Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
www.elsevier.com/locate/cma
On the theory of fourth-order tensors and their applications in
computational mechanics
Mikhail Itskov *
Institut fur Statik und Dynamik, Ruhr-Universit
at Bochum, Universit
atsstraûe 150, 44780 Bochum, Germany
Received 1 December 1998; received in revised form 29 November 1999
Dedicated to Professor Dr. -Ing. Yavuz Bas ar on the occasion of his 65th anniversary
ß
Abstract
Many problems concerned with the mathematical treatment of fourth-order tensors still remain open in the literature. In the present
paper they will be considered in the framework of a complete theory involving a set of notations and de®nitions, a tensor operation
algebra, di€erentiation rules, eigenvalue problems, applications of fourth-order tensors to isotropic tensor functions and some other
relevant aspects. A tensor is understood to be an invariant quantity with respect to any co-ordinate system transformation, which
justi®es the use of absolute notation preferred in this work. As a most important application ®eld of fourth-order tensors, elastic
moduli are formulated in a material and a spatial description and given for various hyperelastic material models. Ó 2000 Elsevier
Science S.A. All rights reserved.
Keywords: Fourth-order tensors; Tensor algebra; Eigenvalue problems; Isotropic tensor functions; Tangent moduli
1. Introduction
Fourth-order tensors as a mathematical object have found in the last 20 years a wide use in computational mechanics and especially in the ®nite element method. Their well-known applications are tangent
(elastic or elasto-plastic) moduli as well as damage tensors playing an important role in the formulation of
constitutive and evolution equations. The use of fourth-order tensors remains, nevertheless, extremely
complicated because of the absence of a closed theory embracing many relevant aspects. In spite of active
e€orts in recent times (see [3±9,11,21,22,24,27±30,37,38,41,42]), there are many problems concerned with
the treatment of fourth-order tensors, which are not completely solved and still remain open in the literature.
· How can a fourth-order tensor be constructed from two second-order ones?
· How can algebraic operations with fourth-order tensors be given in the absolute notation?
· What does the derivative of a second-order with respect to another one mean? There is no uni®ed definition for the derivative with respect to a tensor. Is there a di€erence in the de®nition of the derivative
with respect to a symmetric second-order tensor?
· How can the eigenvalue problem be formulated for fourth-order tensors and how can principal invariants and eigenvalues of a fourth-order tensor be calculated? How many irreducible invariants does a
fourth-order possess?
*
Present address: Department of Applied Mechanics and Fluid Dynamics, University of Bayreuth, D-95440 Bayreuth, Germany.
E-mail address: [email protected] (M. Itskov).
0045-7825/00/$ - see front matter Ó 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 0 4 5 - 7 8 2 5 ( 9 9 ) 0 0 4 7 2 - 7
420
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
The goal of the present paper is to try to solve these problems in a framework of the complete theory and
to demonstrate then its application for the derivation of tangent moduli. The important characteristic
feature of the development is the absolute notation used for tensor quantities throughout the paper. This is
not only a notation manner but the basic concept of the paper. A tensor is understood to be an invariant
quantity with respect to any co-ordinate system transformation and can be neither covariant, nor contravariant, nor mixedvariant in contrast to its components. In this connection, for example, the widespread
use of the metric tensor as an argument of a function or as a di€erentiation parameter seems to be
meaningless, since the metric tensor represents in the end the identity tensor. Such notations always have to
be accompanied by an indication of the component variance to be used in the present operation and will be
avoided here.
As a further crucial point of the paper it should be noted that in contrast to many other works we deal
with arbitrary non-symmetric fourth-order tensors and understand them as a linear mapping of one arbitrary second-order tensor into another one (see also [19,29]). This considerably simpli®es the treatment of
fourth-order tensors, which can be seen especially clearly in consideration of the eigenvalue problem. The
symmetry properties of fourth-order tensors, which are relevant, for instance, for tangent moduli, can be
enforced without loss of generality at the last stage of derivation.
The starting point of the development is the set of notations and de®nitions given in Section 2. To have
more freedom in construction of fourth-order tensors we propose a new tensor product of two second-order
ones. The crucial point of the presented theory lies in a new de®nition of the derivative of one second-order
tensor with respect to another one …2;2 †. Due to this de®nition, the well-known product derivative rule valid
for scalar values is proved to hold also for second-order tensors. The double contraction of a fourth-order
tensor with a second-order one (4:2) is also rede®ned to be consistent with the di€erentiation rule intro_
duced, such that the important rate relation A…B†
ˆ A;B : B_ is satis®ed. Further, it is shown that the derivative with respect to a symmetric tensor is not unique and requires a slight correction in the de®nition.
Special attention is paid in Section 2 to the transposition operations for fourth-order tensors and the resultant de®nition of a symmetric and a super-symmetric tensor. In this context, we introduce a special
transposition operation bringing a fourth-order tensor to the standard de®nition of the tensor derivative. In
the following this will permit to verify some important results by comparing them with those available in
the literature.
The indispensable part of the proposed theory is the algebra of fourth-order tensors given in Section 3.
Simple, double, quadruple contractions and other operations with fourth-order tensors constructed from
second-order ones are presented in absolute notation and can easily be proved using the de®nitions of the
previous section. The following important topic is the formulation and proof of the di€erentiation rules
such as the product derivative and the chain rule. On this basis, we then obtain the derivative of the power
tensor function for positive as well as negative integer exponents.
Section 4 is devoted to the formulation of eigenvalue problems and spectral decomposition of fourthorder tensors. Using matrix representation the eigenvalue problem of a fourth-order tensor is reduced to
that of a matrix and can then be solved by a standard procedure. For a symmetric fourth-order tensor this
yields nine real eigenvalues and nine corresponding eigentensors. A complete analogy with the eigenvalue
problem of a second-order tensor can be observed. On the basis of the other possible de®nitions for the
double contraction 4:2 at whole 27 principal invariants and 27 eigenvalues of the fourth-order tensor can be
received in this way. The discussed procedure is illustrated by spectral decompositions for a number of
important fourth-order tensors like the identity, the trace projection and the transposition projection
tensor.
Application of the proposed theory to isotropic tensor functions is the topic of Section 5. Attention is
focused on the representation of the derivative of an isotropic tensor-valued tensor function with respect to
its argument, which is of major importance for the formulation and numerical calculation of tangent
moduli in large strain elasticity and elasto-plasticity. We start with the consideration of a special class of
isotropic tensor functions, which can be expanded in tensor power series. In the case of a symmetric tensor
argument we receive for the derivative in question two closed-form representations. The ®rst one is expressed in terms of eigenvectors of the tensor argument and corresponds to the well-known result (see
[9,27]). The second representation is given through the eigenvalue bases and is advantageous, as it avoids
the numerically expensive computation of eigenvectors. In contrast to the analogous result by Miehe [24],
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
421
this representation does not use the derivatives of the eigenvalue bases and is obtained in a simpler and
more compact form.
The tensor algebra and di€erentiation rules developed are applied in Section 6 to formulate tangent
moduli in a material and a spatial description. It is proved that the tangent moduli presented are symmetric
fourth-order tensors and permit consequently the spectral decomposition. The important question to be
treated then is how the isotropy can be identi®ed by an elastic modulus. Using the spectral decomposition
we prove that the elastic modulus corresponds to an isotropic material in the natural state if only it can be
represented in the form of the St.Venant±Kirchho€ model involving two material parameters (see also
[14,15,17,19,20,29,31] and references therein). Elastic moduli for this and some other hyperelastic material
models such as Mooney±Rivlin and Ogden are given in Section 7.
