Journal of Elasticity 62: 171–201, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
171
On the Comparison of Two Strategies to Formulate
Orthotropic Hyperelasticity
A. MENZEL and P. STEINMANN
Chair of Applied Mechanics, University of Kaiserslautern, P.O. Box 3049,
D-67653 Kaiserslautern, Germany. E-mail: {amenzel;ps}@rhrk.uni-kl.de
Received 23 March 2000; in revised form 7 February 2001
Abstract. The main goal of this work is to clarify the relation between two strategies to formulate constitutive equations for orthotropic materials at large strains. On the one hand, the classical
approach is based on the incorporation of structural tensors into the free energy function via an
enriched set of invariants. On the other hand, a fictitious isotropic configuration is introduced which
renders an anisotropic, undeformed reference configuration via an appropriate linear tangent map.
This formulation results in a reduced (with respect to the more general setting based on structural
tensors) but nevertheless physically motivated set of invariants which are related to the invariants defined by structural tensors. As a main conceptual advantage standard isotropic constitutive equations
can be applied and moreover, due to the reduced set of physically motivated invariants, the numerical
treatment within a finite element setting becomes manageable.
Mathematics Subject Classifications (2000): 74E10, 74A20, 74B20.
Key words: anisotropy, structural tensors, fictitious configurations, large strains.
1. Introduction
The phenomenological description of anisotropic materials has been an active research subject in the last decades. Within the setting of geometrically linear and
nonlinear kinematics one common strategy is the incorporation of structural tensors
into the free energy function. This contribution highlights an alternative approach
based on an additional, fictitious configuration which is related to the standard reference configuration via a linear tangent map. Anisotropy comes into the picture if
this mapping is nonspherical. Consequently, we deal with a reduced but physically
motivated set of invariants in terms of prespecified combinations of the invariants
of the structural tensor approach. Without loss of generality this formulation holds
for elastic and inelastic processes. Nevertheless, in order to clarify concepts, we
place emphasis on the case of hyperelasticity in this paper.
The noncoaxiality of the stress and strain tensor for the general anisotropic case
results usually in a delicate numerical setting, since a computation in terms of
eigenvalues similar to isotropic hyperelasticity is not available. More precisely, the
formulation of anisotropic constitutive equations is directly related to the theory of
172
A. MENZEL AND P. STEINMANN
invariants. For a general survey with respect to continuum mechanics we refer to
Spencer [1]. Naturally, nonlinear anisotropy is related to group theoretical settings.
In this context Hackl and Schmidt [2] give a framework of crystal elastoplasticity
at finite strains. Recently Hackl [3] presented a general formulation of anisotropic
hyperelasticity based on symmetric irreducible tensors.
The formulation of anisotropic elasticity at small strains has been highlighted by
various authors, see, e.g., the overview article of Ting [4] with emphasis on physical
effects like cracks and wedges for the two dimensional case. The common backbone of three dimensional, linear elasticity rests on the possible decomposition of
second order tensors into a spherical and a deviatoric part. On this basis Mehrabadi
and Cowin [5] constructed the eigentensors of the anisotropic symmetry classes
as well as their invariants, which are also denoted proportional invariants. Within
this framework Sadegh and Cowin [6] placed special emphasis on orthotropy and,
moreover, the underlying restrictions on the stored energy function for several
symmetry classes are outlined in [7].
For background information on constitutive equations incorporating specific
material symmetries at large strains, we refer to the early work of Pipkin and Rivlin
[8]. Moreover, general surveys on the modelling of anisotropic materials are, e.g.,
given by Boehler [9] and Smith [10]. Throughout this paper we consider symmetric
second order tensors representing anisotropy. Indeed, more complex theories are
possible incorporating constitutive anisotropy tensors of arbitrary but even order. In
this context Betten [11] generated the set of irreducible invariants of a fourth order
tensor and, in view of the coupling to a second order tensor like stress or strain,
additionally the corresponding simultaneous invariants. A further framework on
anisotropy at finite strains has been pointed out by Elata and Rubin [12]. Here, in
particular the strain tensor is associated with a finite number of orientations which
in turn bring anisotropy into play.
The idea of a fictitious configuration was originally established for the formulation of anisotropic failure criteria of inelastic processes like creep and damage,
see [13, 14]. Recently Park and Voyiadjis [15] discussed in detail the underlying kinematics with application to damage. Likewise, the concept of a fictitious,
undeformed configuration has been advocated in the light of continuum damage
mechanics by Steinmann and Carol [16]. Moreover, all possible combinations of
either isotropic or anisotropic hyperelasticity coupled to either quasi isotropic or
anisotropic damage evolution have been considered by Menzel and Steinmann
[17]. In fact, the proposed introduction of a fictitious configuration represents a
specific subclass of orthotropy (or transversal isotropy as a special application)
which itself stands for a particular case of general anisotropy. Nevertheless, these
two categories are of major interest in engineering applications.
For an outline of anisotropic plasticity see, e.g., [18–22] among others, and,
with special emphasis on the construction of anisotropic yield functions [23].
The paper is organised as follows: For convenience of the reader Section 2 summarizes the classical theory of anisotropic hyperelasticity in terms of structural ten-
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
173
sors. Subsequently, Section 3 presents the framework of anisotropic hyperelasticity
based on a fictitious, isotropic configuration. The relations between these two formulations are highlighted in Section 4. Specifically, we compare for various cases
of anisotropy the invariants implied by the fictitious configuration concept with
the invariants that are obtained by the introduction of structural tensors. Finally,
the homogeneous simple shear deformation is discussed within the framework of
a fictitious configuration in Section 5 and, in addition, a finite element setting is
highlighted in Appendix A which underlines the practicability of the proposed
formulation.
2. Anisotropic Hyperelasticity Based on Structural Tensors
To set the stage this section reiterates the essentials of nonlinear continuum mechanics and introduces the notation used in the sequel. Thereby the anisotropy
approach is based on classical structural tensors whereby we will later on restrict
ourselves to the cases of orthotropy and transverse isotropy for the sake of transparency. Due to the anisotropic character it is convenient to develop the formulation
in the material setting in terms of an appropriate set of invariants defined by a
kinematic strain variable and the structural tensors.
2.1. LARGE STRAIN KINEMATICS
The reference configuration of the considered body is denoted by B0 ⊂ R3 and
x = ϕ(X) represents the standard nonlinear deformation map of material points
X ∈ B0 onto spatial points x ∈ B ⊂ R3 in the current configuration. The corresponding linear tangent map F = ∂X ϕ ∈ GL+ : T B0 → T B transforms tangent
vectors to material curves into tangent vectors to spatial curves and is denoted as
deformation gradient, see Figure 1 for a graphical representation.
In terms of convected coordinates θ i with x = x(θ i = θ i (X)) and i = 1, 2, 3
the natural base vectors read
Gi = ∂θ i X ∈ T B0 ,
gi = ∂θ i x ∈ T B,
Gi = ∂X θ i ∈ T B0 ,
g i = ∂x θ i ∈ T B.
Figure 1. Nonlinear deformation map ϕ and linear tangent map F .
(1)
174
A. MENZEL AND P. STEINMANN
They define the covariant () and contravariant (
) metric tensors (j = 1, 2, 3)
G
G
g
g
=
=
=
=
Gij Gi ⊗ Gj
Gij Gi ⊗ Gj
gij g i ⊗ g j
g ij gi ⊗ gj
∈ T B0 × T B0 ,
∈ T B0 × T B0 ,
∈ T B × T B,
∈ T B × T B,
Gij
Gij
gij
g ij
= Gi · Gj ,
= Gi · Gj ,
= gi · gj ,
= gi · gj ,
(2)
compare [24] with respect to the notation. The deformation gradient denoted by
the two field tensor F is now defined via the dyads
F = gi ⊗ Gi ∈ T B × T B0 .
