1
Nearly incompressible and nearly inextensible finite hyperelasticity
2
Adam Zduneka,∗, Waldemar Rachowiczb , Thomas Erikssonc
a
3
4
b Institute
5
Swedish Defence Research Agency FOI, Aeronautics and systems integration, SE-164 90 Stockholm, Sweden
for Computer Modelling, Section of Applied Mathematics, Cracow University of Technology, Pl 31-155 Cracow, Poland
c Department of Engineering Science, Institute of Biomedical Engineering, University of Oxford, UK
6
Abstract
7
A novel approach to computational strongly transversely isotropic and nearly incompressible finite hypere-
8
lastic fibre mechanics is introduced. It relies on using an equivalent generalised right Cauchy-Green stretch
9
tensor which is volume preserving and simply-stretch free in the limit of incompressibility and inextensibility.
10
In other words, its third and fourth principal invariants become trivial. Otherwise it represents volume change
11
and fibre stretch by aid of point-wise equivalent auxiliary measures in the continuous case. The generalised
12
kinematics implies the common orthogonal spherical – deviatoric decomposition of the stresses. The devia-
13
toric stresses are further orthogonally decomposed into axial fibre- and ground substance-stresses. The novel
14
approach implies that the deviatoric ground substance stresses are trivial in the fibre direction as opposed to
15
the current standard formulation. The approach is also able to represent exact inextensible fibres which is a
16
problem recently addressed in the literature using an additive volumetric - isochoric decoupled strain energy
17
density function, relying on volume preserving stretch. The formulation is corroborated by a couple of simple
18
numerical examples using a preliminary finite element setting.
19
1. Introduction
20
This work presents a computational framework for the phenomenological theory of transversely isotropic
21
finite hyperelastic materials. The approach is especially aimed for materials which are nearly incompress-
22
ible and may become nearly inextensibile at finite strains. Typical applications can be found in soft tissue
23
biomechanics and in the mechanics of fibre-reinforced rubberlike materials. The boundary value problems in
24
these applications have to be solved by numerical methods like the finite element method (FEM). The pro-
25
vided framework extends the current displacement, pressure and dilatation formulation for isotropic nearly
26
incompressibile finite hyperelastic solids to the anisotropic case.
27
Figure 1 on page 4 shows the extensional collagen fibre stiffness for a human artery normalised by the
28
artery ground substance shear stiffness µ as a function of the volume preserving part of the fibre stretch λ̄. A
29
representative ratio between the artery ground substance bulk stiffness κ and its shear stiffness is κ/µ = 1000.
30
Figure 1 shows that the normalised extensional fibre stiffness reaches this value at the realistic fibre strain
∗ Corresponding
author: phone: +46855503226, mobile: +46739472819
Email addresses: zka@foi.se (Adam Zdunek), wrachowicz@pk.edu.pl (Waldemar Rachowicz),
Preprint
submitted to Elsevier (Thomas Eriksson)
thomas@tseeriksson.se
May 15, 2014
31
of 16%. These stiffness ratios will penalise volume change and fibre extension, respectively. They call
32
for special computational methods in finite precision arithmetic. The problem of competing imposition of
33
the corresponding volume- and extensibility-constraint is neither recognised nor previously addressed to our
34
knowledge. An appropriate setting of the problem is given here.
35
The augmenting volumetric and extensibility constraints coming into play in the limiting case imply
36
stability requirements on the FEM formulation, in the Ladyzhenskaya Babuska Brezzi (LBB) sense [1].
37
The LBB criterion is also known as the inf-sup condition Bathe [2]. Stable discrete formulations for the
38
incompressible case alone exists and are today extensively used. The construction and validation of a stable
39
FEM formulation for the finite hyperelastic (near) inextensible case is on the other hand, to our knowledge,
40
essentially an open problem. It needs an appropriate strong formulation in the first place. A proper setting is
41
provided here. A preliminary higher order FEM formulation is used to corroborate the formulation.
42
Existing efficient formulations for nearly incompressible isotropic materials rely on the split into isochoric
43
and volumetric deformation measures due to Flory [3]. The third invariant of the isochoric Cauchy-Green
44
stretch is trivial. By definition it is incapable to measure volume changes. The material (constitutive) as-
45
sumptions used in computational practice are adapted to fit the isochoric volumetric decoupled form. It is
46
noteworthy(!) that this adaptation is also used for strongly anisotropic materials. The split has become stan-
47
dard in finite element implementations for near incompressibility. Among early contributions that appeared
48
independently are: Simo, Taylor and Pister [4] and Simo and Lubliner [5], Zdunek and Bercovier [6] and
49
Sussman and Bathe [7]. Our starting point is the mixed three-field (displacement, dilatation and pressure)
50
Hu-Washizu formulation developed by Simo and Taylor [8].
51
The founding developments of the continuum theory of finite deformations of elastic materials reinforced
52
by cords or fibres were contributed by Adkins and Rivlin and are described by Green and Adkins [9]. The
53
approach followed here is the resulting phenomenological theory proposed by Spencer [10, 11]. Spencer
54
developed a description where the fibres are characterized by a unit vector field that defines the fibre direction
55
and which deforms with the material. Spencer’s approach has close connection to the theory of anisotropic
56
tensor representations involving the use of so-called structural tensors initiated by Boehler [12] and later
57
developed by Zheng [13]. Recently the theory has been applied extensively to soft tissue materials, see for
58
example Holzapfel and Ogden [14].
59
It is a common practice to use the displacement, dilatation and pressure formulation [8] also for trans-
60
versely isotropic biological materials with exponentially stiffening fibres, see for example Weiss et al. [15],
61
Holzapfel [16], Gasser et al. [17], Mortier et al. [18], Crane et al. [19], Boerboom et al. [20], Dal et al. [21]
62
and Göktepe [22]. In other words, the near inextensibility of the fibres is not addressed currently in general.
63
This work provides an appropriate setting for the combined nearly incompressible and nearly inextensible
64
case. A generalised Cauchy-Green stretch measure is constructed. In the limit of exact incompressiblility and
2
65
exact simple inextensibility its third and fourth principal invariants are trivial. Normally the third invariant
66
measures the volume change and fourth invariant measures the fibre stretch, respectively. The construction
67
involves an exact and general extraction procedure of the actual fibre stretch. The mixed three-field (dis-
68
placement, dilatation and pressure) Hu-Washizu Simo, Taylor and Pister [4] (STP) formulation is extended
69
by an additional auxiliary variable for the fibre stretch and an associated auxiliary fibre tension variable in
70
the transversely isotropic case. The STP formulation provides the orthogonal decomposition of the stresses
71
into spherical and deviatoric parts. The extended formulation put forward provides a further orthogonal de-
72
composition of the deviatoric stresses into ground substance- and axial fibre-stresses respectively. The near
73
inextensible behaviour of the fibres is thus modelled by a separate deviatorically completely decoupled con-
74
stitutive behaviour, see further regarding the constitutive description below. The mean pressure and fibre
75
tension responses are in general coupled.
76
The constitutive description of soft tissue is a large field by its own, and it is rapidly developing. See
77
for example the review article on arterial continuum mechanics by Holzapfel and Ogden [23] the pioneering
78
work on arteries by Holzapfel, Gasser and Ogden [24] (HGO) and more recently for myocardium [25].
79
Common macroscopic phenomenological material models of soft tissues are usually described as hyperelastic
80
(rubber-like) nearly incompressible materials with a strongly anisotropic contribution from fibres. A model
81
intended for numerical implementation is usually stated in the volumetric isochoric decoupled form. The
82
strain energy constitutive equation is then a sum of the volumetric, fibre and ground substance contributions.
83
There are some special issues associated with this way of constitutive description, see Helfenstein et al. [26]
84
Sansour [27] Annaidh et al. [28], Vergori et al. [29]. The following are dealt with; (a) giving the fibre
85
contribution in terms of the volume-preserving stretch prevents the description of exact fibre inextensibility
86
at finite volume change, and (b) giving the ground substance contribution in terms of the volume preserving
87
Cauchy-Green stretch provides an inappropriate ground substance contribution in the fibre direction. Both
88
these issues are resolved in this work using a generalised form of the so-called Standard Reinforcing Material
89
(SRM) model, see [30, 31]. Finally a correction to the HGO model is proposed. The proposed framework is
90
verified by a couple of simple numerical examples using a preliminary FEM formulation.
91
1.0.0.1. Notation. The theory herein is presented using an engineering continuum mechanics notation co-
92
herent with [32]. Vectors and tensors are boldface. Tensors are upright, vectors are italic. Eulerian vectors
93
and tensors are lower case letters, Lagrangean vectors and tensors are denoted by capital letters. Scalars
94
are light-face. Here the mean pressure is denoted p̄ and is defined with the opposite sign, cf. [32, (6.91)].
95
Hydrostatic pressure is denoted p. Here the subscript (iso ) means “isotropic”, in [32] it means “isochoric”
96
which we abbreviate (isoch ) subscripted. Stress tensors without spherical component are here denoted with
97
a subscript (dev ) meaning “deviatoric”. A superscripted tiny plus-sign (+ ) is also used to denote deviatoric
98
stress occasionally.
3
4000
3000
2000
1000
0
1
1,05
1,1
1,15
1,2
Figure 1: Normalised fibre extensional stiffness as function of isochoric fibre stretch λ̄. Material parameters for Left Anterior Descending
(LAD) coronary human artery adventitia represented by a HGO material model [24] with parameters µ = 2.7 [kPa], k1 = 5.1 [kPa]
and k2 = 15.4 [-] according to [33]. The normalised fibre extensional stiffness exceeds the normalised bulk stiffness for λ̄ > 1.156.
99
2. Kinematics
100
The material configuration denoted B of a continuum body is an open bounded subset in R3 . A material
101
particle P ∈ B is identified with its position X ∈ R3 . A smooth transplacement is a non-linear one-to-one
102
mapping:
ϕ(X) : B → S ⊂ R3 ,
X 7→ x = ϕ(X),
(1)
where x ∈ S denotes a point in the spatial configuration S = ϕ(B). Further, let Γ : R → B and γ : R → S
be material and spatial curves parametrised by the same variable S ∈ R on B and S , respectively. Moreover, the spatial curve is the image of the material curve in the transplacement ϕ, i.e. γ(S) = ϕ(Γ(S)). The
material and spatial tangents T := Γ0 (S) and t := γ 0 (S) are vectors in the tangent spaces TX B and Tx S ,
endowed with inner products, respectively. Using the chain rule we obtain,
F(ϕ) := ∇X ϕ(X) : TX B → Tx S ,
T 7→ t = ∇X ϕ(X) T ,
(2)
J(ϕ) := det(F(ϕ)) > 0.
103
The usual volume-preserving deformation gradient denoted F̄ is defined as,
F̄ := J −1/3 F,
det F̄ ≡ 1,
4
(3)
104
where the identity det(αT) = α3 det T is used in the equivalence statement. An auxiliary measure for
105
dilatation is introduced as,
.
J˜ = J(ϕ),
(4)
106
.
where the symbol (=) designates point-wise equality in the following. It is valid in the continuous formula-
107
tion. Simo, Taylor and Pister [4] introduced the so-called generalised deformation gradient using the auxiliary
108
dilatation measure as,
F̃J˜ := J˜1/3 F̄,
˜
det F̃J˜ = J.
(5)
109
They made a Hu-Washizu formulation in terms of (5) imposing Eq. (4) as a generalised compressibility
110
constraint using a Lagrange multiplier representing the mean pressure. We will generalise the formulation
111
[4] with respect to extensibility in a preferred direction.
112
113
To that end, for a unit direction M in TX B we determine the unit direction m in Tx S and obtain the
actual stretch λ(ϕ) in the preferred direction M as,
q
λ(ϕ) := M · F(ϕ)T F(ϕ)M > 0,
m
:= λ−1 FM ,
(6)
|m| = 1.
114
2.0.0.2. Fibre stretch extraction. Consider a simple extension λ in the material direction M ∈ TX B with
115
lateral contraction α = α(λ),
F = λα(I − m ⊗ M ) + λm ⊗ M ,
α > 0,
J = det F = α2 λ3 > 0.
(7a)
116
Recalling the polar decomposition F = VR = RU the simple extension is rotation free, R = I. In other
117
words, m = M and F = V = U is a pure stretch. Using variables J and λ̄ = J −1/3 λ; by (7a)3 we find
118
α = λ̄−3/2 , and it may be eliminated from (7a)1 , resulting in,
F = F̄J 1/3 I
with
F̄ = λ̄−1/2 (I − m ⊗ m) + λ̄m ⊗ m.
(7b)
119
Using the identity, T−1 T = I, it is clear that the volume preserving part of the fibre stretch λ̄ and the volume
120
change J associated with F may be extracted, or annihilated by composition. To that end we introduce the
121
fibre stretch extraction mappings f̄ ext ∈ Tx S × Tx S and δ ∈ Tx S × Tx S such that,
f̄ ext (λ̄; a) := λ̄−1/2 (1 − a) + λ̄a,
a := m ⊗ m,
f ext = δ f̄ ext ,
δ(J) := J 1/3 1,
(7c)
wherein a is the so-called Eulerian structural tensor, and where 1 is the spatial identity mapping. The tensor
a is idempotent a2 = a with the trace a : 1 = 1. The extraction mappings have the following properties,
m · f̄ ext (λ̄; a)m = λ̄,
det δ(J) = J,
det f̄ ext (λ̄; a) = 1,
(7d)
f̄ −1
ext (λ̄; a)
= f̄ ext (λ̄
−1
; a) and δ
5
−1
(J) = δ(J
−1
),
122
respectively. Finally, it is stressed that the fibre stretch λ = λ(ϕ) and the volume ratio J = J(ϕ) used to
123
extract, are the quantities measured in the actual deformation ϕ(X).
124
Remark 1 (Lateral fibre contraction in extension). It is noteworthy that the lateral contraction ratio α
125
disappears from the description of the simple extension in the form Eqs. (7c). It is a consequence of the
126
volumetric - isochoric decomposition Eq. (7b).