2. Basic notations and de®nitions
A second-order tensor (a bold capital letter):
A ˆ Aij gi gj ˆ Aij gi gj ˆ Aij gi gj ˆ …2:1†
A fourth-order tensor (a bold italics capital letter):
k
gi gj gk gl ˆ D ˆ Dijkl gi gj gk gl ˆ Dijkl gi gj gk gl ˆ Dijl
…2:2†
The simple contraction of arbitrary order tensors:
m
g i gl gm gn :
AD ˆ …Aij gi gj †…Dklnm gk gl gm gn † ˆ Aij Djln
…2:3†
Tensor products of two second-order tensors are de®ned as follows:
D ˆ A B ˆ …Aij gi gj † …Bkl gk gl † ˆ Aij Bkl gi gj gk gl ;
…2:4†
D ˆ A B ˆ …Aij gi gj † …Bkl gk gl † ˆ Aij Bkl gi gk gl gj :
…2:5†
It can easily be con®rmed the tensor products satisfy the distributive rule:
A …B ‡ C† ˆ A B ‡ A C;
A …B ‡ C† ˆ A B ‡ A C:
…2:6†
Transposition operations … †T and … †t for fourth-order tensors:
T
k
k j
gi gj gk gl † ˆ Dijl
g gi gl gk ;
DT ˆ …Dijl
…2:7†
k
k
gi gj gk gl †t ˆ Dijl
gi gk gj gl :
Dt ˆ …Dijl
…2:8†
T
t
Remark. One can easily prove that the transposition operations … † and … † are not commutative with
each other: DTt 6ˆ DtT .
According to the above de®nitions the transposition applied to the tensor products (2.4) and (2.5) yields:
T
…A B† ˆ AT BT ;
T
…A B† ˆ B A;
t
…A B† ˆ A BT :
…2:9†
A symmetric fourth-order tensor (cf. [19,29])
DT ˆ D:
…2:10†
A super-symmetric fourth-order tensor
ET ˆ E
and
E t ˆ E:
…2:11†
422
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
By means of the operation … †0
1
E 0 ˆ …E ‡ E t †;
2
…2:12†
a symmetric fourth-order tensor can be transformed into a super-symmetric one.
Remark. For a fourth-order tensor some other transposition operations can be de®ned and particularly:
k
k
gi gj gk gl †R ˆ Dijl
g i gl g j gk ;
… †R : DR ˆ …Dijl
…2:13†
which possess the important property
…A B†R ˆ A B:
…2:14†
Double contractions of a fourth-order tensor with a second-order one …4 : 2† and …2 : 4† are de®ned by
k
k
gi gj gk gl † : …Amn gm gn † ˆ Dijl
Ajk gi gl ;
D : A ˆ …Dijl
…2:15†
k
k j
gi gj gk gl † ˆ Ail Dijl
g gk :
A : D ˆ …Amn gm gn † : …Dijl
…2:16†
The double contraction of a fourth-order tensor with another fourth-order one (4:4):
k
n
s
ik jr
n
l
gi gj gk gl † : …Amr
D : A ˆ …Dijl
ns gm g gr g † ˆ Djl Ank gi g gr g :
…2:17†
Evidently, the double contraction of fourth-order tensors is not commutative …D : A 6ˆ A : D†, but can easily
be proved to ful®l the associative rule:
…A : B† : C ˆ A : …B : C†:
…2:18†
In contrast to the double contraction the scalar product (quadruple contraction) of two fourth-order
tensors …4 :: 4†
k
k jl
gi gj gk gl † :: …Amnsr gm gn gr gs † ˆ Dijl
Aik
D :: A ˆ …Dijl
…2:19†
obey the commutative rule:
D :: A ˆ A :: D:
…2:20†
According to the de®nitions (2.15)±(2.17) and (2.19) the operations …2 : 4†, …4 : 2†, …4 : 4† and …4 :: 4† satisfy
the distributive rule e.g.:
A : …B ‡ C† ˆ A : B ‡ A : C;
…A ‡ B† : C ˆ A : C ‡ B : C;
…2:21†
A : …B ‡ C† ˆ A : B ‡ A : C;
…A ‡ B† : C ˆ A : C ‡ B : C;
…2:22†
A :: …B ‡ C† ˆ A :: B ‡ A :: C; …A ‡ B† :: C ˆ A :: C ‡ B :: C:
…2:23†
The second-order identity tensor:
I ˆ g i gi :
…2:24†
The fourth-order identity tensor:
I ˆ I I ˆ gi gi gj gj :
…2:25†
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
423
The trace of a fourth-order tensor is de®ned by
tr D ˆ D :: I ˆ I :: D ˆ Diijj :
…2:26†
Powers of fourth-order tensors:
D0 ˆ I;
D1 ˆ D;
D2 ˆ D : D; . . .
Dn ˆ D
: D‚{z‚‚‚‚‚‚‚‚
: D
|‚‚‚‚‚‚‚‚
‚} :
…2:27†
n
An inverse fourth-order tensor:
D : Dÿ1 ˆ Dÿ1 : D ˆ I:
…2:28†
The matrix representation of a fourth-order tensor … † is de®ned through its mixedvariant components by
2 11
2
3
1
2
3
1
2
3 3
D11 D111
D111
D121
D121
D121
D131
D131
D131
1
2
3
1
2
3
1
2
3 7
6 D112
D112
D112
D122
D122
D122
D132
D132
D132
7
6
6 D1 1 D1 2 D1 3 D1 1 D1 2 D1 3 D1 1 D1 2 D1 3 7
6 13
13
13
23
23
23
33
33
33 7
6 21
2
3
1
2
3
1
2
3 7
7
6 D11 D211
D211
D221
D221
D221
D231
D231
D231
7
6 21
22
23
21
22
23
21
22
23 7
6
…2:29†
D ˆ 6 D12 D12 D12 D22 D22 D22 D32 D32 D32 7:
6 D2 1 D2 2 D2 3 D2 1 D2 2 D2 3 D2 1 D2 2 D2 3 7
6 13
13
13
23
23
23
33
33
33 7
6 31
2
3
1
2
3
1
2
3 7
7
6 D11 D311
D311
D321
D321
D321
D331
D331
D331
7
6
4 D3 1 D3 2 D3 3 D3 1 D3 2 D3 3 D3 1 D3 2 D3 3 5
12
1
D313
12
2
D313
12
3
D313
22
1
D323
22
2
D323
22
3
D323
32
1
D333
32
2
D333
32
3
D333
The partial derivative of a scalar-valued tensor function a…A† with respect to its second-order argument A
…0;2 † is given by
a…A†;A ˆ
oa
oa i
g gj
ˆ
o…Aij gi gj † oAij
…2:30†
and possesses the following important properties:
1. The derivative is independent of the choice of the component variance used in the di€erentiation. Since
Akl ˆ Akm gml , we have
oa i
oa oAkl i
oa jl i
oa i
g
g
ˆ
g gj ˆ
g g gj ˆ
g gl :
j
oAij
oAkl oAij
oAil
oAil
Similarly it can easily be proved also for the covariant components.
