Kintzel Fourth Order Tensors Part2

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ZAMM Z. Angew. Math. Mech., 1 23 (2005) /DOI 10.1002/zamm.200410243

Fourth-order tensors tensor differentiation

with applications to continuum mechanics.

Part II: Tensor analysis on manifolds

O. Kintzel

1 Institute for Structural Mechanics, Ruhr University Bochum, Bochum, Germany

Received 29 May 2004, revised 24 April 2005, accepted 27 June 2005

Published online 19 October 2005

Key words fourth-order tensors, tensor differentiation, tangent operators, anisotropic hyperelasticity.

MSC (2000) 04A25

In Part II of this contributionthe theoryof fourth-ordertensors andtensor differentiationintroduced in Part I will be applied to

finite deformation problems using a mathematical framework which is mainly borrowed from tensor analysis on manifolds.

The main aspects are the explicit consideration of component variance, the introduction of different configurations and its

associated metric tensors. Some relevant problems of continuum mechanics like the construction of tangent operators are

discussed. Special attention is given to the conjugated formulation of the corresponding tangent operator in terms of the

left Cauchy-Green-tensor.

c 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

In the first part of this contribution a tensor formalism for fourth-order tensors has been introduced which has been used in

the process of tensor differentiation with respect to a second-order tensor according to a recent approach ofItskov [8]. The

proposed rules allow an easy application of the tensor differentiation law in absolute tensor notation which is in particular

useful for the linearization of nonlinear functionals, the differentation of isotropic tensor functions e.g. in pow er series or

the construction of tangent operators. In the first part the laws have been applied in a mathematical framework referred to

as classical tensor analysis [1, 16, 11, 12]. However, as far as finite deformation or large strain problems are concerned,

the mathematical framework of tensor analysis on manifolds [2, 13, 24] is more appropriate since it delivers a rigorous and

profound mathematical theory for the study of such kinds of problems. This mathematical framework has attained popularity

especially through the work of Marsden & Hughes [13]. In a number of overview articles the topic of tensor algebra

on manifolds has been addressed in a comprehensive way (see e.g. [4, 5, 22]).

Due to the relevance of this mathematical field the present contribution will focus on the algebra of tensor analysis

on manifolds and aims at a unified presentation considering the theory of fourth-order tensors and the rules of tensor

differentiation. The consideration of large strain problems rests upon the introduction of different configurations of a body

and its associated metric tensors. Following the typical approach of tensor algebra in dual spaces these metric tensors, which

are used to construct invariants of tensors, are introduced as independent argument tensors in the corresponding tensor

functions since the push-forward or pull-back of a metric tensor is not trivially the identity. According to this approach thederivative with respect to a metric tensor is a well-established quantity. We introduce the well-known Doyle-Ericksen-

formula [17, 19], which is given as derivative of the strain energy function with respect to the spatial metric tensor g, and

discuss its conjugated formulation in terms of the left Cauchy-Green-tensor b. The latter problem has been addressed

in earlier contributions [15, 14, 7]. However, we will derive this formulation solely on the basis of the proposed tensor

differentiation rules and are able to show that the obtained result is at variance with those relations proposed earlier. In

addition, we are concerned with the most general case of anisotropic hyperelasticity which has been considered in [14]

only partly. Two equivalent solutions are obtained with a number of useful connections. By making use of the principle of

covariance the application of the chain rule in time derivatives of covariant scalar-valued tensor functions is addressed [18].

This paper is divided into five sections. In the second section some selected concepts from tensor analysis on manifolds

are introduced which are used in the sequel. In the third section the theory of fourth-order tensors and tensor differentiation

presented in Part I will be briefly recalled. Finally, to show the usefulness of the introduced formalism some relevant problems

of nonlinear continuum mechanics are discussed. E-mail: [email protected] , Address: IA 6, Universitatsstrae 150, 44780 Bochum, Germany, Web-page: http://www.sd.rub.de



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2 O. Kintzel: Fourth-order tensors tensor differentiation. Part II: Tensor analysis on manifolds

2 Mathematical preliminaries

The terminology of tensor analysis on manifolds will be adopted and in the sequel there will be distinction between members

of the tangent space called vectors and of the co-tangent space which will be denoted one-forms or simply dual vectors or co-

vectors. Linear vector spaces are introduced by demanding the well-known rules of linearity (see [16]). To measure lengthsand angles an inner vector product is introduced with the well-known properties such that an Euclidean linear vector space

is obtained. Note that in this contribution only three-dimensional Euclidean linear vector spaces are considered. The inner

product of vectors and co-vectors is denoted by < (.), (.) >x and < (.), (.) >x , respectively. The kind of configuration ormetric used is indicated by the subscript attached to the brackets,for example x denoting the current (deformed) configuration

or Xdenoting the reference (undeformed) configuration. Vectors of the deformed configuration are in the sequel denoted by

lower case letters and dual vectors by lower case Greek symbols. For vectors and co-vectors of the reference configuration

upper case letters will be used. The inner vector products of two vectors resp. two co-vectors are therefore given by:

< aaa,bbb >x , < , >x , < AAA,DDD >X , < , >X . (1)

In contrast, the scalar product is based on a pairing of dual quantities. The scalar product is denoted by ( ) and can be givene.g. in the following forms:

AAA , aaa , (2)

where vectors and co-vectors are involved. Vectors (co-vectors) originally have contra-variant (co-variant) components. By

lowering (raising) the components so-called associated tensors are obtained which are denoted by the symbols (. . . ) and(. . . ) according to [13]. A vector can be transformed into a co-vector = aaa by lowering his component which is indicated

by (. . . ). In the same manner a vector DDD = is obtained by raising the component of a co-vector. Applied to higher-ordertensors these symbols indicate the lowering or raising of all indices of the tensor in question.

The identity tensors of the current configuration are given by:

g = gij gi gj , g1 = gij gi gj , i = gi gi , i = gi gi (3)

and the identity tensors of the initial configuration read as:

G = GIJ GI GJ , G1 = GIJ GI GJ , I = GI GI , I = GI GI . (4)

The following identities hold:

gi gj = ij , gij =< gi, gj >x , gij =< gi, gj >x , gikgkj = ij ,

GI GJ = IJ , GIJ =< GI, GJ >X , GIJ =< GI, GJ >X , GIKGKJ = IJ .(5)

The base vectors gi and GI used in (3), (4) and (5) will be introduced as tangent vectors with respect to curvilinear coordinate

lines i and J defined on the manifold Bat the points x and X, respectively, where x related to the current configurationis the deformed image of X, both belonging to the same material point. The associated dual base vectors can be interpreted

as gradients to the coordinate planes i = const. and I = const.. The resulting base vectors read as:

gi =x

i=

xk

iik , GI =

X

I=

XK

IIK ,

gi = grad i = i

x=

i

xkik , G

I = Grad I =I

X=

I

XKIK .

(6)

Using cartesian coordinate systems ik and IK the tangent space could be, in principle, identified with its corresponding

co-tangent space. Thus, the Euclidean inner vector product, which by abuse in notation is also denoted by (), can be usedin (5):

gi gj = ij , gij = gi gj , gij = gi gj , gi = gij gj , gi = gij gj ,

GI GJ = IJ , GIJ = GI GJ , GIJ = GI GJ , GI = GIJ GJ , GI = GIJ GJ .(7)

However, despite this fact the distinction between the tangent and co-tangent space and between the inner vector product

and scalar product will be made further on for formal reasons. Note that in view of (7) the above given tensors (3), (4) are



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ZAMM Z. Angew. Math. Mech. (2005) / www.zamm-journal.org 3

just different representations of the identity tensors i and I, respectively. In the following g, g1 and G, G1 will be used

to represent the metric in absolute notation. The notation g1 or G1 is consistent considering the fact that the matrices of

the metric tensor components [gij ] and [GIJ] are inverse to [gij ] and [GIJ], respectively.Any invariant first-order tensor living on the manifold

Bin the deformed or undeformed configuration with base-point

x or X, respectively, can be, under the assumptions (6) and (7), in principle, decomposed either with respect to co-variant

or contra-variant base vectors. Considering the above given coordinate systems we then obtain the following vectors and

co-vectors aaa, and DDD, , respectively:

aaa = ajgj , = jgj , DDD = DIGI , = JG

J . (8)

In view of (7) their contra-variant and co-variant components are then coupled by means of the metric:

ai = gijj , I = GIJD

J , (9)

The identities (9) can be recast in absolute notation in the form:

aaa = g1 = , = GDDD = DDD . (10)

Transforming a vector or a co-vector into its dual form (10), a relationship between the inner vector product and the scalarproduct can finally be found. Reexamining the expressions (1) delivers:

< aaa,bbb >x= aaa bbb = aaa gbbb = aaa bbb = gaaa bbb ,

< , >x= = g1 = = g1 .(11)

Similar relations hold for tensors related to the reference configuration.