(3)
Similar to F but with both “legs” in one configuration we define two mixed-variant
tensor fields with respect to the material and spatial setting
G = F −1 · F = Gi ⊗ Gi ∈ T B0 × T B0 ,
g = F · F −1 = gi ⊗ g i ∈ T B × T B.
(4)
In the following emphasis is placed on the Green–Lagrange strain tensor E which
represents the difference of the spatial and material covariant metric coefficients
E = Eij Gi ⊗ Gj ∈ T B0 × T B0
1
with Eij = [gij − Gij ].
2
(5)
For a general overview on the underlying kinematics and further details with respect to convected coordinates we refer, e.g., to [25].
2.2. STRUCTURAL TENSORS
The symmetry group G of an anisotropic material is assumed to be a subgroup
of the orthogonal group O(3). Furthermore, let V = {V1,...,k } denote a covariant
structural tensor series with elements of arbitrary order. Now, G ⊂ O(3) is defined
by
(6)
G = Q ∈ O(3) | Q V = V ,
whereby [] represents the appropriate linear operator, compare [26]. Typically, the
elements of V |F =G are functions of the type Vi = Vi (vi ) (e.g., Vi = vi ⊗ vi )
with unit vectors vi = [vi ]j Gj ∈ S 2 . The contravariant notation of these direction
vectors reads vi = [vi ]j Gj ∈ S 2 and thus it follows straightforward that Q ∈ O(3)
is characterized via
T
Q ∈ ± G , ref R vi , vi , rot R(θvj ); vi , vi , vj ∈ S 2 ; θ ∈ R: 0 θ < 2π
(7)
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
175
with i = m, . . . , n and j = o, . . . , p. Thereby, possible definitions of the corresponding reflections ref R(vi ) and rotations rot R(θvj ) which map covariant fields
onto covariant fields take, e.g., the following format
ref
T
R vi , vi = G − 2vi ⊗ vi ∈ O(3),
rot
R θvj = exp −θε · vj
∈ SO(3),
(8)
whereby ε = εi j k Gi ⊗ Gj ⊗ Gk denotes the third order permutation tensor.
2.3. BASIC INVARIANTS
In order to compute scalar-valued isotropic functions in terms of tensor valued variables an irreducible set of invariants has to be computed. The underlying painstaking analysis to obtain irreducible representations of isotropic tensor functions has
been developed over several decades, see, e.g. [27] for an overview on scalarvalued functions and the references cited therein. In the sequel we restrict the
comparison to the case of three symmetric second order covariant tensors A0,1,2
whereby, later on, A1,2 are assumed to stay constant. Thus, we deal with a specific subclass of anisotropy – compare [28] – namely, orthotropy and transversal
isotropy, which are of major interest in engineering applications.
Nevertheless, the representative set of invariants in terms of A0,1,2 takes the
following format:
Ai : G
,
Ai · G
· Ai : G
,
Ai · G
· Ai · G
· Ai : G
,
Ai · G
· Aj : G
,
Ai · G
· Ai · G
· Aj · G
· Aj : G
,
Ai · G
· Ai · G
· Aj : G
,
A0 · G
· A1 · G
· A2 : G
,
(9)
with i, j = 0, 1, 2 and i = j . Please observe that for the definition of the trace
operator we have to keep in mind that the derivatives of the invariants with respect to A0,1,2 are supposed to result in contravariant tensors. In order to be more
specific A0 is chosen to equal the strain metric tensor E and we additionally
.
assume Aζ = Vζ = vζ ⊗ vζ ∈ T B0 × T B0 , ζ = 1, 2, to stay constant during
the deformation process. With these restrictions at hand the invariants of interest
E V12
I1,...,9 = E V12 I1,...,9 (E , V1,2
) are given by
176
A. MENZEL AND P. STEINMANN
E V12
I1 = E : G ,
E V12
I2 = E · G · E : G ,
E V12
I3 = E · G · E · G · E : G ,
E V12
I4 = E · G
· V1 : G
,
E V12
I5 = E · G
· E · G
· V1 : G
,
E V12
I6 = E · G
· V2 : G
,
E V12
I7 = E · G
· E · G
· V2 : G
,
E V12
I8 = V1 · G
· V2 : G
,
E V12
I9 = E · G
· V1 · G
· V2 : G
.
(10)
Further restrictions like the orthogonality of v1 and v2 or a vanishing fibre vector v2
yield classical orthotropic and transversely isotropic behaviour, respectively. The
hyperelastic constitutive law for the second Piola–Kirchhoff stress tensor S =
) as
∂E – see, e.g. [29] – is based on the free energy function = (E , V1,2
discussed in the above and renders
S = 1 G
+ 22 G
· E · G
+ 33 G
· E · G
· E · G
sym
+ 4 G
· V1 · G
+ 25 E , V1
sym
+ 6 G
· V2 · G
+ 27 E , V2
+ 29 V1 , V2 sym .
(11)
Here, we incorporated the abbreviations 1,...,9 = ∂E V
12 I1,...,9
for the derivatives
of the free energy function. The corresponding Hessian as well as the symmetry
operators ·, ·sym and ·, ·sym are given in Appendices B and C, (B.2), (C.2).
Concerning the numerical treatment of anisotropic materials at finite strains
within the structural tensor setting, especially in the case of transverse isotropy
with application to compressible and incompressible materials, we refer to the investigations by Qiu and Pence [30], Weiss et al. [31] and Almeida and Spilker [32].
Another approach towards large strain anisotropy within a finite element setting has
been pointed out by Park and Youn [33]. As the backbone of their formulation the
free energy function in terms of the right Cauchy–Green tensor C = F T · g · F ∈
T B0 × T B0 reads = (C J1,2,3 , Jnij ) with Jnij = Gi · [C ]n · Gj .
Remark 2.1. Usually, isotropic hyperelasticity is formulated in terms of the
three principal invariants C J1,2,3 of the right Cauchy–Green tensor C instead
of the basic invariants E I1,2,3 of the strain metric tensor. Selecting the principal
invariants E J1,2,3 with respect to the strain metric tensor E renders terms that
include the inverse [E ]−1 when the stress tensor S and the corresponding Hessian
are computed. Since for the undeformed state [E ]−1 |F =G is not defined one has
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TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
to re-express the inverse strain metric tensor via the Cayley–Hamilton theorem to
end up in a singularity-free formulation. Here, alternatively, we invoke throughout
this contribution the basic invariants defined by the strain metric tensor because the
fictitious configuration is directly based on the concept of strain energy equivalence
– compare [17, 34] in the light of anisotropic continuum damage mechanics –
whereby basic invariants turn out to be conveniently realizable.
3. Anisotropic Hyperelasticity Based on a Fictitious Configuration
Within this section the continuum mechanical concept of a fictitious isotropic configuration is outlined. The approach allows the interpretation as a multiplicative
composition of the standard deformation gradient and an attached additional anisotropy map. The principle of strain energy equivalence – well-known in continuum
damage mechanics, see, e.g. [13, 14, 34] – renders two sets of invariants with
respect to either the fictitious isotropic configuration or the undeformed anisotropic
reference configuration. Based on these sets standard stress–strain relations within
the hyperelastic context are obtained. Obviously, anisotropy is incorporated as
soon as the fictitious map is nonspherical. In particular, the specific form of the
anisotropy map affects the type of material symmetry. Indeed it turns out that a
reduced formulation of orthotropy is obtained.
