Advances in Extended and Multifield Theories for Continua

Lecture Notes in Applied and Computational Mechanics Volume 59 Series Editors Prof. Dr.-Ing. Friedrich Pfeiffer Prof. D...

Author: Bernd Markert

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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]

Vol. 58 Zavarise, G., Wriggers, P. (Eds.) Trends in Computational Contact Mechanics 354 p. 2011 [978-3-642-22166-8]

Vol. 44: Ibrahim R.A., Babitsky, V.I., Okuma, M. (Eds.) Vibro-Impact Dynamics of Ocean Systems and Related Problems 280 p. 2009 [978-3-642-00628-9]

Vol. 57 Stephan, E., Wriggers, P. Modelling, Simulation and Software Concepts for Scientific-Technological Problems 251 p. 2011 [978-3-642-20489-0]

Vol.43: Ibrahim, R.A. Vibro-Impact Dynamics 312 p. 2009 [978-3-642-00274-8]

Vol. 54: Sanchez-Palencia, E., Millet, O., Béchet, F. Singular Problems in Shell Theory 265 p. 2010 [978-3-642-13814-0] Vol. 53: Litewka, P. Finite Element Analysis of Beam-to-Beam Contact 159 p. 2010 [978-3-642-12939-1] Vol. 52: Pilipchuk, V.N. Nonlinear Dynamics: Between Linear and Impact Limits 364 p. 2010 [978-3-642-12798-4] Vol. 51: Besdo, D., Heimann, B., Klüppel, M., Kröger, M., Wriggers, P., Nackenhorst, U. Elastomere Friction 249 p. 2010 [978-3-642-10656-9] Vol. 50: Ganghoffer, J.-F., Pastrone, F. (Eds.) Mechanics of Microstructured Solids 2 102 p. 2010 [978-3-642-05170-8] Vol. 49: Hazra, S.B. Large-Scale PDE-Constrained Optimization in Applications 224 p. 2010 [978-3-642-01501-4] Vol. 48: Su, Z.; Ye, L. Identification of Damage Using Lamb Waves 346 p. 2009 [978-1-84882-783-7] Vol. 47: Studer, C. Numerics of Unilateral Contacts and Friction 191 p. 2009 [978-3-642-01099-6] Vol. 46: Ganghoffer, J.-F., Pastrone, F. (Eds.) Mechanics of Microstructured Solids 136 p. 2009 [978-3-642-00910-5]

Vol. 42: Hashiguchi, K. Elastoplasticity Theory 432 p. 2009 [978-3-642-00272-4] Vol. 41: Browand, F., Ross, J., McCallen, R. (Eds.) Aerodynamics of Heavy Vehicles II: Trucks, Buses, and Trains 486 p. 2009 [978-3-540-85069-4] Vol. 40: Pfeiffer, F. Mechanical System Dynamics 578 p. 2008 [978-3-540-79435-6] Vol. 39: Lucchesi, M., Padovani, C., Pasquinelli, G., Zani, N. Masonry Constructions: Mechanical Models and Numerical Applications 176 p. 2008 [978-3-540-79110-2] Vol. 38: Marynowski, K. Dynamics of the Axially Moving Orthotropic Web 140 p. 2008 [978-3-540-78988-8] Vol. 37: Chaudhary, H., Saha, S.K. Dynamics and Balancing of Multibody Systems 200 p. 2008 [978-3-540-78178-3] Vol. 36: Leine, R.I.; van de Wouw, N. Stability and Convergence of Mechanical Systems with Unilateral Constraints 250 p. 2008 [978-3-540-76974-3] Vol. 35: Acary, V.; Brogliato, B. Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics 545 p. 2008 [978-3-540-75391-9] Vol. 34: Flores, P.; Ambrósio, J.; Pimenta Claro, J.C.; Lankarani Hamid M. Kinematics and Dynamics of Multibody Systems with Imperfect Joints: Models and Case Studies 186 p. 2008 [978-3-540-74359-0

Advances in Extended and Multifield Theories for Continua

Bernd Markert (Ed.)

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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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94 94 94 96 96 97 99 101 102 103 104 104 106 109 109 112 112 114 116 120 121 122 124 125 129 129 130 130 132 132 134 138 138 139 141

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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

Contents

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XV

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]

XVIII

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

XIX

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]

Continuum Thermodynamic and Rate Variational Formulation of Models for Extended Continua Bob Svendsen

Dedicated to my colleague and friend Wolfgang Ehlers on the occasion of his 60 th birthday.

Abstract. The purpose of this work is the formulation of models for selected extended or generalized continua with the help of continuum thermodynamic and rate variational methods. The current approach is based on energy balance, the dissipation principle, as well as frame-indiﬀerence (i. e., Euclidean and material). Energetics and kinetics are based on the free energy density and a dissipation potential, respectively. More specifically, attention is focused here on the class of generalized continua whose energetic behavior depends on (i) the first- and second-order gradients of the standard deformation field, (ii) a microstructure field and its gradient, (iii) local inelastic internal variables. This is suﬃciently general to include well-known cases such as second-order Mindlin, director, micropolar (Cosserat), microstretch, or micromorphic, continua, as well as gradient inelasticity. Two types of models are identified depending on whether or not the microstructure field involved is modeled as spatial or non-spatial (e. g., intermediate or material) in nature. In particular, this constitutive assumption influences the form of the evolution-field relation for the microstructure field as well as its coupling to standard momentum balance. Given the resulting continuum thermodynamic model relations, the corresponding initialboundary-value problem is then formulated in rate-variational form. This is based on bulk and surface rate potentials determining a rate functional whose stationarity conditions yield the corresponding evolution-field relations and flux boundary conditions of the model.

1 Introduction An ever-increasing number of engineering materials and systems is strongly influenced by an emergent and / or existing material microstructure (e. g., phases, voids, Bob Svendsen Material Mechanics, Jülich Aachen Research Alliance, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 1–18. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

2

B. Svendsen

cracks, dislocations, grains, ...) and the corresponding material heterogeneity. At the continuum / mean-field level, this has provided renewed impetus to examine the extension or generalization of traditional continuum and phenomenological modeling approaches in various ways. In the process, one or more dependencies on spatial and / or temporal lengthscales are obtained which are lacking in the traditional (lengthscale-less) model formulation. Such scale-dependent and multiscale generalizations come in many sizes and flavors. From a theoretical point of view, many such generalized or extended continua have already been introduced in the last century. Perhaps the simplest among these from a conceptual point of view are the elastic non-simple or higher-order continua [e. g., 23, 24, 37, 38] based on a dependence of the material behavior on second- and / or higher-order gradients of the continuum deformation field. The idea here is that the region of influence of neighboring material points on the behavior of a given material point increases with increasing intrinsic material heterogeneity, leading to a loss of local action [56, §26 and §28], something which happens as the system size approaches that of the microstructure. Another class of generalizations involves the introduction of one or more fields besides the continuum deformation field which represent in a strongly idealized “mean-field” fashion the eﬀect of the (deformation) microstructure on the macroscopic material behavior (in a sense relative to the standard “mean” continuum deformation itself). Prominent examples include Cosserat [4] and micropolar [e. g., 30] continua, general oriented continua [54], micromorphic continua [e. g., 10, 11], director models for anisotropic fluids and liquid crystals [e. g., 12–14], or models for porous materials [e. g., 9, 21]. All of these may be categorized under the rubric of elastic continua with elastic microstructure [e. g., 6, 19, 45]. More recently, the focus has been on further development of such approaches in the direction of inelasticity and history-dependence [e. g., 1, 17, 26, 28, 43]; for a recent review, see for example [29]. Although formally a special case of the current formulation, the generalization of first-order-continuum-based extended crystal plasticity [e. g., 25, 27, 53] and continuum dislocation theory [e. g., 31, 41] to second-order is too extensive and multifaceted to do it justice here and is left for the future. The sheer vastness of the literature on extended or generalized continua spanning more than 100 years is so daunting that it almost certainly defies comprehensive review and in any case lies beyond the scope of the current work; attempts in this direction include [56, §98] (before 1965) or more recently [35] celebrating 100 years of Cosserat theory. One purpose of the current work is to investigate the formulation of models combining second-order continua in the sense of [54] or [38] and continua with microstructure in the sense of [54] and [6] generalized to inelastic and generally dissipative materials. In contrast to [54] and [38], however, this is not done by working with variational principles and special variations from the start. Rather, in the spirit of for example [6, 7, 23, 24, 40, 42, 46], this is carried out here in the framework of continuum thermodynamics, the energy balance, the dissipation principle, and invariance arguments. In this framework, the basic physical assumption is that both higher-order “Mindlin” stresses and microstructure evolution result in an additional flux and supply of energy in the material. The latter forms the cornerstone of the formulation of evolution-field relations for the microstructure field via

Continuum Thermodynamic and Rate Variational Formulation of Models

3

Euclidean frame-indiﬀerence requirements, e. g., on the total rate-of-work [6], or more generally on the total energy balance [e. g., 7], a strategy which is also followed in the current work. A number of material theoretic and configurational issues for the current class of models have been examined recently [e. g., 52]. In the current work, attention is focused on the reduced field and constitutive relations for the current class of extended continua satisfying energy balance, the dissipation principle, and indiﬀerence requirements. On this basis, we are then in a position to formulate the corresponding initial-boundary-value problem in variational form; given the non-linear dependence on inelastic and dissipative processes, the rate form of this is employed [51]. Before we begin, a word on notation. Let R, V 3 and Lin(V 3 , V 3 ) represent the spaces of real numbers, three-dimensional Euclidean vectors, and second-order Euclidean tensors, respectively. Elements of V 3 , or mappings taking values in this space, are denoted by bold-face, lower-case a, b, . . . italic letters. Likewise, uppercase A, B, . . . , italic letters denote elements of Lin(V 3 , V 3 ), or mappings taking values in this set. In particular, I ∈ Lin(V 3 , V 3 ) represents the second-order identity tensor. Let a · b ∈ R represent the scalar product on V 3 . The tensor product a ⊗ b of any two a, b ∈ V 3 is interpreted as an element of Lin(V 3 , V 3 ) via (a ⊗ b)c := (b · c)a. The scalar product on V 3 and trace operation on Lin(V 3 , V 3 ) induce the scalar product A · B := tr(AT B) ∈ R of any two A, B ∈ Lin(V 3 , V 3 ). Let sym( A) := 12 ( A + AT ) and skw(A) := 12 ( A − AT ) represent the symmetric and skew-symmetric parts, respectively, of any A ∈ Lin(V 3 , V 3 ). Further, let Sym(V 3 , V 3 ), Skw(V 3 , V 3 ) and Orth(V 3 , V 3 ) represent the sets of all symmetric, skew-symmetric, and orthogonal, elements, respectively, of Lin(V 3 , V 3 ). Third-order Euclidean tensors are interpreted as elements of Lin(V 3 , Lin(V 3 , V 3 )) or of Lin(Lin(V 3 , V 3 ), V 3 ) in this work. Such tensors, or mappings taking values in these sets, are denoted by upper-case slanted sans-serif characters A , B , . . .. For example, the second-order gradient ∇∇v of any vector field v, or the gradient ∇T of any second-order tensor field T, are third-order tensor fields. The scalar product of any two third-order tensors A , B is denoted by A · B ∈ R. For the formulation to follow, the transpose operation BTA · c := A · B c on third-order tensors will be used. Other concepts and definition will be introduced as needed along the way.

2 Energy Balance and Basic Constitutive Assumptions The class of second-order continua with microstructure under consideration here is characterized by a pair (χ, ς) of continuum fields. Here, χ is the macroscopic (mean) deformation field, and ς is a kinematic field whose evolution represents in a mean- or phase-field-like fashion the eﬀect of kinematic or deformation microstructure development on the macroscopic behavior. For example, ς could represent one or more scalar fields (e. g., glide-system shears), or a vector field (e. g., director field), or a tensor field (e. g., Cosserat or micropolar local rotation, micromorphic local deformation). For simplicity, the following treatment is restricted to isothermal processes. As well, all processes are assumed for simplicity to be continuous

4

B. Svendsen

and continuously-diﬀerentiable in space and time, which excludes for example singular surfaces. On this basis, the general form1 ˙ E=

f + B

δ− B

∂B

f ·n−

s = 0

(1)

B

holds for the total energy balance with respect to an arbitrary reference configuration B and its boundary ∂B of (outer) orientation n. Here, f =ψ+k

(2)

is the sum of the free energy density ψ and kinetic energy density k, f represents the total energy flux density, s is the total supply-rate density, and δ 0 is the dissipation-rate density. For the current class of second-order continua with deformation microstructure, the basic constitutive forms f = PT χ˙ + PT ∇ χ˙ + ΦT ς˙ , s=

b · χ˙

+ β · ς˙ ,

k˙ =

m ˙ · χ˙

+ μ˙ · ς˙ ,

(3)

hold for the energy flux, energy supply rate, and kinetic energy rate, densities, respectively. Here, P is the (first-order) first Piola-Kirchhoﬀ stress, P is the corresponding second-order stress, and Φ represents the flux density associated with the evolution of ς. Note the symmetry P · a ⊗ b ⊗ c = P · a ⊗ c ⊗ b. Further, m is the standard momentum density, b represents the corresponding supply-rate density, μ is the microstructural momentum density, and β its supply-rate density counterpart. In the context of (2) and (3), localization of the energy balance (1) yields the form δ = −p · χ˙ + (P + div P ) · ∇ χ˙ + P · ∇∇ χ˙ − π · ς˙ + Φ · ∇ ς˙ − ψ˙

(4)

for the dissipation-rate density, with p := m ˙ − b − div P ,

(5)

π := μ˙ − β − div Φ , the production-rate densities of standard and microstructural momentum, respectively. Given the above relations, we are now in a position to investigate the consequences of the Euclidean frame-indiﬀerence of the energy balance for the formulation, to which we now turn.

1

The surface da and volume dv elements are left out of the notation for simplicity.

Continuum Thermodynamic and Rate Variational Formulation of Models

5

3 Euclidean Frame-Indiﬀerence of the Energy Balance As usual, Euclidean frame-indiﬀerence is based on the transformation properties of the fields f from (2), f and s from (3), and δ from (4), appearing in E from (1) with respect to change of Euclidean observer. In fact, since δ as given by (4) follows directly from the localization of (1), the Euclidean frame-indiﬀerence of (1) is equivalent to that of δ as given by (4). As such, we examine the transformation properties of this latter field in what follows. As usual [e. g., 56, §17, or 46, Chap. 6], a change of Euclidean observer is represented via the corresponding time-dependent, spatially constant translation c and time-dependent, spatially constant orthogonal transformation Q, such that χ (t, x) = c(t) + Q(t) χ(t, x)

(6)

follows for the transformation of the time-dependent deformation field χ between an unprimed and primed observer. Since the results to follow must hold for any observer transformation, assume for simplicity that c(0) = 0 and Q(0) = I at the ˙ time origin t = 0. Then Ω := Q(0) ∈ Skw(V 3 , V 3 ) is skew-symmetric. Further, 3 define ω := c˙ (0) ∈ V , ϕ0 := ϕ|t=0 , and the deviation [[ϕ]] := (ϕ − Q∗ ϕ)|t=0

(7)

of some physical quantity ϕ from being Euclidean frame-indiﬀerent (EFI). This latter is based on the action Q∗ ϕ of Q ∈ Orth(V 3 , V 3 ) on ϕ. For example, Q∗ v ≡ Qv in the case that ϕ ≡ v is spatially vector-valued. In this context, the usual transformation relations [[χ]] = c0 , [[∇χ]] = 0 ,

[[∇∇χ]] = 0 ,

˙ = ω + Ω χ0 , [[χ]]

(8)

˙ = Ω ∇χ0 , [[∇∇ χ]] ˙ = Ω ∇∇χ0 , [[∇ χ]]

hold for χ and its derivatives. In contrast to the case of χ, there are at least two possibilities for the transformation of ς depending on how it is modeled. The first possibility is to model ς as spatial and EFI. For this class of models, it suﬃces to restrict attention here to the case that ς transforms as a spatial vector2 . Perhaps the best known examples of this case are micropolar (i. e., Cosserat) and micromorphic continua. Then ς = Q∗ ς ≡ Qς holds, implying [[ς]] = 0 ,

[[∇ς]] = 0 ,

[[ς]] ˙ = Ω ς0 ,

[[∇ ς]] ˙ = Ω ∇ς0 .

(9)

The second possibility is to model ς as non-spatial (e. g., intermediate, material) and EFI. Examples of this include internal-variable-like fields like FP , glide-system 2

This does not imply that ς itself is vector-valued. For example, the deformation gradient F = ∇χ transforms as a spatial vector.

6

B. Svendsen

shears, or dislocation densities, as in the case of extended crystal plasticity, or any material tensor field. For this class of models, ς = Q∗ ς ≡ ς holds. Then [[ς]] = 0 ,

[[∇ς]] = 0 ,

[[ς]] ˙ =0,

[[∇ ς]] ˙ =0,

(10)

follow instead. For both classes (i. e., spatial and non-spatial) of models, note that all transformations are independent of ω. Generalizing the result of Šilhavý [46], Proposition 6.2.5 for the transformation relation between the standard momentum supply-rate density b and the momentumrate density m ˙ to the corresponding microstructural quantities μ˙ and β, [[ m]] ˙ = [[b]] ,

[[μ]] ˙ = [[β]] ,

(11)

˙ P, P , and Φ are required3 to be EFI, follow. Further, the constitutive quantities ψ, in which case the restrictions ˙ =0, [[ψ]]

[[P]] = 0 ,

[[Φ]] = 0 ,

[[P ]] = 0 ,

(12)

hold. Note that (11) and (12)2 imply [[p]] = 0 from (5)1 . On this basis, [[δ]] + p0 · ω − {(P + div P ) (∇χ)T + P (∇∇χ)T − p ⊗ χ}0 · Ω ⎧ ⎪ ⎪ ⎪ ς spatial ⎪ {Φ (∇ς)T − π ⊗ ς}0 · Ω ⎨ = ⎪ ⎪ ⎪ ⎪ ⎩0 ς non-spatial

(13)

follows for the transformation of δ from (4) for the two model classes under consideration. Note that the combination P (∇χ)T is nothing other than the usual (firstorder) Kirchhoﬀ stress. Analogously, we could interpret P (∇∇χ)T as a kind of second-order Kirchhoﬀ stress. In any case, (13) must vanish as usual for arbitrary observer transformation for the energy balance to be EFI. As above, then, we are free to choose the simplest cases of (i) pure translation (Ω = 0), and (ii) pure orthogonal transformation (ω = 0). In particular, for a pure translation, (13) imply that [[δ]] vanishes identically for arbitrary ω iﬀ p vanishes identically, i. e., p=0

(14)

from (5)1 . In other words, [[δ]] vanishes identically iﬀ there is no momentum production, or alternatively, iﬀ the standard momentum balance holds [e. g., 6, 46, 56]. Analogously, in case (ii), [[δ]] vanishes identically iﬀ

3

Such constitutive quantities are also required to be material frame-indiﬀerent, which is treated below.

Continuum Thermodynamic and Rate Variational Formulation of Models

skw{(P + div P ) (∇χ)T + P (∇∇χ)T } ⎧ ⎪ ⎪ ⎪ ς spatial ⎪ skw{π ⊗ ς − Φ (∇ς)T } ⎨ = ⎪ ⎪ ⎪ ⎪ ⎩0 ς non-spatial

7

(15)

hold. These represent direct generalizations of well-known results for first-order continua [6, 7, 23, 24, 49, 56] to the current second-order case [37, 38, 54, 56]. If we restrict attention to the classical forms m = χ˙ ,

μ = ν ς˙

(16)

for the respective momentum densities based on constant referential mass density and microinertia ν, note that these results induce the better-known angularmomentum-based form Σ˙ = div M + skw{(∇χ) (P + div P )T + (∇∇χ) PT } + B

(17)

of the microstructure balance relation (5)2 via (15) in terms of the spin momentum density Σ := skw(ς ⊗ μ) and couple stress M := skw(ς ⊗ Φ) with skw(ς ⊗ Φ)a := skw(ς ⊗ Φ a), and B := skw(ς ⊗ β).

4 Material Frame-Indiﬀerence of the Free Energy Density Consider next the general constitutive form ψ = ψ(χ, ∇χ, ∇∇χ, ς, ∇ς, )

(18)

for the (equilibrium) free energy density of the current constitutive class for secondorder continua with microstructure depending in particular on a set of historydependent inelastic internal variables. Being constitutive, the basic constitutive form (18) of the free energy density ψ for the current constitutive class is required to be material frame-indiﬀerent (MFI), i. e., EFI and form-invariant [5, 48]. In this case, the restriction ψ(χ, ∇χ, ∇∇χ, ς, ∇ς, ) ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ψ(c + Qχ, Q∇χ, Q∇∇χ, Qς, Q∇ς, ) = ⎪ ⎪ ⎪ ⎪ ⎩ ψ(c + Qχ, Q∇χ, Q∇∇χ, ς, ∇ς, )

ς spatial

(19)

ς non-spatial

on the form of ψ holds for all translations c and all orthogonal transformations Q. For fixed arguments χ, ς, ∇χ, ∇ς, ∇∇χ, , the left-hand side of (19) is constant. As such, the time-derivative of (19)1 at t = 0 yields the result

8

B. Svendsen

0 = {∂χ ψ}0 · ω + {∂χ ψ ⊗ χ + ∂∇χ ψ (∇χ)T + ∂∇∇χ ψ (∇∇χ)T }0 · Ω

(20)

+ {∂ς ψ ⊗ ς + ∂∇ς ψ (∇ς)T }0 · Ω for the case that ς is spatial. An analogous result holds for the case that ς is non˙ = 0 follows, as assumed in (12) . Since (20) holds for spatial. In both cases, [[ψ]] 1 arbitrary observer transformations, we are free to choose the simplest cases of (i) pure translation (Ω = 0), and (ii) pure orthogonal transformation (ω = 0). In case (i), ∂χ ψ = 0 holds identically since c and the time origin t = 0 are arbitrary. This reduces (18) to ψ = ψ(∇χ, ∇∇χ, ς, ∇ς, )

(21)

independent of χ as expected. Analogously, in case (ii), (20) holds identically, i. e., for arbitrary Ω and time origin t = 0, iﬀ skw{∂∇χ ψ (∇χ)T + ∂∇∇χ ψ (∇∇χ)T } ⎧ ⎪ ⎪ T ⎪ ⎪ ⎨ −skw{∂ς ψ ⊗ ς + ∂∇ς ψ (∇ς) } = ⎪ ⎪ ⎪ ⎪ ⎩0

ς spatial

(22)

ς non-spatial

hold. As in case (i), these respresent restrictions on the functional form (21) of ψ. Note the formal similarity of these restrictions to those (15) based on the Euclidean frame-indiﬀerence of δ and the energy balance.

5 Dissipation Principle and Reduced Evolution-Field Relations With the basic results and restrictions (14), (15), (21) and (22) in hand, we now return to the thermodynamic formulation. In particular, substituting the standard momentum balance (14) and the material frame-indiﬀerent form (21) of the free energy density into (4), we obtain the reduced expression δ = (PN + div PN ) · ∇ χ˙ + PN · ∇∇ χ˙ + ΦN · ∇ ς˙ − πN · ς˙ − ∂ ψ · ˙

(23)

for the dissipation-rate density depending on the non-equilibrium quantities PN := P − ∂∇χ ψ + div ∂∇∇χ ψ ,

PN := P − ∂∇∇χ ψ , πN := π + ∂ς ψ , ΦN := Φ − ∂∇ς ψ .