In the sequel the tangent space will be denoted by the symbols TXBor TxBand the co-tangent space by the symbolsTXBor TxB. In what follows the dual (. . . ) and the transpose ( )T of a second-order tensor will be introduced. Thedual C of a tensor C is defined by:

DDD C = CDDD , C Lin(TXB, TXB) , TXB, DDD TXB. (12)As a rule, the dual can be obtained by reversing the order of base vectors:

C = CIJ. G

I GJ C = CIJ. GJ GI , (C1C2) = C2C1 . (13)Since metric tensor components are symmetric (gij = gji and g

ij = gji i.e. g = g and g1 = g) the third and fifthterms of the relations (11) are, under consideration of (12), indeed equivalent.

The definition of the transpose (. . . )T of a second-order tensor, which shall be used only for those tensors which transformvectors onto vectors or co-vectors onto co-vectors, is based on the use of the inner vector product. For two two-point tensors

A3 Lin(TXB, TxB) and A4 Lin(TXB, TxB) and two arbitrary vectors bbb TxBand BBB TXBand co-vectors TXBand TxB, respectively, the following relations hold:

< A3BBB,bbb >x=< BBB, AT3 bbb >X , < A4, >x=< , A

T4 >X , (14)

where to each inner vector product belongs a certain metric. Transformed to the scalar product an explicit relation for thetranspose AT can be inferred,

A3BBB gbbb = BBB A3gbbb = BBB GAT3 bbb , A4 g1 = A4g1 = G1AT4 , (15)in the form:

AT3 = G1A3g , A

T4 = GA

4g

1 . (16)

By raising and lowering the components of A3 or A4 according to (10) it is ensured that the transpose maps vectors onto

vectors or co-vectors onto co-vectors, respectively. For more detail of this special topic please refer to [4, 9] and [22].

The mapping of vectors from the reference onto the current configuration is accomplished by the linear transformation

F, the so-called deformation gradient. In this sense the deformation gradient F is a member of Lin+(TXB, TxB), of thespace of linear transformations ofTX

Bonto Tx

Bwith det(F) > 0. Its definition is given by

F = Grad x =x

X=

x

i

i

X=

x

i

i

JJ

X=

i

Jgi GJ . (17)



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For convective coordinates, which we will use throughout this contribution, the condition i(x, t) = I(X, t0) 1(x, t)holds, where (X, t) denotes the deformation mapping. Then we may, at first, identify all indices in upper case format with

lower case letters in (4)(9), (13), (17), and, secondly, since the Jacobi-matrix i

j=

i

(j(X,t))= ij is the identity

matrix, we obtain the following representations for the deformation gradient and its related forms:

F = gi Gi , F1 = Gi gi , F = Gi gi , F = gi Gi . (18)Note that the use of convective coordinates implies an important fact: Tensorial expressions which are convected by the

deformation gradient have the same component representations. This fact is highlighted in (66). Furthermore, the convection

by any tensor, which is calledpush-forward or pull-backis denoted by A(. . . ) and A(. . . ), which extends the applicability

to thoses cases, where A is a linear isomorphism (see [22] for details) in contrast to the more usual notations (. . . ) and(. . . ) (see [13]), where is assumed to be smooth (i.e. a diffeomorphism). The results of push-forward or pull-back-operations applied to second-order tensors are rather standard and can e.g. be found in [1]. However, since we have used a

new notation ( ) for the dual these important relations are for easy reference given in Table 1.

Table 1 Definition of push-forward and pull-back of second-order tensors and Lie-derivatives holding for convective

coordinates (l = FF1 = FF1, F Lin(TXB, TxB), gi = FGi, gi = FGi).

A1 = A1ij Gi Gj Lin(TXB, TXB)

a1 = a1ij gi gj Lin(TxB, TxB)

A2 = A2ij Gi Gj Lin(TXB, TXB)

a2 = a2ij gi gj Lin(TxB, TxB)

F(A1) = FA1F

1

F(a1) = Fa1F

Lv(a1) = a1 + la1 + a1l = a1ij g

i gj

F(A2) = FA2F

F(a2) = F1a2F

Lv(a2) = a2 la2 a2l = a2ij gi gj

A3 = A3i. j Gi Gj Lin(TXB, TXB)

a3 = a3i. j gi gj Lin(TxB, TxB)

A4 = A4ij. G

i Gj Lin(TXB, TXB)a4 = a4i

j. g

i gj Lin(TxB, TxB)F(A3) = FA3F

1

F(a3) = F1a3F

Lv(a3) = a3 la3 + a3l = a3i. j gi gj

F(A4) = FA4F

F(a4) = Fa4F

Lv(a4) = a4 + la4 a4l = a4ij. gi gj

Symmetry and skew-symmetry-conditions for second-order tensors A1 Lin(TXB, TXB) or A2 Lin(TXB, TXB)are based on the use of the dual, whereas for tensors A3 Lin(TXB, TXB) or A4 Lin(TXB, TXB), which map vectorsonto vectors or co-vectors onto co-vectors, the transpose has to be used. This fact has to be considered to find the correct

form of the fourth-order (skew-)symmetry-transformation tensors S (A) which will be given in Sect. 3. Note that symmetry

holds only for those tensors whose both legs are in the same configuration. In particular, symmetry for A3 implies:

A3 = AT3 A3G = GA3 (GA3) = GA3 , (GA3) Lin(TXB, TXB) . (19)

Thus, the operations (. . . )sym and (. . . )skw are defined as follows:

A1

Lin(TXB

, T

XB) , A

1sym=

1

2(A

1+ A

1)

Sym , A1skw

=1

2(A

1 A

1)

Skw , (20)

A2 Lin(TXB, TXB) , A2sym =1

2(A2 + A

2) Sym , A2skw =

1

2(A2 A2) Skw , (21)

A3 Lin(TXB, TXB) , A3sym = 12

(A3 + AT3 ) Sym , A3skw =

1

2(A3 AT3 ) Skw , (22)

A4 Lin(TXB, TXB) , A4sym =1

2(A4 + A

T4 ) Sym , A4skw =

1

2(A4 AT4 ) Skw . (23)

Remark 2.1 A rotation as part of a rigid body motion does preserve lengths and angles and represents therefore, as far

as the inner vector product is concerned, an invariance operation. The corresponding rotation tensor can be related to one

configuration only or can be a two-point tensor like R

Lin(TX

B, Tx

B) used in the polar decomposition of the deformation

gradient F = RU = vR. For the latter case the following holds:

< RAAA, RBBB >x=< AAA, RTRBBB >X=< AAA,BBB >X (24)



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which postulates:

RTR = I RT = G1Rg = R1 . (25)

Each tensor which fulfills the condition RT

= R1

is said to be orthogonal. Using Table 1 we conclude that the orthogonalrotation tensor R pushes the metric tensor G forward to g and vice versa:

g = R(G) , g1 = R(G

1) . (26)

An orthogonal transformation Q defined in the tangent space of the current configuration is defined by:

Q Orth(TxB, TxB) := {Q Lin(TxB, TxB) | QT = g1Qg = Q1} . (27)This so-called isometry leaves the metric unchanged (g = Q(g) and g

1 = Q(g1)).

