Polyconvex Anisotropic Energies and Modeling of Damage Applied

Daniel Balzani
Polyconvex Anisotropic Energies
and Modeling of Damage
Applied to Arterial Walls
Universität Duisburg-Essen
Ingenieurwissenschaften
Abtl. Bauwissenschaften
Institut für Mechanik
Prof. Dr.-Ing. J. Schröder
Bericht Nr. 2
Polyconvex Anisotropic Energies
and Modeling of Damage
Applied to Arterial Walls
Vom Fachbereich Bauingenieurwesen und Geodäsie
der Technischen Universität Darmstadt
zur Erlangung des akademischen Grades eines
Doktor–Ingenieurs (Dr.–Ing.) genehmigte Dissertation
von
Dipl.–Ing. Daniel Balzani
aus
Köln
Referent:
Prof. Dr.–Ing. habil. J. Schröder
Korreferent:
Prof. Dr.–Ing. habil. D. Gross
Korreferent:
Dr. rer. nat. habil. P. Neff
Tag der Einreichung:
02.12.2005
Tag der mündlichen Prüfung:
07.04.2006
Darmstadt im April 2006
D 17
Editor:
Prof. Dr.-Ing. habil. Jörg Schröder
University of Duisburg-Essen
Faculty for Engineering Sciences
Department of Civil Engineering
Institute of Mechanics
D-45117 Essen
c Prof. Dr.-Ing. habil. Jörg Schröder
Institut für Mechanik
Abteilung Bauwissenschaften
Fakultät für Ingenieurwissenschaften
Universität Duisburg-Essen
Universitätsstraße 15
45117 Essen
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Preface
The work presented in this thesis has been carried out in the three years, when I was
scholarship holder in the “DFG Graduiertenkolleg 853: Modellierung, Simulation und
Optimierung von Ingenieuranwendungen” at the Technical University of Darmstadt. At
the end of this period I feel grateful to a lot of people who accompanied me in this time.
First of all, I want to thank my academic teacher Professor Jörg Schröder for his excellent personal and scientific supervision, his intensive interest for my research and the
permanent positive motivation for further developments. His confiding and dynamic way
of supervision let him become a contact person for me not only with view to my work
but also in private concerns.
My special thanks also go to Professor Dietmar Gross for the scientific support and his
permanent interest for the development of my work. Moreover, I thank him for giving
me the opportunity to participate at international conferences, which enabled me to
present my results continuously.
Further special thanks go to Dr. Patrizio Neff, whose intensive support especially with
respect to generalized convexity conditions gave me detailed insights into the mathematical foundations of elasticity.
I want to thank Professor Gerhard Holzapfel for helpful and high-producing discussions
and for providing experimental data. Then I thank the “Deutsche Forschungsgemeinschaft” for the financial support in the form of the graduate scholarship, which I was
awarded. Next, I address my thanks to all co-workers for their advice and the friendly
working atmosphere. My relation to them has not only been professionally fruitful but
also amicable. Especially, I thank Professor Joachim Bluhm and Professor Reint de
Boer for valuable councils and helpful discussions. To Holger Romanowski I send further special thanks for his highly qualified computer and software support as well as
to Petra Lindner-Roullé, Veronika Jorisch and Angelika Priessnigg, who relieved me of
organizational jobs. I would like to express my gratitude for advice and a constructive
atmosphere to my temporary room mates Oliver Hilgert, Ingo Kurzhöfer and especially
Professor Tim Ricken, who taught me a variety of continuum mechanical foundations in
the beginning of my PhD. I thank Alexander Schwarz and Mark-André Keip for helpful
discussions.
Hearty thanks go to my family, especially my mother, my father, my grandmother and
my deceased grandfather, who did their utmost in supporting me unconditionally in
every sense. They gave me the familiar background for the successful completion of my
PhD as well as my brother Claudio Balzani. I thank my girl-friend for her comprehension
and patience especially in the final stage of this thesis.
Essen, im April 2006
Daniel Balzani
Abstract
In the present thesis a framework for the modeling of anisotropic hyperelasticity in biological soft tissues is developed based on the concept of polyconvexity. An anisotropic
discontinuous damage model is proposed for the description of damage effects observed
in experiments. The developed models are applied in numerical simulations of atherosclerotic arteries. After providing the continuum mechanical fundamentals, the representation theorems for isotropic and anisotropic tensor functions, the generalized convexity
conditions and a simple principle for the construction of polyconvex stored energy functions are presented. Based on this principle a variety of polyconvex transversely isotropic
functions, which automatically satisfy the condition of a stress-free reference configuration, is generated and discussed with view to biological soft tissues. Two polyconvex
functions suitable for the description of arterial tissues are constructed based on the
given construction principle and adjusted to a human abdominal aorta. In another part
of this work an anisotropic damage model for biological soft tissues is developed based on
the concept of internal variables. The main assumption in this model is that the damage
acts mainly in the fiber direction. A scalar-valued damage function is introduced which
is able to account for discontinuous damage. The damage model is applied to one of the
constructed stored energy functions for arterial tissues and some numerical simulations
based on the Finite-Element method document the performance of the proposed models.
Zusammenfassung
In der vorliegenden Arbeit wird im Rahmen der Polykonvexität ein Konzept zur Modellierung anisotroper Hyperelastizität in biologischen Weichgeweben entwickelt. Für die
Beschreibung von Schädigungseffekten, welche in Experimenten beobachtet werden, wird
ein anisotropes diskontinuierliches Schädigungsmodell vorgeschlagen. Die entwickelten
Modelle finden Anwendung in numerischen Simulationen von arteriosklerotischen Arterien. Nach der Bereitstellung der kontinuumsmechanischen Grundlagen, der Darstellungstheoreme für isotrope und anisotrope Tensorfunktionen und der ver-allgemeinerten
Konvexitätsbedingungen, wird ein einfaches Prinzip für die Konstruktion von polykonvexen Energiefunktionen, die automatisch die Bedingung einer spannungsfreien Referenzkonfiguration erfüllen, vorgestellt. Auf Grundlage dieses Prinzips wird eine Vielzahl
polykonvexer transversal isotroper Energiefunktionen generiert und im Hinblick auf biologisches Weichgewebe diskutiert. Es werden zwei für die Beschreibung arterieller Gewebe
geeignete polykonvexe Funktionen mit Hilfe des Konstruktionsprinzips aufgestellt und
an eine menschliche Abdominalaorta angepasst. Basierend auf dem Konzept der internen
Variablen wird in einem weiteren Teil dieser Arbeit ein anisotropes Schädigungsmodell
für biologische Weichgewebe entwickelt. Die wesentliche Annahme in diesem Modell ist,
dass die Schädigung hauptsächlich in Faserrichtung stattfindet. Es wird eine skalarwertige Schädigungsfunktion eingeführt, die in der Lage ist diskontinuierliche Schädigung
zu berücksichtigen. Das Schädigungsmodell wird auf die entwickelten Energiefunktionen für Arteriengewebe angewendet und einige numerische Simulationen mit der FiniteElemente Methode dokumentieren die Funktionsfähigkeit der vorgestellten Modelle.
Table of Contents
I
Contents
1
Introduction and Motivation
1
2
Mechanical Behavior of Arterial Walls
9
2.1 Composition of Arteries
9
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Atherosclerosis and Methods of Treatment . . . . . . . . . . . . . . . . . . . 11
2.3 Mechanical Properties of Arterial Tissues . . . . . . . . . . . . . . . . . . . 13
3
Continuum Mechanical Preliminaries
17
3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Stress Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Balance Principles and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3.2 Balance of Linear Momentum . . . . . . . . . . . . . . . . . . . . . . 24
3.3.3 Balance of Angular Momentum . . . . . . . . . . . . . . . . . . . . . 24
3.3.4 Balance of Mechanical Energy . . . . . . . . . . . . . . . . . . . . . 25
3.3.5 Entropy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Basic Principles of Material Modeling . . . . . . . . . . . . . . . . . . . . . 28
3.5 Representation for Isotropic and Anisotropic Tensor Functions
. . . . . . . 31
3.5.1 Representation Theorems for Isotropic Tensor Functions . . . . . . . 32
3.5.2 Representation of Transverse Isotropy by Isotropic Tensor Functions
4
Finite-Element Method
33
37
4.1 Fundamentals of the Finite-Element Method . . . . . . . . . . . . . . . . . 37
4.1.1 Variational Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.1.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.1.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Special Finite-Element Implementations . . . . . . . . . . . . . . . . . . . . 44
4.2.1 Triangular Element with Quadratic Ansatz Functions . . . . . . . . 44
4.2.2 Surface Load Element with Quadratic Ansatz Functions . . . . . . . 45
4.2.3 Implementing Inertia Terms . . . . . . . . . . . . . . . . . . . . . . . 46
5
Polyconvex Framework for Anisotropic Hyperelasticity
49
5.1 Generalized Convexity Conditions . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.1.2 Quasiconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Table of Contents
II
5.1.3 Polyconvexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.1.4 Rank-1-Convexity and Legendre-Hadamard Ellipticity . . . . . . . . 56
5.2 Stresses and Tangent Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2.1 Stresses and Tangent Moduli for Isotropic Materials . . . . . . . . . 57
5.2.2 Stresses and Tangent Moduli for Transversely Isotropic Materials
. 58
5.2.3 Stress-Free Reference Configuration . . . . . . . . . . . . . . . . . . 60
5.3 Polyconvex stored energy functions
. . . . . . . . . . . . . . . . . . . . . . 60
5.3.1 Isotropic Polyconvex stored energy functions . . . . . . . . . . . . . 60
5.3.2. Transversely Isotropic Polyconvex Functions . . . . . . . . . . . . . . 62
6
7
Parameter Adjustment Based on the Evolution Strategy
69
6.1 Evolution Strategy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Objective Function
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Polyconvex Energies for Soft Biological Tissues
75
7.1 Comparative Study with Respect to Material Stability . . . . . . . . . . . . 75
7.1.1 Analysis of Special Models . . . . . . . . . . . . . . . . . . . . . . . 76
7.1.2 Quantitative Adjustment of the Polyconvex Model . . . . . . . . . . 78
7.1.3 Localization Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.2 A New Polyconvex Model for Arterial Walls
. . . . . . . . . . . . . . . . . 88
7.2.1 Experimental Data of a Human Aortic Layer . . . . . . . . . . . . . 89
7.2.2 Representation of the Arterial Tissue
8
. . . . . . . . . . . . . . . . . 90
Anisotropic Damage Model for Arterial Walls
97
8.1 (1 − D) - Approach and Discontinuous Damage . . . . . . . . . . . . . . . . 97
8.2 The Damage Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.3 Linearization and Algorithmic Implementation . . . . . . . . . . . . . . . . 102
8.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9
Numerical Simulation of Arterial Walls
107
9.1 Residual Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.1.1 Closing the Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.1.2 The Prestressed State . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.1.3 Residual Stresses in Atherosclerotic Arteries . . . . . . . . . . . . . . 113
9.2 Damage in Overexpanded Arteries . . . . . . . . . . . . . . . . . . . . . . . 116
9.2.1 Overexpanded Artery with Conservative Load . . . . . . . . . . . . . 117
9.2.2 Overexpanded Artery with Non-Conservative Load . . . . . . . . . . 118
Table of Contents
III
9.2.3 Overexpanded Artery with Residual Stresses . . . . . . . . . . . . . 119
10 Conclusion and Outlook
121
A Notation
123
B Proof of Convexity.
123
C Polyconvex Functions.
126
D Matrix Notation of Special Tensors.
133
References
141
Introduction and Motivation
1
1
Introduction and Motivation
The importance of the numerical description of material and structural behavior in the
context of computer-oriented mechanics has significantly increased in the last decades.
Especially in the field of biomechanics the materials pose an outstanding challenge for
researchers working in the area of material modeling. It seems to be the first official occurrence of a biomechanical institution, when the Working Group of Biomechanics (J.
Wartenweiler, chairman; J. Vredenbregt, secretary-general) was founded in 1967. Numerous developments enriched this relatively new subject and today the definition of biomechanics varies to such an extent that an exact definition is not possible. At the beginning
it was mostly the analysis of physical movements of the body and body parts, whereas
today even the smallest parts of biological systems as e.g., bones, tissues and even cells are
investigated and modeled with respect to their mechanical properties. Some of the most
important works, which have contributed to the mechanical understanding, description
and modeling of biological components, have been presented by Fung, e.g. the extensive
descriptions in Fung [34], [36], [37]. More recently, the detailed and comprehensive works
of Humphrey [62] and the appropriate chapters in Bronzino [19] provide a comprehensive introduction to the topic of biomechanics.
The present work covers a biomechanical issue, since it mainly deals with the phenomenological modeling of biological soft tissues, especially arterial tissues, and the numerical
simulation of arterial walls. As main topics may be seen the construction of polyconvex
stored energy functions able to describe these tissues and the development of a damage
model which takes into account the fibrous structure of arterial tissues.
In the last decades the number of fatalities caused by cardiovascular disease has augmented
distinctively, so that the interest in the biomechanical description and simulation of the
blood system has exponentially increased. In most cases atherosclerotic degenerations of
the blood vessels leading to the development of atherosclerotic plaques are the reason for
a detraction of the cardiovascular system.
For the treatment of these stenoses the balloon-angioplasty (since 1977) in combination
with stent implants (since 1986) are available and frequently performed nowadays. These
techniques exploit the mechanical properties of arterial walls and lead to an increased
arterial lumen and therefore to a normalized blood circulation. Due to damage effects,
which occur when a balloon-angioplasty or stent implant is performed, the inside of the
artery can be injured. This is followed by biological reactions, which in turn lead in
some cases to so-called re-stenoses a few months after treatment. Furthermore, damage
effects during the balloon-angioplasty may cause the atherosclerotic plaque to rupture and
separate partially from the arterial wall, which blocks a following artery. In case of affected
coronary arteries this usually ends up in a myocardial infarction. Hence, the interest in
the numerical simulation of arterial walls, especially diseased arteries, is quite high in
order to optimize the methods of treatment and to reduce the number of re-stenoses.
Due to the existence of embedded fibers basically reinforcing the material, biological soft
tissues appearing in elastic arteries can be characterized as anisotropic materials undergoing large deformations. For the structure-independent modeling of such materials in
the sense of continuum mechanics a suitable framework is based on the invariant theory.
Representations of anisotropic finite elasticity based on elements of the invariant theory
are given in Ericksen & Rivlin [31] and Doyle & Ericksen [29], although first for-
Introduction and Motivation
2
mulations of the general hyperelastic theory with applications to some special classes of
anisotropy are already found in Murnaghan [89]. For comprehensive overviews to the
contributions published in the 1970’s and 1980’s in the field of non-linear elasticity we
refer to Marsden & Hughes [79], Ciarlet [22], Antman [1], Ogden [91] and Lurie
[76].
Via introducing additional argument tensors, the so-called structural tensors, and by
means of tensor representation theorems the polynomial basis of the constitutive equations, consisting of basic and mixed invariants of the deformation and structural tensors,
can be derived. This polynomial basis reflects the anisotropy of the material. For an introduction to the coordinate-invariant formulation of anisotropic constitutive equations
based on the concept of structural tensors we refer to Spencer [123], Boehler [15], [16],
Betten [13], Schröder [102] and Smith & Rivlin [120], [121]. In this context also
the publications Smith, Smith & Rivlin [122], Smith [117], Wang [135], [136], [137],
[138], Smith [118], [119], Spencer [123] and Zheng & Spencer [146], [147] must be
mentioned. The complete and irreducible representations for tetratropic, hexatropic and
octotropic scalar-, tensor- and vector-valued functions in plane problems with view to fiber
reinforced composites is given in Zheng, Betten & Spencer [145]. More recently, some
different representations are published in the context of finite anisotropic hyperelasticity,
as e.g., Xiao [144], where a new representation for finite elasticity in cubic crystals is
introduced, or Menzel & Steinmann [81] deducing special formats of stress tensors
and tangent moduli in the material and spatial setting. For an overview to most classes
of anisotropy see e.g., Ting [127] or Haupt [44].
In order to concretize the general representations of finite elasticity such that a physically
reasonable material behavior is obtained, there exist various restrictions to the form of
the stored energy. The Baker-Ericksen inequality (Baker & Ericksen [2]) states that in
the direction of the maximum principal stretches the maximum principal Cauchy-stresses
occur, which has to be satisfied for every arbitrary pair of associated principal stresses and
stretches. This condition holds only for isotropic materials and is not able to be applied to
anisotropy. The Coleman-Noll (Coleman & Noll [23]) inequality implies the convexity
of the stored energy with respect to the deformation gradient. In large strain formulations this condition is not reasonable, because it precludes buckling, it is incompatible
with the principle of material frame-indifference and it violates some essential growth requirements. The most suitable condition seems to be the Legendre-Hadamard inequality,
where the existence of traveling waves is investigated by analyzing the ellipticity of the
acoustic tensor. In the non-linear framework the direction with minimal wave speeds is
calculated for the linearization of the nominal stresses at the considered solution points.
This explains why it is difficult to construct physically reasonable constitutive models by
satisfying the Legendre-Hadamard condition for all possible deformation states. Violation
of the Legendre-Hadamard ellipticity leads to material instability of the model and therefore the existence of singular surfaces where the deformations may jump.
Furthermore, there exist restrictions on the stored energy with a view to the existence of
minimizing deformations of the energy potential and given boundary conditions. In order
to ensure the existence of minimizers, the energy has to be sequentially weakly lower semicontinuous and coercive. The concept of quasiconvexity (Morrey [87]) implies together
with some growth condition the sequential weak lower semicontinuity (s.w.l.s.). Unfortunately, the growth condition excludes that the energy becomes infinite for vanishing
determinants of the deformation gradient. Satisfying the quasiconvexity condition is quite
Introduction and Motivation
3
important, because otherwise initially homogeneous bodies could break down in coexisting stable phases, see Krawietz [72], Ball & James [5], Silhavý [112] or Müller
[88]. Due to the fact that the quasiconvexity condition is complicated to handle since it
is essentially an integral inequality, it is not suitable for practical use.
Another important restriction is the notion of polyconvexity in the sense of Ball [3], [4].
This condition can be checked locally and implies directly quasiconvexity. Furthermore,
polyconvexity is a sufficient condition for s.w.l.s. even without growth conditions and represents therefore a powerful restriction with view to the existence of solutions. Due to the
fact that a polyconvex stored energy leads also to material stability in the sense of the
Legendre-Hadamard condition and does not preclude some physically essential properties
which are excluded by convex stored energies, the polyconvexity can be interpreted as the
most suitable restriction known at present for the construction of physically reasonable
models guaranteeing also the existence of minimizers.
For the material modeling of biological soft tissues a variety of interesting works have been
published in the last three decades. We refer to technical literature, as. e.g. Holzapfel
& Ogden [53], Humphrey [60], Humphrey [61] and Cowin & Humphrey [24], for
an overview of models for biological tissues. In Vaishnav, Young & Patel [132] a twodimensional model for a canine aorta is proposed based on three polynomial expressions.
Due to the fact that biological soft tissues appearing in elastic arterial walls are characterized by an exponential stress-strain response, in Fung, Fronek & Patitucci [33] a
first model is introduced for the two-dimensional mathematical description of such arteries reflecting the exponential material behavior in the physiological domain; this model
is generalized to the three-dimensional case in Chuong & Fung [20]. An extension to
this model is given in Fung & Liu [35], where residual stresses occurring in unloaded
configurations of arteries are considered.
The above mentioned models are mainly constructed based on the special cylindrical structure of arterial walls and are therefore not able to describe realistic arteries, especially
diseased arteries, whose geometry differs from the idealization of a cylindrical structure.
By applying the invariant theory and representation theorems for anisotropic tensor functions numerous material models for biological soft tissues are introduced in recent years.
Delfino, Stergiopulos, Moore & Meister [26] propose an isotropic model approximating the anisotropic material response and reflecting the exponential character of the
stress-strain relation in the physiological range of deformations of biological soft tissues.
Another model taking into account the anisotropy of biological soft tissues is introduced in
Weiss, Maker & Govindjee [140]. There, an exponential function is proposed which
is formulated in the first mixed invariant of the deformation and structural tensor.
The models as listed above do not satisfy the polyconvexity condition and hence, neither
the existence of minimizers nor ellipticity and material stability can be guaranteed a priori. The first framework for the description of transverse isotropy by polyconvex stored
energy functions is provided in Schröder & Neff [106], which has been used by Itskov
& Aksel [63] for the description of calendered rubber sheets. In Steigmann [126] this
framework is extended to functions in terms of the right symmetric stretch tensor. In this
thesis it is shown that representative non-polyconvex models, as e.g., the model in [140],
lead to acoustic tensors which are not positive definite, which is in contrast to polyconvex
models. It is shown by comparing some representative models for biological soft tissues,
that although the same stress-strain behavior of a polyconvex and non-polyconvex models
can be obtained for a limited range of stress/strain, only the polyconvex model ensures
Introduction and Motivation
4
positive definite acoustic tensors and thereby a stable material, see Schröder, Neff &
Balzani [107]. A first model able to describe the physiological behavior of arterial tissues
by an exponential function in the first mixed invariant, which satisfies the polyconvexity condition, is proposed in Holzapfel, Gasser & Ogden [50], see also Holzapfel,
Gasser & Ogden [54], Holzapfel, Eberlein, Wriggers & Weizsäcker [48], [49].
The proof of convexity of the transversely isotropic energy of the model in [54] is given
in Balzani, Neff, Schröder & Holzapfel [11]. This model accounts for the special
fiber arrangement in arterial walls mainly consisting of two fiber families by the superposition of two transversely isotropic energy functions. Hereby, a weak interaction between
the fiber families is assumed which seems to be physically reasonable. In order to ensure polyconvexity a special case distinction has to be considered, whereby stresses in the
fibers are solely generated for tension and not for compression. In Holzapfel, Gasser
& Stadler [52] the material model is extended to describe also the visco-elastic behavior
observed much more distinctively in muscular arteries.
Another model for the physiological behavior of biological soft tissues ensuring the polyconvexity condition without the consideration of some kind of case distinction is proposed
in Schröder & Neff [104], [105]. In the present work this model is extended to the
description of arterial layers and adjusted to “virtual experiments”, where the model
represents the same stress-strain relations as some other representative models for soft
biological soft tissues, see also Schröder, Neff & Balzani [107]. Unfortunately, the
model has a high number of parameters and requires therefore optimization procedures for
the adjustment to experiments. Furthermore, the function does not satisfy the condition
of a stress-free reference configuration a priori and therefore additional restrictions for the
material parameters arise.
In the framework of this thesis a simple construction principle for polyconvex stored energies satisfying automatically the stress-free reference configuration is developed. Based on
this principle and motivated by the case distinction occurring in Holzapfel, Gasser
& Ogden [50] a variety of transversely isotropic polyconvex stored energy functions accounting for the natural state condition a priori is generated and already published in
Balzani, Neff, Schröder & Holzapfel [11]. In contrast to the polyconvex model
in [104], i.e. [107], the functions proposed here do not require as many parameters and
are therefore more practicable. As an example, we choose two of these polyconvex transversely isotropic functions and construct some polyconvex models for arterial walls. A
main feature of these models is the additively decoupled structure of the stored energy
into an isotropic part for the matrix material and the superposition of two transversely
isotropic parts for the embedded fiber families. As examples, the polyconvex models are
adjusted for real experiments of a human abdominal aorta “by hand” in order to show
the easy handling of the proposed functions.
The numerical simulation of an arterial overexpansion seems to be an interesting tool for
the optimization of methods of treatment for stenotic arteries. This requires the development of models not only able to describe the physiological range of stresses and strains,
but also reflecting the material properties in unphysiological domains. Following e.g.,
Holzapfel, Gasser & Ogden [50] or Gasser & Holzapfel [38] there are damage
effects observed when overexpanding arterial walls. These damage effects are responsible
for a stiffness reduction of the material and also for some plastic strains remaining in the
natural state. This stiffness reduction combined with plastic strains leads to increased
strains for the physiological loading range and explains why the arterial lumen remains
Introduction and Motivation
5
increased when removing the catheter after a balloon-angioplasty. In [38] a model for the
elastoplastic behavior of arterial walls is proposed based on the multiplicative decomposition of the deformation gradient. This model is able to represent a stiffness reduction
linked with plastic strains. Another possibility to describe the stiffness reduction in arterial walls, is the continuum damage mechanics providing an interesting framework. In
1958 a scalar-valued damage variable able to describe isotropic damage at small strains
has been introduced in Kachanov [65]. The resulting so-called (1 − D)-approach has
been enhanced thenceforward. In the beginning the approach has been generalized to different types of anisotropy classes by introducing damage tensors of second or fourth order.
For an introduction to the damage mechanics at small strains we refer to Lemaitre [74]
and for overviews on this subject please see Skrzypek & Ganczarski [116], Krajcinovic [71] or Lemaitre & Desmorat [75]. In the 1980’s the extension to large strains
has gained scientific interest. Based on the concept of internal variables in Simo [113],
an isotropic damage model for large strains in polymers is given, see also Simo & Ju
[114]. For the description of continuous as well as discontinuous damage at large strains,
Miehe [85] provides some special internal variables. The above mentioned models for
large strains are able to represent isotropic symmetries. Usually, also for the description
of anisotropic damage at large strains second- or fourth-order tensors are necessary. As an
example, in Menzel & Steinmann [80] a framework for anisotropic continuum damage
mechanics at large strains is given using an internal variable tensor of second-order. Here,
the multiplicative split of the deformation gradient often used in finite elasto-plasticity is
adapted for the damage mechanics.
Using such higher-order tensors leads to complicated structures of the damage model. In
the present thesis another main effort is the construction of an anisotropic damage model
for the consideration of the fibrous structure in biological soft tissues without secondor fourth order damage tensors. Due to the special composition of the reinforcing fibers
the damage effects are mainly assumed to occur in the fiber direction. Based on this assumption the special additively decoupled structure of the stored energy describing the
hyperelastic behavior in the physiological loading range enables us to apply the (1 − D)concept to that part of the energy, which is associated with the fiber elasticity. Hereby, we
represent the real three-dimensional anisotropic damage by the “one-dimensional problem” of fiber damage. This is done by introducing the scalar-valued damage variable D(a)
describing the damage in fiber direction a(a) . A function for the damage variable is constructed by introducing an internal variable similar to the one in [85] in order to reflect
discontinuous damage. We use the formalism given in Lemaitre & Chaboche [73] for
the functional dependency of D(a) and ensure thermodynamical consistency. In Balzani,
Schröder & Gross [7], [8], [9] and Schröder, Balzani & Gross [108] steps of the
development of the model are documented.
Numerical simulations which show first steps to realistic simulations of atherosclerotic
arteries are presented in e.g., Holzapfel, Schulze-Bauer & Stadler [51]. There,
the authors obtain a more detailed discretization with the particular tissues appearing in
a representative atherosclerotic plaque. Furthermore, experiments (uniaxial tension tests)
for the different tissues are performed, for which the utilized material model is adjusted
and a balloon-angioplasty is simulated with and without stenting. More extensive experiments are published in Holzapfel, Sommer & Regitnig [55], where the particular
tissues of an atherosclerotic artery are tested for some arteries and altogether 107 specimens are investigated. In Holzapfel, Stadler & Gasser [56] numerical simulations
Introduction and Motivation
6
of atherosclerotic arteries with stents are presented with respect to the form optimization
of stents. The present thesis shows the simulations of residual stresses in atherosclerotic
arteries, which is already published in Balzani, Schröder & Gross [10], and of some
overexpanded atherosclerotic arteries in order to show that the proposed models are generally able to reflect the material behavior of soft biological tissues in arterial walls.
The work at hand is organized as follows. Section 2 provides a detailed description of
the mechanical properties of arterial walls. Therefore, essential information about the
composition of arteries is given firstly, which is necessary for the understanding of the
proposed models in following sections. Furthermore, important facts on atherosclerosis
and its methods of treatment are given. Finally, the mechanical behavior of arterial walls
is explained in detail including the properties in the physiological loading range as well
as the damage effects occurring when arteries are overexpanded.
In Section 3 the basics of the continuum mechanics, i.e. the kinematics, the stress concept,
the balance principles and entropy and the principles of material modeling are recalled as
well as the representation for isotropic and anisotropic tensor functions.
Due to the fact that the presented numerical simulations are performed by using the
Finite-Element method, the fundamentals of this method are described in Section 4.
Starting from the variational problem, the linearization is deduced and the discretization
is explained. Then the utilized Finite-Element, a 6-node triangular element with quadratic
ansatz functions, is described in detail as well as some other special Finite-Element implementations.
Section 5 starts with the definition of important generalized convexity conditions, i.e.
convexity, polyconvexity, quasiconvexity and rank-one convexity, and explains their interconnection and relevance with respect to the existence of solutions of boundary value
problems. Then the formal stresses and tangent moduli are calculated from transversely
isotropic stored energy functions formulated in the framework of the invariant theory. The
main part in Section 5 defines a simple principle for the construction of polyconvex stored
energy functions automatically satisfying the condition of a stress-free reference configuration. This principle is used in order to generate a variety of polyconvex transversely
isotropic functions, which are partly discussed with a view to the material modeling of
biological soft tissues.
In Section 6 the framework of adjusting a material model to experimental data by an optimization scheme is described. For this purpose the evolution strategy is used and explained
firstly. Then a special objective function including a local iteration, which depends on the
kind of experiment performed, is given and will be minimized by the evolution strategy.
Section 7 provides some analysis concerning polyconvex stored energy functions. In a first
investigation a polyconvex model is adjusted such that it represents the same stress-strain
behavior as two non-polyconvex ones. Then localization analyses are performed and the
models are compared with respect to material stability. The second investigation shows
that the construction of polyconvex energies for biological soft tissues can be accomplished
in a simple way when using the proposed construction principle. This is done by adjusting some constructed polyconvex energy functions to uniaxial tension tests of a human
abdominal aorta.
The anisotropic damage model is given in Section 8. Therefore, the fundamentals of the
damage mechanics are described at first and then the proposed damage model is described.
After providing the linearization of the constitutive equation and explaining the algorithmic treatment, a numerical example shows the stress-strain response and especially the
Introduction and Motivation
7
stress softening governed by the damage model.
Section 9 presents some numerical simulations of atherosclerotic arteries. After describing the general framework of the simulations, a first numerical example simulates the
residual stresses inside arterial walls. For this reason a method for the incorporation of
residual stresses is described firstly. As a second numerical example some simulations are
performed with respect to the overexpansion of arterial walls. There, different types of
boundary conditions are compared.
Finally, Section 10 summarizes this work and provides an outlook for further research.
Introduction and Motivation
8
Mechanical Behavior of Arterial Walls
2
9
Mechanical Behavior of Arterial Walls
For the modeling of arterial walls fundamental understanding of the composition, the morphological structure and the complete histology of arteries is necessary. The knowledge
about these issues enables us to comprehend the mechanical properties of arterial tissues
and the overall artery. Therefore, in this section we point out first the main components
of healthy arterial walls and then focus on one of the most important arterial diseases,
namely atherosclerosis, and its methods of treatment. Finally, we explain the basic mechanical characteristics of arterial tissues. It should be noted that here we only provide
basic information necessary for the understanding of further sections. For extensive illustrations please see technical literature from the field of medicine or biomechanics, as e.g.,
Junqueira & Carneiro [64], Fung [37], Humphrey & Delange [62], or the review
of Rhodin [99].
2.1 Composition of Arteries
Based on their location in the body arteries are mainly subdivided into the types:
elastic and muscular arteries. Muscular arteries are located at the periphery of the blood
circulation system, e.g., femoral, celiac and cerebral arteries therefore they are small in
diameter. These arteries undergo only small deformations compared to the elastic arteries
and contain much more muscle cells. Close to the heart we find elastic arteries, e.g., the
aorta, carotid and iliac arteries, which have larger diameters and a much lower amount of
muscle cells. Here, we mainly discover (passive) smooth muscle cells which are not able to
shrink actively, which is in contrast to e.g., muscular arteries. Due to the fact that elastic
arteries are located close to the heart, they receive much more distinctive oscillations
of blood and undergo therefore much larger deformations than muscular arteries. Since
we are interested in such large deformations we concentrate on elastic arteries in this
work. Another important point is that we will mainly focus on the components which
influence the mechanical properties of the arterial wall. For this reason we keep out most
components which have more or less physiological tasks as e.g., vasa vasorum (blood
vessels supplying the artery itself with blood), nerves or perivascular connective tissue.
Membrana elastica interna
Endothel
Intima
Media
Adventitia
Figure 1: Schematic illustration of a healthy elastic artery, taken from Junqueira [64].
Mechanical Behavior of Arterial Walls
10
Healthy elastic Arteries can be divided into three distinct layers, the intima (tunica intima), the media (tunica media) and the adventitia (tunica externa), see Fig. 1.
The innermost layer, the intima, consists of two layers, a single layer endothel, which
covers the internal surface of the vessel resting on a thin basal membrane, and the stratum subendotheliate, which is composed of a very thin connective tissue and sometimes
smooth muscle cells. The thickness of the intima of healthy young individuals is very
small, therefore its contribution to the solid mechanical properties of the arterial wall is
insignificant. With an increase in age the intima thickens and stiffens, so that its mechanical contribution may become significant. This pathological alteration is referred to as
atherosclerosis, the most common disease of arterial walls, and will be described in more
detail in the next section.
The middle layer of the artery, the media, is divided into a varying and with age increasing number of well-defined concentrically fiber-reinforced medial layers separated by
fenestrated elastic laminae. As an example the number of layers of an adult is about
70, while that one of a baby is approximately 40. These layers consist of smooth muscle
cells, thin collagen fibrils (type I, type III) and elastic fibrils. The media is separated
from the intima by the so-called internal elastic lamina and from the adventitia by the
external elastic lamina. The structure of the fenestrated elastic laminae in a cross-section
is wavelike, as shown in Fig. 2a).
a)
b)
c)
Figure 2: a) Wavelike elastic laminae in the media of a healthy elastic artery, b) wavelike orientation of collagen fiber bundles and c) ramified elastic fibers generating oriented
networks; photographs taken from Junqueira [64].
The collagen fibrils are mostly organized in bundled collagen fibers which have also a
wavelike structure, as shown in Fig. 2b), while the elastic fibers generate oriented fiber
networks of ramified fibers consisting of elastin, see Fig. 2c). In the media the amount
of collagen fibrils not bundled in fibers is higher than the amount of bundled fibrils.
The orientation of the elastic and collagen fibrils and the smooth muscle cells between
constitute a continuous fibrous helix running through the artery. This is the reason, why
the media is able to resist both kind of loads, the longitudinal and the circumferential one
and can be interpreted as the most significant layer in a healthy elastic artery.
The outermost layer of the artery, the adventitia, consists mainly of ground substance,
thick bundles of collagen fibers and fibroblasts and fibrocytes, which are cells producing
collagen and elastin. In elastic arteries the adventitia is relatively thin and hardly to
distinguish from the surrounding tissue. Contrary to the media, the amount of wavelike
Mechanical Behavior of Arterial Walls
11
bundles of collagen fibrils is higher than the amount of unbundled fibrils in the adventitia.
Another difference between the two layers is that the fraction of elastin fibrils is higher
in the adventitia than in the media, being in contrast to the fraction of collagen fibrils,
which is higher in the media than in the adventitia. In the adventitia the structure of the
fibers can also be described as a continuous helix reinforcing the arterial wall.
2.2 Atherosclerosis and Methods of Treatment
One of the most common diseases of arterial walls, known as atherosclerosis, takes center
stage in the scientific interest of medical and biochemical researchers for decades. The
malicious trait of this indisposition is that it develops slowly over years or even decades
without any symptom until it becomes manifest in ischemia (undersupply of organs or
tissues caused by deficient blood supply), thrombosis (development of blood clots), angina
pectoris (thoracic pain as a consequence of narrowed blood vessels), myocardial infarction or stroke. In this section we provide the main characteristics of atherosclerosis and its
methods of treatment only briefly. For further information we refer the reader to technical
literature from the field of medicine. Atherosclerosis can be defined as the degenerative
change in the intima accompanied by an accumulation of lipids, complex carbohydrates
and blood components, by a growth of fibrous tissue and by the development and increase of calcifications; in severe stadiums the media may also be infected. The main
factors supporting atherosclerosis are well-known nowadays. As primary risk factors hypercholesterolemia (high cholesterol level), smoking and hypertonia (pathological high
blood pressure) are mentioned while the secondary risk factors are identified to be diabetes mellitus, excess of weight, lack of exercise, adiposity and hyperuricemia (high uric
acid level in the blood).
Due to the complexity of the influencing substances as e.g., endothelial cells, smooth
muscle cells, monocytes, macrophages, thrombocytes, lipoproteins, etc., the cause for the
development and an exact description of the evolution of atherosclerosis is not found yet.
However, two hypothesis have become apparent: the 1) “response to injury” hypothesis
and the 2) “lipoprotein-induced atherosclerosis” hypothesis. The first hypothesis assumes
injuries at the inner side of the artery, which may be e.g., morphological deterioration,
caused by trauma or mechanical lesions, or e.g., biochemical deterioration, caused by
bacterial toxins, viruses or antigen-antibody reactions, to be the reason for atherosclerosis.
The second hypothesis considers the oxidative modification of low density lipid in the
blood as the primary cause for intimal injuries. For the further progress of atherosclerotic
disease the two hypotheses do not differ. Due to intimal injuries the arterial tissue starts to
proliferate, smooth muscle cells migrate from the media into the intima and extracellular
lipid emerges, inducing the characteristic development of the so-called atherosclerotic
plaque.
With growing plaque the arterial lumen decreases, which is referred to as stenosis, and
the arterial wall stiffens as a consequence of the alteration of the arterial tissues in the
plaque. Then the risk of a thrombotic closure of the artery increases significantly. In case
of a coronary artery this would lead to a lack of oxygen supply and the local destruction of
heart muscle tissues ending up in a myocardial infarction. For detailed information about
pathophysiological aspects of coronary artery disease and its consequences see Willerson, Hillis & Buja [142]. The fact, that atherosclerosis has already overtaken cancer
Mechanical Behavior of Arterial Walls
12
Figure 3: Cross-section of an atherosclerotic artery in advanced state with dark areas representing calcifications, taken from the official website of the Brown-University in Providence,
USA (www.brown.edu).
to be the cause of death number one in the industrialized nations, gives outstanding rise
to the interest of research for methods of treatment.
There exist two basic groups of methods of treatment: the bypass operations and catheterbased invasive techniques. In most cases invasive techniques are performed due to its
relatively uncomplicated and undangerous procedure. The interest in the optimization
of the catheter-based methods is one of the fundamental motivations for the computer
simulation of atherosclerotic arteries, therefore, we concentrate on these techniques here.
What these techniques have in common is that a catheter is inserted into the affected
artery, usually through the femoral artery by entering the arterial system in the groin
or the upper thigh, and placed at the atherosclerotic location. Catheter-based invasive
techniques can be subdivided into three methods: the 1) atherectomy, the 2) balloonangioplasty and 3) stenting.
When an atherectomy is accomplished, then the catheter is equipped with a sharp blade
or abrasive material with sandpaper-like properties in order to cut away the plaque. This
procedure is only possible if the plaque is very stiff and it is precluded by high probability
that big parts of the plaque break during the treatment. This would lead to the closure
of following arteries with smaller diameters. When a satisfying result is reached, then the
catheter is removed and leaves behind a larger lumen; if not, then a balloon-angioplasty
and/or stenting is performed afterwards.
In the context of a balloon-angioplasty a catheter with a special balloon at the tip is
inserted and placed at the atherosclerotic vessel segment. Then the balloon is inflated
by an internal pressure of approximately 3–12 bar, which pushes the plaque against the
arterial wall and overexpands the artery such that remanent deformations remain in the
physiological configuration of the artery. This leads to a larger lumen when removing
the balloon from the artery. During this process the atherosclerotic plaque or even the
arterial wall may suffer ruptures and therefore it might be the case that a big part of
the plaque separates from the arterial wall so that a following artery may be blocked.
Due to supposable cracks in the arterial wall the development of e.g., an aneurysm is also
possible. The drawback of performing solely a balloon-angioplasty is, that after months
normally the remanent deformations decrease and the arterial lumen is reduced again;
this resetting of a stenosis is referred to as re-stenosis.
Mechanical Behavior of Arterial Walls
13
The stenting method exists since approximately 20 years and is nowadays the mostly
applied method for treating atherosclerotic stenoses. In most cases it is combined with
a balloon-angioplasty and/or atherectomy. When applying this method a catheter with
a small, extendable wire mesh tube, called stent, is inserted in the affected artery and
placed in the stenosis. Usually a balloon is located in the inside of the stent and by
inflating the balloon the stent and the artery are expanded. Hereby, damage effects in
the arterial wall and/or the plaque lead to remanent deformations in the physiological
configuration of the artery, leaving behind a larger lumen when removing the balloon.
The main advantage of this method is that the positive effects of a balloon angioplasty
are exploited and assisted such that the stent keeps the artery open and avoids re-stenoses
caused by decreasing remanent deformations. Another possibility for the stenting method
without inflating a balloon is to insert a self-extendable stent. These are kept unexpanded
by a cover when inserting the catheter into the affected artery. When reaching the place of
the atherosclerotic stenosis the cover is withdrawn and the stent extends itself, pressing
the artery outwards. There exist also self-expandable stents doing without an imposed
cover, which are constructed of shape-memory alloys. In each case the stent remains in
the artery and becomes a permanent device being a part of the cardiovascular system.
Within one month after the treatment the stent is covered by a thin layer of the artery’s
inner lining cells and is usually not detectable by metal detectors.
Unfortunately, the stent struts often hurt the inner side of the artery when extending the
stent, due to stress concentrations located where the stent is bearing the artery. On one
hand this may cause the plaque to rupture and increase the risk of a separation of plaque
portions which most probably block following arteries with smaller diameter. On the other
hand the intimal injuries caused by the stent induce the intima to grow and therefore,
quite often re-stenoses occur some months after implanting a stent. In order to prevent
lesions in the intima and therefore to reduce the tissue growth around the stent struts,
coated stents can be employed, which are stents covered by coatings, e.g., consisting of
polymer. A special group of such coated stents are the so-called drug eluting stents, which
emit medical substances during the first months after implant. These substances reduce
directly the tendency of the arterial tissue to grow.
2.3 Mechanical Properties of Arterial Tissues
From the mechanical point of view, biological soft tissues occurring in elastic arterial
walls can be described as fiber-reinforced and nearly incompressible materials. In arterial
walls mainly two fiber families can be identified which are embedded in an extracellular
matrix also referred to as ground substance. This ground substance can be treated as
isotropic as well as it follows nearly a linear stress-strain response. It should be mentioned
that the ground substance is much less stiff than the embedded fibers. Furthermore,
viscoelastic effects are observed leading to a viscoelastic response in the overall tissue in
elastic arteries, which is only negligible. The viscoelastic effect plays a more important
role in muscular arteries, where also the (active) muscle cells are responsible for this effect.
Due to the properties and the special structure of the embedded fibers the whole tissue
behaves non-linear in the stress-strain relation, see Fig. 4.
For the physiological range of deformations in elastic arteries (deformations that occur in normal life) the mechanical behavior may be described as (perfectly) elastic, see
Holzapfel, Gasser & Ogden [50]. With respect to the collagen bundles, which are
Mechanical Behavior of Arterial Walls
14
II
σ
I
A
range of physiological
stress
∆l/l0
∆lr,plas /l0
∆lr,phys /l0
Figure 4: Typical uniaxial stress (σ) – strain (∆l/l0 ) diagram for circumferential arterial strips in passive condition, cf. Holzapfel [50]. Remanent strains due to elasto-plastic
material behavior and remanent strains at the edge of physiological stress range after two
overexpansion cycles are denoted by ∆lr,plas and ∆lr,phys , respectively.
aligned in wavelike lines, the contribution of these fibers to the stiffness rather vanishes
for small deformations. With increasing deformations the wavelike fibers are straightened
and the stiffness increases. Concerning the elastic fibers, the mechanical behavior is quite
similar. For small deformations the fiber net of elastin is totally wrinkled and therefore
its participation to the stiffness of the whole tissue is rather zero. If the deformations are
increased then the fiber net straightens and the stiffness of the tissue increases. Since the
deformation, necessary to activate the contribution of the fibers to the stiffness, is much
larger for elastic fibers than for collagen fibers, the collagen fibers may be seen as the
main reinforcing component in arterial walls. In Holzapfel [57] experiments show that
the intima is again stiffer than the media and adventitia. But due to the relatively very
small layer thickness in healthy arteries the intima may be neglected from mechanical
considerations and its function is more founded in physiological tasks.
As a result of the fact that the fraction of collagen fibers is higher than the fraction of
elastic fibers in the media and vice versa in the adventitia, the media can be interpreted
as the most important load carrying layer in the physiological range of internal pressure.
The adventitia represents the layer saving the artery from overstretch and rupture due to
its large amount of elastic fibers.
Since the structure of an artery can be interpreted as a thick-walled tube, increasing
internal pressure induces a higher gain of circumferential deformations in the outside than
in the inside of the arterial wall. So it seems to be that the artery is arranged such that
it levels the circumferential stresses over the arterial wall: in the inside the intima/media
is stiffer and undergoes lower deformations and in the outside the adventitia is less stiff
and undergoes higher deformations. This effect may also be assisted by the existence of
residual stresses appearing in the unloaded configuration of an elastic artery. If an axial
segment of an artery is cut in radial direction, then it usually springs open so there must be
eigenstresses in the uncut state, see Vaishnav & Vossoughi [133]. Assuming the opened
artery to be stress-free, cf. Chuong & Fung [21], Fung [37], Humphrey & Delange
[62] and Ogden [92], we are able to imagine what kind of stress state should be expected
Mechanical Behavior of Arterial Walls
15
by closing the artery. Then of course bending stresses would occur with respect to the
circumferential direction leading to negative stresses in the inside and tensile stresses
in the outside of the arterial wall. The superposition of this stress state and the state
governed by blood pressure would level the circumferential stresses over again. If an axial
segment of an artery is cut in circumferential direction between the media and adventitia
then the opening angles of the two arterial layers are differing, see Greenwald, Moore,
Rachev, Kane & Meister [41]. This implies the existence of further residual stresses
in the unloaded configuration but the biomechanical function is not yet clear.
A circumferential strip of an elastic artery subjected to uniaxial cyclic loading and unloading shows stress softening effects as schematically depicted in Fig. 4. The softening
effect diminishes with repeating loading and unloading cycles until the path is reached
which ends in point I, representing the (perfectly) hyperelastic path. In this state the
material is often referred to as pre-conditioned and reflects what we should expect in an
artery under in vivo conditions with physiological blood pressure. The stress softening
is a result of a combination of continuous and discontinuous damage effects responsible
for a stiffness reduction of the material. Up to a load reached at point I, no plastic deformations occur, which changes when applying further load. Then plastic deformations
are generated, which remain when the material is completely unloaded. Furthermore, the
intensity of damage effects becomes more distinctive, which can be observed in Fig. 4 in
the curve between point I and II. The stress state, obtained after applying the unphysiological load (at point II), is significantly influenced by the stiffness reduction caused by
damage effects. For a better understanding of these effects we have to go more into detail
with respect to the composition of the main reinforcing material in arteries, the collagen
fibers. It is already pointed out in 2.1 that collagen fibers are bundled and separated in
collagen fibrils. In Fig. 5 the extensive composition of the fibrils is shown.
columns
overlaps
microfibril
300 nm
tropocollagen
collagen
fibril
void columns,
dark stripe
overlaps, light stripe
collagen
fiber
bundle of
collagen fibers
≈ 67 nm
Figure 5: Schematic illustration of the composition of collagen fiber bundles, taken from
Junqueira [64].
We see that collagen fibrils are divided into microfibrils, which are again composed of
parallelly oriented tropocollagen. Between these tropocollagens there exist some kind of
cross-bridges linking the particular tropocollagens to the adjoining ones. It stands to
Mechanical Behavior of Arterial Walls
16
reason that if the collagen fiber bundle is expanded in fiber direction over a certain level,
then some of the cross-bridges break, and if all of the cross bridges in one microfibril
are broken, then even the whole microfibril breaks. For each load which is further on
applied more cross-bridges or even microfibrils break so that for increasing load less fibrils
are available to resist load. This implies an oriented stiffness reduction and therefore a
special damage effect in fiber direction. Assuming that the cross-bridges or microfibrils
do not rearrange in some kind and that they are not able to heal themselves in the time
of loading and unloading cycles, leads to a (perfectly) discontinuous damage effect solely
in fiber direction. The rearrangement of cross-bridges during the loading and unloading
cycle could be the reason for further damage in the same loading range after one cycle and
therefore for the continuous damage also observed in Fig. 4. It could be also imaginable
that the microfibrils, collagen fibrils or fibers may be broken away from each other, if the
material were expanded transversely to the fiber direction. But since the ground substance
is much less stiff than the fibers, the ground substance between the fibers is expected to be
damaged and not the fibers themselves. Due to the fact that the fibers can be interpreted
as the main components giving rise to the stability of arterial walls, the damage in fiber
direction may be more important for the modeling of healthy arteries than damage of the
ground substance.
With a view to diseased arteries it is remarked that the mechanical properties of affected
atherosclerotic layers, as the intima or media, differ significantly from a healthy intima
or media. In Holzapfel, Sommer & Regitnig [55] several experiments are performed
with atherosclerotic arteries not only with respect to the intima, media and adventitia,
but also with respect to the plaque components. Due to the complexity of the plaque,
where the constituents differ distinctively from artery to artery, we abstain from a detailed
description of their mechanical properties here, simply because in later sections the models for these components are not adjusted to experiments. It should be only mentioned
that an atherosclerotic plaque is in general stiffer than the surrounding layers media and
adventitia, which leads to a reduction of the flexibility of the whole blood vessel.
The mechanical properties of arterial walls constitute the basis of the functionality of
a balloon-angioplasty and stenting. Especially the damage effects are exploited for the
success of these methods of treatment. If we take a look at Fig. 4, the application of the
further load from point I to point II may be seen as a representation of what happens
during a balloon-angioplasty. When unloading the arterial tissue, i.e. deflating the balloon,
then we follow the lower path (A). If we then repeat this process e.g., one more time,
then we would obtain the remanent strain ∆lr,phys remaining in the arterial wall in the
physiological loading state, increasing the arterial lumen. On one hand this remanent
strain consists of a part governed by the damage directly, because of the stiffness reduction
increasing the strains in the range of physiological stress. On the other hand the remanent
strains ∆lr,phys contain a part which is indirectly induced by damage mechanisms in the
arterial wall, i.e. plastic strains ∆lr,plas . It seems to be irony of fate that especially the
damage effects represents the reason for the risks of the balloon-angioplasty and stenting.
Due to damage phenomena in the plaque it may happen that parts of the plaque separate,
which then block following smaller arteries. It may also happen that damage in the intima
or even media leads to rupture and additional diseases as e.g., aneurysms.
Continuum Mechanical Preliminaries
3
17
Continuum Mechanical Preliminaries
Physical bodies that consist of true materials are from the microscopical view inhomogeneous due to the atomistic composition. The description of these inhomogeneities is
very expensive in macroscopical considerations and from the engineer’s part of view not
necessary in many cases. In this work the method of continuum mechanics is applied in
order to describe the deformations and internal stress states of the considered material in
a phenomenological way. In this context a physical body is approximated as a continuous
medium, in which the complex real structure is replaced by certain field quantities, e.g.
density, temperature and velocity. As a matter of course this approach is not an exact
description of the material itself, but sufficiently accurate with regard to engineering materials. The fundamentals of continuum mechanics are put down on paper in Truesdell
& Toupin [130], Truesdell & Noll [129] and Eringen [32]. More similarly to the
notation used in this contribution an extensive insight into the mechanics of continua is
provided by Silhavý [112] and Holzapfel [47]. Here, we subdivide the section into the
six parts:
• Kinematics, where the configurations, motions and deformation states of the continuum body are described,
• Stress Concept, in which the framework for the description of stress states within
the body is given,
• Balance Principles and Entropy, where the basic equations of continuum mechanics
are outlined,
• Basic Principles for Constructing Material Models, wherein the fundamental principles for the construction of material models are presented and
• Representation for Isotropic and Anisotropic Tensor Functions, where the basics of
the representation theory are outlined.
3.1 Kinematics
A physical body can be interpreted as accumulation of material points, whose location
may be described by position vectors. If the body B ⊂ IR3 is arranged in its undeformed
(reference) configuration at time t = t0 we parametrize it in X, whereas the euclidian
three-dimensional space is denoted by IR3 . Since we are interested in the description of deformations, we have to consider the deformed (actual) configuration, too. Here, the body
of interest S ⊂ IR3 at a fixed time t ∈ IR+ is parametrized in x. The reference configuration
is also referred to as material or Lagrangian configuration, while the actual configuration
is also called spatial or Eulerian configuration. The deformation of the body can be interpreted as the motion of material points, or more precisely the change of distance between
material points. The non-linear, continuous and one-to-one transformation
ϕ(X, t) : B → S
(3.1)
maps at time t ∈ IR+ points X ∈ B of the reference configuration onto points x ∈ S of
the actual configuration. For fixed time t ∈ IR+ we write
ϕt (X) : B → S .
(3.2)
Continuum Mechanical Preliminaries
ϕt (X)
B
G2
G3
Ξ3
18
S
Ξ2
ξ3
g3
Ξ1
G1
ξ2
ξ
e2
E2 X
g2
1
g1
x
e1
E1
e3
E3
a)
b)
Figure 6: Covariant basis vectors and curved coordinates of a) reference - and b) actual
configuration of the considered body.
It should be noted that in the sequel indices of tensor components associated to basis vectors in the reference configuration are indicated by latin capitals and that ones associated
to basis vectors in the actual configuration by latin lowercase letters. For the description
of arbitrary bodies with arbitrarily formed boundaries the usage of special (local) basis
systems might be preferable instead of the right-handed orthogonal, cartesian coordinate
system, which is spanned up by the basis vectors E A and ea with A, a = 1, 2, 3. Then we
have to consider arbitrarily curved coordinates ΞA , A = 1, 2, 3 or ξ a , a = 1, 2, 3, in order
to respect the geometry of the body. Since the basis vectors are tangentially oriented to
these curved coordinates the covariant material and spatial basis vectors are defined by
GA =
∂ X̂(ΞA )
∂ΞA
and ga =
∂ x̂(ξ a )
,
∂ξ a
(3.3)
respectively. An illustration of the covariant basis vectors and the two considered configurations is given in Fig. 6.
Together with the contravariant basis vectors, which are introduced by
GA =
∂ΞA
∂X
and g a =
∂ξ a
,
∂x
(3.4)
the co- and contravariant basis systems set up a dual system and the relations
GA · GB = δ A B
and g a · g b = δ a b
(3.5)
hold. Herein δ A B and δ a b denote the so-called Kronecker-Delta, which is equal to one if
A = B or a = b, and equal to zero if A 6= B or a 6= b. The covariant metric coefficients of
the basis are defined by
GAB = GA · GB
and gab = g a · g b .
(3.6)
One of the most important kinematic quantities in the framework of continuum mechanics
is the deformation gradient
F (X) := Grad[ϕ̂t (X)] ,
with F a A =
∂xa
= xa , A ,
A
∂X
(3.7)
Continuum Mechanical Preliminaries
dA
B
x = ϕt (X)
S
F
dV
dv
da
cof[F ]
dX
19
det[F ]
a)
b)
dx
Figure 7: a) Reference - and b) actual configuration of the considered body.
which is defined as the gradient of the transformation map x̂(X) = ϕt (X) with respect
to the material coordinates X. From (3.7)2 we see directly, that F is a two-field tensor,
because one basis vector is associated to the reference and the other to the actual configuration. The deformation gradient represents a linear transformation map of material
tangent vectors to spatial tangent vectors F : TX B → Tx S with the material tangent space
TX B and the spatial tangent space Tx S, respectively. In order to ensure, that the linear
transformation map is one-to-one, F is not allowed to be singular, thus, the existence of
the inverse of the deformation gradient
F −1 = grad[X] =
∂X
,
∂x
with {F −1 }A a = X A ,a
(3.8)
has to be guaranteed. A sufficient condition for local invertibility is that the determinant
of F has to differ from zero. Together with the continuity of ϕt (X) we postulate the
Jacobi-determinant to be strictly positive
J := det[F ] > 0 .
(3.9)
From the physical point of view this is reasonable, because otherwise the body could interpenetrate itself.
We define the displacement u as the difference vector of the position vector in the reference
configuration and in the actual configuration. Then we obtain the alternative representation for the deformation gradient
F = 1 + Grad[u] .
(3.10)
An infinitesimal line element in the reference configuration can be mapped to an infinitesimal line element in the actual configuration via the deformation gradient
dx = F dX .
(3.11)
Similarly to this relation, the normal to an infinitesimal material area element dA maps
to the normal of an infinitesimal spatial area element da via Nanson’s formula
da = cof[F ] dA
(3.12)
with the cofactor cof[F ] = det[F ] F −T . Herein, the normals of the area elements are given
by
da = n da , and dA = N dA .
(3.13)
Continuum Mechanical Preliminaries
20
Formulating the volume element as the product of an area element and a line element,
e.g. dV = (dX 1 × dX 2 ) · dX 3 , we are able to calculate the actual volume element by
dv = det[F ] dV .
(3.14)
For an illustration of the transformation relations, also referred to as transport theorems,
see Fig. 7. The deformation gradient can be split multiplicatively into the rotation tensor
R and the stretch tensor U , i.e. V , so we obtain
F = RU = V R ,
(3.15)
which is referred to as left and right polar decomposition, respectively. Two important
deformation tensors, the right Cauchy-Green tensor and the Finger tensor (left CauchyGreen tensor),
C := F T F and b := F F T ,
(3.16)
respectively, are governed by the square of the infinitesimal line elements dx, i.e. dX.
Since the rotation tensor is an orthogonal tensor (R−1 = RT ), we are able to transform
the deformation tensors into
)
C = F T F = (RU )T RU = U T RT RU = U T U
,
(3.17)
b = F F T = V R(V R)T = V RRT V T = V V T
and directly see that no rigid body motions are included within these two deformation
tensors. In index notation the right and left Cauchy-Green tensor are given by
CAB = F a A gab F b B
and bab = F a A GAB F b B ,
(3.18)
respectively. The difference of the square of referential and actual infinitesimal line element
(dx · dx − dX · dX) leads to the Green and the Almansi strain tensor