127
It follows from the product rule of determinants that the volume change associated with mappings Eqs. (7c)
128
is equal to det f ext = J. Finally, an extension without lateral contraction is the one-dimensional deformation
129
achieved by setting α = λ−1 in (7a). The associated volume change is J = λ > 0. It is an admissible ex-
130
treme. It implies setting λ̄ = λ2/3 , λ̄−1/2 = λ−1/3 and J 1/3 = λ1/3 in Eqs. (7c). The fibre stretch extraction
131
functions (7c) cover all admissible simple extensions with and without lateral contraction1 . Given λ, J and
132
a the proposed extraction of the fibre-stretch is an exact procedure. 2
133
2.0.0.3. Auxiliary measure of stretch. An auxiliary measure for the stretch in the preferred direction which
134
is point-wise equal to the actual stretch λ(ϕ) is introduced as,
.
λ̃ = λ(ϕ).
(8)
135
Remark 2 (Constraint equivalence). On account of the impenetrability of matter J > 0 and the point-wise
136
equality (4) we have the following equivalence for the statement Eq. (8),
λ̃ = λ
137
⇔
˜ = λ̄
λ̄
where
˜ := J˜−1/3 λ̃
λ̄
and
˜ = λ̄
˜ (λ̃, J)
˜ (λ̃, 1) = λ̃ and λ̄
˜ (1, 1) = 1 .
˜ with λ̄
where it is noted that λ̄
λ̄ := J −1/3 λ.
(9)
2
Finally, we extend the concept of the dilatationally generalised deformation gradient F̃J˜ defined by Eq. (5)
with respect to the extensibility in the preferred direction. The enhanced generalised deformation gradient
which measures dilatation in terms of J˜ and stretch in the preferred direction in terms of λ̃ is defined as,
˜ , J;
˜ a) := f̄ ext (q; a)F̃ ˜ = [qa + q −1/2 (1 − a)]F̃ ˜,
F̃(F̄, λ̄
J
J
˜
det F̃ = J,
138
m · F̃M = λ̃.
˜ /λ̄,
q = λ̄
(10a)
(10b)
In words, we first remove the actual volume-preserving stretch present in the preferred direction in the dilata-
140
tionally modified deformation gradient F̃J˜, see Eq. (5), composing it with f̄ −1
ext (λ̄; m), and then we replace
˜ composing it with f̄ (λ̄
˜ ; m). No reference is made to the so-called
the stretch λ̄ by the auxiliary measure λ̄
141
multiplicative split of the deformation gradient. Only compositions in Tx S are performed. To be precise,
139
ext
1 including
the exactly incompressible case J = 1, i.e. α = λ−3/2 .
6
142
actual dilatation and volume-preserving uniaxial stretch described by F(ϕ) are extracted and replaced by
143
auxiliary measures, by composition.
144
145
146
Properties (10b)1,2 are readily proven. Eq. (10b)1 is proven using the identity det(αT) = α3 det T and
the product rule of determinants. Equation (10b)2 is proven by direct calculation as,
!
˜
λ̄
1/3
˜ = λ̃,
m · F̃M = J˜
λ̄m · m = J˜1/3 λ̄
λ̄
where we used Eqs. (6), (10a)2 and (9)3 . Further we have the equivalence,
˜ , J)
˜ ≡ F,
F̃(F̄, λ̄
147
(11)
subject to J˜ = J
and λ̃ = λ.
(12)
This is readily proven by setting q = 1 in (10a) and using equality Eq. (4) and definition Eq. (3).
9
148
149
We also define tensor F which is volume preserving and stretch-free in the preferred direction by setting
J˜ = λ̃ = 1 in Eqs. (10), i.e.
9
F := F̃(F̄, 1, 1) = [λ̄−1 a + λ̄1/2 (1 − a)]F̄,
9
9
det F ≡ 1
and m · FM ≡ 1.
(13)
Objective kinematics are set by introducing the generalised right Cauchy-Green tensor C̃ := F̃T F̃. The
kinematic basis is summarised as,
˜ , J;
˜ A) := J˜2/3 q −1 [C̄ − (1 − q 3 )λ̄−2 C̄AC̄],
C̃(C̄, λ̄
det C̃ = J˜2
˜ , J;
˜ A) ≡ C,
C̃(C̄, λ̄
150
˜ /λ̄
q := λ̄
and
C̄ := J −2/3 C,
and C̃ : A = λ̃2 ,
subject to J˜ = J
(14)
and
λ̃ = λ,
wherein A = λ2 F−1 aF−T is the Lagrangean structural tensor corresponding to a. It is defined as,
A := M ⊗ M
such that
A2 = A
and
A : A = 1.
(15)
Specialising for simultaneous volume-preservation and inextensibility we define the volume-preserving and
9
simply stretch-free right Cauchy-Green tensor C by setting J˜ = λ̃ = 1 in Eqs. (14),
9
C(C̄; A) = λ̄[C̄ − (1 − λ̄−3 )λ̄−2 C̄AC̄],
9
det C ≡ 1
and
and
C̄ := J −2/3 C,
9
C : A ≡ 1,
9
C(C̄; A) := C̃(C̄, 1, 1) ≡ C subject to
det C = 1 and
(16)
C : A = 1.
151
Hence, using C̃ we can describe any deformation including the case with an incompressibility constraint
152
and/or with inextensibility in a preferred direction specified by a strutural tensor A. The concept put forward
153
may readily be generalised to several preferred directions.
7
154
3. Generalised transversely isotropic finite hyperelasticity
155
Recall that the stress constitutive equation of a hyperelastic solid is fully described specifying an ad-
156
missible objective scalar valued strain energy density function of the right Cauchy-Green stretch tensor C,
157
henceforth denoted Ψ = Ψ (C). Here Ψ is given per unit undeformed volume, i.e. in [Nm/m3 ] in SI-
159
units. Using the generalised right Cauchy-Green stretch C̃ Eqs. (14) and assuming strain energy equivalence
.
Ψ (C̃) = Ψ (C) we use Spencer’s phenomenological theory [11] for a hyperelastic solid with one preferred
160
direction specified by the Lagrangean structural tensor A. This implies, that Ψ becomes a function of the
161
joint principal invariants of C̃ and A. In other words,
158
.
Ψ (C) = Ψ (C̃) = Ψ (I˜1 , I˜2 , I˜3 , I˜4 , I˜5 ),
(17)
where,
I˜1 := C̃ : I,
I˜2 := 21 [I˜12 − C̃2 : I],
I˜3 = det C̃ = J˜2 ,
(18)
I˜4 := C̃ : A = λ̃2
and
I˜5 := C̃2 : A.
162
A dependence on I˜5 provides the means to model a shear coupling between the ground substance and the
163
fibres. This coupling is second order in C̃ as I˜5 := C̃2 : A.
164
4. Hu-Washizu stress constitutive equations
166
˜ and J˜ in the
We will now derive explicit constitutive stress equations for the mixed situation where λ̄
˜ , J;
˜ A) given by Eqs. (14) are auxiliary kinematic
expression for the generalised strain tensor C̃ = C̃(C̄, λ̄
167
independent variables with the deviatoric fibre stress τ + and the mean pressure p̄ as energy conjugate gen-
168
eralised Lagrange multipliers, respectively. Actually, for extensible and compressible materials τ + and p̄
165
169
170
determine the constitutive functions for deviatoric axial fibre stress and for the pressure respectively. A for˜ , p̄, J)
˜ is constructed. Our
mulation of the Hu-Washizu type [34, 35] in the independent variables (ϕ, τ + , λ̄
172
˜ formulation [4] for the isotropic case. The
formulation generalises the Simo, Taylor and Pister (ϕ, p̄, J)
˜ in our constitutive ansatz are matched by appending the
auxiliary dilatation and stretch variables J˜ and λ̄
173
generalised volume and extensibility constraints, Eqs. (4) and (8), respectively. After noting Remark 2 our
174
ansatz choice2 is,
171
p
√
˜ , J;
˜ ),
˜ A)) − π( det C − J)
˜ + %( C̄ : A − λ̄
Ψ := Ψ (C̃(C̄, λ̄
2 the
choice of the generalised stretch constraint is not unique.
8
(19)
175
where π and % are temporary scalar Lagrangean multipliers. Given Eq. (19) the hyperelastic stress constitutive
176
equation is advantageously derived using the Clausius-Planck form of the second law of thermodynamics in
177
the dissipation-free form:
Dint = 21 S : Ċ − Ψ̇ = 0,
(20)
where Dint is the internal dissipation, S is the second (symmetric) Piola-Kirchhoff stress tensor, and where
the over-dot ( ˙ ) denotes the material time derivative. Differentiating our ansatz Eq. (19) with respect to time
we obtain,
!
1
∂Ψ
(
C̃)
−1
] : Ċ
Ψ̇ =
−πJC + J −2/3 Dev [ %λ̄−1 A + 2
2
∂ C̄
!
!
p
√
∂Ψ (C̃)
∂Ψ (C̃) ˜˙
'
˜ ).
− % λ˙ + π̇(J˜ − det C) + %̇( C̄ : A − λ̄
J+
+ π+
˜
∂ J˜
∂ λ̄
178
As usual the Lagrangean deviator
Pdev : [•] = Dev [ • ] is identified from the tensor gradient ∂
C
I
∂ C̄
= J −2/3 [ − 13 (C ⊗ C−1 )] = J −2/3
∂C
179
180
181
182
183
C̄ as,
(22)
I is the fourth order symmetric identity mapping [I]IJKL = 1/2(δIK δJL + δILδJK ). Further, the
identity B : AT : C = (A : B) : C for the transpose of a fourth-order tensor A is used. Moreover, the
where
˜ , J;
˜ A). Inserting the expression
indicated derivatives in Eq. (21) need to be expanded since C̃ = C̃(C̄, λ̄
Eq. (21) in Eq. (20) and using standard arguments yields the following set of stress constitutive equations,
S = −πJC−1 + J −2/3 Dev [ %λ̄−1 A + S̃ : ∂ C̃ ],
∂ C̄
∂Ψ (C̃)
(23)
π =−
,
∂ J˜
∂Ψ (C̃)
,
% =
˜
∂ λ̄
together with the imposed constraints,
J˜ =
184
PTdev .
(21)
√
det C
and
˜=
λ̄
p
C̄ : A.
(24)
Further, in Eq. (23)1 ,
S̃ := 2
∂Ψ (C̃)
,
∂ C̃
(25)
185
defines the generalised stress S̃, i.e. the hyperelastic constitutive equation using the generalised right Cauchy-
186
Green stretch C̃.
187
188
Firstly, the abstract pressure Lagrange multiplier π in Eq. (23)1 is related to the physically based mean
pressure p̄ in the Cauchy sense, defined in the Lagrangean setting as,
p̄ := − 31 J −1 (S : C),
9
(26)
189
using the trace-less property of the deviator. That is, for arbitrary Lagrangean tensor T,
Dev [ T ] : C = 0,
190
191
⇒
π = p̄.
(27)
Secondly, the temporary Lagrange multiplier % is related to the physically based deviatoric fibre stress in
the Kirchhoff sense, defined in the Lagrangean setting as,
τ + := Dev [ S ] : λ−2 CAC.
192
(28)
To this end, taking the deviator of the left- and right hand-sides of Eq. (23)1 yields,
Dev [ S ] = %λ̄−1 J −2/3 Dev [ A ] + J −2/3 Dev [ S̃ :
P2dev
=
∂ C̃
].
∂ C̄
(29)
Pdev of the deviatoric operator and the orthogonality property
193
where the idempotent property
194
Dev [ C−1 ] = 0 are used. Further we collect the identities,
Dev [ A ] : CAC = 32 λ4 ,
and
PTdev : CAC = CDev [ A ]C,
(30)
where the dyadic nature A = M ⊗ M is used. Proceeding, we introduce the short-hand notation,
#
"
T
−2/3 ∂ C̃
˜
and
Iso [ T ] := iso : T,
iso := J
∂ C̄
P̃
P̃
P̃
where the identity B : AT
T:
195
196
T
iso
:N=N:
: C = (
P̃iso : T = (P̃iso : T) : N := Iso [ T ] : N.
A : B) : C for the transpose of a fourth-order tensor A is used.
Introducing Equations (29), (30) and (31) in Eq. (28) yields,
τ + = 32 %λ̄ + λ−2 r2/3 Iso [ S̃ ] : CDev [ A ]C,
197
(31)
˜
r := J/J.
(32)
Finally, by the novel fundamental orthogonality property Eq. (33)1 (Appendix A, Theorem 1), Eq. (32) yields,
Iso [ S̃ ] : CDev [ A ]C = 0
⇒
% = 32 τ + λ̄−1 .
(33)
198
In words, Eq. (33)1 says that the part denoted Tiso = Iso [ T ] of any Lagrangean tensor T is trivial in the
199
actual fibre direction, by construction (from C̃ defined by Eqs. (14)). Using Eqs. (27)2 and (33)2 , the stress
200
constitutive equations can be written down on a form with physically motivated phenomenological response
201
functions, for the mean pressure, the deviatoric axial fibre stress and for the deviatoric ground substance
202
stress, p̄, τ + and Dev [ Iso [ S̃ ] ] respectively:
S = −p̄JC−1 + J −2/3 Dev [ S̄ ],
∂Ψ (C̃)
,
p̄
=−
∂ J˜
2 ∂Ψ (C̃)
,
τ + = 3 λ̄
˜
∂ λ̄
10
(34a)
203
where the generalised fictitious stress S̄ in the last term of Eq: (34a)1 is determined as,
S̄ := 23 τ + λ̄−2 A + J˜2/3 Iso [ S̃ ],
where Iso [ • ] :=
(34b)
P̃iso : [•] is endowed with property Eq. (33)1, see Theorem 1 in Appendix A. In Eq. (34b)
S̃ is the generalised stress tensor3 defined by Eq. (25). The new orthogonal decomposition of the fictitious
stress response Eq. (34b) into fibre extensional and ground substance stresses is another direct consequence
of the use and construction of C̃ as function of C̄, see Eqs. (14) and Appendix A for further details. The
generalised fictitious stress Eq. (34b) may thus be rewritten in decomposed form as,
S̄ext := 32 τ + λ̄−2 A
S̄ = S̄ext + S̄iso ,
and S̄iso := J˜2/3 Iso [ S̃ ],
(35a)
where S̄iso : CDev [ A ]C = 0.
