Lecture Notes in Applied and Computational Mechanics. Edited by ... Mechanics
of Microstructured Solids 2 ... Institute of Applied Mechanics (Civil Engineering).
Lecture Notes in Applied and Computational Mechanics Volume 59 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers
Lecture Notes in Applied and Computational Mechanics Edited by F. Pfeiffer and P. Wriggers Further volumes of this series found on our homepage: springer.com Vol. 59 Markert, B., (Ed.) Advances in Extended and Multifield Theories for Continua 219 p. 2011 [978-3-642-22737-0]
Vol. 45: Shevchuk, I.V. Convective Heat and Mass Transfer in Rotating Disk Systems 300 p. 2009 [978-3-642-00717-0]
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Advances in Extended and Multifield Theories for Continua
Bernd Markert (Ed.)
123
PD Dr.-Ing. Bernd Markert University of Stuttgart Institute of Applied Mechanics (Civil Engineering) Chair of Continuum Mechanics Pfaffenwaldring 7 70569 Stuttgart, Germany E-Mail:
[email protected]
ISBN: 978-3-642-22737-0
e-ISBN: 978-3-642-22738-7
DOI 10.1007/ 978-3-642-22738-7 Lecture Notes in Applied and Computational Mechanics
ISSN 1613-7736 e-ISSN 1860-0816
Library of Congress Control Number: 2011932814 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 9876543210 springer.com
This volume is dedicated to the 60 th birthday of Professor Wolfgang Ehlers.
Preface
Extended models of continuum mechanics as well as the macroscopic description of coupled multi-field and multi-physics problems are, without a doubt, of the utmost importance in Engineering and Materials Science. The steady improvement in computational power and in efficiency of numerical algorithms in recent years rendered the treatment of realistic scenarios of practical relevance, as opposed to simplified academic benchmark problems, increasingly possible. It is beyond question that, for many decades now, thanks to computational techniques such as the Finite Element Method (FEM), continuum theories have been successfully exploited for numerous applications in science and technology. However, the underlying models, independent of their degree of sophistication, are commonly based on standard continuum mechanics and the so-called CauchyBoltzmann continuum. This incorporates only the classical thermodynamic degrees of freedom at material points, namely displacement and temperature, in combination with various constitutive material laws for small and finite deformation. By their very nature, these models are restricted to the description of so-called simple materials as local grade-one and single-phase continua, including only dissipative ODE evolutions of local internal variables describing viscosity, plasticity etc. Although this general modus operandi is still of great importance, it increasingly appears that material properties stemming from microstructural phenomena have to be considered. This is particularly true for inhomogeneous load and deformation states, where lower-scale size effects begin to affect the macroscopic material response; something for which standard continuum theories fail to account. Such states are typical for materials with distinct microtopologies associated with characteristic length scales that influence shear zone localization, local material degradation (damage, fracture) or are responsible for the well-known size effect among others. It is obvious that the situation becomes even more delicate if the material consists of multiple constituents, components or phases, which gives rise to internal flow, diffusion and phase-transformation processes driven by additional state quantities and dependent upon microstructural properties. Following this idea, it is evident that standard continuum mechanics has to be augmented to capture lower-scale structural and compositional phenomena, and to
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make this information accessible to macroscopic numerical simulations. In general, extended continuum models, in the sense of higher-order and multi-field continua, are associated with an increase in independent field variables to account for activatable microstructural effects and coupled internal processes that influence the macroscopic material response. The conceivable spatio-temporal extensions can be roughly classified into those associated with (1) external (observable) quantities via additional fields of continuum degrees of freedom involving new state variables, higher gradients or adding affine micro-deformations, and (2) internal (nonobservable) quantities by considering, for instance, additional phase and order fields, non-local dissipative PDE evolution relations or non-classical diffusion approaches. As the referenced works in this present volume reveal, many researchers have worked in the field of extended and multi-field continuum mechanics and valuably contributed to its principle understanding. However, the classical works commonly follow individual and purely phenomenological approaches. Only recently has the mechanics community started to perceive the need for a more unified approach that accounts for the underlying physics of microstructured and multiphasic materials. Of course, there are still many challenges to overcome, but recent advances in multiscale and multi-physics modeling and simulation appear promising and will allow us to gain new insights into the mechanisms taking place on lower scales, thus paving the way for future research in this exciting topic. In this context, an ambitious guiding principle may be formulated as: “From individual phenomenological approaches towards integrated physics-based extended and multi-field continuum models.” This perception actually motivated the present multi-author book, which brings together renown experts in the field. The volume gives an overview of the state of the art of extended and multi-field continuum approaches in ten review-like contributions each addressing a particular subject in a self-contained manner. The topics range from micromorphic and Cosserat theories, via phase-field models and multi-phase porous media approaches to experimental investigations and parameter optimization and model reduction methods. The book is intended for researchers working in the general fields of Computational Mechanics as well as Engineering and Materials Science, from PhD students, who want to get into the subject, to senior scientists, who want to obtain a synoptic survey. August 2011
Bernd Markert, Stuttgart Stefan Diebels, Saarbrücken
Acknowledgements
The editor most gratefully acknowledges the support and assistance of the Stuttgart Research Centre for Simulation Technology and the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart during the preparation of this book and the organization of a scientific symposium where the book has been solemnly presented to Professor Wolfgang Ehlers.
