Weak or strong [Elektronische Ressource] : on coupled problems in continuum mechanics / vorgelegt von Bernd Markert

Weak or strong [Elektronische Ressource] : on coupled problems in continuum mechanics / vorgelegt von Bernd Markert

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Weak or StrongOn Coupled Problems in Continuum MechanicsVom „Stuttgart Research Centre for Simulation Technology“und der Fakultät Bau- und Umweltingenieurwissenschaftender Universität Stuttgart genehmigte Habilitationsschriftvorgelegt vonDr.-Ing. Bernd MarkertausMayen/EifelHauptberichter: Prof. Dr.-Ing. Wolfgang Ehlers1. Mitberichter: Prof. Dr.-Ing. Rainer Helmig2. Mitberichter: Prof. dr. ir. René de BorstTag der mündlichen Prüfung: 9. Juni 2010Institut für Mechanik (Bauwesen) der Universität StuttgartLehrstuhl für KontinuumsmechanikProf. Dr.-Ing. W. Ehlers2010Report No. II-20Institut für Mechanik (Bauwesen)Lehrstuhl für KontinuumsmechanikUniversität Stuttgart, Germany, 2010Editor:Prof. Dr.-Ing. W. Ehlersc Dr.-Ing. Bernd MarkertInstitut für Mechanik (Bauwesen)Lehrstuhl für KontinuumsmechanikUniversität StuttgartPfaffenwaldring 770569 Stuttgart, GermanyAll rights reserved. No part of this publication may be reproduced, stored in a retrievalsystem, or transmitted, in any form or by any means, electronic, mechanical, photocopy-ing, recording, scanning or otherwise, without the permission in writing of the author.ISBN 3–937399–20–8(D93 – Habilitation, Universität Stuttgart)PREFACEThe work described in this habilitation thesis was carried out between the years 2005and 2010 at the Institute of Applied Mechanics (Civil Engineering), Chair of ContinuumMechanics, University of Stuttgart.

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Weak or Strong
On Coupled Problems in Continuum Mechanics
Vom „Stuttgart Research Centre for Simulation Technology“
und der Fakultät Bau- und Umweltingenieurwissenschaften
der Universität Stuttgart genehmigte Habilitationsschrift
vorgelegt von
Dr.-Ing. Bernd Markert
aus
Mayen/Eifel
Hauptberichter: Prof. Dr.-Ing. Wolfgang Ehlers
1. Mitberichter: Prof. Dr.-Ing. Rainer Helmig
2. Mitberichter: Prof. dr. ir. René de Borst
Tag der mündlichen Prüfung: 9. Juni 2010
Institut für Mechanik (Bauwesen) der Universität Stuttgart
Lehrstuhl für Kontinuumsmechanik
Prof. Dr.-Ing. W. Ehlers
2010Report No. II-20
Institut für Mechanik (Bauwesen)
Lehrstuhl für Kontinuumsmechanik
Universität Stuttgart, Germany, 2010
Editor:
Prof. Dr.-Ing. W. Ehlers
c Dr.-Ing. Bernd Markert
Institut für Mechanik (Bauwesen)
Lehrstuhl für Kontinuumsmechanik
Universität Stuttgart
Pfaffenwaldring 7
70569 Stuttgart, Germany
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopy-
ing, recording, scanning or otherwise, without the permission in writing of the author.
ISBN 3–937399–20–8
(D93 – Habilitation, Universität Stuttgart)PREFACE
The work described in this habilitation thesis was carried out between the years 2005
and 2010 at the Institute of Applied Mechanics (Civil Engineering), Chair of Continuum
Mechanics, University of Stuttgart. During this period of my further academic and sci-
entific development as postdoc and senior lecturer, it was my intention to elucidate the
notion coupled problem from a more general perspective. This idea was also inspired by
theobservation thatthe‘multi’ words, such asmulti-field, multi-physics, multi-scale, etc.,
which naturally imply some sort of coupling or interaction, find more and more their way
into the computational mechanics community, not least because of the increased com-
puter power. The profound expertise at the institute, particularly in the field of coupled
multi-phase and porous media mechanics, was the self-evident starting point to set up
a fundamental theoretical framework which comprises most of the coupled formulations
conceivable in continuum physics without loosing the link to real application problems.