204
where J˜2/3 Iso [ • ] is the auxiliary notation for the transpose of the tensor gradient ∂C̄ C̃, see Eq. (31). For
205
important implications confer Eqs. (34) with [32, Eqs. (6.90) and (6.91)], and see Remarks 3a and 3b at the
206
end of this section.
207
Thus, we have shown that Eqs. (14) enables the volumetric - deviatoric form Eq. (34a)1 of the stress con-
208
stitutive equation for a transversely isotropic finite hyperelastic solid, where the deviatoric fictitious stresses
209
Eq. (34b) are orthogonally decomposed into a fibre extensional and a ground substance part, see Eqs. (35a,b).
210
The concept can be generalised to other classes of anisotropy involving several fibre families.
211
4.1. Combined incompressibility and inextensibility
212
This limiting case is obtained by setting J˜ = λ̃ = 1 in Eq. (14) and in the strain energy ansatz Eq. (17)2 .
213
It implies imposition of the classical constraints J = 1 and λ = 1. Moreover, the generalised Cauchy-Green
214
tensor C̃(C̄, 1, 1; A) transforms into the volume preserving and simply stretch-free tensor C(C̄; A) given by
215
Eqs. (16). Most importantly constitutive Equations (34a)2,3 for p̄ and τ + are removed from the formulation
216
and replaced by scalar Lagrange multiplier fields p = p(x) and τ + = τ + (x), respectively. The remaining
217
equation for the total stress is,
9
−1
S = −pJC
+J
−2/3
Dev [
3
2 λ̄
−2
+
τ A+
Piso : S ],
9
9
Piso
9
"
9
∂C
:=
∂ C̄
9
#T
9
,
9
S := 2
∂Ψ (C)
9
∂C
,
(36)
Piso : S is given
9
9
218
where S is the fibre tension-less work-performing stress tensor. The expression for S̄iso =
219
by Eq. (38)3 below, see also Eq. (A.11). In computational mechanics it is custom to keep factors J and J −2/3
220
in front of the volumetric and deviatoric terms of Eq. (36) although J = 1. We extend this custom and keep
3 The
pressure and fibre tension constitutive equations can be given in terms of the generalised stress S̃ by aid of the standard
volumetric operator P̃vol and a new extensional operator corresponding to P̃iso respectively.
11
221
λ understanding that it also is trivial. By definition the work-performing stress response is completely decou-
222
pled from the reaction stresses. Moreover, in our formulation the reaction stresses are mutually orthogonal,
223
and the work-performing stress is orthogonal with respect to both stress reactions.
224
4.2. Strong anisotropy combined with a slight compressibility
For computational purposes the limiting form described in Section 4.1 is replaced by a regularised for-
225
226
mulation reinstating the constitutive stress response functions Eqs. (34a)2,3 for p̄ and τ + , respectively.
The fictitious ground substance stress response function S̄iso is in the same form as the extra stress of an
227
229
incompressible and inextensible material, see Eq. (36) above. As for the fully constrained case we evaluate
9
˜ , J)|
˜ ˜
the generalised stretch tensor as C̃(C̄, λ̄
= C. The appropriate form of the resulting strain energy
230
ansatz Eq. (17)2 is the sum of a volumetric contribution Ψvol describing the overall compressibility, a contri-
231
bution describing the fibre extensibility Ψext , and a contribution describing the (isotropic) ground substance
232
response Ψiso ,
228
J=λ̃=1
˜ I˜4 , I 5 ) = Ψvol (J)
˜ + Ψext (I˜4 ) + Ψiso ( I 1 , I 2 ),
Ψ ( I 1 , I 2 , J,
9
233
234
235
9
9
9
9
Ψ (3, 3, 1, 1, 1) = 0,
˜ , J}
˜ we compute the fourth
where it is recalled that I˜4 (C̃; A) = λ̃2 . Using the kinematic variables {ϕ, λ̄
˜ 2 . In passing it is noted that I˜ (J,
˜ )| ˜ = λ̄
˜ 2 becomes volume
˜ λ̄
principal invariant I˜4 (C̃; A) as I˜4 = J˜2/3 λ̄
4
J=1
˜ 2 ) is then
preserving and that the standard volumetric – isochoric decoupled formulation Ψ | ˜ = Ψ (λ̄
ext J=1
9
236
(37)
ext
9
9
retrieved. The three non-trivial joint principal invariants of C̃|J=
˜ λ̃=1 = C and A are denoted I 1 , I 2 and
9
9
9
9
237
I 5 , respectively. In other words, I 3 = I 4 = 1 meaning that C preserves volume and length in the preferred
238
direction. In order to simplify matters it is henceforth assumed that the shear coupling between the ground
239
substance and the fibres may be neglected, i.e. the dependence on I 5 is neglected.
9
The Hu-Washizu constitutive equation for the total stress Eq. (34a)1 corresponding to the additive strain
energy density ansatz Eq. (37) becomes,
S = −p̄JC−1 + J −2/3 Dev [ S̄ext + S̄iso ],
S̄iso =
Piso
9
(38)
S̄ext = 32 τ + λ̄−2 A,
h9
i
9
9
9
9
: S = λ̄ S − ιA + 2(1 − λ̄−3 ) λ̄−4 (C̄AC̄ : S)A − λ̄−2 12 (SC̄A + AC̄S) ,
9
9
9
ι = 23 λ̄−4 (C̄Dev [ A ]C̄ : S),
Piso is the auxiliary notation for ∂
S := 2
∂Ψiso (C)
9
∂C
,
9
9
C, see also Eq. (A.11). The ground substance
240
where it is recalled that
241
part of the stress, denoted S, is tension-less in the preferred direction, i.e. S :
242
admissible S by Corollary 1 in Appendix A. That is, the ground substance and axial fibre stress responses
243
S̄iso and S̄ext are orthogonal (decoupled).
9
C̄
9
9
12
PisoT : C̄Dev [ A ]C̄ = 0 for any
9
4.2.0.4. Pressure and fibre tension. The Hu-Washizu stress constitutive equations for the pressure and fibre
244
tension Eqs. (34a)2.3 corresponding to the additive strain energy density ansatz Eq. (37) become,
i
∂ h
˜ )) ,
˜ + Ψext (I˜4 (J,
˜ λ̄
p̄ = −
Ψvol (J)
∂ J˜
˜ ))
˜ λ̄
Ψext (I˜4 (J,
τ + = 32 λ̄
.
˜
∂ λ̄
The pressure and fibre tension constitutive equations evidently become coupled.
(39a)
(39b)
The consistently linearised elasticity tensor for the mixed Hu-Washizu formulation is derived in
245
246
Section 7.1, page 20.
247
5. Material model
248
In this section we provide explicit strain energy expressions in Eq. (37), i.e. for the ground substance,
249
˜ respectively.
axial fibre response and for the volumetric contributions; Ψiso ( I 1 , I 2 ), Ψext (I˜4 ) and Ψvol (J),
250
In fact ansatz Eq. (37) generalises the so-called Standard Reinforcing Material (SRM) model originally
251
stated by Triantafyllidis and Aybeyarante [30], and used by Merodio and Ogden, [31]. The volumetric -
252
isochoric decoupled SRM model used here is defined as:
9
9
Definition 1 (Volumetric - isochoric decoupled Standard Reinforcing Material [SRM]).
˜ λ̄, C̄; A) = κΨ̄vol (J)
˜ + µΓΨ̄ext (I¯4 ) + µΨ̄iso (I¯1 ),
ΨSRM (J,
˜ = J˜ − 1 − ln J,
˜
Ψ̄vol (J)
Ψ̄ext (I¯4 ) = 21 (I¯4 − 1)2 ,
Ψ̄iso (I¯1 ) = 12 (I¯1 − 3),
(40)
0
Ψ̄vol (1) = Ψ̄vol
(1) = 0,
00
Ψ̄vol
(1) = 1,
0
Ψ̄ext (1) = Ψ̄ext
(1) = 0,
00
Ψ̄ext
(1) = 1,
Ψ̄iso (3) = 0,
∂C Ψ̄iso (3) = 0,
253
where κ is the bulk modulus and µ is the shear modulus in the reference configuration with units [N/m2 ] in
254
the SI-system, respectively. The dimensionless parameter Γ ≥ 0 [-] controls the strength of the anisotropy.
255
The prime and double prime denote the first and second derivative of the function with respect to its argument.
256
257
2
Using the same parameters as in Definition 1, a simple Generalised SRM form of Eq. (37) is proposed as:
Proposition 1 (Generalised Standard Reinforcing Material [GSRM]).
9
˜ , C;
˜ )) + µΨ̄ ( 9
˜ λ̄
˜ + µΓΨ̄ext (I˜4 (J,
˜ λ̄
ΨGSRM (J,
A) = κΨ̄vol (J)
iso I 1 ),
˜ = J˜ − 1 − ln J,
˜
Ψ̄vol (J)
Ψ̄ext (I˜4 ) = 12 (I˜4 − 1)2 ,
9
0
Ψ̄vol (1) = Ψ̄vol
(1) = 0,
0
Ψ̄ext (1) = Ψ̄ext
(1) = 0,
9
Ψ̄iso ( I 1 ) = 12 ( I 1 − 3),
Ψ̄iso (3) = 0,
13
00
Ψ̄vol
(1) = 1,
00
Ψ̄ext
(1) = 1,
∂C Ψ̄iso (3) = 0.
˜2,
I˜4 = J˜2/3 λ̄
(41)
258
259
260
2
In order to make it simple we model the ground substance by an incompressible and simply inextensible
neo-Hookean material.
261
Classically the neo-Hookean material is considered as incompressible. Analytically due to the subsidiary
262
condition, J = 1, its constitutive stress equation cannot respond to volume change, i.e. C is subject to
263
J = 1. The standard computational formulation achieves this by formulating the constitutive stress response
264
using the volume-preserving measure, C̄, with det C̄ ≡ 1 by construction. Generalising by adding a simple
265
inextensibility we also remove its constitutive stress response in axial tension in the preferred direction. Using
266
the volume preserving and simply stretch free deformation measure C with det C ≡ 1 and C : A ≡ 1 by
267
construction, recall Eqs. (16), we achieve this goal.
9
268
9
9
The incompressible and inextensible neo-Hookean ground substance contribution is completely specified
9
269
by the strain energy function, µΨ̄iso ( I 1 ) in Eq. (41), i.e.
9
Ψiso ( I 1 ) =
270
µ 9
( I 1 − 3),
2
9
9
I 1 := C : I.
(42)
It yields the simple generalised fictitious stress,
9
9
S=2
271
∂Ψiso ( I 1 )
9
∂C
= µI.
(43)
The complete set of Hu-Washizu stress constitutive equations specialised for the GSRM material become,
S = −p̄JC−1 + J −2/3 Dev [ S̄ext + S̄iso ],
0
0
˜ − µΓ 2 J˜−1 I˜4 Ψ̄ext
(44)
p̄
= −κΨ̄vol
(J)
(I˜4 ),
3
τ + = µΓ 4 J˜2/3 λ̄2 Ψ̄ 0 (I˜4 ) q,
ext
3
9
S̄ext = 32 τ + λ̄−2 A and
S̄iso = µ Iso [ I ],
(45)
272
.
.
˜ /λ̄ =
wherein q = λ̄
1. Inserting Eq. (44)2 into Eq. (44)1 we encounter the ratio J/J˜ = 1. The latter simpli-
273
fication can also be made inserting Eq. (44)3 into Eq. (45)1 . The explicit expression for Iso [ I ] :=
274
is given by Eq. (38)3 with S = I. It is readily shown that the total stress vanishes, i.e. S = 0, in the reference
275
configuration C = I. The sum of the fibre and ground substance stresses is evidently trace-less by construc-
276
tion, Dev [ S̄ext + S̄iso ] : C = 0. Finally, by Corollary 1 in Appendix A the fictitious ground substance stress
277
S̄iso has no contribution in the fibre direction, S̄iso : CDev [ A ]C = 0. The material part of the neo-Hookean
278
generalised fictitious elasticity tensor is trivial,
9
Piso : [I]
9
9
C = 4J −4/3 ∂∂ CΨiso2 = O,
2
9
9
14
(46)
O is a fourth order zero tensor.
279
where
Equations Eq. (44) and Eq. (45) together with Eq. (46) define the
280
Hu-Washizu mixed form total stress Eq. (34a)1 and the consistently linearised elasticity tensor for the nearly
281
incompressible and nearly inextensible GSRM model, see Eq. (72). These quantities are later used in the
282
weak form shown in Eq. (71a).
Further the linearised forms of Eqs. (44)2,3 are needed in the weak formulation, see Eq. (84b). To this
end we obtain,
∂ 2 Ψvol
∂ 2 Ψext
00
00
0
= κΨ̄vol
= µΓ 29 J˜−2 I˜4 2I˜4 Ψ̄ext
,
− Ψ̄ext
,
∂ J˜2
∂ J˜2
q
(47)
∂ 2 Ψext
∂ 2 Ψext
∂ 2 Ψext
0
0
00
00
=
= µΓ 43 J˜−2/3 I˜4 (Ψ̄ext
= µΓ2J˜2/3 Ψ̄ext
+ I˜4 Ψ̄ext
+ 2I˜4 Ψ̄ext
,
),
˜
˜ ∂ J˜
˜2
˜ λ̄
∂ J∂
∂ λ̄
∂ λ̄
283
which are used in the weak forms Eqs. (71g), Eqs. (71e) and (71h).
284
Proposition 2 (Modified Holzapfel-Gasser-Ogden [HGO] model).