Contents
Continuum Thermodynamic and Rate Variational Formulation of Models for Extended Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bob Svendsen 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Energy Balance and Basic Constitutive Assumptions . . . . . . . . . . . . 3 Euclidean Frame-Indifference of the Energy Balance . . . . . . . . . . . . 4 Material Frame-Indifference of the Free Energy Density . . . . . . . . . 5 Dissipation Principle and Reduced Evolution-Field Relations . . . . . 6 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Lattice Models to Extended Continua . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan Diebels, Daniel Scharding 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Lattice Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Honeycomb Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Effective Shear Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Reference Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Extended Continuum Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Linear Cosserat Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analytical Solution for Shear . . . . . . . . . . . . . . . . . . . . . . . . . 4 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gradient-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Evolution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 3 5 7 8 10 13 15 19 19 21 21 23 25 26 27 28 30 30 32 34 35 38
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Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elena Pasternak, Hans-Bernd Mühlhaus, Arcady V. Dyskin 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Non-standard Continua for Modeling Materials with Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Homogenization of 1D Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Homogenization by Differential Expansion . . . . . . . . . . . . . 3.2 Homogenization by Integral Transformation (Non-local Cosserat Continuum) . . . . . . . . . . . . . . . . . . . . . . 3.3 Harmonic Waves in 1D Structures . . . . . . . . . . . . . . . . . . . . . 4 Homogenization by Differential Expansions in 3D . . . . . . . . . . . . . . 5 Cosserat Model of Layered Materials with Sliding Layers and Stress Concentrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Path-Independent Integrals in Cosserat Continuum . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Micromorphic vs. Phase-Field Approaches for Gradient Viscoplasticity and Phase Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . Samuel Forest, Kais Ammar, Benoît Appolaire 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Thermomechanics with Additional Degrees of Freedom . . . . . . . . . 2.1 General Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Micromorphic Model as a Special Case . . . . . . . . . . . . . . . . . 2.3 Phase-Field Model as a Special Case . . . . . . . . . . . . . . . . . . . 3 Constitutive Framework for Gradient and Micromorphic Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction of Viscous Generalized Stresses . . . . . . . . . . . . 3.2 Decomposition of the Generalized Strain Measures . . . . . . . 4 Phase-Field Models for Elastoviscoplastic Materials . . . . . . . . . . . . 4.1 Coupling with Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Partition of Free Energy and Dissipation Potential . . . . . . . . 4.3 Multi-phase Approach for the Mechanical Contribution . . . 4.4 Voigt/Taylor Model Coupled Phase-Field Mechanical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometrically Nonlinear Continuum Thermomechanics Coupled to Diffusion: A Framework for Case II Diffusion . . . . . . . . . . . . . . . . . . . . . . . . Andrew T. McBride, Swantje Bargmann, Paul Steinmann 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries: Notation and Key Concepts . . . . . . . . . . . . . . . . . . . . .
47 47 48 52 53 54 57 58 59 61 64 65 69 69 71 71 73 74 75 75 77 78 80 81 83 85 86 86 89 89 91
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3
Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conservation of Solid Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conservation of Diffusing Species Mass . . . . . . . . . . . . . . . . 3.3 Balance of Linear and Angular Momentum . . . . . . . . . . . . . 3.4 Balance of Internal Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Balance of Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Constitutive Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Temperature Evolution Equation . . . . . . . . . . . . . . . . . . . . . . 4 Key Features of the Helmholtz Energy Required to Reproduce Case II Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Energy Associated with Viscoelastic Effects . . . . . . . . . . . . . 4.2 Energy Associated with Mixing . . . . . . . . . . . . . . . . . . . . . . . 5 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Effective Electromechanical Properties of Heterogeneous Piezoelectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marc-André Keip, Jörg Schröder 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Boundary Value Problems on the Macro- and the Mesoscale . . . . . . 2.1 Macroscopic Electro-Mechanically Coupled BVP . . . . . . . . 2.2 Mesoscopic Electro-Mechanically Coupled BVP . . . . . . . . . 3 Effective Properties of Piezoelectric Materials . . . . . . . . . . . . . . . . . 4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Invariant Formulation and Material Parameters . . . . . . . . . . 4.2 Investigation of the “Wolfgang Ehlers 60” Mesostructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled Thermo- and Electrodynamics of Multiphasic Continua . . . . . . . Bernd Markert 1 Mixture and Porous Media Theories . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Macroscopic Mixture Approach . . . . . . . . . . . . . . . . . . . 1.2 Volume Fractions, Saturation and Density . . . . . . . . . . . . . . . 2 Kinematical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Mixture Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Deformation and Strain Measures . . . . . . . . . . . . . . . . . . . . . 3 Some Aspects of Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Macroscopic Maxwell Equations . . . . . . . . . . . . . . . . . . 3.3 Fusion of Electrodynamics and Thermodynamics . . . . . . . .