3.1. KINEMATICS OF THE FICTITIOUS CONFIGURATION
Here, in addition to B and B0 we consider a fictitious isotropic configuration
which is in general incompatible. The corresponding tangential space is denoted by
T B and in analogy to the standard deformation gradient F let the linear tangent
map F ∈ GL+ : T B 0 → T B0 transform fictitious tangent vectors into tangent
vectors to material curves. Note that this fictitious configuration is, in analogy
to the intermediate configuration of multiplicative elastoplasticity theory, a nonEuclidian space. In other words, the correlated metric tensors define a nonvanishing
Riemann–Christoffel tensor and hence the conditions of compatibility are generally
not satisfied, see, e.g. [35]. Figure 2 gives a symbolic graphical representation of
the multiplicative composition of the linear tangent maps F and F .
Accordingly, we introduce the base vectors in the fictitious configuration which
are not derivable from position vectors but are rather defined by the nonholonomic
Pfaffian map F , see, e.g. [36]
Gi ∈ T B 0
and
i
G ∈ T B0,
i = 1, 2, 3.
(12)
Then the corresponding metric tensors of the fictitious configuration follow as
usual
i
j
G = Gij G ⊗ G ∈ T B 0 × T B 0 ,
ij
G = G Gi ⊗ Gj ∈ T B 0 × T B 0 ,
Gij = Gi · Gj ,
ij
i
j
G =G ·G ,
(13)
178
A. MENZEL AND P. STEINMANN
Figure 2. Linear tangent maps F and F .
with j = 1, 2, 3. Therefore, the linear tangent or rather anisotropy map reads
i
F = Gi ⊗ G ∈ T B0 × T B 0 .
(14)
Similar to the standard Green–Lagrange strain tensor E a fictitious strain metric
tensor is introduced as
i
j
E = E ij G ⊗ G ∈ T B 0 × T B 0 ,
(15)
whereby standard push-forward and pull-back relations based on the linear tangent
map F hold, thus E ij = Eij .
3.2. ENERGY METRIC TENSORS
Next, for the computation of the scalar-valued free energy function , we introduce
the contravariant energy metric tensors B and B in addition to the covariant strain
metric tensors E and E ij
B = B Gi ⊗ Gj ∈ T B 0 × T B 0 ,
B = B ij Gi ⊗ Gj ∈ T B0 × T B0 .
(16)
ij
.
Thereby, we set B = B ij = const. Again push-forward and pull-back operations
in terms of F can be applied. Note that B is supposed to substitute the contravariant metric tensor G
in the free energy function. As the underlying assumption of
the proposed formulation the fictitious configuration is isotropic, thus the fictitious
energy metric tensor has to be spherical. Therefore, throughout this contribution
.
we set B = G . Consequently F may be interpreted as a prestretch of a fictitious
isotropic hyperelastic material.
Finally, the scalar-valued free energy function is restricted to remain invariant
under superposed diffeomorphism defined by F , thus we obtain
(17)
E , B = (E , B )
179
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
which is established in the context of continuum damage mechanics as the principle
of strain energy equivalence.
To underline the heart of the proposed framework we stress at this stage that
(17) includes all assumptions of the material modelling. Nevertheless we clearly
deal with a reduced representation of anisotropy which is highlighted in Section 4.
As an advantage, note that besides the introduction of the fictitious configuration
no further assumptions or additional material parameters have to be included since
standard isotropic free energy functions can be applied to model anisotropic behav.
iour. Of course isotropy is incorporated for B = β0 G
, β0 ∈ R+ and anisotropy
enters the formulation if B is a nonspherical tensor.
3.3. BASIC INVARIANTS
Since T B 0 is the tangent space to a fictitious isotropic configuration a set of merely
three basic invariants E B I1,2,3 defined in terms of the fictitious strain and energy
metric tensors E and B is given as
I1 = E : B ,
E B
I2 = E · B · E : B ,
E B
I3 = E · B · E · B · E : B .
E B
(18)
Next, the set of invariants E B I1,2,3 with respect to the anisotropic reference con
figuration B0 is obtained by expressing E and B in terms of E and B via the
pull-back operation ϕ̄ ∗ with F
T
E = ϕ̄ ∗ E = F · E · F ,
B = ϕ̄ ∗ B = F
−1
· B
· F
−T
,
(19)
compare Appendix B. Straightforward calculations result in the following set of
basic invariants:
E B I1 = E : B ,
E B I2 = E · B · E : B ,
E B I3 = E · B · E · B · E : B .
(20)
Thus the hyperelastic constitutive law for the second Piola–Kirchhoff stress tensor
is expressed as
(21)
S = 1 B + 22 B · E · B + 33 B · E · B · E · B .
.
Again we defined abbreviations 1,2,3 = ∂E B I for the derivatives of the free
1,2,3
energy function. Remarkably, this formula has the identical structure compared to
the hyperelastic constitutive law of standard isotropy.
180
A. MENZEL AND P. STEINMANN
4. Relations between Structural Tensors and the Fictitious Configuration
The main goal of this section is to highlight the relations between the two frameworks based on either structural tensors or on the advocated fictitious configuration,
respectively. Therefore, one has to compare the arguments of the free energy functions, i.e., the corresponding sets of invariants. Conceptually speaking, the task is
to compute E B I1,2,3 = E B I1,2,3 (E V12 I1,...,9 ).
4.1. INCORPORATION OF STRUCTURAL TENSORS INTO THE FICTITIOUS
TANGENTIAL MAP
In order to incorporate an analogue to structural tensors into the framework of a
fictitious configuration we make a specific ansatz for the fictitious base vectors. In
i
particular, we give a representation of G in T B0 which reads
.
G = F · G = α0 G i +
αζ vζi vζ
i
T
2
i
(22)
ζ =1
and emphasizes the heart of the proposed formulation. Conceptually speaking, (22)
i
. Now, the two field tensor
defines G in terms of Gi and the fibre directions v1,2
i
F = Gi ⊗ G takes the following format with respect to T B0 :
.
F = α0 G +
αζ Vζ
2
with Vζ = vζ
⊗ vζ .
(23)
ζ =1
Remarkably, beside this ansatz no further assumptions to model anisotropy are
required. Obviously, the introduction of F as a symmetric quantity with respect to
G is no severe restriction since the energy metric tensor B is generally symmetric.
.
Next, straightforward calculations render with B = ϕ̄∗ B and B = G the
energy metric tensor in B0
B = β0 G
+ β1 V1
+ β2 V2
+ 2β3 V1
, V2
sym ,
(24)
whereby we incorporated the abbreviated notations
β0 = α02 ,
β1 = 2α0 α1 + α12 ,
β2 = 2α0 α2 + α22 ,
β3 = α1 α2 .
(25)
Consequently, the relations between structural tensors and the fictitious configuration with respect to the proposed ansatz can easily be verified by the incorporation
of (24) into (20) and comparing the obtained result to (10).
Remark 4.1. The parameters α0,1,2 have to be chosen such that F ∈ GL+ .
Therefore, we consider the following cases:
(i) α0 = 0, α1 = α2 = 0: This isotropic situation ends up in the restriction
α0 > 0 which is assumed to hold throughout.
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TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
(ii) α0 > 0, α1 = 0, α2 = 0: Application of the Sherman–Morrison–Woodbury
theorem, see, e.g. [37], results in the constraint α1 > −α0 for this transversely
isotropic case.
