FOR A ANDA

INTEGRITY BASIS FOR A SECOND-ORDER AND A FOURTH-ORDER TENSOR

JOSEF BETTEN

Rheinisch-Westflische Technische Hochschule Aachen Templergraben 55

D-5100 Aachen

Federal Republic of Germany (Received July 8,

1981)

ABSTRACT. In this paper a scs!ar-valued isotropic tensor function is considered, the variables of which are constitutive tensors of orders two and four, for instance, characterizing the anisotropic properties of a material. Therefore, the system of of irreducible invariants of a fourth-order tensor is constructed. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are found. In a similar way one can construct an integrity basis for a tensor of order grea.ter than four, as shown in the paper, for instance, for a tensor of order six.

KEY WORDS AND PHRASES. Theory of ^{algebraic inviants,} integry basis under ^a subgroup, otropic tensor functions, representation- theory irreducible basic and principal invariants of ^a foh-order ^tensor, constitutive tensors, cho er-

istic polynomial, alternation process, integrity basis for ^{a tensor} of ^{ord greater}

than fo, bilinear operator, construction of simultaneous or joint invaria, Hamiton-Cayley’ s theorem, isotropic constitutive tensors.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 53A45, 73B.

i. INTRODUCT ION.

In many branches of mathematics and physics, for instance, in continuum mechanics, the central problem is: For a given set of tensors which are not neces- sarily of the same order, and a given group of transformations, find an integrity basis, the elements of which are algebraic invariants.

Many

mathematicians have studied the theory of algebraic invariants in detail.

The results are published, for instance, by GRACE and YOUNG [i], GUREVICH

[2],

WEITZENBCK ^{[3 ],}

WEYL

[4].

Very extensive accounts of algebraic invariant theory from the point of view of ^its application to modern continuum mechanics are pre- sented, for example, by SPENCER

[5],

TRUESDELL and NOLL

[6].

It is convenient to employ these results of the theory of invariants in the mechanics of isotropic and anisotropic materials

[7]. In

the theory of algebraic invariants the central problem is:

For

a given set of tensors which are not nec- essarily of the same order, and a given group of transformations, find an integrity basis, the elements of which are algebraic invariants. An

integrity

^{basis is} a set of polynomials, each invariant under the group of transformations, such that any polynomial function invariant under the group ^is expressible as a polynomial in elements of the integrity basis.

In continuum mechanics, a constitutive expression may be a polynomial function which is appropriate for the description of the response of an anisotropic material.

The representation of such an expression is based upon an integrity basis.

In this paper the system of irreducible (basic and principal) invariants of a fourth-order tensor is found. Furthermore, the joint or simultaneous invariants of a second-order and a fourth-order tensor are constructed.

The mentioned systems of invariants cannot be found in the cited literature.

Therefore,

the results of this paper are important, for instance, for a reader who is working in the field of theoretical continuum mechanics and who is familiar with tensor calculus.

2. INTEGRITY BASIS UNDER A SUBGROUP.

Let F

F(o)

^{be a} scalar-valued function of a second-order tensor, for instance, of the stress tensor

o.

This function is said to be isotropic if the condition

F(aip

^a.jq ^opq

^F(o

ij

^(2.1)

is fulfilled under any orthogonal transformation

(aik _ajk oij),

where the summa- tion convention is utilized, and represents

KRONECKER’s

tensor.

For example, the function F in

(2.1)

may be the plastic potential. Then, from

the theory of isotropic tensor functions

[5,

7,

8],

it is evident that in an iso- tropic medium the plastic potential F can be expressed as a single-valued function of the irreducible basic invariants

S tr 1,2,3

(2.2)

or,

alternatively, of the irreducible

principal

inya.ria.nts

Jl- ii J2 -i[i] j[j] J3- i[i] oj[j] Ok[k] ^(2.3a,b,c)

of the stress tensor

o,

^{that is}

F

F[S()]

^{or F}

F[J(o)]

^1,2,3

^(2.4a,b)

respectively. In

(2.3)

the operation of alternation is used. This process is indicated by placing square brackets around those indices to which it applies, that is, the bracketed indices i...k are permuted in all possible ways, while indices which are excluded from the alternation are not bracketed. They keep their posi- tions. Thus, we obtain

!

terms. The terms corresponding to even permutations are given a plus sign, those which correspond to odd permutations a minus sign, and they are then added and divided by

!.