1
1
E := 2 (C − G) , with EAB = 2 (CAB − GAB ) , and 
,
(3.19)

e := 1 (g − b−1 ) , with e = 1 (g − {b−1 } )
ab
2
2
ab
ab
which do not include any rigid body motions. Since we are interested in history-dependent
non-linear deformation processes the time derivatives of the kinematic quantities are evaluated here. We denote the velocity in the material description by V̄ (X, t) and in the spatial
description by v̄(x, t). Both of them can be transformed into each other by a composition
with the transformation map ϕ, then
v̄(x, t) = V̄ (X, t)
(3.20)
holds. The material velocity is defined by
V̄ (X, t) =
∂ϕ(X, t)
= ẋ
∂t
(3.21)
and the velocity in the actual configuration yields
v̄(x, t) = V̄ (ϕ−1 (x, t), t) .
(3.22)
Continuum Mechanical Preliminaries
21
Herewith, the spatial velocity gradient l is introduced as the derivative of the spatial
velocity with respect to the spatial coordinate x and we obtain
l := grad[v̄] ,
with la b = v̄ a ,b .
(3.23)
The spatial velocity gradient can be separated into a symmetric and a skew symmetric
part. Then the two parts are defined as the spatial strain velocity gradient
d := sym[l] = 12 (l + lT ) ,
with dab = 12 (gac lc b + gbc lc a )
(3.24)
with wab = 12 (gac lc b − gbc lc a ) .
(3.25)
and the spatial spin tensor
w := skew[l] = 21 (l − lT ) ,
The time derivative of the deformation gradient is defined as the material velocity gradient
∂
∂ ẋ
∂ϕ(X, t)
∂ ∂ϕ(X, t)
=
=
= Grad[v̄]
(3.26)
Ḟ =
∂t
∂X
∂X
∂t
∂X
and the connection between the spatial and the material velocity gradient is computed as
l=
∂ ẋ ∂X
= Ḟ F −1 .
∂X ∂x
(3.27)
If the Jacobi-determinant is differentiated with respect to time, we obtain
∂det[F ] ∂F
J˙ =
:
= JF −T : Ḟ = J tr[l] = J div[v̄] ,
∂F
∂t
(3.28)
i.e. in index notation J˙ = J v a ,a . In an analogous way the time derivatives of the infinitesimal line element, area element and volume element are calculated to


dẋ = ldx


T
.
(3.29)
dȧ = div[v̄] da − l da



dv̇ = div[v̄] dv
3.2 Stress Concept
If a physical body is loaded by external forces, an internal loading state is generated, which
is generally described by the notion of stresses. It is easy to imagine that in an arbitrary
cross-section within a continuous body the resulting force, referred to as stress vector, is
depending on the orientation of the cross-section. This orientation can be characterized by
a normal vector, which is perpendicular to the cross-section. In the actual configuration we
denote the stress vector by t and the normal vector by n̄. Of course, more expedient would
be a quantity that is not depending on the orientation of the cross-section. Therefore, the
stress tensor is introduced, which maps the normal vector to the resulting force vector,
i.e. the Cauchy theorem
t = σn
(3.30)
holds. The Cauchy stresses σ represent the real stresses inside the body and are associated
to the actual configuration. For the covariant basis, the Cauchy stresses σ are given by
σ = σ ab g a ⊗ g b .
(3.31)
Continuum Mechanical Preliminaries
22
Relating to the reference configuration we consider the referential stress vector
T = P N̄
(3.32)
with the referential normal vector N̄ and the 1. Piola-Kirchhoff stress tensor P , which is
generally unsymmetric and also referred to as nominal stress tensor. Utilizing equations
(3.12), (3.30) and T dA = t da we obtain the 1. Piola-Kirchhoff stresses
with P aA = Jσ ab {F −1 }A b .
P = JσF −T ,
(3.33)
Weighting the Cauchy stresses by the volume ratio J (compare the next section) we obtain
the Kirchhoff stresses, which are defined by
τ := Jσ ,
with τ ab = Jσ ab .
(3.34)
Applying a pull-back-operation to the Kirchhoff stresses, we obtain the 2. Piola-Kirchhoff
stress tensor S, which has to be understood as a pure numeric quantity and for which
there is no possibility of a physical interpretation. The second Piola-Kirchhoff stresses are
then calculated by
S = F −1 P , with S AB = {F −1 }A a P aB
(3.35)
or with respect to the Kirchhoff stresses
with S AB = {F −1}A a τ ab {F −1 }B b .
S = F −1 τ F −T ,
(3.36)
It should be noted that contrary to the 1. Piola-Kirchhoff tensor, S is symmetric, which
is a result of the balance of angular momentum as pointed out in Section 3.3.3. The rate
of the internal mechanical work is defined as the internal stress power Pint and can be
written down for the actual configuration as
Z
Pint = σ : l dv .
(3.37)
S
Since the Cauchy stress tensor is also regarded as symmetric the spatial velocity gradient
l can be replaced by its symmetric part (3.24) and we obtain
Z
Pint = σ : d dv .
(3.38)
S
With respect to the reference configuration the internal stress power can be reformulated
with (3.14) and (3.34) through
Z
Pint = τ : d dV .
(3.39)
B
Considering other work-conjugated variables the internal stress power can equally be
calculated as
Z
Z
Pint = P : Ḟ dV = S : Ė dV .
(3.40)
B
B
The quantities (τ , d), (P , Ḟ ) and (S, Ė) are referred to as work-conjugated pairs.
Continuum Mechanical Preliminaries
23
3.3 Balance Principles and Entropy
In this section the material-independent fundamental relations of continuum mechanics
are explained. These relations have axiomatic character, which means, that they represent basic laws, that can not be deduced from other laws, and that no counter-examples
exist. They have to be immediately acceptable and have to agree with experience without
restriction. It should be remarked that we only consider the framework of one-component
continuums and closed systems. Then the particular balance principles and theorems of
thermodynamics are
• conservation of mass
• balance of linear momentum
• balance of angular momentum
• balance of mechanical energy (1. fundamental theorem of thermodynamics)
• entropy inequality (2. fundamental theorem of thermodynamics).
In the following subsections the main mechanical results deducing from each principle are
presented.
3.3.1 Conservation of Mass The first balance principle is the conservation of mass,
saying that in a closed system the mass m remains unaltered by the deformation process.
We obtain the global formulation
m :=
Z
ρ̂0 (X) dV =
B
Z
ρ̂(x, t) dv = const.
(3.41)
S
with the referential and actual density ρ0 and ρ, respectively. Utilizing the transport
theorem (3.14) we calculate the 1. local form of mass conservation
J=
ρ0
,
ρ
(3.42)
which explains that the Jacobi-determinant can be interpreted as volume ratio. The condition m = const. indicates that the time derivative of mass must be zero, i.e.
d
ṁ =
dt
Z
S
d
ρ dv =
dt
Z
B
d
ρ J dV =
dt
Z
˙ dV = 0 .
(ρ̇ J + ρ J)
(3.43)
B
Inserting equation (3.28) and transforming back to the actual configuration leads to the
2. local form of mass conservation
ρ̇ + ρ div[v] = 0 .
(3.44)
Continuum Mechanical Preliminaries
24
3.3.2 Balance of Linear Momentum The axiom of balance of linear momentum
postulates that the time derivative of linear momentum L is equal to the influence of all
externally acting forces f
L̇ = f .
(3.45)
Herein, the linear momentum is defined by
Z
L := ρ ẋ dv
(3.46)
S
and the vector of externally applied load reads
Z
Z
f := ρ b̄ dv + t da
(3.47)
∂S
S
with the volume acceleration b̄ and the stress vector t. By inserting equations (3.14), (3.28)
and the 2. local form of mass conservation (3.44), we compute the total time derivative
of linear momentum
Z
Z
Z
d
ρ ẋ dv = (ρ̇ ẋ J + ρ ẋ div[v] J + ρ ẍ J) dV = ρ ẍ dv
(3.48)
dt
S
B
S
R
R
Utilizing the Cauchy theorem (3.30) and the divergence theorem ∂S σ n da = S div[σ] dv
we may reformulate the load vector and the balance of linear momentum reads
Z
Z
ρ ẍ dv = (ρ b̄ + div[σ]) dv .
(3.49)
S
S
This leads directly to the local form of the balance of linear momentum
ρ ẍ = ρ b̄ + div[σ] .
(3.50)
3.3.3 Balance of Angular Momentum The balance of angular momentum states
that the total time derivative of the angular momentum h(0) , associated to the fixed point
of origin (0), is equal to the sum of all externally acting moments m(0) , i.e. the equation
ḣ(0) = m(0)
holds. Herein, the angular momentum is defined by
Z
h(0) := x × ρẋ dv
(3.51)
(3.52)
S
and the externally applied moments in the considered Boltzmann continuum read
Z
Z
m(0) = x × ρb̄ dv + x × t da .
(3.53)
S
∂S
Continuum Mechanical Preliminaries
25
If the transport theorem (3.14), the time derivative of J (3.28) and the 2. local form of
mass conservation (3.44) are inserted, the total time differential of the angular momentum
yields
Z
Z
d
x × ρẋ dv = x × ρẍ dv .
(3.54)
dt
S
S
R
R
Applying the Cauchy theorem (3.30) and the divergence theorem ∂S x×σ n da = S (x ×
div[σ] + ǫ : σ T ) dv the balance of angular momentum reformulates to
Z
S
x × ρẍ dv =
Z
x × ρb̄ dv +
S
Z
(x × div[σ] + ǭ : σ T ) dv ,
(3.55)
S
with the third-order permutation tensor


 1 for even permutations (e.g. ǭ123 = 1)

−1 for uneven permutations (e.g. ǭ321 = −1)
ǭabc =
.


0 else
(3.56)
Z
(3.57)
Inserting the local form of the balance of linear momentum (3.50) yields the expression
ǭ : σ T dv = 0 ,
S
which is only fulfilled if the Cauchy stress tensor is symmetric, i.e.
σ = σT .
(3.58)
Here it is remarked, that the symmetry of the Kirchhoff and the 2. Piola-Kirchhoff stress
tensor follows directly from (3.58).
3.3.4 Balance of Mechanical Energy The fourth axiom, the balance of mechanical
energy, states that the time-dependent change in total energy of a physical body is equal to
the increase of total energy. Here, we consider only mechanical and thermal energies, thus,
the increase of total energy consists of the mechanical and thermal power. We subdivide
the total energy into the internal and kinetic energy. The kinetic energy represents the
energy, which is provided by kinematic causes, and is required not to disappear. However,
it is qualified to disperse over the atoms of the body, i.e. it transforms to the kinetic energy
of the disordered atom movement and the potential energy between the atoms. From the
continuum mechanical point of view the kinetic energy is in this case not observable
anymore and we say that the kinetic energy passes into internal energy. The internal
energy is tangible and can be measured as heat, thus, the temperature is a suitable
measure for the medium kinetic energy of the atoms, i.e. the internal energy.
The internal energy E is specified by
E :=
Z
S
ρ e dv
(3.59)
Continuum Mechanical Preliminaries
26
with the specific internal energy e = ê(x, t), which is defined per unit reference mass.
Its time derivative is given with (3.14), (3.28) and the 2. local form of mass conservation
(3.44) by
Z
Z
Z
d
E˙ =
ρ e dv = (ρ̇ e J + ρ ė J + ρ e J div[ẋ]) dV = ρ ė dv .
(3.60)
dt
S
B
S
For the kinetic energy K we obtain the formulation
Z
1
ρ ẋ · ẋ dv
K :=
2
(3.61)
S
and the total time differential of K computes analogously to
Z
Z
d
1
K̇ =
ρ ẋ · ẋ dv = ρ ẋ · ẍ dv .
2
dt
S
S
The mechanical power PM is given by
Z
Z
PM = ρ ẋ · b̄ dv + ẋ · t da
Q=
Z
S
(3.63)
∂S
S
and the thermal power Q reads
(3.62)
ρ r dv −
Z
q · n da ,
(3.64)
∂S
with the external heat source r = r̂(x, t) per unit reference mass and the heat flow vector
q = q̂(x, t). Since the temporary change of total energy must be equal to the increase of
the total energy, the balance of mechanical energy states
Ė + K̇ = PM + Q .
(3.65)
With (3.60), (3.62), (3.63), (3.64), the Gauss integral theorem and the local form of the
balance of linear momentum we obtain the local form of the balance of mechanical energy
ρ ė = σ : d + ρ r − div[q] .
(3.66)
3.3.5 Entropy Inequality Trusting in the words of Rudolf Julius Emmanuel Clausius
(1822–1888) “1. the energy of the universe is constant, 2. the entropy of the universe tends
to a maximum”, which is also known as the first and second law of thermodynamics, the
heat can only transmit from a body of higher temperature to a body of lower temperature
without compensation. It seems to be plausible, because otherwise it would be possible to
produce mechanical work by cooling down a body. These considerations are represented
by the entropy inequality, also known as Clausius-Duhem-Inequality. It postulates, that
the temporary change of the entropy Γ is greater or equal than the total entropy supply
caused by internal heat production minus the entropy supply caused by heat flow over
the surface of the body:
Z
Z
q·n
ρr
dv −
da .
(3.67)
Γ̇ ≥
ϑ
ϑ
S
∂S
Continuum Mechanical Preliminaries
27
Herein, ϑ = ϑ(x, t) denotes the absolute temperature. The entropy characterizes the
measure for that part of the thermal energy, which can not be transformed into mechanical
work, because of its uniform distribution over all molecules. With the notion of the specific
entropy η = η̂(x, t), defined per unit reference mass, the entropy of the body is given by
Z
Γ := ρ η dv .
(3.68)
S
Analogously to the sections before, we calculate the total time differential of the entropy
Z
Z
d
Γ̇ =
ρ η dv = ρ η̇ dv
(3.69)
dt
S
S
and obtain by transforming the surface integral to a volume integral the local form of the
entropy inequality
ρr
q
≥ 0.
(3.70)
ρ η̇ −
+ div
ϑ
ρ
Reformulating (3.70) and inserting the local balance of mechanical energy (3.66) leads to
the alternative formulation of the local form of the entropy inequality
ρ (ϑ η̇ − ė) + σ : d −
1
q · grad[ϑ] ≥ 0 .
ϑ
(3.71)
The postulate of the existence of a scalar-valued free Helmholtz energy ψ̃ := e−ϑ η, which
is defined per unit reference mass, yields with (3.71) the final formulation
1
˙
D := −ρ (ψ̃ − ϑ̇ η) + σ : d − q · grad[ϑ] ≥ 0 .
ϑ
(3.72)
Since this work is interested in isothermal processes, i.e. ϑ̇ = 0 and q = 0, the local form
of the Clausius-Duhem-inequality reduces to
˙
D := σ : d − ρ ψ̃ ≥ 0
(3.73)
in the spatial setting. If the stored energy ψ is defined per unit reference volume we obtain
the relationship ψ = ρ0 ψ̃, which seems to be reasonable, because ψ then represents all
material properties including the referential density ρ0 . Considering the work-conjugated
pairs we are able to write down the Clausius-Duhem-Inequality in the reference configuration:
D := S : Ė − ψ̇ ≥ 0 .
(3.74)
For the pure elastic case we compute the time derivative of ψ = ψ̂(E) and obtain with
(3.74)
D := S : Ė − ∂E ψ : Ė ≥ 0 .
(3.75)
Applying the standard argument of rational continuum mechanics, saying that the latter
equation has to be satisfied for each imaginable process, the constitutive equation for the
stresses is governed by
∂ψ
∂ψ
, with S AB =
.
(3.76)
S=
∂E
∂EAB
Continuum Mechanical Preliminaries
28
The latter equation plays an important role in the framework of finite elasticity. The
constitutive relations for other stresses can be deduced from (3.76). It is remarked that
the notion of constitutive equations is explained in the following section. If the stored
energy ψ := ψ̂(C) and Ė = 12 Ċ are considered, the Clausius-Duhem-Inequality (3.75)
can be reformulated to
D := 12 S : Ċ − ∂C ψ : Ċ ≥ 0
(3.77)
and the constitutive equation for the second Piola-Kirchhoff stresses yields
S=2
∂ψ
,
∂C
with S AB = 2
∂ψ
.
∂CAB
(3.78)
Considering the spatial setting, we assume a stored energy being also a function of the
spatial metric ψ = ψ̂(C) = ψ̂(Ĉ(F , g)), then the Kirchhoff stresses can be directly
computed by the so-called Doyle-Ericksen formula
τ =2
∂ψ
,
∂g
with τ ab = 2
∂ψ
,
∂gab
(3.79)
see e.g. Marsden & Hughes [79].
3.4 Basic Principles of Material Modeling
For the description of the thermodynamic behavior of elastic solids the balance principles,
outlined in the sections above, are available. In detail, there is 1 equation coming from
the conservation of mass (3.44), 3 equations which are included in the balance of linear
momentum (3.50) and 1 equation is obtained by the balance of mechanical energy (3.66).
Here, it is remarked that the balance of angular momentum is taken into account by the
symmetry of the Cauchy-stress tensor σ = σ T , which consists therefore of 6 components.
Totally, 5 equations are given for solving initial- or boundary value problems describing
states of equilibrium within e.g. deformation analysis of thermodynamic solid continuums.
Since the volume acceleration b̄ and the heat source r are given quantities, the unknown
variables appearing in the system of equations are the deformation map ϕ, the stresses
σ, the density ρ, the free energy ψ̄ (or stored energy ψ), the specific entropy η, the
temperature ϑ and the heat flow vector q. Totally, 16 (= 3 + 6 + 1 + 1 + 1 + 1 + 3)
unknown quantities have to be calculated by 5 equations, which is not possible, of course.
In order to close the system of equations 11 (= 16 − 5) additional equations are necessary,
which are known as the constitutive equations. Generally, the symmetry of Cauchy stresses
σ = σ T , the free energy ψ̄, the specific entropy η and the heat flow q are said to be the
constitutive quantities, which are computed by 11 (= 6 + 1 + 1 + 3) additional equations.
Herewith, we obtain a closed system of equations with 16 unknown quantities and 16
(= 5 + 11) equations.
Due to the fact that this thesis deals with isothermal processes, we are able to reduce
the system of equations, because there is no need to take into account the balance of
mechanical energy. Therefore, we consider 4 (= 1 + 3) equations, in which the deformation
map ϕ, the symmetric stresses σ, the density ρ and thus a total amount of 10 (= 3+6+1)
unknown quantities occur. If the stress tensor is defined as a constitutive quantity, we
obtain 6 additional (constitutive) equations and are able to close the system of equations
with 10 equations and 10 variables.
Continuum Mechanical Preliminaries
29
The main difficulty in material modeling is to construct these constitutive equations in
such a way, that they represent the properties of the considered material properly and
fulfill some mathematical conditions to avoid unnecessary numerical complications coming
from material failure. The entropy inequality can be understood as a helpful advice for
constructing material models. It provides for example the condition that the stresses can
be calculated by the free energy, cf. Eq. (3.76). Hence, it is only necessary to construct a
scalar-valued function for the free energy in order to obtain the constitutive equations for
the stresses.
In order to avoid the construction of physically unreasonable material models, there exist
several principles which provide useful advice, see Truesdell & Toupin [130] and
Truesdell [128]. Some of the principles in [130] and [128] are also explained in an
attractive way in e.g. Gummer [39] and de Souza Neto et al. [28]. Especially in
Holzapfel [47] big attention is paid to the principle of material frame indifference,
which is often referred to as principle of objectivity, too. Some of the most important
basic principles of material modeling are explained in the sequel:
• principle of consistency: any constitutive description has to be constructed in such
a way that it does not violate one of the balance principles given above. If a material model fulfills the second law of thermodynamics, too, then it is said to be
thermodynamically consistent.
• principle of equipresence: the principle of equipresence, defined in [130], demands
the same set of functional arguments for all constitutive laws. Herewith, a deductive
proceeding is recommended, which makes sure that no important functional dependencies are omitted when constructing complex material models. The evaluation of
other principles can reduce the number of independent variables.
• principle of causality: this principle assumes the motion and temperature of each
material point to be the agent for the behavior of the body. In the framework of
thermomechanical processes these two field variables are understood as independent
process variables. The dependent variables are then the stresses σ, the heat flow
vector q, the specific energy e and the specific entropy η.
• principle of local agency: here, it is stated that the constitutive variables of material
points in large distance to the considered point do not take effect on the constitutive
quantities of the considered point. Therefore, the constitutive equations can be
reduced in such a way that they become independent of the place. These materials
are referred to as “simple bodies”.
• principle of thermodynamically compatible determinism: the principle postulates
that the thermomechanical quantities in one material point are determined by the
history of motion ϕ and temperature ϑ of all material points. Hereby, the influences arising from future events are precluded as well as the existence of further
independent process variables except for ϕ and ϑ.
• principle of fading memory: this principle states that the constitutive quantities in
large temporary distance do not affect the actual ones.
Continuum Mechanical Preliminaries
30
• principle of material frame indifference: the principle of material frame indifference
requires observer-invariant constitutive laws. This means that if the deformed body
S is placed in the configuration S + by applying an arbitrary rigid body motion Q,
which is an orthogonal transformation, no alterations in the constitutive quantities
may occur. The considered configurations are shown in Fig. 8.
S+
B
x+
X
Q
ϕ(X)
x
S
Figure 8: Rigid body motion applied to the actual configuration.
The transformed actual position vector x+ = Qx leads to the deformation gradient F + = QF . The principle of material frame indifference means that e.g. the
constitutive quantity W satisfies
Ŵ (F ) = Ŵ (F + ) ∀ Q ∈ SO(3), F ,
(3.80)
where W denotes the stored energy depending of F and SO(3) represents the special
orthogonal group, which represents the group of all arbitrary rigid body motions
with detQ = 1 and QT = Q−1 . Since (3.80) has to be satisfied for arbitrary rigid
body motions Q we choose Q = RT and consider the polar decomposition of F
given in (3.15), then we obtain
√
Ŵ (F ) = Ŵ (QF ) = Ŵ (QRU ) = Ŵ (U ) = Ŵ ( C) = ψ̂(C) .
(3.81)
This means that the reduced form of the energy ψ = ψ̂(C) satisfies a priori the
principle of material frame indifference, i.e.
ψ̂(C + ) = ψ̂(C) ∀ Q ∈ SO(3), C = C + ,
(3.82)
with C + = F +T F + = F T QT QF = C. For this reason we concentrate on formulations in terms of the right Cauchy-Green tensor in this thesis.
• principle of material symmetry: the principle of material symmetry, also referred
to as isotropy of space, demands that no rigid body rotations Q ∈ Gk applied to
the reference configuration are allowed to take effect on the constitutive quantities.
Herein, Gk identifies the symmetry group of the considered body and characterizes
Continuum Mechanical Preliminaries
B
31
ϕ(X)
X
QT
x
S
X∗
ϕ(X ∗ )
B∗
Figure 9: Rigid body rotation applied to the reference configuration.
the symmetry properties of the material. For an illustration of the superposed rigid
body rotation see Fig. 9.
With X ∗ = QT X ∀ Q ∈ Gk we obtain F ∗ = F Q, i.e. C ∗ = QT CQ. Then the
isotropy of space postulates with respect to a tensor-valued constitutive quantity,
e.g. the second Piola-Kirchhoff stresses S, that the equation
QT Ŝ(F )Q = Ŝ(F ∗ ) ∀ Q ∈ Gk , F
(3.83)
holds. If the symmetry group Gk is equal to the special orthogonal group SO(3),
then the material is said to be isotropic.
3.5 Representation for Isotropic and Anisotropic Tensor Functions
In the framework of continuum mechanics the description of nonlinear material behavior
is mainly determined by scalar-valued and tensor-valued tensor functions. In this context the so-called representation theorems of isotropic tensor functions play an important
role. Herein, the underlying constitutive tensor functions are represented by scalar-valued
coordinate-independent quantities, which are called invariants, and tensor-valued quantities referred to as tensor generators. The classical invariant theory is described in e.g.
Grace & Young [40], Elliott [30], Turnbull [131], Weyl [141] and Gurevich
[42]. In Schur [110] the so-called integrity basis is defined as that finite subsystem of
the whole invariant system of a group, so that any further invariant can be formulated
as a polynomial of this special subsystem. The Hilbert Theorem, cf. Gurevich [42] and
Spencer [123], establishing that for a finite system of vectors and tensors there exists
an integrity basis consisting of a finite number of invariants, legitimates searching the
irreducible set of invariants, which is required for the description of general isotropic
or anisotropic materials. Introductive contributions to this subject are put down on paper in Boehler [16], [17] and a concise compendium is given in Betten [14]. For the
application and representation of isotropic and anisotropic tensor functions we refer to
Rychlewski & Zhang [101], Schröder [102], Smith [117], [118], [119], Smith et al.
[120], [121], [122], Spencer [124], [125], Ting [127], Wang [135], [136], [137], [138] and
Continuum Mechanical Preliminaries
32
Koorsgaard [69],[70]. Also very interesting contributions to the representation theory
are found in the publications Zheng & Spencer [146], [147] and Zheng & Boehler
[148]. A unified invariant approach is introduced in Zheng [149].
Exemplarily, the reason for introducing an invariant basis is shown in the sequel. The
representation of such a scalar-valued tensor function by scalar-valued invariants has not
only the advantage that it will become more manageable to investigate a function depending on scalar-valued quantities. In the last section we have seen that the constitutive
function ψ := ψ̂(C) is able to satisfy the principle of material frame indifference automatically. Applying a special orthogonal transformation Q ∈ SO(3) to the referential basis
we obtain the relation
C ∗ = (F ∗ )T F ∗ = QT F T F Q = QT CQ ,
(3.84)
and directly notice that the principle of material symmetry is not generally satisfied for
the anisotropic case, i.e.
ψ̂(C) 6= ψ̂(C ∗ ) ∀ Q ∈ SO(3), C
(3.85)
If the stored energy is formulated in terms of the invariants of the right Cauchy-Green
deformation tensor, then the principle of material symmetry is satisfied automatically for
the isotropic case.
3.5.1 Representation Theorems for Isotropic Tensor Functions Let us consider
the arbitrary constitutive scalar-valued tensor function f , the vector-valued tensor function f and the second-order tensor-valued tensor function F . The set of whose arguments
consists of a finite number of first-order tensors (vectors) v j , a finite number of secondorder tensors T i and a finite number of tensors of higher order, which are not denoted for
simplicity. Then we obtain the definitions

ˆ j , T i , ...) 
f := f(v


.
(3.86)
f := f̂ (v j , T i , ...)