(24)

Continuum Thermodynamic and Rate Variational Formulation of Models

9

In terms of these, for example, note that the coupling conditions and restrictions (15) become skw{(PN + div PN ) (∇χ)T + PN (∇∇χ)T } ⎧ ⎪ ⎪ T ⎪ ς spatial ⎪ ⎨ skw{πN ⊗ ς − ΦN (∇ς) } = ⎪ ⎪ ⎪ ⎪ ⎩0 ς non-spatial

(25)

via the restrictions (22) on the form of the free energy density due to material frameindiﬀerence. In the current thermodynamic context, non-equilibrium quantities like (24) are modeled constitutively via the relations PN = ∂∇ χ˙ d − div ∂∇∇ χ˙ d ,

PN = ∂∇∇ χ˙ d , −πN = ∂ς˙ d ,

(26)

ΦN = ∂∇ ς˙ d , −∂ ψ = ∂ ˙ d , in terms of the dissipation potential4 ˙ ∇∇ χ, ˙ ς, d = d(. . . , ∇ χ, ˙ ∇ ς, ˙ ˙ )

(27)

for the current constitutive class. By definition, this potential is non-negative and convex and in the rate arguments shown. In this case, the corresponding form δ = ∂∇ χ˙ d · ∇ χ˙ + ∂∇∇ χ˙ d · ∇∇ χ˙ + ∂ς˙ d · ς˙ + ∂∇ ς˙ d · ∇ ς˙ + ∂ ˙ d · ˙

(28)

of the dissipation-rate density is non-negative for all thermodynamic processes, i. e., δ d 0, and so satisfies the dissipation principle suﬃciently. In terms of the above results, the basic momentum balance relations (5) can now be expressed in thermodynamic form. To this end, note first that (14), (24) and (26) imply m ˙ − p − div P = δ2χ˙ ri , μ˙ − π − div Φ = δς˙ ri ,

4

(29)

Although not exploited here, the form of this potential is, like that of the free energy density above, restricted by material frame-indiﬀerence.

10

B. Svendsen

of the corresponding (internal) rate potential density ri := f˙ + d

(30)

determined by the dissipation potential and the dynamic stored energy rate density f˙ = m ˙ · χ˙ + ∂∇χ ψ · ∇ χ˙ + ∂∇∇χ ψ · ∇∇ χ˙ + (μ˙ + ∂ς ψ) · ς˙ + ∂∇ς ψ · ∇ ς˙ + ∂ ψ · ˙

(31)

from (2) and (3)2,3 , with δ x ϕ := ∂ x ϕ − div ∂∇ x ϕ ,

(32)

δ2x ϕ := ∂ x ϕ − div ∂∇ x ϕ + div div ∂∇∇ x ϕ , variational derivatives of first- and second-order, respectively. Recall that the vanishing of ∂χ˙ ψ˙ = ∂χ ψ and ∂χ˙ d is necessary (but not suﬃcient) to fulfill material frame-indiﬀerence. Consequently, (5) reduce to the compact form δ2χ˙ ri = b ,

(33)

δς˙ ri = β , in terms of these derivatives. Analogous to these is the “variational” form ∂ ˙ ri = 0

(34)

for the implicit evolution of from (26)5 . In particular, (33) represents the dynamic generalization of the quasi-static special case formulated in [52]. On this basis, the evolution relations for all independent constitutive fields have been expressed in terms of variational derivatives. This suggests that it is possible to formulate the corresponding initial-boundary-value problem in variational form, our next task.

6 Variational Formulation As has been recognized and exploited in earlier work [e. g., 51, 52], the physical modeling of energetic and kinetic eﬀects as based on thermodynamic potentials (21) and (27) leading to evolution-field relations like (33) expressed in variational form facilitates a (rate) variational formulation of the initial-boundary-value problem (IBVP) for the fields χ, ς and internal variables . For simplicity, attention is restricted to loading environments of the generalized displacement-traction type [e. g., 46, §13.3] here; other such conditions (e. g., unilateral or bilateral generalized contact) are possible. As usual, in this case, the boundary ∂B is divided into kinematic and flux parts. By analogy with the case of the deformation or displacement

Continuum Thermodynamic and Rate Variational Formulation of Models

11

˙ as well as their variations, gradient in the context of elasticity [38, 54], χ˙ and ∇ χ, are not necessarily independent on ∂B. Indeed, by analogy with the flux itself, only ˙ n can be considered so. Consequently, consider the the normal part ∇n δ χ˙ = (∇ χ) split ∇ χ˙ = ∇n χ˙ ⊗ n + ∇s χ˙

(35)

˙ n and tangential or in-surface ∇s χ˙ parts, of ∇ χ˙ on ∂B into normal ∇n χ˙ = (∇ χ) ˙ ∇n χ, ˙ and respectively. On this basis, the independent kinematic fields on ∂B are χ, ς. ˙ The thermodynamically conjugate boundary normal flux densities are t, s, and φ, respectively. These determine the total boundary normal energy flux density on the flux part of ∂B via f · n = t · χ˙ + s · ∇nχ˙ + φ · ς˙

(36)

from (3)1 . Given these boundary conditions, the variational formulation of the corresponding IBVP is based on rate-potential based forms f · n = t · χ˙ + s · ∇nχ˙ + φ · ς˙ = −∂χ˙ rf · χ˙ − ∂∇ χ˙ rf · ∇nχ˙ − ∂ς˙ rf · ς˙ , n

(37)

s = b · χ˙ + β · ς˙ , = −∂χ˙ rs · χ˙ − ∂ς˙ rs · ς˙ ,

for the total normal energy flux and supply-rate densities, respectively, from (3) are relevant. Here, t = −∂χ˙ rf ,

s = −∂∇nχ˙ rf ,

φ = −∂ς˙ rf ,

b = −∂χ˙ rs ,

β = −∂ς˙ rs ,

(38)

hold for the boundary normal flux densities and supply-rate densities, respectively. In terms of the combined rate potential r := ri + rs

(39)

from (30), the variational formulation of the IBVP for the current case of secondorder continua with microstructure can be based on the rate functional ˙ ς) ˙ ∇∇ χ, ˙ ς, ˙ ∇nχ, R= ˙ . (40) r(. . . , ∇ χ, ˙ ∇ ς, ˙ ˙ ) + rf (. . . , χ, B

∂B

12

B. Svendsen

To see this, consider its first variation δR = ∂χ˙ r · δ χ˙ + ∂∇ χ˙ r · δ∇ χ˙ + ∂∇∇ χ˙ r · δ∇∇ χ˙ B ∂ς˙ r · δ ς˙ + ∂∇ ς˙ r · δ∇ ς˙ + ∂ ˙ r · δ ˙ + B ∂χ˙ rf · δ χ˙ + ∂∇nχ˙ rf · δ∇nχ˙ + ∂ς˙ rf · δ ς˙ + ∂B

= + + +

B

δ2χ˙ r · δ χ˙ + δς˙ r · δ ς˙ + ∂ ˙ r · δ ˙

∂B t

(41)

{∂χ˙ rf + δ∇nχ˙ r + (divs n) ∂∇n∇nχ˙ r − divs (∂∇n∇ χ˙ r)} · δ χ˙

∂Bs ∂Bφ

{∂∇nχ˙ rf + ∂∇n∇nχ˙ r} · δ∇nχ˙ {∂ς˙ rf + ∂∇nς˙ r} · δ ς˙

in the rates, with δ∇nχ˙ r := (δ∇ χ˙ r) n , ∂∇n∇nχ˙ r := (∂∇∇ χ˙ r) n ⊗ n , ∂∇nς˙ r := (∂∇ ς˙ r) n .

(42)

On this basis, the corresponding stationarity conditions are then δ2χ˙ r = 0 ,

δς˙ r = 0 ,

δ ˙ r = 0 ,

(43)

i. e., (33) and (34), in B, and t = δ∇ χ˙ r + (divs n) ∂∇ ∇ χ˙ r − divs (∂∇ ∇ χ˙ r) , n

s = ∂∇ ∇ χ˙ r ,

n n

n

(44)

n n

φ = ∂∇ ς˙ r , n

on ∂B t , ∂B s, and ∂Bφ , respectively. As in the case of first-order continua with microstructure [51], note also that the upper bound E= ∂χ˙ r · χ˙ + ∂∇ χ˙ r · ∇ χ˙ + ∂∇∇ χ˙ r · ∇∇ χ˙ B ∂ς˙ r · ς˙ + ∂∇ ς˙ r · ∇ ς˙ + ∂ ˙ r · ˙ + (45) B ∂χ˙ rf · χ˙ + ∂∇nχ˙ rf · ∇nχ˙ + ∂ς˙ rf · ς˙ + ∂B

R

Continuum Thermodynamic and Rate Variational Formulation of Models

13

on R from (1) holds as well in the current case of second-order continua with microstructure. Along with the variational formulation itself, such bounds provide the basis for stability and other considerations which are the focus of current research on such continua.

7 Discussion The continuum thermodynamic and rate variational-based approach pursued in this work is but one among many which can be used to formulate models for extended or generalized continua with microstructure. As hinted at in the introduction, there are at least two types of approaches in this regard in the literature. The first is based on some variational or variational-like principle from the start, for example spacetime action [e. g., 4, 54], virtual work [e. g., 17, 18, 37, 38], or virtual power [e. g., 20, 22, 34]. The second type of approach, to which the current work belongs, follows basically the work of [23, 24] and many other authors after them [e. g., 6, 46], is based on continuum thermodynamics, the energy balance, the dissipation principle, and material theory. Representing perhaps the principle of choice in mechanics and the one in most widespread use, consider for example the principle of virtual work in the current context. This was applied for example by [37, 38] to the case of second- and higherorder hyperelastic continua without microstructure, and more recently in many other works involving generalized continua such as strain-gradient plasticity [17, 18]. Restricting attention here to first-order continua for simplicity, the generalized form of this principle [e. g., 55, §232] in the current context is the assertion that χ¨ · δχ + ν ς¨ · δς B ≡ t · δχ + φ · δς + b · δχ + β · δς (46) B ∂B P · ∇δχ − π · δς + Φ · ∇δς − B

for arbitrary variations δχ and δς. Alternatively, as discussed in [55, §232], one can follow Piola by neglecting the flux terms in (46) and restrict attention to special variations. In particular, with respect to rigid virtual translations (∇ δχ = 0), the corresponding principle yields the momentum balance. Likewise, with respect to rigid virtual displacements (sym(∇F δχ) = 0), one obtains the angular momentum balance. Here, ∇F δχ := (∇ δχ)F−1 is the push-forward of the referential gradient to the current configuration. A further principle exploited for the formulation of models for generalized continua is that of virtual power (mainly in the French community) [e. g., 20, 22, 34]. Formally speaking, this can be “obtained” from the principle of virtual work (46) simply by replacing the variations by their rates, yielding P(c) + P(e) + P(i) + P(k) = 0 ,

(47)

14

B. Svendsen

with

P(c) ≡ P(e) ≡ P(i) ≡ − P

(k)

t · δ χ˙ + φ · δ ς˙ ,

∂B

b · δ χ˙ + β · δ ς˙ ,

B

(48)

B

≡−

P · ∇δ χ˙ − π · δ ς˙ + Φ · ∇δ ς˙ , χ¨ · δ χ˙ + ν ς¨ · δ ς˙ ,

B

represent the power of contact forces, the power of external forces, the power of internal forces, and the power of kinetic forces, respectively, with respect to the variations δ χ˙ and δ ς. ˙ Again formally, one can proceed by analogy with Piola in the case of virtual work by neglecting the flux terms in (48) and restrict attention to special variations. Again by analogy, one obtains the linear momentum balance with respect to rigid virtual translation velocities (∇ δ χ˙ = 0). Further, one obtains angular ˙ = 0). momentum balance with respect to rigid virtual velocities (sym(∇F δ χ) In the second type of approach mentioned above [e. g., 6, 23, 24, 46], the balance relations are derived from the Euclidean frame-indiﬀerence of the energy balance rather than from some variational principle. From this point of view, rather than being “axiomatic” in nature, the variational form of the model is clearly derived from the continuum thermodynamic one. This is the point of view taken in the current work [see also 51, 52]. It is basically in agreement with that expounded for example by5 Truesdell and Toupin [55], §231 that variational “principles” represent derived entities. More to the point, the purpose of the variational formulation of the model here is the solution of the IBVP and determination of thermodynamic equilibrium states as well as their stability. Following many previous works [46, Chap. 13–14], a distinction is made in the current work between thermodynamic equilibrium (i. e., states of zero velocity, uniform temperature, and no dissipation) and the more general steady-state or dynamic equilibrium, which are dissipative due to, e. g., gradients in temperature and velocity. In particular, both equilibrium and steady-states are generally dependent on the environment in which the system finds itself. Philosophical or metaphysical diﬀerences notwithstanding, results obtained via application of one type of approach can also be obtained via the other. To underscore this point, consider for example the work of [54] in current terms. He based his approach6 to the formulation of generalized materials with microstructure on 5

6

In fact, one gains the impression that [55] are of the opinion that variational “principles” represent little more that “the formal problem of setting up expressions such that the vanishing of their first variation is equivalent to Cauchy’s laws.” Metaphysical issues aside, I would hazard to claim that, at least from a mathematical point of view, the elegance and utility of variational formulations for the solution of initial-boundary-value problems and optimization problems is unquestioned. Strictly speaking, the approach of [54] is formally more general than the current approach in the sense that he formulates a space-time action principle and derives the energy balance via time-translation invariance (i. e., explicit time-independence) of his Lagrangian. If we restrict ourselves to the context of classical physics from the start, however, as tacitly done here, both approaches are physically equally general.

Continuum Thermodynamic and Rate Variational Formulation of Models

15

continuum mechanics and an action principle generalizing the Cosserats’ “Euclidean action”. More specifically, [54] considered two cases: (i) first-order hyperelastic continua with spatial vector (director) microstructure, and (ii) secondorder hyperelastic continua without microstructure. In current terms, his results are obtained via the particular forms7 m = χ˙ ,

μ = ν ς˙ ,

ψ = ψ(∇χ, ς, ∇ς) ,

d =0,

(49)

for the momemtum density, the free energy density, and the dissipation potential, respectively, in case (i), and those m = χ˙ ,

ψ = ψ(∇χ, ∇∇χ) ,

d =0,

(50)

in case (ii). Like the referential mass density , the microinertia ν is (assumed) constant. Among other things, analogous to the Euclidean frame-indiﬀerence of the energy balance, he requires his space-time action density to be “Euclidean invariant”, a special case of (6) in which c and Q are constant, and shows that this density is Euclidean invariant iﬀ linear momentum, moment of momentum, and energy are balanced. These and other parallels exist in both approaches. Although the modeling of the microstructure field involved as (i) spatial or (ii) non-spatial has been treated in general terms in the current work, the focus in particular cases above has been on case (i). As mentioned in the introduction, perhaps the most recent application of the concepts of extended continua is that of gradient plasticity and especially gradient crystal plasticity [2, 15, 16, 25, 27, 32, 33, 36, 50, 53] for which case (ii) generally, but not always, applies. This issue, the consequences of second-order, and the formulation of configurational field and balance relations based on the Eshelby stress, for the case of extended or generalized crystal plasticity, all represent work in progress and will be reported on in the future. This also holds for much more sophisticated approaches that the current “mean-field” approaches for extended or generalized continua such as those based on distribution functions [e. g., 3, 39, 49] or even statistical mechanics itself [e. g., 8, 44, 47], which promise even further insight into the structure of such models and the interpretation of their fields.

References [1] Aifantis, E.C.: On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology 106, 326–330 (1984) [2] Bardella, L.: A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 54, 128–160 (2006) [3] Blenk, S., Muschik, W.: Orientation balances for nematic liquid crystals. Journal of Non-Equilibrium Thermodynamics 16, 67–87 (1991) 7

[54] actually considered multiple director fields d1 , d2 , . . .. For our purposes here, it is suﬃcient to restrict attention to one ς ≡ d.

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[4] Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et fils, Paris (1909); Theory of deformable bodies, NASA TT F-11 561 (1968) [5] Bertram, A., Svendsen, B.: On material objectivity and reduced constitutive relations. Archives of Mechanics 53, 653–675 (2001) [6] Capriz, G.: Continua with microstructure. Springer Tracts in Natural Philosophy, vol. 37. Springer, Heidelberg (1989) [7] Capriz, G., Virga, E.: On singular surfaces in the dynamics of continua with microstructure. Quarterly Journal of Applied Mathematics 52, 509–517 (1994) [8] Dahler, H.S., Scriven, L.E.: Theory of structured continua. I. General considerations of angular momentum and polarization. Proceedings of the Royal Society of London A 275, 505–527 (1964) [9] Ehlers, W., Volk, W.: On theoretical and numerical methods in the theory of porous media based on polar and non-polar elastoplastic solid materials. International Journal of Solids and Structures 35, 4597–4617 (1998) [10] Eringen, A.C.: Mechanics of micromorphic materials. In: Gortler, H. (ed.) Proceedings of the 11th Congress of Applied Mechanics, pp. 131–138. Springer, Heidelberg (1964) [11] Eringen, A.C.: Microcontinuum Field Theories. I: Foundations and Solids. Springer, Heidelberg (1999) [12] Ericksen, J.L.: Theory of anisotropic fluids. Archive for Rational Mechanics and Analysis 4, 231–237 (1960) [13] Ericksen, J.L.: Conservation laws for liquid crystals. Transactions of the Society of Rheology 4, 23–24 (1961) [14] Ericksen, J.L.: Liquid crystals with variable degree of orientation. Archive for Rational Mechanics and Analysis 113, 97–120 (1991) [15] Ertürk, I., van Dommelen, J.A.W., Geers, M.G.D.: Energetic dislocation interactions and thermodynamical aspects of strain gradient crystal plasticity theories. Journal of the Mechanics and Physics of Solids 57, 1801–1814 (2009) [16] Evers, L.P., Brekelmanns, W.A.M., Geers, M.G.D.: Non-local crystal plasticity model with intrinsic SSD and GND eﬀects. Journal of the Mechanics and Physics of Solids 52, 2379–2401 (2004) [17] Fleck, N.A., Hutchinson, J.W.: Strain gradient plasticity. Advances in Applied Mechanics 33, 295–361 (1997) [18] Fleck, N.A., Hutchinson, J.W.: A reformulation of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 49, 2245–2271 (2001) [19] Fried, E.: Continua described by a microstructural field. Zeitschrift für Angewandte Mathematik und Physik 47, 168–175 (1996) [20] Forest, S.: The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics 135, 117–131 (2009) [21] Goodman, D.C., Cowin, S.: A theory of granular materials. Archive for Rational Mechanics and Analysis 44, 249–266 (1972) [22] Germain, P.: Cours de Mécanique des Milieux Continus. Masson et Cie (1973) [23] Green, A.M., Rivlin, R.S.: Simple force and stress multipoles. Archive for Rational Mechanics and Analysis 16, 325–353 (1964) [24] Green, A.M., Rivlin, R.S.: Multipolar continuum mechanics. Archive for Rational Mechanics and Analysis 17, 113–147 (1964) [25] Gurtin, M.E.: A theory of viscoplasticity that accounts for geometrically necessary dislocations. Journal of the Mechanics and Physics of Solids 50, 5–32 (2002) [26] Gurtin, M.E.: On a framework for small-deformation viscoplasticity: Free energy, microforces, strain gradients. International Journal of Plasticity 19, 47–90 (2003)

Continuum Thermodynamic and Rate Variational Formulation of Models

17

[27] Gurtin, M.E.: A finite-deformation, gradient theory of single-crystal plasticity with free energy dependent on densities of geometrically necessary dislocations. International Journal of Plasticity 24, 702–725 (2008) [28] Gurtin, M.E., Anand, L.: A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part II: finite deformations. International Journal of Plasticity 21, 2297–2318 (2005) [29] Hischberger, B., Steinmann, P.: Classification of concepts in thermodynamically consistent generalized plasticity. ASCE Journal of Engineering Mechanics 150, 156–170 (2009) [30] Kafadar, C.B., Eringen, A.C.: Micropolar media: I. The classical theory. International Journal of Engineering Science 9, 271–305 (1971) [31] Kröner, E.: Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334 (1960) [32] Kuroda, M., Tvergaard, V.: On the formulation of higher-order strain gradient crystal plasticity models. Journal of the Mechanics and Physics of Solids 56, 1591–1608 (2008) [33] Levkovitch, V., Svendsen, B.: On the large-deformation- and continuum-based formulation of models for extended crystal plasticity. International Journal of Solids and Structures 43, 7246–7267 (2006) [34] Maugin, G.A.: Method of virtual power in continuum mechanics: application to coupled fields. Acta Mechanica 35, 1–70 (1980) [35] Maugin, G.A., Metrikine, A.V.: Mechanics of generalized continua. Advances in Mechanics and Mathematics, vol. 21. Springer, Heidelberg (2010) [36] Menzel, A., Steinmann, P.: On the continuum formulation of higher gradient plasticity for single and polycrystals. Journal of the Mechanics and Physics of Solids 48, 1777– 1796 (2000) [37] Mindlin, R.D.: Microstructure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 54–78 (1964) [38] Mindlin, R.D.: Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures 1, 417–438 (1965) [39] Muschik, W., Ehrentraut, H., Papenfuss, C.: Mesoscopic continuum mechanics. In: Maugin, G.A. (ed.) Geometry, Continua and Microstructure, Collection Travaux en Cours, vol. 60, pp. 49–60. Herrman, Paris (1999) [40] Noll, W.: La mécanique classique, basée sur un axiome d’ objectivité. In: La Méthode Axiomatique dans les Mécaniques Classique et Nouvelles (Colloque International à Paris, 1959), pp. 47–56. Gauthier-Villars, Paris (1963) [41] Noll, W.: Material uniform simple bodies with inhomogeneities. Archive for Rational Mechanics and Analysis 27, 1–32 (1967) [42] Noll, W.: Lectures on the foundations of continuum mechanics and thermodynamics. Archive for Rational Mechanics and Analysis 52, 62–92 (1973) [43] Peerlings, R.H.J., Geers, M.G.D., de Borst, R., Brekelmanns, W.A.M.: A critical comparison of nonlocal and gradient-enhanced softening continua. International Journal of Solids and Structures 38, 7723–7746 (2001) [44] Pitteri, M.: On a statistical-kinetic model for generalized continua. Archive for Rational Mechanics and Analysis 111, 99–120 (1990) [45] Segev, R.: A geometric framework for the statics of materials with microstructure. Mathematical Models and Methods in the Applied Sciences 4, 871–897 (1994) [46] Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Springer, Heidelberg (1997)

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[47] Svendsen, B.: A statistical mechanical formulation of continuum fields and balance relations for granular and other materials with internal degrees of freedom. In: Wilmanski, H., Hutter, K. (eds.) Kinetic and Continuum Mechanical Approaches to Granular and Porous Materials, CISM, vol. 388, pp. 245–308. Springer, Heidelberg (1999) [48] Svendsen, B., Bertram, A.: On frame-indiﬀerence and form-invariance in constitutive theory. Acta Mechanica 132, 195–207 (1999) [49] Svendsen, B.: On the continuum modeling of materials with kinematic structure. Acta Mechanica 152, 49–80 (2001) [50] Svendsen, B.: Continuum thermodynamic models for crystal plasticity including the eﬀects of geometrically-necessary dislocations. Journal of the Mechanics and Physics of Solids 50, 1297–1329 (2002) [51] Svendsen, B.: On the thermodynamic- and variational-based formulation of models for inelastic continua with internal lengthscales. Computer Methods in Applied Mechanics and Engineering 48, 5429–5452 (2004) [52] Svendsen, B., Neﬀ, P., Menzel, A.: On constitutive and configurational aspects of models for gradient continua with microstructure. Zeitschrift für Angewandte Mathematik und Mechanik 89, 687–697 (2009) [53] Svendsen, B., Bargmann, S.: On the continuum thermodynamic rate variational formulation of models for extended crystal plasticity at large deformation. Journal of the Mechanics and Physics of Solids 58, 1253–1271 (2010) [54] Toupin, R.A.: Theories of elasticity with couple stress. Archive for Rational Mechanics and Analysis 17, 85–112 (1964) [55] Truesdell, C., Toupin, R.: The classical field theories. In: Flügge, S. (ed.) Handbuch der Physik III/1, Springer, Heidelberg (1960) [56] Truesdell, C., Noll, W.: The non-linear field theories of mechanics, 2nd edn. Springer, Heidelberg (1992)

From Lattice Models to Extended Continua Stefan Diebels and Daniel Scharding

Abstract. The mechanical behavior of cellular structures is dominated by the underlying micro-topology and therefore has to be denoted very complex. For that reason the question of the computational treatment of such structures is not completely cleared yet. One way to do computations for materials with underlying microstructure is to treat them as homogeneous bodies in the framework of extended continuum theories. These theories consist of an extended set of balance equations, extended kinematic strain measures and extended constitutive equations with additional material parameters. Mostly the physical interpretation of those additional material parameters is not demonstrated vividly. So it is very diﬃcult to quantify them by parameter identification based on experimental reference data. It will be shown how virtual reference data from microscopic computations can be used to determine the extended set of macroscopic material parameters for the linear Cosserat theory.