Comparing

(2.2)

and

(2.3),

we find the relations:

Jl SI J2 ^($2 SI)/2 J3-- ^(2S3- 3S2SI ⁺

^S

^{)/6 (2.5a,b,c)}

The irreducible (basic or principal) invariants are the elements of the integrity basis for the orthogonal group:

An

_integrity basis is a set of polynomials, each

invariant under the group of transformations, such that any polynomial function invariant under the group is expressible as ^a polynomial in the elements of the integrity basis

[4, 5, 9].

The invariants

(2.2)

or, alternatively,

(2.3a,b,c)

form the integrity basis for the stress tensor

o

^{under the proper} orthogonal group, i.e.

aij

+i, and this integrity basis also forms ^a functional basis.

The representations

(2.4a,b)

imply

isotropy.

In the anisotropic case the restriction

(2.1)

of F is less severe. Then, the function F is merely required to be invariant under the group of transformations

(SikSjk ij)

associated with the symmetry properties of the material

[8],

where

s

^{is a subgroup of the} orthogonal group

a. ^In

^{other words,} the symmetry properties

of the material impose restrictions upon the manner in which the function F depends on the stress components [i0].

For a particular crystal class

[ii]

the potential F may be represented as a polynomial in the stresses which is invariant under the subgroup

s

^{of transform-}

tions associated with the symmetry properties of the crystal class considered.

The function F is then expressible as a polynomial in these invariants, which form a functional basis.

It is shown by PIPKIN and RIVLIN

[12]

and PIPKIN and

WINEMANN [13]

that an integrity basis will also form a functional basis if the group of transformations is finite; then all invariants can be expressed as functions of the invariants of an integrity basis.

3. INTEGRITY BASIS FOR A FOURTH-ORDER TENSOR.

Instead of the representation by an integrity basis under a subgroup anisotro- pic behaviour may be characterized by a function

F

F(oij Aij Aijkl Aijklmn ^(3.1)

in which

Aij, Aijkl

^{etc. are} the components of constitutive tensors characterizing the anisotropic properties of the material. Then, by analogy ^to

(2.1),

we have the invariance condition

o

;...

a

qakralsA

p ...)

F(ij; Aij

F(aipaj

q pq

ipaj qrs’

kl

(3.2)

and the central problem is: to construct an irreducible integrity basis for the tensors o

ij

Aij Aijkl

^{Together with} the invariants of the single argument tensors

oij Aij Aijkl

^{etc., like}

^(2.2)

^or

^(2.3),

^{we have to consider the system}

of simultaneous or

joint

^invariants found in section 4.

Let us first construct the irreducible principal invariants of a fourth-order tensor

Aijkl,

^{which may be a linear operator, i.e.,}

Yij Aijkl_ Im

^or

Y mA^oXo ^(3.3a,b)

where i,j,k,l 1,2,3 or

a,B

1,2

9,

respectively.

In (3.3b) the operator

A

^{defines a} linear transformation on a 9-dimensional vector space V

9, which is a correspondence that assigns to every vector

X

^{in V}₉ ^a

vector

AX

ⁱⁿ

Vg,

^{in such a way that}

A(alXI ⁺ a2X ²⁾ al I + a2

2 identically in the vectors

X _I

^and

X

₂ ^{and the scalars a}

_I

^{and a}2.

Let

X; ^1,2,...,9;

^{be the} components of an arbitary vector of unit magni- tude which we call direction vector or simply a direction. We then ask: For what directions does the linear transformation

A

^{yield a vector}

Y

^{according to}

^(3.3)

which is in the same direction as

X?

^{That is,}

(0)

_or

(A8

^%A

⁽⁰⁾

(Aijkl %ijkl)l Oij _a8 )X8 O ^(3.4a,b)

(o) A(O)

where is a real scalar to be determined,

aijkl ikjl

^or

are the components of the unit tensor

A ⁽⁰⁾

^{whereas O}_ij ^or

Oa

^are the components of the zero tensor O.