285
In soft tissue biomechanics the HGO model [24, (2000)] is frequently used. It is a member of the SRM
286
class of models [31]. Its ground substance response is neo-Hookean. It is given for computational purposes
287
in the volumetric - deviatoric decoupled form, by using the isochoric invariants I¯1 and I¯4 in place of I1 and
288
I4 . The constitutive equation for fibre extension is of the exponential Fung type, active in tension-only.
289
Simply by changing the fibre extension function Ψ̄ext (I˜4 ) in the nearly incompressible and nearly inexten-
290
sible GSRM model Eq. (41) it becomes a modified HGO model with fully decoupled ground substance and
291
axial fibre responses. Moreover, it fulfils the formal requirement to model near inextensibile fibres, λ → 1,
292
independent of the dilatation magnitude.
293
For a modified HGO type model use,
1
exp[k2 (I˜4 − 1)2 ], λ̃ ≥ 1,
2k
˜
2
Ψ̄ext (I4 ) =
0,
λ̃ < 1.
(48)
294
In Eq. (48) k2 is a dimensionless parameter [-]. Further, by setting µΓ = k1 in Eq. (44) the specification of
295
the proposed modified HGO model is completed. Finally, when the fibres become inactive in compression,
296
9
the ground substance part should be degenerated to an isotropic compressible neo-Hooke model ( I 1 → I¯1 ),
297
i.e. S̄ → I for λ̃ < 1. 2
298
Remark 3 (On the standard volumetric–isochoric decoupled theory). It is observed that the standard rep-
299
resentation with decoupled free energy in the additive form of Eq. (49) displays a number of anomalies
300
when used in situations with strong anisotropy in the preferred direction, see [26, 27, 28, 29]. We address
301
and propose remedies to two issues (a) ground substance contribution in the preferred direction in the nearly
302
inextensible case, and (b) inability to enforce inextensibility at non-trivial dilatations.
15
303
(a) Ground substance contribution in the preferred direction. It is noteworthy to compare the deviatoric
304
fibre stress response obtained using the proposed formulation with a common standard characterisation
305
used for computational purposes. To that end we use the standard characterisation of a transversely
306
isotropic hyperelastic material with decoupled free energy, according to Holzapfel [32, Eq. (6.214)]
307
for a transversely isotropic solid;
Ψ = Ψ (C; A) = Ψvol (J) + Ψisoch (I¯1 , I¯2 , I¯4 , I¯5 ),
(49)
where Ψvol and Ψisoch are the volumetric and isochoric contributions to the hyperelastic response. The
principal invariants I¯i , i = 1, 2, 4, 5 are the non-trivial joint invariants of (note!) C̄ and A. The
expression for the standard fictitious stress, denoted S̄ [32, Eq. (6.215)] is,
S̄ =
∂Ψisoch (I¯1 , I¯2 , I¯4 , I¯5 )
∂ C̄
(50)
= γ̄1 I + γ̄2 C̄ + γ̄4 A + γ̄5 (M ⊗ C̄M + M C̄ ⊗ M ),
308
with the response coefficients,
γ̄i = 2
∂Ψisoch
,
∂ I¯i
i = 1, 2, 4, 5.
309
Denoting the deviatoric fibre stress in the Kirchhoff sense predicted by S̄ given by Eq. (50) τ̄ + defined
310
as,
τ̄ + := (dev[τ̄ ] : a) = (τ̄ : dev[a]) = J 2/3 λ̄−2 (S̄ : C̄Dev [ A ]C̄).
311
(51)
Inserting Eq. (50)2 in Eq. (51) yields,
J −2/3 23 τ̄ + = 2I¯4
∂Ψisoch
+ (γ̄1 + γ̄2 I¯4 + 2γ̄5 I¯4 )I¯4 .
∂ I¯4
(52)
312
Constructing the corresponding deviatoric stress contribution denoted τ̃ + using S̄ as defined by the
313
generalised formulation Eq. (34b) yields on the other hand,
∂Ψ (C̃; A)
J −2/3 32 τ̃ + = 2I˜4
,
∂ I˜4
314
(53)
where I˜4 = C̃ : A and where we have used the relaxed constraint Eq. (24)2 and the identity,
˜ ∂Ψ = 2I˜ ∂Ψ .
λ̄
4
˜
∂ I˜4
∂ λ̄
315
According to our understanding of the phenomenological theory for anisotropic hyperelastic materials
316
[11] it is doubtful to have a stress contribution from the ground substance in the preferred (fibre)
317
direction, as predicted by the second term in the standard formulation Eq. (52). There is no length-scale
318
or volume-fractions in Spencer’s theory. There are only fibres in the fibre direction. It is a homogenised
16
319
description. The matter becomes evident in the further fibre - ground substance decomposed reduced
320
form,
Ψisoch = Ψaniso (I¯4 ) + Ψ̂isoch (I¯1 )
(54)
321
frequently used in practice for the nearly inextensibile case [24, cf. Eq. (63) adapted for transverse
322
isotropy].
323
As shown by Eq. (53) this flaw is corrected using the fibre stress predicted by Eqs. (34) based on the
324
generalised Cauchy-Green tensor C̃, i.e. Eqs. (14).
325
(b) Inability to enforce inextensibility at non-trivial dilatations. Note that the materials in the classs
326
defined by Eq. (49) are compressible. By definition compressibility may imply volume changes J 6= 1.
327
Recall that, inextensibility in the preferred direction is defined by the simple kinematic constraint λ = 1
328
or as I4 (C; A) = 1. The term Ψaniso (I¯4 ) in the further decoupled strain energy form Eq. (54) is
329
intended to play an analogous role as Ψvol (J) in the volumetric – isochoric decoupled theory. We have
330
resolved the decoupling issue in the anisotropic case. The argument in the contribution Ψaniso (·) is
331
I¯4 := J −2/3 I4 6= I4 for J 6= 1. Hence, at J 6= 1 this isochoric strain energy contribution cannot
332
enforce the inextensibility constraint I4 = 1 in the penalty sense no matter how large extensional
334
stiffness is used. Again the remedy is to use the generalised Cauchy-Green tensor C̃, i.e. Eqs. (14). Its
.
fourth principal invariant with A is I˜4 (C̃; A) = λ̃2 is point-wise equal to the fibre stretch, i.e. λ̃ = λ
335
using the Hu-Washizu formulation proposed. The remedy is employed in the proposed Generalised
336
Standard Reinforced Material model Proposition 1 on page 13.
333
337
338
2
To summarise, Equations (34a) and (34b) provide a general form of the stress constitutive equation for a
340
hyperelastic transversely isotropic solid with physically motivated pressure, deviatoric fibre-stress and devi˜ as independent auxiliary kinematic variables.
atoric ground substance response functions with J˜ and λ̄
341
6. The model boundary value problem
339
342
We consider the problem of determining an admissible static deformation x = ϕ(X) for a material
343
body B ⊂ R3 subject to prescribed deformations, ϕ(X) = ϕ̄(X) on the part of the boundary ∂ϕ B and
344
loaded by distributed nominal surface loads P = P̄ (X) on the remainder of the boundary, ∂σ B. That is,
345
∂B = ∂ϕ B ∪ ∂σ B and ∂ϕ B ∩ ∂σ B = ∅. Here, P = PN is the traction vector corresponding to the first
346
Piola-Kirchhoff stress or engineering stress tensor P := FS.
347
In other words, we are looking for a deformation ϕ in the set of admissible configurations,
C = ϕ : B → R3 | det[∇X ϕ] > 0
17
in B
and
ϕ|∂ϕ B = ϕ̄ .
(55)
348
Here N = N (X) denotes the normal to the loaded surface ∂σ B at X.
The model boundary value problem of the mixed type is,
∇X · P + ρ0 B = 0,
PN = P̄
ϕ(X) = ϕ̄(X),
in
B,
(56a)
on
∂σ B ,
(56b)
on
∂ϕ B.
(56c)
349
in which B = b ◦ ϕ(X) is the Lagrangean form of the bodyforce per unit mass, and where ρ0 = ρ0 (X) is
350
the mass density in the reference configuration.
351
7. The weak formulation
352
The weak form corresponding to the boundary value problem (56a)-(56c), is formulated in a standard
353
fashion using test functions δU forming the tangent space4 to the set of admissible variations, denoted TX C ,
TX C = δU : B → R3 | δU (X) = 0 for
X ∈ ∂ϕ B .
(57)
354
The test functions are also called virtual displacements. The functions δu(x) = δU ◦ϕ−1 (x) are superposed
355
in the spatial configuration S = ϕ(B), in the sense ϕ = ϕ + δU , and do not violate the prescribed
356
boundary conditions of place (56c).
357
From a geometrical point of view TX C is the pull-back description of the tangent space to the set of
358
admissible configurations (55) C at the (trial) solution ϕ ∈ C . For a displacement based formulation, it is
359
standard to take TX C = V with,
V = δU ∈ [H 1 (B)]3 : δU = 0|∂ϕ B .
360
361
The weak form of the equilibrium equations is constructed by taking the inner product of equation (56a) with
a test function δU ∈ TX C and integrate it over the material configuration,
Z
Z
GU (ϕ)[δU ] :=
(∇X · P) · δU dV +
δU · ρ0 B dV = 0.
B
(58)
B
def
362
Using the strain equivalence (14)6 in terms of the generalised Green-Lagrange strain Ẽ = 1/2(C̃ − I),
363
integrating the first integral by parts and using the Gauss divergence theorem the weak formulation of me-
364
chanical equilibrium (58), i.e. the principle of virtual work, is obtained as:
365
Find an admissible deformation ϕ ∈ C = V + ϕ̄ (see Eq. (55)) such that,
Z
GU (ϕ)[δU ] = 0
where
GU (ϕ)[δU ] =
DU E(ϕ)[δU ] : S dV + Ḡ(ϕ)[δU ],
B
∀δU ∈ V ,
(59)
4a
differential manifold
18
366
where the bracket notation GU (ϕ)[•] is used to indicate linearity in the indicated argument. In the first term
367
in (59) we have used the relation between the first and second Piola-Kirchhoff stress tensors S = F−1 P and
368
the symmetry of S. Here,
DU E(ϕ)[δU ] :=
d
= 21 (FT ∇X δU + ∇T
E(ϕ + δU )
X δU F),
d
=0
(60)
369
is the directional derivative of the Green-Lagrange strain E = 1/2(C − I). The second term Ḡ(ϕ)[δU ] in
370
(59) is the standard external forcing,
Z
Ḡ(ϕ)[δU ] :=
B
Z
δU · ρ0 B dV +
δU · P̄ dA,
(61)
∂σ B
371
where surface integral is simplified using the Neumann boundary condition (56b) and the fact that δU = 0
372
vanishes on ∂ϕ B. Internal constraints such as incompressibility and/or in-extensibility are not yet taken
373
into account in (59). The stress constitutive equations in the strong Hu-Washizu formulation are given by
374
375
376
Equations (34a). The corresponding weak Hu-Washizu setting for the transversely isotropic case is a five˜ , π, J}
˜ 5 . The weak forms of the
field mixed formulation in the set of primary field variables Φ = {U , %, λ̄
generalised volume constraint (24)1 and the generalised extensibility constraint (24)2 become,
Z
Z
˜ )[δ%] =
˜ ) δ% dV = 0 ∀δπ, δ% ∈ Q, (62)
˜
˜ δπ dV = 0,
GJ˜(U , J)[δπ]
=
(J − J)
Gλ̄˜ (U , λ̄
(λ̄− λ̄
B
377
B
respectively. Here δπ and δ% are taken to be square integrable test functions in the set,
Q = q(x) : S → R | q ∈ L2 (S ) .
There are no boundary conditions associated with the pressure and axial stress test functions. The weak forms
of the pressure and axial stress constitutive equations Eq. (34a)2 and Eq. (34a)3 become,
378
379
380
˜ , J,
˜ π)[δ J]
˜ =
Gπ (U , λ̄
Z ˜ , J,
˜] =
˜ %)[δ λ̄
G% (U , λ̄
Z π+
B
B
∂Ψ (C̃) ˜
δ J dV = 0,
∂ J˜
˜ ∈ Q,
˜ δ λ̄
∀δ J,
(63a,b)
∂Ψ (C̃) ˜
%−
δ λ̄ dV = 0.
˜
∂ λ̄
˜ > 0 are dilatation and axial stretch square integrable test functions in Q.
Here δ J˜ > 0 and δ λ̄
The weak Hu-Washizu formulation of mechanical equilibrium for a transversely isotropic hyperelastic
solid is formulated as follows.
5 which
yield a symmetric Hessian.
19
381
Find Φ ∈ C × Q 4 such that,
˜ , J,
˜
G
(U
,
λ̄
π,
%)[δU
]
U
˜
G
(U
,
λ̄
)[δ%]
˜
λ̄
˜ , J,
˜]
˜ %)[δ λ̄
GHW (Φ)[δΦ] =
= 0,
G% (U , λ̄
˜
GJ˜(U , J)[δπ]
˜
˜ π)[δ J]
˜
Gπ (U , λ̄, J,
(64)
382
def
˜ , δπ, δ J)
˜ ∈ V × Q 4 . Here the orthogonal additive volumetric - deviatoric defor all δΦ = (δU , δ%, δ λ̄
383
composition of the stress Eq. (34a)1 comes into play. The deviatoric generalised fictitious stress S̄ is further
384
orthogonally decomposed into fibre and isotropic ground substance stresses using Eqs. (35).
385
The five-field Hu-Washizu mechanical equilibrium Equations (64) generalise the Simo, Taylor and Pister
386
formulation for quasi incompressibility [4] to the transversely isotropic case. See also [32, Section 8.6] for the
388
˜ π) three-field formulation for near-incompressibilty only. Henceforth we focus on
details regarding the (ϕ, J,
˜ , %) for the nearly in-extensible case. To that end, we
the generalisation associated with the two extra fields (λ̄
389
invoke the nearly incompressible and nearly inextensible form of the GSRM neo-Hooke material suggested
390
in Proposition 1 on page 13 with the stress constitutive equations given by Eqs. (44) and (45).