1 Introduction Polymer and metal foams are established as state-of-the-art engineering materials e. g. in automotive and aerospace industry. The main reasons for that are their high stiﬀness and good damping properties against the background of low weight. Because of the wide-ranging use of foams as construction materials the necessity for cost-saving numerical computations during the process of product development increased in the recent decades. Apart from the computational treatment of cellular materials as homogeneous bodies on the macro-scale there are two other basic approaches.

Stefan Diebels · Daniel Scharding Chair of Applied Mechanics, Department of Materials Science and Engineering, Saarland University, Campus A4 2, 66123 Saarbrücken, Germany e-mail: {s.diebels,d.scharding}@mx.uni-saarland.de B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 19–45. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

20

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The first of them, which is not followed in the present contribution, is to replace the extended set of constitutive equations included in an extended continuum theory by a numerical homogenization scheme. Therefore, an additional boundary value problem is solved which represents the micro-structure in each material point or integration point, respectively, cf. Diebels et al [8], Ebinger et al [12, 13], Jänicke [27], Jänicke et al [28] and Jänicke and Diebels [29, 30]. This numerical homogenization procedure called FE2 was first introduced by Feyel [21, 22] and Feyel and Chaboche [23]. It was successfully applied to polycrystalline materials by Miehe et al [37, 38] and Miehe and Koch [36] and extended to higher order continua e. g. by Geers et al [25]. In the present contribution the focus lies on the second alternative approach which is to regard a full-resolution micro-model. In the case of foams a convenient micro model consists of beam elements. Actually this approach is a way to represent the microstructure’s mechanical behavior in detail but in terms of the finite element method this incorporates a too large number of degrees of freedom for large structures. Thus, in this contribution full-resolution micro-models will only serve to provide virtual experimental reference data for stress and strain quantities to be followed by a parameter identification algorithm in order to solve the inverse problem of determining the unknown set of material parameters incorporated in the extended constitutive equations of the linear Cosserat theory. A common way to recreate full-resolution micro-models for cellular materials or foams, respectively, is to reproduce the struts of e. g. open-cell foam structures by beam elements, cf. Gibson and Ashby [26], Onck et al [41], and Schraad and Triantaﬀylidis [46]. Also granular materials like concrete can be modeled by lattice structures, cf. Schlangen and van Mier [43]. In foams and granular media size eﬀects can be observed which cannot be described by the classical Boltzmann continuum approach but by extended theories like micropolar or micromorphic theories. The Cosserat theory or micropolar theory in turn incorporates additional rotational degrees of freedom for each material point, cf. Cosserat and Cosserat [5], Diebels [7], Eringen [19], Eringen and Kafadar [20] and Steinmann [47]. Hence, the Cosserat theory is able to capture the microstructural behavior observed in the micro-computations, e. g. local rotations and macroscopic size eﬀects. Therefore, the Cosserat theory is often chosen as a physically motivated extended continuum model for foams, cf. Diebels and Steeb [9–11], or granular media, cf. Ehlers et al [14], Ehlers and Scholz [15, 16] and Ehlers and Volk [17, 18]. The main problem about the parameter identification is that reference data derived from uniform experiments results in material parameters which allow a very good representation of the micro structural behavior by a homogeneous model while the agreements for non-uniform experiments are substantially worse, cf. Bigoni and Drugan [3], Forest [24] and Neﬀ et al [39]. Additionally, inhomogeneous experiments are required to activate the additional degrees of freedom incorporated in the Cosserat theory, cf. Ehlers and Scholz [15, 16]. As an example in this contribution the parameter identification will be performed on basis of virtual microscopic reference data derived from inhomogeneous shear tests due to clamped shear bounds.

From Lattice Models to Extended Continua

21

A second important aspect is the anisotropic behavior of the boundary layer eﬀect resulting from the observed inhomogeneity depending on the orientation of the investigated honeycomb structure with respect to the loading direction, cf. Teko˘glu and Onck [50]. The outline of the present contribution is as follows: In Sect. 2 the setup of the micro-models and the inhomogeneous shear tests as well as the extraction of the reference stress and strain values for the subsequent parameter identification will be described. Section 3 will introduce the linear Cosserat theory in detail and show how an analytical solution for the previously mentioned shear tests can be derived. Next, the basic approaches for parameter identification, deterministic and stochastic, as well as the ascertained Cosserat parameters will be presented in Sect. 4. Finally, the results will be summed up in Sect. 5 and an outlook on the subsequent work will be given.

2 Lattice Models In this chapter the set of micro models will be introduced which serves to provide the reference data for the subsequent parameter identification. As already mentioned the increasing computational time needed to perform full resolution micro-scale computations is accepted to replace experimental reference data which is too diﬃcult to obtain in real inhomogeneous experiments. This is the case because extended continuum theories usually incorporate local eﬀects which would have to be reproduced and measured locally in inhomogeneous experiments. The eﬀort of supplying real experimental reference data can be economized by substitution of micro-scale finite element computations to quantify reference stress and strain values. Based on this data a design of real experiments and strategies for the parameter identification can be developed. Furthermore, numerical virtual experiments can help to understand the deformation mechanisms in micro structured continua.

2.1 Honeycomb Structure In this particular case a regular two-dimensional honeycomb structure is examined as a simple example of a foam. In numerical shear tests sets of reference data are generated to identify the extended set of material parameters contained in the linear Cosserat theory. The Cosserat theory introduces additional rotational degrees of freedom for each material point and will be discussed in detail in Sect. 3. For a lattice structure the rotational degrees of freedom can be identified with the rotations of the nodes connecting the individual struts. So the aim is to compute shear stress values as well as values for the nodal rotations and translations of the honeycomb structure. Therefore a honeycomb lattice

22

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consisting of second order Timoshenko beam elements with a square cross-section was implemented in ABAQUSR . The degrees of freedom of each beam element are the translations of it’s end points denoted by u1 and u2 in Fig. 1 and the rotations of it’s cross-section ϕ3 . u2 ϕ3

t = 1.0 mm

u1 l = 10.0 mm Fig. 1 Timoshenko beam for honeycomb structure.

In technical applications foam units are glued between solid plates most often to apply certain loads. This corresponds to clamping boundary conditions at the micromodel’s top and bottom. For the finite element computations this means locking of the nodal rotations at the model’s top and bottom. To prevent softening eﬀects due to free beams at the model’s lateral bounds periodic boundary conditions corresponding to an infinitely long shear layer are applied there, cf. Figs. 2 and 3. The clamping boundary conditions at the model’s top and bottom will cause a boundary layer eﬀect under shear leading to size dependent macroscopic eﬀective properties, cf. Diebels and Steeb [9]. In consequence a boundary layer of constant thickness will develop and cause additional stiﬀness in the virtual foam sample. This stiﬀening eﬀect increases with decreasing model height and vice versa depending on the boundary layer thickness in relation to the height of the sample. A closer look to Figs. 2 and 3 shows that there are several possible orientations of the honeycomb structure with respect to the shear direction, cf. Teko˘glu and Onck [50]. What can be expected is a dependency of the mechanical properties from the hexagons’ orientation. For that reason another configuration in which the hexagons are rotated by 90◦ is examined too, cf. Fig. 3. Note that Gibson and Ashby [26] u1 = h

u2 = 0, ϕ = 0

Fig. 2 Honeycomb micromodel (config. 1) with complete set of boundary conditions for shear test. The size dependent boundary layer eﬀect is caused by clamping boundary conditions ϕ = 0 on top and bottom.

u1 = u1 u2 = u2 ϕ = ϕ

e2 u1 = 0, u2 = 0, ϕ = 0

e1

From Lattice Models to Extended Continua

23 u2 = 0, ϕ = 0

Fig. 3 Honeycomb micromodel (config. 2) with complete set of boundary conditions for shear test. The size dependent boundary layer eﬀect is caused by clamping boundary conditions ϕ = 0 on top and bottom.

u1 = h

u1 = u1 u2 = u2 ϕ = ϕ

e2 u1 = 0, u2 = 0, ϕ = 0

e1

derived isotropic properties for regular honeycombs but their investigations did not take into account any boundary layer eﬀects.

2.2 Eﬀective Shear Modulus To investigate the stiﬀening eﬀect the micro shear tests were performed for samples of varying height. The applied boundary conditions correspond to Figs. 2 and 3 with = 0.02 to keep the limits of the linear theory. At the bottom of the sample the displacements and the rotations are set to zero. A displacement u1 = 0.02 h is applied at the top of the sample while u2 and ϕ3 according to the clamping boundary conditions are set to zero. At the sample’s left- and right-hand side periodic boundary conditions are chosen. The expected observation is that an infinitely tall regular honeycomb sample approaches an eﬀective modulus which can be calculated according to Gibson and Ashby [26]: μGA = 0.57

t 3 l

Es

(1)

Thereby E s = 70000 MPa represents the Young’s modulus of the beam elements forming the lattice and hence μGA = 39.9 MPa is found as the eﬀective shear modulus. An overview of the set of virtual shear samples can be found in Table 1. Plotting the micro-computation results verifies this relation. In Fig. 4 the eﬀective shear stiﬀness is plotted over the samples’ heights. As a result the shear modulus decreases with increasing model height for both of the regarded configurations as expected. The values for the eﬀective shear moduli and the relative error related to μGA = 39.9 MPa are listed in Table 2. While μGA is reached for tall samples a size eﬀect is observed in the form ‘smaller is stiﬀer‘, cf. Diebels and Steeb [9].

24

S. Diebels and D. Scharding Table 1 Honeycomb shear samples. config. 1

config. 2

no. sample

h [no. hex.]

h [mm]

h [mm]

1 2 3 4 5 6 7 8

1 3 5 9 11 23 47 95

20.0 50.0 80.0 140.0 170.0 350.0 710.0 1430.0

17.321 51.962 86.603 155.885 190.526 398.372 814.064 1645.448

The small deviations between the numerical results and μGA obtained by analytical homogenization can be explained by the use of diﬀering beam models. While in Gibson and Ashby [26] Bernoulli beams were used for the homogenization Timoshenko beams are used in the current computations. Note that taller samples show the isotropic behavior predicted in Gibson and Ashby [26] for regular honeycombs. Anisotropic eﬀects can be observed for the smallest samples yielding μ = 44.43 MPa for config. 1 and μ = 58.75 MPa for config. 2. This result indicates that anisotropies are induced by the boundary layers, i. e. constraining the rotations at the top and bottom has a diﬀerent influence on the solution in config. 1 and config. 2. 60.0

μconfig. 1 μconfig. 2 μGA

56.0

μ [MPa]

52.0 48.0 44.0 40.0 0

20

40

60

80

h [no. hex.] Fig. 4 Eﬀective shear modulus for varying sample heights, cf. Table 2.

100

From Lattice Models to Extended Continua

25

Table 2 Eﬀective shear moduli and corresponding relative deviations from μGA . config. 1 no. sample h [no. hex.] 1 2 3 4 5 6 7 8

1 3 5 9 11 23 47 95

μe f f

config. 2

μ [MPa]

error [%]

μ [MPa]

error [%]

44.34 40.93 40.13 39.55 39.43 39.11 38.97 = 38.91

11.13 2.59 0.56 0.88 1.17 1.97 2.33 2.48

58.75 44.10 41.85 40.45 40.15 39.45 39.15 = 39.00

47.24 10.53 4.89 1.38 0.63 1.13 1.88 2.26

μe f f

This verification of the micro-models shows their reliability and allows to extract the required reference data sets for the shear stress as well as the nodal displacements and rotations.

2.3 Reference Data For each sample the shear stress is constant over its height h because of the balance of forces. The results of the micro-structure computations for both configurations can be found in Table 3. As the displacement of the micro-models’ top was applied via a reference node which the top nodes had to follow in the e1 -direction the shear stress could be computed as T 12 =

F re f bt

(2)

with F re f as the reaction force at the reference node, b as the samples’ width and t as their thickness, cf. Fig. 1. Table 3 Reference shear stress values. config. 1 re f

config. 2 re f

no. sample

h [no. hex.]

T 12 [MPa]

T 12 [MPa]

1 2 3 4 5 6 7 8

1 3 5 9 11 23 47 95

0.887 0.819 0.802 0.791 0.787 0.782 0.779 0.778

1.175 0.882 0.837 0.809 0.803 0.789 0.783 0.780

26

S. Diebels and D. Scharding

To extract the reference data for the nodal displacements and rotations a unit cell for the honeycomb micro-structure has to be defined. Therefore the description in Jänicke [27] is followed and hence there are basically two identical unit cells with diﬀering orientation, cf. Fig. 5. Fig. 5 Unit cells for the honeycomb structure with diﬀerent orientations.

To receive reference values for the nodal displacements and rotations the values at the shear sample’s top and bottom are included as well as average values of the nodal displacements and rotations per unit cell. As already mentioned a unit cell consists of three second order Timoshenko beams with three nodes each. This means the average displacement and rotation, respectively, for each unit cell found in the shear samples regardless from it’s orientation is represented by the mean of 7 nodal displacements and rotations each. The resulting data represents the reference data sets for the subsequent parameter identification and the corresponding plots for both orientations of the honeycomb lattice can be found in the appendix.

3 Extended Continuum Theories As outlined in the introduction one big disadvantage of full-resolution micro-models is the large number of degrees of freedom. Another severe disadvantage lies in the modeling of geometrically complex structures which is hardly possible. The aim is to solve this problem by using extended continuum theories to replace an inhomogeneous body by a homogeneous one. Extended continuum theories introduce additional degrees of freedom for each material point. Consequently these theories enable the capture of an inhomogeneous body’s inherent micro-structure. This does not sound like the targeted reduction of degrees of freedom but in the framework of the finite element it is one indeed. In this section the linear Cosserat theory will be described. This well-known extended continuum model introduced by Cosserat and Cosserat [5] contains a rotational field in addition to the classical displacement field and can be regarded as a special case of the more general micromorphic continuum theories, cf. Eringen [19].

From Lattice Models to Extended Continua

27

It will be shown how it can be used to replace the previously introduced honeycomb micro-models.

3.1 The Linear Cosserat Theory As well as the classical continuum theory extended theories consist of a set of balance equations, kinematic assumptions and constitutive equations. The Cosserat theory or micropolar theory belongs to the class of micromorphic theories, cf. Eringen [19]. It is obtained if the micromotion is restricted to a proper orthogonal tensor. Within the framework of the micropolar theory each material point is interpreted as a rigid body on the microscale. In order to that the material points have the usual three (two) translational and now additionally three (one) rotational degrees of freedom in three (two) dimensions. According to Sect. 2 the subsequent analysis will be made for the two dimensional case for simplicity and small deformations to stay in the limits of the linear theory. At first the additional rotational degrees of freedom induce an extension of the classical theory’s set of balance equations. As the translational degrees of freedom are covered by the linear balance of momentum (3) the rotational degrees of freedom will now be covered by the balance of moment of momentum (4) div T + ρb = 0 , div M + I × T + ρc = 0 .

(3) (4)

Herein T represents the Cauchy type stress tensor which is not symmetric any more as in the classical theory. Its asymmetric part is represented by the axial vector t = 12 I × T. The tensor M is the so-called couple stress tensor which arises from the additional rotational degrees of freedom. Subsequently the body forces ρb and the body couples ρc will be neglected. The operator div (·) is the divergence related to the gradient grad (·). The constitutive equations define T and M as functions of the Cosserat strain ε¯ and the curvature κ¯ , cf. de Borst [6]: T = 2μ ε¯ sym + 2μC ε¯ skw + λ (ε : I) I , M = 2μC (lC )2 κ¯ .

(5) (6)

Therein λ and μ are the Lamé constants known from Hooke’s law describing the symmetric stress tensor for elastic material behavior in the classical theory. The further parameters μC and lC are the already mentioned additional Cosserat parameters representing an additional stiﬀness as well as some kind of internal length. Concerning the subsequent parameter identification this means there is the possibility to reproduce the shear samples’ increasing stiﬀness for decreasing sample height via μC . The constant thickness of the boundary layer in turn can be captured by lC , cf. Diebels and Steeb [9].

28

S. Diebels and D. Scharding

To complete the theory strain measures have to be derived from the kinematic assumptions. According to the non-symmetric part of the stress tensor the strain tensor is also non-symmetric within the Cosserat theory and can be split in a symmetric and a skew-symmetric part: 3

ε¯ = grad u + E · ϕ¯ , 1 ε¯ sym = grad u + gradT u = ε , 2 3 1 ε¯ skw = grad u − gradT u + E · ϕ¯ . 2

(7) (8) (9)

The Cosserat strain tensor consists of the displacement gradient and the permutation of ϕ, ¯ cf. equation (7), whereby ϕ¯ represents the vector of the (continuous) microrotations and may not be mixed up with ϕ in Sect. 2 which represents the (discrete) nodal rotations with respect to the finite element micro models. Apparently the symmetric part of the Cosserat strain tensor (8) corresponds to the classical linear strain tensor ε. The micro-rotations in turn only aﬀect the skewsymmetric part of the Cosserat strain tensor, cf. equation (9). The skew part of ε¯ may be interpreted as the diﬀerence rotation related to the skew part of the displacement gradient (continuum rotation) and the additional Cosserat degree of freedom. In the couple stress tensor M another strain measure is included which is called curvature tensor. It is defined as the gradient of the micro-rotations κ¯ = grad ϕ¯ .

(10)

A deviation of these linear strain measures from finite deformation tensors is discussed e. g. in Volk [51]. Thus, the Cosserat theory includes everything needed to describe the behavior of the micro-models investigated in Sect. 2. First the additionally introduced microrotations provide the possibility to describe the internal rotations of the beams in the unit cells of which the micro-models consist. Second the additional Cosserat parameters μC and lC serve to model the stated stiﬀening and boundary layer eﬀect.

3.2 Analytical Solution for Shear Following the investigations of Diebels and Steeb [9] an analytical solution for the simple shear test performed for the micro-models in Sect. 2 will be recreated for the homogeneous model based on the Cosserat theory. The first step to do so is the following ansatz for the displacement field and the rotation field u = u (x2 ) e1 and ϕ¯ = ϕ (x2 ) e3

(11)

representing a simple shear deformation in an infinitely long shear layer, cf. Fig. 6.

From Lattice Models to Extended Continua

29 u2 = ϕ¯ 3 = 0

Fig. 6 Shear test for homogeneous model.

u1 = h u1 = u1 u2 = u2 ϕ¯ 3 = ϕ¯ 3

e2

h

e1 u1 = u2 = ϕ¯ 3 = 0

This ansatz transfers the balance equations (3) and (4), Eqs. (5) and (6) inserted, to a set of two diﬀerential equations μC ∂2 u μC ∂ϕ 1+ +2 = 0, (12) μ ∂x22 μ ∂x2 (lC )2

∂2 ϕ ∂u − 2ϕ − = 0. ∂x2 ∂x22

(13)

The solution for these second order linear and homogeneous diﬀerential equations can be written as u (x2 ) = U1 + U 2 x2 + U 3 exp (+ω x2 ) + U 4 exp (−ω x2 ) , ϕ (x2 ) = Φ1 + Φ2 x2 + Φ3 exp (+ω x2 ) + Φ4 exp (−ω x2 )

(14) (15)

with eigenvalue 1 ω= lC

2μ . μ + μC

(16)

The constants Ui and Φi in the solutions for the displacement field (14) and the rotation field (15) are related by the following expressions arising from the diﬀerential equations (12) and (13): 1 Φ1 = − U 2 , 2 1 μ μ + μC Φ3 = − U3 , lC μC 2μ

Φ2 = 0 , 1 μ Φ4 = lC μC

μ + μC U4 2μ

(17)

as well as the boundary-values corresponding to those of the micro-models u (0) = 0 , ϕ (0) = 0 , u (h) = h , ϕ (h) = 0 .

(18)

From Eqs. (17) and (18) the constants Ui and Φi can be uniquely determined. Additionally the shear stress results in T 12 (x2 ) = (μ + μC )

∂u + 2 μC ϕ . ∂x2

(19)

30

S. Diebels and D. Scharding

The solution for the constants Ui and Φi was generated in MAPLER depending on the boundary conditions. Furthermore the functions for the displacement field and the rotation field as well as all of their first and second order derivatives and the function of the shear stress were implemented in MAPLER . These MAPLER -functions were exported as MATLABR -functions to proceed with the parameter identification. Since the relevant functions for the kinematic and constitutive relations exist in an analytical form now the whole problem becomes accessible for both deterministic and stochastic parameter identification.

4 Parameter Identification Basically there are two ways of practicing parameter identification. On the one hand there are deterministic methods which mostly are gradient-based. These first-order methods usually supply faster convergence than gradient-free or zero-order methods, respectively, cf. Mahnken and Kuhl [31], Mahnken and Stein [32], Mahnken et al [33], Barthold and Stein [1] and Materna and Barthold [34, 35]. But apart from the potentially faster convergence the gradient-based methods are more susceptible for running into local minima instead of finding the global one. On the other hand the parameter identification can be based on stochastic methods as evolution strategies, cf. Beyer [2], Rechenberg [42] and Schwefel [48]. Stochastic methods do not necessarily require an analytical solution for the investigated problem. Instead a missing analytical solution can be replaced by a numerical replacement model. But in contrast to deterministic algorithms stochastic methods can not keep up with respect to convergence especially if the whole problem is ill-posed. In the following the goal of identifying the Cosserat parameters will be performed by evolution strategies as a stochastic method of parameter identification and it will be shown why deterministic or gradient-based methods, respectively, were unrewarding.