F := F̂ (v , T , ...) 
j
i
Isotropic tensor functions are defined as tensor functions which are invariant with respect
to special orthogonal transformations. Hence, if the equations

T
T
ˆ
ˆ
f(v j , T i , ...) = f(Q v j , Q T i Q, ...) 


T
T
T
∀ Q ∈ SO(3), v j , T i , ...
(3.87)
Q f̂ (v j , T i , ...) = f̂ (Q v j , Q T i Q, ...)


QT F̂ (v , T , ...)Q = F̂ (QT v , QT T Q, ...) 
j
i
j
i
hold, then f , f , F are isotropic tensor functions. The fundamental statement of the
representation theory is, that each isotropic tensor function can be expressed by scalarvalued invariants Is , which are the traces of powers of the argument tensors, and tensor
generators f k , F k , so we obtain

∗

ˆ
ˆ
f(v j , T i , ...) = f (I1 , I2 , .., Ip )



X

ϕ̂k (I1 , I2 , .., Ip ) f k 
f̂ (v j , T i , ...) =
.
(3.88)
k

X


F̂ (v j , T i , ...) =
ϕ̂k (I1 , I2 , .., Ip ) Fk 


k
Continuum Mechanical Preliminaries
33
Herein, ϕ̂k (I1 , I2 , .., Ip ) identify scalar-valued functions depending on the irreducible set
of p scalar invariants. For the derivation of (3.88)2,3 via introducing auxiliary vectors and
tensors we refer to Koorsgaard [69], [70]. The existence of a specified irreducible number
of invariants p is ensured by the Hilbert’s Theorem, cf. [42], [123]. Here, we pass on the
deduction of an explicit integrity basis and merely state the well-known and canonical
set of invariants for a scalar-valued function depending on one symmetric second-order
tensor. In the case of isotropic hyperelasticity we consider the stored energy ψ := ψ̂(C)
as a function of the right Cauchy-Green deformation tensor. The invariants of C are
determined by the characteristic polynomial
det[λ̃1 − C] =
3
X
k=0
(−1)k Ik λ̃3−k = I0 λ̃3 − I1 λ̃2 + I2 λ̃ − I3 λ̃0 = 0
(3.89)
with the eigenvalues λ̃ of C and the second-order identity tensor 1. Since I0 = 1 and
λ̃0 = 1 the set of the so-called principal invariants is given by
I1 := tr[C],
I2 := tr[cofC],
I3 := det[C] .
(3.90)
These invariants can also be expressed in terms of the so-called basic invariants, which
are given by
J1 := tr[C], J2 := tr[C 2 ], J3 := tr[C 3 ]
(3.91)
with the relations
J1 = I1 ,
J2 = I12 − 2 I2 ,
J3 = I13 − 3 I1 I2 + 3 I3 .
(3.92)
As an example, we consider the first principle invariant of C and apply a rigid body
rotation to quantities in the reference configuration so that X ∗ = QT X, cf. Fig. 9. By
the algebraic transformation
I1∗ = tr[C ∗ ] = QT CQ : 1 = QQT : C T = I1
(3.93)
we observe that the principle of material symmetry for isotropy ψ̂(I1 ) = ψ̂(I1∗ ) ∀ Q ∈
SO(3) is satisfied.
3.5.2 Representation of Transverse Isotropy by Isotropic Tensor Functions
Anisotropic materials are characterized by the property that arbitrary applied rigid body
rotations on the reference configuration take effect on the constitutive functions. Demonstratively, this means that it is not possible to rotate a body arbitrarily in a fixed coordinate system, keep the deformation axis with respect to the fixed coordinate system
and then expect the stress response to remain unaltered when the body is deformed. For
example, a cubic material, which has two perpendicular oriented preferred directions a1
and a2 in the two-dimensional case, cf. Fig. 10a, would be able to be transformed by
the planar rigid body rotations 90◦ , −90◦ and 180◦ without changing the stress response.
The description of such symmetries is based on rotations and reflections. Considering a
right-handed orthonormal coordinate system spanned by the three vectors w1 , w2 and
w3 the reflections concerning planes oriented normal to w 1 are given by
R(w1 ) = −w 1 ⊗ w 1 + w 2 ⊗ w 2 + w 3 ⊗ w 3 .
(3.94)
Continuum Mechanical Preliminaries
34
Generally, in the case of anisotropy the principal directions of the strains (nC I , nC II ) and
the principal directions of the stresses (nS I , nS II ) are not the same. This fact is illustrated
in Fig. 10, cf. Boehler [16], where the circle characterizes the undistorted reference
configuration and a1 , a2 the preferred directions. In contrast to this the directions of the
principal strains and stresses are identical for the isotropic case, see [16].
a2
a2
a2
nC I
nS II
nC II
a1
a)
nS I
a1
b)
a1
c)
Figure 10: Physical interpretation of the definition of anisotropic material behavior: preferred directions ai , in the a) undistorted reference configuration and principle axis of the
b) strains and c) stresses nC i and nS I in the deformed state, cf. [16].
In order to be able to utilize the representation theorems for isotropic tensor functions the
anisotropic constitutive equations are modified in such a way, that they become isotropic
tensor functions. For this purpose, the concept of structural tensors is applied, which has
been developed in the seventies of the last century. For the introduction of structural
tensors in an attractive way with important applications see Boehler [15]. This concept
works appropriately if the anisotropy can be specified by certain directions, lines or planes.
Transversely isotropic materials are characterized by one preferred direction with special
properties, and rotation-symmetric properties around this preferred direction. Then the
material symmetry group is given by
Gti = {±1, Q̃(α, a) with 0 < α < 2π}
(3.95)
wherein the central inversion is calculated by −1 = R(w 1 )R(w 2 )R(w3 ) according to
(3.94). Rotations around the preferred direction a with the angle α are denoted by
Q̃(α, w). We introduce the structural tensor M of second order and consider the constitutive equation
S := Ŝ(C, M ) .
(3.96)
The principle of material symmetry requires the invariance of the constitutive equation
under orthogonal transformations of the material symmetry group, i.e.
QT Ŝ(C, M )Q = Ŝ(QT CQ, QT M Q) ∀ Q ∈ Gti
(3.97)
holds. The consequence of this is illustrated in Fig. 11, cf. [16].
The goal is now to construct a structural tensor such that an isotropic tensor function is
obtained, i.e. the material symmetry group is extended to the special orthogonal group.
For this purpose we have to consider a structural tensor whose invariant group is equal
to the material symmetry group, i.e.
M = QT M Q ∀ Q ∈ Gti
(3.98)
Continuum Mechanical Preliminaries
ã2
Qa2 a2
ã2
nC I
nS
nC II
35
nC I
Qa2
a2
nS I
Qa1
Qa1
II
nS I
ã1
nC II
ã1
a)
a1
nS II
a1
b)
Figure 11: Physical interpretation of the principle of material symmetry: a) preferred
directions ai and b) transformed preferred directions Qai . The principle axis of strains and
stresses are invariant under the transformation Q with respect to the transformed systems.
holds. Let a with |a| = 1 describe the preferred direction in the reference configuration,
then we are able to introduce the suitable structural tensor
M := a ⊗ a with tr[M ] = 1 ,
(3.99)
QT Ŝ(C, M )Q = Ŝ(QT CQ, M ) ∀ Q ∈ Gti .
(3.100)
QT Ŝ(C, M )Q = Ŝ(QT CQ, QT M Q) ∀ Q ∈ SO(3) ,
(3.101)
who preserves the material symmetry group. Then we obtain an anisotropic tensor function with respect to C and satisfy the equation
By introducing the structural tensor (3.99) we extend the material symmetry group to
the special orthogonal group and satisfy
which is the definition for an isotropic tensor function. Thus, we are able to utilize the
representation theorems for isotropic tensor functions and consider a constitutive equation
which consists of scalar-valued invariants and tensor generators. The invariants for the set
of arguments consisting of the two symmetric second-order tensors C and M are given
by the traces of the powers of the argument tensors up to a finite order. Then we obtain
the invariants (3.90) for C and
J4 := tr[CM ],
J5 := tr[C 2 M ],
J6 := tr[CM 2 ],
J7 := tr[C 2 M 2 ]
(3.102)
for C and M , cf. [17]. It should be remarked, that due to the special property (3.99)2
the invariants of M , e.g. IM 1 = tr[M ] = 1, are equal to 1 and therefore no expedient
invariants for the representation. Because of |a| = 1 we obtain that J6 = J4 and J7 = J5 ,
thus the final polynomial basis
P1 := {I1 , I2 , I3 , J4 , J5 }
(3.103)
provides the construction of transversely isotropic material models and we end up in the
representation
ψ := ψ̂(I1 , I2 , I3 , J4 , J5 ) .
(3.104)
For a physical interpretation of the fourth invariant in (3.103) we let ã = F a describe
the preferred direction in the actual configuration. Then we are able to reformulate J4 to
2
|ã|
J4 = a · Ca = F a · F a =
.
(3.105)
1
Due to the fact that 1 = |a|, we see that J4 represents the square of the stretch in the
direction of a.
Continuum Mechanical Preliminaries
36
Finite-Element Method
4
37
Finite-Element Method
In the framework of computer simulation of a variety of engineering applications numerical
methods play a major role. Especially for nonlinear problems the development of powerful
hardware in the last years has given rise to the importance of numerical methods. An
advantageous concept for solving nonlinear boundary value problems efficiently is the
Finite-Element method. Since this work investigates some numerical simulations by using
this method we briefly outline the basic concept in a first part of this section. The second
part explains the particular finite elements utilized in the numerical analysis of later
sections. Hence, we subdivide this section into the two parts
• Fundamentals of the Finite-Element Method, where we start from variational principles in order to explain the main concept for the general case and
• Special Finite-Element Implementations, where the utilized triangle element with
quadratic ansatz functions is explained as well as the appropriate surface load element, a special spring-interface element and how inertia terms are incorporated.
In this section we only describe the basic concept of the Finite-Element method. For
further information the interested reader is referred to the extensive literature, as e.g.
Zienkiewicz & Taylor [150], Bathe [12] and Wriggers [143].
4.1 Fundamentals of the Finite-Element Method
Firstly, the balance of linear momentum and the balance of angular momentum are considered together with some defined boundary conditions. This leads to a set of partial
differential equations which are not generally able to be solved analytically. Therefore,
variational principles are applied and the so-called weak form of equilibrium is obtained,
which is described in the first part of this section. The second part provides the linearization of the weak form necessary in the context of nonlinear problems. The obtained
formulation represents the starting point for the Finite-Element method, which discretizes
the continuous body by a finite number of elements. This approach is explained in the
third part.
4.1.1 Variational Problem The set of differential equations given by the local form
of the balance of linear momentum (3.50), or in the associated form with the 1st PiolaKirchhoff stresses
Div[P ] + ρ0 b̄ = ρ0 ẍ with Div[P ] = P aA ,A ,
(4.1)
and the displacement and traction boundary conditions
u = ū on ∂Bu
and P N̄ = t̄ on ∂Bσ ,
(4.2)
respectively, is referred to as the strong form of equilibrium. We multiply the strong form
by a suitable vector-valued test function δu = {δu|δu = 0 on ∂Bu } (here the test function
can be interpreted as virtual displacements) and integrate over the volume; then we obtain
Z
G := [Div[P ] + ρ0 (b̄ − ẍ)] · δu dV = 0
(4.3)
B
Finite-Element Method
38
T
with the boundary conditions given above. The
Div[P
R transformation
R ]·δu = Div[P δu]−
T
P : Grad[δu], applying the Gauss theorem B Div[P δu]dV = ∂Bt δu · t̄dA and implementing the static boundary conditions lead to the weak form of equilibrium
G := Gint − Gext = 0 ,
with
G
G
int
ext
=
=
Z
B
Z
(4.4)






P : Grad[δu] dV
δu · t̄ dA +
∂Bt
Z
B
.
(4.5)

δu · ρ0 (b̄ − ẍ) dV 



If we assume the existence of a potential we alternatively obtain the internal part of the
weak form by the variation of the potential of internal energy


Z
Z
Z
∂W
int


: δF dV = P : Grad[δu] dV .
(4.6)
G =δ
Ŵ (F ) dV =
∂F
B
B
B
Since G represents the virtual work, the internal part can be also represented by all
work-conjugated pairs, e.g.
P : Grad[δu] = S : F T Grad[δu] = S : 12 (F T δF + δF T F ) = S : δE
(4.7)
and we obtain the virtual internal work
G
int
=
Z
S : δE dV .
(4.8)
B
4.1.2 Linearization There exist several possibilities for nonlinearities occurring in
continuum mechanics. Using a nonlinear strain measure, as e.g., the Green-Lagrange
strain tensor, will lead to geometrical nonlinearities, while choosing nonlinear constitutive equations, as e.g., necessary for the description of elasto-plastic or visco-plastic
effects or especially for biological soft tissues, result in so-called material nonlinearities.
For detailed explanations of continuum-mechanical nonlinearities we refer to Section 2 in
Wriggers [143]. Due to these nonlinearities the analytical solution of boundary value
problems of such type as (4.4) is not possible. Thus, for a numerical solution the linearization of the underlying equations is required, in order to be able to apply e.g., a
Newton-iteration scheme as the Newton-Raphson procedure. This procedure represents
a very efficient method for solving nonlinear algebraic systems of equations because the
iterative solution converges quadratically in a sufficiently small neighborhood of the real
solution. We obtain the linearization of the weak form of equilibrium at u = ū by
LinĜ(ū, δu, ∆u) := Ĝ(ū, δu) + ∆Ĝ(ū, δu, ∆u) ,
(4.9)
wherein the incremental weak form of equilibrium ∆G is computed by the directional
derivative defined as
d
= D Ĝ(ū, δu) · ∆u .
(4.10)
∆Ĝ(ū, δu, ∆u) = [Ĝ(ū + ǫ∆u, δu)]
dǫ
ǫ=0
Finite-Element Method
39
Herein, Ĝ(ū, δu) is referred to as the residuum. Assuming displacement-independent (conservative) volume forces ρ0 b̄ and traction forces t̄ at the boundary and neglecting inertia terms (ẍ = 0), the directional derivative of the external virtual work becomes zero
(DGext = 0). Since δF = Grad[δu] does not depend on the primary variable u the
linearization of δF is equal to zero, too, and we obtain the incremental virtual work
Z
int
∆Ĝ(ū, δu, ∆u) = D Ĝ (ū, ∆u) · ∆u = δF : ∆P dV
(4.11)
B
with the incremental first Piola-Kirchhoff stresses
∆P = A : ∆F
with ∆F = Grad[∆u] .
(4.12)
Herein, the nominal tangent moduli are defined by A := ∂F P . Finally, we obtain the
linearization of the weak form of equilibrium given by
Z
Z
Z
Z
(4.13)
LinG = P : δF dV −
δu · t̄ dA − δu · ρ0 b̄ dV + δF : A : ∆F dV .
B
∂Bt
B
B
Since the first Piola-Kirchhoff stress tensor is not symmetric, and since we consider constitutive equations of the form S := Ŝ(C) (in order to account for the principle of material
frame indifference), we linearize the alternative weak form of equilibrium (4.8). With the
variation and the linearization of the Green-Lagrange strain tensor
δE = δ[ 12 (C − 1)] = 21 δC
and ∆E = 12 ∆C
we obtain the incremental virtual work
Z
Z
int
1
∆G = DG · ∆u = ∆S : 2 δC dV + S : 21 ∆δC dV .
B
(4.14)
(4.15)
B
Herein, the virtual right Cauchy-Green deformation tensor and the linearization of δC
are given by


δC = δ(F T F )
= δF T F + F T δF
.
(4.16)
∆δC = ∆(δF T F + F T δF ) = δF T ∆F + ∆F T δF 
Inserting the linearization of the second Piola-Kirchhoff stresses
∆S = C : 12 ∆C
with C = 2∂C S
into (4.15) yields the incremental virtual work
Z
Z
1
1
δC : C : 2 ∆C dV + S : 21 ∆δC dV ,
∆G =
2
B
(4.17)
(4.18)
B
with the linearized right Cauchy-Green deformation tensor
∆C = ∆(F T F ) = ∆F T F + F T ∆F .
(4.19)
Finite-Element Method
40
The complete linearization of the weak form of equilibrium in terms of S and C reads
Z
Z
Z
ext
1
1
1
(4.20)
LinG = S : 2 δC dV − G + 2 δC : C : 2 ∆C dV + S : 21 ∆δC dV .
B
B
B
In each iteration step n of Newton-type iteration procedures the incremental displacements
∆un+1 are computed so that LinG = 0 and the total displacements are updated un+1 =
un + ∆un+1 . This iteration step is repeated until the residuum (the actual weak form of
equilibrium Ĝ(u, δu)|u=un+1 ) is equal to zero. In this case the incremental displacements
tend to zero and the incremental virtual work vanishes.
4.1.3 Discretization The analytical solution of LinG = 0 with G of the type given
in (4.13) or (4.20) will most probably be impossible. Therefore, we solve this equation
numerically by using the Finite-Element method. The main idea of this method is to
replace the real physical body by a finite number of polygonal elements, each of them approximating the primary variables, here incremental displacements, inside these elements
via ansatz functions considering the values of the incremental displacements at the corners of the polygonal elements (nodes). Hereby, the infinitesimal number of incremental
displacements located at each point in the real body is reduced to a certain number of
incremental displacements being located at the nodes of the Finite-Element mesh.
Mathematically, the geometric approximation Bh of the real body B may be expressed as
the union of all elemental bodies Be
h
B≈B =
nele
[
e=1
Be ,
(4.21)
where nele represents the number of elements. In Fig. 12 the idea of approximating the
physical body by a finite number of elements is illustrated.
Be
η
ξ
Ωe
I
B
Bh
Figure 12: Approximation of the real body by a finite number of elements, here quadrilateral elements with nen = 4 and nele = 17 (nen and nele represent the number of element
nodes and the number of elements, respectively).
This idea demands of course that the accuracy of the geometric approximation should
increase with increasing number of elements. In this section we exemplarily describe the
two-dimensional Finite-Element formulation within the isoparametric concept, where the
geometry as well as the displacements inside one element are approximated by the same
Finite-Element Method
41
ansatz functions. Since the material models investigated in this work are formulated in the
reference configuration we only provide the Finite-Element formulation in the Lagrangian
setting. The reference configuration, the actual configuration and the isoparametric subspace are linked by transformation maps as shown in Fig. 13 for a quadrilateral element.
η
η
ϕt (X)
Be
X
x
ξ
Se
F = ∇ϕ
ξ
X2
x2
η
X1
J
(1, 1)
X̂(ξ)
x1
j
x̂(ξ)
ξ
(−1,−1)
Ωe
Figure 13: Configurations within the isoparametric concept.
We obtain the geometry in the reference configuration and the actual configuration by
X = X̂(ξ, η) =
nen
X
N̂I (ξ, η)X I
and x = x̂(ξ, η) =
I=1
nen
X
N̂I (ξ, η)xI ,
(4.22)
I=1
respectively, wherein the ansatz functions are denoted by N̂(ξ, η). In case of the quadrilateral element with bilinear ansatz functions the ansatz function reads N̂I (ξ, η) =
1
(1 + ξξI )(1 + ηηI ) and ξ ∈ [−1, 1] and η ∈ [−1, 1] are the natural coordinates. The
4
coordinates and the ansatz functions for a node I are denoted by NI and ξI , ηI , respectively. We define the discrete nodal displacements dI such that the actual position vectors
are computed by
x I = X I + dI .
(4.23)
With given ansatz functions NI we approximate the physical, the virtual and the incremental displacements in the same way as the geometry (4.22) and achieve
u=
nen
X
NI dI
,
δu =
I=1
nen
X
NI δdI
and ∆u =
I=1
nen
X
NI ∆dI .
(4.24)
I=1
Due to the fact that in (4.20) not only the displacements occur, but also derivatives of
the displacements, i.e.
δF = Grad[δu] and ∆F = Grad[∆u] ,
(4.25)
we also need the approximations for these derivatives. We define the natural coordinate
vector ξT = [ξ, η] and obtain the Jacobi matrices, cf. Fig. 13, by
J=
∂X
∂ξ
and j =
∂x
.
∂ξ
(4.26)
Finite-Element Method
42
Then the derivatives of the ansatz functions with respect to the referential coordinates
are governed by
∂NI
∂NI ∂ξ
= J −T
.
(4.27)
Grad[NI ] =
∂ξ ∂X
∂ξ
In order to arrive at a comfortable abbreviation we now turn over to the matrix notation,
where the second-order tensors are denoted by vectors while the fourth-order tensors
are characterized by second-order tensors, cf. Appendix D, where the three-dimensional
matrix representation is given. Then the virtual and incremental right Cauchy-Green
deformations are given by
1
δC
2
=
nen
X
B I δdI
1
∆C
2
and
I=1
=
nen
X
B I ∆dI
(4.28)
I=1
with the B-matrix for the two-dimensional case


F11 NI,1
F21 NI,1


BI = 
F12 NI,2
F22 NI,2
.
F11 NI,2 + F12 NI,1 F21 NI,2 + F22 NI,1
(4.29)
Herein, NI,A denotes the derivative of the ansatz function with respect to the referential
coordinates XA . With the approximations given in (4.28) the discretized weak form of
equilibrium for one element (elemental residuum) reads
e
e
e
Ĝ (d̄ , δd ) =
nen
X
(δdeI )T r eI
I=1
=
nen
X
I=1
(δdeI )T r e,int
− r e,ext
I
I
with the internal and external elemental residual vectors
Z
e,int
rI
:= B TI S dV
r e,ext
I
:=
BZe
NI t̄ dA +
δBte
Z
Be







.
(4.30)
(4.31)

NI ρ0 b̄ dV 




We define the complete elemental vector of virtual nodal displacements
(δde )T = (de1 )T (de2 )T ... (denen )T
and the complete elemental residual vector
r e = r e1 re2 ... r enen
(4.32)
(4.33)
and reformulate the complete elemental residuum by
Ge = (δde )T r e .
(4.34)
The increment of the linearized weak form of equilibrium is given by
Z
Z
1
1
δC : C : 2 ∆C dV + S : 21 ∆δC dV ,
∆G =
2
|B
{z
∆Gmat
}
|B
{z
∆Ggeo
}
(4.35)
Finite-Element Method
43
cf. (4.18). The first integral is often referred to as the material part and the second integral
as the geometric part. With the approximations given in (4.28) we obtain the discretized
form of the material part for one element by
Z
nen X
nen
X
e,mat
e,mat
e T
e
e,mat
(4.36)
∆G
=
(δdI ) K IJ ∆dJ with K IJ := B TI C̄B J dV .
I=1 J=1
Be
It should be noted that here C̄ are the referential tangent moduli in the reduced matrix
notation, cf. Appendix D for the three-dimensional case. For the geometric part of the
incremental virtual work we use the index notation. At first consider the approximations
for the virtual and incremental deformation gradient
δF
a
A
=
nen
X
NI,A δdaI
and ∆F
a
A
=
nen
X
NI,A ∆daI .
(4.37)
I=1
I=1
Then we obtain the approximation for the linearized virtual right Cauchy-Green deformation tensor



1

∆δCAB = 12 (δF a A gab ∆F b B + ∆F a A gab δF b B )

2
!
nen
nen
nen
nen
(4.38)
X
X
X
X

NI,A δdaI gab
NJ,B ∆dbJ +
NI,A ∆daI gab
NJ,B δdbJ 
= 21


I=1
J=1
I=1
J=1
and a further algebraic transformation yields the geometric part for one element
Z
nen X
nen
X
e,a
e,geo
(NI,A gab NJ,B ) S AB dV ∆de,b
∆G
=
δdI
J .
I=1 J=1
(4.39)
Be
|
{z
e,geo
=:kIJ
}
For the two-dimensional case we introduce the elemental stiffness matrix


e,geo
k
0
IJ
,
K e,geo
:= 
IJ
e,geo
0
kIJ
(4.40)
and define the complete incremental displacement vector for one element analogously to
(4.32). Then we obtain the complete elemental stiffness matrix


e
e
e
K 11
K 12 ... K 1 nen




e
e
e
 K 21
K 22 ... K 2 nen 
e

(4.41)
K =



 ...
...
...
...


e
e
e
K nen 1 K nen 2 ... K nen nen
with K eIJ = K e,mat
+ K e,geo
IJ
IJ . By arranging the elemental virtual and incremental displacements to their global form
δd = (δd1 , δd2 , ..., δdnen )T
and ∆d = (∆d1 , ∆d2 , ..., ∆dnen )T
(4.42)
Finite-Element Method
44
the global stiffness matrix is well-defined by the assemble operator
nele
K=
AK
e
,
(4.43)
e=1
and we obtain the linearized system of equations by setting LinG = 0:
δdT (K∆d + r) = 0 .
(4.44)
In each iteration step the global incremental displacement vector is updated, i.e. d ⇐
d + ∆d, and the iteration is repeated until the residuum r is less than a certain tolerance.
It should be noted that the integrals occurring in this section are generally not able to be
solved analytically and therefore numerical integration schemes are required. In this work
we use the Gauss integration procedure, which replaces the integral by a sum of integrand
functions evaluated at certain Gauss points multiplied by certain weighting factors. For
detailed information about the Gauss integration scheme for 1-, 2- or three-dimensional
integrals we refer to e.g., Wriggers [143].
4.2 Special Finite-Element Implementations
In later sections we investigate some heterogeneous numerical simulations where e.g.,
the overexpansion of an atherosclerotic artery is simulated. Therefore, we describe some
special Finite-Element implementations in this section. At first we explain the FiniteElement formulation used in the heterogeneous simulations, which is a triangular element
with quadratic ansatz functions. Then we describe the implemented surface load element
utilized for applying internal pressure inside the artery. The consideration of inertia terms
is as well provided as the description of a special spring interface element.
4.2.1 Triangular Element with Quadratic Ansatz Functions We are interested
in the simulation of biological soft tissues, in particular arterial tissues, thus, we have
to account for the incompressibility constraint. It should be noted that in this section
we restrict ourselves to the two-dimensional case. Elements like the quadrilateral (Q1)
or triangular element (T1) with linear ansatz functions are not able to represent the mechanical response of incompressible materials when bending is applied. For incompressible
materials the Q1-element converges very slowly to the true displacement and does not
converge to the true stresses when increasing the number of elements, cf. e.g., Klaas,
Schröder, Stein & Miehe [66]. This problem is referred to as the incompressible locking. Generally, so-called mixed Finite-Element formulations are used when considering
incompressible materials due to the fact that they overcome the incompressible locking
problem. Here, not only the displacement, also e.g., strains or stresses, are approximated
by ansatz functions. Based on the Hu-Washizu functional in Simo, Taylor & Pister
[115] the fundamental for the Q1P0-element is given, where the displacements are approximated by linear ansatz functions and the pressure is approximated by constant ansatz
functions. Elements of such type usually require the constitutive equation to be additively decoupled into a volumetric and an isochoric part. This is not generally possible
concerning material models governed by polyconvex stored energy functions since some
important terms for the decoupling are not polyconvex, cf. Section 5. Therefore, we will
use triangular elements with quadratic ansatz functions (T2).
Finite-Element Method
45
For the considered 6-node triangular element we use the definition for the node numbering
as shown in Fig. 14a), where the parametric subspace of the element is illustrated.
η
(0,1)
3
η
1
3
6
5
6
1
Ω
1
(0,0)
4
2
(1,0)
5
4
ξ
a)
b)
2
ξ
Figure 14: a) 6-node triangular element and b) quadratic ansatz function N1 .
The quadratic ansatz functions in the 6-node triangular element are given by

N1 = λ(2λ − 1) , N4 = 4ξλ , 


N2 = ξ(2ξ − 1) , N5 = 4ξη ,



N3 = η(2η − 1) , N6 = 4ηλ ,
(4.45)
wherein the abbreviation λ = 1 − ξ − η is used. As an example, the ansatz function for
node 1 is plotted in Fig. 14b). The derivatives of the ansatz functions with respect to the
parametric coordinates are computed by


N1,ξ = 4(ξ + η) − 3 , N1,η = 4(ξ + η) − 3 , 





N2,ξ = 4ξ − 1 ,
N2,η = 0 ,





N3,ξ = 0 ,
N3,η = 4η − 1 ,
(4.46)

N4,ξ = 4(1 − 2ξ − η) , N4,η = −4ξ ,






N5,ξ = 4η ,
N5,η = 4ξ ,




N
= −4η ,
N
= 4(1 − 2ξ − η) . 
6,ξ
6,η
Hereby, we are able to compute the derivatives of the ansatz functions with respect to
the referential coordinates by equation (4.27) and proceed as explained in Section 4.1.3.
It is noted that we apply for the numerical integration the Gauss integration scheme with
three Gauss points located inside the element. For details on this procedure and for the
Gauss point coordinates and weighting factors of the Gauss scheme see Wriggers [143].
4.2.2 Surface Load Element with Quadratic Ansatz Functions To apply hydrostatic pressure to a surface in two-dimensional simulations a one-dimensional surface load
element is required. Since we consider triangular elements with quadratic ansatz functions
for the discretization of the physical body we also use quadratic ansatz functions for the
load line element. In Fig. 15 the definition for the node numbering used in the sequel is
illustrated. The origin of the parametric coordinate ξ is located in the center of the line
element in the subspace Ω, so that ξ ∈ [−1, 1].
Finite-Element Method
1
3
46
t̄
2
Ω
1
3
2
ξ
S
ξ = −1
a)
ξ=0
ξ=1
b)
Figure 15: a) 3-node line element for surface load and b) parametric coordinate and definition of node numbering.
We consider the part for external loads in the weak form of equilibrium
Z
ext1
G
=
δuT t̄ dA ,
(4.47)
∂Bt
cf. (4.5)2 . We account for hydrostatic pressure by replacing the traction vector by t̄ = p n,
where n is the norm vector with respect to the surface and p is the value for the hydrostatic
pressure, and insert the approximation for the virtual displacements (4.24)2 . Then we
obtain the discretization of Gext1 for one element
G
e,ext1
=
3
X
(δdeI )T
I=1
Z
NI p n dA .
(4.48)
∂Bte
For one-dimensional elements the quadratic ansatz functions, given by
N1 = 21 ξ(ξ − 1) ,
N2 = 12 ξ(ξ + 1) and N3 = (1 − ξ 2 ) ,
(4.49)
are inserted in (4.48). We are interested in applying hydrostatic pressure and therefore we
consider an external load which changes with changing deformations of the body. Hence,
the considered load is not conservative and during linearization the increment of Gext1
does not vanish. Here we abstain from the description of the complete linearization as
well as providing details of the Gauss integration scheme for one-dimensional problems
and refer to relevant literature as e.g., Zienkiewicz & Taylor [150], Bathe [12] and
Wriggers [143].
4.2.3 Implementing Inertia Terms In the sections 4.1.2 and 4.1.3 only the quasistatic case is discussed, i.e. the acceleration is neglected (ẍ = 0). In this section we focus
on the consideration of acceleration terms and regard the inertia part of the weak form
of equilibrium
Z
tia
G = ρ0 δuT ẍ dV ,
(4.50)
B
cf. (4.5)2 , and the linearization of Gtia reads
LinG
tia
=G
tia
+ ∆G
tia
with ∆G
tia
=
Z
B
ρ0 δuT ∆ẍ dV .
(4.51)
Finite-Element Method
47
Using the ansatz functions of the considered element formulation, here the triangular
element with quadratic ansatz functions, we approximate the virtual displacement and
the incremental acceleration by
δu =
nen
X
NI δd and ∆ẍ =
I=1
nen
X
NI ∆d̈ ,
(4.52)
I=1
taking into account that the referential position vectors are independent of time and
therefore ẍ = ü. For one element the inertia part of the linearized weak form of equilibrium
reads
nen X
nen
nen
X
X
e
(δdeI )T r e,tia
(4.53)
+
(δdeI )T K e,tia
LinGe,tia =
I
IJ ∆d̈J .
I=1 J=1
I1
R
e,tia
With the abbreviation kIJ
= B ρ0 NI NJ dV the elemental stiffness matrix for the twodimensional case and the elemental residual vector are given by


e,tia
k
0
e
 and r e,tia
 IJ
K e,tia
= K e,tia
d̈ ,
(4.54)
IJ =
I
IJ
e,tia
0
kIJ
respectively. We reformulate LinGe,tia in the complete matrix notation replacing the summations and therefore set
1
2
∆d̈ = (∆d̈ , ∆d̈ , ..., ∆d̈
nen T
)
1
2
and d̈ = (d̈ , d̈ , ..., d̈
nen T
) .
(4.55)
Then we obtain by the assemble operator the global mass matrix and the global dynamic
residual vector
nele
M=
AK
e,tia
and r tia = M d̈
(4.56)
e=1
and with (4.44) the complete system of equations taking into account inertia terms reads
M ∆d̈ + K∆d = −r − M d̈ .
(4.57)
We notice that in (4.57) not only the incremental displacements, but also the incremental
accelerations occur. Such systems of differential equations can be numerically solved by
the Newmark procedure, which is explained in the sequel.
The actual discrete accelerations and velocities for the time tn+1 at the end of a typical
time interval are given by
d̈ =
d − d̃
β̄ ∆t2
˙
and ḋ = d̃ + ∆t γ̄ d̈ .
(4.58)
Herein, β̄ and γ̄ are the Newmark parameter and ∆t = tn+1 − tn is the time increment; d̃
˙
and d̃ are the predictors of d and ḋ, which are defined by

∆t2
(1 − 2β̄)d̈n 
d̃ := dn + ∆t ḋn +
2
.