2. The derivative is independent of the choice of the co-ordinate system. In a transformed co-ordinate system we obtain:
k
k
A ˆ Aij gi gj ˆ Al gk gl ) Al ˆ …gk gi †Aij …gj gl †;
k
oa i
oa oAl i
oa k
oa k
g gj ˆ
…g gi †…gj gl †gi gj ˆ
g gl :
i g gj ˆ
k
k
k
oA
oAij
oAl
oAl
oAl
j
These properties ensure the objectivity of the tensor derivative and justify the use of the absolute notation
a…A†;A .
Remark 1. Properties 1 and 2 and their proofs are not valid for the derivative with respect to the metric
tensor. As mentioned above, the derivative with respect to the metric tensor is not objective since it depends
not only on the component variance but also on the co-ordinate system used in di€erentiation (see [16]).
Remark 2. It is important to note that in the di€erentiation with respect to a tensor its vector base is kept
constant. Disregard of this fact may lead to an incorrect result. As a typical example we consider derivatives
with respect to the Green±Lagrange and Almansi strain tensors
424
and
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
1
1
E ˆ …C ÿ G† ˆ …gij ÿ Gij †Gi Gj ˆ Eij Gi Gj
2
2
1
1
e ˆ …g ÿ bÿ1 † ˆ …gij ÿ Gij †gi gj ˆ eij gi gj ;
2
2
respectively, which are often used for the formulation of constitutive relations. It can be seen that these
tensors have equal covariant components Eij ˆ eij related, nevertheless, to the di€erent bases remaining
constant in the di€erentiation. Thus
!
! ow
ow T
ow
ow
ow
ow
T
F ˆF
ˆ
Gi Gj F ˆ F 2
Gi Gj FT ˆ 2
gi gj 6ˆ
g gj
F
oE
oEij
ogij
ogij
oe oeij i
ˆ ÿ2
ow
ow
g gj ˆ ÿ2
g gj ;
ÿ1 i
oGij i
obij
which evidently contradicts the constitutive equation for the Cauchy stress tensor of the form s ˆ w;e (see
e.g. [32]).
The partial derivative of a second-order tensor C ˆ Cij Gi Gj with respect to another second-order one
A ˆ Akl gk gl (2;2 ) is de®ned by
C;A ˆ
o…Cij Gi Gj †
o…Akl gk
gl †
ˆ
oCij
oAkl
G i gk gl G j
…2:31†
for the tensor bases Gi Gj of C, which are independent of the tensor components Akl . In other respect the
choice of co-ordinate systems and the variance of components of both tensors A and C are irrelevant, which
can easily be proved similarly to the properties of the derivative …0;2 †. This ensures the objectivity of the
de®nition (2.31) and justi®es the use of absolute notation for the tensor C;A .
Remark. The derivative obtained according to the de®nition (2.31) can be transformed to the standard one
R
by means of transposition … † . Using (2.13) we have:
R
…C;A † ˆ
oCij
oAkl
Gi Gj gk gl :
The de®nitions (2.30) and (2.31) become non-unique if the tensor A is symmetric. In this case the di€erentiation cannot still be carried out with respect to nine components of A, since only six of them are independent. This de®ciency can be eliminated by modifying the de®nitions (2.30) and (2.31). Accordingly the
derivatives with respect to the symmetric tensor A ˆ AT can be de®ned by
a…A†;A ˆ
ˆ
C;A ˆ
ˆ
3
3
1X
oa
1X
oa i
…gi gj ‡ gj gi † ˆ
…g gj ‡ gj gi †
2 i;j 6 i oAij
2 i;j 6 i oAij
3
1X
oa i
…g gj ‡ gj gi †;
2 i;j 6 i oAij
…2:32†
3
3
oCij
oCij
1 X
1 X
Gi …gk gl ‡ gl gk † Gj ˆ
Gi …gk gl ‡ gl gk † Gj
2 k;l 6 k oAkl
2 k;l 6 k oAkl
3
oCij
1 X
Gi …gk gl ‡ gl gk † Gj :
2 k;l 6 k oAkl
…2:33†
As can be observed the di€erentiation in (2.32) and (2.33) is accomplished with respect to the six independent components of A and the resulting tensor is then symmetrized.
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
425
3. The algebra of fourth-order tensors
In the following we summarize some important tensor identities, which will be intensively used in further
derivation:
T
A B : C ˆ ACB;
…A B† : C ˆ …B : C†A;
C : A B ˆ AT CBT ˆ …A B† : C;
C : …A B† ˆ …C : A†B;
…3:1†
…3:2†
…A B†C ˆ A …BC†;
…A B†C ˆ …AC† B;
C…A B† ˆ …CA† B;
C…A B† ˆ …CA† B;
…3:3†
…3:4†
…A B† : …C D† ˆ …AC† …DB†;
…A B† : …C D† ˆ …B : C†A D;
T
…A B† : …C D† ˆ …ACB† D;
T
T
…A B† :: …C D† ˆ …A : C †…B : D †;
…A B† :: …C D† ˆ …ACB† : D;
ÿ
ÿ
t
…A B†t : …C D† ˆ ADT CT B ;
T
T
…E : F† ˆ F : E ;
T
t
…3:6†
…A B† :: …C D† ˆ …A : D†…C : B†;
…3:7†
T
T
E : A ˆ A : E;
T
…A B† :: …C D† ˆ A : …C BD †;
t
T
…A B† : …C D† ˆ …AC B† D;
t
…3:8†
…3:9†
…E : F† ˆ E : F ;
…3:10†
T
C : E ˆ …C : E† ;
…3:11†
A…E : C†B ˆ …AEB† : C:
…3:12†
t
E :CˆE:C ;
…3:5†
…A B† : …C D† ˆ A …C BD †;
t
T
T
The above relations directly emanate from the corresponding de®nitions (2.3)±(2.5), (2.7)±(2.8), (2.15)±
(2.17) and (2.19). For example the identity …3:5†1 can easily be proved using (2.17), which leads to
…A B† : …C D† ˆ …Aij Bkl gi gj gk gl † : …Cmn Drs gm gn gr gs †
ˆ Aij Bkl Cjn Drk gi gn gr gl ˆ …AC† …DB†:
By virtue of (2.31) we construct the partial derivative of a second-order tensor with respect to itself yielding
the fourth-order identity tensor:
A;A ˆ
oAij
oAkl
gi gk gl gj ˆ dik dlj gi gk gl gj ˆ gi gi gj gj ˆ I:
…3:13†
The properties of the fourth-order identity tensor I following from (2.25), (3.1) and …3:5†1 can now be
given by
I : B ˆ B : I ˆ B;
I : D ˆ D : I ˆ D:
…3:14†
Using (2.26), (3.7) and (3.8) the validity of the following important identities with the trace of a fourthorder tensor can also be proved:
j
kl
Cij
;
tr‰C : DŠ ˆ tr‰D : CŠ ˆ ‰C : DŠ :: I ˆ C :: D ˆ D :: C ˆ Dikl
…3:15†
tr‰A BŠ ˆ ‰A BŠ :: I ˆ tr Atr B; tr‰A BŠ ˆ ‰A BŠ :: I ˆ A : B;
tr I ˆ I :: I ˆ …I : I†…I : I† ˆ 9; tr …I I† ˆ I I :: I ˆ I : I ˆ 3:
…3:16†
…3:17†
Due to the important property
…I I† : A ˆ A : …I I† ˆ …tr A†I
…3:18†
the tensor I I introduced in …3:17†2 is referred to as the trace projection tensor.