4.1 Gradient-Based Methods A very common approach for gradient-based parameter identification is the Newton method. The first step to apply the Newton method on the present problem of determining the Cosserat parameters in such way that the macro-model’s kinematic and constitutive behavior can reproduce the reference data achieved from the micromodels is to define an error function which returns the deviation between the analytical solution and the reference data. According to Scholz and Ehlers [45] the error function of a Cosserat-model can be defined as

From Lattice Models to Extended Continua

31

⎡ 8 2 ⎤⎥ ⎢⎢⎢ ⎥ re f ⎢⎢⎢ T 12,i (μ, μC , lC )μ = μ − T 12,i ⎥⎥⎥⎥ ⎢⎢⎢ ef f ⎥⎥⎥ μC i = 1 ⎢ ⎥ ⎢ ⎥ F (p) = ⎢⎢ 8 ⎥⎥⎥ , p = l . 2 ⎢⎢⎢ C ⎥⎥⎥ f ⎢⎢⎣ ⎥⎦ ϕi x2,k , μ, μC , lC μ = μ − ϕre i ef f

(20)

i=1

In Eq. (20) p represents the Cosserat parameters and i is the enumerator for the diﬀerent sample-heights investigated in the virtual shear experiments. x2,k are the x2 -coordinates for the centroids of the honeycomb unit-cells contained in each shear sample corresponding to Fig. 5. There is a reference value at each x2,k for a unitcell’s average rotation evaluated in the micro-computations available, cf. Sect. 2.3. By substituting μ = μe f f according to Table 2 the third free material parameter is eliminated. For shear the solution is entirely independent from λ as can be seen in Eqs. (12) and (13). This reduces the number of free material parameters once more. Alternatively, the classical Lamé parameters μ and λ can be determined from homogeneous experiments like uniaxial tension tests, e. g. Chatzouridou and Diebels [4]. The solution for the shear stress is independent from x2 as mentioned in Sect. 2.3. So the error function F can be interpreted as a function of only two remaining free material parameters μC and lC . F becomes accessible to gradient-based methods as the Newton method in the following way pn+1 = pn + Δ, Δ = − J−1 pn · F|pn , (21) ⎡ ∂F 1 ⎢⎢⎢ ⎢⎢⎢ ∂ μ ⎢ C J = ⎢⎢⎢⎢⎢ ⎢⎢⎢ ∂ F2 ⎣ ∂ μC

⎤ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥⎥ ⎥⎥ . ∂ F2 ⎥⎥⎥⎥ ⎦ ∂ lC ∂ F1 ∂ lC

(22)

But as can be seen in the solutions for the displacement field (14) and the rotation field (15) in combination with the eigenvalue (16) and the relations between the constants Ui and Φi (17) the solution is highly sensitive towards the Cosserat parameters μC and lC . This can be demonstrated by performing a sensitivity analysis as suggested in Scholz and Ehlers [44] for the error function F itself. Therefore the derivatives of the error function F with respect to the Cosserat parameters are formed. In contrast to Scholz and Ehlers [44, 45] there are no direct dependencies of F from the Cosserat parameters but only indirect dependencies incorporated via the displacement field (14) and the rotation field (15), cf. Eq. (20). So according to Scholz and Ehlers [44, 45] the derivative of F can be denoted as ⎡ (i) (i) ⎤ ⎢⎢⎢ ∂ F1 ∂ F 1 ∂ T 12 ∂ F1 ∂ F 1 ∂ T 12 ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ + + (i) ∂ μ (i) ∂ l ⎢ ∂ lC ⎥⎥⎥ C C ⎥ ∂ T 12 ∂ T 12 d F ⎢⎢⎢⎢ ∂ μC ⎥⎥⎥ = J. = ⎢⎢⎢ (23) ⎥⎥⎥ d p ⎢⎢⎢ ⎢⎢⎢ ∂ F2 ∂ F 2 ∂ ϕ(i) ∂ F2 ∂ F2 ∂ ϕ(i) ⎥⎥⎥⎥ ⎣ ⎦ + + ∂ μC ∂ lC ∂ ϕ(i) ∂ μC ∂ ϕ(i) ∂ lC

32

S. Diebels and D. Scharding

In (23) the superscript (·)(i) represents the enumerator for the height of the samples. (i) The mentioned high sensitivity towards μC and lC arises from the terms ∂T 12 /∂p j (i) and ∂ϕ /∂p j in (23) which is made comprehensible by a comparison with Eqs. (14) – (16). Since the derivatives with respect to the direct dependencies of F from μC and lC cancel out the sensitivity represents the gradient itself, see Eq. (22). Hence it is obvious that the gradient for the Newton method is too high to perform a stable parameter identification because of the exponential functions in (14) and (15). Even a very small damping factor could not guaranty convergence of the Newton method. In the end this was the reason why the decision fell on parameter identification with stochastic methods.

4.2 Evolution Strategies As mentioned in the preceding section evolution strategies were chosen to perform the parameter identification for the Cosserat theory. The great advantage of the evolution strategy is that it requires either an analytical solution of the investigated problem or alternatively an numerical analysis model, e. g. an approximation of the missing analytical solution by a finite element model. The second advantage which is important for the highly sensitive analytical solution that actually is available here is that evolution strategies are gradient-free. But basically convergence problems could arise if the model’s sensitivity towards the single parameters is highly diﬀering. The basic principles of evolution strategies are mutation, selection and recombination. Mutation is associated with asexual reproduction in genetics. This means oﬀspring parameter sets are generated by normally distributed random variation of the actual parameter sets. Corresponding to sexual reproduction in genetics recombination represents the creation of oﬀspring parameter sets by cross-over of the actual parameter sets. Finally the ‘best‘ parameter sets have to be selected among the actual and the oﬀspring population to continue the evolution. Further information can be found in Beyer [2], Rechenberg [42] and Schwefel [48]. In the particular case of the simple shear test for a Cosserat-model an analytical solution exists and was processed in Sect. 3.2 and implemented in MATLABR . The ’genetic-algorithm’-toolbox in MATLABR oﬀers a reliable implementation of a stochastic optimization algorithm based on evolution strategies. The results for both investigated configurations and orientations of the honeycomb structure respectively can be found in Table 4. Apparently the deviation of the macroscopic shear modulus μ related to it’s reference value determined by the microscopic computations is very small. The only way to induce this is to choose the appropriate starting value μ = μe f f , cf. Table 2. Otherwise a large variety of possible results is conceivable. From the general point of view stochastic methods for parameter identification are able to find the global minimum but in practical application the user is limited with respect to the computation time. Therefore the final parameter set has to be denoted a local minimum.

From Lattice Models to Extended Continua

33

Table 4 Results of the parameter identification for both orientations of the honeycomb structure.

config. 1 config. 2

μ [MPa]

μC [MPa]

lC [mm]

μe f f − μ [%] μe f f

38.853 38.817

59.706 60.412

1.828 4.578

0.146 0.469

The corresponding plots of the translational degree of freedom u = u (x2 ) e1 and the rotational degree of freedom ϕ¯ = ϕ (x2 ) e3 can be found in the appendix. Here the formation of a boundary layer becomes obvious again. For the investigated samples of config. 1 the boundary layer thickness is smaller than the boundary layer thickness of the samples of config. 2. This coherence is reflected by the internal length parameters lC for both configurations. Due to the reference data for the nodal rotations ϕ of the micro-models being discrete values the boundary layer thickness can not be quantified clearly and therefore a relation between the internal length scale parameters can not be derived analytically yet. What can be seen in the plots of the rotational degrees of freedom in the appendix is that after the boundary layer eﬀect is decayed the micro- and the macro-rotations adjust at the level of the continuum rotation ϕ/ = −0.5. The average deviation for both configurations is smaller than 3.0 %. Of course this applies only to samples tall enough to surmount the boundary layer thickness. This means samples 3 − 8 for config. 1 and samples 6 − 8 for config. 2. Besides this good agreement of the 0.90

T 12,mic T 12,mac

0.88

T 12 [MPa]

0.86 0.84 0.82 0.80 0.78 0.76 0.0

0.2

0.4

0.6

0.8

1.0

h h8 Fig. 7 Values of T 12 for the micro-models (config. 1) and the Cosserat-models.

34

S. Diebels and D. Scharding 1.20

T 12,mic T 12,mac

1.15 1.10

T 12 [MPa]

1.05

1.00 0.95 0.90 0.85 0.80 0.75 0.0

0.2

0.4

0.6

0.8

1.0

h h8 Fig. 8 Values of T 12 for the micro-models (config. 2) and the Cosserat-models.

kinematic results the results for the shear stress of the microscopic and the macroscopic models is even better, see Figs. 7 and 8. The maximum relative deviation between the microscopic and the macroscopic results related to the microscopic reference values is 0.11 % for config. 1 and 0.17 % for config. 2.

5 Conclusions and Outlook The aim of the present work was to identify the additional material parameters of the linear Cosserat theory based on reference data derived from full-resolution inhomogeneous shear tests. The results are very satisfying on both counts. On the one hand the micro-models’ kinematic behavior and on the other hand their stress-states under shear loading could be reproduced with a set of physically reasonable material parameters for each of the investigated configurations. The deviations are very small in both cases. An important aspect which becomes clearly visible is the ratio between the macroscopic and the microscopic length scales. For smaller models whose macroscopic dimensions are more and more in the scale of the boundary layer thickness the deviations of the kinematic results for the homogeneous models become larger. A further aspect is the anisotropy of the boundary depending on the honeycomb structures orientation with respect to the loading direction. The Cosserat theory is not able to capture this anisotropic behavior in the present isotropic formulation. A conceivable approach to deal with this phenomenon is an extension of the Cosserat-model. By splitting the couple stress tensor M into its native representation and an extension based on structural tensors capturing the investigated structure’s

From Lattice Models to Extended Continua

35

symmetry in conjunction with an additional internal length parameter the anisotropic behavior may be reproduced. Besides and separate from the further elaboration of this approach the parameter identification will be performed for other inhomogeneous load cases, like bending e. g., to be able to compare the deviation of the derived results.

References [1] Barthold, F.-J., Stein, E.: A continuum mechanical-based formulation of the variational sensitivity analysis in structural optimization. Part I: Analysis. Structural Optimization 11, 29–42 (1996) [2] Beyer, H.G.: The theory of evolution strategies. Springer, Heidelberg (2001) [3] Bigoni, D., Drugan, W.J.: Analytical Derivation of Cosserat Moduli via Homogenization of Heterogeneous Elastic Materials. Journal of Applied Mechanics 74, 741–753 (2007) [4] Chatzouridou, A., Diebels, S.: Identification of material parameters in extended continuum mechanical models. In: Proceedings in Applied Mathematics and Mechanics (2005), doi:10.1002/pamm.200510224 [5] Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et fils, Paris (1909); Theory of deformable bodies, NASA TT F-11 561(1968) [6] de Borst, R.: Simulation of strain localization: a reappraisal of the Cosserat continuum. Engineering Computations 8, 317–332 (1991) [7] Diebels, S.: Mikropolare Zweiphasenmodelle: Formulierung auf Basis der Theorie poröser Medien. Bericht Nr. II-4 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart (2000) [8] Diebels, S., Ebinger, T., Steeb, H., Düster, A., Rank, E.: Modelling materials with lattice microstructures by a higher order FE2 approach. In: Proceedings of: International Conference on Computational Methods for Coupled Problems in Science and Engineering, Santorini, Greece (2005) [9] Diebels, S., Steeb, H.: The size eﬀect in foams and it’s theoretical and numerical investigation. Proceedings of the Royal Society London A 458, 1–15 (2002) [10] Diebels, S., Steeb, H.: Stress and Couple Stress in Foams. Computational Material Science 28, 714–722 (2003) [11] Diebels, S., Steeb, H.: Microscopic and Macroscopic Modelling of Foams. In: Proceedings in Applied Mathematics and Mechanics (2003), doi:10.1002/pamm.2003.10.063 [12] Ebinger, T., Steeb, H., Diebels, S.: Modeling and Homogenization of Foams. Numerical Methods in Continuum Mechanics, Žilina, Slowak Republic (2003) [13] Ebinger, T., Steeb, H., Diebels, S.: Modeling macroscopic extended continua with the aid of numerical homogenization schemes. Computational Material Science 32, 337– 347 (2005) [14] Ehlers, W., Diebels, S., Volk, W.: Deformation and compatibility for elasto-plastic micropolar materials with application to geomechanical problems. Journal de Physique IV France 8 Pr5, 127–134 (1998) [15] Ehlers, W., Scholz, B.: Lecture Notes in Applied and Computational Mechanics, vol. 28. LNACM, pp. 83–112 (2006) [16] Ehlers, W., Scholz, B.: An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material. Archive of Applied Mechanics (2007), doi:10.1007/s00419-007-0162-9

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[17] Ehlers, W., Volk, W.: On shear band localization phenomena of liquid-saturated granular elastoplastic porous solid materials accounting for fluid viscosity and micropolar solid rotations. Mechanics of Cohesive-Frictional Materials 2, 301–320 (1997) [18] Ehlers, W., Volk, W.: On theoretical and numerical methods in the theory of porous media based on polar and non-polar elasto-plastic solid materials. International Journal of Solids and Structures 24, 3486–3506 (1998) [19] Eringen, A.C.: Microcontinuum field theories. Foundations and solids, vol. I. Springer, Heidelberg (1999) [20] Eringen, A.C., Kafadar, C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Mechanics, vol. VI, pp. 1–73. Academic, New York (1976) [21] Feyel, F.: Multiscale FE2 elastoviscoplastic analysis of composite structures. Computational Material Science 16, 344–354 (1999) [22] Feyel, F.: Multiscale non linear FE2 analysis of composite structures: Fibre size eﬀects. Journal de Physique IV France 11 Pr5, 195–202 (2001) [23] Feyel, F., Chaboche, J.L.: FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials. Computational Methods in Applied Mechanics and Engineering 183, 309–330 (2000) [24] Forest, S.: Aufbau und Identifikation von Stoﬀgleichungen für höhere Kontinua mittels Homogenisierungsmethoden. Technische Mechanik 19, 297–306 (1999) [25] Geers, M.G.D., Kouznetsova, V., Brekelmans, W.A.M.: Gradient-enhanced computational homogenization for the micro-macro scale transition. Journal de Physique IV France 11 Pr5, 145–152 (2001) [26] Gibson, L.J., Ashby, M.F.: Cellular solids. Cambridge Solid State Science Series. Cambridge University Press, Cambridge (1999) [27] Jänicke, R.: Micromorphic Media: Interpretation and homogenization. Saarbrücker Reihe Materialwissenschaft und Werkstoﬀtechnik Band 21 (2010) [28] Jänicke, R., Diebels, S.: A numerical homogenisation strategy for micromorphic continuua. Nuovo Cimento della Società Italiana di Fisica - C 31, 121–132 (2009) [29] Jänicke, R., Diebels, S.: Numerical homogenisation of micromorphic continuua. Technische Mechanik 30, 364–373 (2010) [30] Jänicke, R., Diebels, S., Sehlhorst, H.-G., Düster, A.: Two-scale modelling of micromorphic continua. Continuum Mechanics and Thermodynamics 21, 297–315 (2009) [31] Mahnken, R., Kuhl, E.: Parameter identification of gradient enhanced damage models with the finite element method. European Journal of Mechanics A/Solids 18, 819–835 (1999) [32] Mahnken, R., Stein, E.: A unified approach for parameter identification of inelastic material models in the frame of the finite element method. Computational Methods in Applied Mechanics and Engineering 136, 225–258 (1996) [33] Mahnken, R., Stein, E., Bischoﬀ, D.: A stabilization procedure by line-search computation for first-order approximation strategies in structural optimization. International Journal for Numerical Methods in Engineering 35, 1015–1029 (1992) [34] Materna, D., Barthold, F.-J.: Variational design sensitivity analysis in the context of structural optimization and configurational mechanics. International Journal of Fracture 147, 133–155 (2007) [35] Materna, D., Barthold, F.-J.: On variational sensitivity analysis and configurational mechanics. Computational Mechanics 41, 661–681 (2008) [36] Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics (2002), doi:10.1007/s00419-002-0212-2

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[37] Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Material Science 16, 372–382 (1999) [38] Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity Simulation of texture development in polycrystalline materials. Computational Methods in Applied Mechanics and Engineering 171, 387–418 (1999) [39] Neﬀ, P., Jeong, J., Fischle, A.: Stable identification of linear isotropic Cosserat parameters: bounded stiﬀness in bending and torsion implies conformal invariance of curvature. Acta Mechanica 211, 237–249 (2010) [40] Onck, P.R.: Cosserat modeling of cellular solids. Comptes Rendus Mecanique 330, 717–722 (2002) [41] Onck, P.R., Andrews, E.W., Gibson, L.J.: Size eﬀects in ductile cellular solids, Part I: modeling. International Journal of Mechanical Sciences 43, 681–699 (2001) [42] Rechenberg, I.: Evolutionsstrategie 1994, Frommann-Holzboog (1994) [43] Schlangen, E., van Mier, J.G.M.: Simple lattice model for numerical simulation of fracture of concrete materials and structures. Materials and Structures 25, 534–542 (1992) [44] Scholz, B., Ehlers, W.: Sensitivitätsanalyse im Cosserat-Kontinuum. In: Proceedings in Applied Mathematics and Mechanics, vol. 1, pp. 173–174 (2002) [45] Scholz, B., Ehlers, W.: Inverses Rechnen zur Identifikation der Materialparameter des Cosserat-Kontinuums. In: Proceedings in Applied Mathematics and Mechanics (2003), doi:10.1001/pamm.200310132 [46] Schraad, M.W., Triantafyllidis, N.: Scale eﬀects in media with periodic and nearly periodic microstructures, Part I: Macroscopic properties. Journal of Applied Mechanics 64, 751–762 (1997) [47] Steinmann, P.: A micro polar theory of finite deformation and finite rotation multiplicative elasto-plasticity. International Journal of Solids and Structures 31, 1063–1084 (1994) [48] Schwefel, H.P.: Evolution and optimum seeking. In: Sixth-Generation Computer Technology. Wiley Interscience, Hoboken (1995), http://ls11-www.cs.uni-dortmund.de/lehre/wiley/ [49] Tekoˇglu, C., Onck, P.R.: Size eﬀects in the mechanical behaviour of cellular solids. Journal of Material Science 40, 5911–5917 (2005) [50] Tekoˇglu, C., Onck, P.R.: Size eﬀects in two-dimensional Voronoi foams: A comparison between generalized continuua and discrete models. Journal of the Mechanics and Physics of Solids (2008), doi:10.1016/j.jmps.2008.06.007 [51] Volk, W.: Untersuchung des Lokalisierungsverhaltens mikropolarer porser Medien mit Hilfe der Cosserat-Theorie. Bericht Nr. II-2 aus dem Institut fr Mechanik (Bauwesen), Universität Stuttgart (1999)

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Appendix 1.0 0.8

x2 h1

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

u1,mac. ϕ3,mac.

0.5 1.0 u1 ϕ3 , ε h1 ε Fig. 9 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h1 ) and the Cosseratmodels. 0.0

1.0 0.8

x2 h2

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0

u1 ϕ3 , ε h2 ε

0.5

u1,mac. ϕ3,mac. 1.0

Fig. 10 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h2 ) and the Cosseratmodels.

From Lattice Models to Extended Continua

39

1.0 0.8

x2 h3

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

u1,mac. ϕ3,mac.

0.0

1.0 0.5 u1 ϕ3 , ε h3 ε Fig. 11 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h3 ) and the Cosseratmodels.

1.0 0.8

x2 h4

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0.0

u1 ϕ3 , ε h4 ε

0.5

u1,mac. ϕ3,mac. 1.0

Fig. 12 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h4 ) and the Cosseratmodels.

40

S. Diebels and D. Scharding 1.0 0.8

x2 h5

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0.0

u1 ϕ3 , ε h5 ε

u1,mac. ϕ3,mac. 1.0

0.5

Fig. 13 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h5 ) and the Cosseratmodels.

1.0 0.8

x2 h6

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0.0

u1 ϕ3 , ε h6 ε

0.5

u1,mac. ϕ3,mac. 1.0

Fig. 14 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h6 ) and the Cosseratmodels.

From Lattice Models to Extended Continua

41

1.0 0.8

x2 h7

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0.0

u1 ϕ3 , ε h7 ε

0.5

u1,mac. ϕ3,mac. 1.0

Fig. 15 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h7 ) and the Cosseratmodels.

1.0 0.8

x2 h8

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.5

0.0

u1,mac. ϕ3,mac.

1.0 0.5 u1 ϕ3 , ε h8 ε Fig. 16 Normalized values of u1 and ϕ3 for the micro-models (config. 1, h8 ) and the Cosseratmodels.

42

S. Diebels and D. Scharding 1.0 0.8

x2 h1

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

u1,mac. ϕ3,mac.

0.0

1.2 0.4 0.8 u1 ϕ3 , ε h1 ε Fig. 17 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h1 ) and the Cosseratmodels.

1.0 0.8

x2 h2

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

0.0

u1,mac. ϕ3,mac.

0.4 0.8 1.2 u1 ϕ3 , ε h2 ε Fig. 18 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h2 ) and the Cosseratmodels.

From Lattice Models to Extended Continua

43

1.0 0.8

x2 h3

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

u1,mac. ϕ3,mac.

0.4 0.8 1.2 u1 ϕ3 , ε h3 ε Fig. 19 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h3 ) and the Cosseratmodels. 0.0

1.0 0.8

x2 h4

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

0.0

u1,mac. ϕ3,mac.

0.4 0.8 1.2 u1 ϕ3 , ε h4 ε Fig. 20 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h4 ) and the Cosseratmodels.

44

S. Diebels and D. Scharding 1.0 0.8

x2 h5

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

u1,mac. ϕ3,mac.

0.4 0.8 1.2 u1 ϕ3 , ε h5 ε Fig. 21 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h5 ) and the Cosseratmodels. 0.0

1.0 0.8

x2 h6

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

0.0

u1,mac. ϕ3,mac.

1.2 0.4 0.8 u1 ϕ3 , ε h6 ε Fig. 22 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h6 ) and the Cosseratmodels.

From Lattice Models to Extended Continua

45

1.0 0.8

x2 h7

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

u1,mac. ϕ3,mac.

0.0

0.4 0.8 1.2 u1 ϕ3 , ε h7 ε Fig. 23 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h7 ) and the Cosseratmodels.

1.0 0.8

x2 h8

0.6 0.4 0.2

u1,mic. ϕ3,mic.

0.0 −0.4

0.0

u1,mac. ϕ3,mac.

1.2 0.4 0.8 u1 ϕ3 , ε h8 ε Fig. 24 Normalized values of u1 and ϕ3 for the micro-models (config. 2, h8 ) and the Cosseratmodels.

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale Elena Pasternak, Hans-Bernd Mühlhaus, and Arcady V. Dyskin

Abstract. Engineering materials are non-homogeneous and even discrete, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. We review models dealing with the discrete nature of granular materials and with layered materials with sliding layers. Both cases require the introduction of rotational degrees of freedom; for layered materials it is the layer bending that represents a rotational degree of freedom. We consider the eﬀects of the rotational degrees of freedom from apparent strain localization in simple shearing to new fracture modes.