391
Remark 4 (Near incompressibility and near in-extensibility). The weak forms (63a,b) are general. They
387
392
393
394
395
396
397
are used for the unconstrained fully coupled case. For the nearly incompressible and nearly in-extensible
˜ )) replaces Ψ (C̃; A) in (63a) and
˜ + Ψ (I˜ (J,
˜ λ̄
case the strain energy form Eq. (37) is used. Then Ψ (J)
vol
ext
4
˜ )) replaces Ψ (C̃; A) in (63b), so that π = π(J,
˜ ) and % = %(J,
˜ ). Finally, when the pressure
˜ λ̄
˜ λ̄
˜ λ̄
Ψext (I˜4 (J,
˜ can be eliminated altogether as independent
and fibre tension constitutive equations are invertible J˜ and λ̄
variables to yield a reduced {U , π, %} three-field Hellinger-Reissner type formulation. 2
7.1. Linearised weak form – near incompressibility and near inextensibility
The Hu-Washizu weak mechanical equilibrium conditions Eqs. (64) constitute a non-linear set of equations in terms of the set of independent variables Φ. Therefore (consistently) linearised variational equilibrium equations have to be derived as a basis for a numerical method of the Newton-Raphson type. These
linearised equations are obtained using the “perturbed” configuration,
ϕ (X) = ϕ(X) + ∆U (X),
π = π + ∆π
398
399
˜ = λ̄
˜ + ∆λ̄
˜
λ̄
% = % + ∆%,
˜
and J˜ = J˜ + ∆J.
(65)
(66)
def
˜ , ∆π, ∆J}
˜ are small increments that are added to the last known admissible
Here ∆Φ = {∆U , ∆τ, ∆λ̄
˜ , π, J}.
˜ To a first order in terms of these increments a Taylor series expansion of
configuration {ϕ, τ, λ̄
20
400
Equations (64) corresponds to,
D G
D% GU
0
U U
DU G˜
0
Dλ̄˜ Gλ̄˜
λ̄
0
D% G% Dλ̄˜ G%
DU GJ˜
0
0
0
0
Dλ̄˜ Gπ
Dπ GU
0
0
0
Dπ Gπ
˜
˜
0
∆U
GU (U , λ̄, J, π, %)[δU ]
˜
0
∆%
G
(U
,
λ̄
)[δ%]
˜
λ̄
˜
˜
˜
˜
=
−
.
DJ˜G%
∆λ̄
G% (U , λ̄, J, %)[δ λ̄]
˜
∆π
DJ˜GJ˜
GJ˜(U , J)[δπ]
˜
˜
˜
˜
DJ˜Gπ
∆J
Gπ (U , λ̄, J, π)[δ J]
(67)
401
where the right hand-side is linear in the argument within bracket, G[•] and where D[•] denotes the directional
402
derivative with respect to the indicated argument. We consider only boundary value problems leading to a
403
symmetric Jacobian. It is well known that this includes certain configurations with pressure loads (56b)
404
although these are non-conservative in general.
405
406
Using the product rule of differential calculus linearisation of the integrand of the mechanical equilibrium
(59) in the Hu-Washizu form yields,
2
DU (DU E(ϕ)[δU ] : S)[∆U ] = DU E(ϕ)[δU ] : DU S + S : DU
U E(ϕ)[δU , ∆U ],
407
(68)
where we have used the abbreviated notation,
DU S
=
C : DU E(ϕ)[∆U ],
C
=
Cπvol + C%dev ,
2
DU
U E(ϕ)[δU , ∆U ]
=
1
2
(69)
∇TX ∆U ∇X δU + ∇TX δU ∇X ∆U .
408
The explicit expressions for the elasticity tensors Cπvol and C%dev are obtained by aid of Equations (73) and
409
(74). A close connection with a linear finite element implementation is prepared using the total or accumu-
410
lated displacement as primary unknown,
U (X) = ϕ(X) − X.
411
(70)
The linearised system (67) is rewritten as,
a[δU , ∆U ]
+b[δU , ∆τ ]
b[δ%, ∆U ]
˜ , ∆τ ]
d[δ λ̄
˜ , J,
˜ π, %)[δU ]
= −GU (U , λ̄
˜ )[δ%]
= −G (U , λ̄
+c[δU , ∆π]
˜]
+d[δ%, ∆λ̄
˜ , ∆λ̄
˜]
+e[δ λ̄
˜
λ̄
c[δπ, ∆U ]
˜]
˜ ∆λ̄
s[δ J,
˜ ∆π]
+g[δ J,
˜ , ∆J]
˜
+s[δ λ̄
=
˜
+g[δπ, ∆J]
˜
GJ˜(U , J)[δπ]
˜ , J,
˜ π)[δ J]
˜
= −Gπ (U , λ̄
˜ ∆J]
˜
+h[δ J,
˜ , J,
˜]
˜ %)[δ λ̄
G% (U , λ̄
=
(71)
21
where the bi-linear forms on the left hand-side are,
Z
a[δU , ∆U ] =
DU E(ϕ)[δU ] : (Cπvol + C%dev ) : DU E(ϕ)[∆U ] dV
B
Z
+
∇X δu0 : [(∇X ∆U ) S] dV ,
B
Z
b[δU , ∆%] =
λ̄−1 Ādev : DU E(ϕ)[δU ]∆% dV ,
B
Z
c[δU , ∆π] = − JC−1 : DU E(ϕ)[δU ]∆π dV ,
Z B
˜] =
˜ δ% dV,
d[δ%, ∆λ̄
∆λ̄
B
Z
∂ 2 Ψext ˜ ˜
˜
˜
∆λ̄δ λ̄ dV,
e[δ λ̄, ∆λ̄] =
˜2
B ∂ λ̄
Z
˜ =
˜ dV,
g[δπ, ∆J]
∆Jδπ
B
Z
∂ 2 (Ψvol + Ψext ) ˜ ˜
˜
˜
h[δ J, ∆J] =
∆Jδ J dV ,
∂ J˜2
B
Z
∂ 2 Ψext ˜ ˜
˜ , ∆J]
˜ =
δ λ̄∆J dV ,
s[δ λ̄
˜ ∂ J˜
B ∂ λ̄
(71a)
(71b)
(71c)
(71d)
(71e)
(71f)
(71g)
(71h)
412
where Ādev = J −2/3 Dev [ A ] and where S̄iso is the isotropic part of the fictitious ground substance stress
413
given by Eqs. (44),(45) and Eq. (46). The linearised system Eq. (71) is now amenable for discretization.
414
7.2. Elasticity tensor
415
In this section we derive the consistently linearised elasticity tensor corresponding to the stress equation
416
Eq. (38), setting τ + = 32 %λ̄ and p̄ = π. The definition of the volumetric – deviatoric decomposed elasticity
417
tensor for a nearly incompressible hyperelastic solid is recalled from Holzapfel [32, Ch. 6.6, p. 254] as,
∂ Ψ
∂S
C := 4 ∂C
2 = 2 ∂C = Cvol + Cdev .
2
(72)
C
C
419
We can adapt the standard deviatoric and volumetric elasticity tensors dev and vol for the mixed
˜ , π, J)
˜ five-field Hu-Washizu formulation by regarding π and % as independent (constant) Lagrange
(U , %, λ̄
420
multiplier variables.
418
C C
C
The properly adapted elasticity tensor = πvol + %dev is obtained with,
∂Svol π
= −Jπ(C−1 ⊗ C−1 − 2J C−1 ),
vol := 2
∂C π
h
i
∂Sdev %
:=
2
= 23 (J −2/3 S̄ : C) − (C−1 ⊗ S̄dev + S̄dev ⊗ C−1 ) +
dev
∂C %
C
C
I
P̃
(73)
Pdev : M̄% : PTdev .
(74)
421
Here Svol := −πJC−1 and S̄dev := J −2/3 Dev [ S̄ ] is given by Eqs. (38)2 and (38)3 . Due to the orthogonal
422
projection Eq. (35) of the fictitious stress S̄ = S̄ext + S̄iso also the elasticity tensor
423
decomposed as,
C%dev = C%ext + Ciso.
22
C%dev becomes additively
(75)
The definitions of the fourth order tensor operators
P̃ and I
C−1
in Eqs. (73) and (74) are as usual6
[32, cf. Eqs. (6.164),(6.165) and (6.170)],
P̃ := I
C−1
I
424
where [
425
defined as,
C−1
− 13 C−1 ⊗ C−1 ,
and
I
C−1
:= −
∂C−1
= sym{C−1 C−1 },
∂C
−1 −1
−1 −1
]IJKL = 12 (CIK
CJL + CIL
CJK ). The generalised fictitious elasticity tensor
426
2
(77)
Its deviatoric projection is expanded as,
Pdev : M̄% : PTdev = M̄% − 31
427
M̄% in Eq. (74) is
M̄% := 2J −4/3 ∂∂C̄S̄ % = 4J −4/3 ∂∂C̄Ψ2 %.
(76)
M̄% + M̄% : C ⊗ C−1 + 91 (C : M̄% : C) C−1 ⊗ C−1 . (78)
Again the generalised fictitious elasticity tensor M̄% is additively decomposed into a fibre extensional and
C−1 ⊗ C :
428
a ground substance part by introducing the orthogonal projection S̄ = S̄ext + S̄iso Eq. (35) into Eq. (77)
429
yielding,
M̄% = M̄%ext + M̄iso,
430
M̄%ext := 2J −4/3 ∂∂S̄C̄ext % and M̄iso := 2J −4/3 ∂∂S̄C̄iso .
(79)
We are now set to derive the ground substance elasticities Ciso and the fibre extensional elasticities, C%ext .
7.2.0.5. Ground substance elasticities,
Ciso.
Using the definitions (74) and Eq. (79)3 the ground substance
elasticities are obtained as,
2 −2/3
(S̄iso : C) − (C−1 ⊗ S̄+iso + S̄+iso ⊗ C−1 ) +
iso := 3 J
" 9 #T
2
9
9
9
9
−4/3 ∂ C
: S + P iso : C : P T
iso := 2J
iso .
∂ C̄2
C
P̃
Pdev : M̄iso : PTdev .
M̄
(80a)
(80b)
9
9
431
T
is the tensor gradient ∂C̄ C.
where the compact notation S̄+iso = Dev [ S̄iso ] is used in Eq. (80a) and where P iso
432
The fictitious stress S and the material part of the ground substance generalised fictitious elasticity tensor C
433
are computed with the strain energy ansatz Eq. (37) as,
9
9
9
S := 2
∂Ψiso
9
∂C
,
9
C := 4J
−4/3 ∂
2
Ψiso
9
∂ C2
.
(80c)
The second order tensor gradient in Eq. (80b) is finally expanded as,
" 9 #T
9
9
∂2C
−3
−1 1 9
1 −1
3 −4 9
1 −2 9
(
S
⊗
A
+
A
⊗
S)
+
λ̄
(8
λ̄
−
1)
λ̄
S
:
C̄A
C̄
−
λ̄
S
:
C̄
A⊗A
:
S
=
λ̄
2
2
2
2
∂ C̄2
9
9
9
9
+ (1 − 4λ̄−3 )λ̄−3 12 (SC̄A + AC̄S) ⊗ A + A ⊗ (SC̄A + AC̄S)
9
9
+ (λ̄−3 − 1)λ̄−1 A S + S A .
6 In
[32] the tensor product (A B) : T denotes the symmetric part of the tensor product (A B) : T = ATBT .
23
(80d)
9
434
T
In passisng it may be noted that it corresponds to ∂C̄ P iso
. The final expression for the ground substance
435
elasticities is obtained by inserting Eqs. (80) and (38)3 into Eq. (74).
7.2.0.6. Fibre extensional elasticities,
+
with Eq. (38)2 , setting τ =
2 −2/3
3J
= 32 %λ̄
2
3 %λ̄,
C%ext.
The first term on the right hand-side of Eq. (74) is evaluated
as,
P̃ − (C−1 ⊗ Dev [ S̄ext ] + Dev [ S̄ext ] ⊗ C−1)
h
(S̄ext : C)
I
C−1
i
− 13 C−1 ⊗ C−1 − λ̄−2 (C−1 ⊗ Ādev + Ādev ⊗ C−1 )
(81)
436
where Ādev = J −2/3 Dev [ A ]. The second term in the right hand-side of Eq. (74) is evaluated using the
437
definition Eq. (79)2 and Eq. (38)2 with τ + = 23 %λ̄ as,
Pdev : C̄%ext : PTdev = −%λ̄−3Ādev ⊗ Ādev
438
Summing up Eqs. (81) and (82) yields the fibre extensional elasticity tensor for the mixed method,
C%ext = 23 %λ̄ I
439
(82)
C−1
− 13 C−1 ⊗ C−1 − λ̄−2 (C−1 ⊗ Ādev + Ādev ⊗ C−1 ) − 23 λ̄−4 Ādev ⊗ Ādev . (83)
7.3. Volumetric and extensional material stiffnesses
Linearised forms of the pressure and axial fibre tension Hu-Washizu constitutive equations Eqs. (39a,b)
˜ are derived in this section. The arguments and composed form of the strain energy
with respect to J˜ and λ̄
˜ )) are suppressed for transparency,
˜ and Ψext (I˜4 (J,
˜ λ̄
contributions Ψvol (J)
κ := −
440
∂ 2 Ψvol
,
∂ J˜2
σ1 :=
∂ 2 Ψext
,
˜
˜ λ̄
∂ J∂
σ2 :=
∂ 2 Ψext
,
˜2
∂ λ̄
σ3 := −
yields the sought linearised system and the condition for its inverse in the form,
˜
˜
κ
+
σ
σ
∆
J
∆π
∆
J
σ2
3
1
1
det A6=0
=
=
=⇒
det A
˜
˜
∆%
∆λ̄
σ1
σ2
∆λ̄
−σ1
|
{z
}
∂ 2 Ψext
,
∂ J˜2
(84a)
∆π
, (84b)
κ + σ3 ∆%
−σ1
A
441
where det A = (κ + σ3 )σ2 − (σ1 )2 . We anticipate the situation dominated by the volumetric stiffness
442
00
00
Ψvol
Ψext
implying a positive definite system det A > 0 and the reverse situation leading to a negative
443
definite system det A < 0 when (σ1 )2 > (κ + σ3 )σ2 . It is noteworthy that the extensional-volumetric
444
coupling term is of the same order as the axial stiffness, i.e. σ1 ≈ σ2 . Further, the strength of the volumetric
446
00
00
– extensional coupling is essentially governed by the ratio Ψvol
/Ψext
. The standard volumetric – isochoric
˜ ). It can be argued that
decoupled form (with σ = 0) is retrieved by the constitutive postulate Ψ = Ψ (λ̄
447
such a restriction is inconsistent in the presence of a volumetric constitutive contribution Ψvol in the strain
448
energy ansatz, see Remark 3b.