Remark. For the standard de®nition of the tensor derivative the trace projection tensor has the form
ÿ
R
…I I†R ˆ gi gj gj gi ˆ gi gi gj gj ˆ I I:
426
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
Now, we construct the partial derivative of a second-order tensor with respect to its transverse counterpart
delivering the transposition projection tensor T:
A;TA ˆ A;AT ˆ
oAij
oAkl
gj gk gl gi ˆ dik dlj gj gk gl gi ˆ I t ˆ T;
…3:19†
which possesses the following properties:
T : B ˆ gj gi gj gi : Bkl gk gl ˆ Bij gj gi ˆ BT ;
B : T ˆ BT :
…3:20†
For the derivative with respect to the symmetric part symA ˆ …1=2†…A ‡ AT † of a tensor A one receives
under consideration of (2.12) and (2.33)
1
A;symA ˆ A;TsymA ˆ symA;symA ˆ …I ‡ T† ˆ I 0 :
2
…3:21†
Generalizing this result to the derivative of an arbitrary tensor delivers:
B;symA ˆ B;A : A;symA ˆ B;A : I 0 ˆ …B;A †0 :
…3:22†
Now, attention is focused on the tensor di€erentiation rules. We start with the product derivative
…AB†;C ˆ A;C B ‡ AB;C ;
…3:23†
which is well known for scalar values. According to the new de®nition of the derivative …2;2 † (2.31) the
product derivative rule (3.23) also holds for second-order tensors.
Proof.
…AB†;C ˆ
ˆ
o…Aij Bjk gi gk †
o…Clm gl gm †
oAij
oClm
ˆ
o…Aij Bjk †
oClm
gi gl gm gk
Bjk gi gl gm gk ‡ Aij
oBjk
gi gl gm gk ˆ A;C B ‡ AB;C :
oClm
Using this remarkable relation one can easily obtain the derivative of the tensor power function with respect to its argument B;nB …n ˆ 1; 2 . . .†. Considering (3.3), …3:5†1 and (3.15) one gets:
B;2B ˆ B I ‡ I B;
B;3B
…3:24†
2
2
2
nÿ1
X
nÿ1
X
nÿ1ÿr
r
Bnÿ1ÿr Br ˆ
…B I†
: …I B† ;
2
ˆ B I ‡ B B ‡ I B ˆ …B I† ‡ …B I† : …I B† ‡ …I B† ;
B;nB ˆ
rˆ0
B;nB : I ˆ nBnÿ1 ;
…3:25†
…3:26†
rˆ0
…3:27†
where we adopt B0 ˆ I. To obtain the derivative of the inverse function …Bÿ1 †;B we ®rst use the identity:
…BBÿ1 †;B ˆ I;B ˆ 0, which leads by virtue of (3.3) and (3.23) to
…Bÿ1 †;B ˆ ÿBÿ1 Bÿ1 :
…3:28†
By analogue and exploiting (3.28) we can further write for the derivative of the power function with a
negative integer exponent:
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
…Bÿn †;B ˆ ÿ
nÿ1
X
Bÿrÿ1 Brÿn ;
427
…3:29†
rˆ0
…Bÿn †;B : I ˆ ÿnBÿnÿ1 :
…3:30†
Two other important di€erentiation rules can be formulated as follows:
…aA†;B ˆ A a;B ‡aA;B ;
…A : B†;C ˆ A : B;C ‡B : A;C ;
…3:31†
where a denotes a scalar function.
Proof.
…aA†;B ˆ
o…aAij gi gj †
o…Bkl gk gl †
ˆ
o…aAij †
oBkl
oAij
oa i
k
j
gi g gl g ˆ k Aj gi g gl g ‡ a k gi gk gl gj
oBl
oBl
k
j
ˆ A a;B ‡aA;B ;
…A : B†;C ˆ
o…Aij Bij †
o…Ckl gk gl †
ˆ
oAij
oB j
Aij ik ‡ Bij k
oCl
oCl
!
gk gl ˆ A : B;C ‡ B : A;C :
For the derivative of a scalar f …A† and a tensor-valued A…C† tensor function one can prove the following
chain rules:
f …A†;B ˆ
A;B ˆ
i
of oAj k
g gl ˆ f ;A : A;B ;
oAij oBkl
…3:32†
oAij oCkl
m
j
m gi g gn g ˆ A;C : C;B :
oCkl oBn
…3:33†
4. Spectral decomposition, eigenvalues and principal invariants of a fourth-order tensor
The double contraction 4:2 (2.15) establishes the basis for the formulation of the eigenvalue problems
D : M ˆ kM
…4:1†
for a fourth-order tensor and enables then to determine its nine eigenvalues and principal invariants. To
this end, we transform (4.1) as follows
…D ÿ kI† : M ˆ 0:
…4:2†
This equation has a non-trivial solution
11
2
D11 ÿ k
D111
D1 1
2
D112
ÿk
12
det‰D ÿ kIŠ ˆ ...
.
.
.
D3 1
D3 2
13
13
M 6ˆ 0 if the corresponding characteristic equation
3
...
D131
13
D32 ˆ 0;
...
. . . . . . D3 3 ÿ k …4:3†
33
which can also be presented in the form:
ÿk9 ‡ ID k8 ÿ IID k7 ‡ IIID k6 ‡ ‡ IXD ˆ 0;
…4:4†
is ful®lled. In this equation the coecients ID ; IID ; IIID ; . . . ; IXD denote the principal invariants of the tensor
D. On the basis of Newton formula and in view of the identity A : B ˆ A B they can be expressed in terms
of principal traces of D (see also [42]):
428
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
ID ˆ tr D ˆ k1 ‡ k2 ‡ k3 ‡ ‡ k9 ;
i
1h
2
IID ˆ ÿ tr D2 ÿ …tr D† ˆ k1 k2 ‡ k1 k3 ‡ ‡ k8 k9 ;
2
1
3
1
IIID ˆ tr D3 ÿ tr D2 tr D ‡ …tr D†3 ˆ k1 k2 k3 ‡ k1 k2 k4 ‡ ‡ k7 k8 k9 ;
3
2
2
1
4
1
1
2
2
4
IVD ˆ ÿ tr D4 ÿ tr D3 tr D ‡ tr D2 …trD† ÿ …tr D2 † ‡ …tr D†
4
3
2
6
…4:5†
ˆ k 1 k 2 k 3 k 4 ‡ k1 k2 k 3 k 5 ‡ ‡ k6 k 7 k 8 k 9 ;
..
.
IXD ˆ det D ˆ k1 k2 . . . k9 :
In such a way the eigenvalue problem of a fourth-order tensor is transformed into the eigenvalue problem
of its matrix representation and can then be solved by a conventional procedure of the linear algebra.
Remark. The double contraction 4:2 (2.15) used for the formulation of the eigenvalue problem (4.1) is only
one of three possible ones. Each of them would lead to its own matrix representation, eigenvalue problem
and, ®nally, to its nine principal invariants. Thus one can obtain the basis of a fourth-order tensor consisting of its 27 invariants. The question about the integrity and irreducibility of this basis requires a special
consideration (see [5]).