1 Introduction Many engineering applications deal with materials comprised of constituents with absent or weak binder phases and exhibiting mechanical behavior that is strongly aﬀected by rotational degrees of freedom. Examples of such materials include particulate materials, layered and blocky materials. Zones that are strongly aﬀected by rotational degrees of freedom can form in heterogeneous materials where a high degree of damage is produced either by external load or locally by stress concentrators (e. g., cracks). A common feature of such solids is the ability of constituents (grains, blocks, etc.) to rotate relative to each other, the rotations being resisted either by the Elena Pasternak · Arcady V. Dyskin The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia e-mail: [email protected],[email protected] Hans-Bernd Mühlhaus Earth Systems Science Computational Centre, The University of Queensland, St Lucia, QLD 4072, Brisbane, Australia e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 47–67. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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weak binder and/or by the kinematic constraints imposed by the geometry of the neighboring blocks in the presence of compressive stress. These rotations can control the mechanical behavior of such solids, including deformation patterning and fracture propagation, which aﬀects the stability of structures built in or from them. It has been quite a while since internal rotational degrees of freedom were first identified as being important in correctly describing the mechanics of deformable solids. In 1909 the Cosserat brothers [4], conjectured that internal parts of materials can experience relative rotations beyond the ones caused by the displacement field and that the eﬀect thereof warrants investigation. They developed a complex theory, which went unnoticed at the time and was rediscovered half a century later and named the Cosserat theory [11, 12, 23]. However, numerous attempts to detect the Cosserat eﬀects in ordinary materials proved to be inconclusive (see review [41]). In the 1980’s and 90’s a realization came that one should look for the Cosserat eﬀects in non-homogeneous materials. Since then the theory (with various modifications) found its acceptance in the modeling of geomaterials, especially layered [1, 24, 42] and granular matter [3, 24, 26, 31]. The main obstacle to its use, however, is the inability to determine and calibrate the Cosserat elastic moduli from the experiments. The frustration was so great that some considered reverting to the use of standard theories. As a result, the higher order theories were demoted to be used as numerical stabilizers for otherwise mesh-dependent numerical models [40] despite the general understanding that internal grain/block rotations do exist. Conversely, calls were made to abandon continuum mechanics altogether in favor of particle models that automatically reproduce the complex behavior of particulate materials, requiring only a few parameters [5]. The problem is that the direct manifestations of Cosserat eﬀects are hard to see as they require the detection of either microrotations or specific Cosserat wave modes, which exist at very high frequencies [32]. This is diﬃcult, especially in highly heterogeneous materials such as geomaterials. The picture becomes less gloomy if one recalls that the rotational degrees of freedom aﬀect the propagation of the much more easily detectable conventional shear waves, creating dispersion (frequency dependence of the wave velocity). A new technique has been recently developed that uses dispersion to reconstruct the Cosserat moduli from conventional shear wave measurements [34]. A way to overcome the obstacle of determining the Cosserat parameters is to use homogenization methods to build the Cosserat continuum such that the Cosserat moduli are inferred from measurable microstructural parameters. Here we overview these methods.

2 Non-standard Continua for Modeling Materials with Microstructure When the size of redistribution of an external load is comparable with the microstructure size or if the stress gradients at internal contact points (finite contact area, modeled as a contact point) have to be taken into account, the classical

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

49

continuum mechanics approach is insuﬃcient. This is the case when one considers layered materials, especially when the layers can slide, blocky structures, granular or fractured media. There are diﬀerent approaches (or combinations thereof) to take microstructural eﬀects into account. Each approach results in diﬀerent types of standard or non-standard continua [37]. The first step in generalizing the standard continuum theory is the introduction of rotational degrees of freedom in addition to the conventional translational leading to the Cosserat type theories [4, 33]. Further generalization leads to gradient or higherorder gradient theories, resulting in the introduction of additional strain measures and the corresponding stress tensors. Mindlin [22] based his reasoning for the need for the further degrees of freedom on the simultaneous consideration of macro- and micro- displacements within a volume element. Considering a macro-volume in a Cartesian coordinate frame (x1 , x2 , x3 ). Let P be an arbitrary point with macro-coordinates xi . The macro-motion of this point can be described by the macro-displacement vector u(xi ) and macro-rotation vector ϕ = 12 rot u. The deformation measures at this point are the components of the macro-distortion tensor ui j . Its symmetric part is the usual macro-strain tensor εi j = 12 (ui, j + u j,i ). The antisymmetric part of the macro-distortion tensor gives the macro-rotational vector ϕi = − 12 εkl j uk,l , where εkl j is the alternating tensor. It is seen that the macro-rotation vector and the distortion tensor are fully determined by the components of the displacement vector. Next, assume the material point P as a centroid of a micro-volume of the originally inhomogeneous medium. This volume element defines the scale of resolution of the envisaged continuum theory. Eﬀects below this characteristic scale are ignored. This volume element could not be constricted to the point because of the microstructure of the material. We presume that separation of scales holds: the micro-volume size is much larger than the microstructure size to make the micro-volume representative and much smaller than the external size such as the dimension of the problem or a characteristic length of the load redistribution (e. g., wave length), to asymptotically satisfy the requirement for the micro-volume to be infinitesimal. We now introduce another local Cartesian coordinate system with the origin at P. An arbitrary point P of the micro-volume has the micro-coordinates (x1 , x2 , x3 ). The vector u (xi ) connecting P and P is called the micro-displacement vector and characterizes the displacement of the point P within the micro-particle (microvolume element). The displacement of the point P being a point of the macro-volume is given by the sum of the macro-displacement vector u(xi ) and the micro-displacement vector u (xi ). Expanding the components of the micro-displacement vector into the Taylor series at the vicinity of the pointP one gets the correspondent coordinates of the displacement vector of the point P : 1 ui (x) + ui (x ) = ui (x) + ui (0) + ui, j (0)xj + ui, jk (0)xj xk + ... , 2 de f

de f

where ui (0) = 0, x = (x1 , x2 , x3 ), x = (x1 , x2 , x3 ).

(1)

50

E. Pasternak, H.-B. Mühlhaus, and A.V. Dyskin

The underlying assumption behind the equation (1) is that the displacements within the representative volume can be represented by Taylor expansions around point P. For separation of scales to hold, the size of the volume element h must be much smaller than the macro-volume characteristic dimension L and much bigger than the microstructure size a, thus the continuum approximation is a double asymptotic as h/L → 0, a/h → 0. As a result of this asymptotic transition, we have a continuum that permits the usual description based on the established rules of differential geometry. The only diﬀerence from a conventional continuum is that each point may have additional DOF namely the higher order coeﬃcients in (1) enabling the consideration of deviations of the deformation from the mean values within a representative volume element. The term ui, j provides us with nine micro-distortion components: three microrotations and six micro-strains. If for simplicity we take into account only the antisymmetric part, we arrive at a continuum with 6 DOF (three translational DOF represented by the macro-displacements u(xi ), and three rotational ones represented by micro-rotations ϕi ). This is the Cosserat theory or the theory of micro-polar elasticity [8, 12, 23, 33]. The rotational degrees of freedom are very often referred to as the Cosserat rotations giving tribute to the brothers Cosserat who were the first to propose such a theory. Further generalizations can be obtained by including the symmetric part of the microgradients into the model as well and/or by keeping the next term of the Taylor expansion ui, jk bringing the total number of DOFs to 36 [22]. It should be emphasized that the micro-deformations in the expansion (1) are independent in general from the macroscopic deformation gradient. The relationship between the macro- and the micro deformation is established by means of additional constitutive relationships. The higher-order gradient theories necessitate the introduction of additional stress tensors which are conjugate to the additional deformation measures (e. g., couple or moment stresses in the Cosserat type theories; double forces tensor in the Mindlin continuum, etc.). In the elasticity theories, these new stress tensors can normally be obtained by diﬀerentiating the variation of an elastic potential (the elastic energy density) with respect to the variation of the deformation measures. The equations of equilibrium or motion also have to be obtained for additional stress factors in the higher-order gradient theories. It should also be mentioned that the formulation of boundary value problems maybe in terms of displacements complemented by the additional DOFs (for instance, rotations in the Cosserat theory) or in terms of the stress tensors complemented by the conjugates of the additional DOFs (for instance, couple/moment stresses in the Cosserat theory) or in a mixed form. The requirement that the governing equations are translational and rotational invariant yields exactly 2 × 3 balance equations. Translational invariance requires the consideration of linear momentum; rotational invariance requires the consideration of angular momentum. The equations of motion in the standard theories result from the momentum balance, while the moment of momentum balance gives the

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

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symmetry of stress tensor. In the Cosserat theories both momentum balance and moment of momentum balance are used, each contributing three equations. Mindlin [22] used the Hamilton’s principle for independent variations, which were his 12 DOF, and obtained twelve equations of motion from the variational equation of motion. However, this approach is applicable only in elastic materials. An application of the method of virtual power for derivation of the balance equations of micropolar and second gradient continua was discussed in [14, 15, 21]. The second approach (homogenization by integral transformation) involves the introduction of a non-local (integral) constitutive law (to account for long-range interactions between the particles [18]). This in general means that the stress components depend on the strains at all points of the continuum albeit with weight decreasing with distance from the point of interest. In essence, this approach shifts the procedure of homogenization from the definition of deformation measures (by introducing the volume element) to the constitutive relationships. Both approaches can be combined leading to non-local theories with additional degrees of freedom. Non-local homogenization strategies for periodic discrete materials were introduced in [11, 16–20]. The homogenization was performed by trigonometric interpolation of the discrete field of displacements and rotations of the particles. In those theories the particle centers are assumed to be situated at the nodes of a regular grid. This leads to non-local stress-deformation relationships reflecting the fact that the values of interpolating polynomial at a point are sensitive to the values at the other points. The kernels of the non-local relationship are expressed through the Kunin’s analogue of the Dirac-delta function which “remembers” the microstructure size. Specifically, in the case of a three-DOF continuum this homogenization procedure leads to a non-local continuum (with the same three DOF), the non-local stressstrain relationship and the non-local stresses satisfying the conventional equations of equilibrium or motion. In the non-local continua the stresses and couple or moment stresses become pseudo- stresses as they no longer refer to an elementary area, but to a finite volume. This obviously contradicts the Cauchy-Euler principle. The question arises why the equations of equilibrium should necessarily hold in their “conventional”, local form when the interaction between the parts of the body is not of the surface nature; it is transmitted not only through the surface, but through the volume. Granular materials are traditionally the ones where the rotational degrees of freedom are considered. Depending on the packing density, granular materials can behave like solids or like fluids. We concentrate on dense packings when granular material behaves solid-like. (For a discussion of the fluid-like regime see [27].) We concentrate on elastic models for simplicity since the emphasis is on the homogenization procedure and not on constitutive details. In the continuum approach the equations of motion are derived for a volume element, governing equations describing constitutive behavior are formulated by using the continuum stress-strain concepts. Continuum models can be classified as phenomenological and microstructural. Microstructural continuum modeling was extensively developed over the past few years as an alternative or a strategy to

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provide constraints for phenomenological constitutive models. The benefit of the microstructural approach is that it results in rational estimates of the model parameters. The first simple micro-polar (Cosserat) type theories for random packing granulates were developed in [3, 26, 31]. Both stresses and moment stresses were introduced, but the contact particle interaction was less sophisticated than in the more recently developed theories [13]. For example, moment stresses were attributed to the tangential component of the contact force and only resistance to the relative particle displacements was introduced into the contact relations [26].

3 Homogenization of 1D Structures In order to demonstrate how additional rotational degrees of freedom arise naturally from mathematical homogenization of a discrete system and compare the homogenization methods Pasternak and Mühlhaus [37] considered a simple periodic discrete model of identical spheres connected to each other by both rotational and translational springs. It is assumed that the chains do not interact with each other. The grains in a chain are connected by translational shear springs of stiﬀness k and rotational springs of stiﬀness kϕ , r is the sphere radius. The grains in neighboring chains are not connected and move independently. The potential (elastic) energy of a single chain in the system reads U1 =

2 1 1 a k (ui − ui−1 ) + (ϕi + ϕi−1 ) + kϕ (ϕi − ϕi−1 )2 2 i 2 2 i

with the potential energy density referred to the sphere i being: 2 1 a 2 Wi = k (u3i − u3i−1 ) + (ϕ2i + ϕ2i−1 ) + kϕ (ϕ2i − ϕ2i−1 ) . 2ηa3 2

(2)

(3)

Here a is the spacing of the mass centers of neighboring spheres, and a−2 η−2 is the number of chains per unit area of cross-section. We note that the rotational springs are important in this arrangement, since neglecting the resistance to rotation (kϕ → 0) makes the energy lose its positive definiteness. The equations of motion for the spheres are: a m¨u3i − k(u3i+1 − 2u3i + u3i−1 ) − k (ϕ2i+1 − ϕ2i−1 ) = q3i , 2

(4)

a a2 J ϕ¨ 2i + k (u3i+1 − u3i−1 ) + k (ϕ2i+1 + 2ϕ2i + ϕ2i−1 )− 2 4 −kϕ (ϕ2i+1 − 2ϕ2i + ϕ2i−1 ) = M2i ,

(5)

where u3i, ϕ2i are the independent Lagrange coordinates, q3i and M2i are applied load and moment at sphere i respectively, J = 25 mr2 is the inertia moment of the sphere.

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3.1 Homogenization by Diﬀerential Expansion We replace the finite diﬀerence expressions in (3) with corresponding diﬀerential expressions. Truncation of the Taylor expansions in a after the second order terms gives ⎧ 2 2 ⎫ ⎪ ⎪ ⎪ 1 ⎪ ∂u3 ∂ϕ2 ⎨ ∂u3 2 2 2 2⎬ W(x1 ) = k a + 2ka ϕ a + ka ϕ + k a . (6) ⎪ ⎪ 2 ϕ 2 ⎪ ⎪ 3 ⎩ ⎭ ∂x1 ∂x1 ∂x1 2ηa We introduce the Cosserat deformation measures, viz. strains and curvatures, γ13 =

∂u3 + ϕ2 , ∂x1

κ12 =

∂ϕ2 , ∂x1

(7)

Diﬀerentiation of the energy density with respect to these measures gives σ13 = k (ηa)−1 γ13 ,

μ12 = kϕ (ηa)−1 κ12 .

(8)

Introduction of body force f3 and moment m2 and consideration of momentum and angular momentum equilibrium yield ∂σ13 ∂μ12 + ρ f3 = ρ¨u3 , − σ13 + ρm2 = J˜ϕ¨ 2 , (9) ∂x1 ∂x1 where ρ = m a3 η is the density, J˜ = J a3 η is rotational inertia per unit volume. Equations (7)–(9) formally represent a 1D Cosserat continuum whose points have two degrees of freedom represented by the displacement,u3 and independent (Cosserat) rotation, ϕ2 . When all the chains are put together, these equations describe a 3D orthotropic Cosserat continuum (all other components of stress and moment stress tensors and corresponding deformation measures are zero). Formally, these equations can be interpreted as the governing equations of a Timoshenko beam [25] if kϕ a is interpreted as the bending stiﬀness, k/a as the shear modulus, ϕ2 as the rotation of the beam cross-section and u3 as the displacement of its neutral fiber. After substituting constitutive Eqs. (8) into equations of motion (9) we finally obtain the Cosserat equations of motion (the Cosserat equivalent of Lamé equations): ⎛ ⎞ k ⎜⎜⎜ ∂2 u3 ∂ϕ2 ⎟⎟⎟ ⎜⎝ 2 + ⎟⎠ + ρ f3 = ρ¨u3 , ηa ∂x1 ∂x1 (10) ⎛ ⎞ ⎟ 1 ⎜⎜⎜ ∂2 ϕ2 ∂u J ⎜⎝kϕ 2 − k 3 − kϕ2 ⎟⎟⎟⎠ + ρm2 = 3 ϕ¨ 2 . ηa ∂x1 aη ∂x1 We compare these equations with the result of direct homogenization of discrete equations of motion (4). Following [30], these equations can formally be written in a homogenized (continuous) form by replacing the discrete coordinate with a continuous discrete coordinate ai → x introducing continuous functions u3 (x), ϕ2 (x)

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which coincide with u3i and ϕ2i at discrete points ai and assume some values in between. Then by replacing the finite diﬀerences with the Taylor series expansions one can see that the Cosserat theory gives exact terms to the first order in a (see [37] for details). One could anticipate that the terms of order higher than a would be captured by higher order theories. The resolution of the theory is a, i. e. all the “events” smaller than a are not seen (recognized) by the Cosserat continuum. This Cosserat theory has also another length parameter given by the square root of the ratio of the stiﬀness of the rotational spring kϕ to the translational spring stiﬀness k. It follows from the above analysis that the Cosserat equations of motion can be obtained either by the direct homogenization of the discrete equations of motion or by homogenization by diﬀerential expansions of the potential energy density of the discrete system provided that the same order of approximation is maintained in both cases. The homogenization by diﬀerential expansions allows one to formulate the appropriate continuum description of the discrete system without explicitly writing the corresponding discrete system.

3.2 Homogenization by Integral Transformation (Non-local Cosserat Continuum) Homogenization by integral transformations for discrete periodical structures is based on the trigonometric interpolation [18]. For the material consisting of independent periodical chains of grains we have: u u3 (x1 ) 3i =a δ (x − ia) , (11) ϕ2 (x1 ) ϕ2i K 1 i

u3i u (x ) = δK (x1 − ia) 3 1 dx1 , ϕ2i ϕ2 (x1 )

δK (x) = (πx)−1 sin

πx . a

(12)

The application of (11) to the discrete equations of motion (4) yields the nonlocal equations of motion (non-local Lamé equations): +∞ k 2δK (x1 − y1 ) − δK (x1 − y1 − a) − δK (x1 − y1 + a) u3 (y1 )dy1 + (13) −∞ a +∞ +k δK (x1 − y1 − a) − δK (x1 − y1 + a) ϕ2 (y1 )dy1 = q3 (x1 ) − m¨u3 (x1 ) , 2 −∞ a +∞ M2 (x1 ) − J ϕ¨ 2 (x1 ) = k δK (x1 − y1 + a) − δK (x1 − y1 − a) u3 (y1 )dy1 + 2 −∞ a2 +∞ +k δK (x1 − y1 + a) + 2δK (x1 − y1 ) + δK (x1 − y1 − a) ϕ2 (y1 )dy1 + (14) 4 −∞ +∞ +kϕ 2δK (x1 − y1 ) − δK (x1 − y1 + a) − δK (x1 − y1 − a) ϕ2 (y1 )dy1 . −∞

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

55

In order to obtain the constitutive relationship we need the non-local representation of the elastic energy of the chain. Applying (11) to (2) assuming that both displacement and rotation functions and their derivatives decay strongly enough at infinity to make the non-integral terms zero, leads to the following: k U1 = [2C(x − y) − C(x − y − a) − C(x − y + a)]γ13 (x)γ13 (y)dxdy+ 2a k + [2K(x − y) − K(x − y − a) − K(x − y + a)]κ12 (x)γ13 (y)dxdy+ a k + [2K1 (x − y) − K1 (x − y − a) − K1 (x − y + a)]κ12 (x)κ12 (y)dxdy+ 2a k + [C(x − y − a) − C(x − y + a)]γ13 (x)κ12 (y)dxdy+ (15) 2 k + [K(x − y − a) − K(x − y + a)]κ12 (x)κ12 (y)dxdy 2 ka + [2C(x − y) + C(x − y − a) + C(x − y + a)]κ12 (x)κ12 (y)dxdy+ 8 kϕ + [2C(x − y) − C(x − y − a) − C(x − y + a)]κ12 (x)κ12 (y)dxdy , 2a

where C (x) = −δK (x), K (x) = C(x), K1 (x) = −C(x) and the definitions of deformation measures (7) are used. Variation of the energy of the whole body U=N 1 N 2 U 1 , where N 1 , N 2 are numbers of chains in the directions x2 and x3 respectively is +∞ +∞ 2 2 ηa N1 N2 δW(x1 )dx1 = ηa N1 N2 (σδε + μδκ)dx1 , (16) −∞

−∞

where W(x1 ,x2 ,x3 ) is the specific potential energy at point x=(x1 ,x2 ,x3 ), and h1/2 a is the spacing between non-interacting chains of the spatial assembly. Since dU=N 1 N 2 dU 1 the variation of U 1 =U 1 (g13 , k12 ) in (15) with subsequent extraction of dW in (16) and the introduction of the stress and moment stress σ13 (x) =

δW , δγ13 (x)

μ12 (x) =

δW δκ12 (x)

(17)

yield the following expressions for the stress and the moment stress: +∞ E σ13 (x1 ) = √ 2 { [2C(x1 − y) − C(x1 − y − a) − C(x1 − y + a)]γ13 (y)dy− ( +∞ηa) −∞ (18) − [2K(x1 − y) − K(x1 − y − a) − K(x1 − y + a)]κ12 (y)dy+ −∞ a +∞ k + [C(x1 − y − a) − C(x1 − y + a)]κ12 (y)dy} , E = , 2 −∞ a

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1 Ea +∞ μ12 (x1 ) = √ 2 { [C(x1 − y + a) − C(x1 − y − a)]γ13 (y)dy+ ( ηa) 2 −∞ a2 +∞ +E [2C(x1 − y) + C(x1 − y − a) + C(x1 − y + a)]Ω12 (y)dy+ 4 −∞ +∞ +Eϕ [2C(x1 − y) − C(x1 − y − a) − C(x1 − y + a)]Ω12 (y)dy+ −∞ +∞ +E [2K(x1 − y) − K(x1 − y + a) − K(x1 − y − a)]γ13 (y)dy+ −∞ +∞ +E [2K 1 (x1 − y) − K1 (x1 − y + a) − K1 (x1 − y − a)]Ω12 (y)dy+ −∞ +∞ kϕ +Ea [K(x1 − y − a) − K(x1 − y + a)]Ω12 (y)dy} , E ϕ = . a −∞

(19)

The homogenization by integral transformations produces non-local constitutive relationships with oscillating kernels. The origin of this particular type of nonlocality is in the fact that the interpolation function for a given set of u3i , ϕ2i is unique, and hence the alteration of any local value leads to the change of the whole function. Integrating the non-local “Lamé equations” (13) by parts, extracting the expressions (18), (19) and accounting for volume forces and moments yields the following Euler-Lagrange equations: dσ13 (x1 ) q3 (x1 ) m + 3 = 3 u¨ 3 (x1 ) , dx1 a η a η

(20)

dμ12 (x1 ) M2 (x1 ) J − σ13 (x1 ) + = 3 ϕ¨ 2 (x1 ) . dx1 a3 η a η

(21)

The form of the angular momentum balance is standard and consistent with its Cosserat counterpart (cf. (8)). Stresses are interpreted conventionally. However the constitutive relationships are non-local i. e. determined by the deformations of all parts of a chain. It is important that as shown in [37] the kernels in non-local constitutive equations (18), (19) are invertible. Furthermore, the homogenization by integral transformation gives a continuous representation which is isomorphic to the original discrete system. In other words, the solution of non-local equations (13) coincides with the solution of the discrete equations at the nodes ai. It should be noted that while Kunin’s non-local model does give correct results at the nodes, the continuation between the nodes it produces may be devoid of physical meaning. This was demonstrated in [37] through a 1D model of a vertical duct and showed that while the non-local model gives the downwards displacements at the nodal point coinciding with the exact discrete solution, there are values of parameters when in between the nodes the homogenized solution produces unphysical upward displacements. This is obviously an artifact of polynomial interpolation employed in Kunin’s method which can produce oscillatory polynomials.

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

57

(Interestingly, this artifact does not even require high degree polynomials where the oscillatory behavior is mostly expected. Indeed, the example considered in [37] consisted of only 3 nodes.) Furthermore it was demonstrated that opposite to the non-local model, the Cosserat local model obtained by diﬀerential expansions gives relatively accurate and physical results. An important remark should be made about the kernels in the non-local constitutive equations. In general, the choice of the kernels is based on purely mechanistic or phenomenological considerations [9, 10] material symmetry combined with a choice of the size of the domain of influence. These requirements do not suﬃciently constrain the possible variety of kernel forms, especially in 3D, which poses considerable diﬃculties in determining possible kernel forms from experiments. The common feature however that the form of the kernels traditionally chosen, like the bell-shaped, makes the kernels not invertible, which precludes isomorphism between stress and deformation measures even in the linear elastic case. This reduces the non-local theories to the stress singularity removals whereby the non-local operators on strains are used as substitutes for local strains in the damage loading function [6] leading to non-local damage models [37]. Contrary to this the homogenization by integral transformations produces invertible kernels. This is a result of oscillatory behavior of some kernels, which distinguishes them from the traditional non-invertible bell-shaped Gaussian kernels often used in the literature (e. g., [10]). It is interesting that, as shown in [37], the oscillating behavior of the kernels does not disappear after adding some randomness to the chain and describing the behavior of the chain in terms of the first and second statistic moments.

