +T 22 =cO+c1 L1 +c2 L2+ c3L 1L2'
syzygies exist, this is reflected in the form of the generating function.
T 33 = dO + d 1L1 + d 2L2 + d 3L1L2 ,
In some cases, the syzygies are known or may be determined. With the
T 12 = (e1 + e2 L2)S12 + (e3 + e4L2 )S23 S31'
= (£1 + £2L 1)(8 11 -
T11 - T 22
aid of the syzygies, we may establish an irreducible expression. (10.3.18)
822 ) + (£3 + £4Ll)(8~3 - 8§1)'
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
5
T
We
[T 23 , T 31 ] = EgiVi+g6L1V1 +g7L1V2+g8L2V1 i=1 where the cO, ,g8 are polynomials in the invariants K 1,... ,K 6 . The quantities K 1, ,K 6, L1, L2 and V 1,... ,V5 are defined in (10.3.6) and (10.3.15) respectively. The general expression for an nth-order tensor-valued function . (8) which is invariant under D 2d is readily generated. We may 11·" 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 T·
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 ), G ( ... ; R3 ), 2 G3 (···; R3 ) for the number of linearly independent scalar-valued, vectorvalued, symmetric second-order tensor-valued and skew-symmetric second-order tensor-valued functions of the vectors xl'.'" x ' the skewm symmetric second-order tensors AI' ... ' An and the symmetric secondorder tensors 81,... , 8p which are invariant under R3 are given by
which will automatically generate such results for any of the crystallographic 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 for . almost all of the crystallographic groups by Bao [1987]. 11··· 1n
where i = 0, ... ,3; IXjl, lak l, Is£1 10.4 Irreducible Expressions: The Orthogonal Groups R3 , 03
XO(O)
= 1,
Xl (0)
< 1 and
= X3(0) = e iO + 1 + e-iO ,
The general expressions for functions of vectors xl' x2' ... , skewsymmetric second-order tensors AI' A 2 , ... and symmetric second-order tensors 8 1, 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
(10.4.2)
L(8) =
1
o
0
o o
cos 8
-sin 8
sin 8
cos 8
The matrix L(2)(0) in (1004.1) denotes the symmetrized Krollecker
312
[Ch. X
Irreducible Polynomial Constitutive Expressions
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
traceless tensors which are invariant under R3 may be readily deter-
representations of R3 which define the transformation properties of
mined if we are given the generating functions for the number of
scalars, vectors, symmetric and skew-symmetric second-order tensors
linearly independent functions of two-dimensional symmetric 2nth-
respectively. The factor (1 - cos B)dB in (10.4.1) is the volume element
order tensors which are invariant under the two-dimensional unimodular group.
associated with the group R3 . With (10.4.2), we have
Functions which are invariant under the two-dimen-
sional unimodular group are studied in the classical theory of (10.4.3)
det(E6 - sL(2)(9)) = (1- se2i9 )(1_ sei9 )(1- s)2(1- se-i9 )(1- se-2i9 ). The generating functions G O(."; 03)' G 1(... ; 03)' G 2 (···; 03)' G 3 (.. ·; 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'.'" x m ' 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
~Gk(Xl""'Xm'
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.
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.
Integrity basis:
al,· ..,an , sl""'sp; R3) +
+~Gk(-Xl'''''-Xm' al,..·,an , sl""'sP;
l=x·x:=xTx;
R3) (k = 0,2,3);
(10.4.5)
Irreducible expressions. (10.4.4)
Gl(xl""'xm , al,..·,an , sl'''''sP; 03)
We have, in fact, obtained the generating functions
10.4.1 Invariant Functions of a Vector x: R3
Gk (X1'''·'Xm , a1, .. ·,an , sl'·"'sp; 03) =
invariants. The use of generating functions in classical invariant theory
=~Gl(Xl'''''Xm' al,· ..,an ,
The irreducible scalar, vector, sym-
metric second-order tensor and skew-symmetric second-order tensorvalued functions of x which are invariant under R3 are given by
sl""'sP; R3) -~Gl(-Xl'''''-Xm' al,.. ·,an , sl""'sP; R3)· (10.4.6) 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
T(x)
= Ti/x) = ~15ij + a3 xixj'
A(x) = Aij(x) = a4cijkxk
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)). Generating functions.