T
If the tensor D is symmetric, which immediately leads to symmetry of its matrix representation D ˆ …D†
in an orthogonal co-ordinate system, the solution of the eigenvalue problems (4.1) yields nine real eigenvalues kn with the corresponding orthogonal eigentensors Mn …n ˆ 1; 2; . . . ; 9†. In this case the spectral
decomposition of D is given by (the spectral theorem):
Dˆ
9
X
kr M r M r ;
Ms : Mt ˆ dst :
…4:6†
rˆ1
If the eigenvalues kn …n ˆ 1; 2; . . . ; 9† of D are known and do not coincide, its eigenvalue bases Mr Mr can
be presented by the closed formula [29,41]:
M r Mr ˆ
9
Y
D ÿ ks I
s6ˆr
with
kr ÿ ks
n
Y
Ai ˆ A1 : A2 : An :
…4:7†
i
If the tensor D is super-symmetric, which is relevant for the tangent moduli, its matrix representation in an
orthogonal co-ordinate system includes three pairs of equal rows and columns and has the reduced rank 6.
This means that at least three invariants and three eigenvalues of D are identically equal to zero. For the six
eigentensors of D associated with non-vanishing eigenvalues one can obtain by virtue of …2:9†3 and …3:10†2 :
t
…Mr Mr † ˆ Mr MTr ˆ Mr Mr
)
Mr ˆ MTr :
…4:8†
Thus, the spectral decomposition of a super-symmetric fourth-order tensor can be expressed in terms of
only six eigenvalues and corresponding symmetric eigentensors (see also [11,12,21,30,35]):
Dˆ
6
X
kr M r M r ;
rˆ1
Mr ˆ MTr :
…4:9†
The closed formula solution for the eigenvalue bases (4.8) takes in this case the form:
M r Mr ˆ
6
Y
D ÿ ks I 0
:
kr ÿ ks
s6ˆr
…4:10†
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
429
Concluding this section we present for example spectral decompositions of some important fourth-order
tensors I, I I, T and I 0 :
Iˆ
3
X
ÿ
ÿ
ni nj ni nj ;
k1 ˆ k2 ˆ ˆ k9 ˆ 1;
…4:11†
k1 ˆ 3;
…4:12†
i;jˆ1
1
1
I I ˆ 3 p I p I ;
3
3
Tˆ
3
X
… ni ni † … ni ni † ‡
i
ÿ
k2 ˆ k3 ˆ ˆ k9 ˆ 0;
3
X
1 ÿ
1 ÿ
p ni nj ‡ nj ni p ni nj ‡ nj ni
2
2
i;j<i
3
X
1 ÿ
1 ÿ
p ni nj ÿ nj ni p ni nj ÿ nj ni ;
2
2
i;j>i
k1 ˆ k2 ˆ ˆ k6 ˆ 1;
k7 ˆ k8 ˆ k9 ˆ ÿ1;
…4:13†
3
3
X
X
1
1 ÿ
1 ÿ
p ni nj ‡ nj ni p ni nj ‡ nj ni ;
…ni ni † …ni ni † ‡
I 0 ˆ …I ‡ T† ˆ
2
2
2
i
i;j<i
k1 ˆ k2 ˆ ˆ k6 ˆ 1;
…4:14†
k7 ˆ k8 ˆ k9 ˆ 0
with ni …i ˆ 1; 2; 3† being a set of orthogonal unit vectors: ni nj ˆ dij .
5. Application to isotropic tensor functions
A tensor-valued tensor function G…A† is said to be isotropic if it satis®es the relation [39]:
G…QAQT † ˆ QG…A†QT
…5:1†
ÿT
for an arbitrary orthogonal tensor Q ˆ Q .
In the further derivation we shall ®rst deal with tensor power series building a special class of isotropic
tensor functions:
2
3
G…A† ˆ g0 I ‡ g1 …A ÿ aI† ‡ g2 …A ÿ aI† ‡ g3 …A ÿ aI† ‡ ;
…5:2†
where gi and a are scalar-valued constants. These functions are referred here to as analytical tensor
functions. For instance, isotropic tensor functions such as exponential and logarithmic ones being important and often used in computational mechanics are de®ned in the form (5.2) by
exp…A† ˆ I ‡ A ‡
1 2
A ‡ ;
2!
…5:3†
1
1
2
3
ln…A† ˆ …A ÿ I† ÿ …A ÿ I† ‡ …A ÿ I† ‡ 2
3
…5:4†
Our purpose is to obtain the derivative of an isotropic function with respect to its argument, which is of
high importance in computational elasticity and elasto-plasticity for the construction of tangent moduli.
Using the relation (3.26) and setting in (5.2) without loss of generality a ˆ 0, we receive for the derivative of
an analytical tensor function
G…A†;A ˆ g1 I ‡ g2 …A I ‡ I A† ‡ ‡ gn
rˆnÿ1
X
rˆ0
Anÿ1ÿr Ar ‡ …5:5†
430
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
On the basis of expression (5.5) the important identity for the time derivative can be con®rmed as follows
_
‡ ‡ gn
G…A†;A : A_ ˆ g1 A_ ‡ g2 …AA_ ‡ AA†
rˆnÿ1
X
_
_ r ‡ ˆ G…A†:
Anÿ1ÿr AA
…5:6†
rˆ0
The representation (5.5) holds for arbitrary tensors. In the special case of a symmetric tensor argument
A ˆ AT the derivative G…A†;A takes the form:
G…A†;A ˆ g1 I 0 ‡ g2 …A I ‡ I A†0 ‡ ‡ gn
rˆnÿ1
X
…Anÿ1ÿr Ar †0 ‡ …5:7†
rˆ0
and can expressed by means of the spectral decomposition of A
A ˆ k 1 n1 n1 ‡ k2 n2 n2 ‡ k3 n3 n3 ;
by
G…A†;A ˆ
1
X
"
gn n
3
X
a
nˆ1
knÿ1
a …na
…5:8†
na n a n a † ‡
rˆnÿ1
X
3
X
rˆ0
a;b6ˆa
#
knÿ1ÿr
krb …na
a
0
na n b nb † :
…5:9†
It can easily be seen that the derivative (5.9) presents a super-symmetric fourth-order tensor in the spirit of
de®nition (2.11):
T
‰G…A†;A Š ˆ G…A†;A ;
t
‰G…A†;A Š ˆ G…A†;A :
Introducing the abbreviation for the diagonal function [24] G…k† ˆ
representation for G(A)
P1
nˆ0
…5:10†
gn kn allowing the alternative
G…A† ˆ G…k1 †n1 n1 ‡ G…k2 †n2 n2 ‡ G…k3 †n3 n3 ;
…5:11†
the relation (5.9) can be given in the case of distinct eigenvalues ki in terms of eigenvectors ni by
G…A†;A ˆ
3
X
G0 …ka †na na na na ‡
3
X
G…ka † ÿ G…kb †
0
…na na nb nb †
ÿ
k
k
a
b
a;b6ˆa
G0 …ka †na na na na ‡
3
1 X
G…ka † ÿ G…kb †
na …na nb ‡ nb na † nb :
2 a;b6ˆa
ka ÿ kb
a
ˆ
3
X
a
…5:12†
Applying the operation … †R (2.13) to the relation (5.12) we can return G…A†;A to the standard de®nition of
the tensor derivative:
‰G…A†;A ŠR ˆ
3
X
G0 …ka †na na na na ‡
a
3
1 X
G…ka † ÿ G…kb †
na nb …na nb ‡ nb na †;
2 a;b6ˆa
ka ÿ kb
…5:13†
which corresponds to the well known result [9,27].