d̃˙ := ḋn + ∆t(1 − γ̄)d̈n
(4.59)
Finite-Element Method
48
It should be noted that we skip the index (•)n+1 for the actual quantities. We see that the
predictors depend only on quantities at time tn and can therefore be directly computed.
Inserting the predictors (4.59) into (4.58)1 and with the linearization of (4.58)1
∆d̈ =
1
∆d
β̄ ∆t2
(4.60)
we obtain the system of linear equations
K ef f ∆d = −r − M d̈ with K ef f := K +
1
M.
β̄ ∆t2
(4.61)
The choice of values for the Newmark parameters β̄ and γ̄ influence the performance of the
algorithm. It is remarked that the Newmark procedure is stable for 2β̄ ≥ γ̄ ≥ 21 . An often
used set of the parameters is based on the assumption of constant average acceleration;
then the parameters are β̄ = 1/4 and γ̄ = 1/2. For details on the Newmark procedure
and different choices of Newmark parameters we refer to Hughes [59].
Polyconvex Framework for Anisotropic Hyperelasticity
5
49
Polyconvex Framework for Anisotropic Hyperelasticity
In the framework of material modeling the basic principles given in section 3.4 have
to be satisfied as well as other restrictions, which are introduced in order to obtain a
physically reasonable material behavior. E.g., the well-known Baker-Ericksen inequality
postulates the same direction for maximum principal stretches and maximum principle
Cauchy stresses. This condition is developed for isotropic materials and can not be applied to anisotropy. A more suitable condition for anisotropic materials is the LegendreHadamard inequality, which ensures, if satisfied, the existence of real wave speeds inside
the considered material.
Another class of additional restrictions is introduced, which represent the mathematical
foundation for ensuring the existence of deformations minimizing a given hyperelastic potential. It is remarked that whether the minimizer satisfies indeed the weak equilibrium
equations (Euler Lagrange equations) is a question of guaranteeing the smoothness of the
minimizer, cf. Ball [6]. In the sequel we restrict ourselves to the case of finite hyperelasticity, where the material behavior can be represented by a stored energy function. The
aim is the computation of a displacement field u for which the potential
Z
Z
Z
Π̂(u) = ψ̂(∇u) dV − ρ0 f · u dV − t · u dA
(5.1)
B
B
∂B
becomes stationary. Herein, (−ρ0 f · u) is the volume load potential and (−t · u) the
potential of the boundary load. This section deals with the construction of stored energy
functions ψ in the framework of polyconvexity, which automatically fulfill the necessary
condition for the existence of minimizers of (5.1). We subdivide this section into the three
main parts
• Generalized Convexity Conditions, where some important generalized convexity conditions are discussed,
• Stresses and Tangent Moduli, giving the fundamental equations for the stresses and
tangent moduli in the considered polynomial basis,
• Polyconvex Stored Energy Functions, providing a variety of polyconvex transversely
isotropic functions automatically satisfying the stress-free reference configuration.
5.1 Generalized Convexity Conditions
A sufficient condition for
R the existence of1,pminimizers is the sequential weak1,plower semicontinuity (s.w.l.s.) of B Ŵ (F ) dV on W (B) and the coercivity of W ; W (B) means,
that the first derivatives of W exist in the weak sense and are p-times integrable over B.
Herein, the coercivity condition is a sharper form of the general coerciveness inequality
and is given in e.g. Ciarlet [22] by the
Definition of Coercivity: There exist constants K > 0, C, p ≥ 2, q ≥ p/(p − 1), r > 1,
such that
Ŵ (F ) ≥ K [||F ||p + ||CofF ||q + (detF )r ] + C
holds for all F ∈ R3×3 .
Polyconvex Framework for Anisotropic Hyperelasticity
50
A condition implying s.w.l.s. is the convexity of the stored energy with respect to the
deformation gradient. Furthermore, a strictly convex function also guarantees the uniqueness of solutions, which means that a local minimum is always a global minimum, too.
From the numerical point of view this is quite interesting because gradient-based linearization methods would then find global minimizers. A huge drawback of the convexity
condition is that it is physically too restrictive. As an example, convexity of the stored
energy precludes buckling and contradicts the principle of material frame indifference.
Another important restriction is the quasiconvexity condition, which has been introduced
by Morrey [87] and which represents together with polynomial growth conditions a
sufficient condition for the s.w.l.s.. The quasiconvexity condition is, from the physical
point of view, a reasonable requirement, because it postulates that homogeneous bodies
under homogeneous boundary conditions lead to a homogeneous strain field in the body.
Unfortunately, the quasiconvexity condition is an integral inequality and therefore only
conditionally appropriate for the analysis of functions. Furthermore, the growth conditions
exclude the treatment of energies W with Ŵ (F ) → ∞ as detF → 0, which is important
in the context of finite elasticity. It should be remarked that it is quite well established in
the literature that solely satisfying the quasiconvexity condition should be sufficient for
the s.w.l.s.. Since there exists no contradictory example until today, the growth conditions
are more interpreted as “technical” assumption necessary for proving that quasiconvexity
implies s.w.l.s.. For the converse implication in Morrey [87] it is shown that s.w.l.s.
directly implies quasiconvexity.
A more suitable condition for the practical use in this context is the notion of polyconvexity in the sense of Ball [3], [4], which is a sufficient condition for s.w.l.s. also without
growth conditions and which implies directly quasiconvexity. Due to its local character
this condition can be checked pointwise. For a detailed discussion of quasi- and polyconvexity we refer to Ball [3], [4], Krawietz [72], Dacorogna [25], Marsden & Hughes
[79] and Silhavý [112].
Another important convexity condition is the rank-1-convexity, which is associated to the
Legendre-Hadamard ellipticity. In its strong form it ensures wave propagation with real
velocity and is strongly linked with material stability. It should be noted that quasiconvexity is a sufficient condition for rank-1-convexity. The important relations between the
generalized convexity conditions and the existence of minimizers are illustrated in Fig.
16.
It is well known that a convex function is also polyconvex, a polyconvex function is also
quasiconvex and a quasiconvex function is also rank-1-convex. Generally, the converse
implications are not true, see Dacorogna [25]. The polyconvexity seems to be the most
suitable condition since it a priori implies s.w.l.s., which is important for the existence
of minimizers, it implies quasiconvexity and therefore ensures that homogeneous bodies
under homogeneous boundary conditions lead to homogeneous strain fields, it implies
rank-1-convexity, which is linked to the physically reasonable requirement of real wave
speeds, and it does not preclude some important properties in finite strains as the convexity condition.
In order to ensure the existence of minimizers, the polyconvexity condition alone is not
a sufficient one, since it only implies s.w.l.s. but not coercivity. Due to the definition of
coercivity we directly notice, that each additively composed stored energy with positive
additive terms will automatically satisfy the coercivity condition provided that at least one
Polyconvex Framework for Anisotropic Hyperelasticity
51
Convexity
Polyconvexity
s.w.l.s.
with growth cond.:
Ŵ (F ) ≤ k||F ||p + C
Quasiconvexity
with coercivity
Existence of Minimizers
Rank-1-Convexity
Figure 16: Important chains of implication of generalized convexity conditions, sequentially
weakly lower semicontinuity and existence of minimizers.
additive term is coercive. It should be noted that in this work we focus on the construction
of polyconvex energy functions and do not treat the issue of coercivity for each polyconvex
function. When constructing applicable polyconvex functions for biological soft tissues,
we use at least one coercive isotropic function in order to guarantee the existence of
minimizers.
5.1.1 Convexity In the context of the following generalized convexity conditions the
convexity of a function with respect to its arguments is very important and explained in
this section, cf. e.g. Schröder [103]. For a better understanding we consider the special
case of scalar-valued functions at first. The function f := fˆ(x) is (strictly) convex with
respect to x if
ˆ 1 ) + (1 − λ)fˆ(x2 ) with λ ∈ ]0, 1[ , x1 6= x2
fˆ(λx1 + (1 − λ)x2 ) ≤ (<) λf(x
(5.2)
holds. The geometric interpretation is that each point (C), element of the linear connection
of two given points (A) and (B), is always above the corresponding point (D) located on
the function f , see Fig. 17. In Figure 17a) it is illustrated that constant functions are
convex, but not strictly convex.
If the first or second derivative fˆ′ (x) and fˆ′′ (x), respectively, exist for each x, then we
obtain another formulation for the definition of (strict) convexity, i.e.
h
i
′
′
ˆ
ˆ
f (x1 ) − f (x2 ) (x1 − x2 ) ≥ 0 (> 0) for x1 6= x2 or fˆ′′ (x) ≥ 0 (> 0) ∀ x , (5.3)
which is equivalent to (5.2) provided that the functions are smooth enough. In material modeling we are interested in inequalities for constructing the stored energy
W := Ŵ (F ) = ψ̂(C), which is a scalar-valued tensor function. For the latter class of
functions the definition of (strict) convexity reads
Ŵ (λF 1 + (1 − λ)F 2 )(<) ≤ λŴ (F 1 ) + (1 − λ)Ŵ (F 2 ) for λ ∈ [0, 1]; ∀ F 1 , F 2 . (5.4)
Polyconvex Framework for Anisotropic Hyperelasticity
fˆ(x)
fˆ(x)
A
A
C
D
a)
52
x1
∗
x
D
B
x2
C
x
x1 x∗
b)
B
x2
x
Figure 17: a) Convex and b) strictly convex function.
By substitution of F 1 − F 2 =: ∆F and F 2 =: F in equation (5.4) we obtain
Ŵ (F + ∆F )(<) ≤ Ŵ (F ) + λ[Ŵ (F + ∆F ) − Ŵ (F )] .
The case of the limiting value λ → 0 is calculated by the Gateaux derivative
d
d
≤ (<)
Ŵ (F + λ∆F ) Ŵ (F ) + λ[Ŵ (F + ∆F ) − Ŵ (F )] dλ
dλ
λ=0
λ=0
(5.5)
(5.6)
and we obtain the alternative representation for the (strict) convexity definition
Ŵ (F + ∆F )(>) ≥ Ŵ (F ) + P̂ (F ) : ∆F ,
(5.7)
which can be interpreted as condition for local convexity, cf. Hill [46]. Herein, P̂ (F ) :=
∂F Ŵ (F ) is the first Piola-Kirchhoff stress tensor, which represents the gradient of W at
F . The definition of strict convexity in (5.7) states that the energy W for all (F + ∆F )
has to be higher than the energy given by Ŵ (F ) + P̂ (F ) : ∆F , see Fig. 18.
Ŵ (F +∆F )
P̂ (F ) : ∆F
Ŵ (F )
F
F +∆F
Figure 18: Illustration of strict convexity of Ŵ (F ).
If we substitute in (5.7) F = F 2 , ∆F = F 1 − F 2 and F = F 1 , ∆F = F 2 − F 1 , we obtain
the two formulations for (strict) convexity

Ŵ (F 2 ) + P̂ (F 2 ) : (F 1 − F 2 ) − Ŵ (F 1 ) ≤ 0 (< 0) and 
,
(5.8)

Ŵ (F 1 ) + P̂ (F 1 ) : (F 2 − F 1 ) − Ŵ (F 2 ) ≤ 0 (< 0)
respectively. These two inequalities are combined and the alternative representation of
the (strict) convexity definition reads
[P̂ (F 1 ) − P̂ (F 2 )] : (F 1 − F 2 ) ≥ 0 (> 0) ∀ F 1 , F 2 ,
(5.9)
Polyconvex Framework for Anisotropic Hyperelasticity
53
which is equivalent to (5.4) and represents the monotonicity of P . Another condition
for convexity can be deduced if the energy Ŵ (F ) is two-times differentiable. We insert
λ ∈ [0, 1] into the inequality (5.7) and re-substitute F 2 = F , F 1 − F 2 = ∆F , then we
obtain
[P̂ (F + λ∆F ) − P̂ (F )] : ∆F ≥ 0 (> 0) ∀ F 1 , F 2 .
(5.10)
In order to investigate the case λ → 0 we calculate the Gateaux derivative
d
≥ 0 (> 0)
[P̂ (F + λ∆F ) − P̂ (F )] : ∆F dλ
λ=0
(5.11)
and obtain the (strict) convexity inequality
∆F : Â(F ) : ∆F ≥ 0 (> 0) ∀ F , ∆F
with Â := ∂F2 F Ŵ (F ) ,
(5.12)
which comes up to the definition of positive semidefinite (positive definite) nominal tangent moduli A. Although a stored energy, which is convex with respect to F , ensures the
existence of solutions of the underlying boundary value problem and also material stability, it is too restrictive for the description of real materials due to the following reasons,
cf. Marsden & Hughes [79] and Ball [4].
• Strictly convex stored energy functions imply uniqueness of solutions, cf. Hill [46],
and rule out instabilities. Therefore, these functions are not able to describe e.g.,
buckling. Examples for nonuniqueness in elastostatics are given in e.g. Rivlin [96],
[97] and Wang & Truesdell [139].
• Convex stored energy functions satisfying the principle of material frame indifference
lead to physically unreasonable constitutive restrictions, cf. Coleman & Noll
[23], Trousdell & Noll [129] and Ciarlet [22]. Let us consider the convexity
inequality (5.7) and let us assume ∆F = QF − F , then this leads to
Ŵ (F ) + P̂ (F ) : (QF − F ) ≤ Ŵ (QF ) .
(5.13)
The principle of material frame indifference requires the constitutive equations to
be invariant under transformations of the special orthogonal group, i.e. Ŵ (F ) =
Ŵ (QF ) ∀ Q ∈ SO(3), F , then (5.13) reduces with detF > 0 to
σ : (Q − 1) ≤ 0 ∀ Q ∈ SO(3) ,
(5.14)
with the Cauchy stresses σ = P F T (detF )−1 . As can easily be seen (5.14) does not
hold for arbitrary σ and Q ∈ SO(3). As an example, we choose




σ1 0 0
1 0
0
σ =  0 σ2 0  and Q =  0 −1 0  ,
(5.15)
0 0 σ3
0 0 −1
and obtain after evaluation of (5.14)
σ2 + σ3 ≥ 0 .
(5.16)
This inequality would lead to physically unreasonable constitutive conditions implying e.g., that both stresses σ2 and σ3 can not be compressive stresses at the same
time.
Polyconvex Framework for Anisotropic Hyperelasticity
54
λ2
λA
2
W
A
C
A
λ2
λB
2
C
B
λ1 λ2 = 1
B
λ1
λA
1
a)
λB
1
λ1
b)
Figure 19: a) 3-dimensional illustration and b) contour plot of energy W (F ) = det[F ] −
4 ln(det[F ]) in a homogeneous deformation with F = diag[λ1 , λ2 , 1].
• The physically reasonable growth condition Ŵ (F ) → ∞ for detF → 0 can not
be described by convex stored energy functions as well as incompressible materials,
since the domain {F |detF > 0} of W is not convex.
For an illustration of this point we use an example similar to Ball [4]: a homogeneous and isotropic rubber sheet is subjected to a homogeneous deformation,
which is represented by F = diag[λ1 , λ2 , 1]. Due to the approximate incompressibility of rubber a high amount of energy is necessary to deform the material in
order to let detF = λ1 λ2 differ from one. Therefore, banana-shaped contours of
equal energy are observed as shown in Fig. 19 where as an example the energy
W (F ) = det[F ] − 4 ln(det[F ]) is considered. There we see that the point C positioned on the linear connection of points A and B is located below the associated
energy, i.e. Ŵ C = 21 [Ŵ (λA ) + Ŵ (λB )] < Ŵ (λC ). This contradicts the definition of
convexity as illustrated in Fig. 17.
5.1.2 Quasiconvexity The mechanical criterion of material stability states that it is
impossible to release energy from a body made of a stable and homogeneous material
by an isothermal process if the body is fixed at the boundaries, cf. Krawietz [72]. This
condition can be reformulated for the case of hyperelasticity by
Zt Z
t0
P : Ḟ dV dt ≥ 0
(5.17)
B
with homogeneous Dirichlet boundary conditions. Herein, P = ∂F Ŵ (F ) denotes the first
Piola-Kirchhoff stress tensor. We denote the energy at an initial time where we assume
a constant F = F̄ over B by Ŵ (F , t0 ) = Ŵ (F̄ ) and, then time integration yields the
expression
Z
Z
Ŵ (F ) dV − Ŵ (F̄ ) dV ≥ 0 with x = F̄ X on ∂B .
(5.18)
B
B
The first term considers a state at t > t0 where the homogeneous Dirichlet boundary conditions are applied and only the positions of points inside the body are varied. Therefore
Polyconvex Framework for Anisotropic Hyperelasticity
55
we insert F = F̄ + Gradw with the fluctuation field w and w = 0 on ∂B into (5.18) and
obtain the
Definition of Quasiconvexity: The elastic stored energy is quasiconvex whenever for all
B ⊂ R3 , all constant F̄ ∈ R3×3 and all w ∈ C0∞ (B) we have
Z
Z
Ŵ (F̄ + Gradw) dV ≥ Ŵ (F̄ ) dV = Ŵ (F̄ ) × V olumen(B) ,
B
B
which has been introduced by Morrey [87]. It should be noted that in the latter definition
the inequality has to hold for arbitrary fluctuations w, hence, this condition is relatively
difficult to check. One effort of the quasiconvexity condition is that it is together with
satisfying the growth condition Ŵ (F ) ≤ k||F ||p + C a sufficient condition for s.w.l.s.. A
huge drawback of this implication is that the growth condition precludes the property
that Ŵ (F ) → ∞ for detF → 0.
5.1.3 Polyconvexity Since quasiconvexity is difficult to verify due to its character of
an integral inequality, a more tractable condition is required. In the context of hyperelasticity the notion of polyconvexity in the sense of Ball [3], [4] seems to be the appropriate
condition. There, a pointwise requirement on the stored energy function is considered and
therefore, polyconvexity can be checked relatively easily. The condition is given by the
Definition of Polyconvexity: F 7→ Ŵ (F ) is polyconvex if and only if there exists a function
P : R3×3 × R3×3 × R 7→ R (in general non-unique) such that
Ŵ (F ) = P̂ (F , Adj[F ], det[F ])
and the function R19 7→ R, (X̃, Ỹ , Z̃) 7→ P (X̃, Ỹ , Z̃) is convex for all points X ∈ R3 . As an example for the non-uniqueness of P we consider the energy function
(
P1 = 3Z̃ = 3 det[F ]
Ŵ (F ) = P̂ (F , Adj[F ], det[F ]) = P̂ (X̃, Ỹ , Z̃) =
P2 = X̃ Ỹ = hF , Cof[F ]i
.
(5.19)
Observe that P̂2 (X̃, Ỹ , Z̃) = X̃ Ỹ is not a convex function of (X̃, Ỹ ), although it is a
polyconvex function since P2 = hF , Cof[F ]i = 3 det[F ] = P1 and P̂1 (X̃, Ỹ , Z̃) = 3Z̃ is
convex with respect to Z̃ and therefore polyconvex.
A great advantage of polyconvexity is that it represents the necessary condition for the
existence of minimizers. Another merit is that the stored energy function Ŵ (F ) has not
necessarily to be convex with respect to F . Thereby, many drawbacks of convex functions
as described in Section 5.1.1, are bypassed. The probably most straightforward example
for a polyconvex, but not convex (with respect to F ) stored energy function is given by
Ŵ (F ) := det[F ] .
(5.20)
The function Ŵ (F ) = det[F ] is not convex with respect to F as already seen in Section
5.1.1, but Ŵ (X = det[F ]) = X is a convex function with respect to X = det[F ] and
therefore polyconvex.
Polyconvex Framework for Anisotropic Hyperelasticity
56
5.1.4 Rank-1-Convexity and Legendre-Hadamard Ellipticity Less restrictive
than the polyconvexity condition is the rank-1-convexity, where the convexity of a function
is checked by the comparison of two points differing by a rank-1-tensor and we obtain the
Definition of Rank-1-Convexity: the elastic stored energy function W is (strictly) rank-1convex if the function
ˆ = Ŵ (F + λm ⊗ N )
f : R → R, f(λ)
is (strictly) convex for all F ∈ R3×3 and all m, N ∈ R3 \{0}.
It should be noted that quasiconvexity implies rank-1-convexity directly. By ensuring
smooth stored energy functions W , the rank-1-convexity implies Legendre-Hadamard ellipticity, which is described in the sequel.
For dead-load body forces f 0 the linearization of the equation of motion ρ0 ϕ̈t =
Div[P ] + f 0 at an equilibrium state (characterized by F̄ = Grad[ϕ̄t ]) leads to the linearized equation
ρ0 ü = Div[A : Gradu] ,
with the nominal tangent moduli A := DF2 W (F )|F¯ evaluated at F̄ . Here ρ0 denotes
the reference density and u the displacements relative to the considered reference state.
Inserting the traveling wave ansatz u = m φ̂(X · N − c t), φ : IR 7→ IR into (5.1.4) leads
to the characteristic equation
ρ0 c2 m = Q̄m ,
where N denotes the referential direction of wave propagation, m the polarization vector,
c the referential wave speed of propagation and Q̄ the acoustic tensor. Taking the scalar
product of the characteristic equation with m yields
ρ0 c2 ||m||2 = m · Q̄m .
(5.21)
In order to ensure real wave speeds the acoustic tensor Q̄ has therefore to be positive
definite. The expression on the right hand side of (5.21) can be identified with the second
differential DF2 W (F ).(m ⊗ N , m ⊗ N ) evaluated in the rank-1 directions m ⊗ N of the
elastic energy W (F ) with m, N ∈ R3 . Then we obtain the
Definition of Ellipticity: we say that the twice differentiable elastic stored energy W (F ) =
ψ(C) leads to a uniformly elliptic equilibrium system whenever the so-called uniform
Legendre-Hadamard condition
∃ c+ > 0 ∀ F ∈ R3×3 : ∀ m, N ∈ R3 \{0} : DF2 W (F ).(m ⊗ N , m ⊗ N ) ≥ c+ kmk2 kN k2
holds. We say that W gives rise to an (strictly) elliptic system if and only if the LegendreHadamard condition
∀ F ∈ R3×3 : ∀ m, N ∈ R3 \{0} :
is satisfied.
DF2 W (F ).(m ⊗ N , m ⊗ N ) ≥ 0 (> 0)
Polyconvex Framework for Anisotropic Hyperelasticity
57
Since the Legendre-Hadamard condition implies real wave speeds, it is a physically reasonable requirement. The loss of strong ellipticity leads to so-called weak discontinuities, i.e.
states of equilibrium with a continuous displacement field but discontinuous time derivatives of the deformation gradient. In this case the material is said to be unstable. Due
to the implication that a polyconvex function is also a rank-1-convex function, a smooth
polyconvex function also implies that the Legendre-Hadamard condition is satisfied.
5.2 Stresses and Tangent Moduli
In this section we provide the main equations for the stresses and tangent moduli in the
context of the invariant theory. Generally, the second Piola-Kirchhoff stresses are obtained
via the first derivative of the stored energy with respect to the right Cauchy-Green tensor,
cf. Eq. (3.78). Therefore we obtain the abstract formulation
m
X ∂ψ ∂Li
∂ψ
=2
S := 2
∂C
∂Li ∂C
i=1
(5.22)
with Li ∈ P = {I1 , ...Im } representing invariants of the abstract polynomial basis P. The
tangent moduli are computed by
C := 2
∂S
∂C
or in index notation CABCD := 2
∂S AB
.
∂CCD
(5.23)
5.2.1 Stresses and Tangent Moduli for Isotropic Materials
In the case of
isotropy we consider the polynomial basis P0 = {I1 , I2 , I3 } with the invariants given in
(3.90). Inserting the derivatives of the invariants with respect to C, we obtain the explicit
expression for the second Piola-Kirchhoff stresses
S
iso
=2
∂ψ
∂ψ
∂ψ
∂ψ
+
I1 G −
C+
cof[C] ,
∂I1 ∂I2
∂I2
∂I3
(5.24)
with G denoting the metric tensor. Remembering (3.88)3 , the representation of the stresses
is given by a scalar-valued function depending on 3 scalar invariants and a set of tensor
generators Fk . We see that in the case of isotropy this set of tensor generators would be
F iso := {G, C, cof[C]} .
(5.25)
Polyconvex Framework for Anisotropic Hyperelasticity
58
Using some special derivatives with respect to C again we receive the tangent moduli in
index notation by
CABCD
iso
∂2ψ
∂2ψ
GAB GCD +
{I1 G − C}AB {I1 G − C}CD
∂I1 ∂I1
∂I2 ∂I2
∂2ψ
+
{cof[C]}AB {cof[C]}CD
∂I3 ∂I3
∂2ψ
[GAB {I1 G − C}CD + {I1 G − C}AB GCD ]
+
∂I2 ∂I1
∂2ψ
+
[GAB {cof[C]}CD + {cof[C]}AB GCD ]
∂I3 ∂I1
∂2ψ
[{I1 G − C}AB {cof[C]}CD + {cof[C]}AB {I1 G − C}CD ]
+
∂I3 ∂I2
∂ψ
[GAB GCD − GAC GBD ]
+
∂I2
∂ψ
−1 AB
−1 CD
−1 AC
−1 BD
I3 [{C } {C } − {C } {C } ] .
+
∂I3
=4
(5.26)
Here, {C}CD is an abbreviation for the index representation of G−1 CG−1 in order to
arrive at a compact formulation.
5.2.2 Stresses and Tangent Moduli for Transversely Isotropic Materials For
transversely isotropic materials we consider the polynomial basis given in (3.103), i.e.
P1 = {I1 , I2 , I3 , J4 , J5 }, and the second Piola-Kirchhoff stresses have to be extended to
S=S
iso
+S
ti
∂ψ
∂ψ
with S = 2
M+
(CM + M C) ,
∂J4
∂J5
ti
(5.27)
and the set of tensor generators for transverse isotropy is given by
F := {G, C, cof[C], M , CM + M C} .
(5.28)
The explicit expression for the tangent moduli has also to be extended and we obtain
C = Ciso + Cti
(5.29)
Polyconvex Framework for Anisotropic Hyperelasticity
59
with the part for the transverse isotropy
CABCD
ti
=4
∂2ψ
∂2ψ
M AB M CD +
{CM + M C}AB {CM + M C}CD
∂J4 ∂J4
∂J5 ∂J5
∂2ψ
[GAB M CD + M AB GCD ]
∂I1 ∂J4
∂2ψ
+
[GAB {CM + M C}CD + {CM + M C}AB GCD ]
∂I1 ∂J5
∂2ψ
[{I1 G − C}AB M CD + M AB {I1 G − C}CD ]
+
∂I2 ∂J4
∂2ψ
[{I1 G − C}AB {CM +M C}CD + {CM +M C}AB {I1 G − C}CD ]
+
∂I2 ∂J5
∂2ψ
[{cof[C]}AB M CD + M AB {cof[C]}CD ]
+
∂I3 ∂J4
∂2ψ
+
[{cof[C]}AB {CM + M C}CD + {CM + M C}AB {cof[C]}CD ]
∂I3 ∂J5
∂2ψ
[{CM + M C}AB M CD + M AB {CM + M C}CD ]
+
∂J4 ∂J5
∂ψ
AC
BD
AC BD
[G M
+M G ] .
+
∂J5
+
(5.30)
As in the last section {C}CD represents the abbreviation for the index representation
of G−1 CG−1 . G−1 denotes the contravariant metric tensor in the Lagrangian setting,
with the index representation GAB . Terms like {•}AB characterize the contravariant index representations of the individual tensor expressions, e.g. {CM + M C}AB denotes
GAC CCD M DB + M AC CCD GDB , see Schröder & Neff [106].
In order to obtain the spatial formulation, the Kirchhoff stresses τ and the associated
tangent moduli c are governed by a push-forward operation, i.e. in index notation we
have
τ ab := F a A F b B S AB and cabcd := F a A F b B F c C F d D CABCD .
(5.31)
The physical Cauchy stresses are then computed by
σ=
1
τ.
J
(5.32)
Alternatively, the Kirchhoff stresses can directly be calculated by the Doyle-Ericksen
formula (3.79); using the derivatives of the invariants with respect to the spatial metric
we obtain the alternative expression
τ =2
∂ψ
∂ψ
∂ψ 2 ∂ψ
∂ψ
∂ψ
−1
ã ⊗ ã +
+
I1 b −
b +
I3 g +
bã ⊗ bã
∂I1 ∂I2
∂I3
∂I3
∂J4
∂J5
(5.33)
with the preferred direction in the actual configuration ã = F a. The associated tangent
moduli can also be alternatively calculated by the derivative c = 2∂g τ̂ (Ĉ(F , g)). It is to
be remarked that in the sequel we focus on the formulation in the referential setting.
Polyconvex Framework for Anisotropic Hyperelasticity
60
5.2.3 Stress-Free Reference Configuration The requirement that the stresses have
to be zero in the reference configuration, is physically reasonable since in purely hyperelastic materials only deformations should induce stresses. In arterial walls there exist
residual stresses in the unloaded configuration (configuration without blood pressure),
which arise rather from growth than from applied deformations. Therefore, we focus on
material models which satisfy the condition of a stress-free reference configuration and incorporate the eigenstresses in another way, see Section 9.1. The mathematical formulation
for the so-called natural state condition reads
Ŝ(C = 1) = 0 .
(5.34)
Inserting C = 1 into the explicit expression for the second Piola-Kirchhoff stresses given
in (5.27) and including (5.34) leads to the equation
∂ψ
∂ψ
∂ψ
∂ψ
∂ψ
1+
M =0
(5.35)
+2
+
+2
∂I1
∂I2 ∂I3
∂J4
∂J5
It should be noted that generally this equality is not automatically satisfied in the context
of polyconvex functions. In the next section we will introduce some polyconvex functions,
which satisfy the natural state condition a priori. We arrive at these functions because
we consider special case distinctions. If such case distinctions are not included then (5.35)
is usually not fulfilled if solely one polyconvex term is used. If further terms are used
then (5.35) will lead to additional restrictions for the material parameters. But this topic
will be discussed in more detail, when concrete polyconvex functions for arterial walls are
given.
5.3 Polyconvex stored energy functions
In this section some well-known isotropic stored energy functions are recalled which satisfy
the polyconvexity condition. Then some basic transversely isotropic polyconvex stored
energy functions are given, which are proposed in Schröder & Neff [106]. Starting from
these functions a variety of transversely isotropic functions, which automatically satisfy
the stress-free reference configuration, is created via utilizing a proposed construction
principle.
5.3.1 Isotropic Polyconvex stored energy functions In this section we remember some well-known isotropic functions, which satisfy the polyconvexity condition, cf.
Schröder & Neff [106]. The most straightforward functions are the invariants for the
isotropic case themselves, i.e. I1 , I2 and I3 as given in (3.90). The polyconvexity of the
third invariant is trivially satisfied, due to the definition of polyconvexity given in Section
5.1.3. In order to check the polyconvexity of the first and second invariant we have to
proof the convexity of these functions with respect to F and cof[F ], respectively. For
this purpose the second derivative of the functions has to be positive, cf. Section 5.1.1.
In Appendix B the proof of polyconvexity of positive powers of I1 is given; the proof for
powers of I2 is trivially obtained by replacing the cofactor in the derivative. Thus, we
obtain the polyconvex functions
ψ1iso = α1 I1α2
and ψ2iso = α1 I2α2 ,
(5.36)
Polyconvex Framework for Anisotropic Hyperelasticity
61
with α1 > 0 and α2 ≥ 1. In Fig. 20 the invariants I1 and I2 are plotted for an uniaxial
tension test of an incompressible material and we observe that the two functions increase
with increasing or decreasing stretch λ1 .
4
3.9
3.8
I1
3.7
t
x3
x1
3.6
3.5
t
3.4
x2
I2
3.3
3.2
3.1
3
t
a)
b)
t
0.6
0.8
1
λ1
1.2
1.4
1.6
Figure 20: a) Uniaxial unconstrained tension of an incompressible material with preferred
direction oriented parallelly to the stretch direction and b) associated values of individual
polyconvex functions I1 , I2 vs. stretch λ1 = (l0 + ∆l)/l0 ; l0 is the cube length in 1-direction
in the reference configuration and ∆l denotes the difference between actual and reference
length.
For some materials the additive split of the stored energy into a volumetric and an isochoric
part is quite important and the energy takes the general form
Ŵ (F ) = Ŵvol (det[F ]) + Ŵisoch (C̃) .
(5.37)
Herein, C̃ := det[C]−1/3 C, because then the determinant of C̃ is equal to one. For the
first invariant of C this means, that the isochoric parts is calculated by
I1
I˜1 = tr[C̃] = 1/3 ,
I3
(5.38)
whose polyconvexity is proved in e.g. Neff [90] or Schröder & Neff [106]. There, in
Lemma C.1, it is shown, that the function
Ŵ (F ) =
kF kp
det[F ]ᾱ
with
ᾱ + 1
p
≥
ᾱ
p−1
(5.39)
is a polyconvex function. This holds for the case ᾱ = 2/3 and p = 2, which is associated
to the function given in (5.38). The substitution of kF k by kcof[F ]k in (5.39) would lead
to the analogous proof of polyconvexity for the function Ŵ (F ) = det[F ]−ᾱ kcof[F ]kp .
Unfortunately, for the isochoric part of I2 the parameters would be ᾱ = 4/3 and p = 2,
which would not satisfy the condition (ᾱ + 1)/ᾱ ≥ p/(p − 1). Reformulated in direct terms
of the invariants we obtain the polyconvex functions
ψ3iso = α1
I1
1/3
I3
and ψ4iso = α1
I2
1/3
I3
(5.40)
with α1 > 0. For volumetric energy functions formulated in I3 some polyconvex functions
are given by
ψ5iso = α1 I3α2 and ψ6iso = −α1 ln(I3 )
(5.41)
Polyconvex Framework for Anisotropic Hyperelasticity
62
with α1 > 0 and α2 > 1 ∨ α2 < 0. Due to I3 = (detF )2 > 0 the second derivative of the
power function in (5.41)1 is always positive and therefore polyconvex. The same holds for
the second function in (5.41), because the negative natural logarithm is a monotonically
increasing function.
Although the functions given above satisfy the polyconvexity condition, they have a physical drawback: they themselves do not fulfill the stress-free reference configuration, which is
an important requirement for material models. Therefore, we now provide some isotropic
polyconvex functions that satisfy this natural state condition. In Schröder & Neff
[106] some examples for such functions are given by

α3


1
α2
iso

ψ7 = α1 I3 + α2 − 2
with α1 > 0, α2 ≥ α3 ≥ 1 


I3




α2
1



iso
2

with α1 > 0, α2 ≥ 1
ψ8 = α1 I3 − 1


!α3
.
(5.42)
α2

I

1

ψ9iso = α1
with α1 > 0, α2 ≥ 1 α3 ≥ 1 
− 3α2

α2 /3


I3


!α3


3α2 /2

√

I
2

α2
iso

with
α
>
0,
α
≥
1
α
≥
1
3)
ψ10 = α1
−
(3
1
2
3

α2
I
3
For the proof of polyconvexity we refer to Hartmann & Neff [43], where also the
coercivity issue is investigated for special isotropic energies. For further functions see
Appendix C.
In the context of the description of biological soft tissues the function
!
I
1
iso
ψ11
= α1
− 3 , α1 > 0 ,
1/3
I3
(5.43)
is often utilized in the literature, as for example in Holzapfel, Gasser & Ogden
[50], [54] and similarly in Weiss, Maker & Govindjee [140]. Another function for the
isotropic part of soft biological tissues is
!
I
2
iso
− 3 , α1 > 0 .
(5.44)
ψ12
= α1
1/3
I3
The difference between the latter two functions is the usage of I1 and I2 and therewith
the use of terms in C and in cofC, respectively.
5.3.2. Transversely Isotropic Polyconvex Functions As in the previous section we
firstly investigate the invariants for transverse isotropy themselves, because these would
be the most straightforward functions. Reformulating J4 = tr[CM ] = F a : F a = kF ak2
we obtain the polyconvex functions
ψ1ti = α1 J4α2
and ψ2ti = α1
J4α2
1/3
I3
(5.45)
with α1 > 0 and α2 ≥ 1. For the proof of polyconvexity see Schröder & Neff [106]
or also Appendix B for (5.45)1 . It is to be remarked that the function ψ2 for α2 = 1
Polyconvex Framework for Anisotropic Hyperelasticity
63
represents the isochoric part J˜4 of the fourth invariant, thus, it might be a useful function
for volumetrically-isochorically decoupled models. By utilizing the alternative structural
tensor
D := 1 − M with tr[D] = 2
(5.46)
and K2 = tr[CD] = I1 − J4 in Schröder & Neff [106] the polyconvex functions
ψ3ti = α1 K2α2
and ψ4ti = α1
K2α2
(5.47)
1/3
I3
with α1 > 0 and α2 ≥ 1 are given.
Since the functions J4 and K2 are linear in C and possess, therefore, a relatively limited
mapping range, the supply of quadratic terms in C seems to be profitable. Unfortunately,
the fifth invariant J5 = tr[C 2 M ] is not polyconvex, see Merodio & Neff [82], although
the associated isotropic basic invariant J2 = tr[C 2 ] is polyconvex, cf. Schröder & Neff
[106]. In [106] a variety of transversely isotropic polyconvex functions are given based on
the introduction of the function
K1 = tr[cof[C]M ] = J5 − I1 J4 + I2 ,
(5.48)
which is a polyconvex function with non-polyconvex terms J5 and
√ I1 J4 . After a short alge2
braic transformation we obtain K1 = ||cof[F ]a|| and see that K1 controls the change of
area with a unit normal into the preferred direction a. Due to this physical interpretation,
K1 seems to be a reasonable function for the description of transverse isotropy. Replacing
the structural tensor M in K1 by D, we obtain the additional polyconvex function
K3 = tr[cof[C]D] = I1 J4 − J5 .
(5.49)
√
Reformulating K3 = kcof[F ]k2 − kcof[F ]ak2 we notice that K3 controls the deformation
of an area element with normal perpendicular to the preferred direction a. Analogous to
the proof of polyconvexity for (5.45) we obtain the polyconvex functions
ψ5ti = α1 K1α2
,
ψ6ti = α1
K1α2
1/3
I3
,
ψ7ti = α1 K3α2
and ψ8ti = α1
K3α2
1/3
I3
(5.50)
with α1 > 0 and α2 ≥ 1.
In Fig. 20 the values of J4 , K1 , K2 and K3 are illustrated for an uniaxial tension test of an
incompressible material with preferred direction oriented parallelly to the stretch direction. Since J4 characterizes the change of length into the preferred direction it increases
with increasing length of the test material. The decrease of K1 with extending length of
the body seems to be logical because it represents the change of area with normal into
the preferred direction, which would decrease in case of an incompressible material. In
Schröder & Neff [106] a variety of polyconvex functions constructed by additive combinations of polyconvex and non-polyconvex terms, which are summarized in Appendix
C as well as the functions given here.
Regrettably, the functions given so far in this section do not satisfy the stress-free reference configuration. This condition is generally not automatically fulfilled concerning
polyconvex functions. In Schröder & Neff [106], Lemma B.9, it has been recalled,
that a function IRn 7→ IR, X 7→ m̂(P̂ (X)) is convex, if the function P : IRn 7→ IR is convex
Polyconvex Framework for Anisotropic Hyperelasticity
64
3.5
3
K3
a
2.5
t
x3
x1
J4
2
t
K2
1.5
x2
1
0.5
a)
K1
0
t
t
b)
0.6
0.8
1
λ1
1.2
1.4
1.6
Figure 21: a) Uniaxial unconstrained tension of an incompressible material with preferred
direction oriented parallel to the stretch direction and b) associated values of individual
polyconvex functions J4 , K1 , K2 and K3 vs. stretch λ1 = (l0 + ∆l)/l0 ; l0 is the cube length
in 1-direction in the reference configuration and ∆l denotes the difference between actual
and reference length.
and the function m : IR 7→ IR is convex and monotonically increasing, cf. Appendix B.
This motivates a construction principle for the simple construction of polyconvex energy
functions that automatically satisfy the stress-free reference configuration. This principle
is used in Balzani, Neff, Schröder & Holzapfel [11] and is rephrased in words as
the following problem:
Principle 1: find a polyconvex function P̂ (X) which is zero in the reference
configuration and include this function into any arbitrary convex and monotonically increasing function m, whose first derivative with respect to P vanishes
in the origin; then the polyconvex function satisfying the stress-free reference
configuration is given by ψ = m̂(P̂ (X)).
Convex and monotonically increasing functions whose first derivative with respect to P
is equal to zero could be
m̂1 (P ) = P k
and m̂2 (P ) = cosh(P ) − 1
(5.51)
with P ≥ 0 and k > 1. The requirement for positive internal functions P ≥ 0 means that
we have to consider the case distinction
(
m̂(P ) for P ≥ 0
(5.52)
m=
0
for P < 0 .
This seems to be a suitable approach, because P = 0 represents the referential state and
therefore, we obtain a smooth energy function when satisfying m̂(P = 0) = 0. A result of
introducing the case distinction (5.52) may be that discontinuous stress functions and/or
discontinuous tangent moduli are obtained. Therefore we investigate m1 and m2 with
respect to their first and second derivatives. In a first investigation we consider the power
function ψ = m1 and compute the first derivative with respect to the (inner) polyconvex
function P
∂P ψ = k P k−1 with k > 1 .
(5.53)
Polyconvex Framework for Anisotropic Hyperelasticity
65
We notice that (5.53) vanishes at the reference configuration since P is equal to zero at
the natural state. Hence, also the second Piola–Kirchhoff stresses
S = 2 ∂C ψ = 2 ∂P ψ
X ∂P ∂Li
∂Li ∂C
L ∈P
(5.54)
i
are zero in the reference configuration and the case distinction leads to a continuous stress
function. Then we calculate the second derivative
∂P2 P ψ = k (k − 1) P k−2
with k > 2
(5.55)
and observe that the tangent moduli
C = 2∂C S =
4 ∂P2 P ψ
"
#
X ∂P ∂Li
∂ X ∂P ∂Li
∂P
⊗
+ 4 ∂P ψ
∂C L ∈P ∂Li ∂C
∂C L ∈P ∂Li ∂C
(5.56)
i
i
are also equal to zero in the reference configuration for k > 2 since the first and second
derivatives of P are zero in the natural state. Therefore, the tangent moduli are continuous
for k > 2, too, when the case distinction (5.52) is considered.
Now we focus on the hyperbolic cosine and set ψ = m2 . The first derivative of this energy
function with respect to P is given by
∂P ψ = sinh(P )
(5.57)
and we see that ∂P vanishes for P = 0. Hence, the stresses are zero in the reference
configuration and the case distinction (5.52) leads to a continuous stress function. By
computing the second derivative
∂P2 P ψ = cosh(P )
(5.58)
we notice that (5.58) is not zero for P = 0 and therefore, it is not guaranteed that the
tangent moduli vanish in the reference configuration. This would lead to discontinuous
tangent moduli at P = 0 when the case distinction (5.52) is considered.
Due to the fact, that a convex and monotonically increasing function m̂(P̂ (X)), whose
first derivative with respect to P vanishes in the origin, has a global minima in the origin,
we can reformulate Principle 1 as
Principle 1♯ : find a polyconvex function P̂ (X) which is zero in the reference
configuration and include this function into any arbitrary convex and monotonically increasing function m, which attains its global minima in the origin;
then the polyconvex function satisfying the stress-free reference configuration
is given by ψ = m̂(P̂ (X)).
In order to give an example we consider the stored energy function of the cosine hyperbolic
type as proposed in (5.51), i.e.
(
cosh(P ) − 1 for P ≥ 0
(5.59)
ψcosh =
0
for P < 0 .
Polyconvex Framework for Anisotropic Hyperelasticity
66
ψcosh
2.0
1.0
P
-2.0
-1.0
0.0
1.0
2.0
Figure 22: Graph of the energy function ψcosh given in (5.59).
In Fig. 22 the graph of the energy function (5.59) is depicted and we observe that at
P = 0 a global minima is obtained.
As (inner) polyconvex functions P satisfying the stress-free reference configuration we can
use any of the polyconvex functions given above provided that a constant with the value
of the function at the referential state is subtracted in order to normalize P in such a way
that the referential state is reached for P = 0.
In order to obtain an energy-free referential state, which is a physically convenient but
unnecessary condition, constant factors having the value of the function itself in the reference configuration have to be subtracted, i.e. ψ = m̂(P ) − m̂(P = 0). For the considered
power function m1 this is automatically satisfied since m̂(P = 0) = 0.
In addition to the functions obtained by applying Principle 1, we are able to construct
further polyconvex functions a priori ensuring the stress-free reference configuration. Analogously motivated by the fact that a function IRn 7→ IR, X 7→ ĝ(m̂(X)) is convex, if the
function m : IRn 7→ IR is convex and the function g : IR 7→ IR is convex and monotonically
increasing, cf. Appendix B, we obtain the principle:
Principle 2: include any function m̂(P̂ (X)), obtained by applying Principle 1,
into the exponential function ĝ(m) = exp(m); then further polyconvex functions are given by
(
ĝ(m̂(P̂ (X))) for P ≥ 0
ψ=
0
for P < 0 .
Due to the fact that the stresses are computed by replacing ψ in the right hand side of
(5.54) by m and multiplying the formula by the derivative ∂m ψ the stresses governed by
Principle 2 are also zero in the referential state. Analogously, the properties with respect
to continuous tangent moduli of the functions ψ = m1 or ψ = m2 are preserved.
Now we are interested in constructing transversely isotropic polyconvex functions by utilizing the construction principles given above. As a first function we choose the stored
energy function

α2

for J4 ≥ 1
α1 (J4 − 1)
ti
ψ(P 1) =
(5.60)


0
for J4 < 1 ,
Polyconvex Framework for Anisotropic Hyperelasticity
67
with α1 ≥ 0 and α2 > 1. This function fits into the first construction principle, because
P = J4 − 1 is polyconvex and (...)p is convex and monotonically increasing for positive
convex arguments. For the complete proof of convexity see Appendix B. At the referential
state the internal function J4 takes the value 1, therefore, it is subtracted here in order to
normalize P . It should be noted that setting α2 > 2 leads to continuous tangent moduli as
shown above. Due to the fact that J4 represents the square of the stretch in fiber direction
a the distinction of cases in (5.60) seems to be reasonable, because J4 < 1 characterizes
the shortening of the fibers, which is assumed to generate no stresses. Note that replacing
1/3
J4 by its isochoric part J˜4 = J4 /I3 leaves (5.60) polyconvex provided that the casedistinction is adapted accordingly.
Soft biological tissues are characterized by an exponential-type stress-strain behavior in
the fiber direction. A model for the description of these materials, which also satisfies the
stress-free reference configuration, is proposed by Holzapfel, Gasser & Ogden [54]
(firstly in [50]). The transversely isotropic function appears as
ti
ψ(HGO)

k1 2


for J4 ≥ 1
 2k exp k2 (J4 − 1) − 1
2
=



0
for J4 < 1 ,
(5.61)
where k1 ≥ 0 is a stress-like material parameter and k2 > 0 is a dimensionless parameter.
An appropriate choice of k1 and k2 enables the histologically-based assumption that the
collagen fibers do not influence the mechanical response of the artery in the low pressure
domain to be modeled (Roach & Burton [98]). The proof of convexity of (5.61) with
respect to F is, e.g., given in Schröder, Neff & Balzani [107], see also Appendix B.
Note that the replacement of J4 by its isochoric part J˜4 is also possible without violating
the convexity condition. Additionally note that the natural state condition is satisfied.
We see, that this function fits into the second principle, because m = k2 (J4 − 1)2 fits
into the first principle as already shown above and is embedded into the exponential
function as proposed in Principle 2. Since the function exp[k2 (J4 − 1)2 ] is equal to one at
the natural state, 1 is subtracted to satisfy the unnecessary condition of an energy-free
reference configuration.
The replacement of J4 in (5.61) and (5.60) by an arbitrary polyconvex function, provided
that the case-distinction is adapted accordingly, leads to a large amount of polyconvex
functions, which are listed in Appendix C.
As examples we now construct some special transversely isotropic polyconvex functions.
Since the functions given in (5.60) and (5.61) are linear in C and have a relatively limited
mapping range, the supply of quadratic terms in C would be profitable. By utilizing the
function K1 we are able to construct two more stored-energy functions which satisfy the
stress-free reference configuration a priori, i.e.
ti
ψ(P
2)
=

α2

α1 (K1 − 1)


0
for K1 ≥ 1
for K1 < 1 ,
(5.62)
Polyconvex Framework for Anisotropic Hyperelasticity
and
ti
ψ(P
3) =
α 1
2

exp
α
(K
−
1)
−
1
for K1 ≥ 1

3
1
 2α3


0
68
(5.63)
for K1 < 1 ,
ti
with α1 ≥ 0, α2 > 1 and α3 > 0. The first one (ψ(P
2) ) represents the substitution of J4
by K1 in (5.60), while the second one characterizes a slight modification in the model
of Holzapfel, Gasser & Ogden. The proof of polyconvexity for (5.62) and (5.63)
is straightforward, since a convex and monotonically increasing function of a polyconvex
argument is also polyconvex ([106]), cf. Appendix B. Note that the restriction α2 > 2
leads to continuous tangent moduli. K1 and also K2 control the change of area with a
unit normal into the preferred direction in some sense, thus, in a uniaxial tension test of an
incompressible material these two functions increase if the material is shortened, as shown
in Fig. 21. Therefore, any function containing K1 or K2 , that is governed by the given
construction principle (e.g., the functions (5.62) and (5.63)) generate stresses only when
the material is shortened in the preferred direction. This is physically not meaningful for
biological soft tissues since collagen fibers mainly support tensile stresses. Nevertheless,
it might be useful for some cases to activate stresses under such condition; then functions
like (5.62) and (5.63) may be utilized.
For incompressible materials K3 increases when the material is elongated in the direction
a, see Fig. 21. Hence, using K3 for the construction of further polyconvex stored energy
functions is physically meaningful for biological soft tissues, because then stresses are
generated when the fibers are elongated. As examples we obtain the two stored-energy
functions

α2

for K3 ≥ 2
α1 (K3 − 2)
ti
ψ(P 4) =
(5.64)


0
for K3 < 2 ,
and
ti
ψ(P
5) =
α 1
2

exp
α
(K
−
2)
−
1
for K3 ≥ 2

3
3
 2α3


0
(5.65)
for K3 < 2 ,
with α1 ≥ 0, α2 > 1 and α3 > 0. These functions are also polyconvex, the proof of
which is analogous to (5.62) and (5.63), respectively. It is remarked that setting α2 > 2
leads to continuous tangent moduli. Of course other polyconvex functions can be obtained
by replacing J4 , K1 and K3 with any other polyconvex function provided that the case
distinction is adopted accordingly. An extensive list of polyconvex functions governed
by this way is given in Appendix C. It is worth noting that each other monotonically
increasing function, e.g., also the other function given in (5.51), cosh(...), with positive
and polyconvex arguments would lead to a polyconvex function, too. As an example, if
the function proposed by Rüter & Stein [100] is embedded into the proposed case
distinction, i.e.