445
1
ext
24
ext
450
Finally, with the weak formulation in mind, provided det A 6= 0 the increments of the dilatation ∆J˜ and
˜ may be eliminated from the linearised weak form. With a discontinuous
the volume preserving stretch ∆λ̄
451
interpolation this elimination may be performed element-wise.
449
452
453
454
In this section we have linearised the stress constitutive equation Eq. (38) regarding π and % as indepen-
C as the sum of the volumetric elasticities Cπvol, the
ground substance elasticities Ciso and the fibre extensional elasticities C%ext . These are given by Eqs. (73),
dent variables. We obtained the total elasticity tensor
455
Eqs. (80) and Eq. (83), respectively. Finally we linearised the Hu-Washizu pressure and axial fibre tension
456
constitutive equations, Eqs. (39) and obtained the corresponding material elasticities Eqs. (84).
457
8. Finite element approximation
458
In this section we present the preliminary mixed finite elements implemented by us in the hp-adaptive
459
code HP3D developed by Demkowicz and associates [36, 37]. In HP3D the formulation is of the Hellinger-
460
Reissner mixed type with respect to the hydrostatic pressure. This code provides automatic hp-adaptivity.
461
We use it in a restricted manner to obtain a sequence of h-refined meshes improving the discretization error
462
measured in the energy norm. It is known that higher-order displacement based elements are more robust
463
against different locking phenomena as opposed to low-order elements. This seem to hold also for the near
464
inextensible cases investigated here.
465
As reference we use the standard h-version Q1 /P02 tri-linear displacement - constant pressure and con-
466
stant dilatation, and the tri-quadratic displacement - linear pressure and linear dilatation Q2 /P12 elements
467
available in the multi-purpose code FEAP [38]. For the moment the fibre stretch and the fibre tension are
468
displacement based in both HP3D and FEAP. Elements with elementwise interpolation of the fibre variables
469
as described below are under development.
470
The finite dimensional displacement space corresponding to the continuous trial- and testspaces V is
472
denoted Vh . The finite dimensional space corresponding to Q used for each of the four auxiliary fields
˜ , π, J}
˜ is denoted Qh . We consider a partition P of a body B into a number of closed subdomains
{%, λ̄
473
Ω̄e = Ωe ∪ ∂Ωe , i.e. hexahedrons. Using standard notation Pk (Ω̄e ) denotes the space of polynomials of
474
total degree ≤ k with dimension dim Pk (Ω̄e ) = 16 (k + 1)(k + 2)(k + 3) and Qk (Ω̄e ) denotes the space of
475
polynomials which are of degree ≤ k in each variable, with dimension dim Qk (Ω̄e ) = (k + 1)3 .
471
476
The reference element is denoted Ω̂ = [0, 1]3 . The bijective mapping φ : Ω̂ → Ω̄e is used to get the func-
477
tion v ∈ [Qk (Ω̄e )]3 as v = φ−1 ◦ v̂ from v̂ ∈ [Qk (Ω̂)]3 on the reference element. Iso-parametric elements
478
are constructed using φ both for element shape and element displacement.
479
With this notation in hand we set up the following finite element spaces,
3
3
Vh = v h ∈ C 0 (Ω̄) : v h |Ω̄e = v̂ h ◦ φ−1 with v̂ h ∈ Qk (Ω̂) ,
25
∀Ω̄e ∈ P ,
(85a)
480
and
Qh =
481
482
qh ∈ L2 (Ω̄) : qh |Ω̄e ∈ Pk−1 (Ω̄e ),
∀Ω̄e ∈ P
(85b)
and construct mixed elements with C 0 -continuous displacement and with discontinuous auxiliary fields
˜ , π, J}
˜ in L . We use the shorthand notation Q /{P 2 , P 2 } to denote the element type, where
{%, λ̄
2
2
Pk−1
k
= Pk−1 × Pk−1
k−1
k−1
˜ } and the second refers to the pair {J,
˜ p}, respectively.
refers to the pair {%, λ̄
483
the first
484
For example, Q1 /{P02 , P02 } denotes a linear displacement - constant pressure-dilatation and constant axial
485
stress-stretch element.
486
The discretized linearised system (71) is written in a matrix form,
K
UU
K%U
0
KπU
0
KU %
0
KU π
0
K%λ̄˜
0
Kλ̄˜ %
Kλ̄˜ λ̄˜
0
0
0
0
0
KJ˜λ̄˜
KJπ
˜
FU
∆U
0
F
∆%
%
˜
=
Fλ̄˜
∆λ̄
Kλ̄˜ J˜
F
∆π
KπJ˜
π
˜
FJ˜
∆J
KJ˜J˜
0
(86)
487
˜ , ∆π, ∆J}
˜
The discontinuous approximation of the increments of the auxiliary primary variables {∆%, ∆λ̄
488
allows them to be condensed out at the element level and a classical displacement based method is recovered
489
for the nearly incompressible and nearly inextensible case.
490
It is known that the limiting saddle point problem is governed by a discrete inf-sup condition. We use
491
2
standard Qk /{Pk−1
} type discretizations of the nearly incompressible case and simply copy the recipe to
492
handle the near inextensibility in the preferred direction.
493
9. Model examples
494
We compare the deformation and stress responses of the GSRM material put forward in Proposition 1 on
495
Page 13 with the volumetric – isochoric version of the SRM material Definition 1 on Page 13 using a couple
496
of examples. Examples 1-3 illustrate the flaw of having a ground substance contribution in the fibre direction
497
approaching near inextensibility. Examples 1 and 2 are corroborated by closed form analytical solutions. The
498
more complicated analytical solution for the GSRM material in pure torsion, Example 2, is original. The
499
final Example 4 illustrates the inability of a standard volumetric-isochoric decoupled compressible strongly
500
anisotropic SRM type of material to predict strongly anisotropic deformations (cf. Vergori [29]) as opposed
501
to the proposed GSRM material.
Example 1 (Isochoric simple tension). The volume preserving right Cauchy-Green tensor for an isochoric
simple tension in the direction of the fibres M , with A = M ⊗ M is
I¯1 = C̄ : I = 2λ̄−1 + λ̄2 .
C̄ = λ̄−1 (I − A) + λ̄2 A,
26
(87)
502
Piso : [I] is obtained by aid of (A.17) and
The expression for the fictitious isotropic stresses S̄iso = µ
9
503
(A.16) in Appendix A. For the simple extension Eq. (87) it is readily shown that the shear correction defined
504
by Eq.(A.16) is zero. Thus, the expression
505
using Eq.(87)2 and Eq. (A.13)2 as,
Piso : [I] includes only the stretch correction which is determined
9
Piso : [I] = λ̄(I − ιA),
ι = 1 − λ̄−3 .
9
(88)
506
The simple extension is assumed to be isochoric, hence J = 1 and λ̄ = λ. The total Kichhoff stress is readily
507
determined as τ = FSFT using Eqs. (44)+(45) and Eq. (88). It becomes,
τGSRM = −p1 + µ2Γ(λ2 − 1)λ2 dev[a].
(89)
508
Equilibrium requires τ : (1 − a) = 0 which yields p = − 31 µ2Γ(λ2 − 1)λ2 . As expected, the GSRM simple
509
tension consists only of the fibre contribution. The stress in isochoric simple tension for the generalised and
510
standard reinforcing models become,
τGSRM = µ2Γ(λ2 − 1)λ2 a
|
{z
}
and
τSRM = µ
fibre contribution
(λ2 − λ−1 )
| {z }
ground substance contribution
2
−1
+ 2Γ(λ2 − 1)λ2 a .
|
{z
}
(90)
fibre contribution
511
The modelling error , e = µ(λ − λ
512
modelling error, |[τ11 ]GSRM − [τ11 ]SRM |/|[τ11 ]SRM | × 100 in percent, and the axial stress for the GSRM and
513
SRM models are shown in Figures 2a and 2b, respectively. The HP3D results are obtained with a Q2 /P1
514
displacement-pressure element while a Q1 /P0 element is used in FEAP.
515
516
), is the ground substance contribution in the SRM model. The
The absence of ground substance contribution follows directly from Corollary 1 in Appendix A and is
exposed in Eq. (89). The fact can be used experimentally to validate the proposed formulation. 2
517
Example 2 (Pure torsion of a solid circular cylinder). We consider the torsion of a solid circular cylinder
518
of radius R = Ro and length L. The fibre reinforcement is in the Lagrangean direction E Z . The La-
519
grangean structural tensor is, A = E Z ⊗ E Z The deformation is given in cylindrical coordinates as r = R,
520
θ = Θ + γZ and z = Z, where γ is the twist in [rad/m]. Pure torsion is isochoric, i.e. J = 1 so that λ̄ = λ
521
and we drop the over-bar in the following. Without loss of generality we may thus consider our model GSRM
522
material as incompressible. The deformation gradient and right Cauchy-Green tensor become,
F = I + K(eθ ⊗ E Z ),
C = I + K(E Θ ⊗ E Z + E Z ⊗ E Θ ) + K 2 (E Z ⊗ E Z ),
I¯1 = 3 + K 2 , (91)
in the normalised circular cylindrical basis {E R , E Θ , E Z } and where K = γ r, is the equivalent local shear.
The r-dependent fibre stretch is, λ2 = 1 + (γr)2 . The expression for the fictitious isotropic extra stresses
S̄iso =
Piso : [I] is analysed by aid of (A.17) and (A.16) in Appendix A. It becomes,
9
Piso : [I] = µλ
µ
9
h
i
I − ιA + 2(1 − λ−3 )(K̂ 2 A − K̂K) ,
ι = (1 − λ−2 + 32 K̂ 2 ) and
27
K = 21 (E Θ ⊗ E Z + E Z ⊗ E Θ ),
K̂ = Kλ−2 .
(92)
7
125
GSRM
SRM
SRM, FEM - HP3D
GSRM, FEM - HP3D
GSRM, FEM - FEAP
SRM, FEM - FEAP
6
100
Γ = 10
Γ = 100
Γ = 10, FEM - HP3D
Γ = 100, FEM - HP3D
Γ = 100, FEM - FEAP
Γ = 10, FEM - FEAP
5
4
3
Γ = 100
75
50
2
25
1
0
1
1,05
1,1
1,15
0
1,2
Γ = 10
1
1.05
1.1
1.15
1.2
Figure 2: Example 1. Isochoric simple tension. Fibres in the tension direction. Comparison between the GSRM-model and a
SRM-model. Neo-Hookean ground substance, µ shear modulus. Dimensionless anisotropy parameter Γ.
(a) Modelling error,
|[τ11 ]GSRM − [τ11 ]SRM |/|[τ11 ]SRM | × 100 [%] using the SRM-model Γ = 10 and 100. (b) Normalised axial stress τ11 /µ versus
stretch λ̄, Γ = 10 and 100, respectively
It is readily shown that l = er , m = λ−1 (Keθ + ez ) and n = λ−1 (−Kez + eθ ), forms the orthogonal spatial fibre triad of unit directions at (r, θ, z). The left Cauchy-Green tensor is obtained as b = FFT
= 1 − (ez ⊗ ez ) + λ2 a wherein λ2 a = K 2 (eθ ⊗ eθ ) + K(eθ ⊗ ez + ez ⊗ eθ ) + (ez ⊗ ez ). Using Eqs. (44)
and Eq. (45) the deviatoric ground substance Kirchhoff stress is obtained as,
+
τiso
= µdev[λb − ιλ3 a + 2(λ−1 − λ−4 )(K 2 a − Kk)],
523
k = Keθ ⊗ eθ + 12 (eθ ⊗ ez + ez ⊗ eθ ). (93)
In the spatial fibre system {l, m, n} we obtain the simple expression,
+
τiso
= µK 2 λ̄−1 12 (l ⊗ l − n ⊗ n) + µK λ̄−4 (m ⊗ n + n ⊗ m).
524
525
(94)
+
Hence, as expected there is no contribution in the spatial fibre direction, i.e. τiso
: a = 0. The torque M and
axial force N for a solid circular cylinder of radius Ro are determined as [39, sect 57. Family 2],
Z Ro
Z Ro
h
ir=Ro
T (Ro ) = 2π
rτθz rdr,
N (Ro ) = π r2 τrr
+π
(2τzz − τrr − τθθ ) rdr.
r=0
0
(95)
0
The cylinder mantle is assumed traction free. Thus, the first term in the expression for the axial force Eq. (95)2
does not contribute. The needed normal stress differences and the shear stress in the normalised spatial
cylindrical system {er , eθ , ez } for the SRM material are obtained as,
τrr − τθθ = −µK 2 (1 + 2ΓK 2 ),
(96a)
τzz − τrr = µK 2 2Γ,
(96b)
τθz = µK(1 + 2ΓK 2 ).
28
(96c)
The corresponding stress expressions for the GSRM material are obtained after some lengthy algebra as,
i
h
3
K2 K2 + 1 2 − 4
2
K
(97a)
+ √
− 2ΓK 4
τrr − τθθ = µ
3
2 K2 + 1
2 (K 2 + 1)
"
#
√
√
K2 K4 K2 + 1 + K2 K2 + 1 + 4
K2
τzz − τrr = −µ
− 2ΓK 2
(97b)
+ √
3
2 K2 + 1
2 (K 2 + 1)
"
#
√
√
K4 K2 + 1 + K2 K2 + 1 − 2 K2 + 2
2
τθz = µK
(97c)
+ 2ΓK .