The generating functions for the number
Irreducible Polynomial Constitutive Expressions
314
[Ch. X
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 1 21I"
Xk(O) (l-cosO)dO Gk(x; R3 ) = 2 ' 0 '0 ' 11" 0J (1 - xel )(1 - x)(l - xe- I )
I x 1< 1
315
t'....
where the cO' 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
We see that the coefficient of x n in (10.4.13) gives the number of
2 cosO = z + z-l
dO = dz/iz,
Irreducible Expressions: The Orthogonal Groups R • 03 3
(10.4.7)
where k = 0,... ,3 and where XO(O), ... , X3(0) are given by (10.4.2). order to evaluate the integral GO(x; R3), for example, we set z - e iO ,
Sect. 10.4J
In
(10.4.8)
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
in (10.4.7) so as to obtain -1 GO(x;R3)=411"i(1_x)
J z(l-xz)(z-x)' (l-z)2 dz
degree n in x appearing in (10.4.12) are linearly independent. There are (10.4.9)
no redundant terms and thus (10.4.12) is irreducible. We may refer to
I traversed in the counterclockwise
HO(x; R3) as the generating function for the number of monomial terms appearing in (10.4.12).
Ixl<1
C
Iz I =
where the contour C is
direction. The residues at the simple poles inside C at z = 0 and z = x are given respectively by (10.4.10)
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
The value of the integral in (10.4.9) is given by 211"i times the sum of
(10.4.14)
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
HI (x; R3) = (I
Go(x; R3 ) = ~ , I-x I +x 2 G2 (x; R3) =--2' I-x
We set dk=1 (k=0,1,2, ... ) and replace I=x·x by x 2, x by x in (10.4.14) to obtain
(10.4.11)
Similarly, we see that
+ x2 + x4 +...)x = x (1- x2)-1 =
G I (x; R3). (10.4.15) (10.4.16)
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 redundant terms and are irreducible.
[Ch. X
Irreducible Polynomial Constitutive Expressions
316
Sect. 10.4]
Irreducible Expressions: The Orthogonal Groups R3' 03
317
. (10.4.20) F(x, fJ) = det(E3 - xL(fJ)) = (1- xe1fJ )(1- x)(l- xe- ifJ ),
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
1 + XYZ Go(x, y, z,. R3) = --n2-------..:-~-~------
the general irreducible polynomial scalar, vector and symmetric second-
(1 - x )(1 - xy)(l - xz)(l - y2)(1 - yz)(1 - z2) ·
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
The matrix L(9) in (10.4.20)2 is defined in (10.4.2).
k = 0, 1, 2 and G3 (Xj
Syzygy:
03) is O. Thus, with (10.4.4) and (10.4.11), we have GO(Xj
03)=~GO(Xj
2
R3)+!G O(-Xj R3 )=(1-x )-1,
~G1 (-Xj
127-
G (Xj 03) = 1
~G1 (Xj
R3) -
G (Xj 03) = 2
~G2(Xj
2 2 R3) +!G 2(-Xj R3) = (1 +x )(1-x )-1,
2 R3) = x (1 - x )-1,
G (Xj 03) = !G3 (Xj R3) +~G3(-Xj R3) = 3
(10.4.17)
xl
x2
xa
xl
Y1
z1
Y1 zl
Y2 z2
Ya za
x2
Y2
z2
xa
Ya
za
x·x x·y x·z y·x y.y y·z z·x z·y z·z
(10.4.21 )
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
o.
10.4.3 Scalar-Valued Invariant Functions of Three Vectors x, y, z: R3 Integrity basis:
1 ,... ,1 =x.x, x·y, x·z, y.y, y·z, z·z, 6 1
The syzygy (10.4.21) shows immediately that the terms involving Ii (10.4.18)
(i
~ 2) are redundant so that (10.4.22) reduces to (10.4.19).