3.3 Harmonic Waves in 1D Structures In order to further compare the homogenization methods Pasternak and Mühlhaus [37] considered propagation of harmonic waves through these 1D chains in both exact discrete model (and the isomorphic non-local model Cosserat) and the Cosserat approximation. In the exact model two types of waves were identified: rotationalshear and shear-rotational waves. The first one has the wave velocity with asymptotics v p ∼ ξ−1 where ξ is the wave number, while the second one has asymptotics ξ→0

v p ∼ ξ. These are asymptotics of the same type as obtained in [30]. Both waves ξ→0

involve a mixture of displacement and rotation. The first wave becomes pure rotational wave at the long wave limit ξ → 0, while the second one become pure shear wave. This justifies their designation as rotational-shear and shear-rotational waves. The same waves are present in the Cosserat approximation, but the wave velocity becomes noticeably diﬀerent from the ones obtained through the exact (discrete) solution as soon as the wave length becomes comparable with the distance between the ball centers ξa 1. We note that such wave lengths are beyond the resolution of a continuum. These considerations bring us to the conclusion that the Cosserat continuum provides a reasonable compromise between the technical diﬃculty to deal with and the accuracy of approximation.

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4 Homogenization by Diﬀerential Expansions in 3D Pasternak et al [38] considered a three-dimensional assembly of identical spherical grains with diameter D, in permanent contact with each other. Interaction between each pair of neighboring particles is represented by the total contact force F and contact moment M. The contact moment reflects either bending resistance of the binder, or in the absence thereof the fact that the real particles contact over a certain area such that relative particle rotations create non-symmetric distributions of contact forces. It is supposed that the contact force and the moment are linearly dependent upon the relative displacement Δu and rotation Δϕ between the neighboring particles respectively: F = KΔu , M = LΔϕ , (22) K = [K i j ] , L = [Li j ] ,

Ki j = (kn − k s ) ni n j + k s δi j , Li j = kϕn − kϕs ni n j + kϕs δi j .

(23)

Here K, L are the matrices of the translational and rotational spring stiﬀnesses, kn , k s , kϕn , kϕs are the normal and shear (tangential) contact stiﬀnesses of the translational and rotational springs respectively. The indices in (23) refer to a spatially fixed Cartesian coordinate system. Assuming that the particle arrangements are statistically homogeneous and applying the method of homogenization by diﬀerential expansions one obtains the following state and constitutive equations σ ji, j = ρ¨ui ,

μ ji, j + εi jk σ jk = ρ

σ ji = Ci jlm γlm + Cl j γli ,

D2 ϕ¨ i , 10

μ ji = Di jlm κlm + Dl j κli .

(24) (25)

Here σi j , μi j are non-symmetric stresses and moment stresses respectively, γi j , κi j are the classical Cosserat continuum deformation measures - strains and curvature twists (7) and ϕi is the Cosserat rotation. The parameters of the constitutive relationships (25), the elastic moduli Ci jlm , Cl j , Di jlm , Dl j have the form 6ν s 6ν s (kn − k s ) Ai jlm , Cl j = k s Al j , πD πD 6ν s 6ν s Di jlm = kϕn − kϕs Ai jlm , Dl j = k ϕ Al j , πD πD s k Al j = Anl n j dn = δl j , 6 α/2 k Ai jlm = Ani n j nl nm dn = {δi j δlm + δil δ jm + δim δ jl } . 30 α/2 Ci jlm =

(26)

(27)

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

59

Here for isotropic distribution of particle contacts A(r, n) = k/4π, k is the coordination number of contacts, αis the spherical angle, dn = sin θdθdϕ in a spherical coordinate frame (r, ϕ, θ) with the origin at the sphere center. Pasternak et al [38] considered an example of a simple shear of infinite strip made of Cosserat material with constitutive relations (25)–(27). Simple shear was applied by prescribing longitudinal displacements at the strip boundaries, while rotations were assumed to be constrained at the boundaries. If the strip were made of a classical elastic material, the shear strain would be uniform through the layer thickness. The presence of independent microrotations however leads to non-uniformity of the displacement gradients whose distribution peaks at the middle axis of the strip. The importance of this distribution is in the fact that in experiments and observations, the strain measurements are often reduced to measurements of displacement gradient. In that case the eﬀect of microrotations can be confused with strain localization typical for material instability. A similar eﬀect of displacement gradient having maximum at the strip middle axis has been found in [35] in a 1D chain representing a section of the layer when instead of particles the layer is made out of blocks which resist rotation with their shapes. Under pure shear the section forms three parts: the central part is in the classical regime (the block rotations get suppressed by the resistance eﬀected by the block shapes), while the rotations are confined to the peripheral parts.

5 Cosserat Model of Layered Materials with Sliding Layers and Stress Concentrations A specific type of microstructure – layered material, especially with possibility of layer sliding was considered at length in [1, 24, 42]. In this 2D modeling the role of the Cosserat rotation (only one rotation in this case) was played by the rotation of the neutral axis of the layer (the deflection gradient), while the moment stresses were the bending moment per unit area in the cross-sections of the layer. The model of layered materials was then modified in [25, 28, 29] by introducing a diﬀerent rotation measure - the relative deformation gradient. An interesting behavior is exhibited by layered materials whose layers are not glued and can slide freely. This situation was considered in [36]. The independent Cosserat rotation is represented by the gradient of deflection which is independent of the vertical (normal to the layers) displacement since the latter comprises both deflection and the displacement caused by deformation of the material of the layers. Essentially, as follows from Sect. 2.1, deflection is a microscopic quantity referred to the deformation of separate layers. In the macroscopic description, deflection as such is not present. Instead it is represented by the Cosserat rotation. The moment stress corresponds to the bending moment per unit area in the layer cross-section. The role of characteristic (internal) length scale in this case is played by the layer thickness; this is the only length scale in the Cosserat image of the original problem.

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The constitutive equations for this 2D model in the plane of bending (x-axis directed along the layer, the y-axis is normal to the layer) have the form σ xx = A11 γ xx + A12 γyy , σyx = G (γyx + γ xy ) , A11 = A22 =

σyy = A12 γ xx + A22 γyy ,

σxy = G (γyx + γ xy ) + G (γ xy ) ,

μ xz = Bκ xz ,

(28)

(1 − ν) E νE Eb2 , A12 = , B= , (1 + ν) (1 − 2ν) (1 + ν) (1 − 2ν) 12 1 − ν2

where E, ν are the Young’s modulus, and Poison’s ratio of the layer, G = G = E/2(1 + ν), b is the layer thickness, and for the case of sliding layers with full contact, G = 0. The vanishing shear modulus, G = 0, is the consequence of the ability of layers to slide freely. Its eﬀect manifests itself in the non-standard stress concentrations created by defects such as dislocations, disclinations and cracks. Two edge dislocations (for the x-dislocation, the Burgers vector is parallel to x-axis and for the y-dislocation, the Burgers vector is parallel to y-axis) and the disclination (the rotation vector is perpendicular to the xy plane) can exist in the considered 2D medium. Since the introduced Cosserat continuum only presents a homogenized model of the layered material, distances smaller than the volume element size (which is supposed to be much larger than the layer thickness) cannot be distinguished. Hence, only the asymptotic solutions corresponding to the distances from the dislocation tip larger than the layer thickness can be relevant to the original discontinuous material. This implies that the stress concentration produced by a defect should be calculated as a result of the following double asymptotics (Dyskin and Pasternak [7]). Suppose a stress (or moment stress) component σ(r, h)is known at a fixed distance r from the dislocation (or disclination) tip in a certain direction. As the continuum presumes that the layer thickness h is much smaller than the size of the representative volume element, it is equivalent to the asymptotics h/r → 0. (Technically it might be convenient to consider the equivalent asymptotics r/h → ∞ instead.) After that, in the obtained asymptotic solution, the singularity at the dislocation (or disclination) tip can be extracted as the asymptotics r → 0. This is what was performed in [36] and the results are as follows. Due to layer sliding, the dislocation with the Burgers vector parallel to the x-axis (along the layers) does not produce any stresses. The dislocation with the Burgers vector parallel to the y-axis (normal to the layers) produces the following stress singularities on the y-axis: 5/2 Eb1/2 sgny b σdisloc (0, y) = − + O , (29) √ xy 5/2 2 1/4 |y| 8 1−ν 2π3 3 |y| σdisloc (0, y) = yy

Eδ(y) , 2(1 − ν2 )

all other stress components being zero on the y-axis.

(30)

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

61

For the disclination we have σdisclin (0, y) yy

3/2 Eb1/2 1 b =− , +O √ 2 1/4 |y|3/2 4 1−ν 2π3 |y|

(31)

3/2 Eb3/2 31/4 sgny b =− , √ +O 2 |y|3/2 24 1 − ν 2π |y|

(32)

μdisclin (0, y) xz

all other stress components being zero on the y-axis. It is seen that the dislocation and disclination produce non-traditional square root stress singularities, while in a conventional material the dislocation would produce a singularity inversely proportional to the distance from the tip. This unconventional type of singularity is clearly the consequence of free sliding.

6 Path-Independent Integrals in Cosserat Continuum We consider here, following [36], a generalization of path-independent integrals that accounts for rotational degrees of freedom. In Cosserat continuum the elastic energy per unit volume W is a function of the so-called relative deformation gi j (see below for definition) and the gradient of the independent rotations ki j [23, 33]. The position of material points is described by ξm , xi (the reference and current configurations respectfully), where xi = xi (ξm ), ξ = x(t = 0). The variations of the energy density, the displacements and the Cosserat rotations due to the variations of the reference and current coordinates ξm → ξm + δξm , xi → xi + δxi , δxi = δ x xi + xi,m δξm read δW = δ x W + W,m δξm , δui = δ x ui + ui,m δξm , δϕi = δ x ϕi + ϕi,m δξm . Here the subscripted x in δ x xi means that the variation is performed with ξ fixed, and (.),m = ∂/∂ξm . The energy variation of the body occupying a volume V is obtained as: δE = δ W(γ ji , κ ji )dvτ = (δ x W + (Wδξm ),m )dvτ . (33) Vτ

Vτ

The insertion of the stress and moment stress tensors s ji =∂W/∂g ji , m ji =∂W/∂k ji , as well as taking into account that the derivatives of the energy density over the relative deformation and curvature tensors (7) and the moment and angular moment static equilibrium equations (9), (33) yield: δE = {(Wδξm ),m + (−(σ ji ui,m + μ ji ϕi,m )δξm + σ ji δui + μ ji δϕi ), j }dvτ − Vτ (34) − ( fi ui,m + mi ϕi,m )δξm dvτ + ( fi δui + mi δϕi )dvτ . Vτ

Vτ

Application of the Gauss theorem under the assumption that the fields are divergence free within V, i. e. no singularities of the solutions within V, yields:

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E. Pasternak, H.-B. Mühlhaus, and A.V. Dyskin

δE =

S

− V

{(Wnm − T iui,m − Mi ϕi,m )δξm + T i δui + Mi δϕi }ds− ( fi ui,m + mi ϕi,m )δξm dvτ + ( fi δui + mi δϕi )dv ,

(35)

V

where S is an arbitrary closed surface in V (or closed contour in 2D), which for instance can comprise parts of or the whole boundary of the volume V; T i = σ ji n j and Mi = μ ji n j are the stress vector (traction) and moment vector (torsion) respectively. The last two terms in equation (35) represent the work of the externally applied surface forces and moments. We shall be looking for sets of infinitesimal variations δξm , δui, δϕi that do not change the elastic energy, i. e. δE = 0 . (36) The described variational principles can be used to derive path independent integrals which are generalizations of Rice-Cherepanov path-independent integrals. We use the independence of strain energy with respect to rigid body translations and, in the isotropic case, to rigid body rotations. The strain energy has to be invariant with respect to rigid body translations of the body as a whole, i. e. the translation of the origin of the coordinate system must leave the energy unchanged. In this case the components δξm are independent, arbitrary constants and δξi = 0. From (35), (36) it follows that: δE = Πm = (Wnm − T iui,m − Mi ϕi,m )dsτ − ( fi ui,m + mi ϕi,m )dvτ = 0 . (37) δξm Sτ Vτ In the derivation of (37) we have assumed that the volume under consideration is divergence free, i. e. free of singularities. If there is a singularity within V then, in general, the vector Πm 0, whereby the value of the components are independent of the extent of the volume surrounding the singularity. Hence the integrals Πm = (Wnm − T i ui,m − Mi ϕi,m )dΓ − ( fi ui,m + mi ϕi,m )dvΓ (38) Γ

VΓ

are invariant with respect to the choice of volume V. This is a generalization of the correspondent result for classical continuum. If the body forces and moments are absent we arrive at the 3D Cosserat analogue to the well known path independent J-integral of the standard continuum. Since δE = Πm δξm we may interpret Πm as the force acting on the singularity or defect contained within Γ and δE = Πm δξm is the virtual work associated with a translation of the defect by δui . The moment and angular moment equilibrium conditions are recovered if we assume in (35), (36) that δui and δϕi are arbitrary, independent constants and δξi = 0. For isotropic materials [36] showed that the deformation energy also has to be invariant with respect to rigid body rotations. The use of the obtained path-independent integrals will be illustrated on the example of a crack of rotational mode in a Cosserat continuum modeling layered materials with sliding layers, Sect. 5.

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

63

As the Cosserat continuum is characterized by the presence of three additional degrees of freedom corresponding to three components of independent Cosserat rotation, three more crack modes (Modes IV – VI) can exist. These cracks can be envisaged as discontinuities in the corresponding components of the Cosserat rotation and modeled as distributions of disclinations (or disclination loops for 3D cracks). In the 2D case, only one rotational degree of freedom exists corresponding to rotations about the z-axis, which is associated with bending of the layers. Furthermore, as the disclination does not create shear stresses along the axis perpendicular to the direction of layering, we can consider a crack of Mode VI along the y-axis. This crack is not coupled with and is independent of the classical crack modes. It is located along the y-axis and is a line filled with disclinations. As the disclination is a discontinuity in the Cosserat rotation representing layer bending, this crack is called the bending crack. For the layered material with a semi-infinite pure bending crack at x = 0, y > 0 whose faces are loaded by moment stress μ0xz (y) only (free of conventional stresses) the following integral equation can be written [36]: ∞ sgn(y − t)Δϕz (t) Eb3/2 31/4 C dt = μ0xz (y) , y ≥ 0 , C = − (39) √ , 0 |y − t| 24 1 − ν2 2π where Δϕz is the disclination density. For the moment stresses on the crack continuation (x = 0, y < 0) the finite energy eigenmode produces a singularity ϕz (y) ∼ y−3/4 ,

ϕz (y) ∼ y1/4 ,

μ xz ∼ (−y)−1/4 .

(40)

This new fracture mode is characterized by the moment stress intensity factor √ de f Mz = lim μ xz (−y) 4 −y , y→−0

μ xz ∼

Mz , (−y)1/4

y → −0 .

(41)

For a semi-infinite crack the stress intensity factor and energy release rate are [36]: √ 33/4 π2 1 − ν2 2 Mz = − , E= Mz2 . (42) 2πa3/4 4Eb3/2 Γ (1/4)2 We now use path-independent integrals (38) to obtain the criterion of bending crack growth. In the case of the absence of body forces and moments one obtains Π x = Πz = 0 and ∂u ∂v ∂ϕz Πy = − (σ xx + σ xy + μxz )n x dΓ ∂y ∂y ∂y Γ +∞ (43) ∂(ϕ+z − ϕ−z ) ∂(u+ − u− ) ∂(v+ − v− ) =− (σ xx + σ xy + μ xz )dy . ∂y ∂y ∂y 0

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Assuming a Barenblatt-type process zone (43) can be written as: lzone ∂Δu(y) ∂Δv(y) ∂Δϕz (y) Πy = − (σ xx + σ xy + μxz )dy ∂y ∂y ∂y 0 Δu∗ Δv∗ Δϕz∗ =− σ xx (Δu)dΔu − σ xy (Δv)dΔv − μ xz (Δϕz )dΔϕz , 0

0

(44)

0

where Δu = u+ − u− , Δv = v+ − v− , Δϕz = ϕ+z − ϕ−z , Δu∗ , Δv∗ and Δϕz∗ are the maximum displacements and rotation discontinuity respectively. For the bending crack in a layered material (44) assumes the form:

Δϕz∗

Πy = −

μ xz (Δϕz )dΔϕz .

(45)

0

The crack growth criterion is then obtained by comparing the energy release rate expressed in terms of the moment stress intensity factor (42) with (45): 33/4 π2 (1 − ν2 ) M 2 = Πy . 4Eb3/2 (Γ(1/4))2 z

(46)

7 Conclusions Continuum modeling of materials with discrete microstructure based on homogenization by diﬀerential expansions with account for internal rotations gives a reasonable compromise between the accuracy and technical complexity. Homogenization using integral transformation leads to non-local theories with invertible kernels. Such theories are isomorphic to the original discrete equations and thus do not provide any simplification. Furthermore, they sometimes produce unphysical solutions between the points that correspond to the original discrete system. Two particular cases are presented where the rotational degrees of freedom lead to non-trivial and unexpected results. These are particulate materials with rotating particles and layered materials with sliding layers where the layer bending is treated as a manifestation of a rotational degree of freedom. Simple shear of a strip of a Cosserat material leads to non-uniform distribution of displacement gradients with maximum at the middle axis of the strip. As the strain measurements are often based on the displacement gradients, the non-uniformity can be confused with strain localization and lead to wrong conclusions about the material instability. Dislocations, disclinations and cracks in layered materials with sliding layers produce weaker stress singularities than dislocations and cracks in conventional elastic medium. In the Cosserat continuum six path independent integrals exist which are consequence of the invariance of the deformation energy of the continuum with respect to rigid body translations and rotations of the common (reference and current) frame of reference. The translational invariance yields three path-independent P-integrals, which for vanishing internal characteristic length reduce to the familiar J-integrals. The rotational invariance (in the case of isotropy) yields an additional set

Rotational Degrees of Freedom in Modeling Materials with Intrinsic Length Scale

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of three path-independent integrals. Subsequently, the Cosserat continuum allows three additional crack modes corresponding to three additional degrees of freedom. In particular, in 2D layered materials with sliding layers the new fracture modes corresponds to the bending crack which is equivalent to a certain distribution of disclinations – discontinuities in layer bending. The criterion of propagation of bending crack can be expressed through the path-independent integral. In this modeling, the fracture energy is external to the Cosserat continuum and should be determined based on the specific microscopic fracture processes. Acknowledgements. The financial support from the Australian Research Council through the Discovery Grants DP0346148 (EP), DP0988449 (AVD, EP), DP0985662 and DP110103024 (HBM) is acknowledged.

References [1] Adhikary, D.P., Dyskin, A.V.: A Cosserat continuum model for layered materials. Computers and Geotechnics 20, 15–45 (1997) [2] Batchelor, G.K.: Transport properties of two-phase materials with random structure. In: Annual Review of Fluid Mechanics, vol. 6, pp. 227–255. California Annual Reviews Inc., Palo Alto (1974) [3] Chang, C.S., Ma, L.: Elastic material constants for isotropic granular solids with particle rotation. International Journal of Solids and Structures 29, 1001–1018 (1992) [4] Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et fils, Paris (1909); Theory of deformable bodies, NASA TT F-11 561 (1968) [5] Cundall, P.A.: A discontinuous future for numerical modeling in soil and rock. In: Cook, B.K., Jensen, R.P. (eds.) Discrete Element Methods: Numerical Modeling of Discontinua, Reston, pp. 3–4. ASCE, Virginia (2002) [6] de Borst, R., Benallal, A., Peerlings, R.H.J.: On gradient-enhanced damage theories. In: Fleck, N.A., Cocks, A.C.F. (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials, pp. 215–226. Kluwer Academic Publishers, Dordrecht (1997) [7] Dyskin, A.V., Pasternak, E.: Cracks in Cosserat continuum – Macroscopic modelling. In: Maugin, G.A., Metrikine, A.V. (eds.) Mechanics of Generalized Continua: One Hundred Years After the Cosserats. Springer series Advances in Mechanics and Mathematics, vol. 21, pp. 35–42. Springer, New York (2010) [8] Eringen, A.C.: Linear theory of micropolar elasticity. Journal of Applied Mathematics and Mechanics 15, 909–923 (1966) [9] Eringen, A.C.: Non-local polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. IV, pp. 205–264. Academic Press, New York (1976) [10] Eringen, A.C.: Non-local continuum mechanics and some application. In: Barut, A.O. (ed.) Non-linear Equations in Physics and Mathematics, pp. 271–318. D.Reidel Publishing Company, Dordrecht (1978) [11] Eringen, A.C.: Non-local continuum description of lattice dynamics and application. In: Chandra, J., Srivastav, R.P. (eds.) Constitutive Models of Deformation, pp. 59–80. SIAM, Philadelphia (1987) [12] Eringen, A.C., Kafadar, C.B.: Polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. IV, Part I pp. 4–73. Academic Press, New York (1976)

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[13] Fleck, N.A., Cocks, A.C.F. (eds.): IUTAM Symposium on Mechanics of Granular and Porous Materials. Kluwer Academic Publishers, Dordrecht (1997) [14] Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continues. Première partie. Théorie du second gradient. Journal de Mécanique 12, 235–274 (1973) [15] Germain, P.: The method of virtual power in continuum mechanics. Part 2: Microstructure. SIAM Journal on Applied Mathematics 25, 556–575 (1973) [16] Kröner, E.: The problem of non-locality in the mechanics of solids: review of the present status. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, National Bureau of Standards Special Publication 317, vol. II, pp. 729– 736. National Bureau of Standards, Washington (1970) [17] Kröner, E., Datta, B.K.: Non-local theory of elasticity for a finite inhomogeneous medium – A derivation from lattice theory. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, National Bureau of Standards Special Publication 317, vol. II, pp. 737–746. National Bureau of Standards, Washington (1970) [18] Kunin, I.A.: Elastic media with microstructure 1. One-dimensional models. Springer, Berlin (1982) [19] Kunin, I.A.: Elastic media with microstructure 11. Springer, Heidelberg (1983) [20] Kunin, I.A., Waisman, A.M.: On problems of the non-local theory of elasticity. In: Simmons, J.A., de Wit, R., Bullough, R. (eds.) Fundamental Aspects of Dislocation Theory, National Bureau of Standards Special Publication 317, vol. II, pp. 747–759. National Bureau of Standards, Washington (1970) [21] Maugin, G.A.: The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mechanica 35, 1–70 (1980) [22] Mindlin, R.D.: Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 51–78 (1964) [23] Mindlin, R.D., Tiersten, H.F.: Eﬀects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis 11, 415–448 (1962) [24] Mühlhaus, H.-B.: Continuum models for layered and blocky rock. In: Comprehensive Rock Engineering, Analysis and Design Methods, vol. II, pp. 209–230. Pergamon Press, Oxford (1993) (invited chapter) [25] Mühlhaus, H.-B.: A relative gradient model for laminated materials. In: Mühlhaus, H.B. (ed.) Continuum Models for Materials with Microstructure, ch.13, pp. 451–482. John Wiley & Sons, Chichester (1995) [26] Mühlhaus, H.-B., de Borst, R., Aifantis, E.C.: Constitutive models and numerical analyses for inelastic materials with microstructure. In: Beer, G., et al. (eds.) Computing Methods and Advances in Geomechanics, pp. 377–385 (1991) [27] Mühlhaus, H.-B., Hornby, P.: On the reality of antisymmetric stresses in fast granular flows. In: Fleck, N.A., Cocks, A.C.F. (eds.) IUTAM Symposium on Mechanics of Granular and Porous Materials, pp. 299–311. Kluwer Academic Publishers, Dordrecht (1997) [28] Mühlhaus, H.-B., Hornby, P.: A beam theory gradient continua. In: de Borst, R., van der Giessen, E. (eds.) Material Instabilities in Solids, ch.32, pp. 521–532. John Wiley & Sons, Chichester (1998) [29] Mühlhaus, H.-B., Hornby, P.: A relative gradient theory for layered materials. Journal de Physique IV France 8, 269–276 (1998) [30] Mühlhaus, H.-B., Oka, F.: Dispersion and wave propagation in discrete and continuous models for granular materials. International Journal of Solids and Structures 33, 2841– 2858 (1996)