Remark 2.2 A double contraction is based on a pairing of dual quantities. In view of this fact the following holds:

B : C = A(B) : A(C) and D : CB = DB : C = B : DC = BC : D . (28)

Remark 2.3 Material and spatial covariance of a tensor function will be explained in more detail in Sect. 4. A scalar-valued tensor function f(B1, B2) with two argument tensors B1, B2 is said to be covariant if for an invertible tensor A whichmaps vectors onto vectors the invariance relation f(B1, B2) = f(A(B1), A(B2)) is satisfied. Then, by fundamental

algebraic manipulations, the following relation for its time rate f is obtained:

f =2

i=1

f

Bi: Bi =

2i=1

f(A(B1), A(B2))

A(Bi):

A(Bi) =

2i=1

A

f(B1, B2)

Bi

:

A(Bi)

=2

i=1

f

Bi: A

A(Bi)

.

(29)

The Lie-derivative given in the last expression can be understood as a general representation of so-called objective timerates. Conclusively, any kind of time derivative can be used for a covariant scalar-valued tensor function. The Lie-derivative

using the deformation gradient A1 = F in (29) isdenoted byLv(. . . ) = F(

F(. . . )). Its definition is included in Table 1where l Lin(TxB, TxB) represents the velocity gradient. (We note that for convective coordinates this Lie-derivative isidentical to a material time derivative of the tensor components [1]). For the above relation (29) to hold it is essential to

include the metric tensor which has been used to construct invariants as one of its argument tensors (see (85)). Note in this

context that the material time derivative of a metric tensor is zero i.e. G = 0 or g = 0, but its Lie-derivative not.

Remark 2.4 We note that we define a tensorand its corresponding component variancefrom theoutset without indicating

this component decomposition by certain symbols. The symbols (. . . ), (. . . ) or (. . . ) which are used in [14] hereexclusively denote associated tensors (Also, we donnot use the notations (. . . )\ and (. . . )/ which have been introduced in[4]).

Remark 2.5 In [4] and [5] a push-forward and pull-back-relation for mixed-variant tensors has been proposed which

differs from that given in Table 1.As motivation for this new definition the invarianceof the scalar-valued inner vector product

with respect to a push-forward or pull-back has been assumed. However, we have to consider that only the scalar product

is a priori invariant with respect to a push-forward- or pull-back-operation due to duality and that a certain metric tensor is

required to transform an inner vector product into a scalar product (11). Indeed, we obtain for a tensor A Lin(TXB, TXB)and two vectors BBB, DDD TXB:

F(< BBB, ADDD >X) = F(BBB GADDD) = FBBB FGF1F(A)FDDD = BBB GF1F(A)FDDD (30)which implies the relation F(A) = FAF

1 which is in full agreement with Table 1.

3 Theory of fourth-order tensors and tensor differentiation

In the following the theory of fourth-order tensors already presented in Part I of this contribution is briefly recalled. Notethat the dual and the transpose have been distinguished. Since the dual has to be used in all tables which were given in Part

I the important tables (Tables 25) are reiterated.



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Table 2 Simple contractions of fourth- and second-order tensors.

(A B) C = A (B C) C (A B) = (C A) B(A B) C = (A C) B C (A B) = (C A) B(A 2 B) C = A 2 (B C) C (A 2 B) = (C A) 2 B

Table 3 Double contractions of fourth- and second-order tensors.

(A B) : C = A (B : C) C : (A B) = (C : A) B

(A B) q aC = A C B C q a(A B) = A C B

(A B) a qC = A C B C a q(A B) = A C B

(A B) : C = A C B C : (A B) = B C A

(A B) q aC = (B : C) A C q a(A B) = (A : C) B(A B) a qC = (A : C) B C a q(A B) = (C : B) A

(A 2 B) : C = A C B C : (A 2 B) = A C B

(A 2 B) q aC = A C B C q a(A 2 B) = B C A(A 2 B) a qC = B C A C a q(A 2 B) = A C B

Three different tensor products to form a fourth-order tensor have been defined:

D = A B = Aij Bmn Gi Gj Gm Gn , (31)E = A B = Aij Bmn Gi Gm Gn Gj , (32)F = A 2 B = Aij Bmn Gi Gm Gj Gn . (33)

Also, three double contraction rules have been introduced. The classical contraction rule (:) is defined in accord with

common convention. In particular, for the purpose of tensor differentiation two more contraction rules ( q a) and ( a q) have

been introduced. Note that the contraction rule ( q a) is mainly used in applications. However, the contraction rule ( a q) is

useful for the computation of derivatives of traces of second-order tensors.

E : F =Eijmn Gi Gj Gm Gn : Fklrs Gk Gl Gr Gs

= Eijmn Fmnrs Gi Gj Gr Gs , (34)E q aF =

Eijmn Gi Gj Gm Gn

q a

Fklrs G

k Gl Gr Gs

= Eijmn Fjlrm Gi Gl Gr Gn , (35)E a qF =

Eijmn Gi Gj Gm Gn

a q

Fklrs G

k Gl Gr Gs

= Eijmn Fkins Gk Gj Gm Gs . (36)

As has been shown in Part I the set of fourth-order tensors is a ring (G, +, ) in conjunction with each operation ( = :, q a, a q)

i.e. the associative and the distributive law holds and there exist well-defined identity elementsIR

resp.I

. However, sincewe distinguish between different decompositions of a second-order tensor the fourth-order identity tensors are also given in

four different representations depending on the component variance of the second-order tensor considered. For the newly



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Table 4 Double contractions of fourth-order tensors.

(A B) : (C D) = (B : C) A D

(A B) q a(C D) = (A C) (D B)(A B) a q(C D) = (C A) (B D)

(A B) : (C D) = A (D B C) (A B) : (C D) = (A C B) D

(A B) q a(C D) = (A C B) D (A B) q a(C D) = A (C B D)(A B) a q(C D) = C (A D B) (A B) a q(C D) = (C AD) B

(A B) : (C D) = (A D) 2 (B C)

(A B) q a(C D) = (B : C) A D(A B) a q(C D) = (A : D) C B

(A B) : (C 2 D) = A (C B D) (A 2 B) : (C D) = (A C B) D

(A B) q a(C 2 D) = (A C) 2 (D B) (A 2 B) q a(C D) = (A D) 2 (C B)(A B) a q(C 2 D) = (C B) 2 (A D) (A 2 B) a q(C D) = (C A) 2 (B D)

(A 2 B) : (C 2 D) = (A C) 2 (B D)

(A 2

B) q a(C 2

D) = (A D)

(C B)

(A 2 B) a q(C 2 D) = (C B) (A D)

(A B) : (C 2 D) = (A D) (B C) (A 2 B) : (C D) = (A C) (B D)

(A B) q a(C 2 D) = A (D B C) (A 2 B) q a(C D) = (A C B) D(A B) a q(C 2 D) = (C A D) B (A 2 B) a q(C D) = C (B D A)

proposed differentiation rule, here denoted by A,A, (see Part I) the fourth-order identity tensors are given in Table 6. Using

Table 6 the symmetry (skew-symmetry) transformation tensors S (A) are defined by:

S =1

2(A,A +A,A ) , A =

1

2(A,AA,A ) , A Sym(TXB, TXB) , Skw(TXB, TXB) , (37)

S =1

2(A,A +A,A ) , A =

1

2(A,AA,A ) , A Sym(TXB, TXB) , Skw(TXB, TXB) , (38)

S =1

2(A,A +A,AT ) , A =

1

2(A,AA,AT ) , A Sym(TXB, TXB) , Skw(TXB, TXB) , (39)

S =1

2(A,A +A,AT ) , A =

1

2(A,AA,AT ) , A Sym(TXB, TXB) , Skw(TXB, TXB) , (40)

where, using (16) along with the relationship A,AT = A,A q aA,AT

A,AT

= I2

Iq a

G G1

= G

2

G

= G

12

G , A Lin(TXB, TXB) , (41)A,AT = (I

2 I) q a(G1 G) = G 2 G = G 2 G1 , A Lin(TXB, TXB) , (42)



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Table 5 Transposition operations for fourth-order tensors.