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4
Balance Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Stress Concept and Dual Variables . . . . . . . . . . . . . . . . . . . . . 4.2 Master Balance Principle for Mixtures . . . . . . . . . . . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
142 142 144 150 151
Ice Formation in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Joachim Bluhm, Tim Ricken, Moritz Bloßfeld 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Simplified Quadruple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constitutive Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Example 1: Capillary Suction during Freezing . . . . . . . . . . . 4.2 Example 2: Heat of Fusion during Phase Transition . . . . . . . 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Optical Measurements for a Cold-Box Sand and Aspects of Direct and Inverse Problems for Micropolar Elasto-Plasticity . . . . . . . . . . . . . . . . . . . . Rolf Mahnken, Ismail Caylak 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Specimens and Testing Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Materials and Specimen Preparation . . . . . . . . . . . . . . . . . . . 2.2 Experimental Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Uniaxial Compression and Tension Tests . . . . . . . . . . . . . . . . . . . . . . 3.1 SD-Effect and Optical Measurements . . . . . . . . . . . . . . . . . . 3.2 Rate Dependency and Reproducibility . . . . . . . . . . . . . . . . . . 3.3 Influence of Storage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Thermo-Mechanical Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Heat Exchanger Variation for Thermal Loading . . . . . . . . . . 4.2 Mechanical Loading for Different Isothermal Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Triaxial Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Modeling of Micropolar Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Yield Function and Plastic Potential . . . . . . . . . . . . . . . . . . . . 7 Direct and Inverse Problems for Micropolar Solids . . . . . . . . . . . . . . 7.1 Direct Problem: Weak Formulation . . . . . . . . . . . . . . . . . . . . 7.2 Inverse Problem: Constrained Least Squares Problem . . . . . 8 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
153 155 157 157 158 166 167 170 171 172 175 175 178 178 178 180 180 183 183 184 184 186 187 188 188 189 191 191 192 194 195
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Model Reduction for Complex Continua – At the Example of Modeling Soft Tissue in the Nasal Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Annika Radermacher, Stefanie Reese 1 Indroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Model Reduction for Non-linear Structural Mechanics . . . . . . . . . . 2.1 SVD-Based Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Error Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Biomechanical Structural Applications . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Study of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Study of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Human Nose Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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197 197 200 200 203 203 203 205 205 213 215 215
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
List of Authors
Kais Ammar Mines ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France e-mail:
[email protected] Benoît Appolaire LEM, ONERA/CNRS, 29 Avenue de la Division Leclerc, BP 72, 92322 Châtillon, France e-mail:
[email protected] Swantje Bargmann Institute of Mechanics, TU Dortmund University, Leonhard-Euler-Straße 5, 44227 Dortmund, Germany e-mail:
[email protected] Moritz Bloßfeld Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail:
[email protected] Joachim Bluhm Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail:
[email protected] Ismail Caylak Chair of Engineering Mechanics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany e-mail:
[email protected] Stefan Diebels Chair of Applied Mechanics, Department of Materials Science and Engineering, Saarland University, Campus A4 2, 66123 Saarbrücken, Germany e-mail:
[email protected]
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List of Authors
Arcady V. Dyskin Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia e-mail:
[email protected] Samuel Forest Mines ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France e-mail:
[email protected] Marc-André Keip Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail:
[email protected] Rolf Mahnken Chair of Engineering Mechanics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany e-mail:
[email protected] Bernd Markert Institute of Applied Mechanics (Civil Engineering), University of Stuttgart, Pfaffenwaldring 7, 70569 Stuttgart, Germany e-mail:
[email protected] Andrew T. McBride Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany e-mail:
[email protected] Hans-Bernd Mühlhaus Earth Systems Science Computational Centre, The University of Queensland, St Lucia, QLD 4072, Brisbane, Australia e-mail:
[email protected] Elena Pasternak School of Mechanical and Chemical Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia e-mail:
[email protected] Annika Radermacher Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Straße 1, 52074 Aachen, Germany e-mail:
[email protected] Stefanie Reese Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Straße 1, 52074 Aachen, Germany e-mail:
[email protected]
List of Authors
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Tim Ricken Computational Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail:
[email protected] Daniel Scharding Chair of Applied Mechanics, Department of Materials Science and Engineering, Saarland University, Campus A4 2, 66123 Saarbrücken, Germany e-mail:
[email protected] Jörg Schröder Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail:
[email protected] Paul Steinmann Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany e-mail:
[email protected] Bob Svendsen Material Mechanics, Jülich Aachen Research Alliance, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany e-mail:
[email protected]