7
17 = det (x, y, z) = Cijk xi Yj zk T
where x· x = xTx, X = [xl' x2' x3] , and Cijk denotes the alternating
10.4.4 .Scalar-Valued Invariant Functions of Three Vectors x, y, z: 03
symbol. Integrity basis: Irreducible expression:
(10.4.19)
Generating function:
1 GO(x, y, Zj R3) = 211"
11,... ,I6 =x.x, x·y, x·z, y.y, y·z,
Z·Z.
(10.4.23)
Irreducible expression:
J F(x,(1fJ)-F(y, cos 8) d8 fJ) F(z, fJ) , Ix I, 1y I, 1Z I < 1,
211"
o
(10.4.24)
318
[Ch. X
Irreducible Polynomial Constitutive Expressions
Generating function. With (10.4.4) and (10.4.20)3' we have
Sect. 10.4]
Irreducible Expressions: The Orthogonal Groups R 3' 03
319
10.4.5 Invariant Functions of a Symmetric Second-Order Tensor S: R
3
Integrity basis:
(10.4.25) 1
(10.4.29) Irreducible expressions.
The monomial terms contained In the expression (10.4.24) given by
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) are identical with the monomial terms contained in (10.4.27) 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 x m yn zP in the generating
(10.4.30) where the aO"'" a3 are polynomial functions of the invariants 1 ,1 ,1 1 2 3 defined by (10.4.29). There are no vector-valued functions P(S) or skew-symmetric second-order tensor-valued functions A(S) which are invariant under R3 . Generating functions:
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 x 2 , xy, xz, y2,
2
. _...L I1rXk(B)(I-COSB)dB G k (s, R3) - 211" 0 F(s,8)
yz, z2 respectively. Thus, HO(x, y, z; 03)
= 0,1,2,3)
(10.4.31)
where Is 1< 1; XO(B), ... , X3(B) are given by (10.4.2) and
=
= (1 + x 2 + x 4 + )(1 + xy + x2 y 2 +) _
(k
( 1 + z 2 + z4 +
1
.)
F(s,8)
= (1- se2i8 )(1_ sei8)(1-s)2(1-se-iB)(1_se-2iB).
(10.4.32)
(10.4.28)
- (1 - x 2)(1 - xy)(1 - xz)(l - y2)(1 - yz)(1 - z2) . 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 scalarvalued functions of x, y, z which are invariant under R3 is also irreducible.
With (10.4.2), (10.4.31) and (10.4.32), we have Go(s; R3) =
12 , G 1(s· R3 ) - 0 (1 - s)(1 - s )(1 - s3) ,- , (10.4.33)
320
[Ch. X
Irreducible Polynomial Constitutive Expressions
With (10.4.33), we may readily establish the irreducibility of the
Sect. 10.4]
Irreducible Expressions: The Orthogonal Groups R 3' 03
321
A(R, S) = (cO + cl l lO)(RS - SR) + C2(R2S - SR2) + C3(RS2 - S2R)
expressions (10.4.30).
+ C4(R2S2 - S2 R 2) + cs(R2SR - RSR2) + C6(S2RS - SRS2) + C7(R2S2R - RS 2R 2) + cs(S2R2S - SR2S2).
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 secondorder tensor-valued functions of S which are invariant under 03. There
The coefficients aO' aI' a2' b O'·'" b 17 , cO,·'" c8 are polynomial functions of the invariants 11,... ,1 9 defined by (10.4.34). There are no vectorvalued functions P(R, S) which are invariant under 03.
are no vector-valued functions P(S) or skew-symmetric second-order tensor-valued functions A(S) which are invariant under 03. observe that the generating functions Gk(s; 03)
= Gk(s;
R3) for k
We
Generating functions. Let
= 0,
. _ I 2'Tr Xk(8) (1- cos 8) d8 Gk(r, s, R3) - 211" 0 F(r, 8) F(s, 8)
J
1, 2, 3. 10.4.7 Invariant Functions of Symmetric Second-Order Tensors R, S: 03 Integrity basis:
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
11, ,IS=trR, trS, trR2 , trRS, trS 2, 16, ,1 10 = tr R 3, tr R 2S, tr RS 2, tr S3, tr R 2S2. Irreducible expressions.