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[31] Mühlhaus, H.-B., Vardoulakis, I.: The thickness of shear bands in granular materials. Géotechnique 37, 271–283 (1987) [32] Nowacki, W.: Dynamic problems of asymmetrical elasticity. International Applied Mechanics 6, 361–375 (1970) [33] Nowacki, W.: The linear theory of micropolar elasticity. In: Nowacki, W., Olszak, W. (eds.) Micropolar Elasticity, pp. 1–43. Springer, Wein (1974) [34] Pasternak, E., Dyskin, A.V.: Measuring of Cosserat eﬀects and reconstruction of moduli using dispersive waves. In: Maugin, G.A., Metrikine, A.V. (eds.) Mechanics of Generalized Continua: One hundred Years After the Cosserats. Springer series Advances in Mechanics and Mathematics, vol. 21, pp. 35–42. Springer, New York (2010) [35] Pasternak, E., Dyskin, A.V., Estrin, Y.: Deformations in transform faults with rotating crustal blocks. Pure and Applied Geophysics 163, 2011–2030 (2006) [36] Pasternak, E., Dyskin, A.V., Mühlhaus, H.-B.: Cracks of higher modes in Cosserat continua. International Journal of Fracture 140, 189–199 (2006) [37] Pasternak, E., Mühlhaus, H.-B.: Generalised homogenisation procedures for granular materials. Journal of Engineering Mathematics 52, 199–229 (2005) [38] Pasternak, E., Mühlhaus, H.-B., Dyskin, A.V.: On the possibility of elastic strain localisation in a fault. Pure and Applied Geophysics 161, 2309–2326 (2004) [39] Pijaudier-Cabot, G., Bazant, Z.P.: Non-local damage theory. The Journal of Engineering Mechanics 113, 1512–1533 (1987) [40] Sluys, L.J., de Borst, R., Mühlhaus, H.-B.: Wave propagation, localisation and dispersion in a gradient dependent medium. International Journal of Solids and Structures 30, 1153–1171 (1993) [41] Yoon, H.S., Katz, J.L.: Is bone a Cosserat solid? Journal of Materials Science 18, 1297– 1305 (1983) [42] Zvolinskii, N.V., Shkhinek, K.N.: Continual model of laminar elastic medium. Mechanics of Solids 19, 1–9 (1984)

Micromorphic vs. Phase-Field Approaches for Gradient Viscoplasticity and Phase Transformations Samuel Forest, Kais Ammar, and Benoît Appolaire

Abstract. Strain gradient models and generalized continua are increasingly used to introduce characteristic lengths in the mechanical behavior of materials with microstructure. On the other hand, phase-field models have proved to be eﬃcient tools to simulate microstructure evolution due to thermodynamical processes in the presence of mechanical deformation. It is shown that both methods have strong connections from the point of view of thermomechanical field theory. A general formulation of thermomechanics with additional degrees of freedom is presented that encompasses both applications as special cases. It is based on the introduction of additional power of internal forces introducing generalized stresses. The current knowledge in the formulation of physically non-linear constitutive equations is used to develop strongly coupled elastoviscoplastic material models involving phase transformation and moving boundaries.

1 Introduction There are strong links between generalized continuum mechanics and phase field models which are striving in modern field theories of materials. Mindlin’s and Casal’s second gradient model of mechanics and the Cahn-Hilliard diﬀusion theory were developed almost simultaneously. More generally, the necessity of introducing additional degrees of freedom in continuum models arose in the 1960s in order to account for microstructure eﬀects on the overall material’s response. However, generalized continuum mechanics, with paradigms like Eringen’s micromorphic model Samuel Forest · Kais Ammar Mines ParisTech, Centre des Matériaux, CNRS UMR 7633, BP 87, 91003 Evry Cedex, France e-mail: {samuel.forest,kais.ammar}@ensmp.fr Benoît Appolaire LEM, ONERA/CNRS, 29 Avenue de la Division Leclerc, BP 72, 92322 Châtillon, France e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 69–88. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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and Aifantis strain gradient plasticity, developed along an independent track from phase-field approach embodied by Khachaturyan’s views, for instance. The links have been seen recently within the context of plasticity and damage mechanics. The computational mechanics community aimed at introducing the evolution of microstructures into their simulations [2, 38] whereas physicists started introducing plasticity into the thermodynamical setting [19]. Cooperation between these communities becomes necessary when tackling damage mechanics and crack propagation simulation [6, 32]. First attempts to present a general constitutive framework encompassing classical enhanced mechanical and thermodynamical models have been proposed recently [7, 14, 30]. Such an approach is presented in this chapter and extended to sophisticated descriptions of interactions between viscoplasticity and phase transformations. The micromorphic model originates from Eringen’s introduction of microdeformation tensor at each material point that accounts for the changes of a triad of microstructure vectors. In the present chapter, the micromorphic approach denotes an extension of this theory to other variables than total deformation, namely plastic strain, hardening variables, and even temperature and concentration. The gist of the micromorphic model is to associate a microstructure quantity (e. g. microdeformation) to an overall quantity (e. g. macroscopic deformation). The deviation of the microvariable from the macrovariable and the gradient of the microvariable are sources of stored energy and dissipation. They are controlled by generalized stresses which contribute to the power of internal forces. On the other hand, the phase-field approach has proved to be an eﬃcient method to model the motion of interfaces and growth of precipitates based on a sound thermodynamical formulation including non convex free energy potentials [13]. The eﬀect of microelasticity on the morphological aspects and kinetics of phase transformation is classically studied but the occurrence of plasticity is recent [19, 20, 38]. Beyond plasticity, damage and crack propagation are the subject of both generalized continuum and phase-field approaches [6, 17, 31, 32, 39]. Phase-field simulations usually rely on finite diﬀerences or fast Fourier methods. More recently, the finite element method was also used in order to tackle more general boundary conditions [2, 32, 36]. The objective of the present chapter is to formulate a thermomechanical theory of continua with additional degrees of freedom. It is shown in a first part that the theory encompasses available generalized continuum theories and phase-field models provided that well-suited free energy and dissipation potentials are selected. The current strain gradient plasticity models are then extended to account simultaneously for plastic strain gradient and plastic strain rate gradient in order to address viscoplastic instabilities occurring in metal plasticity like dynamic strain aging. The second part of the work exposes how the well-known elastoviscoplastic constitutive framework can be incorporated into the available phase-field approach in order to investigate the coupling between viscoplasticity and phase transformation. An original approach is proposed that resorts to standard homogenization techniques used in the mechanics of heterogeneous materials.

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Intrinsic notations are used throughout this work. In particular, scalars, vectors, 2 3 4

tensors of second, third and fourth ranks are denoted by a, a, a, a, a, respectively. Contractions are written as: a · b = ai b i ,

2

2

a : b = ai j b i j ,

3.3 a .. b = ai jk bi jk

(1)

using the Einstein summation rule for repeated indices. The gradient operator ∇ is introduced as ∂ui u ⊗ ∇ = ui, j ei ⊗ e j , with ui, j = , (2) ∂x j where (ei )i=1,2,3 is a Cartesian orthonormal basis. For the sake of brevity, the analysis is limited to the small deformation framework throughout this work. Also most situations are considered under isothermal conditions.

2 Thermomechanics with Additional Degrees of Freedom 2.1 General Setting The displacement variables of mechanics can be complemented by additional degrees of freedom (DOF), φ, that can be scalars as well as tensor variables of given rank: DOF = {u,

φ} .

(3)

A first gradient theory is built on the basis of this set of degrees of freedom: 2

S T RAIN = {ε,

φ,

∇φ} .

(4)

2

The strain tensor, ε, is the symmetric part of the gradient of the displacement field. The main assumption of the proposed theory is that the gradient of the additional degrees of freedom contribute to the work of internal forces in the energy equation, in contrast to internal variables and concentration in diﬀusion theory. Depending on the invariance properties of the variable φ, it can itself contribute to the work of internal forces together with its gradient. It is not the case for the displacement itself which is not an objective vector. The virtual power of internal forces is then extended to the virtual power done by the additional variable and its first gradient: P(i) (u˙ , φ˙ ) = − p(i) (u˙ , φ˙ ) dV , D

2

p(i) (u˙ , φ˙ ) = σ : ∇u˙ + aφ˙ + b · ∇φ˙ ,

(5)

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where D is a subdomain of the current configuration Ω of the body. Stars denote 2

virtual fields. The Cauchy stress tensor is σ and a and b are generalized stresses associated with the additional DOF and its first gradient, respectively. Similarly, the power of contact forces must be extended as follows: (c) ˙ ˙ P (u , φ ) = p(c) (u˙ , φ˙ ) dV , p(c) (u˙ , φ˙ ) = t · u˙ + ac φ˙ , (6) D

where t is the traction vector and ac a generalized traction. In general, the power of forces acting at a distance must also be extended in the form: (e) ˙ P (u˙ , φ ) = p(e) (u˙ , φ˙ ) dV, p(e) (u˙ , φ˙ ) = ρf · u˙ + ae φ˙ + be · ∇φ˙ , (7) D

where ρf accounts for given simple body forces and ae for generalized volume forces. The power of inertial forces also requires, for the sake of generality, the introduction of an inertia I associated with the acceleration of the additional degrees of freedom: (a) ˙ P (u˙ , φ ) = p(a) (u˙ , φ˙ ) dV , p(a) (u˙ , φ˙ ) = −ρu¨ · u˙ − I φ¨ φ˙ . (8) D

Following [21], given body couples and double forces working with the gradient of the velocity field, could also be introduced in the theory. The generalized principle of virtual power with respect to the velocity and additional DOF, is formulated as ˙ (9) P(i) (u˙ , φ˙ ) + P(e) (u˙ , φ˙ ) + P(c) (u˙ , φ˙ ) + P(a) (u˙ , φ˙ ) = 0 , ∀D ⊂ Ω, ∀u˙ , φ. The method of virtual power according to [27] is used then to derive the standard local balance of momentum equation: 2

∇· σ + ρ f = ρ u¨ ,

∀x ∈ Ω

(10)

and the generalized balance of micromorphic momentum equation: ∇· (b − be ) − a + ae = I φ¨ ,

∀x ∈ Ω .

(11)

The method also delivers the associated boundary conditions for the simple and generalized tractions: 2

t = σ · n,

ac = (b − be ) · n ,

∀x ∈ ∂D .

(12)

The local balance of energy is also enhanced by the generalized power already included in the power of internal forces (5): ρ˙ = p(i) − ∇· q + ρr ,

(13)

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73

where is the specific internal energy, q the heat flux vector and r denotes external heat sources. The entropy principle takes the usual local form: −ρ(ψ˙ + ηT˙ ) + p(i) −

q · ∇T ≥ 0 , T

(14)

where it is assumed that the entropy production vector is still equal to the heat vector divided by temperature, as in classical thermomechanics. Again, the enhancement of the theory goes through the enriched power density of internal forces (5). The entropy principle is exploited according to classical continuum thermodynamics to derive the state laws. At this stage it is necessary to be more specific on the depen2

dence of the state functions ψ, η, σ, a, b on state variables and to distinguish between dissipative and non-dissipative mechanisms. The introduction of dissipative mechanisms may require an increase in the number of state variables. These diﬀerent situations are considered in the following subsections.

2.2 Micromorphic Model as a Special Case The micromorphic model as initially proposed by Eringen [12] and Mindlin [33] amounts to introducing a generally non compatible microdeformation field: 2

φ ≡ χ, 2

where χ is a generally non-symmetric second order tensor defined at each material point. When the microdeformation reduces to its skew symmetric part, the Cosserat model is retrieved [10, 16]. The microdeformation is to be compared to the deformation gradient: 2

2

e = u⊗ ∇ −χ.

(15)

2

If the internal constraint e ≡ 0 is enforced, the microdeformation coincides with the deformation and the micromorphic model reduces to Mindlin’s second gradient theory. The free energy density depends of the following state variables: 2

2

S T AT E = {ε,

e,

3

2

K := χ ⊗ ∇,

T,

α} ,

where α denotes the set of internal variables required to represent dissipative mechanical phenomena. The Clausius-Duhem inequality (14) becomes, in the isothermal case, 2

(σ − ρ

∂ψ 2

∂ε

2

2

) : ε˙ + (a − ρ

∂ψ 2

∂e

2

3

) : e˙ + (b − ρ

∂ψ .. 3˙ ∂ψ ∂ψ ) . K − (ρη + ρ )T˙ − ρ α˙ ≥ 0 , (16) 3 ∂T ∂T ∂K

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2

where a was taken as the stress conjugate to the relative deformation rate e˙ in the power of internal forces, which corresponds to an alternative form for (5). The state laws for micromorphic media are obtained by assuming that the first four contribution are non-dissipative: 2

σ=ρ

∂ψ 2

∂ε

2

,

a=ρ

∂ψ 2

∂e

∂ψ

3

,

b=ρ

,

3

η=−

∂K

∂ψ . ∂T

(17)

Elastoviscoplastic micromorphic media are then obtained by a specific choice of the internal variables α and their evolution rules [16].

2.3 Phase-Field Model as a Special Case Enhancing the mechanical power in the energy balance is plausible in the presence of microstructure induced mechanical phenomena, as proposed by Eringen. However, this is also possible in other contexts, namely when the DOF φ has a more general meaning of an order parameter. Fried and Gurtin [18, 22] suggested to consider the following reduced state space: 2

S T AT E = {ε,

φ,

∇φ,

T,

α}

(18)

η=−

∂ψ , ∂T

(19)

and the following state laws 2

σ=ρ

∂ψ 2

∂ε

,

b=ρ

∂ψ , ∂∇φ

so that, in the isothermal case, the dissipation rate reduces to av φ˙ + X α˙ ≥ 0

with av = a − ρ

∂ψ , ∂φ

X = −ρ

∂ψ . ∂α

(20)

The choice of a convex potential Ω(av , X) providing the evolution laws φ˙ =

∂Ω , ∂av

α˙ =

∂Ω ∂X

(21)

ensures the positivity of the dissipation rate. As an illustration, let us consider a quadratic contribution of av to the dissipation potential. We are led to the following relationships φ˙ =

1 v 1 ∂ψ a = (a − ρ ) , β β ∂φ

(22)

where β is a material parameter. The latter equation can be combined with the balance law (11), in the absence of volume or inertial forces, and the state law (19) to derive

Micromorphic and Phase-Field Elastoviscoplasticity

75

∂ψ ∂ψ β φ˙ = ∇· ρ −ρ , ∂∇φ ∂φ

(23)

which corresponds to a general Ginzburg-Landau equation. The authors in [38] have combined the micromorphic approach and the Cahn-Hilliard approach to diﬀusion in order to derive an alternative equation to Cahn-Hilliard.

3 Constitutive Framework for Gradient and Micromorphic Viscoplasticity We now exploit the established general structure to propose a constitutive framework for elastoviscoplastic materials exhibiting plastic strain gradient. The attention is focused on an isotropic elastoviscoplastic medium characterized by the cumulated plastic strain, p. The proposed formulation encompasses Aifantis-like strain gradient plasticity models and introduces additional strain rate gradient eﬀects. The total 2

2

2

strain is split into its elastic and plastic parts: ε = εe + ε p . In this context, the additional DOF φ has the meaning of a microplastic strain [15] to be compared with p itself. Two variants of the constitutive framework are considered which handle in a slightly diﬀerent way the dissipative contribution due to the generalized stresses.

3.1 Introduction of Viscous Generalized Stresses The free energy density is assumed to depend on the following state variables: 2

S T AT E = {εe ,

e := φ − p,

p,

K := ∇φ} .

(24)

The isothermal Clausius-Duhem inequality take the form: 2

(σ − ρ

∂ψ 2

∂εe

2

) : ε˙ e + (a − ρ

∂ψ ∂ψ ˙ 2 2 p ∂ψ ) e˙ + (b − ρ ) · K + σ : ε˙ + a p˙ − ρ α˙ ≥ 0 . (25) ∂e ∂K ∂T

The following state laws are adopted: 2

σ=ρ

∂ψ 2

∂εe

,

R=ρ

∂ψ . ∂p

(26)

To ensure the positivity of the dissipation rate associated with the generalized stress a and b, we adopt the viscoelastic constitutive equations a=ρ

∂ψ + β˙e , ∂e

b=ρ

∂ψ ˙, + κK ∂K

(27)

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where β and κ are generalized viscosity coeﬃcients. This viscoelastic formulation amounts to splitting the generalized stresses a and b into elastic (reversible) and 2

viscous parts. Regarding viscoplastic deformation, a viscoplastic potential Ω(σ, a − R) is chosen such that 2 ∂Ω ∂Ω ε˙ p = , p˙ = . (28) 2 ∂a − R ∂σ In order to evidence the kind of gradient elastoviscoplastic models we aim at, we illustrate the case of a quadratic free energy potential: ρψ = 2

1 2e 4 2e 1 1 1 ε : C : ε + R0 p + H p2 + Hφ e2 + AK · K , 2 2 2 2 4

2

σ = C : εe ,

R = R0 + H p ,

a = Hφ e + β˙e ,

˙. b = AK + κK

(29) (30)

The viscoplastic potential is based on the yield function that introduces the equivalent stress measure σeq and a threshold 2

Ω(σ, a − R) =

σeq + a − R n+1 2 p ∂σeq σeq + a − R n K , ε˙ = p˙ , p˙ = , (31) 2 n+1 K K ∂σ

where · denotes the positive part of the quantity in brackets, and K and n are usual viscosity parameters. The decomposition (27) and the generalized balance (11) become ˙ . a = Hφ (φ − p) + β(φ˙ − p) ˙ = ∇· (AK + κK) (32) We finally obtain the following linear partial diﬀerential equation, under the condition of plastic loading, in the absence of volume and inertial forces: Hφ φ − AΔφ + βφ˙ − κΔφ˙ = Hφ p + κ p˙ ,

(33)

where Δ is the Laplace operator. When the viscous parts are dropped in (27), the Helmholtz type equation used in strain gradient plasticity and damage [11, 14, 35] is retrieved. It is classically used for the regularization of strain localization phenomena. The rate dependent part in the previous equation is expected to be useful in the simulation of strain rate localization phenomena which occur for instance in strain aging materials [29]. Under plastic loading, the equivalent stress can then be decomposed into the following contributions: σeq = R − a + K p˙ 1/n = R0 + H p − AΔφ − κΔφ˙ + K p˙ 1/n .

(34)

If κ = 0, the micromorphic model is retrieved. If, furthermore, the constraint φ ≡ p is enforced, Aifantis well-known strain gradient plasticity model is recovered.

Micromorphic and Phase-Field Elastoviscoplasticity

77

3.2 Decomposition of the Generalized Strain Measures It is proposed now to consider the decomposition of the additional DOF and its gradient into elastic and plastic parts: φ = φe + φ p ,

K = Ke + K p .

(35)

The decomposition of φ itself is allowed only if it is an objective quantity. This 2

would not apply for instance for φ ≡ R, the Cosserat microrotation. But it is allowed for a strain variable [16]. Such generalized kinematic decompositions were proposed in [16] for strain gradient, Cosserat and micromorphic media, also at finite deformation. It is generalized here for more general DOFs, possibly related to physically coupled phenomena. The selected state variables then are 2

S T AT E = {εe ,

φe ,

Ke ,

p} ,

(36)

which leads to the following Clausius-Duhem inequality: 2

(σ − ρ

∂ψ 2

∂εe

2

) : ε˙ e + (a − ρ

∂ψ e ∂ψ ˙e )φ˙ + (b − ρ e ) · K ∂φe ∂K 2

2

p

˙ − R p˙ ≥ 0 . (37) + σ : ε˙ p + aφ˙ p + b · K The retained state laws are 2

σ=ρ

∂ψ 2

∂εe

,

a=ρ

∂ψ , ∂φe

b=ρ

∂ψ . ∂Ke

(38)

The residual dissipation then is 2

2 ˙ p − R p˙ ≥ 0 . σ : ε˙ p + aφ˙ p + b · K

(39)

A simple choice of dissipation potential is ma +1 mb +1 σeq + a − R n+1 Ka beq K |a| Kb Ω(σ, R, a) = + + , n+1 K ma + 1 Ka mb + 1 Kb 2

where beq is a norm of b and from which the evolution rules are derived 2 ∂σeq σeq + a − R n ∂Ω p ε˙ = p˙ , p˙ = − = , 2 ∂R K ∂σ ma mb beq ∂beq ∂Ω |a| ∂Ω p ˙ ˙φ p = = p˙ + sign a , K = = . ∂a Ka ∂b Kb ∂b

(40)

(41) (42)

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The time variation of the additional DOF therefore deviates from the cumulated plastic strain rate by a viscous term characterized by the material parameters Ka and ma . The residual dissipation rate becomes mb beq ∂beq |a|ma+1 (σeq − R + a) p˙ + · b ≥ 0, (43) ma + Kb ∂b Ka which is indeed always positive. Let us illustrate the type of partial diﬀerential equation provided by such a model. For that purpose, a simple quadratic free energy potential is chosen: ρψ =

1 2e 4 2e 1 1 1 ε : C : ε + R0 p + H p2 + Hφ φe2 + AKe · Ke . 2 2 2 2

(44)

As a result, the corresponding state laws can be combined with the extra-balance equation (11): a = Hφ φe = ∇· b = ∇· (AKe ) , (45) which leads to the following partial diﬀerential equation, under the condition of material homogeneity: Hφ (φ − φ p ) = AΔφ − A∇· K p .

(46)

If Ka = ∞ (infinite viscosity), Eq. (42) shows that φ p coincides with p. If, furthermore, Kb = ∞, the plastic part of K vanishes. The Eq. (45) then reduces to the Helmholtz-type equation (33) where β and κ are set to zero. An alternative expression of (45) can be worked out by taking the viscous laws into account ˙p, a = Ka (φ˙ p − p) ˙ = ∇· b = ∇· Kb K

(47)

which leads to the following partial diﬀerential equation: ˙p. Ka (φ˙ p − p) ˙ = Kb ∇· K

(48)

When the elastic contributions φe and Ke are neglected, the previous equation reduces to Ka (φ˙ − p) ˙ = Kb Δφ˙ , (49) which is identical to (33) after taking A = Hφ = 0.