3
2 (K 2 + 1)
The expression for the torque, Eq. (95)1 , for the SRM material is obtained as,
iso
ext
TSRM (Ro ) = TSRM
(Ro ) + TSRM
(Ro ),
iso
TSRM
(Ro ) = µπRo3
(γRo )
,
2
ext
TSRM
(Ro ) = µπRo3 2Γ
(98)
(γRo )3
.
3
For the GSRM material the corresponding expressions are,
iso
ext
TGSRM (Ro ) = TGSRM
(Ro ) + TGSRM
(Ro )
iso
TGSRM (Ro ) = µπRo
3 −3
x2 + 1
2
(99)
√
log x2 + 1 + x2 + 1 x6 − 3 x4 − 12 x2 − 8 + 14 x4 + 19 x2 + 8
,
3 x3 (x2 + 1)2
ext
ext
TGSRM
(R) = TSRM
(R),
where x = γRo . The axial force expression for the SRM material is obtained as,
iso
ext
NSRM (Ro ) = NSRM
(Ro ) + NSRM
(Ro )
iso
NSRM
(Ro ) = −µ
πRo 2
(γRo )2
4
ext
NSRM
(Ro ) = −Γµ
(100)
πRo 2 1
( 3 (γRo )4 − (γRo )2 ).
4
For the GSRM material the corresponding expressions are,
iso
ext
NGSRM (Ro ) = NGSRM
(Ro ) + NGSRM
(Ro ),
√
iso
NGSRM (Ro ) = −µπRo
2
(101)
x2 + 1 x6 − 3 x4 − 12 x2 − 8 + 11 x4 + 16 x2 + 8
,
x2 (x2 + 1)2
ext
ext
NGSRM
(Ro ) = NSRM
(Ro ),
526
in which x = γRo .
527
Analytical and numerical results for torque versus twist, and for axial force versus twist are shown in
528
Figures (4a,b), space respectively. The SRM and GSRM material parameters are µ = 0.1 [MPa] and Γ = 10.
529
The bulk modulus κ = 1000µ [MPa] is used in the numerical analysis emulating the assumed incompress-
530
ibility in the analytical solution. The difference in the torque and axial force responses between the GSRM
29
Figure 3: Example 2. Pure torsion of a solid circular cylinder. Brick element mesh. Cylinder length L = 400 [mm], radius
Ro = 100 [mm]. Fibres in the axial direction. Total number of elements N el = 1536. Lower end is built-in. Upper end is twisted.
The outer mantle is traction-free.
531
and SRM models grows with increasing twist. The SRM model is, as expected, the stiffer one due to the
532
additional ground substance contribution. The HP3D results are obtained with Q2 /P1 displacement-pressure
533
elements while Q1 /P0 elements are used in FEAP.
30
200000
150000
5000
SRM
GSRM
SRM, FEM - FEAP
GSRM, FEM - FEAP
SRM, FEM - HP3D
GSRM, FEM - HP3D
4000
3000
γ = 10
100000
SRM
GSRM
SRM, FEM - FEAP
GSRM, FEM - FEAP
SRM, FEM - HP3D
GSRM, FEM - HP3D
γ = 10
2000
50000
0
0
1000
0,1
0,2
0,3
0
0
0,4
0,1
0,2
0,3
Figure 4: Example 2. Pure torsion of a solid circular cylinder as defined in Fig. 3. Comparison between the GSRM-model and a SRMmodel. Anisotropy parameter Γ = 10. Neo-Hookean ground substance shear modulus, µ = 0.1 [MPa]. (a) Torque T versus twist γRo
in [rad]. (b) Axial force N versus twist γRo in [rad]
31
0,4
20.8
0.0
Figure 5: Example 3. Pressurisation of a tube. (a) Nominal brick element mesh used in HP3D and FEAP. Length L = 200 [mm],
inner radius Ri = 80 [mm] and outer radius Ro = 100 [mm]. Total number of elements N el = 4096. Four element layers across the
thickness. Lower end is built-in. Upper end carries symmetry boundary conditions. Internal pressure, p = 2 [MPa]. (b) HP3D contour
plot of the displacement ur . The mesh is adaptively refined towards the built-in end of the tube in HP3D.
534
Example 3 (Pressurisation of a tube). We consider a circular cylindrical tube with inner radius Ri = 80,
535
outer radius Ro = 100 and length L = 200 [mm]. The tube is clamped at both ends and loaded by an internal
536
pressure P̄ = 2 [MPa]. The SRM and GSRM material parameters are µ = 0.1 [MPa] and Γ = 10. The
537
bulk modulus κ = 1000µ [MPa] is used in the numerical analysis emulating near incompressibility. We
538
exploit the inherent symmetry of the problem and model only 1/8th of the tube using symmetry boundary
539
conditions. The tube is reinforced by a single family of fibres in the azimuthal direction. In this case we do
540
not have a closed form analytic solution. The tube is considered thick-walled. Numerical solutions obtained
541
with the HP3D and FEAP codes are presented. The HP3D and the FEAP results are obtained with Q2 /P1
542
displacement-pressure elements. (using at least a tri-linear hydrostatic pressure approximation improves the
543
situation considerably). The nominal mesh used in HP3D and the contour plot of the radial displacement on
544
the deformed configuration are shown in Figures 5a and 5b, respectively. An adaptive mesh refinement is
545
made in HP3D visible in Figure 5b. The radial displacement along the inner tube mantle and the axial stress
546
along the axial coordinate are shown in Figures 6a and 6b respectively. Finally, the hoop stress on the inner
547
mantle of the tube along the axial coordinate is shown in Figure 6c. The larger hoop strain in the GSRM
548
material dictates the larger hoop stress in the GSRM material compared to the SRM material. The same
549
relation is observed for the axial stress in Figure 6b.
32
0,05
30
SRM, FEM - FEAP
GSRM, FEM - FEAP
SRM, FEM - HP3D
GSRM, FEM - HP3D
0
25
-0,05
20
-0,1
15
10
5
0
0
-0,15
SRM, FEM - FEAP
GSRM, FEM - FEAP
SRM, FEM - HP3D
GSRM, FEM - HP3D
-0,2
γ = 10
-0,25
γ = 10
-0,3
50
100
2
150
200
0
50
100
150
200
γ = 10
1,5
SRM, FEM - FEAP
GSRM, FEM - FEAP
SRM, FEM - HP3D
GSRM, FEM - HP3D
1
0,5
0
0
50
100
150
200
Figure 6: Example 3. Pressurisation of a tube. Fibres in the azimuthal direction. Comparison between the GSRM-model and a SRMmodel. (a) Radial displacement ur at r = ri along the circular cylindrical tube. (b) Axial stress σzz at r = ri along the circular
cylindrical tube. (c) Azimuth stress σθθ at r = ri along the circular cylindrical tube
33
88.4
0.0
Y
Z
X
Figure 7: Example 4. Brick element mesh and deformed configurations of a compressible axially inextensible solid circular cylinder.
Length L = 200 [mm], radius Ro = 100 [mm]. Total number of elements N el = 800. White cylinder – undeformed configuration.
Blue cylinder – deformed configuration GSRM model. Red cylinder – deformed configuration SRM model. Strain energy contribution
in the preferred direction modelled in terms of the isochoric stretch λ̄2 = E z · C̄E z . The SRM model violates the axial inextensibility
λ = E z · CE z = 1, while the GSRM model obeys it.
550
Example 4 (Isostatic loading of an inextensibile solid cylinder). We consider a compressible solid circu-
551
lar cylinder with radius Ro = 100 and length L = 200 [mm] reinforced with virtually inextensibile fibres
552
in the axial direction subjected to isostatic loading P̄ = 2 [MPa]. The shear modulus µ = 0.1 [MPa], bulk
553
modulus κ = 10µ [MPa] and the dimensionless anisotropy parameter Γ = 100 in order to achieve the almost
554
inextesibility in the numerical computation. It is recalled that we use a preliminary finite element setting us-
555
ing a displacement based interpolation of the fibre stretch. Symmetry is exploited, 1/8th of the solid cylinder
556
is meshed using 10 elements axially, 10 elements azimuthally and 8 elements radially. This example is only
557
computed with FEAP using Q1 /P0 elements. The mesh, the undeformed and the deformed configurations
558
are shown in Figure 7. The resulting radial- and axial stretch as function of the applied isostatic loading are
559
finally shown in Figures 8a and 8b respectively. The results shown in Figures 7 and 8 clearly show that the
560
additive volumetric-isochoric strain energy representation used in the SRM model Eq. (40) cannot predict
561
compressible anisotropic deformations. In fact, the stretch in the radial and axial directions are identical for
562
the SRM model.
34
1,8
1,5
SRM, FEM - FEAP
GSRM, FEM - FEAP
1,4
1,6
1,4
Γ = 100
µ = 0.1 [MPa]
κ = 1.0 [Mpa]
Pmax = 2 [MPa]
1,3
1,2
1,2
1
0
SRM, FEM - FEAP
GSRM, FEM - FEAP
Γ = 100
µ = 0.1 [MPa]
κ = 1.0 [MPa]
Pmax = 2 [MPa]
1,1
1
0,2
0,4
0,6
0,8
1
0
0,2
0,4
0,6
0,8
1
Figure 8: Example 4. Isostatic loading of a solid circular cylinder. Radius, R = 100 [mm]. Inextensible fibres in the axial direction.
Comparison between the GSRM-model and a SRM-model. Anisotropy parameter Γ = 100. Neo-Hookean ground substance shear
modulus, µ = 0.1 [MPa] and bulk modulus κ = 1.0 [MPa]. (a) Radial stretch, λr versus isostatic load, P̄/P̄max . (b) Axial stretch,
λz versus isostatic load, P̄/P̄max .
35
563
10. Discussion
564
Computational investigations in the field of soft tissue biomechanics rely on solving complex boundary
565
value problems of real life cases numerically. The biggest challenge has undoubtedly been to find sufficiently
566
good yet simple material descriptions whose parameters can be determined by well controlled measurements.
567
This is an ongoing effort outside the scope of this work. The focus here is on the computational methodology.
568
The currently most frequent computational approach uses a phenomenological homogenised Spencer type
569
continuum description. A mixed formulation is employed to handle the experimentally corroborated near
570
incompressibility. For soft tissue it results in a description of the ground substance that is rubber-like. In
571
other words isotropic and (nearly) incompressible. The entry level model is the neo-Hookean, given in terms
572
of the volume-preserving Cauchy-Green tensor and with an assumed tiny compressibility. Soft tissue often
573
contains fibres which introduces an exponentially Fung-type developing anisotropy providing a reinforcement
574
of the material in tension. The fibres become almost inextensibile at finite strains. The reinforcement is in
575
the first approximation described as an uncoupled additive strain energy contribution in terms of the volume-
576
preserving fibre stretch. The resulting material description is a SRM-type model, see Eq. (40), save for a
577
tension-only response of the fibres. This phenomenological point of view is the point of departure of this
578
work.
579
Our main focus concerns the development of a proper computational framework that can handle com-
580
bined near incompressibility and near inextensibility. This goal is reached by generalising the three-field Hu-
581
Washizu STP-formulation [4] dealing with the near incompressibility. The generalisation is logically straight
582
forward. In the transversely isotropic case it implies replacing the fourth joint invariant of the Cauchy-Green
583
tensor and the fibre dyadic7 by an auxiliary fibre stretch measure. In the inextensibile limit the fourth invariant
584
becomes trivial. This is in complete analogy with the behaviour of the third invariant in the STP-formulation
585
in the incompressible limit. It is shown that the extraction and replacement of the actual fibre stretch is a
586
general and exact procedure, see Remark 1. Constructing the novel formulation we revealed a general logical
587
flaw in the standard volumetric - isochoric decoupled approach. Namely that the ground substance stress
588
contributes in the fibre direction, see Remark 3a. The introduction of the stretch free Cauchy-Green tensor,
589
Eq. (16), removes this contribution. Inextensibility guides us; the work-performing stress and the reactive
590
fibre tension have to be uncoupled.
591
A second general error in the standard volumetric - isochoric decoupled formulation is dealt with. It
593
concerns the inability to represent exact inextensibility at finite volume change (J 6= 1). The remedy is to
˜ using the
write the strain energy contribution from the fibre stretch in terms of auxiliary stretch λ̃ = J˜1/3 λ̄
594
˜ as independent variables, see Remark 3b.
auxiliary volume ratio J˜ and auxiliary volume preserving stretch λ̄
592
7I
4 (C)
= C : A.
36
595
Corrections to the SRM class of models are suggested, see Proposition 1. Stable finite element constructs
596
in the discrete inf-sup condition sense for near inextensibility combined with near incompressibility in finite
597
hyperelasticity is essentially an open problem. This work provides an appropriate framework for the study
598
and development of such constructs.
599
Using the isochoric - volumetric decomposition of the deformation gradient induces the orthogonal spher-
600
ical - deviatoric decomposition of the stresses. Our additional extensional decomposition induces a further
601
decomposition of the deviatoric stresses into an axial fibre part and a part which is orthogonal to the fibre ten-
602
sion. The condition for the determination of a fibre tension that is energetically conjugate to the fibre stretch
603
implies the additional decomposition. The ground substance stresses become trivial in the fibre direction by
604
construction. This decomposition is another original contribution to our knowledge. It is readily generalised
605
to several fibre families, i.e. to other forms of anisotropy.
606
The developed framework is implemented in a preliminary fashion into the standard h-version FEM code
607
FEAP [38] and into the hp-adaptive FEM code HP3D [36, 37]. Differences between a today commonly
608
used model and our new presented model are shown with four numerical examples. The flawed contribution
609
from the ground substance to the response in the fibre direction is as expected quantitatively small, especially
610
approaching near inextensibility. Qualitatively it is wrong and should not appear in a proper formulation
611
of the nearly inextensible case. The inability of the standard decoupled volumetric-isochoric formulation
612
to predict compressible strongly anisotropic deformations is made evident. A working remedy is proposed
613
keeping volume ratio and volume preserving stretch as independent kinematic variables.