Go(r, Sj 03) = GO(r, Sj R3) = (1 + r 2s 2 + r 4s4)/ K(r, s),
(10.4.34)
G 1(r, Sj 03) = ~GI (r, Sj R3) - ~GI (r, Sj R3) = 0,
The irreducible scalar-valued, sym-
G2(r, s; 03)
metric second-order tensor-valued and skew-symmetric second-order
+ (b 7 + b Sl lO )R2 + (b g + b lO l lO )(RS + SR) + (b n + b 1211O )S2 + (b 13 + b 1411O )(R2S + SR2) + (b iS + b 1611O )(RS2 + S2R) + b 17(R2S2 + S2R2),
= G2(r, s; R3)
+ r + s + r 2 + rs + s2 + r 2s + rs 2 + 2r2s2 + r 3s2 + r 2s3 + r 4s2 + r 3s3 + r 2s4 + r 4s3 + r 3s4 + r4s4)f K(r, s),
variant under 03 are given by
= (b O+ b l l lO + b 21Io)E3 + (b 3 + b 411O )R+ (b S + b 611O)S
where K(r, s) = (1 - r)(1 - r 2)(I- r 3)(1 - s)(1 - s2) · (1 - s3)(1 - rs)(1 - r 2s)(1 - rs 2 ).
(10.4.35)
(10.4.37)
= (1
tensor-valued functions W(R,S), T(R,S) and A(R,S) which are in-
T(R, S)
(10.4.36)
(10.4.38)
Since the invariants 11,... ,1 10 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
Irreducible Polynomial Constitutive Expressions
322
i i 10 W(R, S) = W(I 1,.. ·, 110 ) = c· . 1 1 ... 110 . 11"· 110 1
[Ch. X
(10.4.39)
Sect. 10.5]
Traceless Symmetric Third-Order Tensor: R3 , 03
323
Similarly, the number of monomial terms of degree m, n in R, S appearing in the expressions al (11'·'" 19)1 10 and a2(1 1,... , 19 )110 are given by the coefficient of r m sn in the expressions obtained by multiplying (10.4.44) by r 2 s2 and by r 4 s4 respectively. The sum of
We may also write (10.4.39) as (10.4.40)
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
where the cO' c1' c2'." are polynomials in the invariants 11,... ,1 9. Smith [1973] has shown that
aO + al l lO + a2 110 where ai
= ai(1 1,... , 19 ).
We have (10.4.45)
(10.4.41 ) With (10.4.37)1 and (10.4.45), we see that HO(r, s; 03) = GO(r, s; 03). where the f3 ' f3 , f3 2 are polynomials in the invariants 11,... ,1 9. With 0 1 (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
Hence the expression W(R, S) = aO + al l lO + a2110 in (10.4.35) contains no redundant terms and is irreducible. The details of the argument leading to the result that any
(10.4.35)1' i.e.,
symmetric second-order tensor-valued polynomial function of R, S 2
W(R, S) = aO + a1 I 10 + a2 I 10
(10.4.42)
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].
where the aO' aI' a2 are polynomials in 11,... ,1 9. The monomial terms contained in the polynomial expression aO(I 1,... ,I 9), say, are identical with the monomial terms contained in (1
2 2 2
+ II + 11 + ... )(1 + 12 + 12 + ...)... (1 + 19 + 19 +...).
It is also shown there that
the generating function H 2(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.
(10.4.43)
The number of monomial terms of degree ill, n in R, S appearing in (10.4.43) is given by the coefficient of r m sn in the expression obtained 223223 . from (10.4.43) by replacIng 11,1 2, ... , 19 by r, s, r , rs, s ,r , r s, rs , s . This is given by
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 invariant under R3 . The components F ijk of F satisfy the relations
(10.4.44) 1
(10.5.1) F··· =0 =l/K(r,s).
IJJ
'
F··· =0 JIJ
'
F··· =0. JJI
324
[Ch. X
Irreducible Polynomial Constitutive Expressions
There are seven independent components of F which are given by
Sect. 10.6]
Traceless Symmetric Fourth-Order Tensor: R3
325
are invariant under 03 is given by (see Spencer [1970])
(10.5.2) (10.5.6) 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 This indicates that an integrity basis for functions of a third-order
det(E 7 -£L(O))=
(10.5.3)
= (1 - fe 3iO )(1 - fe 2iO )(1 - fe iO )(1 - f)(l - fe- iO )(l - fe- 2iO )(1 - fe- 3iO ).