α1 cosh (J4 − 1) for J4 ≥ 1
ti
(5.66)
ψ(P 6) =


0
for J4 < 1 ,
then this would give a polyconvex function.
Parameter Adjustment Based on the Evolution Strategy
6
69
Parameter Adjustment Based on the Evolution Strategy
In this section the issue of determining the values of the parameters of a material model is
discussed. In this problem a set of parameters is sought which lets the model describe some
experimental data as well as possible. In our investigations this means that a measured
stress-strain relation should be represented by the considered model using a certain set of
parameters. Generally, there exist four methods in the field of parameter identification,
see the survey in Mahnken [77]:
1. the “hand fitting” method,
2. the “trial and error” method,
3. Neural networks and
4. the least-squares method.
Normally the “hand-fitting” method is applied when some of the material parameters can
be directly related to experimental data. If no material parameter has a direct meaning
of a quantity that can be measured, this method may be also reasonable for material
models where a direct answer in the stress-strain response is observed when a parameter
is changed. This method seems to be a very comfortable procedure if the material model
has a low amount of parameters. If more parameters occur and a strong coupling between
these exists, then a more sophisticated method is necessary.
Very similar to the latter method is the “trial and error” method, where a set of parameters is estimated without knowledge about any direct meaning of the parameters, and
then the model-response is compared to that of the experiment. Even if this method is not
very complicated to implement since no inverse problem has to be solved, it is of course
a very expensive one, especially when the values of the parameters have totally differing
orders of magnitude.
The third identification method, the neural networks, is based on flow simulation of information between biologic nerves, the neurons. This method has the main advantage
that it can also be applied to inverse problems. For further information about the general
method see Haykin [45].
The last one is the least-squares method, where the issue of parameter identification is
considered as an optimization problem. Here, the sum of squares of differences of measured
quantities and quantities computed by the model is set as the so-called objective or error
function, which has to be minimized. For this optimization task generally two types of algorithms exist: the first one considers only function evaluations (zero-order methods), the
second one requires additionally the gradients (first-order methods). Another possibility
of a classification is grouping into deterministic and stochastic methods. Examples for the
gradient-based deterministic methods are the Gauss-Newton or the Levenberg-Marquard
method. A gradient-free deterministic method is the Simplex method. As examples for the
stochastic methods we only mention the Monte-Carlo method and the evolution strategy.
When convex problems are regarded then the gradient based procedures are advantageous
since they find the optimum much faster than stochastic methods, because they use additional information governed by the gradient. A huge drawback of these methods is that if
non-convex problems are considered, the gradient-based methods will probably find multiple optimums when the starting points are not inside a locally convex area around the
Parameter Adjustment Based on the Evolution Strategy
70
optimum solution. In the context of parameter adjustment, the magnitude of each parameter is not even roughly known when starting the optimization procedure, and hence the
starting point may normally not be close to the optimum solution. The evolution strategy
distinguishes itself by finding the global optimum solution with a very high probability,
although it might take some time. Especially for highly non-convex problems which will
occur when highly non-linear material models are regarded, the evolution strategy is quite
useful to find the optimum.
In this work we focus on the least-squares method and utilize the so-called evolution
strategy for the optimization problem. Therefore, we subdivide this section into two main
parts
• Evolution Strategy, where the fundamental idea of the evolution strategy is explained, and
• Objective Function, where the error function for the considered type of experiments
is given.
6.1 Evolution Strategy
In billions of years an endless number of species almost fitted perfectly in their environment
developed. This optimization process allowing for the survival of the best fitted ones of
a specie only, is known as evolution and provides a very interesting approach for finding
reliable optimization algorithms; these are referred to as evolutionary algorithms. Such
algorithms imitate, contrary to the so-called genetic algorithms, the impact of genetic
procedures for the phenotype. The basic requirement for these methods is a sufficiently
strong causality, which means that small changes in the cause may lead to small changes
in the impact. From the author’s knowledge the evolution strategy has been developed
in the 1960’s by I. Rechenberg and first published in Rechenberg [94]. Afterwards it
has been enhanced by many others in the 1970’s and 1980’s, especially by H.P. Schwefel.
For a detailed description of the evolution strategy we refer to Schwefel [111] and
Rechenberg [95].
In nature an existing population of a specie, each of them carrying different genetic material, procreates in such a way that the genetic material of two parents is recombined.
Additionally, during this process a mutation of the combined genetic material takes place
and new individuals are born by a rather random way. Then a selection determines which
descendants are fitted accurately into their environment and thereby, which are allowed
to reproduce or even to survive. These selected new individuals become also parents and
procreate again. In Fig. 23 the process of the evolution strategy is schematically illustrated
and we notice the strong accordance with nature.
Generally, in optimization issues there exists an objective function that has to be minimized. The problem is then formulated as
min[ê(α1 , ...αn )] with αi ∈ R̄ , i = 1...n , n ≥ 1.
(6.1)
Herein, R̄ is a sub-group of the real numbers R, representing additional restrictions for
n objective variables αi , condensed in the vector α. It is remarked that especially the
parameter adjustment of polyconvex functions leads to a large number of additional restrictions, which have to be fulfilled. The objective function e considered in 6.1 may here
Parameter Adjustment Based on the Evolution Strategy
Estimation
• Recombination
71
Initialization
Selection
• Mutation
Figure 23: Scheme of the evolution process.
be minimized by using the evolution strategy, which is a special sub-group of evolutionary
algorithms. In this context at first a number of sets of objective variables α is estimated
randomly; this step is called initialization. The set of variables and some strategy parameters can be interpreted as an individual occurring in the natural evolution process. This
individual can be represented by the tuple
(α, δ1 , ...δm ) ,
m≥1
(6.2)
with the strategy parameters δ. As the second step the selection denotes the method of
choosing a parent, which is done randomly concerning the evolution strategy. The next
step, the recombination, decides how one or more descendants are generated by one or
more parents, i.e. how the new set of objective variables is generated. There exist many
modes of recombination, which characterize mainly the type of evolution strategy. The
simplest would be the case that one parent generates one descendant. During this step
the new population is not only generated by combining properties of the parents, but also
by mutation. This is called the mutation step, wherein usually a small number is added
(or subtracted) to the value of the objective variable obtained after recombination. The
intensity of deviation of the mutated individual compared to the original one is referred
to as variation and determines also the type of evolution strategy. Normally, a Gaussian
distributed number with median 0 and a certain deviation is added, cf. the chapters about
standard statistics in Bronshtein, Semendyayev, Musiol & Muehlig [18]. In nature the recombination could be equated with the sexual election of potential partners
and with the transfer of their genetic material to their descendants, while the mutation
represents the modification of the genetic material. After these steps the individuals have
to be estimated, which is done by allocating a so-called fitness-function to each descendant. Usually, this fitness function consists of the objective function e and decides which
individual is allowed to assemble the next generation. This step is associated to the main
argument in the evolution theory saying that only the “best” survive. When the estimation is evaluated the process starts again with the selection and is repeated until a certain
number of generations is reached, the optimization results are not improving anymore or
a combination of these two conditions is satisfied.
There exist many types of evolution strategies: the so-called (1 + 1)−evolution strategy
(ES) considers each population to consist of only one new individual. Hence, no selection
Parameter Adjustment Based on the Evolution Strategy
72
is required. In detail the procedure of this type means that the parent generates a descendant by mutation, then the descendant receives also a fitness value and finally the fitter
one of both (parent and descendant) generates the new descendant. Another type is the
(µ + 1)−ES, where one parent from µ individuals is selected and one new individual is
created. This new individual obtains also a fitness value and the less fit one of the whole
population is canceled. The (µ + λ)−ES accounts for a population consisting of µ individuals, creating λ descendents. In this process the number of descendants should exceed
the number of parents. Then the fitness for all new individuals is computed and the µ
fittest ones constitute the new population. Due to the fact that descendants and parents
are taken into account when deciding which one of them should survive, it results that the
quality of each new population can not be worse than that of older generations. Because
these two strategies possess the tendency to find only local optimums, the (µ, λ)−ES has
been developed. Here, µ individuals per generation are considered creating λ descendants.
The difference to the (µ + λ)−ES is, that here directly the parents are erased and only
the fittest ones of the descendants constitute the new population. This means that no
individual is allowed to survive more than one generation.
The basic analogy of the latter strategies is, that they do not use the recombination, i.e.
the descendants stem only from one parent, which is for most of the species not possible in
nature. This means that the properties of the parent is copied and only changed via mutation. Contrary to these strategies, the (µ/p#λ)−ES combines properties of one parent
and others (altogether p parents) in order to generate the properties of the descendant.
The method of recombination can for example be the computation of the average of the
objective variables of two parents.
An example for a more complex strategy is the (µ′ , λ′)(µ, λ)y −ES, where µ′ parents are
λ′ -times copied and these copies are isolated for y generations. In each of these y generations the isolated population creates λ descendants. After calculating the fitness the
µ best ones survive and the others inside this generation are canceled. After regarding
the y generations the best λ′ population is chosen and copied again λ′ -times for the new
generation.
The main advantage of the evolution strategy compared to other evolutionary algorithms
is, that the algorithm is able to adapt itself dynamically to the problem. This is done by
adjusting dynamically the intensity of how the mutation modifies the objective variables.
In this context the so-called 1/5−success-rule, cf. Rechenberg [95], plays an important
role. This rule states that the quotient of successful mutations (mutations that lead to
a better fitness of the descendants) and the total number of mutations should be 1/5. If
this quotient is lower than 1/5, then the variation has to be increased and vice versa. The
quality of the mutation strategy decides if the optimization will be successful or not: too
large variations will lead to the strong probability of finding less fit solutions, and too small
variations would exclude detecting other local valleys where the global optimizer could
exist. Therefore, the mutation step can be seen as the major operator in the evolution
strategy.
6.2 Objective Function
As mentioned above we use the least-squares method for the parameter adjustment, therefore, we consider an objective function that has to be minimized. We are interested in the
modeling of soft biological tissues and therefore we have not only to adjust the model for
Parameter Adjustment Based on the Evolution Strategy
73
stresses measured in some experiments. Additionally we have to account for the quasiincompressibility constraint. As a first step we consider the sum of squares of differences
between measured stresses and computed stresses and obtain the error function
( 3 3
)
h
i2
X X
X
(i)
(i)
S
ê∗ (α) =
ωAB
S̃AB − ŜAB (C̃, α)
.
(6.3)
i
A=1 B=1
(i)
We denote all experimental quantities by (˜
•), i.e. the stress values S̃ are the experimental stresses for each evaluation point i. Because of the varying range of the particular
components of the stress tensor and hence of the particular errors, the weighting factors
S
ωAB
are included in order to level the accuracies of particular error components. Then 6
independent stresses S are calculated by the model taking into account 6 experimental
right Cauchy-Green deformations C̃ (note that S and C are symmetric). Usually, experiments are performed in which not all of the stress components differ from zero. As an
example, the unconstrained uniaxial tension test induces only stresses into the tension
direction and all other components vanish. In order to account for the constraint that
some stress components are restricted to zero, we have to implement an iteration scheme
which is performed on the components of the right Cauchy-Green deformation associated
to the restricted stress components. This fact provides a good possibility to include the
incompressibility condition by introducing an additional term in the error function taking
into account the deviation of the determinant of C from 1. Then we obtain
( 3 3
)
h
i2
X X
X
(i)
(i)
(i)
S
C
2
ê(α) =
ω
S̃ − S̄ˆ (C̄, α) + ω (det C̄ − 1)
(6.4)
AB
i
AB
AB
A=1 B=1
with the weighting factor for the incompressibility error ω C . In (6.4) special deformations
and stresses C̄ and S̄, respectively, are introduced containing the restricted stresses. For an
abbreviated notation we use the matrix notation as given in Appendix D, i.e. the tensors
of second order are represented by vectors and the fourth-order tensors are rearranged in
matrices. Then we obtain the stress and deformation vector
Sc
Cc
S̄ :=
and C̄ :=
.
(6.5)
S0
C0
Herein, the computed stress S c and the constraint S 0 = 0 are associated to C c and C 0 ,
respectively, and linked via the tangent moduli such that
Ccc Cc0
1
S̄ = 2
C̄
(6.6)
C0c C00
holds. For our considerations the components assembled in C c are experimental right
Cauchy-Green deformations. As an example, for the unconstrained tension test the constraint stresses would be S 0 = (S22 , S33 , S12 , S23 , S13 )T and for the constrained tension
test, where one direction apart from the tension direction is fixed, we regard the constraint
stresses S 0 = (S33 , S12 , S23 , S13 )T . Restricting S 0 equal to zero and considering C 0 to
be the unknown deformations we express S 0 in a Taylor series
(j)
(j+1)
S0
(j)
(j)
= S 0 + 12 C00 ∆C 0 + ... = 0 with C00 = 2
∂S 0
(j)
∂C 0
,
(6.7)
Parameter Adjustment Based on the Evolution Strategy
74
with the superscripts denoting the local iteration numbers. We neglect higher order terms
and arrive at the increment
(j) −1 (j)
(6.8)
∆C 0 = −2C00 S 0 .
Then we compute the deformations for the next local iteration step by

(j)

(j+1)
(j)

Cc
= C c = C̃ c
.
(j+1)
(j)
(j) 

C0
= C 0 + ∆C 0
(6.9)
The local iteration has to be repeated until the restricted stresses are approximately equal
to zero, which can be checked by the condition
(j)
kS 0 k < ǫ ,
(6.10)
where ǫ is a small parameter. In Fig. 24 the local iteration procedure is schematically
illustrated.
1) Start values from equilibrium of last exp. (i-1):
(j)
(i−1)
C0 = C0
2) Actual stresses and tangent moduli:
C (j) ⇒ S (j) , C(j)
3) Check restricted stresses:
(j)
4) Update deformations:
if ||S 0 || < ǫ, then goto 6)
(j+1)
C0
5) Increase local iteration counter:
(j)
(j)−1
= C 0 − 2C00
(j)
S0
j = j + 1, goto 2)
6) Store history terms for next exp. (i+1):
C j0 ⇒ History
Figure 24: Local iteration algorithm.
The iteration as shown in this section is very similar to that one occurring when threedimensional material models are utilized in Finite-Element-Formulations as e.g., beam
and shell elements, where some stress components are restricted to zero, cf. Klinkel &
Govindjee [67].
As given in (6.4) we use the second Piola-Kirchhoff stresses and the right Cauchy-Green
deformations, which are not measured directly in real experiments. Therefore, before
starting the optimization scheme, the deformation gradient has to be computed from the
measured displacements and then the right Cauchy-Green deformation tensor and the
second Piola-Kirchhoff stresses are computed via (3.18)1 and (3.36).
Polyconvex Energies for Soft Biological Tissues
7
75
Polyconvex Energies for Soft Biological Tissues
In this section we focus on some special three-dimensional constitutive models for soft
biological tissues. Biological tissues are composed of a so-called groundsubstance, which
can be seen as an isotropic material, and embedded collagen fibers, which are characterized
by a transversely isotropic material behavior. The orientation of the collagen fibers plays
an important role and differs greatly in the different types of tissues. It is well-known,
that biological tissues adapt to loading conditions in order to obtain an optimized state,
e.g. tissues tend to grow when they are loaded higher than normal. The same holds for
the direction of the fibers inside, which are normally arranged in such a way that the load
can be resisted optimally. As examples in ligaments or tendons the fibers are basically
oriented in one direction while the collagen fibers in arterial walls are mainly oriented
in two directions. In order to represent the above explained characteristics we consider a
stored energy function of the general structure
iso
ψ := ψ̂ (I1 , I2 , I3 ) +
n
X
(a)
(a)
ψ̂ ti,(a) (I1 , I2 , I3 , J4 , J5 ) .
(7.1)
a=1
Herein, the groundsubstance is described by the isotropic part ψ iso and the fibers by
the superposition of the transversely isotropic parts ψ ti,(a) for each fiber orientation. The
(a)
(a)
invariants J4 = tr[CM (a) ] and J5 = tr[cof[C]M (a) ] are computed by the structural
tensors M (a) = a(a) ⊗ a(a) for the fiber directions a(a) . This seems to be an appropriate
approach, because we assume a weak interaction between the fiber directions and the same
material behavior for each fiber direction itself. This structure of the energy is also used
in e.g. Holzapfel, Gasser & Ogden [50]. It should be noted that setting n = 1 we
obtain an energy able to describe e.g., tendons or ligaments, while an energy for arterial
layers is achieved for n = 2. This section is mainly subdivided into the two parts:
• Comparative Study with Respect to Material Stability, where a special polyconvex
model is adjusted to the stress-strain response of two non-polyconvex models and a
localization analysis is performed in order to show that contrary to the polyconvex
model the non-polyconvex ones indeed lose ellipticity although they have nearly the
same stress-strain behavior in some restricted curves, and
• Easy Construction of Polyconvex Models for Arterial Walls, wherein a special polyconvex model is constructed and adjusted to a human abdominal aorta “by hand”
in order to show that the new proposed polyconvex functions can be easily used for
constructing a model for soft biological tissues.
7.1 Comparative Study with Respect to Material Stability
In this section we want to illustrate one of the advantages of polyconvex energy functions, namely that a polyconvex energy automatically satisfies the Legendre-Hadamard
condition and guarantees therefore a stable material. In contrast to this, non-polyconvex
functions might easily lose ellipticity.
For this purpose we consider two non-polyconvex models from the literature and a polyconvex model. The stress-strain behavior of the polyconvex model is adjusted to that one
of the other models in two restricted experiments using an optimization method; here,
Polyconvex Energies for Soft Biological Tissues
76
the evolution strategy is used. Then a localization analysis is accomplished and the loss
of ellipticity is observed when using the non-polyconvex models.
7.1.1 Analysis of Special Models As a first example we consider a model which
is able to describe the material behavior of transversely isotropic soft biological tissues,
as e.g. ligaments or tendons. For this purpose we choose the model of Weiss, Maker
& Govindjee [140] and extend the model by a term for being able to satisfy the incompressibility constraint det[F ] = 0. Then we obtain the isotropic part of the reference
model 1 given by
!
!
I
I
1
2
iso
ψ(R1)
= C1
− 3 + C2
− 3 + ǫ[I3 − ln (I3 )] ,
(7.2)
1/3
2/3
I3
I3
and the transversely isotropic part
"
ti
ψ(R1)
= C3 exp
J4
1/3
I3
−1
!
−
J4
1/3
I3
#
.
(7.3)
It is to be remarked that the second additive term in (7.2) and (7.3) are non-polyconvex.
iso
ti
Neglecting the ǫ-term, ψ(R1)
+ ψ(R1)
is identical to the one proposed in [140], where the
incompressibility constraint is enforced via an Augmented Lagrange Scheme. For the
representation of a soft biological tissue we use the parameters given in [140], see Table
1. The parameter ǫ is chosen in such a way that det[F ] = 1.0 ± 1%.
C1
C2
C3
ǫ
[kPa] [kPa] [kPa]
[kPa]
10.00 10.00 100.00 10,000.00
Table 1: Material parameters of reference model ψ(R1) ; except for ǫ see [140].
The second example considers a model for arterial tissues, which is quite similar to that
one proposed by Holzapfel, Gasser & Ogden [50]. The isotropic part in [50] is
extended by an additional polyconvex penalty term for the incompressibility constraint
and we obtain the part of the energy for the groundsubstance
!
µ
1
I1
ǫ2
iso
ψ(R2) =
− 3 + ǫ1 I3 + ǫ2 − 2 .
(7.4)
2 I31/3
I3
Since arterial walls possess fibers being arranged in mainly two directions, this model
takes into account the case n = 2. Due to the fact that the fibers are more or less
identical in both directions it is suitable to assume that the material behavior in each
fiber direction remains the same. A superposition of two transversely isotropic models with
identical material parameters reflects the latter observation. This is an appropriate model
for orthotropic materials with weak interactions between the two preferred directions a(1)
and a(2) . The full anisotropic part reads



!2 
2
(a)


X
J4
k1
f iber
−1 ,
−
1
exp k2
(7.5)
ψ(R2)
=
1/3


2k
2
I
3
a=1
Polyconvex Energies for Soft Biological Tissues
77
with the parameters given in Table 2. The penalty parameters ǫ1 and ǫ2 are chosen such
a way that the deviation of the incompressibility constraint is again less than 1 %. The
other parameters for the media and adventitia of the carotid artery of a rabbit are taken
from [50].
µ
ǫ1
ǫ2
k1
k2
[kPa] [kPa]
[-]
[kPa]
[-]
Media
3.000 23.000 29.000 2.363 0.839
Adventitia
0.300 23.000 29.000 0.562 0.711
Table 2: Material parameters of reference model ψ(R2) ; except ǫ1 and ǫ2 see [50].
The energy (7.5) is similar to the one proposed by Holzapfel et al. [50]; the only
(a)
difference is that in [50] this part is only activated for J4 ≥ 1. The basic improvement
of this condition is that hereby, the function is convex with respect to F .
It should be noted that the parameters ǫ1 in (7.2) and ǫ1 and ǫ2 in (7.4), which control the incompressibility constraint, are not necessary in more sophisticated variational
approaches.
As an example for a polyconvex model we consider the model proposed in Schröder &
Neff [104], see also [105] and [107]. The isotropic part is given by the function
I2
1
I1
α5
iso
ψ(P OLY ) = α1 1/3 + α2 1/3 − α3 ln (I3 ) + α4 I3 + α5 − 2 .
(7.6)
I3
I3
I3
The fourth term in (7.6) is able to enforce approximately the incompressibility constraint.
Furthermore, it has the advantage that it itself fulfills a priori the requirement of a stressfree reference configuration; cf. Section 5. In order to ensure this natural state condition for
the whole isotropic stored energy, the logarithmic part (ln[I3 ]) has to be taken into account.
Pn
ti,(a)
f iber
The explicit expression for the anisotropic part is given by ψ(P
=
a=1 ψ(P OLY ) with
OLY )
(a)
J α8
(a)
(a)
(a)
ti,(a)
(a)
(a)
+ α10 J4 α11 .
ψ(P OLY ) = α6 J5 − I1 J4 + I2 + α7 4 1/3 + α9 I1 J4 − J5
I3
(7.7)
Setting n = 1 leads to a transversely isotropic response function while n = 2 characterizes
an orthotropic material model with weak interactions between the fiber directions. In order
to ensure polyconvexity we have to satisfy α5 ≥ 1, α8 ≥ 1, α11 ≥ 1 and the remaining
parameters have to be greater than zero. For the comparison with reference model 2
the anisotropic part is constructed by two identical superposed transversely isotropic
functions with different preferred directions. In Section 5 a variety of polyconvex functions
satisfying automatically the stress-free reference configuration are shown based on special
case distinctions. If these case distinctions are not applied we need a larger amount of
polyconvex terms in order to account for the natural state condition. Hence, the first
term in (7.7) is necessary. The other terms are responsible for the characteristic highly
non-linear stress response in fiber direction. We have to enforce S = 0 at the natural
state, which is characterized by F = 1. Then the invariants have the values
(a)
(a)
I1 = 3, I2 = 3, I3 = 1, J4 = J5 = tr[M (a) ] = 1 .
(7.8)
Polyconvex Energies for Soft Biological Tissues
78
Consequently the stress condition for the natural state, i.e. S(1) = 0, leads to the equation
!
n
X
∂ψ
∂ψ
∂ψ
∂ψ
∂ψ
+ 2 (a) M (a) = 0 for n = 1 or n = 2 . (7.9)
1+
+2
+
(a)
∂I1
∂I2 ∂I3
∂J4
∂J5
a=1
Assuming the same material parameters for each fiber direction, as described above, the
latter equation leads to two independent restrictions. It implies that two of the material
parameters appearing in (7.6) and (7.7) have to depend on the others, because the terms
in front of the independent tensor generators 1 and M (a) have to be zero for arbitrary
structural tensors M (a) . The evaluation of (7.9) leads with (7.8) to


α3 − α2 − 2 n α9 − n α10 α11


α7 =

for n = 1
n (α8 − 1/3)
(7.10)

or n = 2 .
α3 − α2 − 2 n α9 − n α10 α11


α8 + α9 + α10 α11 
α6 =
n (α8 − 1/3)
The dependent parameters α7 and α6 must be elements of IR+ to ensure the overall polyconvexity. We remark that in case of isotropy a minimum number of 4 parameters (2
classical Lamé constants, 1 tuning parameter, 1 dependent parameter to adjust to the
stress-free reference configuration) are needed in order to match the standard representation of linearized elasticity, see Ciarlet [22]. For the highly non-linear polyconvex model
we choose a set of 11 parameters (5 classical constants for transverse isotropy, 4 tuning
parameters, 2 dependent ones for fulfilling the stress-free reference configuration).
7.1.2 Quantitative Adjustment of the Polyconvex Model Here, we adjust the
polyconvex model given in the last section for the two reference models in order to represent the same material behavior as the reference models do. For this purpose a cube
consisting of a transversely isotropic material modeled by the two reference models is
exposed to two experiments: homogeneous and constrained tension, as shown in Figure
25.
t
x3
x1
t
x1
x2
t
Exp. 1
x3
t
t
x2
t
t
Exp. 2
t
Figure 25: Experiment 1: homogeneous tension, Experiment 2: constrained tension.
In both virtual experiments a cube of the test material, fixed in x1 -direction at x1 = 0,
is deformed in x1 -direction by applying the nodal forces t. In the first experiment the
deformation of the cube is free in x2 - and x3 -direction, differing from Experiment 2,
where the cube is fixed in x2 -direction (representing plain-strain conditions). For both
experiments we consider Finite-Element simulations, where the cubes are discretized by
Polyconvex Energies for Soft Biological Tissues
79
one hexahedron finite element with linear ansatz functions; see e.g., Wriggers [143]. To
the knowledge of the author the considered reference models have been adjusted to onedimensional experiments; due to this fact we also consider only comparable experiments in
our investigations. The polyconvex model is adjusted for the (calculated) “experimental”
data generated by the reference models. This is performed by utilizing the optimization
procedure as shown in Section 6 with the objective function given therein. We use the
so-called Multi-Evolution-Algorithm implemented in the program OptimA in Schwefel
[111]. It should be remarked that for a more reliable parameter adjustment a set of linear
independent experiments would be necessary. Because of the large amount of parameters
and the non-linear coupling of the individual functions in our proposed polyconvex model
an iterative optimization algorithm has to be utilized. This is a drawback with respect to
the considered reference models.
At first we adjust the polyconvex model for the reference model 1 given in (7.2), (7.3).
Herein, both experiments, Exp. 1 and Exp. 2, are considered simultaneously. We apply the
optimization scheme as pointed out in 6.2 and after finding the minimum of the objective
function the material parameters in Table 3 are obtained.
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10
α11
[kPa] [kPa] [kPa] [kPa]
[-]
[kPa] [kPa]
[-]
[kPa] [kPa]
[-]
10.042 4.826 17.825 8.902 16.528 8.987 0.707 5.889 4.247 0.351 1.637
Table 3: Material parameters of adjusted polyconvex model.
In Fig. 26 the second Piola-Kirchhoff stresses are shown for Experiment 2. We observe
slight deviations of the polyconvex model compared to the reference model for the first
component of the second Piola-Kirchhoff stresses S11 for 1.1 < λ1 < 1.5 and for the the
second component S22 for λ1 > 1.3. We conclude that the polyconvex model is able to
represent the same stress-strain response in this restricted setting as the reference model
1.
S11
S22
1000
ref
prop
ψ(R1) vs. ψ(P OLY )
500
100
ref
prop
ψ(R1) vs. ψ(P OLY )
50
0
0
a
-500
t
x3
x1
-1000
a
-50
t
t
x3
x1
x2
t
x2
-100
t
-1500
t
t
t
-150
0.4
a)
0.6
0.8
1
λ1
1.2
1.4
1.6
0.4
b)
0.6
0.8
1
1.2
1.4
1.6
λ1
Figure 26: Comparison of reference model 1 and the polyconvex model showing the second
Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment 2 versus stretch λ1 . The preferred
direction is set to a = (1.0 , 0.0 , 0.0)T .
The second example compares the stress-strain response of the reference model 2 and
Polyconvex Energies for Soft Biological Tissues
80
the adjusted polyconvex model. After the optimized adjustment we obtain the material
parameters for the media and adventitia of the carotid artery as given in Table 4.
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10
α11
[kPa] [kPa] [kPa] [kPa]
[-]
[kPa] [kPa]
[-]
[kPa] [kPa]
[-]
Media
1.502 0.0015 0.328 6.019 10.223 0.310 0.0325 7.339 0.027 0.041 1.077
Adventitia
0.107 0.0002 0.132 4.278 5.940 0.066 0.0002 1.477 0.0003 0.012 5.381
Table 4: Material parameters of adjusted polyconvex model
It is to be remarked that the restrictions (7.10) are considered for the transversely isotropic
case (n = 1), because this is in line with the experimental setup used for the adjustment,
as well as the case n = 2, because at the end the polyconvex model should be able to
represent arterial layers where two fiber orientations are present.
S11
S22
100
ref
prop
ψ(R2) vs. ψ(P OLY )
50
10
ref
prop
ψ(R2) vs. ψ(P OLY )
5
0
0
-5
a
-50
t
x3
-10
t
x1
-100
a
t
t
x1
x2
-15
Media
x3
Media
t
x2
t
t
t
-150
-20
0.4
0.6
0.8
1
1.2
1.4
1.6
λ1
a)
0.4
0.6
0.8
1
1.2
1.4
1.6
λ1
b)
Figure 27: Comparison of reference model 2 (media) and the polyconvex model showing
the second Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment 2 versus stretch λ1 .
S11
S22
15
ψ(R2) vs. ψ(P OLY )
10
2
ref
prop
ref
prop
ψ(R2) vs. ψ(P OLY )
1.5
1
5
0.5
0
0
-0.5
a
-5
t
x3
x1
-10
Adventitia
a
t
t
-1
x2
x1
Adventitia
-1.5
t
x3
t
x2
t
t
t
-15
-2
0.4
a)
0.6
0.8
1
λ1
1.2
1.4
1.6
0.4
b)
0.6
0.8
1
1.2
1.4
1.6
λ1
Figure 28: Comparison of reference model 2 (adventitia) and the polyconvex model showing
the second Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment 2 versus stretch λ1 .
When the material parameters given in Table 4 are used together with the polyconvex
model, then this leads to a stress-strain response shown in Fig. 27 for the media and Fig.
Polyconvex Energies for Soft Biological Tissues
81
28 for the adventitia. Here, the results of the adjustment are shown, thus, we only consider
one preferred direction. If we take a look at Fig. 27 and 28, we notice that the polyconvex
model is able to describe the same material behavior as reference model 2. In arterial
layers mainly two fiber directions occur, therfore, we have to consider the superposition
of two transversely isotropic models, even though we only regard one direction for the
adjustment. Therefore, we have to check if the material behavior of the polyconvex model
is also the same as the reference model for the case of real arterial layers, where two distinct
preferred directions a(1) and a(2) occur. For this purpose we investigate the experiment
shown in Fig. 29, where also the assumed relative orientation of the fibers in a healthy
carotid artery of a rabbit are shown. The angles for the fiber directions are taken from
Holzapfel, Gasser & Ogden [50].
2β
a(2)
t
β
a(1)
x3
a(1)
t
x1
Media
29.0000o


0.8746
 0.4848 
0.0000


0.8746
 −0.4848 
0.0000
x2
a(2)
t
t
Adventitia
62.0000o


0.4695
 0.8829 
0.0000


0.4695
 −0.8829 
0.0000
Figure 29: Assumed orientation of fiber directions of a healthy carotid artery from a rabbit.
Then we use the polyconvex model with the parameters given in Table 4 and obtain also
a quite similar stress-strain response of the polyconvex model compared to that of the
reference model 2. In Figs. 30 and 31 the Kirchhoff stresses are illustrated for the media
and adventitia, respectively.
τ11
τ22
100
ref
prop
ψ(R2) vs. ψ(P OLY )
80
20
ref
prop
ψ(R2) vs. ψ(P OLY )
15
10
60
a(2)
40
t
x3
x1
20
5
a(1)
t
0
x2
-5
t
0
a(2)
t
t
-10
-20
Media
x3
x1
Media
-15
a(1)
t
x2
t
t
-40
-20
0.4
a)
0.6
0.8
1
λ1
1.2
1.4
1.6
0.4
b)
0.6
0.8
1
1.2
1.4
1.6
λ1
Figure 30: Comparison of reference model 2 (media) and the polyconvex model showing
the Kirchhoff stresses a) τ11 and b) τ22 in Experiment 2 versus stretch λ1 .
Hereby, it is shown that it is possible to adjust for a transversely isotropic model with
one preferred direction, in order to obtain the parameters for the model consisting of two
transversely isotropic models. This seems to be obvious remembering that the same material behavior for each fiber direction is assumed and therefore, no additional independent
Polyconvex Energies for Soft Biological Tissues
τ11
4
82
τ22
4
ref
prop
ψ(R2) vs. ψ(P OLY )
ref
prop
ψ(R2) vs. ψ(P OLY )
3
2
2
1
0
0
a(2)
t
-2
x3
x1
Adventitia
-4
a(2)
a(1)
-1
t
t
x1
-2
x2
Adventitia
-3
t
x3
a(1)
t
x2
t
t
t
-4
0.4
a)
0.6
0.8
1
λ1
1.2
1.4
1.6
0.4
b)
0.6
0.8
1
1.2
1.4
1.6
λ1
Figure 31: Comparison of reference model 2 (adventitia) and the polyconvex model showing
the Kirchhoff stresses a) τ11 and b) τ22 in Experiment 2 versus stretch λ1 .
information is included in the experiment considering directly two fiber directions. It is to
be remarked that concerning an adjustment for a real experiment with an arterial layer
the case of one preferred direction can not be considered, because it does not occur in reality. For our goal, to obtain a polyconvex model representing the same material behavior
as some reference models do, the adjustment performed here is sufficient.
Regarding the modeling of materials with mainly two fiber families we assumed a weak
interaction between the two fiber families and arrived at the superposition of two transversely isotropic models. In a general formulation of orthotropy and especially for fiberreinforced materials with non-perpendicular preferred directions more invariants occur.
Furthermore, a canonical representation of this type of materials would require ansatz
functions which take into account the direct coupling between the mixed invariants associated to the individual directions. In order to give a more detailed explanation of the
superposition of two transversely isotropic functions we consider the elasticity matrix C̄ at
F = 1. For the polyconvex stored energy function for the media of an artery with the material parameters given in Table 4 and the fiber orientations a(1) = (0.8746 , 0.4848 , 0.0)T
and a(2) = (0.8746 , −0.4848 , 0.0)T we obtain