It is worth mentioning that during the years numerous people contributed in many ways
to the realization of this work – all their support is most gratefully acknowledged. First
of all, I particularly want to thank my academic mentor, Professor Wolfgang Ehlers, for
giving me the opportunity to prepare the thesis in the institute and for granting me all
the scientific freedom to make this work possible. He supported me in an open-minded
and unreserved fashion, which led to many valuable and fruitful discussions. Besides the
scientificsupport,Ialsothankhimforthefriendlyandfamiliaratmosphereattheinstitute
aswell asthe companionable relationship that grew over the past 15 years. I am also very
grateful to Professor Rainer Helmig from the Institute of Hydraulic Engineering, Chair of
Hydromechanics and Modeling of Hydrosystems, University of Stuttgart and to Professor
RenédeBorst, Dean oftheDepartment ofMechanical Engineering, Eindhoven University
of Technology for taking the co-chairs in my habilitation procedure. Moreover, I want to
thank all of my current and former colleagues, who always shared their experience and
knowledge with me. In this regard, also the effort of my countless student assistants,
diploma, and master students is hereby acknowledged with thanks, as they allowed me to
follow new ideas, test unconventional methods, and carry out time-consuming numerical
simulationsbesidesthedailyuniversity business. Finally, Iliketodeeplythankmyfamily,
who really had a hard time during the writing process of my habilitation thesis and often
had to share me with the computer at evenings and weekends. Actually, they provided
me with the strength to bring this work to a successful end.
Stuttgart, June 18, 2010 Bernd Markert
Science is facts; just as houses are made of stones, so is science made of facts;
but a pile of stones is not a house and a collection of facts is not necessarily science.
Jules Henri Poincaré (1854–1912)Contents
Deutsche Zusammenfassung VII
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII
Themenkomplex und Problemstellung . . . . . . . . . . . . . . . . . . . . . . VIII
Gliederung der Arbeit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XI
I Introduction and Theoretical Fundamentals 1
1 Introduction to Coupled Problems 3
1.1 What is a Coupled Problem? . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Solution of Coupled Problems . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Coupled Problems at a Glance . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 Classification of Coupled Problems . . . . . . . . . . . . . . . . 9
1.3.2 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.3 Differential and Algebraic Coupling . . . . . . . . . . . . . . . . 13
1.3.4 Stiff and Higher-Order Differential Equations . . . . . . . . . . 15
1.4 Partitioning and Splitting . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 Spatial Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.2 Time Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.3 Two Illustrative Examples . . . . . . . . . . . . . . . . . . . . . 27
1.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2 Fundamentals of Volume-Coupled Formulations 43
2.1 Mixture and Porous Media Theories . . . . . . . . . . . . . . . . . . . . 43
2.1.1 The Macroscopic Mixture Approach . . . . . . . . . . . . . . . . 43
2.1.2 Volume Fractions, Saturation, and Density . . . . . . . . . . . . 44
2.2 Kinematical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
III References
2.2.1 Mixture Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Deformation and Strain Measures . . . . . . . . . . . . . . . . . 48
2.3 Some Aspects of Electrodynamics . . . . . . . . . . . . . . . . . . . . . 52
2.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.2 The Macroscopic Maxwell Equations . . . . . . . . . . . . . . . 54
2.3.3 Fusion of Electrodynamics and Thermodynamics . . . . . . . . 55
2.4 Balance Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.1 Stress Concept and Dual Variables . . . . . . . . . . . . . . . . 56
2.4.2 Master Balance Principle for Mixtures . . . . . . . . . . . . . . 58
2.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
II A Selection of Coupled Problems 71
3 Introductory Notes to Part II 73
3.1 The Strongly Coupled Problem of Poroelastodynamics . . . . . . . . . 73
3.2 Intrinsic Interactions in Hydrated Multi-Component Bio-Tissues . . . . 74
3.3 Surface Interactions in Porous Media Contact . . . . . . . . . . . . . . 75
3.4 Materially Coupled Phenomena in Cartilage . . . . . . . . . . . . . . . 76