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 - .l 2'lr (1 - cos 0) dO _ 1 + f15 0(' 3) - 211" 0Jdet(E - £L(O)) - (1- £2)(1 - £4)(1 - £6)(1- £10) , 7
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-
(10.5.4)
variant under R3. The components V· 1· 1..1 of V satisfy the 4! 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.
(10.6.1 )
Spencer [1970] has obtained the same result upon employing a procedure based on results from classical invariant theory. The form of the
11
234
where (0, (3, " b) is any of the 4! permutations of (1, 2, 3, 4) together with the relations
generating function (10.5.4) indicates that there are five elements 11,... ,1 5 of degrees 2, 4, 6, 10, 15 comprising the integrity basis, and further, that the irreducible expression for a scalar-valued polynomial
(10.6.2)
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 invariants 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' (10.6.3) V1222' V1223' V2222' V2223·
Irreducible Polynomial Constitutive Expressions
326
[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
REFERENCES
about the x3 axis. We may show that (10.6.4) The number of linearly independent polynomial functions of degree n in V which are invariant under R is given by the coefficient of v n in the 3
expansion of the generating function GO(v; R3 ) where
o
'3
o
(1 - cos 9) d9 4 ·k9 (I-vel )
n
k=-4
Adkins, J .E. [1960b]: Further Symmetry Relations for Materials. Arch. Rational Mech. Anal. 5, 263 - 274.
Bao, G. [1987]: Application of Group and Invariant -Theoretic Methods to the Generation of Constitutive Equations. Ph.D. Dissertation, Lehigh University.
(10.6.S)
Billings, A.R. [1969]: Tensor Properties of Materials. York. Birss, R.R. [1964]: Symmetry and Magnetism. Amsterdam.
where we take Iv I < 1. The form of the generating function indicates that the elements of an integrity basis would consist of invariants 12 , 13 , 14 , IS' 16 , 17, IS' 19 , 110 of degrees 2,. 3, 4, S, 6, 7, S, 9, 10 in V. We may further conjecture that the irreducible expression for W(V) is given by
Wiley-Interscience, New
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Boehler, J.P. [1977]: On Irreducible Representations for Isotropic Scalar Functions. Z.A.M.M. 57, 323 - 327. Boehler, J.P. [1978]: Lois de Comportement Anisotrope des Milieux Continus. J. de Mecanique 17 (2), 153 - 190. Boerner, H. [1963]: Representations of Groups. John Wiley and Sons, New York. Elliott, E.B. [1913]: Algebra of Quantics. Oxford University Press, London. Ericksen, J .L. and R.S. Rivlin [1954]: Large Elastic Deformations of Homogeneous Anisotropic Materials. J. Rational Mechanics and Analysis 3 (3), 281 - 301.
(10.6.6) where the aO' ... , a4 are polynomial functions of the invariants 12, 13 , 14 , IS' 16 , 17. We emphasize that the above statements are conjectures. We note from (10.6.S) that the number of linearly independent invariants of degrees 2, ... , 10 in V are given by 1, 1, 2, 2, 4, 4, 7, S, 12 respectively.
Transversely Isotropic
Bao, G. and G.F. Smith [1990]: Syzygies, Orbits and Constitutive Equations. Yielding, Damage and Failure of Anisotropic Solids, ed. J. P. Boehler, 147 -154. Mechanical Engineering Publications Ltd., London.
21r
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Adkins, J .E. [1960a]: Symmetry Relations for Orthotropic and Transversely Isotropic Materials. Arch. Rational Mech. Anal. 4, 193 - 213.
Fieschi, R. and F.G. Fumi [1953]: High-Order Matter Tensors in Symmetrical Systems. II Nuovo Cimento 10, 865 - 882. Franklin, F. [1880]: On the Calculation of the Generating Functions and Tables of Groundforms for Binary Quantics. Amer. J. Math. 3, 128 - 153. Fumi, F.G. [1952]: Matter Tensors in Symmetrical Systems. 1-18.