(iii) α0 > 0, α1,2 = 0: Within this general case the determinant is again computed via the Sherman–Morrison–Woodbury theorem and takes the following
format:
α2
α1 α2
2
E V12
−
I8 >0
˙
det F = α0 [α0 + α1 ] 1 +
α0 α0 + α1 /α0
with E V12 I8 = V1 : [V2 ]T . Thus, possible solutions can be obtained by the
restrictions
α02 [α0 + α2 ]
,
α1 > max − α0 ,
α02 α2 E V12 I8 − α0 − α2
α0 [α02 + α1 ]
.
α2 >
α02 α1 E V12 I8 − α02 − α1
4.2. TWO ARBITRARY FIBRES
In the following we discuss in detail the most general case of two arbitrary me
chanically not equivalent (α1 = α2 ) fibres v1,2 . Within this setting the first invariant
E B I1 in terms of E
E B I1 = γ10
V 12
E V12
I1,...,9 reads
I1 + γ11 E
V 12
I4 + γ12 E
V 12
I6 + γ13 E
V 12
I9
(26)
together with the scalar-valued coefficients γ1n(βi )
γ10 = β0 ,
γ11 = β1 ,
γ12 = β2 ,
γ13 = 2β3 .
(27)
Similar computations for the second basic invariant which is quadratic in the strain
metric tensor end up in
E B I2 = γ20 E
V 12
I2 + γ21 E
V 12
I5 + γ22 E
V 12
I42
+ γ23 E V12 I62 + γ24 E V12 I7 + γ25 E V12 I92 E
+ γ26 E V12 I4 E V12 I6 E V12 I8 + E V12 I92
+ γ27 E
+ γ29
V 12
E V12
I4 E
V 12
1/2
I8
I9 + γ28 E
E V12
V 12
I6 E
V 12
V 12
I8−1
I9
Ired ,
(28)
and the corresponding coefficients γ2n (βi ) result in
γ20
γ23
γ26
γ29
= β02 ,
= β22 ,
= 2β32 ,
= 4β0 β3 ,
γ21 = 2β0 β1 , γ22 = β12 ,
γ24 = 2β0 β2 , γ25 = 2β1 β2 ,
γ27 = 4β1 β3 , γ28 = 4β2 β3 ,
(29)
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A. MENZEL AND P. STEINMANN
with E V12 Ired = v1
·E ·G
·E ·v2
– see Remark 4.2. The invariants E V12 I4,...,9 of the
structural tensors as given in (10) only include terms up to order two with respect
to the strain metric tensor. Thus we have to apply the Cayley–Hamilton theorem
for the comparison of the third invariant. Thereby the relations V1,2 : G = 1 have
been taken into account and moreover, the determinant of the strain metric tensor
γdet E comes into play which reads in terms of the basic invariants as follows:
γdet E =
1 E V 2 12 I3 − 3 E V12 I1 E V12 I2 + E V12 I13 .
6
(30)
After some tedious algebra we obtain the result
E B I3 = γ30E V12 I3 + γ31 E V12 I4 E V12 I5 + γ32 E V12 I43
1 + γ33 E V12 I1 E V12 I5 − E V12 I12 − E V12 I2 E V12 I4 + γdet E 2
E V12 E V12
I6
I7 + γ35 E V12 I63 + γ36 E V12 I4 E V12 I92 E V12 I8−1
+ γ34
−1/2
E V12
+ γ37E V12 I6 E V12 I92 E V12 I8−1 + γ38 E V12 I9 E V12 I8
Ired
E V E V 1 E V 2 E V
12 I
12 I −
12 I −
12 I E V12 I + γ
+ γ39
1
7
2
6
det E 1
2
E V12 2 E V12
I4
I9 + γ311E V12 I62 E V12 I9
+ γ310
+ γ312E V12 I8 E V12 I5 E V12 I6 + E V12 I4 E V12 I7
−1/2 E V 12 I
+ 2E V12 I9 E V12 I8
red
+ γ313 E V12 I4 E V12 I92 E V12 I8−1 + E V12 I42 E V12 I6
+ γ314 E V12 I6 E V12 I92 E V12 I8−1 + E V12 I4 E V12 I62
−1/2 E V12
Ired + E V12 I5 E V12 I9
+ γ315 E V12 I4 E V12 I8
−1/2 E V12
Ired + E V12 I7 E V12 I9
+ γ316 E V12 I6 E V12 I8
+ γ317 E V12 I4 E V12 I6 E V12 I9 + E V12 I93 E V12 I8−1
1/2 + γ318 E V12 I1 E V12 I8 E V12 Ired
1 − E V12 I12 − E V12 I2 E V12 I9 + 2γdet E E V12 I8
2
1 + γ319 E V12 I4 E V12 I6 E V12 I8 E V12 I9 + E V12 I93 .
3
The scalarvalued coefficients γ3n (βi ) are defined by the following formulae:
(31)
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
γ30 = β03 ,
γ33 = 3β02 β1 ,
γ36 = 3β12 β2 ,
γ39 = 3β02 β2 ,
γ312 = 3β0 β32 ,
γ315 = 6β0 β1 β3 ,
γ318 = 6β02 β3 ,
γ31 = 2β0 β12 ,
γ34 = 3β0 β22 ,
γ37 = 3β1 β22 ,
γ310 = 6β12 β3 ,
γ313 = 3β1 β32 ,
γ316 = 6β0 β2 β3 ,
γ319 = 6β33 .
γ32 = β13 ,
γ35 = β23 ,
γ38 = 6β0 β1 β2 ,
γ311 = 6β22 β3 ,
γ314 = 3β2 β32 ,
γ317 = 6β1 β2 β3 ,
183
(32)
with α1 = α2 in order to keep the
So far we chose two arbitrary vectors v1,2
relation to the fibre directions as transparent as possible. In the sequel, emphasis is
placed on certain dependencies of these vectors which results, e.g., in classical or
– isotropy is included.
thotropy or transverse isotropy. Trivially – for vanishing v1,2
Moreover, the energy metric tensor B is generally symmetric such that its spectral
decomposition can be applied which is pointed out in Section 4.7.
Remark 4.2. Please note that
. −1/2 E V12 E V12
I9
Ired
E · V1
· E · V2
· E : G
= E V12 I8
is not an invariant but reducible. For an outline on the derivation we refer to [38],
(33).
4.3. TWO EQUIVALENT FIBRES
As a special application of constitutive equations that incorporate two structural
, with identical
tensors we consider the case of two different fibre orientations v1,2
.
characteristics, thus α2 = α1 . With respect to (24) the corresponding scalars to
compute the energy metric tensor B read
β0 = α02 ,
β1 = 2α0 α1 + α12 ,
β2 = β1 ,
Next, in view of (26) the first invariant E
γ10 = β0 ,
γ11 = β1 ,
B
γ12 = γ11 ,
β3 = α12 .
(33)
I1 is related to structural tensors via
γ13 = 2β3 .
(34)
Taking (28) into account, we find that the correlated scalars γ2n for two mechanically equivalent fibres are
γ20
γ23
γ26
γ29
= β02 ,
= γ22 ,
= β32 ,
= 4β0 β3 ,
γ21 = 2β0 β1 ,
γ24 = γ21 ,
γ27 = 4β1 β3 ,
γ22 = β12 ,
γ25 = γ21 ,
γ28 = γ27 ,
(35)
184
A. MENZEL AND P. STEINMANN
and furthermore with respect to (31) the computation of the third invariant ends up
with
γ30 = β03 ,
γ33 = 3β02 β1 ,
γ36 = 3γ32 ,
γ39 = γ33 ,
γ312 = 3β0 β32 ,
γ315 = 6β0 β1 β3 ,
γ318 = 6β02 β3 ,
γ31 = 3β0 β12 ,
γ34 = γ31 ,
γ37 = γ36 ,
γ310 = 6β12 β3 ,
γ313 = 3β1 β32 ,
γ316 = γ315,
γ319 = 6β33 .