(A

B

)

T

=A

B

(A

B

)

D

=B

A

(A B)T = B A (A B)D = B A

(A 2 B)T = B 2 A (A 2 B)D = A 2 B

(A B)ti = A 2 B (A B)dl = A B(A B)ti = A B (A B)dl = B 2 A(A 2 B)ti = A B (A 2 B)dl = B A

(A

B)to = B 2

A (A

B)dr = A

B

(A B)to = A B (A B)dr = A 2 B(A 2 B)to = B A (A 2 B)dr = A B

(A B)t = B A (A B)d = A B

(A B)t = A B (A B)d = B A(A 2 B)t = B 2 A (A 2 B)d = B 2 A

Table 6 Identity tensors of fourth-order.

new convention

A = Aij Gi Gj A = Aij Gi Gj

A,A = I I

A,A = I2 I

A,A = I2 I

A,A = I I

A,A = I IA,A = I 2 IA,A = I 2 IA,A = I I

A = Ai. j Gi Gj A = Aij. Gi GjA,A = I IA,A = I 2

I

A,A = I 2 IA,A = I

I

A,A = I I

A,A = I2

I

A,A = I 2 IA,A = I I

holds. The operations of basis rearrangement are defined as follows:

(A B)L = A B , (A B)L = A 2 B , (A 2 B)L = A B ,(A B)R = A 2 B , (A B)R = A B , (A 2 B)R = A B . (43)

The derivative A;A consistent with the old convention can be derived from A,A by means of the above introduced basis

rearrangement operations such that A;A = (A,A )R

and A,A = (A;A )L

holds. Note that only the differentiation rule

here denoted by a comma satisfies the important product rule of differential calculus (for more information on this see

Part I). To set the stage for Sect. 4 where attention is focussed on the construction of tangent operators the following

definitions for second-order derivatives of scalar-valued tensor functions with respect to an arbitrary second-order tensor

b Lin(TxB, TxB) are recalled:

W,(bb) =2W

bb=

2W

bijbklgi gk gl gj ,

W;(bb) = 2W

bb

R=

2W

bijbklgi gj gk gl . (44)

Furthermore, the transposition operations introduced in Part I applied to the tensors E = G q aC q aH and F = K : D : Lof fourth-order read as:

ET

= HT

q aCT

q aGT

, FD

= LD

: DD

: KD

,

Eti

= G q aC q aHti

, Fdr

= K : D : Ldr

,

Eto

= Gto

q aC q aH , Fdl

= Kdl

: D : L .

(45)

4 Application to relevant problems of continuum mechanics

In this section the theory of fourth-order tensors as introduced in Part I of this work and recalled in Sect. 3 will be applied

to relevant problems of continuum mechanics by using the tensor formalism which has been presented in Sect. 2.



9/23


Isotropic tensor functions and the principle of covariance. Relying on the Euclidean structure of space the principle

of frame-indifference and the principle of material objectivity require form invariance of tensor functions with respect to

superposed rigid body movements i.e. spatial isometries. Material anisotropies can be considered by means of structural

tensors ai which satisfy the material symmetry group

GGG. In case of a scalar-valued isotropic tensor function W(c, d, ai) the

above mentioned principles imply the following constraint:

W(c, d, ai) = W(Q(c), Q(d), Q(ai)) , ai GGG , (i = 1,..,n) , Q Orth , (46)postulating the invariance ofW with respect to a change of observer. On manifolds the more general principle of covariance

is used which demands form invariance with respect to diffeomorphisms [13] , or, like in the present context, with respect

to isomorphisms A, a, A1, a1 Lin, which map vectors onto vectors, [22] such that e.g.W(c, d, C, D) = W(a(c), a(d), A(C), A(D)) , a, A Iso , (47)

holds. Depending on the kinds of configurational spaces which are transformed, we speak of the principle of spatial covari-

ance (using a) or of the principle of material covariance (using A) including two-point tensors in place of a, A as well.

In the following W now represents a strain energy function. By considering isotropic finite strain elasticity the argumenttensors are the deformation gradient F Lin(TXB, TxB), the metric of the current configuration g and the metric of thereference configuration G or rather G1:

W = W(F, g, G1) . (48)

Applying the principle of material covariance with A = F yields:

W(F, g, G1) = W(FF1, g, F(G1)) = W(i, g, b) = W(g, b) (49)

with the left Cauchy-Green-tensor b = FG1F = FFTg1 Sym(TxB, TxB). Note that F represents a two-pointtensor and therefore only the right leg of F is affected by the push-forward-operation. The resulting identity tensor i has

been ignored in the potential W. Finally, by means of a spatial back-transformation with a1 = F due to the principle of

spatial covariance, the material representation ofW is obtained:

W(g, b) = W(F(g), F(b)) = W(C, G1) . (50)

Accordingly, as a well-known fact in case of isotropic material behaviour the constitutive laws can be either given in terms

of the right Cauchy-Green-tensor C = FgF = GFTF Sym(TXB, TXB) and the material metric G1 or in termsof the left Cauchy-Green-tensor b and the spatial metric g. Note that in classical tensor analysis C is derived from b by

using R as transformation tensor. Using Table 1, this fact can be indeed confirmed since

C = R(b) = R1bR = R1FG1FR

= R1RUG1URR = UG1U = UUG1 = G1CG1(51)

holds, in which, additionally, the polar decomposition F = RU and the symmetry of U Sym(TXB, TXB) (U =GUG

1

) have been used. Note that the associated tensor C

ofC is obtained whose indices have been raised by means ofthe metric tensor G1. Using this identity and the rule G = R(g) an alternative material representation ofW reads as:

W(g, b) = W(R(g), R(b)) = W(G, C) (52)

which can be transformed to (50) by means of raising or lowering indices.

By using the representation theorem for isotropic tensor functions the strain energy function W can be defined in terms

of the three invariants of C and b:

IC = tr(CG1) = Ib = tr(gb) ,

IIC =1

2

(tr(CG1))2 tr(CG1)2 = IIb = 1

2

(tr(gb))2 tr(gb)2 ,

IIIC = det(CG1

) = IIIb = det(gb)

(53)

bearing the identity tr(CG1)n = tr(GC)n = tr(G1C)n = tr(CG)n in mind.



10/23


Construction of tangent operators. Here we will consider the following hyperelastic constitutive law ofNeo-Hookean

type [20]:

W =

1

2 (ln J)

2

+

1

2 J2/3 IC 3 , (54)with a volumetric-deviatoric-split involving the Jacobi-determinant J =

IIIC, the bulk modulus and the shear modulus

as material constants. Since C is a symmetric tensor the symmetry-transformation tensor S takes by using Table 6 the

following form:S = 12

(C,C +C,C ) =12

(I I + I 2 I). As example let us compute the derivative of the second-orderinvariant tr(CG1)2 = tr(G1C)2. By using Tables 24 this derivative can be obtained easily:

tr(CG1)2 = tr(CG1CG1) = (CG1CG1) a qI = I q a(CG1CG1) ,

tr(CG1)2,C = I q a(I G1CG1 + CG1 G1) q aS (55)

= 2 (GCG) q aS = (G1CG1 + GCG) = 2 G1CG1 ,

considering the symmetry of C and G1 (C = C, G1 = G). The derivative of a trace of an arbitrary tensor powerwith positive exponent n can be inferred from (55) and is given by:

tr(CG1)n,C = n

G1Cn1

G1 . (56)

Using this result along with J,C =12JC

, the first- and second-order derivatives of the potential (54) finally read as:

S = 2 W,C = ln JC1 + J2/3

G1 1

3IC C

1

, (57)

CCC = 4 W,(CC) =

+

2

9 J2/3 IC

C1 C1

23

J2/3

G1 C1 + C1 G1

ln J 13

J2/3 IC

C1 C1 + C1 2 C1 , (58)

where S denotes the second Piola-Kirchhoff stress tensor. The symmetry ofC and G1 has been considered by using

the symmetry transformation tensor S in the differentiation procedure and by setting C = C and G = G1. It isobservable from (58) that C represents a supersymmetric fourth-order tensor such that by using the identities from Table 5

and (45)C = Cti

= Cto

= CT

holds.

Remark 4.1 With some experience it will be more effective to avoid the explicit use of the symmetry transformation

tensor S by using already derived formulas such as C1,C = 12 (C1 C1 + C1 2 C1) (ifC is symmetric) from theoutset (see Part I) or by following the simplifying procedure explained at the end of the paper (see last example).