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Grace, J.H. and A. Young [1903]: The Algebra of Invariants. Cambridge University Press, London. Kearsley, E.A. and J.T. Fong [1975]: Linearly Independent Sets of Isotropic Cartesian Tensors of Ranks up to Eight. J. Res. Nat. Bur. Stds. 79B, 49 - 58.
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Littlewood, D.E. [1950]: The Theory of Group Characters and Matrix Representations of Groups, 2nd ed. Oxford University Press, London.
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Smith, G.F. [1967]: Tensor and Integrity Bases for the Gyroidal Crystal Class. Q. Applied Math. 25, 218 - 221.
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von Mises, R. [1928]: Mechanik der Plastischen Formanderung von Kristallen. Z.A.M.M. 8, 161 - 185. Murnaghan, F.D. [1937]: On the Representations of the Symmetric Group. Amer. J. Math. 59, 437 - 488. Murnaghan, F.D. [1938a]: The Theory of Group Representations. The Johns Hopkins Press, Baltimore. Murnaghan, F.D. [1938b]: The Analysis of the Direct Product of Irreducible Representations of the Symmetric Groups. Amer. J. Math. 60, 44 - 65. Murnaghan, F.D. [1951]: The Analysis of Representations of the Linear Group. An. Acad. Brasil. Ci. 23, 1-19. Pipkin, A.C. and R.S. Rivlin [1959]: The Formulation of Constitutive Equations in Continuum Physics. I. Arch. Rational Mech. Anal. 4, 129 - 144. Pipkin, A.C. and R.S. Rivlin [1960]: The Formulation of Constitutive Equations in Continuum Physics. II. A rch. Rational M echo Anal. 4, 262 - 272. Rivlin, R.S. [1955]: Further Remarks on the Stress-Deformation Relations for Isotropic Materials. Arch. Rational Mech. Anal. 4, 681- 702. Rivlin, R.S. and J .L. Ericksen [1955]: Stress-Deformation Relations for Isotropic Materials. J. Rational Mechanics and Analysis 4 (2), 323 - 425. Rivlin, R. S. and G. F. Smith [1970]: Orthogonal Integrity Bases for N Symmetric
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Smith, G.F. [1968b]: On the Generation of Integrity Bases, Parts I and II. Academia Nazionale dei Lincei 9, 51 - 101. Smith, G.F. [1970]: The Crystallographic Property Tensors of Orders 1 to 8. Annals of the New York Academy of Sciences 172, 57 -106. Smith, G.F. [1971]: On Isotropic Functions of Symmetric Tensors, Skew-Symmetric Tensors and Vectors. Int. J. Engng. Sci. 9, 899 - 916. Smith, G.F. [1973]: On Symmetric Tensor-Valued Isotropic Functions of Two Symmetric Tensors. Q. Applied Math. 31, 373 - 376. Smith, G.F. [1982]: On Transversely Isotropic Functions of Vectors, Symmetric Second - Order Tensors and Skew - Symmetric Second - Order Tensors. Q. Applied Math. 34, 509 - 516. Smith, G.F. and G. Bao [1988]: Constitutive Relations for Transversely Isotropic Materials. Composite Material Response: Constitutive Relations and Damage Mechanisms, eds. G. C. Sih et aI., 71 - 90. Elsevier Applied Science Publishers, New York. Smith, G.F. and G. Bao [in press]: On Irreducible Constitutive Equations. ceedings of the Symposium on Nonlinear Effects in Solids and Fluids.
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Smith, M.M. and G.F. Smith [1992]: Group-Averaging Methods for Generating Constitutive Equations. Mechanics and Physics of Energy Density, eds. G. C. Sih and E. E. Gdoutos, 167 - 178. Kluwer Academic Publishers, The Netherlands.