In terms of the eigenvalue bases Mi ˆ ni ni …i ˆ 1; 2; 3†, which can be obtained by means of the closedformula solution [2,26,36], the derivative (5.13) is expressible in the form:
G…A†;A ˆ
3
X
a
G0 …ka †Ma Ma ‡
3
X
G…ka † ÿ G…kb †
0
…Ma Mb † :
k
ÿ
k
a
b
a;b6ˆa
…5:14†
The last representation has the advantage, that it avoids a numerically expensive determination of eigenvectors. Comparing with the analogous solution by Miehe [22,24] it is observable that the representation
(5.14) does not include the derivatives of the eigenvalue bases and has a simpler and more compact form.
The special cases of two or three equal eigenvalues ki require a special treatment. Considering in (5.14)
the limit case ka ÿ kb ˆ D ! 0, one can obtain the following results:
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
431
1. Triple coalescence of eigenvalues: k1 ˆ k2 ˆ k3 ˆ k; A ˆ kI,
G…A†;A ˆ G0 …k†I 0 :
…5:15†
2. Double coalescence of eigenvalues: ka 6ˆ kb ˆ kc ˆ k; A ˆ ka Ma ‡ k…I ÿ Ma †,
G…ka † ÿ G…k†
0
‰Ma …I ÿ Ma † ‡ …I ÿ Ma † Ma Š
ka ÿ k
0
‡ G0 …k†‰…I ÿ Ma † …I ÿ Ma †Š :
G…A†;A ˆ G0 …ka †Ma Ma ‡
…5:16†
According to the representation theorem [39] an arbitrary isotropic tensor function, which cannot
generally be presented via power series (5.2), is given in the form:
G…A† ˆ u0 …A†I ‡ u1 …A†A ‡ u2 …A†A2 ;
…5:17†
where ui …A† ˆ ui …QAQT † are scalar-valued isotropic tensor functions of A. By virtue of (3.13), (3.24) and
(3.31) we construct the derivative G…A†;A by
G…A†;A ˆ u1 …A†I ‡ u2 …A†…A I ‡ I A† ‡ I u0 …A†;A ‡A u1 …A†;A ‡A2 u2 …A†;A ;
…5:18†
where exploiting the isotropy of ui : ui …A† ˆ ui …IA ; IIA ; IIIA † their derivatives can be given by
ui …A†;A ˆ
oui oui
oui T
oui
IA I ÿ
A ‡
IIIA AÿT :
‡
oIA oIIA
oIIA
oIIIA
…5:19†
Using the result (5.18) and (5.19) and by virtue of (3.1) and (3.2) it can easily be proved that the relation:
…AG…A†;A † : lA ˆ …G…A†;A A† : Al ˆ AG…A†;A : I A : l ˆ I A : G…A†;A : A I : l;
…5:20†
where l presents an arbitrary second-order tensor, holds for symmetric tensors A ˆ AT .
For the time derivative of an arbitrary isotropic tensor function we obtain by using (3.1), (3.2), (3.14),
(5.18) and (5.19) the well-known relation:
_
_
‡ Iu_ 0 …A† ‡ Au_ 1 …A† ‡ A2 u_ 2 …A† ˆ G…A†:
G…A†;A : A_ ˆ u1 …A†A_ ‡ u2 …A†…AA_ ‡ AA†
…5:21†
To complete this section we give the derivatives of some isotropic tensor functions, which cannot be expanded in power series (5.2):
‰tr…A†IŠ;A ˆ I …trA†;A ˆ I I;
1
1
…devA†;A ˆ A ÿ …trA†I ;A ˆ I ÿ I I ˆ P
3
3
…5:22†
…5:23†
with P being the deviatoric projection tensor: A : P ˆ P : A ˆ devA.
6. Tangent moduli
One of the most important applications fourth-order tensors ®nd in computational elasticity and elastoplasticity, where they appear as tangent or elastic moduli. The algebra operations and di€erentiation rules
presented above now enable to formulate tangent moduli in absolute notation and discuss then their
properties. The formulation succeeds in spatial as well as material descriptions. Important notations to be
exploited in this context are:
432
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
w ˆ w…C†
F ˆ gi Gi
C ˆ FT F
b ˆ FFT
E ˆ 12 …C ÿ I†
S
s ˆ FSFT
C; c
_ ÿ1
l ˆ FF
d ˆ …1=2†…l ‡ lT †
Lm a ˆ a_ ÿ la ÿ alT
energy density function (per unit volume of the undeformed body),
deformation gradient,
right Cauchy-Green tensor,
left Cauchy±Green tensor (Finger tensor),
Green±Lagrange strain tensor,
second Piola±Kirchho€ stress tensor,
Kirchho€ stress tensor,
tangent(elastic) moduli in material and spatial description respectively,
spatial gradient of velocity,
rate of deformation tensor,
Lie derivative (Oldroyd rate) of a spatial tensor a ˆ aij gi gj .
We start with the general de®nitions of the tangent moduli. Using the Doyle±Ericksen formula [13] and
by virtue of relations (3.1) and …3:12†2 the tangent moduli are constructed such that they satisfy the corresponding constitutive rate equations.
Material description:
ow ow
ˆ
;
oC oE
o2 w
o2 w
ˆ
Cˆ4
oCoC oEoE
oS
1
1
_
: C_ ˆ C : C_ ˆ C : E:
S_ ˆ 2
oC 2
2
Sˆ2
…6:1†
…6:2†
…6:3†
Spatial description:
s ˆ 2F
ow T
F ;
oC
…6:4†
o2 w
: …FT F† …cabcd ˆ FaA FbB FcC FdD CABCD †;
oCoC
o2 w _
o2 w T
T
_
: E FT ˆ 4F
F : …FT dF† ˆ c : d:
Lm s ˆ FSF ˆ F 4
oCoC
oCoC
c ˆ …F FT † :
…6:5†
…6:6†
The expression of the tangent modulus in the spatial formulation (6.5) coincides with the well-known
representation [33]:
cˆ2
os
o2 w
ˆ4
;
og
ogog
…6:7†
if only the derivation in (6.7) is carried out with respect to the covariant components of the metric tensor
g ˆ gij gi gj in the current con®guration and it is assumed that the covariant vector bases of s ˆ sij gi gj
are independent of gij . This leads to
cˆ
os osij
oS T
o2 w
ˆ
F : …FT F† ˆ …F FT † :
: …FT F†;
gi gk gl gj ˆ F
og ogkl
oC
oCoC
…6:8†
which is evidently identical with (6.5). Such a derivative of a tensor, where its basis is kept constant, is
analogical to the Lee derivative. Otherwise, the di€erentiation with respect to the metric tensor in the spirit
of the de®nition (2.31) would lead to a meaningless result, since g presents the identity tensor.