γ32 = β13 ,
γ35 = γ32 ,
γ38 = 2γ31 ,
γ311 = γ310 ,
γ314 = γ313 ,
γ317 = γ310 ,
(36)
4.4. TWO ORTHOGONAL FIBRES
Next, we consider the case of two orthogonal fibres v1,2
with generally different
mechanical characteristics, often denoted as classical orthotropy. This orthogonal ity assumption results in E V12 I8,9 = 0 and moreover the energy metric tensor B can be computed via
β0 = α02 ,
β1 = 2α0 α1 + α12 ,
β2 = 2α0 α2 + α22 ,
β3 = 0.
(37)
The first basic invariant is characterised by
γ10 = β0 ,
γ11 = β1 ,
γ12 = β2
(38)
and γ13 = 0. The second basic invariant correlates to
γ20 = β02 ,
γ23 = β22 ,
γ21 = 2β0 β1 ,
γ24 = 2β0 β2 ,
and γ2[6,...,9] = 0. Finally,
equations
γ30
γ33
γ36
γ39
= β03 ,
= 3β02 β1 ,
= 3β12 β2 ,
= 3β02 β2
E B γ22 = β12 ,
γ25 = 2β1 β2
(39)
I3 is related to structural tensors via the following
γ31 = 3β0 β12 ,
γ34 = 3β0 β22 ,
γ37 = 3β1 β22 ,
γ32 = β13 ,
γ35 = β23 ,
γ38 = 6β0 β1 β2 ,
(40)
and γ3[10,...,19] = 0.
4.5. TRANSVERSE ISOTROPY
For a transversely isotropic material with only one fibre we have
and therefore the coefficients to compute B read
β0 = α02 ,
β1 = 2α0 α1 + α12 ,
β2 = 0,
β3 = 0.
E V12
I6,...,9 = 0
(41)
185
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
These relations render E
γ10 = β0 ,
B
I1 in terms of
γ11 = β1
(42)
and γ1[2,3] = 0. The second basic invariant is obtained via
γ20 = β02 ,
γ21 = 2β0 β1 ,
γ22 = β12
(43)
and γ2[3,...,9] = 0. Moreover, exploiting (41) results in
γ30 = β03 ,
γ31 = 3β0 β12 ,
γ32 = β13 ,
γ33 = 3β02 β1
(44)
and γ3[4,...,19] = 0.
4.6. ISOTROPY
For completeness we finally consider the case of an isotropic material whereby
obviously via E V12 I4,...,9 = 0 no structural tensors are taken into account. The
computation of the energy metric tensor B and the isotropic invariants E B I1,2,3
is defined by the following formulae
γ10 = β0 ,
γ20 = β02 ,
γ30 = β03
(45)
and γ1[1,2,3] = γ2[1,...,8] = γ3[1,...,19] = 0.
4.7. SPECTRAL REPRESENTATION
Since the energy metric tensor B is of second order, positive definite and symmetric, the appropriate spectral decomposition reads
B =
3
B
λi B ni ⊗ B ni
i=1
= η0 G
+ η1 B n
1 ⊗ B n
1 + η2 B n
2 ⊗ B n
2
(46)
with B λi ∈ R: B λi > 0, η0 = B λ3 , η1 = B λ1 − B λ3 , η2 = B λ2 − B λ3
and B ni = [B ni ]j Gj ∈ S 2 in T B0 , j = 1, 2, 3 – compare [20]. The relations
to classical orthotropy, transverse isotropy and isotropy are self evident – compare
(24) – and read
.
ηi = βi
and
B
nζ
.
⊗ B nζ = Vζ
with i = 0, 1, 2; ζ = 1, 2
(47)
.
and β3 = 0. For two mechanically equivalent fibres (α2 = α1 ) with arbitrary
orientations these relations are not that obvious. Without loss of generality the
j
following relations between the vectors vζ
= vζ Gj ∈ S 2 in T B0 which define
186
A. MENZEL AND P. STEINMANN
the structural tensors and the eigenvectors
considerations into account
B
nζ hold by taking basic geometrical
v + v2
,
= 1
2 cos ϕ
v − v2
B
,
n2 = 1
2 sin ϕ
B
n1
v1
= cos ϕ B n
1 + sin ϕ B n
2 ,
(48)
v2
=
cos ϕ B n1
−
sin ϕ B n2 ,
1/2
whereby 2ϕ = arccos(E V12 I8 ) denotes the angle between the two fibres – compare [38]. The underlying idea is based on the fact that v1
and v2
have to be
interchangeable for two mechanically equivalent fibres and hence B n
ζ denote
admissible reflections ref Rζ = G − 2B n
ζ ⊗ B nζ . With these relation at hand
the symmetric part V1
, V2
sym of (24) reads
V1
, V2
sym
=
1
2 cos2 ϕ B n
1 ⊗ B n
1 − 2 sin2 ϕ B n
2 ⊗ B n
2
2
(49)
and the computation of the coefficients η0,...,2 yields
η0 = α02 ,
η1 = 4α1 α0 cos2 ϕ + α1 cos4 ϕ ,
η2 = 4α1 α0 sin2 ϕ + α1 sin4 ϕ .
(50)
Furthermore, with these modified scalars – which as a matter of fact bring the
invariant E V12 I8 into the picture – the standard formulae of classical orthotropy as
pointed out in (38), (39), (40) hold.
Remark 4.3. A computation based on the proposed framework with respect to
the fictitious configuration is numerically much less expensive than the standard
approach in terms of structural tensors. Equations (11), (21) of the corresponding
stress tensors and especially the Hessians in (C.2), (C.4) underline this fact most
impressively. It is our belief that the general structural tensor approach with no further assumptions results in an almost unmanageable numerical setting, especially
within the computation of inelastic materials. Contrary, the proposed framework
based on a fictitious configuration ends up with nearly identical numerical costs
than standard isotropy.
Remark 4.4. Within geometrically linear anisotropic elasticity based on structural tensors, transverse isotropy and orthotropy contain five and nine independent
material parameters, respectively. Contrary, the formulation based on the fictitious
configuration incorporates three independent material parameters in the case of
a transversely isotropic material and four independent parameters for orthotropy
(thereby the two Lamé constants are taken into account, and thus the additional
isotropic parameter α0 is not independent.) This underlines that we deal with a
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
187
reduced formulation. For the sake of clarity, we consider, e.g., a linear elastic
material of St. Venant–Kirchhoff type ((E , B ) = λ/2E B I12 + µE B I2 ) which
results in the following tangent operator;
E B LE E = ∂E2 ⊗E = λB ⊗ B + µ B ⊗ B + B ⊗ B .
(51)
Now, referring to a Cartesian frame ei , we choose in view of (46) or (24), respec.
tively, the specific case of B = η0 I + η1 e1 ⊗ e1 + η2 e2 ⊗ e2 (thus ei denote the
principal axes of B). Adopting Voigt’s notation, the coefficients of the Hessian read


L1 L2 L3 0
0
0
0
0 
 L2 L4 L5 0


0
0 
 L3 L5 L6 0
voi
Lij = 
(52)

0 
0
0 L7 0
 0
 0
0
0
0 L8 0 
0
0
0
0
0 L9
whereby L1,...,9 are not entirely independent but are defined via four (independent)
material parameters
L1
L3
L5
L7
L9
= [λ + 2µ][η0 + η1 ]2 ,
= λη0 [η0 + η1 ],
= λη0 [η0 + η2 ],
= µ[η0 + η1 ][η0 + η2 ],
= µη0 [η0 + η1 ].