Now the spatial representation of the above given constitutive law is examined. By using the relation tr(CG1)n =tr(gb)n for the trace of a tensor power (CG1)n and (gb)n (see (53)), respectively, the spatial representation of the strainenergy function reads as:

W =1

2 (ln J)2 +

1

2

J2/3 Ib 3

, (59)

where the Jacobi-matrix J =

IIIb has been used.

Doyle-Ericksen-formula. The well-known Doyle-Ericksen-formula [17, 19] originally proposed for Cauchy-

stresses (see [13]) leads to the following formula for the Kirchhoff stress tensor :

= 2W

g . (60)



11/23


Remark 4.2 For some authors (see e.g. [8]) the differentiation with respect to g is not admissible since in their view of

classical tensor analysis the tensor g represents an identity tensor. Also they argue, that by virtue of the invariance theorem g

and g1 are the same tensor, but, as can be shown, the derivative W,g differs from W,g1 which contradicts itself. However,

we note that by resorting to tensor algebra in dual spaces g and g1 are not the same tensor. In the present context relation

(60) is just the push-forward of relation (57) ( = F(S) = 2 F(W(C, G1),C ) = 2 W(g, b),g) using solely theco-variant representation of the identity tensor, the so-called metric tensor g = gij g

i gj .By means of the formulas J,g =

12

Jg, g1,g = 12 (g1 g1 + g1 2 g1) and by considering the symmetryofg and b (setting g = g and b = b) we obtain in analogy to (57) and (58):

= 2 W,g = ln Jg1 + J2/3

b 1

3Ib g

1

, (61)

ccc = 4 W,(gg) =

+

2

9 J2/3 Ib

g1 g1

23

J2/3

b g1 + g1 b

ln J 13

J2/3 Ib

g1 g1 + g1 2 g1 . (62)

Remark 4.3 In the derivation of (61) and (62) the metric tensor g has been treated as an independent argument tensor.

This maysubstitute for a more complicated procedureby pulling thespatial representation back to thematerialrepresentation,

differentiating with respect to C and pushing the result again forward to the current configuration.

Push-forward and pull-back of a fourth-order tensor. Let us introduce at this particular state the push-forward and

pull-back of a fourth-order tensor. In analogy to (28) and in case of a fourth-order tensor ccc Lin((TxB, TxB), (TxB, TxB))the following identity holds:

( ) q accc q a( ) = ( ) q aF(CCC) q a( )

= (F

() F

())q aCCC q a

(F

() F

())= (F F) q aCCC q a(F F)= (F( )F) q aCCC q a(F( )F)= ( ) q a(F F) q aCCC q a(F F) q a( ) ,

(63)

where Table 3 has been used. For two relevant cases we obtain:

CCC = Cijkl

Gi Gj Gk Gl ccc = F(CCC) = (F F) q aCCC q a(F F) ,

CCC = F(ccc ) = (F1 F) q accc q a(F F1) , (64)

CCC=C

ijkl G

i

Gj

Gk

Gl

ccc = F(CCC

) = (F

F1

)q aCCC q a

(F1

F

) ,

CCC = F(ccc ) = (F F) q accc q a(F F) . (65)Remark 4.4 For convective coordinates the right Cauchy-Green-tensor is given by: C = F(g) = gij F

gi gjF = gij G

i Gj . Analogously, the expression for the left Cauchy-Green-tensor b = F(G1) = Gij FGi GjF

= Gij gi gj is obtained. Accordingly, in this special case the deformation gradient F transforms only the basevectors without affecting the component matrix. In view of this fact, the component relations of the above given expressions

for the stresses and tangent operators (57), (58) and (61), (62), respectively, are given in a unified form by:

Sij = ij = ln J gij + J2/3(Gij 13

(gmnGnm)gij) ,

Cijkl

= c ijkl = ( +2

9

J2/3 (gmnGnm))gilgjk

2

3

J2/3(Gilgjk + gilGjk)

( ln J 13

J2/3(gmnGnm))(gijgkl + gikgjl) .

(66)



12/23


Remark 4.5 It is worthwhile to transform the above given tangent operator (62) by using the basis rearrangement

operation (. . . )R into a form consistent with the old convention.

(ccc )R = 4 W;(gg) = +2

9 J2/3 Ib g1 g

1

23

J2/3

b g1 + g1 b

ln J 13

J2/3 Ib

g1 2 g1 + g1 g1 . (67)

The conjugated formulation in terms of the left Cauchy-Green tensor. Attention is now drawn to the conjugated

formulation of the constitutive relations which in the old convention is usually known as:

= 2W

bb = 2 b

W

b, (ccc )R = 4 b (W;(bb) )b . (68)

These formulations are credited to [23, 15]. The associated formula for the Kirchhoff stresses can be obtained easily.

However, by using the proposed tensor differentiation rules it has not been possible to confirm the above given formulationfor the tangent operator (68.2). Also, a result recently proposed by Menzel and Steinmann [14], who considered the

general case of anisotropy by incorporating structural tensors, is not satisfying since the derived tangent operator does not

fulfill minor or major symmetries, which will be confirmed in the following. Here a completely new approach will be used

by analyzing the first- and second-order derivatives of W and their relationships.

Note that for isotropic material behaviour the hyperelastic strain energy function in the spatial representation depends on

the invariants ofb in essentially two different forms:

W = W(gb) = W(bg) . (69)

Analyzing (69.1) the differentiation ofW with respect to g and b yields:

W

g

=W

(gb)

q a (gb)

g ,2W

gg

= (gb)

g T

q a

2W

(gb)(gb)

q a

(gb)

g

, (70)

W

b=

W

(gb)q a

(gb)

b

,

2W

bb=

(gb)

b

Tq a

2W

(gb)(gb)q a

(gb)

b, (71)

where (gb),g = i b and (gb),b = g i. Accordingly, the derivatives with respect to b and g are coupled by means of

the derivatives with respect to (gb). By eliminating these coupling terms we end up with the following equation:

(g i) q a 2W

ggq a (g i) = (i b) q a

2W

bbq a (i b) , (72)

which by means of a double contraction with (g i)1 = g i from the left and its transposed form (. . . )T fromthe right finally leads to

2W

gg=

g i q a (i b) q a 2Wbb

q a (i b) q a g1 i = g b q a 2Wbb

q a

g1 b (73)

and

W

g=

W

bq a(g1 i) q a(i b) = W

bq a

g1 b = g W

bb . (74)

For the second representation W = W(bg) equivalent but quite different relations are obtained:

2W

gg=

i g q a (b i) q a 2Wbb

q a (b i) q a i g1 = b g q a 2Wbb

q a

b g1

(75)

and



13/23


W

g=

W

bq a(i g1) q a(b i) = W

bq a

b g1 = b W

bg . (76)

Remark 4.6 Note that the symmetry ofg and b has not to be considered in the differentiation procedure. Therefore (73)

and (75) hold in this form for non-symmetric tensors g and b. However, to account for the symmetry of g the desired minor

symmetry of the tangent operator is enforced finally by means of a contraction with the symmetry transformation tensor

S = 12

(i i + i 2 i) from the right and its transposed form ST from the left which leads to the following two distinctresults:

2W

gg=

1

2

b 2 g1 + g b q a 2W

bbq a

1

2

g1 2 b + g1 b ,

2W

gg=

1

2

g1 2 b + b g q a 2W

bbq a

1

2

b 2 g1 + b g1 .

(77)

That (77) leads to the same solution as represented by (62) can be confirmed if we consider the following second-order

derivative of (59) with respect to b:

4 W,bb = + 29

J2/3 Ib b b 2

3 J2/3

g b + b g

2 ln J 23

J2/3 Ib

b 2 b . (78)

Remark 4.7 By an element-wise pull-back (F(. . . )) i.e. by replacing g and b by C and G1, respectively, thecorresponding relations are also valid in the material representation.