614
11. Summary and Conclusions
615
We provide a new original computational framework for nearly incompressible and strongly transversely
616
isotropic finite hyperelasticity. It is a five field Hu Washizu type formulation of Spencer’s phenomenological
617
theory. It generalises the Simo-Taylor-Pister formulation for nearly incompressible isotropic materials by
618
adding an auxiliary fibre stretch variable and a work conjugate fibre tension. The framework relies on a new
619
generalised right Cauchy-Green stretch tensor. Its third and fourth principal invariants together with the fibre
620
dyadic are trivial by construction in the limit of incompressibility and inextensibility.
621
Two errors in the standard volumetric - isochoric so-called decoupled theory are highlighted, addressed
622
and corrected. First, the ground substance contribution to the stresses in the fibre direction is shown and
623
removed. Second, the inability to represent exact inextensibility or even strong anisotropy at finite volume
624
change is corrected. Corrections to the class of so-called Standard Reinforcing Materials, including the
625
popular so-called Holzapfel-Gasser-Ogden model, are suggested.
626
The new generalised Cauchy-Green stretch tensor induces the standard orthogonal spherical - deviatoric
627
decomposition of the stresses. It also induces a new decomposition of the deviatoric stresses into an axial
37
628
fibre extension part and a part that is orthogonal to the fibre tension. The ground substance stress is trivial in
629
the fibre direction, by construction.
630
A five-field finite element formulation for the transverse isotropic case is proposed. The preliminary
631
numerical results obtained corroborate the novel formulation. An appropriate framework for the investigation
632
of the stability in the the inf-sup condition sense of plausible finite element constructs is provided.
633
Acknowledgements
634
635
W.R. acknowledges the financial support via grant Nr: UMO-2011/01/B/ST6/07306 recieved from the
Polish National Center of Science.
38
636
Appendix A. The projection perpendicular to the extensibility constraint normal
637
In order for the article to be self-contained a number of derivations are supplied in this appendix.
638
First an explicit expression for a tentative operator designated
639
640
641
642
P̂iso with property Eq. (33)1. is constructed.
P̂iso is a projection. The explicit expression for the Lagrangean projection of stresses along
the material fibre-direction, denoted P̂ext 8 , i.e. the perpendicular projection to P̂iso is first constructed. The
projection P̂iso itself is then obtained as P̂iso = I − P̂ext . For this purpose the definition of the deviatoric
It is deduced that
fibre stress in the Kirchhoff sense is used in the Lagrangean setting,
τ + = dev[τ ] : a = τ : dev[a] = S : λ−2 CDev [ A ]C,
643
(A.1)
see Eq. (28). The first term in Eq. (34b), can now be rewritten using identity Eq. (A.1) as,
3 + −2
A
2τ λ
= A ⊗ 32 λ−4 (CDev [ A ]C) : S :=
P̂ext : S.
(A.2)
Hence, we obtain,
P̂ext := A ⊗ 32 λ−4(CDev [ A ]C),
P̂ext : A = A
and
and
P̂iso = I − P̂ext
(A.3)
P̂iso : A = 0.
644
Properties Eqs. (A.3)3,4 provide further motivation for the naming of these operators. Using the identity
645
A : CDev [ A ]C = 32 λ4 it is readily proven that Eq. (33)1 holds with
P̂iso in the form Eq. (A.3)3.
Secondly, we investigate if the tensor gradient of the function C̃ Eq. (14b) with respect to C̄ provides an
operator introduced in Eq. (31) as
P̃iso that has the property Eq. (33)1. Using Eq. (14) we find after some
lengthy but simple algebra,
P̃iso
Q1 =
"
#T
h
−2/3 ∂ C̃
˜
:= J
= q −1
∂ C̄
λ̄
−4
A ⊗ (C̄AC̄) ,
P̂iso + 2(1 − q3) Q1 − Q2
Q2 =
i
,
(A.4)
λ̄−2 12
I (AC̄) + (AC̄) I ,
P̂iso is given by Eq. (A.3)2.
646
˜ 1/3 λ̃/λ and where
where q = (J/J)
647
Eqs. (24) implies setting q = 1. It can be shown that it implies stress-equivalence with an unconstrained
648
pure displacement based formulation. Thus, the basic isotropic projection operator
649
correct stress determination in the unconstrained case. It is rewritten,
3 −4
λ̄ (C̄Dev [ A ]C̄) .
iso :=
iso q=1 = I − A ⊗
2
P̂
650
Relaxing the generalised constraints
It is readily verified that
8 Here
P̃
P̂iso is idempotent, (P̂iso)2 = P̂iso and that P̂iso : [A] = 0.
the subscript “ext” is an abbreviation for extension.
39
P̂iso is sufficient for
(A.5)
651
Theorem 1. For an arbitrary tensor T, and with C̃ defined by Eq. (14), the tensor gradient ∂C̄ C̃ has the
652
orthogonality property,
T : J˜−2/3 ∂C̄ C̃ : N = T :
653
654
P̃Tiso : N = 0,
N = C̄Dev [ A ]C̄.
(A.6)
2
Proof 1. The proof consists of showing,
P̂Tiso : N = 0
T
(b) T : QT
1 − Q2 : N = 0.
(a) T :
655
656
657
Using A = M ⊗ M it is readily verified that N : A = 23 λ̄4 .
658
Part (a): The identity follows by simple expansion,
T : N − 32 λ̄−4 N(N : A) = T : N − N = 0.
Part (b):The
(A.7)
Q1-part is expanded as,
λ̄−4 T : (C̄AC̄ ⊗ A) : N = λ̄−4 T : (C̄AC̄)(N : A) = T : 23 (C̄AC̄)
659
660
Q
Using Eq. (A.4)4 and the rule (P Q)T = (PT QT ) the 2 -part is expanded as,
h
i
h
i
λ̄−2 T : 21 I (AC̄)T + (C̄A) I : N = λ̄−2 T : C̄(AC̄A)C̄ − 13 λ̄2 C̄AC̄ = T : 23 (C̄AC̄). (A.9)
Here, the projection (AC̄A) = λ̄2 A which is readily proven writing the tensor product,
C̄A = C̄LM (L ⊗ M ) + C̄M M (M ⊗ M ) + C̄N M (N ⊗ M ),
661
(A.10)
in the orthonormal basis {L, M , N } where M is the preferred direction, i.e. C̄M M = λ̄2 .
Eqs. (A.8) and (A.9) are equal. Thus Part (b) is also proven. Since each part vanishes independently the
662
663
(A.8)
theorem is proven. 2
9
664
Appendix A.1. The tensor gradient ∂C̄ C.
Formally, setting J˜ = λ̃ = 1 in Eq. (A.4) yields the expression,
" 9 #T
h
∂C
9
˜
=
= λ̄ iso + 2(1 − λ̄−3 )
iso :=
iso J=
λ̃=1
∂ C̄
665
P
P̃
P̂
Q1 − Q2
i
.
(A.11)
9
666
Corollary 1. By Theorem 1 the expression Eq. (A.11) for the tensor gradient ∂C̄ C obtained form Eq. (16)
667
provides orthogonality property,
9
T : ∂C̄ C : N = T :
668
2
40
PTiso: N = 0.
9
(A.12)
9
669
Appendix A.2. The expression [∂C̄ C]T : I.
9
The contraction [∂C̄ C]T : I comes into play for a neo-Hooke ground substance (matrix) material. Eq. (A.11)
is recalled and its contributions are rewritten,
Q1 := λ̄−4A ⊗ C̄AC̄,
Q2 := 21 λ̄−2 I (AC̄) + (AC̄) I
P̂iso := I − 23 A ⊗ λ̄−4C̄Dev [ A ]C̄
670
Their contractions with I become,
P̂iso : I = I − ιA,
671
and
ι = 21 [3(1 + γ̂12 + γˆ2 2 ) − λ̄−2 I¯1 ]
Q1 : I = λ̄−4(C̄AC̄ : I)A,
Q2 : I = λ̄−2 12 (C̄A + AC̄),
(A.13)
(A.14)
respectively. The coefficients γ̂1 and γ̂2 in Eq. (A.13)2 are defined in Eq. (A.16). The tensor products AC̄
and C̄A are written in the orthonormal basis {L, M , N } where M is the preferred direction as,
C̄A = C̄LM (L ⊗ M ) + C̄M M (M ⊗ M ) + C̄N M (N ⊗ M ),
AC̄ = C̄M L (M ⊗ L) + C̄M M (M ⊗ M ) + C̄M N (M ⊗ N ),
(A.15)
C̄AC̄ : I = γ12 + λ̄4 + γ22 ,
where C̄M M = λ̄2 and where C̄LM = CM L = γ1 and C̄N M = C̄M N = γ2 are shear strains in the
transversely isotropic plane. The summed-up result is,
Q1 − Q2) : I = (γ̂12 + γ̂22)A − K,
(
K=
1
2
γ̂1 (L ⊗ M + M ⊗ L) + γ̂2 (M ⊗ N + N ⊗ M ) ,
γ̂1 := λ̄−2 γ1
and
(A.16)
γ̂2 := λ̄−2 γ2 ,
672
where the coefficients γ̂1 and γ̂2 are the shear corrections normalised by the fibre stretch λ̄2 . Hence, the neo-
673
Hookean type normalised Lagrangean ground substance stress I is corrected for stretch in the fibre direction
674
and for any associated shears in the transversely isotropic plane according to Eq. (A.11). The A proportional
675
term in Eq. (A.16) corrects for the strains in the fibre direction caused by the shears. The result is written in
676
the fibre basis as,
Piso : I = λ̄
9
h
io
n
I − ιA + 2(1 − λ̄−3 ) (γ̂12 + γ̂22 )A − K ,
(A.17)
677
where the correction K and coefficients γ̂1 and γ̂2 are defined by Eq. (A.16). The correction can here be
678
understood as a decoupling of the fibre and ground substance (matrix) stress response.
41
679
680
681
682
683
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44
764
765
List of Figures
1
Normalised fibre extensional stiffness as function of isochoric fibre stretch λ̄. Material pa-
766
rameters for Left Anterior Descending (LAD) coronary human artery adventitia represented
767
by a HGO material model [24] with parameters µ = 2.7 [kPa], k1 = 5.1 [kPa] and k2 = 15.4
768
[-] according to [33]. The normalised fibre extensional stiffness exceeds the normalised bulk
769
stiffness for λ̄ > 1.156. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
770
2
Example 1. Isochoric simple tension. Fibres in the tension direction. Comparison between
771
the GSRM-model and a SRM-model. Neo-Hookean ground substance, µ shear modulus. Di-
772
mensionless anisotropy parameter Γ. (a) Modelling error, |[τ11 ]GSRM − [τ11 ]SRM |/|[τ11 ]SRM | × 100
773
using the SRM-model Γ = 10 and 100. (b) Normalised axial stress τ11 /µ versus stretch λ̄,
774
Γ = 10 and 100, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
775
3
L = 400 [mm], radius Ro = 100 [mm]. Fibres in the axial direction. Total number of el-
777
ements N el = 1536. Lower end is built-in. Upper end is twisted. The outer mantle is
778
traction-free.
4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
the GSRM-model and a SRM-model. Anisotropy parameter Γ = 10. Neo-Hookean ground
781
substance shear modulus, µ = 0.1 [MPa]. (a) Torque T versus twist γRo in [rad]. (b) Axial
782
force N versus twist γRo in [rad]
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
FEAP. Length L = 200 [mm], inner radius Ri = 80 [mm] and outer radius Ro = 100 [mm].
785
Total number of elements N el = 4096. Four element layers across the thickness. Lower end
786
is built-in. Upper end carries symmetry boundary conditions. Internal pressure, p = 2 [MPa].
787
(b) HP3D contour plot of the displacement ur . The mesh is adaptively refined towards the
788
built-in end of the tube in HP3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
30
31
Example 3. Pressurisation of a tube. (a) Nominal brick element mesh used in HP3D and
784
789
28
Example 2. Pure torsion of a solid circular cylinder as defined in Fig. 3. Comparison between
780
783
[%]
Example 2. Pure torsion of a solid circular cylinder. Brick element mesh. Cylinder length
776
779
4
32
Example 3. Pressurisation of a tube. Fibres in the azimuthal direction. Comparison between
790
the GSRM-model and a SRM-model. (a) Radial displacement ur at r = ri along the circular
791
cylindrical tube. (b) Axial stress σzz at r = ri along the circular cylindrical tube. (c) Azimuth
792
stress σθθ at r = ri along the circular cylindrical tube . . . . . . . . . . . . . . . . . . . . .
45
33
793
7
Example 4. Brick element mesh and deformed configurations of a compressible axially inex-
794
tensible solid circular cylinder. Length L = 200 [mm], radius Ro = 100 [mm]. Total num-
795
ber of elements N el = 800. White cylinder – undeformed configuration. Blue cylinder –
796
deformed configuration GSRM model. Red cylinder – deformed configuration SRM model.
797
Strain energy contribution in the preferred direction modelled in terms of the isochoric stretch
798
λ̄2 = E z · C̄E z . The SRM model violates the axial inextensibility λ = E z · CE z = 1,
799
while the GSRM model obeys it.
800
8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
Example 4. Isostatic loading of a solid circular cylinder. Radius, R = 100 [mm]. Inex-
801
tensible fibres in the axial direction. Comparison between the GSRM-model and a SRM-
802
model. Anisotropy parameter Γ = 100. Neo-Hookean ground substance shear modulus,
803
µ = 0.1 [MPa] and bulk modulus κ = 1.0 [MPa]. (a) Radial stretch, λr versus isostatic load,
804
P̄/P̄max . (b) Axial stretch, λz versus isostatic load, P̄/P̄max . . . . . . . . . . . . . . . . .
46
35
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