L2
L4
L6
L8
= λ[η0 + η1 ] + 2µ[η0 + η2 ],
= [λ + 2µ][η0 + η2 ]2 ,
= [λ + 2µ]η02 ,
= µη0 [η0 + η2 ],
Obviously, we deal with a subclass of rhombic symmetry – compare, e.g., [35] –
and the corresponding symmetry group G is defined by Q = {±I ,
ref
R(π e1 ), ref R(π e2 )}, with respect to a Cartesian frame (see (7) for the notation).
) incorporating
Nevertheless, the general linear constitutive equation (E , V1,2
two orthogonal fibres in terms of structural tensors
= λ/2 E V12 I12 + µ E V12 I2
+ δ1 E V12 I4 + δ2 E V12 I6 E V12 I1 + 2µ1 E V12 I5 + 2µ2 E V12 I7
+ δ3 /2 E
V 12
I42 + δ4 /2 E
V 12
I62 + δ5 E
V 12
I4 E
V 12
I6 ,
as given, e.g., by Spencer [38], results with respect to a Cartesian frame and V1 =
e1 ⊗ e1 , V2 = e2 ⊗ e2 in
L1
L3
L5
L7
L9
= λ + 2δ1 + δ3 + 2µ + 4µ1 ,
= λ + δ1 ,
= λ + δ2 ,
= 2[µ + µ2 ],
= 2[µ + µ1 + µ2 ],
L2
L4
L6
L8
= λ + δ1 + δ2 + δ5 ,
= λ + 2δ2 + δ4 + 2µ + 4µ2 ,
= λ + 2µ,
= 2[µ + µ1 ],
188
A. MENZEL AND P. STEINMANN
whereby nine independent material parameters are incorporated. Thus, the coefficients L1,...,9 within the framework of a fictitious configuration represent a
specific reduced form but with identical symmetry properties. The derivation of
the corresponding relations for transversal isotropy is straightforward. Note that
the determination of the formulation based on the fictitious configuration in terms
of structural tensors is always possible while the opposite does not hold.
5. Example
To discuss the implications of anisotropic material behaviour we first point out
an example of homogeneous deformation in simple shear in detail (Section 5.2).
Subsequently a three-dimensional finite element computation underlines the applicability and the numerical efficiency of the proposed anisotropic framework (see
Appendix A). To set the stage first the utilized material type is highlighted.
5.1. HYPERELASTIC MATERIALS
The free energy function of an isotropic, hyperelastic material is generally defined
by
∞
p q cpqr E I1 E I2 E I3r ,
= E =
(53)
p,q,r=0
compare Ogden (1997). Thereby two restrictions have to be obeyed, namely
E F =G = 0 ⇒ c000 = 0,
(54)
∂E E F =G = 0 ⇒ c100 = 0.
In the sequel we extend a nonlinear constitutive equation in the spirit of Kauderer
[39] in terms of the first and second anisotropic invariant; = (E B I1,2 ) =
∞
E B p E B q
I1
I2 . Consequently, the second Piola–Kirchhoff stress tensor
p,q=0 cpq
reads
2 dev E B .
vol
I1 B + 2Gκ dev B · E · B ,
(55)
S = 3Kκ − Gκ
3
whereby K and G denote the constant compression and shear moduli, respectively. Note that this structure coincides with a St. Venant–Kirchhoff material type,
whereby nonlinearities are introduced via the scalar-valued dimensionless functions κ vol and κ dev . In the following we choose a polynomial ansatz
κ vol(ιvol ) = 1 + κ1vol ιvol + κ2vol ι2vol + · · · ,
κ dev ι2dev = 1 + κ2dev ι2dev + κ4dev ι4dev + · · · ,
(56)
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
189
with
ιvol =
1 E B I1
3
and
ι2dev =
4 E B I2 −
3
1 E B 2
I1
3
.
The incorporation of constant, linear and quadratic terms – κivol = 0, i = 3, . . . , ∞
and κjdev = 0, j = 6, . . . , ∞ – renders the following identification of constants:
c20 = 21 K − 31 G,
c30 = 91 Kκ1vol ,
c01 = G,
c40 = 2 Gκ2dev + 1 Kκ2vol ,
27
12
16
dev
c60 = − 729 Gκ4 ,
c02 = 32 Gκ2dev ,
c21 = − 4 Gκ2dev ,
9
16
c41 = 81 Gκ4dev ,
16 Gκ dev ,
c03 = 27
4
16 Gκ dev ,
c22 = − 27
4
(57)
and cpq = 0 otherwise. Moreover, neglecting additionally the second order terms
vol
dev
= 0 and κi\2
= 0) results in the following coefficients:
(κi\1
c20 = 1 K − 1 G,
2
3
2
dev
c02 = 3 Gκ2 ,
c01 = G,
2 Gκ dev ,
c40 = 27
2
c30 = 1 Kκ1vol ,
9
c21 = − 94 Gκ2dev ,
(58)
with cpq = 0 otherwise. Recall that the applied anisotropic constitutive function,
as, e.g., given in (55), can be expressed in terms of structural tensors. The opposite – to represent a constitutive equation based on structural tensors within the
framework of a fictitious configuration – is obviously not generally possible.
For an anisotropic constitutive equation the stress and the strain metric tensor
are generally not coaxial, i.e., the nonsymmetric part of their product does not
vanish
. (59)
G · S · E − E · S · G = E S W = 0.
(a)
(b)
Figure 3. Stereo graphic projection and spherical coordinates.
190
A. MENZEL AND P. STEINMANN
To visualize this noncoaxiality the method of stereo graphic projection is applied,
which is well known from crystallography and represents the homomorphism
SO(3) → SU (2), see, e.g. [40]. Conceptually speaking, the eigenvectors of symmetric second order tensors – interpreted as elements of the unit sphere S 2 – are
projected onto the equatorial plane by viewing from the south pole, see Figure 3(a).
Moreover, to define the axes of anisotropy (denoted by the unit vectors vζ
) we apply
spherical coordinates. For the sake of simplicity we refer to a Cartesian frame ei ,
thus one possible parametrization of vζ = vζi ei results in vζ1 = sin ϑζ1 sin ϑζ2 , vζ2 =
cos ϑζ2 and vζ3 = cos ϑζ1 sin ϑζ2 , see Figure 3(b).
5.2. SIMPLE SHEAR
In the sequel we consider a homogeneous deformation in simple shear whereby
we refer to a Cartesian frame. Thus, the deformation gradient reads F = I +
γ e1 ⊗ e2 with I = δij ei ⊗ ej . The nonlinear constitutive law outlined in (55) is
applied whereby the subsequent material parameters are taken into account: K =
8.3333 × 104 , G = 3.8461 × 104 , κ1vol = κ2dev = 0.5 and κ2vol = κ4dev = 0.25.
The energy metric tensor is constructed by two orthogonal unit vectors vζ which
are defined via the following spherical coordinates ϑ11 = 5/6π , ϑ12 = 1/6π , ϑ21 =
1/3π , ϑ22 = 1/2π , compare Figure 3(b). Moreover, the corresponding scalars are
set to α0 = 1.0, α1 = 0.25 and α2 = 0.5.
To highlight the noncoaxiality of the strain metric tensor and the stress tensor
the stereo graphic projection of the corresponding principal directions for three
different deformation states are given in Figure 4. Next, as a measure of the degree
of anisotropy we propose to compute a scalarvalued quantity δ in terms of the
skewsymmetric tensor E S W in a normalized format
E S W ,
δ=
S E (60)
whereby the appropriate norm operator [·] is given in Appendix B. Figure 5
underlines that the anisotropy measure δ reflects a strong dependence on the shear
number γ .