For a nontrivial solution of

(3.4a,b)

we must have

det

(Aijkl Xijkl ,(o)

^{0 or det}

(Aa ^XA(0) a ^{=o (3.5a,b)}

in order to determine the principal or proper values % of the linear transforma- n

t ion A.

In order to construct the principal invariants of a fourth-order tensor

A,

^we

note that

t-terminant ^(3.5)

^{is an invariant, and we therefore consider the}

characterist i-

’.ynomial

(0) = ^%n-

Pn(%) det(Ai3kl- ^{%ijkl J(A) (3.6)}

in which, as we see from

(3.5a,b),

the first ^ind,x pair (ij) a characterizes the rows and the second one

(kl) B

the columns of a n x n matrix

A,

in general n 9.

The principal invariants

J

ⁱⁿ

^(3.6)

^can be determined, analogous to

(2.3),

by the operation of alteration:

=A

A (3 7)

(-i)^n-9

J9

^a

_l[al Aa2[a 2] av[av]

where (-i)ⁿ

J0

^i. The right hand side in

(3.7)

is equal to the sum of all

(n) _!(n

^n!

_)!

^{principal minors of order < n, where i and n lead to}

trA

^{and det}

^A,

respectively.

Assuming the usual symmetry conditions

Aijkl Ajikl Aijlk lij ^(3.8a)

or alternatively exgressed by

1,2

⁶

^(3.8b)

the zero power tensor of fourth-order ⁱⁿ

(3.6)

is given by:

A (0)

A.

^A

^(-I) 1/2(6 +

k

A! ^I)

^A

ijkl :jpq pqkl

ik6j I il6j

:3Pq

pqkl’ (3.9)

as we can see from

(3.3)

for Y X.

Expanding

(3.7)

with n 6, we find the irreducible principal invariants of a fourth-order ^tensor by the alternation process already applied in

(2.3):

Jl Aijij

^tr

^A

J2 -2-(AijijIkl

1

Aijkllij)

J3 -- ^{.v (Aijij AklklAmnmn +} 2AijklAklmnAmnij 3AijkllijAmnmn J4 ^{__l 4!(Aijij} lkl Amnmn ^{Aopop +}

8

Aijkl Akln Amnij Aopop

⁶

Aijkl Aklij

^Amnmn ^Aopop 6

Aijkl Aklmn Amnop Aopij ⁺

³

Aijkl lij Amnop Aopmn) J5 5!(Aijij

i

ikl Aqrqr ⁺

J6 -.v(Aijij Aklkl Astst ^{+ "’’)}

^det

^A

We see that the principal invariants

(3.10)

of a fourth-order tensor can be determined uniquely by polynomial relations from the irreducible basic invariants

S

^tr

^A

^A

_{ilJ li2J} ^{A A.. (3.11)}

2

i2J 2i3J

³

l3ilJ I

that is, by analogy to

(2.5),

J P

(S

1 ^S2

Sv) ^(3.12)

) )

Both the set of the six quantities

(3.10)

or, alternatively, the six quantities

(3.11)

form an irreducible integrity basis for the fourth-order symmetric tensor

(3.8)

under the orthogonal group.

In a similar way we can construct an integrity basis for a tensor of order greater than four. For instance, we consider a tensor

A

^{of order six as a linear}

operator, i.e.

Uij

k

Aijklmn Tlmn ^(3.13)

Then we can also use the representation

(3.3b)

where

,8 1,2,...,27,

and find by analogy of

(3.4):

(0) (3 14)

(Aijklmn- ijklmn)Tlmn Oij

k In

(3.14)

the zero power tensor

,(o)

ijklmn

^{of order six is given by}

iljmkn

^and

the principal invariants can be found by the operation

(3.7)

where n 27.

In continuum mechanics a tensor A or order six is often used as a bilinear operator, that is, instead of

(3.13)

we have bilinear transformations, like

Uij Aij klmnSklTmn ^(3.5)

which appear in constitutive expressions

[14].