4 Monolithic vs. Splitting Solutions in Porous Media Dynamics 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Theoretical Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Theory of Porous Media (TPM) . . . . . . . . . . . . . . . . . . 81
4.2.2 Mixture Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2.3 Governing Balance and Constitutive Relations . . . . . . . . . . 82
4.3 Weak Formulation and Spatial Discretization . . . . . . . . . . . . . . . 85
4.3.1 Governing Weak Formulations . . . . . . . . . . . . . . . . . . . 85
4.3.2 Finite Element Discretization . . . . . . . . . . . . . . . . . . . 87
4.4 Time Discretization of the Coupled Problem . . . . . . . . . . . . . . . 87References III
4.4.1 The Time-Continuous Coupled Problem . . . . . . . . . . . . . 88
4.4.2 Implicit Monolithic Time-Integration Scheme . . . . . . . . . . . 90
4.4.3 Semi-Explicit-Implicit Splitting Scheme . . . . . . . . . . . . . . 94
4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4.5.1 Saturated Poroelastic Column under Harmonic Load . . . . . . 100
4.5.2 Two-Dimensional Wave Propagation . . . . . . . . . . . . . . . 106
4.6 Summary and Discussion of Results . . . . . . . . . . . . . . . . . . . . 115
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5 Continuum Biomechanics of Tissues 123
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.2 The TPM Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2.1 Immiscible Components and Volume Fractions . . . . . . . . . . 124
5.2.2 Miscible Components and Molar Concentrations . . . . . . . . . 125
5.2.3 Constituent Balance Relations . . . . . . . . . . . . . . . . . . . 125
5.3 Swelling Media as Biphasic, Four-Component Aggregates . . . . . . . . 127
5.3.1 Restrictions Obtained from the Entropy Inequality . . . . . . . 128
5.3.2 The Fluid Components . . . . . . . . . . . . . . . . . . . . . . . 131
5.3.3 Ion Diffusion and Fluid Flow. . . . . . . . . . . . . . . . . . . . 131
5.3.4 Electrical Potential . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.5 The Solid Skeleton . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.6 Swelling Media . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
5.4 Avascular Growth as Energy-Controlled Phase Transition . . . . . . . . 137
5.4.1 Biphasic Porous Media Growth Model . . . . . . . . . . . . . . 138
5.4.2 Thermodynamical Restrictions and Constitutive Setting . . . . 139
5.4.3 FE Simulation of Finite 3-D Growth of a Tumor Spheroid . . . 141
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142IV References
6 Fluid Penetration Effects in Porous Media Contact 147
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.2 Saturated Porous Body with Discontinuity Surface. . . . . . . . . . . . 149
6.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.2.2 Introduction of a Singular Surface . . . . . . . . . . . . . . . . . 150
6.2.3 Modified Reynolds’ Transport Theorem . . . . . . . . . . . . . . 151
6.2.4 Modified Constituent Master Balances . . . . . . . . . . . . . . 152
6.3 Contact Boundary Conditions for Saturated Porous Media . . . . . . . 154
6.3.1 Solid-Skeleton Contact . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.2 Distinct Balances and Jump Conditions . . . . . . . . . . . . . 154
6.3.3 Simplifying Assumptions . . . . . . . . . . . . . . . . . . . . . . 156
6.3.4 Derivation of the Contact Conditions . . . . . . . . . . . . . . . 157
6.4 Application to Specific Contact Situations . . . . . . . . . . . . . . . . 158
6.4.1 Quasi-Static Contact . . . . . . . . . . . . . . . . . . . . . . . . 158
6.4.2 Contact of Identical Porous Bodies . . . . . . . . . . . . . . . . 159
6.4.3 Contact of a Porous Body with a Bulk Medium . . . . . . . . . 159
6.5 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 160
6.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7 An Extended Biphasic Model for Charged Hydrated Tissues 165
7.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . 165
7.2 Theory of Porous Media (TPM) . . . . . . . . . . . . . . . . . . . . . . 168
7.3 Balance Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.4 Constitutive Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.4.1 Saturation Constraint and Entropy Inequality . . . . . . . . . . 172
7.4.2 Effective Stress of the Fluid and Solid Constituent . . . . . . . . 172
7.4.3 Inelastic Solid Kinematics . . . . . . . . . . . . . . . . . . . . . 173
7.4.4 Entropy Principle for Viscoelastic Solid Skeleton . . . . . . . . . 174
7.4.5 Dissipation of the Viscous Solid Skeleton . . . . . . . . . . . . . 176