Finally, we discuss the acoustic tensor q ∈ T B × T B within the proposed
anisotropic framework based on the fictitious configuration, see Appendix D for a
short outline on the underlying theory. Within an isotropic linear elastic St. Venant–
Kirchhoff material the determinant of the acoustic tensor is independent of the
wave propagation direction and hence constant. In particular one has
lin,iso
2 3
G+K ,
(61)
=G
det q
4
see, e.g. [24]. In the context of nonlinear hyperelasticity and large strain kinematics
det(q ) remains no longer constant. Figure 6 highlights this dependence on the
wave propagation direction n ∈ S 2 in T B for the isotropic (det(q )iso, α1 = α2
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
191
Figure 4. Simple shear: Stereo graphic projection due to the principal directions of strain E:
◦ and stress S: • with respect to a Cartesian frame.
Figure 5. Simple shear: Scalarvalued anisotropy measure δ in terms of the shear number γ .
= 0.0) and the anisotropic setting (det(q )ani , α1 = 0.25, α2 = 0.5) at γ = 0.25.
Thereby, the spherical coordinates ϑ i define the wave propagation directions with
respect to a Cartesian frame, compare Figure 3(b).
6. Summary and Conclusions
The main goal of this work was the comparison of two approaches to formulate
orthotropic materials, namely the incorporation of structural tensors and the introduction of a fictitious configuration. Obviously, the underlying motivation for the
latter strategy relies in the consideration of an energy metric tensor which allows
the interpretation as a fictitious Finger tensor characterizing a prestretched material.
The corresponding fictitious linear map was assumed to be constant and moreover we selected a specific ansatz to introduce an analogue to structural tensors.
The comparison of the incorporated sets of invariants underlines that the framework within the fictitious configuration deals with a reduced set of invariants and
generators compared to the structural tensor approach. It is our belief that this con-
192
A. MENZEL AND P. STEINMANN
Figure 6. Simple shear: Determinant of the acoustic tensor for γ = 0.25 within the isotropic
and anisotropic setting.
tinuum mechanically motivated reduction makes the formulation and especially
the correlated numerical treatment of a variety of anisotropic materials manageable. Moreover, standard isotropic constitutive equations can be applied to model
orthotropy which is a significant benefit of the formulation.
Finally, based on a simple nonlinear constitutive prototype equation we discussed two numerical examples. Firstly, for a homogeneous deformation in simple
shear typical effects like the noncoaxiality of the stress and the strain metric tensor and additionally the influence on the acoustic tensor have been highlighted.
Secondly, a three dimensional finite element example – which is attached in the
Appendix – underlines the large scale applicability of the proposed formulation.
Summarizing, this contribution gives a framework for the formulation of a
subclass of orthotropic materials at large strains. This is an alternative approach
compared to the incorporation of structural tensors and turns out to be capable to
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
193
characterize some anisotropy classes of major interest. As a main advantage the
numerical costs are comparable to those of standard isotropy.
Appendix A. Cook’s Problem
In this finite element example we investigate a three dimensional version of the
classical two-dimensional Cook’s membrane problem. Thus the standard discretization in the e1,2 plane is extended into the e3 direction. The geometry as well as the
boundary and loading conditions are visualized in Figure 7 whereby we choose the
following parameters: L = 48, H1 = 44, H2 = 16, T = 4. Moreover, the material
parameters coincide with those in the previous Section 5.2 due to the homogeneous
deformation in simple shear.
The discretization consists of 16 × 16 × 4 eight node bricks, whereby we invoke
enhanced elements (Q1E9) as advocated by Simo and Armero [41]. Furthermore,
the conservative force F is considered as the resultant of a continuous shear stress
with respect to the undeformed reference geometry. For the sake of brevity we do
not further comment on the applied nonlinear finite element setting and thus refer,
e.g., to the books of Oden [42], Hughes [43], Crisfield [44], and Bonet and Wood
[45] for more detailed background information.
Since the orthogonal axes of anisotropy vζ do not lie in the e1,2 plane we consequently observe a severe out of plane deformation. Figure 8 shows the deformed
mesh for F = F = 1.28×105 . Moreover, we study the displacement u = x −X
of the mid point node at the top corner, which is highlighted in Figure 9. In order
Figure 7. Cook’s problem: Geometry, boundary and loading conditions; discretization with
16 × 16 × 4 eight node bricks.
194
A. MENZEL AND P. STEINMANN
Figure 8. Cook’s problem: anisotropic (Q1E9); three different views on the deformed mesh
for F = 1.28 × 105 .
Figure 9. Cook’s problem: anisotropic (Q1E9); displacement curves of the mid point node at
the top corner.
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
(a)
195
(b)
Figure 10. Cook’s problem: (a) isotropic (Q1E9), (b) anisotropic (Ql); displacement curves
of the mid point node at the top corner.
Table I. Cook’s problem: residual norm for the load step F :
[0, 5.12 × 104 ]
Iter. step
Q1E9, anisotropic
Q1E9, isotropic
Q1, anisotropic
1
2
3
4
5
6
7
8
9
10
11
12
13
2.35695E+04
1.70059E+05
3.71217E+03
2.59480E+06
1.16144E+04
1.32396E+04
3.76553E+03
3.93106E+03
4.76101E+02
1.17589E+02
5.68824E−01
1.48105E−04
1.02527E−08
2.35695E+04
2.35936E+05
5.65492E+03
1.24521E+02
5.51955E−02
1.89174E−08
2.35695E+04
1.57264E+05
3.81568E+03
6.76394E+04
7.16359E+03
5.46791E+00
1.47482E+00
3.25775E−07
1.95022E−08
to compare these results to an isotropic setting incorporating a spherical energy
metric tensor sph B we demand
. sph B = sphβ0 G
with sph B = B .
(A.1)
In view of (37) of Section 4.4 this evident assumption renders
1/2
sph
β0 = β02 + [β0 + β1 ]2 + [β0 + β2 ]2
,
(A.2)
β2 = 2α0 α2 +
Figure 10 shows the correwith β0 = α0 , β1 = 2α0 α1 +
sponding displacement curves of the mid point node at the top corner, which are
α12 ,
α22 .
196
A. MENZEL AND P. STEINMANN
obviously within the same range as those of the anisotropic setting. Additionally
we highlight the results of a computation with standard trilinear eight node bricks
(Ql), which shows a more stiff behaviour compared to the previous computation
based on the enhanced elements (Q1E9).
In addition, Table I summarizes the convergence of the computations within the
three different settings. Thereby the residual norm of the corresponding Newton–
Raphson iteration steps are tabulated for the rather large load step F : [0, 5.12 ×
104 ].
Appendix B. Notation
Throughout this contribution push-forward and pull-back operations are applied
especially to second order covariant and contravariant tensors with respect to the
i
fictitious linear tangent map F = Gi ⊗ G . With a small abuse of notation we
will denote these operations by ϕ̄∗ and ϕ̄ ∗ despite the fact that there does not
exist any continuous map ϕ̄. For convenience of the reader (B.1) summarizes these
transformations with respect to a second oder tensor Y . Thereby the notation [·]−1
represents inversion and [·]T characterizes transposition:
ϕ̄∗ Y = F
−T
·Y ·F
ϕ̄∗ Y = F · Y · F
−1
T
ij
= Y Gi ⊗ Gj = Y ,
i
T
ϕ̄ ∗ Y = F · Y · F
ϕ̄∗ Y = F
−1
· Y
· F
= Y ij Gi ⊗ Gj = Y ,
j
= Yij G ⊗ G
−T
=Y ,
(B.1)
= Y ij Gi ⊗ Gj = Y .