Now by analogy to

(3.8a,b),

the usual symmetry conditions are given by

(3.16)

where

, 8, Y

1,2 6. From

(3.15),

in connection with

Uzj %SipT

pj

6ik6

^jn

⁶ ImSklTmn ^(3.17)

and the symmetry

Uij Uji

^i.e.,

SipTpj (SipTpj ⁺ TipSpj)

^or

^{ST TS,}

^{we find}

instead of

(3.4b)

the homogeneous system

(AB- %A(O))VB

B

_0"

^{1 2}

^6;

^{B 1 2 21}

(3.z8)

where

AB

^{is a 6 x} 21 rectangular matrix, and V

B ^with B 1,2

,21

is the image of a fourth-order symmetric tensor product

VB

Skl

^Tmn

Slk

^Tmn

Skl

^Tnm ^Smn

Tkl

Considering the symmetry relations

(3.16),

the unit matrix

A ⁽⁰⁾

(3.19)

in

(3.18)

can be represented by

1

m

ijklmn

(6i16j

kn

imjlkn

im j k in

+ 6injl6km ⁺ iljnkm ⁺

+ 6

ik j m

In +

ik j n im

+ +

in

6

jk im

(3.20)

Because

6 < 21 a nontrivial solution of the homogeneous equation

(3.18)

exists.

4. CONSTRUCTION OF SIMULTANEOUS INVARIANTS.

In the case of several tensor variables, the term simultaneous or joint invariant is used

[6].

This term is used not only for scalar-valued isotroplc functions of several second-order tensor variables, but also for scalar-valued functions of any set of tensors of any order, e. g. of the argument tensors

gij’

Aij’ Aijkl’ Aijklmn

ⁱⁿ

^(3.1).

Such invariants are, for instance,

OijAji’ OijAjki’ gikOkjAji OikOkjAjlAli ^(4.1)

gijAijklgkl gijAijpq ^AvwklOkl, OipOpj Aij klOkl Aij klOikOj

i

(4.2)

Aij

klmn

Oij klmn’ Aij pqrsitupqAmnrstu ^Oij klOnm’

Aij

k

iron

il

gj mOkn

(4.3)

and they have great importance in the theory of anisotropic materials.

To construct a set of sumultaneous invariants of the stress tensor

o..

and the fourth-order constitutive tensor

Aijkl

^{we start} from the following theorem:

A scalar-valued function

f(.,T)

^of one n-dimensional vector

v

and one symmetric second-order tensor

T

^{is an orthogonal invar-}

iant, ^i.e, invariant under the orthogonal group if and only if it can be expressed as a function of the 2n special invariants

2

Tn-lv

JI(T) Jn(T),

^v

^z’Tzv

^v-

^(4.4)

This theorem is valid for arbitrary dimension n

[6].

Assuming the symmetry condi- tions

(3.8)

and using the HAMILTON-CAYLEY theorem, it means that

T

^{n and all higher}

powers

T

^{n+h can be expressed} in terms of

T T

^{2 n}

T~ ^-I

^{we find in general-}

izing of

(4.4)

the set of 15 simultaneous invariants

(D)

o"

ii[] ^{_() (2)}

H

^j

^ijAijkl

^{kl’ 3}

^ijAijklkl

H]

^gij

^(2),() ijkl ^0.(2)

kl ^{i 2 5}

(4.5)

where in a square bracket denotes a label to indicate several invariants,

while 9 in a round bracket is an exponent.

Some of these simultaneous invariants can be constructed in the following way.

Consider the second-order tensor

D o..Ao.

then, for instance, the basic Pq 1313^Pq

A

(2)

O invariant D D can be expressed by the simultaneous invariant

i’"i-’k133 ^kl’

Pq qP

which is contained in the system

(4.5).

Furthermore, the cubic basic invariant D D D can be expressed by the simultaneous invariant

Ai-’klmni-’klmn’33

^which

pq qr rp

is contained in the set

(4.3).

As another example we can consider the second-order tensor E

o(2)A

_{and flnd,} for instance, the invarlant E E 0

(2)-(2)_

pq ij ijpq pq qp ij

Aijklkl

which is contained in

(4.5),

too. Finally, ^{we can} also form the simultaneous invariants

D O

A o

D 0

(2)

0

(2)

pq pq

ij

ijpq

pq’

pq pq

oijAijpq

pq E

o

o

(2)

E 0

(2)

0

(2)

0

(2)

pq pq ij

AijpqOpq’

pq pq ij

Aijpq

pq and obtain simultaneous invariants of the system

(4.5).