5040.0 5031.0 5031.0 0.0
0.0
0.0


 5031.0 5038.0 5031.0 0.0
0.0
0.0 




 5031.0 5031.0 5038.0 0.0

0.0
0.0


C̃(F 1 ) = 
(7.11)
.
 0.0

0.0
0.0
4.184
0.0
0.0




 0.0

0.0
3.221
0.0
0.0
0.0


0.0
0.0
0.0
0.0
0.0 3.457
For the index arrangement of the elasticity tensor in matrix notation see Appendix D. A
satisfactory existence and uniqueness theory for linearized elasticity is strongly related to
the L-H-ellipticity at F = 1, see Ciarlet [22], Section 5.10, and the references therein.
It can easily be seen that the elasticity matrix (7.11) is positive definite. Furthermore it is
obvious that the chosen formulation represents a restricted class of general orthotropy. The
elasticity matrix (7.11) at F = 1 is close to the one of an isotropic model. A significant
Polyconvex Energies for Soft Biological Tissues
83
anisotropic effect occurs only with increasing deformations. This phenomenon can be
studied by comparison of the individual terms of the spatial moduli c = 2 ∂g τ (C(F , g)),
which are given by the push-forward of C, i.e. cabcd = F a A F b B F c C F d D CABCD . Considering
a homogeneous tension test, Figure 25, the complete spatial elasticity tensor for F 2 =
diag[1.6, 0.62, 1.01] appears in matrix notation as follows


0.0
0.0
5574.0 5025.0 4998.0 0.0



 5025.0 5041.0 5033.0 0.0
0.0
0.0





 4998.0 5033.0 5041.0 0.0
0.0
0.0


(7.12)
c̄(F 2 ) = 
.
 0.0
0.0
0.0
31.77 0.0
0.0 





 0.0
0.0
3.606
0.0
0.0
0.0


0.0
0.0 3.219
0.0
0.0
0.0
Let Eitan = 1/(c−1 )iiii (no sum over i) be the spatial tangential stiffness in xi -direction,
then an indicator of anisotropy is defined by the ratio E1tan /E2tan . For the homogeneous
tension test explained above we obtain
E1tan E1tan ≈ 1.1 and
≈ 39.0
(7.13)
E2tan F 1
E2tan F 2
and observe a significant anisotropic effect with increasing deformation. Similar characteristics are observed for the reference model 2.
7.1.3 Localization Analysis In Section 5 we have seen that a polyconvex function
implies that it satisfies also the Legendre-Hadamard condition, which is a physically reasonable requirement since it ensures real wave speeds. In order to check this we investigate
the so-called acoustic tensor, which has to be positive definite. Remembering the strict
Legendre-Hadamard condition requiring that
DF2 Ŵ (F ).(m ⊗ N , m ⊗ N ) > 0 ∀ F ∈ R3×3 ; m, N ∈ R3 \{0}
(7.14)
and defining the nominal tangent moduli by A = ∂F2 Ŵ (F ) we obtain the equivalent
inequality
ˆ (F , N ) · m > 0 ∀ F ∈ R3×3 ; m, N ∈ R3 \{0} .
m · Q̄
(7.15)
Herein, the acoustic tensor is defined by
Q̄ab := Aa A b B NA NB
(7.16)
in index notation. Since we consider a stored energy being a function of the right CauchyGreen deformation tensor, i.e. ψ̂(C) = Ŵ (F ), and hence knowing only S and the associated moduli C, we are interested in a transformation formula which computes the nominal
tangent moduli. Setting P = F S and having in mind that P = DF Ŵ (F ) and C = 2∂C S
we compute the nominal tangent moduli by
AāA b B =
ā
∂S CA ∂CDE ā
∂P āA
CA ∂F C
=
F
+
S
C
∂F b B
∂CDE ∂F b B
∂F b B
=
1 CABE c
C
F E F ā C gbc
2
+
1 CADB d
C
F D F ā C gdb
2
(7.17)
+S
BA ā
δ
b
Polyconvex Energies for Soft Biological Tissues
84
with the Kronecker symbol δ. The nominal tangent moduli in (7.16) can then be calculated
by lowering index a, i.e. Aa A b B = AāA b B gāa .
Notes on algorithmic implementation:
We investigate the acoustic tensor by simulating some experiments using the FiniteElement-Method. In the context of algorithmic implementations normally a cartesian
coordinate system is used and thus we do not have to differ between covariant and contravariant indices. Besides, the spatial metric tensor is replaced by the Kronecker symbol.
Usually, the symmetries of the stress- and strain tensors are exploited in order to obtain
a more efficient algorithmic structure. In our case we consider the second Piola-Kirchhoff
stresses, the right Cauchy-Green deformation tensor and the associated tangent moduli C.
Then the tangent moduli are replaced by the tangent moduli matrix C̄, see Appendix D.
Considering (7.17) and a cartesian coordinate system we obtain for the nominal tangent
moduli the formulation
AaAbB = 21 CCABE FbE FaC + 21 CCADB FbD FaC + SBA δab
= 21 (CCABD + CCADB ) FbD FaC + SBA δab .
(7.18)
The square bracket in (7.18) represents exactly the same arithmetic average included in
the algorithmic tangent moduli matrix C̄, as pointed out in Appendix D. Therefore, we
are able to use the algorithmic tangent moduli C̄ for computing the nominal tangent
moduli.
We investigate the positive definiteness of the acoustic tensor, which means that all main
minors of Q̄ have to be greater than zero. This is satisfied if the localization measure
q := sign[min[q1 , q2 , q3 ]] |q3 | > 0 ,
(7.19)
introduced in [107], is greater than zero. Herein, q1 = Q̄11 , q2 = Q̄11 Q̄22 − Q̄12 Q̄21 and
q3 = det[Q̄] are the main minors of Q̄. If the criterion given in (7.19) is not satisfied
for at least one deformation state (one A) or one direction of the cross-sectional area
represented by N , then the material is called instable and the system looses its ellipticity.
This would lead to states of equilibrium with discontinuous transformation maps ϕ,
see Knowles & Sternberg [68]. For further information about the ellipticity of the
acoustic tensor we refer to Marsden & Hughes [79], Antman [1] and the recent
publications Merodio & Ogden [83], [84] and Merodio & Neff [82].
x3
β̄
N
x2
ᾱ
x1

sin(β̄) cos(ᾱ)




N (ᾱ, β̄) = 
sin(
β̄)
sin(
ᾱ)


cos(β̄)
Figure 32: Parametrization of the vector N in spherical coordinates.
Polyconvex Energies for Soft Biological Tissues
85
The normal vector characterizing each possible cross-section in the considered body is
defined in spherical coordinates, as shown in Fig. 32. In our analysis the localization
measure q is calculated for two experiments: constrained compression and homogeneous
shear, see Fig. 33. In the first experiment the cube is fixed in x2 -direction and compressed in x1 -direction via nodal forces t. Tension is not incorporated in this experiment
because it would not contribute any qualitative differences to the observations of each
considered model with respect to localization analysis. In the second computational experiment we consider a cube, which is discretized by 8 finite hexahedron elements and
fixed in x2 -direction. At the bottom surface all displacements are fixed. The experiment
is deformation controlled in order to obtain homogeneous response, see Figure 33.
undeformed
deformed
a(2)
t
x3
x1
a(1)
u1
u1
x3
t
x1
x2
a(1)
x2
a(2)
t
Exp. 1
t
Exp. 2
Figure 33: Setup of two experiments considered in localization analysis, Experiment 1:
constrained compression, Experiment 2: homogeneous shear.
In Section 7.1.1 we have given two non-polyconvex reference models and one polyconvex
model for the description of soft biological tissues and in Section 7.1.2 we adjusted the
polyconvex model to the two reference models. Now we analyze the acoustic tensor for
these three models and compare the localization measure of the non-polyconvex reference
models with the polyconvex model.
Initially, Experiment 1 is investigated and we use the reference model 1 with the parameters given in Table 1. In order to show the progression of localization Fig. 34 (left hand
side) shows the localization measure for three deformation states: at the top we see q in
the reference configuration, below the cube is compressed up to a stretch of λ1 = 0.6
and at the bottom up to λ1 = 0.4. Here, the stretch is λ1 = (l0 − ∆l)/l0 with the length
of the body in 1-direction in the reference configuration l0 and the change of length ∆l.
For λ1 = 0.6 we observe small red areas increasing when the body is further compressed.
These red areas show angles ᾱ and β̄ for which the localization measure becomes negative,
which means the system looses its ellipticity for these directions. As an example when
λ1 = 0.6 then the Legendre-Hadamard condition is violated for ᾱ = π and β = π/2.
Then we simulate the same experiment by using the polyconvex model and the material
parameters given in Table 3, remembering that the stress-strain response of the polyconvex model is the same in the restricted setting as that of the reference model 1. The results
of the localization analysis are shown in Fig. 34 (right hand side) and we do not observe
negative values of the localization measure. This means that the material is stable, which
should be the case since we know that polyconvex energies imply the fulfillment of the
Legendre-Hadamard condition.
In order to show that non-polyconvex models can lead generally to instable materials and
not only for special ones, we investigate reference model 2, too. For this purpose we use
Polyconvex Energies for Soft Biological Tissues
Reference Model 1 ψ(R1)
86
Polyconvex Model ψ(P OLY )
λ1 = 1.0
q
q
2.5E+08
0
5E+08
-2.5E+08
2.5E+08
-5E+08
0
0
0.785
β̄
1.57
1.57
2.355
2.355
0
0
0
0.785
0.785
β̄
ᾱ
0.785
1.57
1.57
2.355
2.355
ᾱ
3.14 3.14
3.14 3.14
λ1 = 0.6
q
q
2.5E+08
0
5E+08
-2.5E+08
2.5E+08
-5E+08
0
0
0.785
β̄
1.57
1.57
2.355
2.355
0
0
0
0.785
0.785
β̄
ᾱ
0.785
1.57
1.57
2.355
3.14 3.14
2.355
ᾱ
3.14 3.14
λ1 = 0.4
q
q
2.5E+08
0
5E+08
-2.5E+08
2.5E+08
-5E+08
0
0
0.785
β̄
0.785
1.57
1.57
2.355
2.355
ᾱ
0
0
0
0.785
β̄
0.785
1.57
1.57
2.355
3.14 3.14
2.355
ᾱ
3.14 3.14
negative ←→
| positive
-5.8E+08 -4.7E+08 -3.7E+08 -2.6E+08 -1.6E+08 -5.1E+07 5.4E+07 1.6E+08
Figure 34: Localization measure q for the reference model 1 and for the polyconvex model
in three deformation states observed in Experiment 1. The preferred direction is set to
a = (1.0 , 0.0 , 0.0)T .
Polyconvex Energies for Soft Biological Tissues
87
the parameters given in Table 2 and the preferred directions as shown in Fig. 29 and
assume real soft biological tissues; here the media and adventitia of a carotid artery of a
rabbit. At first we investigate Experiment 1 and increase the compression up to a stretch
of λ1 = 0.2. The results of the localization analysis are shown in Fig. 35 and we notice
that negative values of the localization measure q occur for the media as well as for the
adventitia, hence, the system loses ellipticity considering the reference model 2. If we use
the polyconvex model together with the material parameters given in Table 4 then no
negative values of q are observed and we obtain a stable material.
Reference Model 2 ψ(R2)
Polyconvex Model ψ(P OLY )
Media
q
q
0
2E+06
-5E+07
-1E+08
0
0
0.785
β̄
0.785
1.57
1.57
2.355
2.355
0
0
0
0.785
ᾱ
β̄
0.785
1.57
1.57
2.355
neg. ←→
| pos.
3.14 3.14
2.355
ᾱ
3.14 3.14
-1.0E+08 -8.5E+07 -7.0E+07 -5.6E+07 -4.1E+07 -2.6E+07 -1.1E+07 3.5E+06
q
Adventitia
q
0
0
0
0.785
β̄
0.785
1.57
1.57
2.355
2.355
3.14 3.14
ᾱ
0
0
0
0.785
β̄
0.785
1.57
1.57
2.355
neg. ←→
| pos.
2.355
ᾱ
3.14 3.14
-5.9E+04 6.0E+04 1.8E+05 3.0E+05 4.2E+05 5.3E+05 6.5E+05 7.7E+05
Figure 35: Localization measure q for the reference model 2 and for the polyconvex model
in the media and adventitia of the carotid artery of a rabbit observed in Experiment 1 at
λ1 = 0.2. The preferred direction is set as shown in Fig. 29.
Secondly, as a last example, we compare the reference model 2 with the polyconvex model
in Experiment 2, where homogeneous shear is applied. We also use the parameters given
in Table 2 and 4 and the preferred directions as shown in Fig. 29 in order to obtain a
real soft biological tissue. Here, we only investigate the media, because we have already
seen that the loss of ellipticity considering the non-polyconvex model does not occur only
Polyconvex Energies for Soft Biological Tissues
88
for special sets of parameters since we have investigated not only one type of tissue.
We increase the deformation such that the shear component of the deformation gradient
becomes F13 = 1.0. In Fig. 36 the localization measure for both models is shown and we
notice that contrary to the polyconvex model the non-polyconvex model (left hand side)
leads to an instable material.
Reference Model 2 ψ(R2)
Polyconvex Model ψ(P OLY )
q
q
150000
1E+06
100000
0
0
0
0.785
β̄
0.785
1.57
1.57
2.355
2.355
3.14 3.14
ᾱ
50000
0
0
0.785
β̄
0.785
1.57
1.57
2.355
neg. ←→
| pos.
2.355
ᾱ
3.14 3.14
-5.6E+05 -2.2E+05 1.1E+05 4.5E+05 7.8E+05 1.1E+06 1.5E+06 1.8E+06
Figure 36: Localization measure q for the reference model 2 and for the polyconvex model
in the media of a carotid artery of a rabbit observed in Experiment 2 at F13 = 1. The
preferred direction is set as shown in Fig. 29.
Summarizing this section, we realize that although the same stress-strain behavior is
observed for some restricted situations, a completely different localization behavior occurs
considering a non-polyconvex and a polyconvex model. In this section it is illustrated that
a polyconvex model ensures material stability since polyconvexity is a sufficient condition
for the Legendre-Hadamard condition if a smooth energy function is analyzed, see Section
5.1.4 or Schröder, Neff & Balzani [107]. It is to be remarked that the energy W
utilized in the localization analysis includes all the internal energy that exists in the body.
Hence, if a special FE-approach as e.g., starting from a potential of the Hu-Washizu type,
is used, then the differing internal energy has to be taken into account.
7.2 A New Polyconvex Model for Arterial Walls
In Section 7.1 a special polyconvex model (7.6), (7.7) is analyzed, which does not require any case distinction. This model is polyconvex for each value of the invariants.
Unfortunately, it does not satisfy the natural state condition a priori and two additional
restrictions for the material parameters occur, i.e. (7.10). These two restrictions are responsible for a strong dependency between the particular additive terms in the model. If a
material behavior has to be modeled that has an intensely differing stiffness in fiber direction compared to the other directions, i.e. a very strong decoupling between the isotropic
and transversely isotropic part, then this model can not be utilized.
Besides, the large amount of material parameters, which is required to adjust for the
Polyconvex Energies for Soft Biological Tissues
89
stress-free reference configuration, lets the model become manageable such a way that it
is impossible to adjust for experimental data “by hand”. For that reason in Section 7.1.2
an optimization procedure is necessarily utilized.
In Section 5.3 a variety of transversely isotropic polyconvex functions are constructed
based on the construction principle given there. These functions require to take into
account a special case distinction, but have the great advantage that they themselves
satisfy the stress-free reference configuration automatically. In this section we want to
show that the modeling of soft biological tissues by polyconvex stored energy functions
is also possible by considering relatively manageable functions that do not essentially
require any kind of optimization procedure. For this purpose we illustrate the way of
constructing a concrete energy function for human arterial layers and adjust the model
for experimental data without using an optimization scheme.
7.2.1 Experimental Data of a Human Aortic Layer
To give an example of
handling the polyconvex functions provided in section 5.3 we adjust some of these functions to a biological material. As an example, we consider an abdominal aorta from a
human cadaver (male, 40 years, primary disease: congestive cardiomyopathy), which has
been excised during autopsy within 24 hours after death. The arterial wall was separated
anatomically into the three layers, i.e. intima, media and adventitia. In the present work
we focus on the media (i.e. the middle layer of the artery), because from the mechanical perspective the media is the most significant layer in a healthy artery. It consists of
smooth muscle cells, collagen fibers, elastin in form of fenestrated elastic lamellae, and
ground substance. The structured arrangement of these constituents gives the media high
strength, resilience and the ability to resist loads in both the longitudinal and circumferential directions. From the media, strip samples with axial and circumferential orientations
were cut out so that two specimens were obtained, as illustrated in Fig. 37 (for representative tissue samples see, for example, Fig. 4 in Holzapfel, Sommer & Regitnig
[55]).
σ[kPa]
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
a)
σ[kPa]
β
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
a(1)
(1)
(2)
a(2)
(1)
(2)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
0.3
b)
Experiment1
Eng. curve1
Experiment2
Eng. curve2
(1)
(2)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
0.3
Figure 37: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the a) experimental tension tests (loading and unloading) of a circumferentially (1) and longitudinally (2) oriented strip extracted
from the media of a human abdominal aorta and b) the considered associated reference
curves. l0 is the reference length of the stripes while ∆l is denoting the difference between
actual and reference length.
Prior to testing, pre-conditioning was achieved by executing five loading and unloading
cycles at a constant crosshead speed of 1 mm/min for each test to obtain repeatable stressstrain curves. Subsequently, the stripes underwent uniaxial extension tests (loading and
Polyconvex Energies for Soft Biological Tissues
90
unloading) in 0.9 % NaCl solution at 37◦ C with continuous recording of tensile force, strip
width and gage length at a constant crosshead speed of 1 mm/min. For details on the
customized tensile testing machine the reader is referred to Schulze-Bauer, Regitnig
& Holzapfel [109]. The results of experiment (1) (tension in circumferential direction)
and experiment (2) (tension in longitudinal direction) for the loading and unloading process are illustrated in Fig. 37a). We observe a slight viscoelastic effect when comparing
the curve for loading and unloading and therefore, consider a reference curve located in
the center of the loading and unloading curve, see Fig. 37b). This reference curve represents the assumed hyperelastic behavior of the artery. Additional experimental data for
uniaxial extension tests for the Intima and Adventitia are given in Holzapfel [57].
7.2.2 Representation of the Arterial Tissue For the description of the material
behavior as shown in Fig. 37 we compare three polyconvex models. Since we consider
arterial layers which consist mainly of an isotropic groundsubstance and two fiber families,
we take into account stored energies being of the form given in (7.1) with n = 2.
As a first example we consider the model of Holzapfel, Gasser & Ogden [50], which
consists of the polyconvex isotropic part (5.43) and the convex transversely isotropic part
(5.61). For the analysis of this model we incorporate the quasi-incompressibility through
a special Finite-Element approach. The parameters for the best fit to the experimental
data are shown in Table 5.
c1 = µ/2
k1
k2
[kPa]
[kPa]
[-]
10.2069 0.00170 882.847
Table 5: Material parameters of the model of Holzapfel, Gasser & Ogden [50]. The
angle between the (mean) fiber direction and the circumferential direction in the media was
predicted to be 43.39◦. The fiber angle acts here as a phenomenological parameter.
In Fig. 38 the Cauchy stresses are depicted for the circumferentially and longitudinally
oriented strip when using the constitutive model of Holzapfel, Gasser & Ogden.
The angle between the (mean) fiber directions is assumed to be β = 43.39◦.
σ[kPa]
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
a)
σ[kPa]
70
Experiment1
Constitutive Model
Experiment2
Constitutive Model
60
2β
a1
t
x3
x1
2β
50
a2
a1
t
t
x3
40
x1
x2
a2
t
x2
30
t
t
t
t
20
ψ(HGO)
ψ(HGO)
10
0
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
b)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 38: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment and the constitutive
model of Holzapfel, Gasser & Ogden [50]: a) circumferentially and b) longitudinally
oriented stripes. l0 is the reference length of the strip and ∆l the change of length.
As can be seen, the match is quite good, even though the strong exponential character
is underestimated. The (exponential) stiffening effect at higher loads may be described
Polyconvex Energies for Soft Biological Tissues
91
with higher accuracy by introducing one additional dimensionless parameter ranging between zero and one, as recently proposed in Holzapfel, Stadler & Gasser [56],
Holzapfel, Sommer, Gasser & Regitnig [58]. The additional parameter is then a
measure of anisotropy. For zero the function reduces to an isotropic (rubber-like) model,
similar to that proposed in Demiray [27], while for one the function reduces to the model
proposed in [50].
Concerning the second and third model considered later in this section we adjust these
models for the experimental data “by-hand”, because we want to show their applicability.
Therefore, in order to have an objective measure for the quality of adjustment of each
model we introduce the relative error
r :=
|σ exp − σ mod |
.
exp
|σmax
|
(7.20)
Herein, σ exp and σ mod denote the experimental stresses (as illustrated in Fig. 37b as the
solid lines) and the stresses computed by the constitutive model, respectively. Note that r
should be as low as possible and would become zero for a perfect matching. The benchmark
of the adjustment for a complete experiment can be accomplished by the definition of the
quantity
v
u n
1 u
1 X exp
r̄ := exp t
(σi − σimod )2 ,
(7.21)
|σmax | n i=1
wherein the total number of the experimental data-points i is denoted by n. In Fig. 39 the
relative error is shown for the two experiments. For the circumferentially oriented strip
we obtain r̄ = 0.08 and for the longitudinally oriented strip we receive r̄ = 0.05.
0.25
r
0.25
ψ(HGO)
2β
a1
0.2
x3
x1
0.15
a1
t
x3
t
x2
0.15
t
0.1
x1
t
a2
t
x2
t
0.1
0.05
t
0.05
0
a)
ψ(HGO)
2β
0.2
a2
t
r
0
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
b)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 39: Relative error r vs. strain ∆l/l0 using the constitutive model of Holzapfel,
Gasser & Ogden [50]: a) circumferentially (r̄ = 0.08) and b) longitudinally oriented stripes
(r̄ = 0.05).
In this section we are interested in the easy fitting of polyconvex stored energies to soft
biological tissues. As an example, we consider another polyconvex function, whose parameters can easily be adjusted. Therefore, we do not use any optimization procedure
for the adjustment here and obtain the material parameters by “hand-fitting”. For the
second polyconvex model we keep the isotropic part of the Holzapfel, Gasser &
Ogden-model and add the function (5.42)1 with α3 = 1.0 in order to consider the quasiincompressibility constraint via a penalty function. Then the isotropic part of the stored
Polyconvex Energies for Soft Biological Tissues
92
energy reads
iso
ψ(BIO
1)
= c1
I1
1/3
I3
−3
!
1
ǫ2
+ ǫ1 I3 + ǫ2 − 2 , c1 > 0 , ǫ1 > 0 , ǫ2 > 1 .
I3
(7.22)
It is remarked that this isotropic function satisfies the coercivity condition and therefore
iso
each additive combination with ψ(BIO
will be also coercive. For the description of the
1)
material behavior in fiber direction we account for the transversely isotropic part given
in (5.64) and we obtain the complete anisotropic part

2 h
α2 i
X


(a)
(1)
(2)


α
K
−
2
for K3 ≥ 2 ∧ K3 ≥ 2
1
3



a=1






α


(1)
(2)
α1 K (1) − 2 2
for K3 ≥ 2 ∧ K3 < 2
3
aniso
(7.23)
ψ(BIO1 ) =



α2


(2)
(1)
(2)

α
K
−
2
for K3 < 2 ∧ K3 ≥ 2

1
3








(1)
(2)
0
for K3 < 2 ∧ K3 < 2 .
Herein, the case distinction is necessary in order to ensure that the internal function
(a)
(K3 −2) is positive. The complete polyconvex stored energy for the considered biological
soft tissue reads
iso
aniso
ψ(BIO1 ) = ψ(BIO)
+ ψ(BIO
.
(7.24)
1)
In this model c1 scales the isotropic stress response which is seen in the nearly linear
behavior of the curve below ∆l/l0 ≈ 0.15 in Fig. 37. The parameters ǫ1 and ǫ2 control the
volumetric deformation; this is important in order to satisfy the quasi-incompressibility
constraint. These two parameters are set such that the incompressibility condition is
violated by maximal 1.0 %, so that detF = 1 ± 1.0 %. The parameter α2 accounts for the
level of curvature in fiber direction while the parameter α1 is introduced in order to scale
iso
is coercive, also ψ(BIO1 ) is coercive
this response. It should be noted that since ψ(BIO
1)
and since ψ(BIO1 ) is polyconvex the sufficient condition for the existence of minimizers is
satisfied.
We adjust the polyconvex model ψ(BIO1 ) to the experimental data “by hand” in order to
show its uncomplicated handling and obtain the material parameters given in Table 6.
c1
[kPa]
17.5
ǫ1
ǫ2
α1
[kPa] [-]
[kPa]
100.0 50.0 5 · 105
α2
[-]
7.0
Table 6: Material parameters of the model ψ(BIO1 ) . The angle between the (mean) fiber
direction and the circumferential direction in the media was predicted to be 43.39◦. The
fiber angle acts here as a phenomenological parameter.
Fig. 40 presents the results by comparing the Cauchy stresses of the experiment with
the stresses governed by the model ψ(BIO1 ) . We observe deviations of the model from
the experiment with a circumferentially oriented strip for 0.2 < ∆l/l0 < 0.25. When
Polyconvex Energies for Soft Biological Tissues
σ[kPa]
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
a)
93
σ[kPa]
70
Experiment1
Constitutive Model
Experiment2
Constitutive Model
60
50
2β
2β
a1
a1
a2
t
x3
x1
40
t
x3
t
x1
x2
20
t
t
t
ψ(BIO1 )
0
0.05
t
x2
30
t
a2
0.1
0.15
∆l/l0
0.2
0.25
10
ψ(BIO1 )
0
b)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 40: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment and the constitutive
model ψ(BIO1 ) : a) circumferentially and b) longitudinally oriented stripes.
0.25
r
0.25
ψ(BIO1 )
2β
a1
0.2
t
x3
x1
a1
t
x3
t
x2
0.15
t
0.1
x1
t
a2
t
x2
t
0.1
0.05
t
0.05
0
a)
ψ(BIO1 )
2β
0.2
a2
0.15
r
0
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
b)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 41: Relative error r vs. strain ∆l/l0 using the constitutive model ψ(BIO1 ) : a) circumferentially (r̄ = 0.04) and b) longitudinally oriented stripes (r̄ = 0.06).
comparing with the experiment of the longitudinally oriented strip we notice a differing
curve mainly for ∆l/l0 > 0.22.
In Fig. 41 the measure r is depicted for the circumferentially and the longitudinally oriented stripes. We observe the maximum values of r for the first experiment at ∆l/l0 = 0.25
and for the second experiment at ∆l/l0 = 0.27. These results reflect the observations
found when comparing the stress-strain relation of the model with the experiments. We
conclude that the stress-strain response for the two experiments is represented quite accurately, even though the exponential character for the longitudinally oriented strip is
underestimated.
For the circumferentially and longitudinally oriented stripes we calculate r̄ = 0.04 and
r̄ = 0.06, respectively. By comparing this with the results obtained for the model of
Holzapfel, Gasser & Ogden, we notice that ψ(BIO1 ) leads to a measure r̄, which is
lower for the circumferentially and higher for the longitudinally oriented stripe. Here, it
should be recalled that ψ(BIO1 ) is adjusted “by hand”, contrary to ψ(HGO) , and therefore,
an objective comparison of the adjustment accuracy of the two models is not possible.
One of the main characteristics of the experimental data is that the curves for the investigated stripes differ right from the beginning of the deformation. A slight drawback
of the previous two models becomes obvious when the stress-strain response of the two
experiments is depicted in the same diagram. In Fig. 42a) we see that the curve of the
experiments start to differ at approximately ∆l/l0 ≈ 0.15 and the curve in Fig. 42b)
seems to differ earlier but not as much as the curves of the experiments. Therefore, the
Polyconvex Energies for Soft Biological Tissues
94
two models seem not to be able to represent the fact that the experimental curves differ
right from the start.
σ[kPa]
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
a)
σ[kPa]
(1)
(2)
Experiment1
Constitutive Model
Experiment2
Constitutive Model
(1)
ψ(HGO)
(2)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
b)
(1)
(2)
Experiment1
Constitutive Model
Experiment2
Constitutive Model
(1)
ψ(BIO1 )
(2)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 42: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the two experiments and the constitutive models a) ψ(HGO) and b) ψ(BIO1 ) .
To overcome this issue we additionally consider a third polyconvex energy. We keep the
energy ψ(BIO1 ) and extend it by the transversely isotropic function given in (5.60), then
we obtain for two fiber families the case distinction

2 h
α4 i
X


(a)
(1)
(2)


α3 J4 − 1
for J4 ≥ 1 ∧ J4 ≥ 1



a=1






α


(1)
(2)
α3 J (1) − 1 4
for J4 ≥ 1 ∧ J4 < 1
4
aniso
(7.25)
ψ(BIO2 ) =


 α4


(2)
(1)
(2)

α
J
−
1
for J4 < 1 ∧ J4 ≥ 1

3
4








(1)
(2)
0
for J4 < 1 ∧ J4 < 1 .
The complete polyconvex model for the description of the medial layer of a human abdominal aorta reads
iso
aniso
aniso
ψ(BIO2 ) = ψ(BIO)
+ ψ(BIO
+ ψ(BIO
.
(7.26)
1)
2)
aniso
As before, ψ(BIO
describes the exponential-type behavior in the fiber direction, while
1)
aniso
ψ(BIO2 ) takes care of the different curves for the circumferentially and longitudinally oriented stripes in the low loading domain. In order to show the easy handling the model
is adjusted to the experimental data by “hand-fitting” again. The chosen parameters are
summarized in Table 7.
c1
[kPa]
12.0
ǫ1
ǫ2
α1
[kPa] [-]
[kPa]
100.0 40.0 8 · 107
α2
α3
[-] [kPa]
10.0 70.0
α4
[-]
2.05
Table 7: Material parameters of ψ(BIO2 ) for the media of a human abdominal aorta.
The response of the model ψ(BIO2 ) is compared to the experimental data in Fig. 43. Firstly,
we see that the exponential character of the stress-strain behavior of the circumferentially
Polyconvex Energies for Soft Biological Tissues
95
oriented strip is no longer underestimated and the curve fits the experimental data quite
well. Secondly, ab initio the deviating curves for the circumferentially and longitudinally
oriented stripes, as seen in the experiment, may be described accurately. Unfortunately,
the curve of ψ(BIO2 ) underestimates the stress response for ∆l/l0 > 0.23.
σ[kPa]
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
(1)
(2)
Experiment1
Constitutive Model
Experiment2
Constitutive Model
(1)
ψ(BIO2 )
(2)
0
0.05
0.1
0.15
0.2
∆l/l0
0.25
Figure 43: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment ((1) circumferentially
and (2) longitudinally oriented strip). The fit is based on the constitutive model ψ(BIO2 ) .
Analyzing the relative error, see Fig. 44, we see that the model fits the experimental
data of the circumferentially oriented strip quite accurately. For the circumferentially and
longitudinally oriented strip we obtain r̄ = 0.03 and r̄ = 0.07, respectively.
0.25
r
0.25
ψ(BIO2 )
2β
a1
0.2
a2
t
x3
x1
0.15
r
a1
t
0.15
t
0.1
x1
t
a2
t
x2
t
0.1
0.05
t
0.05
0
a)
x3
t
x2
ψ(BIO2 )
2β
0.2
0
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
b)
0
0.05
0.1
0.15
∆l/l0
0.2
0.25
Figure 44: Relative error r vs. strain ∆l/l0 using the constitutive model ψ(BIO2 ) : a) circumferentially (r̄ = 0.03) and b) longitudinally oriented stripes (r̄ = 0.07).
In this work we are interested in the simulation of arterial walls, whose mechanical properties are also mainly effected by the adventitia. Therefore, we adjust the model ψ(BIO1 ) for
the adventitia of a human abdominal aorta, too. For this purpose we use the experimental
data published in Holzapfel [57] and obtain the parameters shown in Table 8.
c1
[kPa]
7.5
ǫ1
ǫ2
α1
[kPa] [-]
[kPa]
100.0 20.0 15 · 109
α2
[-]
20.0
Table 8: Material parameters of ψ(BIO1 ) for the adventitia of a human abdominal aorta.
In Fig. 45 the Cauchy stresses for the circumferentially and longitudinally oriented strip
are shown. Here, the angle between the (mean) fiber directions is estimated as β = 49.0◦ .
Polyconvex Energies for Soft Biological Tissues
96
σ[kPa]
70
Experiment1
(1) Constitutive
Model
Experiment2
(2) Constitutive
Model
60
50
(2)
ψ(BIO1 )
40
30
20
(1)
10
0
0
0.05
0.1
0.15
0.2
∆l/l0
0.25
0.3
0.35
Figure 45: Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment ((1) circumferentially
and (2) longitudinally oriented strip of the adventitia of the considered human abdominal
aorta). The fit is based on the constitutive model ψ(BIO1 ) .
Concluding this section, we notice that the construction of polyconvex energies for soft
biological tissues can be done in a relatively easy way. For this task in Section 5 polyconvex
functions are derived and in Appendix C listed. Additionally, the functions are quite
manageable, which means that the model can be adjusted for experimental data in an
artless way. This is shown by not using an optimization procedure for the adjustment.
Anisotropic Damage Model for Arterial Walls
8
97
Anisotropic Damage Model for Arterial Walls
In Section 2 it has been pointed out that in biological soft tissues damage effects appear,
when the tissues are overexpanded such that deformations occur larger than physiological
ones. Therefore, this section provides an anisotropic damage model for the description of
damage in biological soft tissues. At first we explain the (1 − D) - approach, the effective
stress concept based on the strain equivalence principle and the consequences of discontinuous damage to the stress-strain response of materials. Then we deduce the anisotropic
damage model able to describe the discontinuous damage effects in soft biological soft tissues and provide its algorithmic implementation. Finally, a numerical example compares
the response of the damage model with the qualitative response of discontinuous damage.
It is to be remarked that in this section the basic items of damage mechanics are only
described very briefly. For further information about this subject the extensive surveys
found in Skrzypek & Ganczarski [116] and Krajcinovic [71] and with respect to
engineering applications in Lemaitre & Desmorat [75] may be consulted. For an introduction to damage mechanics we refer to Lemaitre [74], de Souza Neto, Perić &
Owen [28] and for a geometrically motivated explanation of general anisotropic damage
to Betten [13].
8.1 (1 − D) - Approach and Discontinuous Damage
Generally, damage may be interpreted as the breaking of atomic bonds or plastic enlargement of microcavities. For the description of such damage the (1 − D) - approach is
introduced in Kachanov [65]. There, the scalar-valued damage variable D is defined as
the maximum value of area of microcavities per total area of all cross-sections along one
direction (n) in a representative volume element (RVE) extracted from a physical body
D = D̂(n) := sup
x
"
δ ÂD (x, n)
δ Â(x, n)
#
with D ∈ [0, 1] ,
(8.1)
cf. Fig. 46. In this definition x̃ denotes the position vector of the considered point 0 in
the RVE and x describes the position of the considered cross-section along n. It should
be remarked that for isotropic materials the dependency on n vanishes.
n
x
0
e2
x̃
e3
e1
δAD
δA
Figure 46: Micro-meso definition of one-dimensional damage variable introduced by
Kachanov [65], cf. Lemaitre [74].
Anisotropic Damage Model for Arterial Walls
98
Let n now describe the axis of a one-dimensional body, then the damage variable reduces to D := AD /A. Based on the strain equivalence principle, which is introduced by
Rabotnov [93], the effective stress concept considers three states, see Fig. 47.
σ
σ
σ̃
AD
A
σ
virgin state
σ
damaged state
σ̃
pseudo-undamaged
state
Figure 47: Three states of the one-dimensional effective stress concept, cf. Skrzypek &
Ganczarski [116].
The virgin state describes the undamaged state, where the damage variable takes the value
0. The damaged state represents the state in which the damage has taken place (D 6= 0)
and therefore, the material is only able to resist the stresses with a reduced cross-section
area, i.e. (A − AD ). This state can be replaced by a pseudo-undamaged state considering
the effective stresses σ̃ and D = 0. The true (measurable) stresses in the damaged state
are then computed by
σ = (1 − D) σ̃ ,
(8.2)
wherein the effective stresses are governed by the constitutive law for elasticity.
With respect to biological soft tissues one has to take into account that these tissues
are fiber-reinforced and therefore not isotropic materials. In Section 2 the composition
of arterial layers and the structure of the bracing collagen fibers are explained. There,
it is mentioned that if arteries are over-expanded, damage effects are observed in experiments by a reduction of the material stiffness, see e.g., Holzapfel, Gasser &
Ogden [50] or Gasser & Holzapfel [38]. These damage effects can be subdivided
into continuous-type- and discontinuous-type-damage, where the evolution of continuoustype-damage decreases with increasing number of loading and unloading cycles and the
discontinuous-type-damage lowers the material stiffness only during the first cycle. In Fig.
48 the discontinuous damage effect is schematically illustrated.
In a first loading cycle increasing the strain up to ε1 we follow path A and damage the
material. Hereby, we reduce the material stiffness and obtain a lower path (B) when unloading the material. We only assume the existence of damage evolution if the deformation
accomplished before is exceeded. Therefore, we generally do not obtain any damage evolution during unloading processes. For the reloading process this means that if we apply the
deformation ε1 again we do not damage the material and receive the same stress-strain
curve (B). A gain of the damage state is governed when the deformation is increased such
that it exceeds ε1 , e.g., up to ε2 , which is illustrated in Fig. 48 by curve C. After reaching
ε2 and unloading the material we again follow a lower path (D), because damage has
Anisotropic Damage Model for Arterial Walls
σ
E
ε
C
ε2
A
ε1
B
ε1
a)
ε2
E
1
0
0
1
0
1
C
00
11
00
11
1
0
A
D
0
99
B
D
ε
t
b)
Figure 48: Qualitative diagram showing a) stress (σ) vs. strain (ε) and b) associated strain
vs. time (t) for discontinuous damage.
taken place while the deformation was increased from ε1 to ε2 . This procedure is repeated
for each further loading and unloading cycle.
In biological soft tissues we assume, that the damage evolution in the range of deformations which have not been reached before is much higher, and therefore concentrate on
the discontinuous damage. This can reasonably be understood by interpreting the damage mainly as the breakage of collagen crossbridges induced by larger deformations than
occurring in the physiological state of the tissues. The disrupted structure is assumed not
to heal itself in a certain temporal interval and therefore, the damage state remains in
loading and unloading cycles of deformations accomplished before. This physical interpretation of how damage acts in biological soft tissues leads to the assumption that damage
exists mainly in fiber direction.
8.2 The Damage Model
For the modeling of damage explained above we let the effective anisotropic hyperelasticity
of the considered tissue be described by a stored energy function of the type given in
Section 7 and apply the (1 − D) - approach to that part of the energy which is associated
to the description of the fibers. It is remarked that the application of the (1−D) - approach
to a special part of the stored energy is already performed in Simo [113] or Miehe &
Keck [86] for isotropic damage. Hereby, the “one-dimensional problem” of fiber damage
(considering only the fiber direction) represents the “three-dimensional” anisotropy of
the damage in biological soft tissues by using scalar-valued damage variables. This is in
contrast to the general topic of anisotropic damage mechanics, where usually damage
tensors of second or fourth order are employed.
Based on the physical considerations as explained above we assume, that the damage
occurs mainly in the fiber direction, cf. Balzani, Schröder & Gross [7] or Balzani,
Schröder & Gross [9]. This leads to the additively decoupled structure of the stored energy into an undamaged isotropic part for the ground substance and a damaged anisotropic
part, consisting of two superimposed transversely isotropic models. We introduce a scalarvalued damage variable D(a) associated to the fiber family (a), characterizing with increasing values a reduction in the stiffness of the fibers. Due to the fact that we assume same
properties for all fiber families, cf. Section 7, the parameters for the damage in each fiber
Anisotropic Damage Model for Arterial Walls
100
direction will be the same. Then the structure of the stored energy is defined by
(a)
ψ̂(I1 , ...J5 , D(a) )
2 h
i
X
(a)
ti,0
= ψ̂ (I1 , I2 , I3 ) +
(1 − D(a) )ψ̂(a) (I1 , ...J5 ) ,
iso
(8.3)
a=1
where ψ iso and ψ ti,0 denote the isotropic energy and the so-called effective (transversely
isotropic) energy. It is remarked that the transversely isotropic effective energy has to
ti,0
increase for tension in fiber direction. Furthermore, ψ(a)
has to be zero for compression
in fiber direction. This reflects the assumption that the fibers act mainly as wires, i.e.
they are only able to resist tensile stresses and are only damaged if tension is applied.
It should be mentioned that these requirements are fulfilled for the functions listed in
Appendix C, except for the functions containing K1 , because these functions are positive
for compression and zero for tension in fiber direction. The isotropic energy should be of
course polyconvex, too, and therefore, the functions given in 5.3.1 may be utilized.
In accordance with thermodynamic consistency the second law of thermodynamics has to
be satisfied and we consider the Clausius-Duhem inequality for isothermal conditions
1
S : Ċ − ψ̇ ≥ 0 .
2
D=
(8.4)
Remember that the stored energy ψ is defined per unit reference volume. By timedifferentiation of the stored energy the dissipation inequality yields
D=
1
S − ∂C ψ
2
: Ċ +
2 X
ti,0
ψ(a)
Ḋ(a) ≥ 0 .
(8.5)
a=1
The standard argument of rational continuum mechanics, requiring the latter condition to
be satisfied for all imaginable values of the free variables, yields the constitutive equation
for the stresses in the decoupled form
S = 2 ∂C ψ = S
iso
+
2
X
S ti(a) .
(8.6)
a=1
Herein, the isotropic and transversely isotropic part are given by
ti,0
S iso = 2 ∂C ψ iso and S ti(a) = 1 − D(a) S ti,0
with S ti,0
(a)
(a) = 2 ∂C ψ(a) ,
(8.7)
respectively. It should be noted that S ti,0
(a) are the effective transversely isotropic 2nd
Piola-Kirchhoff stresses. By setting (8.6) the reduced dissipation inequality appears as
D red =
2 X
ti,0
ψ(a)
Ḋ(a) ≥ 0 .
(8.8)
a=1
When constructing the constitutive function for the damage this inequality has to be
satisfied in order to ensure the thermodynamical consistency. For the functional dependency of the damage variable the formalism described in Lemaitre & Chaboche [73]
is applied. There, a stored energy is considered as a function of a strain measure (free
variable) and a dependent variable V describing the dissipative effect, i.e. ψ := ψ̂(ε, V ).
In our considerations the free variables are the right Cauchy-Green deformations and the
Anisotropic Damage Model for Arterial Walls
101
dependent variables are the damage variables D(a) for a = 1 and a = 2. In (8.6) we
see that the derivative of ψ with respect to the free variables, i.e. S = ∂C ψ, drives the
hyperelasticity. In an analogous manner the so-called thermodynamic force is defined as
∂ψ
ti,0
= −ψ(a)
,
∂D(a)
A(a) :=
(8.9)
driving the dissipative effect. We postulate the existence of a dissipation potential being
a continuous and convex scalar-valued function of the flux variable, i.e. ϕ := ϕ̂(Ḋ(a) ),
which has to be zero at Ḋ(a) = 0. The normality property yields the work-conjugated
ti,0
variable ψ(a)
= ∂Ḋ(a) ϕ. We apply the Legendre-Fenchel transformation and obtain the
dual dissipation potential
h
i
ti,0
ϕ̂∗ (ψ(a)
) = sup −A(a) Ḋ(a) − ϕ .
(8.10)
Ḋ(a)
If ϕ∗ is differentiable the dual work-conjugated variable reads
−Ḋ(a) =
∂ϕ∗
∂A(a)
⇔
Ḋ(a) =
∂ϕ∗
ti,0 .
∂ψ(a)
(8.11)
ti,0
ti,0
Remember that ψ(a)
= ψ̂(a)
(Ĉ(X, t), M (a) ), then time integration of Ḋ(a) and evaluation
at actual time leads to
Z t̄ ∂ ϕ̂∗ (ψ ti,0 )
(a)
dt .
(8.12)
D(a) =
ti,0
∂ψ(a)
0
This shows that the damage variable is a function of the effective transversely isotropic
ti,0
energy, i.e. D(a) = D̂(a) (ψ(a)
). Hereby, we are able to construct the damage function
"
ti,0
D̂(a) (ψ(a)
) = γ1 1 − exp
ti,0
−β̂(a) (ψ(a)
)
γ2
!#
with γ1 ∈ [0, 1], γ2 > 0 ,
(8.13)
cf. Miehe [85]. It should be remarked that the material parameters γ1 and γ2 do not
differ with the fiber direction.
In order to reflect the discontinuous damage we define a specific internal variable as the
maximum value of the effective transversely isotropic energy reached up to the actual
time, cf. Simo [113]. With the Macauley-bracket h(•)i = 12 [(•) + |(•)|] we obtain
D
E
ti,0
ti,0
ti,0
β̂(a) (ψ(a)
) = sup ψ(a),s
− ψ(a),ini
.
(8.14)
0≤s≤t
ti,0
It must not necessarily be the case that the internal variable β(a) is zero when ψ(a)
= 0; this
would characterize the initiation of the damage for the undeformed reference configuration.
Especially for arterial tissues it is reasonable to consider a initial damage state at which the
damage evolution begins, because no damage evolution should occur in the physiological
range of deformations. Therefore, we introduce the transversely isotropic effective energy
ti,0
at this initial damage state ψ(a),ini
in (8.14).
For ensuring the thermodynamical consistency we have to satisfy (8.8). Due to the special
definition of the internal variable taking into account the discontinuous damage, the time
Anisotropic Damage Model for Arterial Walls
102
derivative of β(a) is always positive or zero, i.e. β̇(a) ≥ 0. Therefore, the time derivative of
the damage variable
∂D(a)
−β(a)
∂D(a)
γ1
Ḋ(a) =
=
exp
≥0
(8.15)
β̇(a) with
∂β(a)
∂β(a)
γ2
γ2
ti,0
is also positive or zero. Since ψ(a)
≥ 0 we conclude that the reduced dissipation inequality
(8.8) is satisfied.
In order to provide an activation condition for the evolution of damage we define the
damage criterion
D
E
ti,0
ti,0
φ(a) := ψ(a) − ψ(a),ini − β(a) ≤ 0 ,
(8.16)
inducing no damage evolution for φ(a) < 0.
Remark: It should be noted that if we consider a damage variable for a fixed time step and
use a polyconvex effective energy, then the complete energy will also be polyconvex since
the damage variable for fixed time can be interpreted as a constant factor manipulating
solely the material parameters of the stored energy.
8.3 Linearization and Algorithmic Implementation
For the representation of biological soft tissues we use highly nonlinear stored energy functi,0
tions ψ iso and ψ(a)
. Thus the constitutive equation for the stresses (8.6) is also nonlinear
and has to be linearized. Computing the consistent linearization of S for the case φ(a) = 0
we obtain
2
X
iso
∆S = ∂C S : ∆C +
∂C S ti(a) : ∆C + ∂D S ti(a) ∆D(a) .
(8.17)
a=1
With the linearization of the damage variable
∆D(a) =
∂D(a) ti,0 1
∂D(a)
∆β(a) =
S : ∆C
∂β(a)
∂β(a) (a) 2
(8.18)
we are able to extract 12 ∆C in (8.17) and reformulate the linearized stresses by
∆S = CeD : 12 ∆C .
(8.19)
The total tangent moduli are also additively decoupled and determined as
eD
C
iso
=C
+
2
X
Cti(a) + CD
(a)
a=1
(8.20)
with the isotropic part given as the standard tangent moduli for hyperelasticity of the
isotropic part of the energy
∂ 2 ψ iso
Ciso = 4
,
(8.21)
∂C∂C
the transversely isotropic part
Cti(a)
= 1 − D(a) 4
ti,0
∂ 2 ψ(a)
∂C∂C
,
(8.22)
Anisotropic Damage Model for Arterial Walls
103
representing the standard moduli of the damaged transversely isotropic energy, and the
damage evolution part