Remark 4.8 It is an easy task to transform the above given relations (77) by means of the process of basis rearrangement

(. . . )R (43) into a form consistent with the old convention :

W;(gg) = 12

b g + g 2 b : 2Wbb

R: 1

2

g1 b + g1 2 b ,W;(gg) =

1

2

g1 b + b 2 g1 :

2W

bb

R:

1

2

b g + b 2 g .

(79)

If we consider the component representation of (73) and (75) in the old convention:2W

gg

R=

g 2 b :

2W

bb

R:

g1 2 b (80)= goibjpgqtbvr W; (b b)opqr gi gj gt gv , (81)

2W

gg R

= b2

g1 :

2W

bbR

: b 2g (82)

= boigjpbqtgvr W; (b b)opqr gi gj gt gv , (83)we observe that W;(gg) cannot be transformed into a form which is in agreement with (68.2) even if symmetry of b andg or major and minor symmetries of W;(bb) would be considered. (But note that minor symmetries of W;(bb) haveactually not to be enforced, an observation also recently made in [7] (see (78) for W,(bb))).

For a comparison of our result with the solution in [14] we have to consider the equivalence relations ( ) and( 2 ). The solution presented in [14] then reads as:

W;(gg) = (b g1) : W;(bb) : (g1 b) = (b 2 g1) : W;(bb) : (g1 2 b) . (84)Note that the tangent operator given above does not satisfy minor symmetries (. . . ) = (. . . )dl = (. . . )dr of the second-orderderivative W;(gg) (considering (45)). Even major symmetry (. . . ) = (. . . )

D is not fulfilled.

Anisotropic hyperelasticity. In the following a more general form is discussed incorporating a complete set of structural

tensors A1j Lin(TXB, TXB), A2i Lin(TXB, TXB), A3k Lin(TXB, TXB) and A4l Lin(TXB, TXB) as argument



14/23


tensors in W. By using the representation theorem for isotropic tensor functions W can be constructed using invariants of

all tensors (see [3]). Since invariants of a second-order tensor have to be formed in mixed-variant representation using a

certain metric tensor for this purpose the following functional relationship is implied:

W(C, A1j , A2i, A3k, A4l, G1

) = W((CG1

), (A1jG1

), (GA2i), (GA3kG1

), (A4l))= W((G1C), (G1A1j), (A2iG), (A3k), (G

1A4lG)) . (85)

By invoking the principle of material covariance the corresponding spatial representation of (85) is obtained in the form:

W = W((gb), (a1jb), (b1a2i), (b

1a3kb), (a4l)) = W((bg), (ba1j), (a2ib1), (a3k), (ba4lb

1)) .

(86)

As can be observed from (86) g is coupled through the functional relationship in W with the structural tensors a1j , a2i,

a3k, and a4l by means of the left Cauchy-Green-tensor b. Now the same method is used as described before. For the

first-order derivative the results read as:

W

g =

W

bq aB

+

m1

j=1

W

a1jq aA

ij +

m2

i=1

W

a2iq aA

2i +

m3

k=1

W

a3kq aA

3k

= gW

bb

m1j=1

ga1j

W

a1j+

m2i=1

gW

a2ia2i +

m3k=1

g

W

a3ka3k a3k

W

a3k

, (87)

and

W

g=

W

bq aB+

m1j=1

W

a1jq aA1j +

m2i=1

W

a2iq aA2i +

m4l=1

W

a4lq aA4l

= bW

b

g

m1

j=1

W

a1ja1j g

+m2

i=1

a2i

W

a2ig +

m4

l=1

a4lW

a4l W

a4la4l g . (88)

After lengthy algebra, the second-order derivative can be obtained in the following form:

2W

gg= B

Tq a

2W

bbq aB+G

+m1j=1

B

Tq a

2W

ba1jq aA1j +A1

Tj

q a

2W

a1jbq aB

+m2i=1

B

Tq a

2W

ba2iq aA2i +A2

Ti

q a

2W

a2ibq aB

+m3k=1

BT q a

2

Wba3k

q aA3k +A3Tk q a 2

Wa3kb

q aB

+

m4l=1

B

Tq a

2W

ba4lq aA4l +A4

Tl

q a

2W

a4lbq aB

+m1j=1

m2i=1

A2

Ti

q a

2W

a2ia1jq aA1j + A1

Tj

q a

2W

a1ja2iq aA2i

+

m1j=1

m3k=1

A3

Tk

q a

2W

a3ka1jq aA1j + A1

Tj

q a

2W

a1ja3kq aA3k

+m1j=1

m4l=1

A4

Tl

q a

2W

a4la1jq aA1j + A1

Tj

q a

2W

a1ja4lq aA4l



15/23


+m2i=1

m3k=1

A3

Tk

q a

2W

a3ka2iq aA2i + A2

Ti

q a

2W

a2ia3kq aA3k

+

m2i=1

m4l=1

A4Tl

q a

2

Wa4la2iq a

A2i + A2Tiq a

2

Wa2ia4lq a

A4l

+

m1i,k=1

A1Ti

q a

2W

a1ia1kq aA1k +

m2j,l=1

A2Tj

q a

2W

a2ja2lq aA2l

+m3

i,k=1

A3Ti

q a

2W

a3ia3kq aA3k +

m4j,l=1

A4Tj

q a

2W

a4ja4lq aA4l , (89)

where the fourth-order transformation tensorsB,A1j ,A2i,A3k ,A4l, andG are defined according to Table 7. Obviously

W,(gg) fulfills in view of (45) major symmetry (. . . ) = (. . . )T.

Remark 4.9 As already mentionedthe symmetry ofb, a1j , a2i, a3k,and a4l hasnot to be considered in thedifferentiationprocedure. The solution for a symmetric second-order tensor g is subsequently obtained if we symmetrize the fourth-order

tensor (89) by means of a right-hand side-multiplication with S and a left-hand side multiplication with ST

.

Remark 4.10 Haupt and Tsakmakis [6] have introduced two sets of strain- and stress-pairs dual to each other. For

one set the inverse of the right Cauchy-Green-tensor C1 is actually coupled with a corresponding stress tensor. This

Table 7 Fourth-order transformation tensors.

W((gb), (a1jb), (b1a2i), (b

1a3kb), (a4l)) W((bg), (ba1j), (a2ib1), (a3k), (ba4lb

1))

B (g1 b) (b g1)

A1j (a1jg1 i) (i g1a1j)

A2i (g1 a2i) (a2i g1)

A3k (a3kg1 i) + (g1 a3k) 0

A4l 0 (i g1a4l) + (a4l g1)

G

m1

j=1

g2 g

a1

j

W

a1j

m1

j=1

g2

W

a1j a1

j g

+m1j=1

ga1j

W

a1j2 g +

m1j=1

W

a1ja1j g

2 g

+m3k=1

g 2 ga3kW

a3k+

m4l=1

W

a4la4l g

2 g

+

m3k=1

ga3k

W

a3k2 g +

m4l=1

g 2 Wa4l

a4l g

m3k=1

ga3k 2 g

W

a3k

m4l=1

W

a4lg 2 a4l g

m3k=1

g Wa3k

2 ga3k m4l=1

a4l g

2 W

a4lg



16/23


gives rise to the relationship:

W((b1g1), (a1jb), (b1a2i), (b

1a3kb), (a4l)) = W((g1b1), (ba1j), (a2ib

1), (a3k), (ba4lb1)) .

(90)

The following relations hold for the non-symmetrized solution only:

W

g1=

W

gq a

g

g1=

W

gq a (g g) = g W

gg ,

2W

g1g1= (g g) q a

2W

ggq a(g g) g 2 W

g1 W

g12 g .