In order to achieve a rather compact form of the Hessian L
E E = ∂E2 ⊗E of the
free energy function with respect to the strain metric tensor E we use different
symmetry operators denoted by sym and SYM which are defined by
.
Y1 , Y2 sym =
.
Y1 , Y2 sym =
1 G · Y · G · Y · G + G · Y T · G · Y T · G ,
1
2
2
1
2
1
T
T
2 Y1 · G · Y2 + Y2 · G · Y1 ,
(B.2)
.
Y1 , Y2 SYM = 21 [G · Y1 · G] ⊗ [G · Y2 · G] + [G · Y2 · G] ⊗ [G · Y1 · G] ,
.
Y1 , Y2 SYM = 21 [Y1 ⊗ Y2 + Y2 ⊗ Y1 ],
.
whereby Y1,2 denote two arbitrary second order tensors. Furthermore, let G = G
. if Y1,2 are covariant and G = G if Y1,2 are contravariant, respectively. Typically
the norm of a second order tensor Y1 reads
1/2
Y1 = [Y1 · G] : [G · Y1 ] .
(B.3)
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
197
Beside the standard dyadic product ⊗ two nonstandard dyadic products ⊗ and ⊗
are applied which yield, e.g., for two contravariant tensors of second order
jl
Y1
⊗Y2
= Y1ik Y2 Gi ⊗ Gj ⊗ Gk ⊗ Gl ,
(B.4)
jk
Y1
⊗Y2
= Y1il Y2 Gi ⊗ Gj ⊗ Gk ⊗ Gl .
Now, in addition to the standard dyadic products G ⊗ G and G
⊗ G
, two fourth
order identity tensors of minor and major symmetry can be constructed:
G =
1 G ⊗G + G ⊗G
2
and
G
=
1 G ⊗G
+ G
⊗G
2
(B.5)
by additive composition of the nonstandard dyadic products in terms of the covariant and contravariant metric tensor G and G
.
Appendix C. Hessian of the Free Energy Function
The classical formulation based on structural tensors as pointed out in Section 2
renders for the case of two arbitrary fibres v1,2 a set of nine invariants E V12 I1,...,9 ,
whereby eight of them depend on the strain metric tensor E , compare (10).
Straightforward but tedious calculations end up in the rather lengthy expression
for the corresponding Hessian LE E ∈ T B0 × T B0 × T B0 × T B0 as given in
(C.2). Thereby, for notational simplicity the abbreviations
.
.
(C.1)
i = ∂E V and ij = ∂E2 V E V with i, j = 1, . . . , 9
12 Ii
12 Ii
12 Ij
are applied.
E V12
L
E E
= 11 G
⊗ G
+ 422 G
· E · G
⊗ G
· E · G
+ 933 G
· E · G
· E · G
⊗ G
· E · G
· E · G
+ 44 G
· V1 · G
⊗ G
· V1 · G
sym sym
⊗ E , V1
+ 455 E , V1
+ 66 G · V2 · G ⊗ G
· V2 · G
sym sym
⊗ E , V2
+ 477 E , V2
sym sym
+ 499 V1 , V2
⊗ V1 , V2
SYM
SYM
+ 412 E , G
+ 1223 E · G
· E , E SYM
sym SYM
+ 634 E · G
· E , V1
+ 245 2 E , V1
, V1
sym SYM
sym SYM
+ 256 2 E , V1
, V2
+ 267 2 E , V2
, V2
SYM
SYM
+ 613 E · G
· E , G
+ 424 E , V1
198
A. MENZEL AND P. STEINMANN
sym SYM
SYM
+ 635 E · G
· E , 2 E , V1
+ 246 V1 , V2
sym sym SYM
+ 257 2 E , V1
, 2 E , V2
sym sym SYM
+ 279 2 E , V2
, 2 V1 , V2
SYM
sym SYM
+ 214 V1 , G
+ 425 E , 2 E , V1
SYM
sym SYM
+ 636 E · G
· E , V2
+ 247 2 E , V2
, V1
sym SYM
sym SYM
+ 269 2 V1 , V2
, V1
+ 215 2 E , V1
,G
SYM
sym SYM
+ 426 E , V2
+ 637 E , 2 E , V2
sym SYM
SYM
+ 259 2 E , V1
, 2 V1 , V2
+ 216 V2 , G
sym SYM
sym SYM
+ 427 E , 2 E , V2
+ 249 2 V1 , V2
, V1
sym SYM
+ 217 2 E , V2
,G
sym SYM
+ 639 E · G
· E · G
· E , 2 V1 , V2
sym SYM
+ 429 E · G
· E , 2 V1 , V2
sym SYM
+ 219 G , 2 V1 , V2
+ 22 G
+ 33 E
E
E + 25 E
V 1
E + 27 E
V 2
E
(C.2)
Furthermore, the additional simplifying notations E ∈ T B0 ×T B0 ×T B0 ×T B0
are defined by the following equation
E [·]
E =
1 G ⊗ G · [·] · G
+ G
⊗ G
· [·] · G
2
+ G
· [·] · G
⊗G
+ G
· [·] · G
⊗G
.
(C.3)
Within the framework of a fictitious configuration – see Section 3 – the reduced
set of invariants E B I1,2,3 introduced in (20) renders the following comparatively
concise format of the Hessian:
E B L
E E = 11 B ⊗ B + 422 B · E · B ⊗ B · E · B + 933 B · E · B · E · B ⊗ B · E · B · E · B + 412 B · E · B , B SYM
+ 1223 B · E · B · E · B , B · E · B SYM
+ 613 B · E · B · E · B , B SYM
+ 22 B + 33 E
B
E ,
(C.4)
TWO STRATEGIES TO FORMULATE ORTHOTROPIC HYPERELASTICITY
whereby two further abbreviation are introduced which read
B = 1 B ⊗B + B ⊗B ,
2
E B E = 21 B ⊗ B · E · B + B ⊗ B · E · B + B · E · B ⊗B + B · E · B ⊗B .
199
(C.5)
Obviously, the general formulation in terms of structural tensors ends up in enormous analytical and numerical costs. In contrast, the approach based on a fictitious
configuration is much less expensive. Actually, the computational effort is in the
same range as that for standard isotropy.
Appendix D. The Acoustic Tensor
As point of departure we consider the incremental equation of motion in the absence of body forces
(D.1)
Div δ F · S = Div δF · S + F · δS = ρ0 ∂t2t δx,
whereby ρ0 denotes the initial mass density and t represents time, compare, e.g.,
.
[46, 47]. Typically a homogeneous state is examined (F = const), moreover the
following incremental relations hold
T (D.2)
F · δS = † L : δF , † L = F · LE E : F T · g ⊗ G
with † L = † Lij k l gi ⊗Gj ⊗g k ⊗Gl ∈ T B ×T B0 ×T B ×T B0 , whereby the minor
symmetry of the Hessian L
E E has been taken into account. For an overview on
Eulerian tangent operators see [48]. Next, we adopt the usual wave propagation
ansatz
.
δx = m
f n · x − ct ,
c ∈ R+ , t ∈ R, m
∈ T B, n ∈ S 2
in T B,
f ∈ C2.
(D.3)
Straightforward calculations with δF = ∂δx/∂X render the acoustic tensor q =
qji gi ⊗g j ∈ T B ×T B occurring in the following eigenvalue problem for the wave
speed c in the propagation direction n with corresponding polarization vector m
:
q · m
= † q + N · S · N g · m
= ρ0 c2 m
with N = F T · n ∈ T B0 and
† q = g ⊗ N : † L · N ∈ T B × T B.
(D.4)
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