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Spencer, A.J.M. [1965]: Isotropic Integrity Bases for Vectors and Second-Order Tensors, Part II. A rch. Rational M echo Anal. 18, 51 - 82. Spencer, A.J.M. [1970]: On Generating Functions for the Number of Invariants of Orthogonal Tensors. M athematika 17, 275 - 286. Spencer, A.J .M. [1971]: Theory of Invariants, Part III of Continuum Physics, v. I, ed. C. Eringen. Academic Press, New York and London. Spencer, A.J.M. and R.S. Rivlin [1959a]: The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua. A rch. Rational M echo Anal. 2, 309 - 336. Spencer, A.J .M. and R.S. Rivlin [1959b]: Finite Integrity Bases for Five or Fewer Symmetric 3 x 3 Matrice&. A rch. Rational M echo Anal. 2, 435 - 446. Spencer, A.J .M. and R.S. Rivlin [1962]: Isotropic Integrity Bases for Vectors and Second-Order Tensors, Part I. A rch. Rational M echo Anal. 9, 45 - 63. Sylvester, J.J. [1879a]: Tables of the Generating Functions and Ground-Forms for the Binary Quantics of the First Ten Orders. Amer. J. Math. 2, 223 - 251. Sylvester, J.J. [1879b]: Tables of the Generating Functions and Ground-Forms for Simultaneous Binary Quantics of the First Four Orders, Taken Two and Two Together. Amer. J. Math. 2, 293 - 306. Sylvester, J.J. [1882]: Tables of Generating Functions, Reduced and Representative, for Certain Ternary Systems of Binary Forms. Amer. J. Math. 5, 241 - 250. Van der Waerden, B.L. [1980]: Group Theory and Quantum Mechanics. SpringerVerlag, Berlin. Wang, C.C. [1969]: On Representations for Isotropic Functions, Parts I and II. A rch. Rational M echo Anal. 33, 249 - 287. Weitzenboch, R. [1923]: In varianten - Theorie. P. Noordhoff, Groningen. Weyl, H. [1946]: The Classical Groups, Their Invariants and Representations. Princeton University Press, Princeton. Wigner, E.P. [1959]: Group Theory and its Application to the Theory of Atomic Spectra. Academic Press, New York.
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Zia-ud-Din, M. [1935]: The Characters of the Symmetric Group of Order II!. Proc. London Math. Soc. (2) 39, 200 - 204. Zia-ud-Din, M. [1937]: The Characters of the Symmetric Groups of Degrees 12 and 13. Proc. London Math. Soc. (2) 42, 340 - 355.
INDEX Abelian group 16 Adjoint matrix 21 Alternating symbol 6 Alternating tensor 6 Axial vector 5, 7 - transformation rules 7
- D6 h
194
- T 273, 275, 277 - Th 281 -° 286, 288, 291 286,287,288,291
- Td - 0h 295 - T1 261 - T2 263 - R3 238, 242, 254 - 0 3 257 Basis, tensor-valued functions - C s 169 - C 3 195 - R3 245,248 Binomial coefficient 93
Basic quantities 164 Basic quantity tables 146, 164 - C i , C s ' C 2 167 - C 2 h' C 2V ' D 2 170 - D 2h 171 - 84' C 4 172 - C 4 h 174 - D 4 , C 4V ' D 2 d 175 - D 4 h 177 - C 3 180 - C 3V ' D 3 181 - C 3 i' C 3 h' C 6 182, 183 - D 3 h' D 3 d' D 6 , C 6V 186 - C 6 h 189 - D 6 h 194 - T 270 - Th 279 - 0, T d 283 - 0h 294 - T1 154,260 - T2 156,262 Basis, scalar-valued functions - Ci' C s , C 2 168,169 - C 2 h' C 2V ' D 2 170 - D 2 h 171 - 84' C 4 173 - C 4 h 174 - D 4 , C 4V ' D 2d 176 - D 4 h 178 - C 3 180 - C 3V ' D 3 182 - C 3 i' C 3 h' C 6 184 - D 3 h' D 3 d' D 6 , C 6V 187 - C 6 h 190
Cayley-Hamilton identity 203 - generalized 205 Character - of a representation 28 - tables 33 -- Sn(n = 2,... ,8) 104-108 Characters, orthogonality properties 29 Class - of a group 18 - order of 18 Complete set of tensors 55 - D 2 h 90 - 0h 92 - 0 3 94 - R3 96 - T1 97 Constitutive equations 1, 11 - non-polynomial 11 Conjugate elements 18 Coset 18 Decomposition - of matrix representations - of physical tensors 114
333
24
Index
334