(91)

Remark 4.11 Note that only by considering co-co-variant or contra-contra-variant tensors the symmetry of a tensor is

preserved during push- and pull-operations:

A1 = A1 a1 = FA1F1 = a1 , A2 = A2 a2 = FA2F = a2 , (92)

which is not true for mixed-variant tensors since by using the transpose (16) we obtain:

A3 = AT3 F(A3) = a3 = F(G1A3G) = ba3b1 = g1a3g , (93)

A4 = AT4 F(A4) = a4 = F(GA4G1) = b1a4b = ga4g1 . (94)

However, by considering the symmetry ofA1j = A1j , A2i = A2

i , A3k = A3

Tk , and A4l = A4

Tl directly in relation (85)

we get:

W(C, A1j , A2i, A3k, A4l, G1) = W((CG1), (A1jG

1), (GA2i), (A3k), (A4l))

= W((G1

C), (G1

A1j), (A2iG), (A3k), (A

4l)) . (95)

Since the push-forward of a dual is the dual of the push-forward of a tensor i.e. F(( )) = (F( )) we then end upwith the following relationship in the spatial representation by applying the principle of material covariance:

W = W((gb), (a1jb), (b1a2i), (a

3k), (a4l)) = W((bg), (ba1j), (a2ib

1), (a3k), (a4l)) . (96)

Since any a3k has the same component variance as any a4l and likewise for a4

l and a3k , the same relationships (87) and

(88) hold under the assumption that we now consider the tensors a3k as belonging to the class of tensors a4l and, along

similar lines, the tensors a4l as belonging to the class of tensors a3k.

Example 4.12 For a first application of the above derived rules we consider an anisotropic material by including,

additionally, a second-order tensor A2 Sym(TXB, TXB) as structural tensor in the strain energy function:

We = We(C, A2, G1) = We(g, a2, b) with A2 Sym(TXB, TXB) . (97)

Using the definition for the strain rate d = 12 (gl + lg) = 12 Lv(g) and Lv(b) = F(G

1) = 0 the Lie-derivative of theKirchhoff stress tensor is then obtained in the two forms:

Lv() = 42We

ggq a

1

2Lv(g) = 4

2We

ggq ad

=

(g1 b + b 2 g1) q a

2We

bbq a(g1 b + g1 2 b)

+ (g1 a2 + a2 2 g1) q a 2We

a2a2q a(g1 a2 + g1 2 a2)

+ (g1

b + b 2

g1) q a

2We

ba2

q a(g1

a2 + g

12

a2)

+ (g1 a2 + a2 2 g1) q a 2We

a2bq a(g1 b + g1 2 b)

q ad ,

(98)



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where V = F1v denotes the so-called material velocity. If we write:

Kt =1

2t (V V) : C = 1

2t (V V) q aC , (108)

the derivative Kt,F can be obtained easily by means of the above rules and Table 3:

Kt,F = Kt F + 12

t (V V) q a(I 2 gF + Fg I) = Kt F + t gFV V . (109)

Example 4.16 To put the above rules into practice in a more complex example let us consider a strain energy function

for a transversely isotropic material with one privileged direction N. Using the representation theorem for isotropic ten-

sor functions [3] the tensor A2 representing the structural tensor is incorporated into We. We assume that We depends

quadratically on the Green-Lagrange-strain tensor E = 12 (C G). The strain energy function used in the subsequentcomputations is proposed in the form:

We =

2

tr(EG1)

2+ tr(EG1EG1) + 1 tr(GA2EG

1)tr(EG1)

+ 2 tr(GA2EG1)2 + 3 tr(GA2EG1EG1) .(110)

Using Table 3, the derivatives of the above appearing traces with respect to E, A2, and G1 are computed as:

(EG1 : I),E = (I G1) a qI = IIG = G , (EG1 : I),G1 = (E I) a qI = EII = E ,

(EG1EG1 : I),E = (I G1EG1 + EG1 G1) a qI

= IGEG + GEIG = 2 GEG ,

(EG1EG1 : I),G1 = (E EG1 + EG1E I) a qI= EIGE + EGEII = 2 EGE ,

(GA2EG1 : I),E = (GA2

G1) a qI = A2G

IG = A2 ,

(GA2EG1 : I),A2 = (G EG1) a qI = GIGE = E ,

(111)

(GA2EG1 : I),G1 = (G GA2EG1 + GA2E I) a qI

= GIGEA2G + EA2GII = 0 ,(GA2EG

1EG1 : I),E = (GA2 G1EG1 + GA2EG1 G1) a qI= A2G

IGEG + GEA2GIG = A2E

G + GEA2 ,

(GA2EG1EG1 : I),A2 = (G EG1EG1) a qI = GIGEGE = EGE ,

(GA2EG1EG1 : I),G1 = (G GA2EG1EG1 + GA2E EG1 + GA2EG1E I) a qI

= GIGEGEA2G + EA2GIGE + EGEA2GII= EA2E

. (112)

We simplify and consider symmetry properties of second- or fourth-order tensors at the end of the derivation process.

Recognizing this, we obtain for the first- and second-order derivative of We with respect to E the results:

We

E= tr(EG1) G + 2 GEG + 1

tr(GA2EG

1) G + tr(EG1) A2

+ 2 2 tr(GA2EG1) A2 + 3

A2E

G + GEA2

(113)

and

CCC =2We

EE = G G + 2 G 2 G + 1 G A2 + A2 G

+ 2 2 A2 A2 + 3

A2 2 G + G 2 A2

.

(114)



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At this state the symmetry of the second-order tensors G1, E and A2 is considered. Then minor symmetries of the fourth-

order tensor We,(EE) are enforced. Let us introduce a simplifying procedure: A look at Table 5 reveals that for the symmetric

second-order tensors here considered (G1 = G, E = E, A2 = A2) a fourth-order tensor constructed by means of the

tensor product()

is invariant if we imposeCCC=CCC

ti

=CCC

to

=CCC

t. In contrast, a fourth-order tensor constructed by means

of ( 2 ) or () is not invariant. Thus, as a rule, if major symmetry CCC = CCCT is already fulfilled and the last mentionedtensor products appear, they are replaced each by a sum of ( 2 ) and () multiplied by one half. This avoids the explicituse of the symmetry-transformation tensor S and delivers:

We

E= tr(EG1) G1 + 2 G1EG1 + 1

tr(GA2EG

1) G1 + tr(EG1) A2

+ 2 2 tr(GA2EG1) A2 + 3

A2EG

1 + G1EA2

(115)

and

2We

EE

= G1

G1 + G

1

G1 + G1 2

G1 + 1 G

1

A2 + A2

G1

+ 2 2 A2 A2 + 3 12

A2 G1 + A2 2 G1 + G1 A2 + G1 2 A2

.

(116)

Finally, taking into account the dependance ofE on C by means ofE,C =12S, which is just the identity multiplied by one

half, we obtain:

S = 2We

C= tr(EG1) G1 + 2 G1EG1 + 1

tr(GA2EG

1) G1 + tr(EG1) A2

+ 2 2 tr(GA2EG1) A2 + 3

A2EG

1 + G1EA2

(117)

and

S = 42We

CCq aE =

G1 G1 + G1 G1 + G1 2 G1

+ 1

G1 A2 + A2 G1

+ 2 2 A2 A2

+ 31

2

A2 G1 + A2 2 G1 + G1 A2 + G1 2 A2

q aE .

(118)

Now we consider the conjugated formulation. Making things more complicated we have a dependance of E on the metric

tensor G itself leading to the derivative E,G1 =12

(G G) (without making use ofS). Using this relation, Table 3 and(113) gives

We

Eq aE,G1 =

1

2

tr(EG1) G + E + 11

2 tr(GA2EG

1) G + tr(EG1) GA2G

+ 2 tr(GA2EG

1) GA2G + 3

1

2(GA2E

+ EA2G) , (119)

which finally yields:

We

G1= tr(EG1) (E +

1

2G) + (2 EGE + E)

+ 1 (tr(GA2EG1)(E +

1

2G) + tr(EG1)

1

2GA2G

)

+ 2 tr(GA2EG1) GA2G

+ 3 (EA2E

+1

2(GA2E

+ EA2G)) .

(120)

Furthermore, we obtain:

We

A2= 1 tr(EG

1) E + 2 2 tr(GA2EG1) E + 3 E

GE . (121)



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Using (87) in the corresponding material representation and the definition E = 12 (C G) delivers:We

C= C

We

G1G + C

We

A2A2

= tr(EG1) C 12 CG + C(C G)GEG + CEG

+ 1 (tr(GA2EG1) C

1

2CG + tr(EG1)

1

2CGA2) + 2 tr(GA2EG

1) CGA2

+ 31

2(C(C G)A2EG + CGA2EG + CEA2)

+ 1 tr(EG1) C

1

2(C G)A2 + 2 tr(GA2EG1) C(C G)A2

+ 31

2C(C G)GEA2

= tr(EG1)1

2G + (GEG) + 1

1

2(tr(GA2EG

1) G + tr(EG1) A2)

+ 2 tr(GA2EG1) A2 + 3

1

2(A2E

G + GEA2) .