Material and spatial formulations presented are valid both for isotropic and anisotropic materials. In
literature there exists also a spatial representation of the elastic modulus in term of the left Cauchy±Green
tensor b proposed by Miehe [23] for isotropic materials
c ˆ 4b
o2 w
b;
obob
Lm s ˆ c : d:
…6:9†
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
433
In the following we show, that the de®nition of the material tensor …6:9†1 leads to the rate relation …6:9†2 if
only one assumes that the tensor b is not symmetric while derivating with respect to b. To prove this we ®rst
transform the elastic modulus …6:9†1 to the proposed de®nition (2.31) of the tensor derivative. Then
c ˆ 4…b I† :
o2 w
o2 w
: …I b† ˆ 4…I b† :
: …b I†;
obob
obob
…6:10†
such that
o2 w
c ˆ 4 b brl i k gm gk gr gj
obj obl
mi
R
!R
ˆ 4bmi brl
o2 w
o2 w
b:
g gj gk gr ˆ 4b
i
k m
obob
obj obl
Using the constitutive relation for isotropic materials in the form s ˆ 2w;b b ˆ 2bw;b and by virtue of
(2.33), (5.20) and (3.11) one can obtain:
h
i
i
h
0
0
Lm s ˆ s;b : b_ ÿ ls ÿ slT ˆ …2I w;b † ‡ 2bw;bb : …lb† ‡ …2w;b I† ‡ 2w;bb b : …blT † ÿ ls ÿ slT
1
1
ˆ 4…I b† : w;bb : …b I† : d ‡ blT w;b ‡w;b lb ÿ …ls ‡ slT † ˆ c : d ‡ blT w;b ‡w;b lb ÿ …ls ‡ slT †:
2
2
On the contrary, assuming b is not symmetric in the di€erentiation and using the corresponding de®nition
of the tensor derivative (2.31) we receive:
Lm s ˆ ‰…2I w;b † ‡ 2bw;bb Š : …lb† ‡ ‰…2w;b I† ‡ 2w;bb bŠ : …blT † ÿ ls ÿ slT
ˆ 4…I b† : w;b : …b I† : d ‡ 2lT bw;b ‡2w;b bl ÿ …ls ‡ slT † ˆ c : d:
It can easily be proved that the tangent moduli C (6.2) and c (6.5) are super-symmetric according to their
de®nitions. For C it directly follows from (5.10). As to c we have by using (2.9), (3.9) and (3.10)
2 T
T
o2 w
ow
: …FT F† ˆ …FT F†T :
: …F FT †T ˆ c;
cT ˆ …F FT † :
oCoC
oCoC
"
#t
t
t
o2 w
o2 w
T
T
T
: …F F† ˆ …F F † :
: …F F†
c ˆ …F F † :
oCoC
oCoC
2
tt
ow
T
T
: …F F† ˆ c:
ˆ …F F † :
oCoC
…6:11†
T
t
…6:12†
Similarly, the symmetry can also be proved for the other elastic modulus
Aˆ
o2 w
;
oFoF
A ˆ AT ;
…6:13†
which is used in literature, too [33].
The super-symmetry of the elastic moduli C (6.2) and c (6.5) permits their spectral decomposition in the
form (4.9). For example we get for C
Cˆ
6
X
rˆ1
kr M r M r ;
Mr ˆ MTr :
…6:14†
Now, the question is what properties the elastic modulus corresponding to an isotropic material possesses
and how the isotropy can be identi®ed by tangent moduli. This problem can easily be solved for the natural
state, where b ˆ C ˆ I. In this case the tensor function dS…dE† ˆ C : dE should satisfy the isotropy condition:
QdSQT ˆ C : …QdEQT †
…6:15†
434
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
for an arbitrary strain variation dE and an arbitrary orthogonal tensor Q ˆ QÿT . Using the decomposition
of dE in terms of Mr :
dE ˆ
6
X
dEr Mr ! QdEQT ˆ
rˆ1
dS ˆ
dEr …QMr QT †
…6:16†
rˆ1
we obtain then from (6.14):
6
X
6
X
!
kr M r Mr
6
X
:
rˆ1
!
dEs Ms
sˆ1
ˆ
6
X
kr dEr Mr :
…6:17†
rˆ1
Under consideration of (6.17) the condition (6.15) takes then the form:
6
X
kr dEr QMr QT ˆ
rˆ1
6 X
6
X
rˆ1
kr dEs Mr : …QMs QT † Mr :
…6:18†
sˆ1
Contracting the left- and right-hand sides with Mt yields
6
X
6
6
X
X
kr dEr Mt : …QMr QT † ˆ
kt dEr Mt : …QMr QT † !
…kt ÿ kr †dEr Mt : …QMr QT † ˆ 0
rˆ1
rˆ1
8dEr
and
rˆ1
ÿT
8Q ˆ Q :
…6:19†
Since eigentensors Mt are symmetric the condition (6.19) can be satis®ed only in two following cases:
1: k1 ˆ k2 ˆ k6 ˆ 2l ! C ˆ 2lI 0 ;
1
2: k1 ˆ 3k; M1 ˆ p I; k2 ˆ k6 ˆ 0 ! C ˆ kI I;
3
…6:20†
…6:21†
where 2l and k denote material constants. Generally we can state that an elastic modulus corresponds to an
isotropic material in the natural con®guration, if and only if C can be represented via linear combination of
the tensors I 0 and I I in the form of the St. Venant±Kirchho€ model (see also [14,15,17,19,20,29,31]):
C ˆ 2lI 0 ‡ kI I:
…6:22†
The analogous statement for the constitutive relations has been proved by Ciarlet [10].
7. Examples of elasticity tensors
Using the results (6.1), (6.2), (6.4) and (6.5) of the previous section as well as tensor algebra operations
and di€erentiation rules (3.5), (3.6), (3.9), (3.10), (3.13)±(3.15), (3.22), (3.23), (3.28), (3.31)±(3.33), (5.21) and
(5.22), we derive here tangent tensors in material and spatial descriptions for a number of hyperelastic
material models intensively used in continuum mechanics.
1. St.Venant±Kirchhoff material model:
1
2
2
w ˆ j…tr E† ‡ l tr…devE† ;
2
…7:1†
k and l being the compression and shear moduli respectively.
Material description:
2
ow
o‰tr…dev E† Š o…dev E†
ˆ j…tr E†I ‡ l
:
ˆ j…tr E†I ‡ 2l dev E;
oE
o…dev E†
oE
oS
1
ˆ jI I ‡ 2l I 0 ÿ I I :
Cˆ
oE
3
Sˆ
…7:2†
…7:3†
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
435
Spatial description:
1
s ˆ FSFT ˆ j…tr b ÿ 3†b ‡ lb dev b;
2
1
0
T
T
c ˆ …F F † : C : …F F† ˆ jb b ‡ 2l …b b† ÿ b b :
3
…7:4†
…7:5†
2. Mooney±Rivlin material model [25]:
w ˆ C1 …IC ÿ 3† ‡ C2 …IIC ÿ 3† ˆ C1 …Ib ÿ 3† ‡ C2 …IIb ÿ 3†;
…7:6†
where C1 and C2 are material constants.
