CONSTITUTIVE EQUATIONS FOR ANISOTROPIC AND ISOTROPIC MATERIALS
MECHANICS AND PHYSICS OF DISCRETE SYSTEMS
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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 == (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 (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.
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
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
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
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 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 ,, ¢>', .
-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 3(r)
= (tr r)3 + 3 tr r
tr r 2 + 2 tr
:r3.
(9.1.5)
The expression for 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
, 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= 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