− ∂D(a) S ti,0 ⊗ S ti,0 , if φtrial > 0
(a)
(a)
∂β(a) (a)
(8.23)
CD
(a) =

trial
0
, if φ(a) ≤ 0 .
Herein, the trial value of the damage criterion is defined by
D
E
ti,0
ti,0
trial
φ̂(a) (tn+1 ) = ψ̂(a) (tn+1 ) − ψ(a),ini − β̂(a) (tn ) ,
(8.24)
where tn+1 and tn denote the actual and the last time step, respectively. Such an algorithmic quantity is introduced in order to have the possibility to check if the Macauley-bracket
becomes greater than any effective energy reached up to the actual time (defined as the
internal variable). In this case the damage criterion is violated, i.e. the trial value of the
damage criterion is greater than zero, and the actual internal variable is set equal to
D
E
ti,0
ti,0
β̂(a) (tn+1 ) = ψ̂(a)
(tn+1 ) − ψ(a),ini
.
(8.25)
Then the damage variable, the stresses and the complete tangent moduli can be evaluated
and the damage criterion is satisfied (φ(a) = 0).
1)
2)
Initialize: D(1) = D(2) = 0
Check if initial damage state not reached:
if t < tini then goto 9)
3)
9)
Do 4) - 8) for a = 1, 2
4) Check:
ti,0
ti,0
if t = tini then set ψ(a),ini
= ψ(a)
and store to history
ti,0
else
read ψini from history
5) Read β(a),n from history
6) Compute φtrial
(a)
7) Check algorithmic damage criterion:
If φ̂trial
(a) (tn+1 ) > 0 then compute β̂(a) (tn+1 ) and write to history
8) Compute D(a)
Compute and return S, CeD
Figure 49: Algorithmic computing of the stresses and tangent moduli for the damage model.
In Fig. 49 the complete procedure of computing the stresses and tangent moduli is summarized. Note, that for quantities of the actual time step we skip the index (•)n+1 .
Anisotropic Damage Model for Arterial Walls
104
8.4 Numerical Example
In order to show that the damage model is quantitatively able to describe discontinuous
damage as schematically illustrated in Fig. 48, we provide a numerical example. For this
purpose we consider a cycled displacement-driven unconstrained tension test of a cube
as shown in Fig. 50a). The displacement-driven deformation is accomplished by applying
the nodal displacement u to one side face of one hexahedron Finite-Element whereas the
opposite side face is fixed in the direction of u.
∆l/l0
a(2)
u
x3
x1
2β
∆l2 /l0
a(1)
∆l1 /l0
u
u
a)
1
0
∆lini /l0 A
x2
0
1
00
11
00
11
C
B
tini
u
D
t
b)
Figure 50: a) Setup of displacement-driven experiment and b) and schematic illustration
of the applied strain - time diagram with ∆l = |u|. The initial damage state is characterized
by tini .
We investigate two loading and unloading cycles illustrated in Fig. 50b), where the applied
strain ∆l/l0 is plotted versus time. ∆l = |u| and l0 denote the change of the cube length
and the reference length, respectively. Here, the two strains defining the turning point of
loading and unloading are set to ∆l1 /l0 = 0.325 and ∆l1 /l0 = 0.35.
For the example we consider the polyconvex hyperelastic stored energy function ψ(BIO1 )
given in (7.24) and apply the damage model explained above to the transversely isotropic
part of ψ(BIO1 ) . Then the complete stored energy reads
ψ(BIO) =
iso
ψ(BIO)
+
2
X
a=1
ti,0
1 − D(a) ψ(a)
(8.26)
with the isotropic stored energy defined in (7.22) and the transversely isotropic part
α2
( (a)
(a)
α
K
−
2
for K3 ≥ 2
1
ti,0
3
ψ(a) =
,
(8.27)
(a)
0
for K3 < 2
cf. (7.23). For the material parameters of the hyperelastic stored energy and the damage
parameters we choose the values as given in Table 9.
The initial damage state is accomplished when the strain reaches the value ∆l/l0 = 0.25,
so no damage occurs before this deformation is reached. In order to provide an indication
of the anisotropy effect of the damage model we take into account two virtual experiments:
the first one (Experiment 1) represents the tension test of the cube where the fibers are
arranged as shown in Fig. 50a) with the fiber angle β = 43.39◦ . In Experiment 2 we
consider the same cube being rotated 90◦ around the x3 -axis before applying the displacement u in x1 -direction. These two simulations are analogous to the tension tests of the
Anisotropic Damage Model for Arterial Walls
c1
[kPa]
0.5
105
ǫ1
ǫ2
α1
α2 γ1
γ2
[kPa] [-]
[kPa] [-] [-] [kPa]
100.0 50.0 5000.0 4.0 0.9 260.0
Table 9: Considered material parameters for Model ψ(BIO) taking into account the proposed
damage model.
circumferentially and the longitudinally oriented arterial strips used for the quantitative
adjustment in Section 7.2. The results of the virtual experiments are shown in Fig. 51,
where the Cauchy stresses are depicted versus the strain.
σ[kPa]
σ[kPa]
4000
4000
3500
3500
2β
2β
a1
a2
3000
u
x3
x1
2500
2000
u
x3
u
x1
2500
x2
2000
u
u
1500
a2
u
x2
u
u
1500
1000
1000
ψ(1)
500
ψ(1)
500
0
a)
a1
3000
0
0
0.1
0.2
0.3
∆l/l0
0.4
0.5
0.6
b)
0
0.1
0.2
0.3
∆l/l0
0.4
0.5
0.6
Figure 51: Cauchy stress σ [kPa] vs. strain ∆l/l0 for a) Experiment 1 and b) Experiment
2 using the constitutive model given by (7.24). The initial damage state is set to tini , where
the strain is equal to 0.25.
In Fig. 51a) we observe a distinct influence of discontinuous damage in the stress - strain
curve and notice that the proposed damage model is generally able to describe the same
characteristics as in the schematic illustration Fig. 48. The anisotropy effect becomes
obvious when having a look at the results of Experiment 2, see Fig. 51b). Here the discontinuous damage effect is considerably less distinctive. Due to the fact that we consider
a fiber angle of β = 43.39◦ the fibers are stronger aligned into the tension direction in
Experiment 1 than in Experiment 2. Therefore, it is quite evident that the damage model
influences the stiffness reduction much more in Experiment 1, since the proposed model
assumes damage only in fiber direction.
Hereby, it is shown that the damage model is able to describe anisotropic discontinuous damage in materials, having the same characteristics as biological soft tissues, by a
scalar-valued damage variable. Furthermore, an initial damage state differing from the
undeformed reference configuration can be taken into account.
Anisotropic Damage Model for Arterial Walls
106
Numerical Simulation of Arterial Walls
9
107
Numerical Simulation of Arterial Walls
In order to provide numerical examples for the polyconvex stored energies and the
anisotropic damage model, some numerical simulations of arterial walls are given in this
section. For this purpose we have to account for a special arrangement of the two fiber
families, mainly occurring in arterial walls. In Fig. 52a) the directions of the fibers in a
healthy artery are schematically illustrated in the two main layers media and adventitia.
fiber net of
adventitia
helically arranged fibers
of media
Media
2βM
⇒
2βA
Adventitia
a)
b)
Figure 52: Schematic illustration of fiber orientation in a) real arteries and b) the computer
model, cf. Holzapfel et al. [50]. Here, the fiber angles are set to βM = 43.39◦ and
βA = 49.0◦ .
This fiber orientation can be represented by two superposing fiber directions as shown in
Fig. 52b). The fiber angle β describes the deviation of the fiber direction from the circumferential direction and in our investigations we consider for the media and adventitia
the fiber angles βM = 43.39◦ and βA = 49.0◦ , respectively.
Due to the fact that healthy arteries are in the physiological state almost perfect cylinders, such arteries are not suitable in order to provide heterogeneous numerical examples.
Caused by atherosclerotic plaques, evolving from accretion of fatty substances at the inner side of the arterial wall and from atherosclerotic intimal change, a reduction of the
arterial lumen occurs. Furthermore, such atherosclerotic plaques lead to the hardening of
the arterial wall which will influence the mechanical properties significantly. We focus on
such stenotic arteries, because hereby we provide heterogeneous numerical examples. Fig.
53a) shows an axial segment of a human stenotic iliac artery, taken from Holzapfel,
Sommer & Regitnig [55], where the heterogeneous character of diseased arteries is
obvious.
The model representation shown in Fig. 52 leads to the structure of the stored energy
as given in Section 7. Therefore, we use the energy function ψ(BIO) given in (8.26) for
the numerical simulation of the media and adventitia. Herein, the damage variable D(a)
represents the stiffness reduction in fiber direction and the damage model as described in
Section 8 is applied.
Generally, atherosclerotic plaques may contain fibrous, fibro-lipidic and calcified-necrotic
substances, which we refer to as plaque, and also extracellular lipid and strongly calcified
regions. The extracellular lipid and the calcification are expected to be isotropic, because
these components are not stiffened by fibers. For the plaque the assumption of isotropy
may also be acceptable due to the relatively incoherent and unorganized fiber arrangement of the fibrous substances. Since the proposed damage model provides damage only
Numerical Simulation of Arterial Walls
Arterial tissue
Adventitia
Media
Plaque
Calcification
a)
b)
Extracellular
lipid
108
Utilized model
Polyconvex model
Polyconvex model
Neo Hooke model,
no damage
Neo Hooke model,
no damage, stiff
Neo Hooke model,
no damage
Figure 53: a) Axial segment of human stenotic iliac artery, taken from Holzapfel, Sommer & Regitnig [55], and b) used models for the particular components.
in fiber direction, we do not assume any damage acting in the atherosclerotic plaque components. This seems to be reasonable for the calcification, because it can be interpreted as
nearly rigid with respect to the other components, and therefore no damage is expected.
The extracellular lipid is much softer than the other tissues and therefore here damage
plays a minor rule. Regarding the plaque, it is known that here damage effects cause a
significant decrease of the heterogeneous material stiffness leading to the rupture between
particular plaque components or delamination from the surrounding arterial tissues. In
order to describe also these damage phenomena in the plaque the damage model has to
be extended.
For the description of the atherosclerotic plaque components we use an isotropic stored energy function and choose the volumetrically-isochorically decoupled Neo-Hookean model,
which is given by
ψ(N EO) =
µ
κ
tr[(detF )−2/3 F F T ] − 3
[ln(J)]2 +
2
2
2
with κ = λ + µ .
3
(9.1)
In Fig. 53b) the utilized models for the particular components are tabularly summarized.
In the following sections we analyze human abdominal aortas with an average diameter of
approximately 1 centimeter and choose for the hyperelastic polyconvex model the parameters obtained by adjusting to experimental data in Section 7.2. For the damage model
we choose γ1 = 0.8 and γ2 = 100.0 kPa and the complete set of parameters for the media
and adventitia is reads as given in Table 10.
c1
ǫ1
ǫ2
α1
[kPa] [kPa] [-]
[kPa]
Media
17.5 100.0 50.0 5 · 105
Adventitia
7.5 100.0 20.0 15 · 109
α2
[-]
γ1
[-]
γ2
[kPa]
7.0
0.8
100.0
20.0 0.8
100.0
Table 10: Material parameters of the stored energy ψ(BIO) , see 8.26. The angle between
the (mean) fiber direction and the circumferential direction in the media is set to 43.39◦ and
in the adventitia to 49.0◦ .
Numerical Simulation of Arterial Walls
109
For the atherosclerotic plaque components, the extracellular lipid, the calcification and
the plaque, we choose the parameters as shown in Table 11.
Extracellular lipid
Calcification
Plaque
κ
[kPa]
6.67 · 105
1.67 · 109
1.33 · 106
µ
[kPa]
1334.2
3.33 · 106
2668.4
Table 11: Material parameters of the stored energy ψ(N EO) , see (9.1).
The Finite-Element simulations described in this section are mainly based on the 6-node
triangular element with quadratic ansatz functions as explained in Section 4.2.1. We
subdivide this section basically into the two parts:
• Residual Stresses, where a method for incorporating residual stresses inside arterial
walls is explained and in a numerical example verified, and
• Damage in Overexpanded Arteries, performing three different numerical simulations
for the analysis of damage distributions as a result of an arterial overexpansion
occurring e.g., when a balloon angioplasty is accomplished.
For the Finite-Element simulations in this section we take into account inertia terms as
described in Section 4.2.3 and the Newmark parameters are set to β̄ = 1/4 and γ̄ = 1/2.
Since the considered biological soft tissues consist mainly of water we assume a density
of 1.0 kg/dm3 . By taking into consideration inertia terms no displacement boundary
conditions have to be applied to the artery, which is reasonable since the kind of fixation
of the abdominal aorta in the body is quite complicated and therefore still ambiguous from
the mechanical point of view. The loading conditions are applied very slowly, because the
influence of inertia is of minor interest since we do not consider real time-dependent
material models.
Remark: It should be clearly emphasized that the numerical calculations presented in
this section should not be interpreted as realistic simulations of arterial walls, although
the hyperelastic polyconvex model is adjusted to the media and adventitia of a human
abdominal aorta. Note that neither the Neo-Hookean model for the plaque components nor
the damage model is adjusted to experimental data. Furthermore, the considered FiniteElement discretizations do not represent real arterial cross-sections, because they are not
governed by any magnetic-resonance-imaging (MRI), computer-tomographical (CT) or
intravascular ultrasound (IVUS) information. The examples given in this section should
be seen as illustration that the proposed models and methods provided in this work are
qualitatively able to describe the hyperelastic behavior, the influence of damage effects
and residual stresses in arterial walls.
9.1 Residual Stresses
This section explains a method for the incorporation of residual stresses in arteries, which
is introduced in Balzani, Schröder & Gross [10]. There are two phenomenons observed in arteries when they are slit: if they are cut transverse to the axial direction they
Numerical Simulation of Arterial Walls
110
shrink in axial direction and when they are sliced in radial direction they spring open, cf.
Vaishnav & Vossoughi [133] where the opening effect is found firstly. Therefore, it is
impossible that the unsliced (closed) unloaded configuration is stress-free, because otherwise the artery would not deform after cutting. It is often stated in the literature, that
the open artery obtained by cutting in radial direction can be regarded as the stress-free
configuration, see e.g., Chuong & Fung [21], Fung [37], Humphrey & Delange [62]
and Ogden [92]. More recently, in Vossoughi, Hedjazi & Borris [134] and Greenwald, Moore, Rachev, Kane & Meister [41] it is pointed out that additional cuts
in circumferential directions at the border between the particular layers (e.g., media and
adventitia) are necessary in order to eliminate further residual stresses. Here, it is emphasized that in our investigations only the radial cut is considered in order to keep the
analysis simple, but generally the proposed method can also be applied to circumferential
cuts. The approximations described above lead to the two-dimensional model representation as shown in Fig. 54.
(A)
B
B0
p
S
(B)
stress-free conf.
(fictitious)
unloaded conf.
(eigenstresses)
loaded conf.
(e.g., blood pressure)
Figure 54: Considered configurations of arteries.
There, the three considered configurations, the stress-free (fictitious) configuration, the
configuration of the artery without blood pressure differing from the stress-free configuration, and the loaded configuration are depicted.
9.1.1 Closing the Gap We assume the open artery governed by a radial cut to be
stress-free and obtain the unloaded configuration in which only the residual stresses occur
by closing the artery. Then an internal pressure p can be applied and the emerging stress
state is affected by the residual stresses and the stresses arising from the internal pressure.
A displacement-driven Finite-Element simulation prescribing one open cut surface of the
artery to move to the opposite cut surface could close the artery but would unfortunately
not generally lead to a smooth stress distribution at the adjacency of the two surfaces.
For this reason, we introduce springs between the two cut surfaces such that each node
of one cut surface in the Finite-Element discretization is connected to one node of the
opposite surface by a spring element. It should be noted that one has to make sure that
at each cut surface the same number of nodes exist when the mesh is generated; Fig. 55
illustrates this requirement.
The relative displacement of two corresponding nodes (i and j) urel
(i,j) considered in the
spring element is related to the force by constant spring stiffness. For the special case
of closing the gap between the two cut surfaces in arteries we enforce the relative displacement of the corresponding nodes, e.g., urel
(i,j) , to be equal to the relative referential
Numerical Simulation of Arterial Walls
111
cut surface (A):
i
(A)
i+n
nodes i, ... i + n
cut surface (B):
nodes j, ... j + n
j+n
(B)
j
corresponding nodes:
(i, j), ... (i + n, j + n)
Figure 55: Pairs of corresponding (associated) nodes.
coordinates of two associated nodes, e.g., X rel
(i,j) = X i −X j . Due to the large deformations
occurring when closing the artery we introduce a factor λ ∈ [0, 1] in order to apply the
prescribed displacement stepwise and consider the constraint
ûrel
(i+k,j+k)(λ) = λ (X i+k − X j+k )
for k = 0, ...n .
(9.2)
When the Finite-Element simulation is performed λ is increased until the state is reached
where the artery is completely closed, i.e. λ = 1. It is remarked that the spring constant
should be interpreted as a penalty parameter, which has to be set high enough to provide
that the two cut surfaces coincide.
If we contract the cut surfaces by this way the resulting unloaded configuration would
not really be a physical one although the residual stresses are included, because the
Finite-Element mesh is still open and only held together by spring interface elements. For
further Finite-Element simulations, as e.g., applying internal pressure, the appearance of
such spring interface elements would have some drawbacks with respect to the numerics.
The main goal at this point is to obtain a modified reference configuration for such further
simulations considering a Finite-Element mesh discretizing the closed artery and applying
the same residual stresses without considering spring elements anymore. Therefore, we
firstly perform the simulation explained above and obtain the unloaded configuration
taking into account the residual stresses. Then the actual nodal coordinates and the
deformation gradient in each Gauss point are stored in a file. The next subsection describes
how this stored data is used in order to incorporate the residual stresses in a second FiniteElement simulation where no spring elements are included.
9.1.2 The Prestressed State We use the stored actual nodal coordinates in order to
generate a new mesh considering the same node-/element-connectivity but changing the
nodal coordinates. It is to be remarked that this new mesh has not included any spring
elements. Then, at the beginning of the new Finite-Element simulation the deformation
gradient at each Gauss point is read from the file, stored into the element history and
applied internally.
Numerical Simulation of Arterial Walls
112
For applying this deformation gradient we consider the body in the unloaded configuration
B, which is described by the position vectors X, and the body in the loaded (actual)
configuration S, where the position vectors are denoted by x. The transformation of
vectors from B to S is performed by the deformation gradient F := Gradx. Let us now
consider the stress-free configuration to be a virtual prior configuration and denote the
body in this configuration by B0 with the infinitesimal line elements dX 0 , see Fig. 56.
B0
F0
dX 0
fictive
stress-free conf.
B
F
dX
unloaded conf.
(eigenstresses)
S
dx
loaded conf.
Figure 56: Mapping of infinitesimal line elements between considered configurations.
We denote the deformation gradient, which maps dX 0 from the assumed stress-free configuration to the reference configuration (unloaded configuration) by F 0 . This is the deformation gradient chosen as our “history” variable governed by the first simulation. With
the relations
dX = F 0 dX 0 , dx = F dX → dx = F F 0 dX 0
(9.3)
we define the multiplicatively decoupled deformation gradient as
F̃ := F F 0 ,
(9.4)
which maps infinitesimal line elements from the stress-free configuration to the actual
configuration. Then we compute the associated right Cauchy-Green deformation tensor
T
C̃ := F̃ F̃
(9.5)
and evaluate the constitutive equation for the stresses and the tangent moduli
S̃ := Ŝ(C̃) and C̃ := Ĉ(C̃) ,
(9.6)
respectively. These are the stresses, which include the residual stresses as well as the
stresses governed by the actual deformation described by F . Due to the fact that the
deformation gradient F̃ is placed between the stress-free and the actual configuration, the
stress tensor S̃ is defined in the stress-free configuration. In the present simulation the
geometry of the unloaded configuration is considered, therefore, the appropriate second
Piola-Kirchhoff stress tensor would be the push-forward of S̃ by F 0
S = F 0 S̃F T0 ,
(9.7)
which is defined in the reference configuration (unloaded configuration). Analogously, the
appropriate tangent moduli are computed by the push-forward operation.
Since F 0 leads to large deformations (this gradient is obtained by closing the artery) it is
Numerical Simulation of Arterial Walls
113
in general not possible to apply the complete gradient in one step. Therefore, we introduce
the algorithmic factor λ̄ and obtain
with F 0,λ̄ := 1 + λ̄ (F 0 − 1) .
(9.8)
F̃ = F F 0,λ̄ λ̄=1
The factor λ̄ enables us to interpolates between the second order identity tensor 1 (λ̄ = 0),
representing the case that no residual stresses are applied, and the deformation gradient
F 0 (λ̄ = 1), which is associated to the complete residual stress state. λ̄ is then increased
from zero to one in order to apply F 0 stepwise.
9.1.3 Residual Stresses in Atherosclerotic Arteries As a first numerical simulation of arterial walls we focus on the residual stresses appearing in (unloaded) configurations where no internal pressure, e.g., blood pressure, exists. For this purpose we consider
the (stress-free) open artery to be the reference configuration.
(A)
(B)
Adventitia
Media
Plaque
PP
PP
PP
Calcification
ExtracellularLipid
Figure 57: Finite-Element mesh of open cross-section of the (undeformed) artery.
Fig. 57 shows the Finite-Element mesh of the considered cross-section of the human abdominal aorta containing 1100 6-node triangular elements and the particular components.
We have to define the preferred direction describing the fiber orientation in this (undeformed) reference configuration depending on the position in the artery. This is done in the
upper half of the cross-section by the segmental partition of the artery into domains with
equal fiber orientation and defining the preferred direction for each domain. In the lower
half of the arterial cross-section we assign an ellipsoidal function which approximates the
circumferential direction and then compute the preferred direction with the given fiber
angles βM or βA . At each Gauss point in the artery it is checked in which domain the
Gauss point is located and then the preferred direction is set. The result of this procedure
is illustrated in Fig. 58, where the vectors of the first preferred direction a(1) are plotted.
As described in the previous sections special spring-interface elements between the two cut
surfaces (A) and (B), see Fig. 57, are implemented, but not shown in the figures because
these elements do not represent real physical components of the artery. Increasing the
factor λ from 0 to 1 enforces the artery to close and for λ = 1 the gap between (A) and
(B) should be completely closed.
Numerical Simulation of Arterial Walls
114
zoom 58b)
b)
a)
Figure 58: Distribution of preferred direction a(1) in a) reference configuration of considered
artery and b) enlarged cut-out.
vonMis
150.00
50.00
15.00
12.50
10.00
7.50
5.00
2.50
1.00
0.50
0.25
0.00
λ = 0.0
λ = 0.5
λ = 1.0
Figure 59: Von Mises Cauchy-stresses σv for λ = 0.0, λ = 0.5 and λ = 1.0.
In Fig. 59 the distribution of the von Mises Cauchy stresses given by
q
2
2
2
2
2
2 1/2
σv = σ11
+ σ22
+ σ33
− σ11 σ22 − σ11 σ33 − σ22 σ33 + 3 σ12
+ 3 σ13
+ σ23
)
(9.9)
is shown for three states during this simulation. We see, that for λ = 0.0 no stresses
occur and for increasing values of λ the artery closes. For λ = 1.0 the artery is completely
closed and we consider this state to represent the unloaded configuration in arterial walls
containing residual stresses. We notice a stress concentration at the left and right edging
of the atherosclerotic plaque, which is comprehensible due to the higher stiffness of the
plaque compared to the media and adventitia in this range of deformations.
In order to obtain an imagination of the transmural stresses Fig. 60 shows the distribution
of the stress components σ11 and σ22 . For x1 = 0 in Fig. 60a) and for x2 = 0 in Fig. 60b)
we may analyze the circumferential stresses. Expectedly, we observe negative stresses near
the inner and tensile stresses near the outer boundary of the arterial wall.
For the closed artery we store the deformation gradient at each Gauss point in a file as well
as the nodal coordinates. These coordinates are used for the generation of a new mesh also
containing 1100 6-node triangular elements, see Fig. 61. It should be noted that in this
Numerical Simulation of Arterial Walls
x2
x2
x1
a)
115
S1
10.00
7.50
5.00
2.50
1.75
1.00
-1.00
-1.75
-2.50
-5.00
-7.50
-10.00
x1
S2
10.00
7.50
5.00
2.50
1.75
1.00
-1.00
-2.50
-5.00
-7.50
-10.00
-25.00
b)
Figure 60: Cauchy-stresses a) σ11 and b) σ22 after closing the artery via interface spring
elements.
mesh no spring-interface elements are implemented. We use this mesh for a new FiniteElement simulation, where the deformation gradient is applied stepwise as described in
the previous section. Since the constitutive equation is evaluated in the virtual stress-free
configuration, cf. (9.6), where the preferred directions a(a) are defined, we have to use the
same preferred directions for each Gauss-point as set for the first simulation.
Adventitia
Media
Plaque
Calcification
Extracellular
Lipid
Figure 61: Finite-Element mesh of closed cross-section of the (undeformed) artery.
When the complete deformation gradient F 0 is applied, i.e. λ̄ = 1, we expect the same
stress distribution as obtained by accomplishing the first simulation, where the artery
is closed. In Fig. 62 the results of this second simulation are shown by depicting the
distribution of the stress components σ11 and σ22 .
Here no additional loads, as e.g., internal pressure, are applied in order to have the opportunity to compare the stresses governed by closing the gap with the stresses obtained
via the procedure described in 9.1.2. If we compare the stresses in Fig. 62 with Fig. 60 we
notice that we obtain the same stress distribution from the second simulation as governed
by the first simulation. Also the other components of the stress tensor are the same for
both calculation procedures. This indicates that the method is generally able to take into
account residual stresses without using spring-interface elements and without considering
the open mesh of the artery. It is remarked that it is still necessary to simulate previously
Numerical Simulation of Arterial Walls
x2
x1
x2
S1
10.00
7.50
5.00
2.50
1.75
1.00
-1.00
-1.75
-2.50
-5.00
-7.50
-10.00
a)
116
x1
S2
10.00
7.50
5.00
2.50
1.75
1.00
-1.00
-2.50
-5.00
-7.50
-10.00
-25.00
b)
Figure 62: Cauchy-stresses a) σ11 and b) σ22 after applying stepwise F 0 .
the closing of the artery for obtaining the deformation gradient at each Gauss point. But
for further simulation, where the artery is loaded and much larger deformations appear,
no interface elements are included and the closed artery is considered.
9.2 Damage in Overexpanded Arteries
For the treatment of atherosclerotic stenosis a balloon-angioplasty in combination with
stenting is performed in most cases nowadays. During this procedure the artery is overstretched by inflating an inserted dilatation-catheter by an internal pressure of 3 - 12
bar, which is up to 50 times of the physiological internal pressure inside arteries. In this
section we focus on the numerical simulation of such an arterial overexpansion. For this
purpose we investigate three simulations, where an internal pressure is applied to the
inner side of the arterial cross-section. This direct impact of the forces to the artery is
not life-like because in reality the internal pressure is applied to the catheter and the
catheter transfuses the force to the arterial wall. As an example, some shear stresses between the artery and the catheter may arise caused by friction when the balloon is not
nestled perfectly to the arterial wall before inflating the balloon. Therefore, we investigate
two possibilities of load: a) conservative load, where the direction of the nodal forces is
set such that the applied forces reflect hydrostatic pressure only in the reference configuration, i.e. the directions do not change with deformation, and b) non-conservative load,
where the direction depends on the actual displacement field and is updated for each time
step. The two types of Neumann boundary conditions may be interpreted as limitations
for numerical simulations where no catheter is discretized, so reality would be represented
by a combination of these two boundary conditions mainly influenced by the mechanical
properties of the catheter. In our analysis we firstly apply an internal pressure of p = 24.0
kPa = 180.0 mmHg, which may be interpreted as the edge of physiological loading range.
This state is defined to be the initial damage state and damage evolution begins. Then
the internal pressure is increased up to p = 500.0 kPa = 3750.0 mmHg characterizing the
overexpansion of the artery and a significant increase of the damage is expected.
Numerical Simulation of Arterial Walls
117
9.2.1 Overexpanded Artery with Conservative Load In this first simulation of
an overexpanded artery we investigate a simulation where a conservative load is applied.
This means that the internal hydrostatic pressure is applied by defining the directions of
the replacement nodal forces in the reference configuration and by prescribing them to be
constant with increasing deformation of the artery.
Adventitia
Media
Plaque
Calcification
Extracellular
Lipid
Figure 63: Finite-Element mesh of the (undeformed) artery, cf. Holzapfel et al. [51].
We take into account a mesh of 2084 6-node triangular elements, see Fig. 63, where also the
identification of the particular components is given. In this (undeformed) configuration the
preferred directions are defined at each Gauss point in the media and adventitia based on
an orbital function describing approximately the circumferential direction in the arterial
wall and the fiber angles βM and βA . We apply the internal pressure of firstly 24.0 kPa
and set the initial damage state. Then the pressure is increased up to 500.0 kPa and the
damage evolution proceeds.
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
a)
10000.0
9285.7
8571.4
7857.1
7142.9
6428.6
5714.3
5000.0
4285.7
3571.4
2857.1
2142.9
1428.6
714.3
0.0
b)
Figure 64: Deformed artery at p = 3750.0 mmHg under conservative load without residual
stresses; distribution of a) damage variable D(1) and of b) von Mises stresses σv .
In Fig. 64 the distribution of the damage variable D(1) and the von Mises-Cauchy stresses
σv is depicted in the deformed artery. We observe a maximum value of the damage variable
of approximately D(1) = 0.7 at the left and right side of the artery close to the plaque.
There also the maximum value of approximately σv = 10000.0 kPa occurs. Furthermore, it
is observed that the span of the arterial lumen is increased more in the direction between
Numerical Simulation of Arterial Walls
118
the plaque and the opposite side of the artery than transverse to this direction. This
is a rational result due to the ellipsoidal form of the arterial lumen in the undeformed
configuration and as a consequence thereof the overestimation of the internal load in the
specific directions. This could be also the reason why the maximum value of the damage
and the von Mises stresses is located in the certain position, because there the fibers would
be expanded mostly.
It is to be remarked that in this simulation the method for incorporating the residual
stresses is not applied and therefore, the influence of these residual stresses is unconsidered.
9.2.2 Overexpanded Artery with Non-Conservative Load As a second example
we consider the overexpansion of an artery by applying a non-conservative load. For this
purpose we implement surface load elements, as described in 4.2.2, at the inner side of the
artery. By this way we obtain a loading condition representing real hydrostatic pressure
in each deformation step. For this simulation we consider the Finite-Element mesh shown
in Fig. 61 containing 1100 6-node triangular elements. This mesh represents another discretization of a typical slightly diseased artery. The preferred directions are defined in this
(undeformed) configuration by the same strategy as in 9.2.1.
When the internal pressure of p = 500.0 kPa is reached we obtain the deformed configuration of the artery depicted in Fig. 65. Remember that the initial damage state is achieved
by an internal pressure of p = 24.0 kPa.
dam(1)
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
a)
vonMis
10000.0
9285.7
8571.4
7857.1
7142.9
6428.6
5714.3
5000.0
4285.7
3571.4
2857.1
2142.9
1428.6
714.3
0.0
b)
Figure 65: Deformed artery at p = 3750.0 mmHg under non-conservative load without
residual stresses; distribution of a) damage variable D(1) and of b) von Mises stresses σv .
As can be seen in Fig. 65a) the maximum value of the damage variable D(1) = 0.7 is
located at the analogous place as in the first overexpansion simulation. The same holds
for the von Mises stresses, cf. Fig. 65b) with Fig. 64b). The main difference between
the first and second simulation of an arterial overexpansion is observed in the different
distribution of damage at the opposite side of the plaque. Here the damage acts more in
the adventitia than in the media for the second simulation. Another difference between
the two simulations becomes obvious by comparing the deformed cross-section of the two
simulations. In the second simulation the span of the lumen has increased more uniformly,
and not primarily in the direction between the plaque and the opposite side of the artery.
Numerical Simulation of Arterial Walls
119
These differences between the two simulations result of course from the differing type
of loading and not from the modified mesh of the artery, since both meshes represent
typical discretizations of slightly diseased arteries. Nevertheless, the main characteristics
of the damage distribution, i.e. the value and location of maximum damage and von Mises
stresses, are nearly the same for both loading conditions.
9.2.3 Overexpanded Artery with Residual Stresses In the last example of an
overexpanded artery we incorporate residual stresses into the numerical simulation. For
this reason we firstly perform the calculation as described in 9.1.3 and apply then the
internal pressure. In detail this means that at first the open artery shown in Fig. 57 is
considered and then closed by the displacement-driven process explained in 9.1.3, where
special spring-interface elements are used. Then the new mesh (Fig. 61) without interface
elements is generated and the proposed method for incorporating residual stresses is
applied. Then we end up in the residual stress state shown in Fig. 62 and use this as
a starting point for the actual simulation. Here, we implement the surface load elements
given in 4.2.2 at the inner side of the artery in order to obtain a real internal hydrostatic
pressure for each deformation step. We define the initial damage state to be reached at
an internal pressure of p = 24.0 kPa and increase the pressure up to p = 500.0 kPa.
dam(1)
0.50
0.46
0.43
0.39
0.36
0.32
0.29
0.25
0.21
0.18
0.14
0.11
0.07
0.04
0.00
a)
vonMis
5000.00
4642.86
4285.71
3928.57
3571.43
3214.29
2857.14
2500.00
2142.86
1785.71
1428.57
1071.43
714.29
357.14
0.00
b)
Figure 66: Deformed artery at p = 3750.0 mmHg under non-conservative load with residual
stresses; distribution of a) damage variable D(1) and of b) von Mises stresses σv .
The results of this simulation are shown in Fig. 66, where the deformed configuration of the
arterial cross-section is shown with the distribution of the damage variable D(1) and the
von Mises Cauchy stresses σv . We notice that the maximum value of the damage variable
of approximately D(1) = 0.5 is less than in the simulations before. The same holds for the
maximum value of the von Mises stresses, which is approximately σv = 5000.0 kPa. This
must be a result of the influence of the residual stresses, which seems to be reasonable due
to the assumption often stated in the literature, that arteries induce residual stresses by
growth processes such that stress concentrations are reduced. We further on observe for the
present simulation that the von Mises stresses are more uniformly distributed, especially
with respect to the atherosclerotic plaque. This may be also seen as a consequence of the
residual stresses due to the biological assumption mentioned above.
Numerical Simulation of Arterial Walls
120
Another important difference between the actual simulation and the simulations before
is, that here the maximum value of the damage and von Mises stresses is reached at the
outer edge of the media nearly all over the arterial wall at the opposite side of the plaque.
A reason for this could be that the assumption to obtain the stress-free configuration of
arterial walls only by a radial cut is not sufficient. There could be also significant residual
stresses between the two layers media and adventitia, as already stated in Vossoughi,
Hedjazi & Borris [134] and Greenwald, Moore, Rachev, Kane & Meister
[41], which are released by additional circumferential cuts. These residual stresses could
smooth out the damage and stress distribution such that the concentrations between the
media and adventitia are reduced.
Finally, we conclude that the influence of residual stresses as well as the influence of damage effects seem to be significant for the stress distribution inside overexpanded arteries.
Different types of loading conditions (conservative and non-conservative load) lead not
to completely different stress and damage distributions inside the arterial wall, but the
deformation of the artery differs significantly. Due to the fact that it is not really clear
which kind of loading condition is the realistic one, the catheter should be discretized,
too, in order to obtain a realistic simulation of a balloon-angioplasty.
Conclusion and Outlook
10
121
Conclusion and Outlook
This work dealt with the modeling of anisotropic hyperelasticity and anisotropic damage in
arterial walls. As one main effort the development of a variety of polyconvex transversely
isotropic stored energy functions is mentioned. The construction of a damage model,
taking into account the special composition of biological soft tissues, represented the
other major part. The proposed concepts were realized numerically in the framework of
the Finite-Element method and numerical simulations were performed.
After a short introduction to the mechanical behavior of arterial walls and a description
of the fundamentals of the non-linear continuum mechanics, the representation theory
for anisotropic tensor functions and the Finite-Element method, some important generalized convexity conditions were defined. As a result it was shown that the polyconvexity
condition seems to be a quite suitable condition for the construction of energy functions
guaranteeing the existence of minimizers. In addition it was stated that the material stability is ensured automatically by using a polyconvex stored energy function, which was
checked by comparing two non-polyconvex functions with a polyconvex one. For this purpose a localization analysis was performed and material stability was only observed for
the polyconvex model, although the polyconvex functions and the non-polyconvex ones
reflected the same stress-strain behavior in some special tests. This identical stress-strain
response of the models was achieved by adjusting the polyconvex model to the other two
models by applying evolution strategies to find the minimum of an error function of leastsquare type.
It has been mentioned, that in the framework of polyconvexity the stress-free reference
configuration is in general not able to be satisfied automatically. Thus, a simple construction principle was proposed based on the property that external convex and monotonically increasing functions of internal polyconvex functions are again polyconvex. This
construction principle required further on the internal function to be zero in the reference
configuration in order to ensure the natural state condition a priori. Then a variety of
transversely isotropic energy functions was constructed, using the construction principle
and accounting special case distinctions providing that the internal function was zero in
the reference configuration. These functions were discussed with respect to the description
of biological soft tissues. As an example, two functions were composed of the proposed
polyconvex transversely isotropic functions and adjusted for experimental data of uniaxial tension tests of a human abdominal aorta. Hereby, the polyconvex framework for the
description of the anisotropic hyperelasticity in biological soft tissues was given.
Then, this work focussed on the development of a damage model, suitable to describe the
damage behavior observed in arterial tissues. After a short recapitulation of the fundamental idea of the damage mechanics, i.e. the (1 − D)-approach, and a detailed explanation of
discontinuous damage, the special requirements for the damage model caused by the particular properties of arterial tissues were biomechanically motivated. The basic assumption
of the proposed damage model was, that the damage happens mainly in fiber direction.
The realization of this assumption was accomplished easily by the fact, that the stored
energy was formulated in the context of the invariant theory and additively decoupled in
an isotropic and an anisotropic part. The main idea of the damage model was the introduction of a scalar-valued damage variable and the application of the (1 − D)-approach to
the anisotropic part of the energy, which was associated to the description of the hyperelasticity of the fibers. Contrary to usual formulations for anisotropic damage where tensors
Conclusion and Outlook
122
of second or fourth order are necessary, the anisotropic damage of the three-dimensional
material could be represented by the “one-dimensional problem” of fiber damage. After
providing the total tangent moduli including damage evolution terms and explaining the
algorithmic implementation, the discontinuous behavior and the anisotropy effect could
be verified in a numerical example.
Finally, the proposed concepts were used for the numerical simulation of atherosclerotic
arteries. At first the numerical analyses were concentrated on residual stresses occurring
in unloaded configurations of arteries. For this purpose a method for the incorporation of
such residual stresses was described, which considered a split into two simulations. The
first simulation assumed an opened artery to be stress-free and accounted for special spring
interface elements in order to close the artery. Hereby, the distribution of the deformation
gradient responsible for the eigenstresses was obtained. In the second simulation this
distribution of deformation gradients was applied stepwise without using any interface
elements. A simulation of the residual stress state in an atherosclerotic artery verified
the proposed method. Then, the damage model was applied in numerical simulations,
where the overexpansion of an artery was considered. By comparing results governed by
simulations without residual stresses with a simulation incorporating residual stresses a
significant influence of the residual stresses became obvious.
The concepts presented in this work provide numerous possibilities of extensions. Concerning the construction of polyconvex anisotropic functions, the construction principle could
also be applied for any other class of anisotropy when internal polyconvex functions, representing e.g., orthotropic or cubic symmetry, are developed. With a view to the damage
model the application of the (1 − D)-approach to the isotropic part would be reasonable
in order to describe also damage in the ground substance. This could give an indication
for positions where delamination mechanisms, transversely to the fiber directions, may
occur. Another suggestive extension of the proposed concept could be the incorporation
of a model suitable to describe the plastic deformations, which also account for remanent
strains remaining in the physiological state of arterial walls when a balloon-angioplasty is
performed. Concerning the simulation of residual stresses the proposed method could also
be applied for eigenstresses released by circumferential cuts between particular layers, e.g.,
media and adventitia, of axial arterial segments. Important improvements, which should
be done in order to obtain a more realistic simulation of overexpanded atherosclerotic
arteries, are the extension to three-dimensional analysis, the detailed material modeling
of the plaque components and the adjustment of the material parameters of the damage
model to appropriate experiments.
Appendix
A
123
Notation
1/2
For a, b ∈ R3 we let ha, biR3 denote the scalar product on R3 with norm kakR3 = ha, aiR3 .
We denote with R3×3 the set of real 3 × 3 matrices and by skew(R3×3 ) the skewsymmetric real 3 × 3 matrices. The standard Euclidean scalar product on R3×3 is
given by hH, BiR3×3 = tr [H · B T ] and subsequently we have kHk2R3×3 = hH, HiR3×3 .
adj[H] denotes the adjugate matrix of transposed cofactors cof[H] such that adj[H] =
det[H]H −1 = cof[H]T if H ∈ GL(3, R), where GL(3, R) characterizes the set of all
invertible 3 × 3-tensors. The identity matrix on R3×3 will be denoted by 11 or 1, so
that tr[H] = hH, 11i = H : 1. The index notation of A : H is e.g. AAB H AB and of
Ha = H · a is e.g. HAB aB . From this point onwards we skip the index R3 , R3×3 when
there is no danger of confusion. Furthermore, DF Ŵ (F ).H and DF2 F Ŵ (F ).(H, H) denote
Frechet derivatives.
B
Proof of Convexity.
Convexity of I1k :
We consider the function I1k , which can be reformulated by Ŵ (F ) = tr[F T F ]k = (||F ||2 )k ,
and compute the piecewise second derivative. After calculating
DF [W ].H = 2kkF k2k−2 hF , Hi
we obtain the second derivation
DF2 [W ].(H, H) = 2k(kF k2k−2 hH, Hi + (2k − 2)kF k2k−4 hF , Hi2 ) > 0 ,
which is greater than zero for k ≥ 1.
Convexity and monotone composition:
We say that P : Rn → R is a convex function and m : R → R is a convex and monotonically increasing function, then the function Rn → R, X → m̂(P̂ (X)) is also a convex
function. The proof of convexity is performed by a direct check of the convexity condition,
see Schröder & Neff [106].
Convexity of powers of polyconvex functions:
For the proof of polyconvexity we consider the function Ŵ (X) = m̂(P̂ (X)) with m̂(P ) =
P k and P̂ (X) = X and show convexity and monotone increase of m since convexity of P
is trivially satisfied. We directly notice that m̂(P ) is monotonically increasing for positive
P . For the proof of convexity we compute the second derivative
∂P2 P [m] = k(k − 1)P k−2 ,
which is greater than zero for k > 1 ∨ k < 0.
Convexity of J4k : For the proof of convexity of J4k we prove that the second derivative of
the function Ŵ (F ) = (kF k2 )k is positive. For this purpose we compute the first derivate
DF [W ].H = 2kkF ak2k−2 hF a, Hai
and obtain the second derivative
DF2 F [W ].(H, H) = 4k(k − 1)kF ak2(2k−2) kF akhF a, Hai + 2kkF ak2k−2 hH, a, i,
Appendix
124
which is positive for k ≥ 1.
Convexity of (5.60):
For proving convexity of (5.60) we compute the piecewise second differential of the nonzero branch of (5.60), i.e. W1ti (F ) = (||F a||2 − 1)p . Since
DF [W1ti ].H = p(||F a||2 − 1)p−1 2 hF a, Hai
we obtain
DF2 [W1ti ].(H, H) = 2 p [(p − 1)(||F a||2 − 1)p−2 2 hF a, Hai2
+(||F a||2 − 1)p−1 hHa, Hai] .
For ||F a||2 → 1 the second differential tends to zero and is positive for ||F a||2 > 1
and each p ≥ 1, thus, the function is convex with respect to F and therefore LegendreHadamard-elliptic.
Convexity of (5.61):
Neglecting the constant terms in (5.61), which do not contribute to the derivatives of the
energy, we show that for any p > 2
(
p
exp (kF ak2 − 1) for kF ak2 ≥ 1 ,
ti
W (F ) =
0
for kF ak2 < 1 ,
is convex with respect to F . For this purpose, we compute the piecewise second differential.
Since
h
i
p−1
p
p
2hF a, H ai ,
DF exp (kF ak2 − 1) .H = exp (kF ak2 − 1) p (kF ak2 − 1)
we obtain for the non-zero branch of W ti
h
i2
p−1
p (kF ak2 − 1)
2hF a, H ai
h
p
p−2
2
+exp (kF ak − 1) 2 p (p − 1) (kF ak2 − 1)
2hF a, H ai2
i
p−1
+ (kF ak2 − 1) hH a, H ai .
DF2 [W ti ](F ).(H, H) = exp (kF ak2 − 1)
p
This formula tends continuously to zero for kF ak2 → 1 and is positive for kF ak2 ≥ 1 .
Hence, the complete second differential is always positive and continuous. By continuity,
we obtain that W ti is convex for p = 2, too. Furthermore, convexity of W ti implies
Legendre-Hadamard ellipticity. It is clear that any additive composition of W ti with an
isotropic elliptic energy will also remain Legendre-Hadamard elliptic.
Polyconvexity of (5.63), (5.65):
For the proof of polyconvexity we show that for any p > 2 and constant c
(
exp (K − c)p for K ≥ c ,
W2ti (K) =
0
for K < c ,
(B.1)
is monotonically increasing and convex with respect to K, (in general K will be a polyconvex function). For this purpose, we compute the first derivative of W2ti
∂K [W2ti ] = exp[(K − c)p ] [p(K − c)p−1 ]
Appendix
125
and see that W2ti is positive for K ≥ c and therefore altogether monotonically increasing.
In order to show convexity we compute the second derivative of W2ti
2
2
∂KK
[W2ti ] = exp[(K − c)p ] [p(K − c)p−1]
+exp[(K − c)p ] p (p − 1)(K − c)p−2 .
This formula tends continuously to zero for K → c and is positive for K ≥ c. Hence, the
second derivative is always positive and continuous. By continuity, we obtain that W2ti is
convex also for p = 2.
Polyconvexity of (5.62), (5.64):
For the proof of polyconvexity we show that for any p > 1 and constant c
(
(K − c)p for K ≥ c ,
ti
W3 (K) =
0
for K < c ,
(B.2)
is monotonically increasing and convex with respect to K, (in general K will be a polyconvex function of F ). For this purpose, we compute the first derivative of W3ti
∂K [W3ti ] = p(K − c)p−1
and notice that W3ti is positive and therefore monotonically increasing for K ≥ c. For
showing convexity we compute the second derivative
2
∂KK
[W3ti ] = p (p − 1)(K − c)p−2 .
For K → c the second derivative tends to zero and is positive for K > c and each p > 1,
thus, the function is polyconvex, since K is polyconvex.
Appendix
C
Polyconvex Functions.
Polyconvex function
Restrictions
Isotropic functions using
I1 = tr[C],
Stress-free r.c.
I2 = tr[cof[C]],
I3 = det[C]
1.) α I α2
1 1
α1 > 0, α2 ≥ 1
no
1.) α I α2
1 2
α1 > 0, α2 ≥ 1
no
1.) α I1
1 1/3
I3
α1 > 0
no
2
1.) α I1
1 1/3
I3
α1 > 0
no
1.) α I2
1 1/3
I3
α1 > 0
no
2
1.) α I2
1 1/3
I3
α1 > 0
no
α1 > 0, α2 ≥ 1, I3 > 0
no
1
I3
α1 > 0
no
1.) −α ln(I )
1
3
α1 > 0
no
α1 > 0
no
α1 > 0
yes
α1 > 0
yes
α1 > 0, α2 ≥ α3 ≥ 1
yes
α
I3 − 1 2
α1 > 0, α2 ≥ 1
yes
− ln(I3 )]
α1 > 0
yes
1.) α I α2
1 3
1.) α
1
√
1.) −α ln(
1
1.) α
1
1.) α
1
1.) α
1
I3 )
1
I3 +
I3
1.) α (I
1 3
√
− 1)2
I3α2
1.) α [I
1 3
1.)
126
+
1
I3α2
α3
−2
cf. e.g. Schröder & Neff [106]
Appendix
Polyconvex function
Restrictions
127
Stress-free r.c.
Isotropic functions (continued)
1.) α
1
I3 − ln(I3 ) + [ln(I3 )]2
I1α2
1.) α
1
α /3
I3 2
1.) α
1
1.) α
1
(
1.) α
1
(
2.) α
1
3.) α
1
1.)
α1 > 0
yes
α1 > 0, α2 ≥ 1 α3 ≥ 1
yes
α1 > 0, α2 ≥ 1 α3 ≥ 1
yes
α1 > 0, α2 ≥ 1, α3 ≥ 1
yes
α1 > 0, α2 ≥ 1, α3 ≥ 1
yes
!
α1 > 0
yes
!
α1 > 0
no
− 3α2
3α /2
I2 2
I3α2
!α3
√
− (3 3)α2
exp
"
α /3
I3 2
exp
"
I2 2
I3α2
I1
1/3
I3
I2
1/3
I3
I1α2
− 3α2
3α /2
−3
−3
!α3
!α3 #
√
− 3( 3)α2
)
−1
!α3 #
)
−1
cf. e.g. Schröder & Neff [106]
cf. e.g. Holzapfel, Gasser & Ogden [50] or Schröder, Neff & Balzani [107]
3.)
cf. e.g. Schröder & Neff [104], Schröder & Neff [105] or Schröder, Neff & Balzani [107]
2.)
Appendix
Polyconvex function
Restrictions
128
Stress-free r.c.
Transversely isotropic functions using I1 = tr[C], I2 = tr[cof[C]], I3 = det[C],
J4 = tr[CM ], J5 = tr[cof[C]M ], K1 = J5 − I1 J4 + I2 , K2 = I1 − J4 , K3 = I1 J4 − J5 :
1.) α J α2
1 4
1.) α
1
J4α2
1/3
I3
1.) α K α2
1 1
1.) α
1
1.) α
1
K1α2
1/3
I3
K12
2/3
I3
1.) α K α2
1 2
1.) α
1
1.) α
1
K2α2
1/3
I3
K22
2/3
I3
1.) α K α2
1 3
1.) α
1
K3α2
1/3
I3
1.) α
1
I12 + J4 I1
1.) α
1
2I12 − K2 I1
1.) α
1
I22 + K2 I2
1.) α
1
2I22 + K3 I2
α1 > 0, α2 ≥ 1
no
α1 > 0, α2 ≥ 1
no
α1 > 0, α2 ≥ 1
no
α1 > 0, α2 ≥ 1
no
α1 > 0
no
α1 > 0, α2 ≥ 1
no
α1 > 0, α2 ≥ 1
no
α1 > 0
no
α1 > 0, α2 ≥ 1
no
α1 > 0, α2 ≥ 1
no
α1 > 0
no
α1 > 0
no
α1 > 0
no
α1 > 0
no
1.) α (α I
1
2 1
− α3 J4 )
α1 > 0, α2 ≥ α3 > 0
no
1.) α (α I
1
2 2
− α3 K1 )
α1 > 0, α2 ≥ α3 > 0
no
1.)
see Schröder & Neff [106]
Appendix
Polyconvex function
129
Restrictions
Str.-fr. r.c.
Transversely isotropic functions (continued), power functions
1.) α (J α3
1
4
1.) α
1
− 1)α2
J4α3
1/3
I3
1.) α (K α3
1
1
1.) α
1
1.) α
1
1.) α
1
1.) α
1
1/3
I3
K12
2/3
I3
1.) α
1
1.)
−1
1/3
I3
K22
2/3
I3
!α2
α1 > 0, α2 ≥ 1, α3 ≥ 1,
!α2
α1 > 0, α2 ≥ 1,
−1
−1
− 2α3
!α2
!α2
−4
− 2α3 )α2
K3α3
1/3
I3
−
α1 > 0, α2 ≥ 1, α3 ≥ 1,
2α3
J4α3
1/3
I3
≥1
α1 > 0, α2 ≥ 1, α3 ≥ 1, K1 ≥ 1
− 2α3 )α2
K2α3
1.) α (K α3
1
3
!α2
− 1)α2
K1α3
1.) α (K α3
1
2
α1 > 0, α2 ≥ 1, α3 ≥ 1, J4 ≥ 1
!α2
K12
2/3
I3
K1α3
1/3
I3
≥1
α1 > 0, α2 ≥ 1, α3 ≥ 1, K2 ≥ 2
α1 > 0, α2 ≥ 1,
K22
2/3
I3
K2α3
1/3
I3
≥ 2α3
α1 > 0, α2 ≥ 1, α3 ≥ 1, K3 ≥ 2
cf. Balzani, Neff, Schröder & Holzapfel [11]
yes
yes
yes
yes
yes
≥4
α1 > 0, α2 ≥ 1, α3 ≥ 1,
yes
yes
≥1
α1 > 0, α2 ≥ 1, α3 ≥ 1,
yes
K3α3
1/3
I3
≥ 2α3
yes
yes
Appendix
Polyconvex function
130
Restrictions
Stress-free r.c.
Transversely isotropic functions (continued), hyperbolic cosine
α1 [cosh (J4α2 − 1) − 1]
"
α1 cosh
J4α2
1/3
I3
!
−1
α1 > 0, α2 ≥ 1, J4 ≥ 1
#
−1
α1 > 0, α2 ≥ 1,
α1 [cosh (K1α2 − 1) − 1]
"
K1α2
"
K12
α1 cosh
α1 cosh
1/3
I3
2/3
I3
!
−1
!
−1
K2α2
#
−1
α1 > 0, α2 ≥ 1,
#
α1 > 0,
−1
"
K22
α1 cosh
α1 cosh
1/3
I3
2/3
I3
− 2α2
!
−4
!
α1 cosh
K3α2
1/3
I3
−
2α2
#
−1
#
K12
2/3
I3
K1α2
1/3
I3
≥1
K22
2/3
I3
K2α2
1/3
I3
≥ 2α2
−1
α1 > 0, α2 ≥ 1,
yes
yes
yes
yes
yes
≥4
α1 > 0, α2 ≥ 1, K3α2 ≥ 2
#
yes
yes
≥1
α1 > 0, α2 ≥ 1,
α1 > 0,
−1
!
≥1
α1 > 0, α2 ≥ 1, K2 ≥ 2
α1 [cosh (K3α2 − 2α2 ) − 1]
"
1/3
I3
α1 > 0, α2 ≥ 1, K1 ≥ 1
α1 [cosh (K2α2 − 2α2 ) − 1]
"
J4α2
yes
K3α2
1/3
I3
≥ 2α2
yes
yes
Appendix
Polyconvex function
131
Restrictions
Str.-fr. r.c.
Transversely isotropic functions (continued), exponential functions
1.)
i
o
h
α1 n
exp α2 (J4α3 − 1)2 − 1
2α2