(122)

Likewise, considering (88) the same result would be obtained. To continue, consulting Tables 4 and 6 and starting from

(120) and (121) the second-order derivatives of We with respect to G1 and A2 are computed as:

2We

G1G1= ( 1

2tr(EG1) G 2 G + (E + 1

2G) E) + 2 E 2 E

+ 11

2(tr(GA2EG1) G 2 G + GA2G E tr(EG1) (G 2 GA2G

+ GA2G2 G)) 2 tr(GA2EG1) (G 2 GA2G + GA2G 2 G)

3 12

(G 2 GA2E + EA2G 2 G) +

(tr(EG1) I 2 I

+ (E +1

2

G)

G) + (2 (I 2

GE + EG 2

I) + I 2

I)

+ 1 (tr(GA2EG1) I 2 I + (E + 1

2G) A2 +

1

2GA2G

G)

+ 2 GA2G

A2 + 3 (I 2 A2E + EA2 2 I +1

2(GA2 2 I + I 2 A2G)

q a

1

2(G G) (123)

= (E +1

2G) (E + 1

2G) + (2 E 2 E + G 2 E + E 2 G + 1

2G 2 G)

+ 11

2(GA2G

(E + 12

G) + (E +1

2G) GA2G

tr(EG1) (G 2 GA2G + GA2G 2 G))

+ 2 (tr(GA2EG1) (G 2 GA2G + GA2G 2 G) +1

2GA2G

GA2G)

+ 31

4(GA2G

2 G + G 2 GA2G) , (124)

2We

G1A2= 1((E

+1

2G) E + 1

2tr(EG1) G 2 G)

+ 2 (tr(GA2EG1) G 2 G + GA2G E)

+ 3 (E2 E + 1

2(G 2 E + E 2 G)) , (125)

2We

A2A2= 2 2 E

E . (126)



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Then, using (89) the fourth-order tensor (114) can be alternatively obtained by evaluating the following formula:

CCC = 4 (C G) q a 2We

G1G1q a(C1 G1)

CCC1

+ 4 (C A2) q a2We

A2A2q a(C1 A2)

CCC2

+ 4 (C A2) q a2We

A2G1q a(C1 G1)

CCC3

+ 4 (C G) q a 2We

G1A2q a(C1 A2)

CCCT

3

. (127)

For the sake of offering practical exercises the four fourth-order tensors which appear in (127) are evaluated. Using Table 4

and the definition E = 12 (C G), the results read as:

CCC1 = 1

4G G + 1

2G 2 G + 1 ( 1

4(CGA2 G + G CGA2)

1

2

tr(EG1)(C 2

CGA2 + C

GA2 2

C))

+ 2 (tr(GA2EG1)(C 2 CGA2 + CGA2 2 C) +1

2CGA2 CGA2)

+ 31

4(CGA2 2 C + C 2 CGA2) , (128)

CCC2 = 21

2(A2 A2 + CGA2 CGA2 CGA2 A2 A2 CGA2) , (129)

CCC3 = 11

2(

1

2(A2 G CGA2 G) + tr(EG1) C 2 CGA2)

+ 2 (tr(GA2EG1) C 2 CGA2 +

1

2(A2 CGA2 CGA2 CGA2))

+31

4(G 2 A2 C 2 CGA2) . (130)

Of course, by combining (127), (128), (129) and (130) we obtain (114). The prove of this is left to the reader. As last

step, the minor symmetries of (127) can be enforced following the above explained simplifying procedure. Considering the

privileged direction N the structural tensor is given by A = N N. However, in the above example we have to considerthe component decomposition A2 = (N Gi)(N Gj) Gi Gj Sym(TXB, TXB). But by virtue of (87), (88), (89) andTable 7 a similar result would be obtained if e.g. A1 = (N Gi)(N Gj) Gi Gj Sym(TXB, TXB) were used.

Remark 4.17 At the end of this application section, two important points are once more called to mind. Firstly, take

care of existing contractions of tensors. If we have e.g.:

F = B q aA q aB , A = const. , (131)

the tensor B,B in F,B has to be transposed on the left-hand side to account for the contraction already involved in (131):

F,B = (B,B )T

q aA q aB + B q aA q a(B,B ) . (132)

Remark 4.18 Secondly, recall that invariants are always formed in mixed-variant representation. If we e.g. consider

the deviatoric part of a tensor b Lin(TxB, TxB), which is given by

dev(b) = b 13

tr(bg) g1 , (133)

note that the construction of any invariant, in this case the determinant, implies two equivalent representations that is:

det(dev(b)) = det(bg 13

tr(bg) i) = det(gb 13

tr(bg) i) . (134)

As corollary, the determinant of the so-called metric tensor g1 is not identical to the determinant of its metric tensorcomponents det([gij ]) = 1g .



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5 Conclusion

In this second part of our contribution the tensor formalism for fourth-order tensors and the tensor differentiation rules

with respect to a second-order tensor, which have been introduced in Part I, have been employed in a certain framework

of continuum mechanics which is mainly borrowed from tensor analysis on manifolds or differential geometry. Since therules of differential geometry are often employed in research works it of great interest for the expected reader of this

contribution to see how the tensor formalism, which has been highlighted in Part I, is actually used in relevant applications

of continuum mechanics considering tensor algebra on manifolds. The ultimate goal of both contributions has been to

simplify and accelerate the analytical derivation process, especially when tensor differentiation is involved. We have tried

to maintain as much coherence with respect to notation which is currently used in literature as possible by adding, on the

other hand, only such new notations or rules, which have no existing counterpart in the current literature. The latter holds

in particular for the notation of transposition rules for fourth-order tensors and the basis rearrangement operations which

are essentially new. The basis rearrangement operations allow an easy transformation of fourth-order tensors which resulted

from the new differentiation rules into those familiar ones. Both arrangements have been studied alongside in detail such

that the application of the new rules is in control in every step of the process. Also, the component variance of a tensor has

not been indicated by certain symbols. Instead we have defined a tensor and its corresponding component variance from the

outset which leads to more concise formulations.To summarize, we have introduced a tensorformalism such that theconcept of tensordifferentiation adopted from Itskov

can be employed in the most coherent and self-contained way. In the present contribution, some basic results of continuum

mechanics have been presented and in particular the so-called conjugated formulation with respect to the left Cauchy-

Green-tensor b has been derived in a way different to earlier approaches solely on the basis of tensor differentiation rules.

The representation for the first- and second-order derivatives has been obtained and important restrictions and differences

to related results in literature have been pointed out. In addition, the case of anisotropic hyperelastic constitutive laws has

been considered. In each single case we have obtained two equivalent solutions for the first- and second-order derivatives.

The power of the principle of covariance became apparent throughout this contribution. As a side-effect, a metric tensor,

which is required to form invariants of argument tensors, has itself to be included as an independent argument tensor in tensor

functions, since the push-forward or pull-back of a metric tensor is not trivially the identity. Thus, the kind of component

variance i.e. the process of raising or lowering of indices by means of metric tensors has not to be borne in mind in implicit

form [1], which would be especially cumbersome if push- or pull-operations are used, but now becomes explicit. In this

context we note that we prefer the notations G1 and g1 to the more usual ones G and g to indicate that in a differentiationprocess the derivative of the inverse has to be formed.

On balance, the main advantage of the tensor formalism, which has been proposed in both parts, is that analyticalderivations involving tensor differentiation are simplified considerably and can be carried out in a much shorter time than bymeans of the usual approach.The authors hope that the proposed tensor formalism will be advantageous for many researchersin continuum mechanics.

Acknowledgements The financial support of the National Science Foundation (DFG) within the collaborative research center (SFB)

398 is gratefully acknowledged.

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Kintzel Fourth Order Tensors Part2

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