The examples mentioned above and the complete system

(4.5)

can be expressed by the simultaneous invariants

(GpqHqp)

_h

I %2-

^{< 2}

lal+la ^-<

⁵

ij "’ijkl

Ukl ^(4.6)

where

o(1) ⁽⁾ A(:,)()

ij

Aijpq’ (Hqp)

pqkl kl

(4.7a,b)

(Gpq)%

,i

2

^,2

are second-order tensors.

The isotropic special case, for instance, can be expressed by the isotropic constitutive tensor

A()

ijkl

a6 ij6kl ⁺ b(6ik6j

1

^{+ 6}

i

16

jk

^{(4 8)} Iv]

from

(45)

are equal of power

1,2,

Then the simultaneous invariants

H

2

i 1

to the principal invariant

J2(o)~

^for

a ^---2 b

^{and equal to the basic}

invariant

$2(o)

^{for a 0}

b

^{1 Similarly, the invariants}

^]

^from

^(4.5)

^are

1 1

3(o)/6-

for a

b

^{and equal to the cubic}

equal to

J3() Jl

2 ^basic

invariant

$3()

^if

a ^0, b

Furthermore, the invariants

H ^] i]Ai]klkl ())

are equal to

Jl()

^S

I()

^if

3a ⁺ 2b--

^i.

REFERENCES

i.

GRACE,

J.H. and

YOUNG, A.,

The

Algebra

of Invariants, Cambridge Univ.

Press,

London and New York, 1903.

2.

GUREVICH, G.B.,

Foundations of the

Theory

of Algebraic Invariants,

P.

Noordhoff, Groningen, 1964.

3.

WEITZENBOCK, R.,

Invariantentheorie, P.

Noordhoff,

Groningen, 1923.

4.

WEYL, H.,

The Classical

Group.s

Their Invariants and Representation, Princeton Univ.

Press,

Princeton and New Jersey, 1946.

5.

SPENCER, A.J.M.,

Theory of Invariants, in: Continuum

Physics.

Edited by A.C. Eringen, Vol. i Mathematics, Academic

Press,

New York and London, 1971.

6.

TRUESDELL,

C. and

NOLL, W.,

The non-linear Field Theories of Mechanics, in:

Handbuch der

Phvsik.

(Edited by S.

Flugge),

Vol.

111/3,

Springer-Verlag,

Berlin/Heidelberg/New

York, 1965.

7.

BETTEN, J.,

Ein Beitrag zur Invariantentheorie in der Plastomechanik anisotro- per

Werkstoffe,

Z.

Ansew.

Math. Mech. 56

(1976),

557-559.

8.

SHITH, G.F.,

On the Yield Condition for Anisotropic Materials,

Quart.

Appl.

Math. 20

(1962),

241-247.

9.

SPENCER,

A.J.M. and

RIVLIN, R.S.,

The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua, Arch. Rational Mech.

Anal. 2

(1958/59),

309-336.

i0.

WINEMAN,

A.S. and

PIPKIN, A.C.,

Material Symmetry Restrictions on Constitutive Equations, Arch. Rational Mech. Anal. 17

(1964),

184-214.

ii.

SMITH, G.F., SMITH, M.M.,

^and

RIVLIN, R.S.,

Integrity

Bases

for a Symmetric Tensor and a Vector- The Crystal Classes, Arch. Rational Mech. Anal. 12

(1963),

93-133.

12.

PIPKIN,

A.C. and

RIVLIN, R.S.,

The Formulation of constitutive equations in continuum physics. Arch. Rational Mech. Anal. 4

(1959),

129-144.

13.

PIPKIN,

A.C. and

WINEMAN, A.S.,

Material Symmetry Restrictions on Non-Poly- nomial Constitutive Equations, Arch. Rational Mech. Anal. 12

(1963), 420-426.

14.

BETTEN, J.,

Plastische Stoffleichungen inkompressibler anisotroper

Werkstoffe,

Z.

Ans.ew.

Math. Mech.

57 ^(1977),

^671-673.

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