α
1.) 1
exp α2
2α2 
1.)



α
1.) 1
exp α2
2α2 



α
1.) 1
exp α2
2α2 
1/3
I3
K12
2/3
I3


−1

!2 
−1


−1

!2 
−1
K2α3
1/3
I3
K22
2/3
I3

!2 

− 2α3  − 1


!2 

−4 −1

i
o
h
α1 n
exp α2 (K3α3 − 2α3 )2 − 1
2α2



α
1.) 1
exp α2
2α2 
1.)
K1α3
i
o
h
α1 n
exp α2 (K2α3 − 2α3 )2 − 1
2α2



α
1.) 1
exp α2
2α2 
1.)

!2 

−1 −1

i
o
h
α1 n
exp α2 (K1α3 − 1)2 − 1
2α2



α
1.) 1
exp α2
2α2 
1.)
J4α3
1/3
I3
K3α3
1/3
I3

!2 

− 2α3  − 1

α1 > 0, α2 > 0, α3 ≥ 1, J4α3 ≥ 1
α1 > 0, α2 > 0, α3 ≥ 1,
J4α3
yes
≥1
yes
α1 > 0, α2 > 0, α3 ≥ 1, K1α2 ≥ 1
yes
α1 > 0, α2 > 0, α3 ≥ 1,
α1 > 0, α2 > 0,
K12
2/3
I3
1/3
I3
K1α2
1/3
I3
≥1
yes
≥1
α1 > 0, α2 > 0, α3 > 1, K2 ≥ 2
α1 > 0, α2 > 0, α3 ≥ 1,
α1 > 0, α2 > 0,
K22
2/3
I3
K2α2
1/3
I3
≥ 2α3
≥4
cf. Balzani, Neff, Schröder & Holzapfel [11]
yes
yes
yes
α1 > 0, α2 > 0, α3 ≥ 1, K3 ≥ 2
α1 > 0, α2 > 0, α3 ≥ 1,
yes
K3α3
1/3
I3
≥ 2α3
yes
yes
Appendix
Polyconvex function
132
Restrictions
Str.-fr. r.c.
Transversely isotropic functions, special models



1.) α1
exp α2
2α2 
"
X
1
1
2.)
µr
4 r
αr
1
+
βr
J4
1/3
I3


−1

!2 
−1
α1 > 0, α2 > 0, α3 ≥ 1,
J4
1/3
I3
≥1
yes
!
X (r)
αr
(
wi Ii ) − 1
i
!
X (r)
βr
(
wi Ji ) − 1
i
1 −γr
−1
I
+
γr 3
µr ≥ 0, αr ≥ 1, βr ≥ 1, γr ≥ − 21
yes
with Ii = tr[CLi ], tr[cof[C]Li ]
and L1 = M , L2 = 21 D
3.)
X µ̃m m
1.)
γm
γ /2
J4 m
1/2
− 1 − µ̃m lnJ4
see Holzapfel, Gasser & Ogden [50]
see Itskov & Aksel [63]
3.)
see Markert, Ehlers & Karajan [78]
2.)
X
m
µ̃m ≥ 0, µ̃m (γm − 2) ≥ 0, J4 ≥ 1
yes
Appendix
D
133
Matrix Notation of Special Tensors.
Generally, tensors can be represented by matrices, whereby a more compact notation is
achieved. Here, we only consider a cartesian coordinate system, so co- and contravariant
indices do not play a role. For the second Piola-Kirchhoff stresses, the right Cauchy-Green
deformation and the associated tangent moduli we write
S=
C=












C=










h
h
S11 S22 S33 S12 S23 S13 S21 S32 S31
iT
C11 C22 C33 C12 C23 C13 C21 C32 C31
,
iT
(D.1)
,
C1111 C1122 C1133 C1112 C1123 C1113 C1121 C1132 C1131
C2211 C2222 C2233 C2212 C2223 C2213 C2221 C2232 C2231
C3311 C3322 C3333 C3312 C3323 C3313 C3321 C3332 C3331
C1211 C1222 C1233 C1212 C1223 C1213 C1221 C1232 C1231
C2311 C2322 C2333 C2312 C2323 C2313 C2321 C2332 C2331
C1311 C1322 C1333 C1312 C1323 C1313 C1321 C1332 C1331
C2111 C2122 C2133 C2112 C2123 C2113 C2121 C2132 C2131
C3211 C3222 C3233 C3212 C3223 C3213 C3221 C3232 C3231
C3111 C3122 C3133 C3112 C3123 C3113 C3121 C3132 C3131
(D.2)












,










(D.3)
respectively. Exploiting the symmetries of S AB = S BA and CAB = CBA we arrive at a
further, more compact notation. For the second Piola-Kirchhoff stresses and the right
Cauchy-Green deformations we obtain
S̄ =
C̄ =
h
h
S11 S22 S33 S12 S23 S13
iT
C11 C22 C33 2 C12 2 C23 2 C13
,
iT
(D.4)
.
(D.5)
Considering the special symmetries of CABCD = CCDAB arising from the derivation rule
∂2ψ
∂2ψ
=
,
∂CAB ∂CCD
∂CCD ∂CAB
we assume the tangent moduli matrix given in (D.3) being identical to the form


C(1) C(2) C(3)




C =  C(2) C(4) C(5)  .


C(3) C(5) C(6)
(D.6)
(D.7)
Now we take into account S AB = 21 (S AB + S BA ) and CAB = 12 (CAB + CBA ), which means
that we have to summarize the second and third row, divide this new row by 2 and do
Appendix
134
the same for the second and third column. Then this leads to a tangent moduli matrix
given by the form



C̄ = 

C(1)
(2)
1
(C
2
1
(C(2)
2
(3)
+C )
(4)
1
(C
4
+ C(3) )
(5)
+ 2C
(6)
+C )

.

(D.8)
If (D.8) is used, of course a loss of information is induced, so we can not expect to be
able to deduce the real complete tangent moduli from C̄. If we extracted the form given
in (D.8) again we would obtain the tangent moduli matrix



C(1)



˜ =
C̄
 1 (C(2) + C(3) )
 2



1
(C(2) + C(3) )
2
1
(C(2)
2
+ C(3) )
1
(C(2)
2
+ C(3) )





(4)
(5)
(6)
(4)
(5)
(6) 
1
1
(C
+
2C
+
C
)
(C
+
2C
+
C
)

4
4



(4)
(5)
(6)
(4)
(5)
(6)
1
1
+
2C
+
C
)
+
2C
+
C
)
(C
(C
4
4
(D.9)
which represents some kind of arithmetic averages. When rebuilding the associated tangent moduli tensor of fourth order we would observe the additional symmetries
˜
˜
˜
˜
C̄
ABCD = C̄BACD = C̄ABDC = C̄BADC .
(D.10)
List of Figures/Tables
135
List of Figures
1
Schematic illustration of a healthy elastic artery, taken from Junqueira
[64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2
a) Wavelike elastic laminae in the media of a healthy elastic artery, b)
wavelike orientation of collagen fiber bundles and c) ramified elastic fibers
generating oriented networks; photographs taken from Junqueira [64]. . . 10
3
Cross-section of an atherosclerotic artery in advanced state with dark areas
representing calcifications, taken from the official website of the BrownUniversity in Providence, USA (www.brown.edu). . . . . . . . . . . . . . . 12
4
Typical uniaxial stress (σ) – strain (∆l/l0 ) diagram for circumferential
arterial strips in passive condition, cf. Holzapfel [50]. Remanent strains
due to elasto-plastic material behavior and remanent strains at the edge
of physiological stress range after two overexpansion cycles are denoted by
∆lr,plas and ∆lr,phys , respectively. . . . . . . . . . . . . . . . . . . . . . . . 14
5
Schematic illustration of the composition of collagen fiber bundles, taken
from Junqueira [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6
Covariant basis vectors and curved coordinates of a) reference - and b)
actual configuration of the considered body. . . . . . . . . . . . . . . . . . 18
7
a) Reference - and b) actual configuration of the considered body. . . . . . 19
8
Rigid body motion applied to the actual configuration. . . . . . . . . . . . 30
9
Rigid body rotation applied to the reference configuration. . . . . . . . . . 31
10
Physical interpretation of the definition of anisotropic material behavior:
preferred directions ai , in the a) undistorted reference configuration and
principle axis of the b) strains and c) stresses nC i and nS I in the deformed
state, cf. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11
Physical interpretation of the principle of material symmetry: a) preferred
directions ai and b) transformed preferred directions Qai . The principle
axis of strains and stresses are invariant under the transformation Q with
respect to the transformed systems. . . . . . . . . . . . . . . . . . . . . . 35
12
Approximation of the real body by a finite number of elements, here quadrilateral elements with nen = 4 and nele = 17 (nen and nele represent the
number of element nodes and the number of elements, respectively). . . . . 40
13
Configurations within the isoparametric concept. . . . . . . . . . . . . . . . 41
14
a) 6-node triangular element and b) quadratic ansatz function N1 . . . . . . 45
15
a) 3-node line element for surface load and b) parametric coordinate and
definition of node numbering. . . . . . . . . . . . . . . . . . . . . . . . . . 46
16
Important chains of implication of generalized convexity conditions, sequentially weakly lower semicontinuity and existence of minimizers. . . . . 51
17
a) Convex and b) strictly convex function. . . . . . . . . . . . . . . . . . . 52
18
Illustration of strict convexity of Ŵ (F ). . . . . . . . . . . . . . . . . . . . 52
List of Figures/Tables
136
19
a) 3-dimensional illustration and b) contour plot of energy W (F ) =
det[F ] − 4 ln(det[F ]) in a homogeneous deformation with F =
diag[λ1 , λ2 , 1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
20
a) Uniaxial unconstrained tension of an incompressible material with preferred direction oriented parallelly to the stretch direction and b) associated
values of individual polyconvex functions I1 , I2 vs. stretch λ1 = (l0 +∆l)/l0 ;
l0 is the cube length in 1-direction in the reference configuration and ∆l
denotes the difference between actual and reference length. . . . . . . . . . 61
21
a) Uniaxial unconstrained tension of an incompressible material with preferred direction oriented parallel to the stretch direction and b) associated
values of individual polyconvex functions J4 , K1 , K2 and K3 vs. stretch
λ1 = (l0 + ∆l)/l0 ; l0 is the cube length in 1-direction in the reference
configuration and ∆l denotes the difference between actual and reference
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
22
Graph of the energy function ψcosh given in (5.59).
23
Scheme of the evolution process.
24
Local iteration algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
25
Experiment 1: homogeneous tension, Experiment 2: constrained tension. . . 78
26
Comparison of reference model 1 and the polyconvex model showing the
second Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment 2 versus
stretch λ1 . The preferred direction is set to a = (1.0 , 0.0 , 0.0)T . . . . . . 79
27
Comparison of reference model 2 (media) and the polyconvex model showing the second Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment 2
versus stretch λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
28
Comparison of reference model 2 (adventitia) and the polyconvex model
showing the second Piola-Kirchhoff stresses a) S11 and b) S22 in Experiment
2 versus stretch λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
29
Assumed orientation of fiber directions of a healthy carotid artery from a
rabbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
30
Comparison of reference model 2 (media) and the polyconvex model showing the Kirchhoff stresses a) τ11 and b) τ22 in Experiment 2 versus stretch
λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
31
Comparison of reference model 2 (adventitia) and the polyconvex model
showing the Kirchhoff stresses a) τ11 and b) τ22 in Experiment 2 versus
stretch λ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
32
Parametrization of the vector N in spherical coordinates. . . . . . . . . . . 84
33
Setup of two experiments considered in localization analysis, Experiment
1: constrained compression, Experiment 2: homogeneous shear. . . . . . . . 85
34
Localization measure q for the reference model 1 and for the polyconvex
model in three deformation states observed in Experiment 1. The preferred
direction is set to a = (1.0 , 0.0 , 0.0)T . . . . . . . . . . . . . . . . . . . . 86
. . . . . . . . . . . . . 66
. . . . . . . . . . . . . . . . . . . . . . . 71
List of Figures/Tables
137
35
Localization measure q for the reference model 2 and for the polyconvex
model in the media and adventitia of the carotid artery of a rabbit observed
in Experiment 1 at λ1 = 0.2. The preferred direction is set as shown in Fig.
29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
36
Localization measure q for the reference model 2 and for the polyconvex
model in the media of a carotid artery of a rabbit observed in Experiment
2 at F13 = 1. The preferred direction is set as shown in Fig. 29. . . . . . . 88
37
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the a) experimental tension tests
(loading and unloading) of a circumferentially (1) and longitudinally (2)
oriented strip extracted from the media of a human abdominal aorta and
b) the considered associated reference curves. l0 is the reference length of
the stripes while ∆l is denoting the difference between actual and reference
length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
38
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment and the constitutive model of Holzapfel, Gasser & Ogden [50]: a) circumferentially
and b) longitudinally oriented stripes. l0 is the reference length of the strip
and ∆l the change of length. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
39
Relative error r vs. strain ∆l/l0 using the constitutive model of
Holzapfel, Gasser & Ogden [50]: a) circumferentially (r̄ = 0.08) and
b) longitudinally oriented stripes (r̄ = 0.05). . . . . . . . . . . . . . . . . . 91
40
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment and the constitutive model ψ(BIO1 ) : a) circumferentially and b) longitudinally oriented
stripes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
41
Relative error r vs. strain ∆l/l0 using the constitutive model ψ(BIO1 ) : a)
circumferentially (r̄ = 0.04) and b) longitudinally oriented stripes (r̄ =
0.06). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
42
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the two experiments and the
constitutive models a) ψ(HGO) and b) ψ(BIO1 ) . . . . . . . . . . . . . . . . . 94
43
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment ((1) circumferentially and (2) longitudinally oriented strip). The fit is based on the constitutive model ψ(BIO2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
44
Relative error r vs. strain ∆l/l0 using the constitutive model ψ(BIO2 ) : a)
circumferentially (r̄ = 0.03) and b) longitudinally oriented stripes (r̄ =
0.07). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
45
Cauchy stress σ [kPa] vs. strain ∆l/l0 of the experiment ((1) circumferentially and (2) longitudinally oriented strip of the adventitia of the considered human abdominal aorta). The fit is based on the constitutive model
ψ(BIO1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
46
Micro-meso definition of one-dimensional damage variable introduced by
Kachanov [65], cf. Lemaitre [74]. . . . . . . . . . . . . . . . . . . . . . 97
47
Three states of the one-dimensional effective stress concept, cf. Skrzypek
& Ganczarski [116]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
List of Figures/Tables
138
48
Qualitative diagram showing a) stress (σ) vs. strain (ε) and b) associated
strain vs. time (t) for discontinuous damage. . . . . . . . . . . . . . . . . . 99
49
Algorithmic computing of the stresses and tangent moduli for the damage
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
50
a) Setup of displacement-driven experiment and b) and schematic illustration of the applied strain - time diagram with ∆l = |u|. The initial damage
state is characterized by tini . . . . . . . . . . . . . . . . . . . . . . . . . . 104
51
Cauchy stress σ [kPa] vs. strain ∆l/l0 for a) Experiment 1 and b) Experiment 2 using the constitutive model given by (7.24). The initial damage
state is set to tini , where the strain is equal to 0.25. . . . . . . . . . . . . . 105
52
Schematic illustration of fiber orientation in a) real arteries and b) the
computer model, cf. Holzapfel et al. [50]. Here, the fiber angles are set
to βM = 43.39◦ and βA = 49.0◦ . . . . . . . . . . . . . . . . . . . . . . . . . 107
53
a) Axial segment of human stenotic iliac artery, taken from Holzapfel,
Sommer & Regitnig [55], and b) used models for the particular components. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
54
Considered configurations of arteries. . . . . . . . . . . . . . . . . . . . . . 110
55
Pairs of corresponding (associated) nodes. . . . . . . . . . . . . . . . . . . 111
56
Mapping of infinitesimal line elements between considered configurations. . 112
57
Finite-Element mesh of open cross-section of the (undeformed) artery.
58
Distribution of preferred direction a(1) in a) reference configuration of considered artery and b) enlarged cut-out. . . . . . . . . . . . . . . . . . . . . 114
59
Von Mises Cauchy-stresses σv for λ = 0.0, λ = 0.5 and λ = 1.0.
60
Cauchy-stresses a) σ11 and b) σ22 after closing the artery via interface spring
elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
61
Finite-Element mesh of closed cross-section of the (undeformed) artery. . . 115
62
Cauchy-stresses a) σ11 and b) σ22 after applying stepwise F 0 .
63
Finite-Element mesh of the (undeformed) artery, cf. Holzapfel et al.
[51]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
64
Deformed artery at p = 3750.0 mmHg under conservative load without
residual stresses; distribution of a) damage variable D(1) and of b) von
Mises stresses σv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
65
Deformed artery at p = 3750.0 mmHg under non-conservative load without
residual stresses; distribution of a) damage variable D(1) and of b) von Mises
stresses σv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
66
Deformed artery at p = 3750.0 mmHg under non-conservative load with
residual stresses; distribution of a) damage variable D(1) and of b) von
Mises stresses σv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
. . 113
. . . . . . 114
. . . . . . . 116
List of Figures/Tables
139
List of Tables
1
Material parameters of reference model ψ(R1) ; except for ǫ see [140]. . . . . 76
2
Material parameters of reference model ψ(R2) ; except ǫ1 and ǫ2 see [50]. . . 77
3
Material parameters of adjusted polyconvex model. . . . . . . . . . . . . . 79
4
Material parameters of adjusted polyconvex model
5
Material parameters of the model of Holzapfel, Gasser & Ogden [50].
The angle between the (mean) fiber direction and the circumferential direction in the media was predicted to be 43.39◦. The fiber angle acts here
as a phenomenological parameter. . . . . . . . . . . . . . . . . . . . . . . . 90
6
Material parameters of the model ψ(BIO1 ) . The angle between the (mean)
fiber direction and the circumferential direction in the media was predicted
to be 43.39◦. The fiber angle acts here as a phenomenological parameter. . 92
7
Material parameters of ψ(BIO2 ) for the media of a human abdominal aorta.
8
Material parameters of ψ(BIO1 ) for the adventitia of a human abdominal
aorta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9
Considered material parameters for Model ψ(BIO) taking into account the
proposed damage model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10
Material parameters of the stored energy ψ(BIO) , see 8.26. The angle between the (mean) fiber direction and the circumferential direction in the
media is set to 43.39◦ and in the adventitia to 49.0◦ . . . . . . . . . . . . . 108
11
Material parameters of the stored energy ψ(N EO) , see (9.1).
. . . . . . . . . . . . . 80
94
. . . . . . . . 109
List of Figures/Tables
140
References
141
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Curriculum Vitae
Name
Daniel Balzani
Geburtsdatum
22. März 1976
Geburtsort
Köln
Staatsangehörigkeit
deutsch
1982 – 1986
Grundschule Waldschule, Moers
1986 – 1995
Gymnasium in den Filder Benden, Moers
1995 – 1996
Zivildienst bei ARCHE e.V., Duisburg
1996 – 2003
Studium der Fachrichtung Bauingenieurwesen an
der Universität Duisburg-Essen
2003 – 2005
Promotionsstudium der Fachrichtung Mechanik
im Rahmen des Graduiertenkollegs Modellierung,
Simulation und Optimierung von Ingenieuranwendungen, Technische Universität Darmstadt
seit 2005
Wissenschaftlicher Mitarbeiter am Institut für
Mechanik, Fachbereich Bauwissenschaften, Universität Duisburg-Essen

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