- of sets of property tensors 56, 88 - of representations (m1". m p ) . (n1." n q ) 231 - of representations (n1". n p ) x (m) 232 Determinant of a matrix 6 Dimension of a representation 21 Direct - product of groups 61 - product of representations 61 - sum of representations 24 Equivalent - coordinate systems 7 - matrix-valued functions - reference frames 2 - representations 23 Frame 62 Function - invariant under a group - basis 12
209
Inner product - property and physical tensors 76 Integrity basis 13, 43, 159 - irreducible 14, 297 see basis, scalar-valued functions Invariant 10, 43 - element of volume 38 - integral 38 -- over 0 3 40 -- over R3 39 -- over T} 40 -- over T2 40 Invariants - sets of symmetry type (n1." n p ) 217 Irreducible - integrity basis 14, 297 - representation 23 Irreducible constitutive expressions 14, 298 - D 2d
10
Generating functions 300 Group - characters 28 - class of 18 - continuous 36 - defining material symmetry 7 - definition of 15 - generators of 18 - manifold 37 - order of 16 - representations 20 Group averaging methods 109 - scalar-valued funcitons 109 - tensor-valued functions 117 - generation of property tensors 128 Hermitian matrix
Index
25
Identities - Cayley-Hamilton 203 - - generalized 205 - relating tensors 94, 96, 99 - relating 3 x 3 matrices 202, 207 Identity matrix 4
304
- functions of vectors: R3 , 0 3 313-318 - functions of symmetric tensors: R3 03 319-326 Irreducible representation tables - Ci' C s , C 2 167 - C 2 h' C 2V ' D 2 170 -D 2h 171 - S4' C 4 172 - C 4 h 173 - D 4 , C 4V ' D 2 d 175 - D 4h
- C3
176
180
- C 3V ' D 3
181
- C 3 i' C 3 h' C 6
182
-
D 3h , D 3 d' D 6 , C 6V C 6 h 188 D 6h 192 T 270 T h 279 - 0, T d 282 - 0h 292
- T} 260 - T2 262 Isomer of a tensor Isotropic - functions 233
56
,
- - of two symmetric matrices - tensors 94
208
Kronecker - delta 4 - product 112 - square 71 - nth power 54 - - symmetrized 111 Material symmetry 7 Matrix - 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 18 Orthogonal - matrix 4, 21 - groups 36, 37 Orthogonality relations - for irreducible representations - for characters 29, 30
Polarization process 51 Polynomial basis 13 Product tables 144 - for D 3 144, 199 Proper orthogonal group 36 Property tensors 53 - complete set of 55 - decomposition of sets of 56 - sets of symmetry type (n1n2 ...) 88 Reducible - matrix product 209 - trace of matrix product 209 Reference frame 3 - equivalent 2, 7 Representations - dimension of 21 - direct products of 61 - equivalent 23 - group 20 - irreducible 23 - matrix 21 - reducible 23 - regular 34 Rivlin-Spencer procedure 207
27
185
Partition 62 Peano's theorem 51 Permutations 19 - cycle 19 - - structure 19 - class of 20 - products of 19 Physical tensors 53 - decomposition of 114 - outer products of 53 - of symmetry class (n1n2 ...)
335
69
Schur's Lemma 24 - application of 133 Skew-symmetric tensor 5 Subgroup 18 Summation convention 3 Symmetric group 19 - character tables 104 - order of a class of 20 Symmetric tensor 5 Symmetry - group 7, 10 - transformations 7 Symmetry class - physical tensor 69 - products of physical tensors Symmetrized Kronecker - nth power 111 - square 72 Syzygy 14, 43, 297
79
57,
336
Tableau 62 - standard 63 - - ordering 63 Tensors - 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 ...) - skew-symmetric 5 - symmetric 5 - transformation properties of 3
Index
Trace of a matrix 28 Transpose of a matrix 21 Transverse isotropy groups 153, 259 Typical basis elements 162 Unitary matrix
37, 38,
21
Vector - absolute 5 - axial 5, 7 - polar 5 57
Weight function
38, 39
Young symmetry operators - properties of 66
64, 220