Material description:
ow
ˆ 2C1 I ‡ 2C2 …IC I ÿ C†;
oC
oS
ˆ 4C2 …I I ÿ I 0 †:
Cˆ2
oC
Sˆ2
…7:7†
…7:8†
Spatial description:
ow
b ˆ 2C1 b ‡ 2C2 …IC b ÿ b2 †;
ob
h
i
0
c ˆ …F FT † : C : …FT F† ˆ 4C2 b b ÿ …b b† :
s ˆ FSFT ˆ
3. Ogden material model [27]:
X lr ÿ a
k1r ‡ ka2r ‡ ka3r ÿ 3 ;
w…ki † ˆ
ar
r
…7:9†
…7:10†
…7:11†
where ki denote the eigenvalues of the right stretch tensor U and ar ; lr are material parameters.
Material description:
S ˆ 2 …aI ‡ aII IC †I ÿ aII C ‡ aIII IIIC Cÿ1 ;
h
C ˆ 4 …cII : ‡ 2cI II IC ‡ cII II I2C ‡ bII †I I ‡ cII II C C ‡ IIIC …bIII ‡ cIII III IIIC †Cÿ1 Cÿ1
ÿ …cI II ‡ cII II IC †…I C ‡ C I† ‡ IIIC …cI III ‡ cII III IC †…I Cÿ1 ‡ Cÿ1 I†
i
0
ÿ IIIC cII III …C Cÿ1 ‡ Cÿ1 C† ÿ bII I 0 ÿ IIIC bIII …Cÿ1 Cÿ1 † :
Spatial description (cf. [18]):
s ˆ 2 …aI ‡ aII Ib †b ÿ aII b2 ‡ aIII IIIb I ;
h
c ˆ 4 …cII : ‡ 2cI II Ib ‡ cII II I2b ‡ bII †b b ‡ cII II b2 b2 ‡ IIIb …bIII ‡ cIII III IIIb †I I
ÿ …cI II ‡ cII II Ib †…b b2 ‡ b2 b† ‡ IIIb …cI III ‡ cII III Ib †…b I ‡ I b†
i
0
ÿ IIIb cII III …b2 I ‡ I b2 † ÿ bII …b b† ÿ IIIb bIII I 0 :
…7:12†
…7:13†
…7:14†
…7:15†
Using the alternative representation form of the energy function w ˆ w…IC ; IIC ; IIIC † the unknown coecients appearing in (7.12)±(7.15)
a K ˆ bK ˆ
ow
;
oK
cKL ˆ
o2 w
;
oKoL
K; L ˆ IC ; IIC ; IIIC
…7:16†
can be specialized for an arbitrary isotropic material model. For the Valanis±Landel hypothesis [40] and its
special case Ogden model (7.11) these coecients have been determined by Bas ar and Itskov [1] for the
ß problems.
cases of distinct as well as equal eigenvalues without explicit solving the eigenvalue
436
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
4. Material models with decoupled volumetric±isochoric response. For these models the energy function is
assumed to be additively split into volumetric and isochoric parts:
…7:17†
w ˆ U …J † ‡ W …C†;
where
J ˆ …IIIC †1=2 ;
C ˆ J ÿ2=3 C:
…7:18†
Accordingly the corresponding stress tensors and elastic moduli permit the additive decomposition:
S ˆ Svol ‡ Siso ;
s ˆ svol ‡ siso ;
C ˆ Cvol ‡ Ciso ;
c ˆ cvol ‡ ciso
…7:19†
…7:20†
and take the following form:
Material description:
Svol ˆ 2
oU
ˆ U 0 J Cÿ1 ;
oC
…7:21†
Siso ˆ 2
ÿ
oW
oW oC
S
iso ;
:
ˆ2
ˆ Siso : p ˆ dev Siso C Cÿ1 ˆ Cÿ1 dev C
oC
oC oC
…7:22†
ÿ
0
oSvol ÿ 00 2
ˆ U J ‡ U 0 J Cÿ1 Cÿ1 ÿ 2U 0 J Cÿ1 Cÿ1 ;
oC
…7:23†
Cvol ˆ 2
0
ÿ
oSiso
oSiso ÿ
ˆ 2Cÿ1 P0 : C
‡ I Siso
: p ÿ 2…Cÿ1 Cÿ1 †0 dev C Siso
oC
oC
h
0 i
ÿ
0
ÿ
ÿ1
: p ÿ 2 Cÿ1 Siso ;
ˆ pT : Ciso : ‡ 2 C : Siso
Ciso ˆ 2
…7:24†
where
Siso ˆ 2
oW
;
oC
Ciso ˆ 2
oSiso
o2 W
;
ˆ4
2
oC
oC
…7:25†
oC
oJ ÿ2=3
1
‡ J ÿ2=3 I 0 ˆ ÿ C Cÿ1 ‡ J ÿ2=3 I 0 :
ˆC
pˆ
oC
oC
3
Spatial description:
svol ˆ FSvol FT ˆ U 0 J I;
…7:26†
T
siso ˆ FSiso F ˆ devsiso ;
…7:27†
T
T
T
T
00 2
0
0
0
cvol ˆ …F F † : Cvol : …F F† ˆ …U J ‡ U J †I I ÿ 2U J I ;
0
ciso ˆ …F F † : Ciso : …F F† ˆ P : ‰ciso ‡ 2…I siso † Š : P ÿ 2…I siso †
2
2
ˆ P0 : ciso : P0 ‡ tr siso P0 ÿ …siso I ‡ I siso †;
3
3
…7:28†
0
…7:29†
where
T
siso ˆ F Siso F ;
T
T
ciso ˆ …F F † : Ciso : …F F†;
F ˆ J ÿ1=3 F:
R
…7:30†
It can easily be seen that after the transformation … † the expressions (7.23) and (7.24) coincide with
those given by Simo and Taylor [34]. In the spatial description we have forgone to use the de®nition (6.9)
M. Itskov / Comput. Methods Appl. Mech. Engrg. 189 (2000) 419±438
437
for the isochoric moduli ciso . For this reason the representation (7.26)±(7.29) holds in contrast to the
analogous result by Miehe [23] not only for isotropic but also for anisotropic materials.
8. Conclusion
In the paper we have presented a theory of fourth-order tensors. The theory enables to solve many
relevant problems in the treatment and application of fourth-order tensors, which are still open in the
literature.
Starting with the set of notations and de®nitions we have presented in absolute notation the tensor
algebra and di€erentiation rules, which are applied then to isotropic tensor functions. The important result
to be emphasised is a closed-formula solution for the derivative of an isotropic tensor function, which is
given in terms of eigenvalue bases for the cases of distinct and equal eigenvalues. This solution has a simpler
and more compact form in contrast to those known in the literature.
The next important topic are the eigenvalue problems and the spectral decomposition of a fourth-order
tensor. It is shown that the eigenvalue problem of a fourth-order tensor can be reduced to that one of its
matrix representation and then easily solved. This leads to a spectral decomposition involving nine eigenvalues and nine corresponding eigentensors. In the case of super-symmetry characterizing for example
some tangent moduli, the spectral decomposition can be given through only six symmetric eigentensors.
Concerning the invariants of a fourth-order tensor we have come to their total number of 27, which can be
obtained using various contraction rules with a second-order tensor.
A most important application of the fourth-order tensors are tangent moduli. The tensor algebra and
di€erentiation rules presented simplify considerably the formulation and derivation of tangent moduli,
which has been demonstrated on examples of some hyperelastic material models. The elastic moduli are
obtained in a material as well as in a spatial description and compared with the results available in the
literature.
Acknowledgements
The ®nancial support of the German National Science Foundation (DFG) is gratefully acknowledged.
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