4 Phase-Field Models for Elastoviscoplastic Materials In this section, the additional degree of freedom is a phase-field variable. We show how the constitutive framework for elastoviscoplastic materials can be embedded in the existing phase-field approach which combines diﬀusion and phase field equations to model the motion of boundaries between phases. The migration of interfaces

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79

and growth of precipitates are strongly influenced by the mechanical behavior of the phases. One observes in current literature a strong endeavor to develop microstructure evolution simulation schemes coupled with complex mechanical material behavior ranging from heterogeneous elasticity to general elastoviscoplasticity. The main diﬃculty of such a task lies in the tight coupling between the complex interface evolutions and the fields, common to many moving boundary problems. The phase-field approach has emerged as a powerful method for easily tackling the morphological evolutions involved in phase transformations. Phase-field models have incorporated elasticity quite early [40] and have succeeded in predicting some complex microstructure evolutions driven by the interplay of diﬀusion and elasticity. It is only very recently that some phase-field models have been enriched with nonlinear mechanical behavior, extending the range of applications and materials which can be handled by the phase-field approach [3, 19, 20, 38]. There are essentially two ways of introducing linear and nonlinear mechanical constitutive equations into the standard phase-field approach: 1. The material behavior is described by a unified set of constitutive equations including material parameters that explicitly depend on the concentration or the phase variable. Each parameter is usually interpolated between the limit values known for each phase. This is the formulation adopted in the finite element simulations of Cahn-Hilliard like equations coupled with viscoplasticity in [38, 39] for tin-lead solders. The same methodology is used in [19, 20] to simulate the role of viscoplasticity on rafting of γ’ precipitates in single crystal nickel base superalloys under load. 2. One distinct set of constitutive equations is attributed to each individual phase k at any material point. Each phase at a material point then possesses its own 2

2

2 2

stress/strain tensor σk , εk . The overall strain and stress quantities σ, ε at this material point must then be averaged or interpolated from the values attributed to each phase. This is particularly important for points inside the smooth interface zone. At this stage, several mixture rules are available to perform this averaging or interpolation. This approach makes possible to mix diﬀerent types of constitutive equations for each phase, like hyperelastic nonlinear behavior for one phase and conventional elastic-plastic model with internal variables for the other one. No correspondence of material parameters is needed between the phase behavior laws. This is the approach proposed in [37] for incorporating elasticity in a multiphase-field model. For that purpose, the authors resort to a well-known homogeneous stress hypothesis taken from homogenization theory in the mechanics of heterogeneous materials [8]. In the present work, we propose to generalize this procedure to nonlinear material behavior and to other mixture rules also taken from homogenization theory. It must be emphasized that the latter procedure is very similar to what has already been proposed for handling diﬀusion in phase-field models by [26]. Two concentration fields cα and cβ are indeed introduced, and the real concentration field is obtained by a mixture rule together with an internal constraint on the diﬀusion

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potentials. Introducing two concentration fields gives an additional degree of freedom for controlling the energy of the interface with respect to its thickness. If this possibility is not obvious when mechanics is introduced, adding a degree of freedom for describing the stresses/strains within a diﬀuse interface could be valuable to get rid of some spurious eﬀects due to unrealistic interface thickness.

4.1 Coupling with Diﬀusion In the context of mass diﬀusion and phase-field evolution, the local form of the energy principle is 2

2

e˙ = σ : ε˙ + aφ˙ + b · ∇φ˙ .

(50)

2e

2

The total strain is partitioned into the elastic strain ε , the eigenstrain ε due to 2

phase transformation and the plastic strain ε p : 2

2

2

2

ε = εe + ε + ε p .

(51)

According to the thermodynamics of irreversible processes, the second law states that the variation of entropy is always larger than or equal to the rate of entropy flux induced by diﬀusion: T η˙ − ∇ · (μJ) 0 , (52) where J is the diﬀusion flux and μ is the chemical potential. The conservation law for mass diﬀusion is then c˙ = −∇ · J . (53) Accordingly, the fundamental inequality containing first and second principles in the isothermal case is written as 2

2

−ρψ˙ + σ : ε˙ + aφ˙ + b · ∇φ˙ + μ˙c − J · ∇μ 0 .

(54)

Assuming that the free energy density depends on the order parameter φ and its 2

gradient, the concentration c, the elastic strain εe and the set of internal variables Vk associated to material hardening1: S T AT E = {φ,

∇φ,

c,

2

εe ,

Vk } .

The Clausius-Duhem inequality now becomes:

1

∂ψ ˙ ∂ψ ∂ψ ˙ a−ρ φ+ b−ρ · ∇φ + μ − ρ c˙ ∂φ ∂∇φ ∂c ⎛ ⎞ 2 ∂ψ ⎟⎟⎟⎟ 2 e 2 ∂ψ ˙ ⎜⎜ 2 + ⎜⎜⎜⎝σ − Vk 0 . (55) ⎟ : ε˙ − J.∇μ + σ : ε˙ p − ρ 2 ⎠ ∂Vk ∂εe

In this section, the notation for internal variables is changed to (Vk )k∈{α,β} since α is now an index denoting one phase.

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81

The following reversible mechanisms and corresponding state laws are chosen: b=ρ

∂ψ , ∂∇φ

μ=ρ

∂ψ , ∂c

2

σ=ρ

∂ψ 2

∂εe

,

Ak := ρ

∂ψ . ∂Vk

The residual dissipation then is 2 ∂ψ 2 a−ρ φ˙ − J · ∇μ + σ : ε˙ p − Ak V˙ k 0 . ∂φ

(56)

(57)

Three contributions appear in the above residual dissipation rate. The first is the phase-field dissipation, associated with configuration changes of atoms and related to the evolution of the order parameter: Dφ = av φ˙

with

av = a − ρ

∂ψ , ∂φ

(58)

where av is the chemical force associated with the dissipative processes [22]. The second contribution is the chemical dissipation due to diﬀusion, associated with mass transport. The last contribution is the mechanical dissipation, as discussed earlier. An eﬃcient way of defining the complementary laws related to the dissipative processes and ensuring the positivity of the dissipation for any thermodynamic pro2

cess is to assume the existence of a dissipation potential Ω(av , ∇μ, σ, Ak ), which is a convex function of its arguments: φ˙ =

∂Ω , ∂av

J=−

∂Ω , ∂∇μ

∂Ω V˙ k = − , ∂Ak

2

ε˙ p =

∂Ω 2

∂σ

.

(59)

These equations represent the evolution law for the order parameter, the diﬀusion flux as well as the evolution laws for the internal variables.

4.2 Partition of Free Energy and Dissipation Potential The total free energy is postulated to have the form of a Ginzburg-Landau free energy functional accounting for interfaces through the square of the order parameter gradient. The free energy density ψ is then split into a chemical free energy density ψch , a coherent mechanical energy density ψmech , and the square of the order parameter gradient: 2

2

ρψ(φ, ∇φ, c, εe , Vk ) = ρψch (φ, c) + ρψmech (φ, c, ε, Vk ) +

A ∇φ · ∇φ . 2

(60)

The irreversible part of the behavior is described by the dissipation potential, which can be split into three parts related to the three contributions in the residual

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S. Forest, K. Ammar, and B. Appolaire

dissipation in Eq.(57): the phase-field part Ωφ (φ, c, av) , the chemical part 2

Ωc (φ, c, ∇μ) and the mechanical dissipation potential Ωmech (φ, c, σ, Ak ): 2

2

Ω(av , ∇μ, φ, c, σ, Ak ) = Ωφ (c, φ, av) + Ωc (c, φ, ∇μ) + Ωmech (φ, c, σ, Ak ) .

(61)

The chemical free energy density ψch of a binary alloy is a function of the order parameter φ and of the concentration field c. The coexistence of both phases α and β discriminated by φ is possible if ψch is non-convex with respect to φ. Following [25], ψch is built with the free energy densities of the two phases ψα and ψβ as follows: ψch (φ, c) = h(φ)ψα (c) + (1 − h(φ))ψβ (c) + Wg(φ) .

(62)

Here, the interpolating function h(φ) is chosen as h(φ) = φ2 (3 − 2φ), and g(φ) = φ2 (1 − φ)2 is the double well potential accounting for the free energy penalty of the interface. The height W of the potential barrier is related to the interfacial energy σ and the interfacial thickness δ as W = 6Λσ/δ. Assuming that the interface region ranges from θ to 1 − θ, then Λ = log((1 − θ)/θ). In the present work θ = 0.05 [2, 25]. The densities ψα and ψβ are chosen to be quadratic functions of the concentration only: kβ kα ρψα (c) = (c − aα )2 and ρψβ (c) = (c − aβ )2 , (63) 2 2 where aα and aβ are the unstressed equilibrium concentrations of both phases which correspond respectively to the minima of ψα and ψβ in the present model. kα and kβ are the curvatures of the free energies. Quadratic expressions are chosen for the chemical dissipation, which ensures the positivity of the dissipation rate: Ωφ (av ) =

1 (1/β)av2 2

and Ωc (∇μ) =

1 L(φ)∇μ · ∇μ , 2

(64)

where av is given by Eq. (58), β is inversely proportional to the interface mobility and L(φ) is the Onsager coeﬃcient, related to the chemical diﬀusivities Dα and Dβ in both phases by means of the interpolation function h(φ) as L(φ) = h(φ)Dα /kα + (1 − h(φ))Dβ /kβ .

(65)

The state laws and evolution equations for the phase-field and chemical contributions can be derived as b = A∇φ ,

μ=ρ

∂ψch ∂ψmech +ρ , ∂c ∂c

˙φ = 1 av = 1 a − ρ ∂ψch − ρ ∂ψmech , β β ∂φ ∂φ

J = −L(φ)∇μ .

(66)

(67)

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83

Substituting the previous equations into the balance equations for generalized stresses and mass concentration, the Ginzburg-Landau and usual diﬀusion equations are retrieved, which represent respectively the evolution equations for order parameter and concentration: ∇· b − a = −βφ˙ + ∇· (A∇φ) − ρ

∂ψch ∂ψmech −ρ = 0, ∂φ ∂φ

∂ρψch ∂ρψmech c˙ = −∇ · (−L(φ)∇μ) = −∇ · −L(φ) ∇ +∇ . ∂c ∂c

(68)

(69)

Note the coupling of mechanics and diﬀusion and phase-field evolution through the partial derivatives of the mechanical free energy with respect to concentration and order parameter.

4.3 Multi-phase Approach for the Mechanical Contribution The second contribution to the free energy density is due to mechanical eﬀects. Assuming that elastic behavior and hardening are uncoupled, the mechanical part of the free energy density ρψmech is decomposed into a coherent elastic energy density ρψe and a plastic part ρψ p as 2

2

ρψmech (φ, c, ε, Vk ) = ρψe (φ, c, ε) + ρψ p (φ, c, Vk ) .

(70)

Moreover, the irreversible mechanical behavior, related to the dissipative processes, 2

is obtained by a plastic dissipation potential Ωmech (φ, c, σ, Ak ). It is assumed to be a function of order parameter, concentration, Cauchy stress tensor as well as the set of thermodynamic force associated variables Ak in order to describe the hardening state in each phase. In the diﬀuse interface region where both phases coexist, we propose to use wellknown results of homogenization theory to interpolate the local behavior. The homogenization procedure in the mechanics of heterogeneous materials consists in replacing an heterogeneous medium by an equivalent homogeneous one, which is defined by an eﬀective constitutive law relating the macroscopic variables, namely 2

2

macroscopic stress σ and strain ε tensors, which are obtained by averaging the corresponding non-uniform local stress and strain in each phase. Each material point within a diﬀuse interface can be seen as a local mixture of the two abutting phases α and β with proportions given by complementary functions of φ. The strain and stress at each material point are then defined by the following mixture laws which would proceed from space averaging in a conventional homogenization problem, but which must be seen as arbitrary interpolations in the present case: 2

2

2

ε = χ εα + (1 − χ) εβ

2

2

2

and σ = χ σα + (1 − χ) σβ ,

(71)

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S. Forest, K. Ammar, and B. Appolaire 2

2

2

2

where εα , εβ are local fictitious strains and σα , σβ are local fictitious stresses in α and β phases respectively and χ(x, t) is a shape function which must take the value 0 in the β-phase and 1 in the α-phase. The following choice is made in the phase field context: χ(x, t) ≡ φ(x, t) . (72) The partition hypothesis, already used for the eﬀective total strain tensor in Eq. (51), requires, in a similar way, a decomposition of the total strain in each phase into elastic, transformation and plastic parts: 2

2

2

2

2

εα = εeα + εα + εαp

2

2

2

p and εβ = εeβ + εβ + εβ ,

(73)

where each point may depend on the local concentration c, but not on order parameter φ. In the proposed model, the elastoplastic and phase-field behaviors of each phase are treated independently and the eﬀective behavior is obtained using homogenization relation (71). It is assumed that the mechanical state of α and β phases at a given time are completely described by a finite number of local state 2

variables εek , Vk defined at each material point. The set of internal variables Vk , of scalar or tensorial nature, represents the state of hardening of phase k: for instance, a scalar isotropic hardening variable, and a tensorial kinematic hardening variable. According to the homogenization theory, the eﬀective elastic and plastic free energy densities are given by the rule of mixtures as follows: 2

2

2

ρψe (φ, c, ε) = φ ρψeα (c, εeα ) + (1 − φ)ρψeβ (c, εeβ ) ,

(74)

ρψ p (φ, c, Vk ) = φ ρψ pα (c, Vα ) + (1 − φ)ρψ pβ (c, Vβ) .

(75)

Similarly, a mixture rule is used to mix the dissipation potentials of the individual phases: 2

2

2

Ωmech (φ, c, σ, Ak ) = φ Ωmechα (c, σα , Aα ) + (1 − φ) Ωmechβ (c, σβ , Aβ ) ,

(76)

where the Aα,β are the thermodynamic forces associated with the internal variables attributed to each phase. Knowing the free energy and dissipation potentials, the evolution of all variables can be computed. The remaining questions is the way of estimating the previously 2

2

defined fictitious stress and strain tensors εα,β , σα,β from the knowledge of the stress 2

2

and strain tensors ε and σ. Several homogenization schemes exist in the literature that can be used to define these new fictitious variables. The most simple schemes are the Voigt/Taylor and Reuss/Static models. We develop the Voigt/Taylor scenario in the sequel.

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4.4 Voigt/Taylor Model Coupled Phase-Field Mechanical Theory According to Voigt’s scheme, the fictitious strains are not distinguished from the local strain. The local stress is then computed in terms of the fictitious stress tensors by averaging with respect to both phases weighted by the volume fractions: 2

2

2

σ = φ σα + (1 − φ) σβ , 2

2

2

2

ε = εα = εβ .

(77)

2

The stresses of both phases σα and σβ are given by Hooke’s law for each phase: 2

4

2

2

2

4

4

2

σα = Cα : (εα − εα − εαp ) ,

2

2

2

σβ = Cβ : (εβ − εβ − εβp ) ,

(78)

4

where Cα and Cβ are respectively the tensor of elasticity moduli in α and β phases. As a result, 2

4

2

2

4

2

2

2

2

σ = φ Cα : (εα − εα − εαp ) + (1 − φ)Cβ : (εβ − εβ − εβp ) .

(79)

From the above relation, it follows that the strain-stress relationship in the homogeneous eﬀective medium obeys Hooke’s law with the following equation: 2

4

2

2

2

σ = Ceﬀ : (ε − ε p − ε ) , 4

where the eﬀective elasticity tensor Ceﬀ is obtained from the mixture rule of the elasticity matrix for both phases: 4

4

4

Ceﬀ = φ Cα + (1 − φ)Cβ 2

(80)

2

and the eﬀective eigenstrain ε and plastic strain ε p vary continuously between their respective values in the bulk phases as follows: 2

4

4

2

4

2p

4

4

2 εαp

4

2

ε = C−1 ε ε eﬀ : (φ Cα : α + (1 − φ)Cβ : β ) , ε =

C−1 eﬀ

: (φ Cα :

+ (1 − φ)Cβ :

(81)

2 εβp ) . 2

2

In the case of non-homogeneous elasticity, it must be noted that ε and ε p are not the average of their respective values for each phase. The proposed approach diﬀers from the one most commonly used in phase-field models, as popularized by Khachaturyan and co-workers, e. g. [24]. The latter rely on mixture laws for all quantities within the interface, including the elastic moduli, the transformation and plastic strain. The eﬀect of these diﬀerent choices on the simulation of moving phase boundaries has been tested in [3] and [4]. In particular, the impact of plasticity on the kinetics of precipitate growth has been evidenced.

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5 Conclusion The general thermomechanical setting for modeling size eﬀects in the mechanics and thermodynamics of materials is based on the main assumption that microstructure eﬀects can be accounted for by the introduction of additional degrees of freedom in addition to displacement, temperature and concentration. The additional DOF and its gradient are expected to contribute to the power of internal forces of the medium and to arise in the energy local balance equations and/or entropy inequality. They induce generalized stresses that fulfill an additional balance equation with associated extra boundary conditions. A clear separation between balance equations and constitutive functionals is adopted in the formulation. Constitutive equations derive from the definition of a specific free energy density and dissipation potential. The crossing of mechanical and physical approaches turns out to be fertile in providing motivated coupling between both kinds of phenomena. As an example, we have shown that the mechanics of heterogeneous materials can be useful to develop a sophisticated and flexible constitutive framework of coupled viscoplasticity and diﬀusion. It was not possible to address applications that already exist in this context. In particular, the presented models predict that viscoplasticity aﬀects the morphology and kinetics of precipitate growth in metals or during oxidation [2, 4, 20]. Special attention must now be dedicated to more precise description of coherent vs. incoherent interfaces [5, 23, 34], and the associated specific interface conditions that can be deduced from asymptotic analysis of phase-field models. On the other hand, the targeted applications of strain gradient plasticity are crystal plasticity and grain boundary migration [1, 9, 28], whereas strain rate gradients are thought to be relevant for aging materials [29].

References [1] Abrivard, G.: A coupled crystal plasticity–phase field formulation to describe microstructural evolution in polycrystalline aggregates. PhD, Mines ParisTech (2009) [2] Ammar, K., Appolaire, B., Cailletaud, G., Feyel, F., Forest, F.: Finite element formulation of a phase field model based on the concept of generalized stresses. Computational Materials Science 45, 800–805 (2009) [3] Ammar, K., Appolaire, B., Cailletaud, G., Forest, S.: Combining phase field approach and homogenization methods for modelling phase transformation in elastoplastic media. European Journal of Computational Mechanics 18, 485–523 (2009) [4] Ammar, K., Appolaire, B., Cailletaud, G., Forest, S.: Phase field modeling of elastoplastic deformation induced by diﬀusion controlled growth of a misfitting spherical precipitate. Philosophical Magazine Letters (2011) [5] Appolaire, B., Aeby-Gautier, E., Teixeira, J.D., Dehmas, M., Denis, S.: Non-coherent interfaces in diﬀuse interface models. Philosophical Magazine 90, 461–483 (2010) [6] Aslan, O., Forest, S.: Crack growth modelling in single crystals based on higher order continua. Computational Materials Science 45, 756–761 (2009)

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[7] Aslan, O., Forest, S.: The micromorphic versus phase field approach to gradient plasticity and damage with application to cracking in metal single crystals. In: de Borst, R., Ramm, E. (eds.) Multiscale Methods in Computational Mechanics. LNACM, vol. 55, pp. 135–154. Springer, Heidelberg (2011) [8] Besson, J., Cailletaud, G., Chaboche, J.L., Forest, S., Blétry, M.: Non-Linear Mechanics of Materials. Series: Solid Mechanics and Its Applications, vol. 167, p. 433. Springer, Heidelberg (2009) [9] Cordero, N., Gaubert, A., Forest, S., Busso, E., Gallerneau, F., Kruch, S.: Size eﬀects in generalised continuum crystal plasticity for two–phase laminates. Journal of the Mechanics and Physics of Solids 58, 1963–1994 (2010) [10] Ehlers, W., Volk, W.: On theoretical and numerical methods in the theory of porous media based on polar and non–polar elasto–plastic solid materials. International Journal of Solids and Structures 35, 4597–4617 (1998) [11] Engelen, R., Geers, M., Baaijens, F.: Nonlocal implicit gradient-enhanced elastoplasticity for the modelling of softening behaviour. International Journal of Plasticity 19, 403–433 (2003) [12] Eringen, A., Suhubi, E.: Nonlinear theory of simple microelastic solids. International Journal of Engineering Science 203, 189–203, 389–404 (1964) [13] Finel, A., Le Bouar, Y., Gaubert, A., Salman, U.: Phase field methods: Microstructures, mechanical properties and complexity. Comptes Rendus Physique 11, 245–256 (2010) [14] Forest, S.: The micromorphic approach for gradient elasticity, viscoplasticity and damage. ASCE Journal of Engineering Mechanics 135, 117–131 (2009) [15] Forest, S., Aifantis, E.C.: Some links between recent gradient thermo-elasto-plasticity theories and the thermomechanics of generalized continua. International Journal of Solids and Structures 47, 3367–3376 (2010) [16] Forest, S., Sievert, R.: Nonlinear microstrain theories. International Journal of Solids and Structures 43, 7224–7245 (2006) [17] Frémond, M., Nedjar, B.: Damage, gradient of damage and principle of virtual power. International Journal of Solids and Structures 33, 1083–1103 (1996) [18] Fried, E., Gurtin, M.: Continuum theory of thermally induced phase transitions based on an order parameter. Physica D 68, 326–343 (1993) [19] Gaubert, A., Finel, A., Le Bouar, Y., Boussinot, G.: Viscoplastic phase field modellling of rafting in ni base superalloys. In: Continuum Models and Discrete Systems CMDS11, pp. 161–166. Mines Paris Les Presses (2008) [20] Gaubert, A., Le Bouar, Y., Finel, A.: Coupling phase field and viscoplasticity to study rafting in ni-based superalloys. Philosophical Magazine 90, 375–404 (2010) [21] Germain, P.: La méthode des puissances virtuelles en mécanique des milieux continus, première partie : théorie du second gradient. Journal de Mécanique 12, 235–274 (1973) [22] Gurtin, M.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1996) [23] Johnson, W.C., Alexander, J.I.D.: Interfacial conditions for thermomechanical equilibrium in two-phase crystals. Journal of Applied Physics 9, 2735–2746 (1986) [24] Khachaturyan, A.: Theory of Structural Transformations in Solids. John Wiley & Sons, New York (1983) [25] Kim, S., Kim, W., Suzuki, T.: Interfacial compositions of solid and liquid in a phase– field model with finite interface thickness for isothermal solidification in binary alloys. Physical Review E 58(3), 3316–3323 (1998) [26] Kim, S., Kim, W., Suzuki, T.: Phase–field model for binary alloys. Physical Review E 60(6), 7186–7197 (1999)

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[27] Maugin, G.: The method of virtual power in continuum mechanics: Application to coupled fields. Acta Mechanica 35, 1–70 (1980) [28] Mayeur, J., McDowell, D., Bammann, D.: Dislocation-based micropolar single crystal plasticity: Comparison of multi- and single criterion theories. Journal of the Mechanics and Physics of Solids 59, 398–422 (2011) [29] Mazière, M., Besson, J., Forest, S., Tanguy, B., Chalons, H., Vogel, F.: Numerical aspects in the finite element simulation of the portevin-le chatelier eﬀect. Computer Methods in Applied Mechanics and Engineering 199, 734–754 (2010) [30] Miehe, C.: A multi-field incremental variational framework for gradient-extended standard dissipative solids. Journal of the Mechanics and Physics of Solids 59, 898–923 (2011) [31] Miehe, C., Welchinger, F., Hofacker, M.: A phase field model of electromechanical fracture. Journal of the Mechanics and Physics of Solids 58, 1716–1740 (2010) [32] Miehe, C., Welchinger, F., Hofacker, M.: Thermodynamically–consistent phase field models of fracture: Variational principles and multifield FE implementations. International Journal for Numerical Methods in Engineering 83, 1273–1311 (2010) [33] Mindlin, R.: Micro–structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 51–78 (1964) [34] Murdoch, A.I.: A thermodynamical theory of elastic material interfaces. The Quarterly Journal of Mechanics and Applied Mathematics 29, 245–275 (1978) [35] Peerlings, R., Geers, M.: Borst, R., Brekelmans, W. critical comparison of nonlocal and gradient–enhanced softening continua. International Journal of Solids and Structures 38, 7723–7746 (2001) [36] Rajagopal, A., Fischer, P., Kuhl, E., Steinmann, P.: Natural element analysis of the Cahn-Hilliard phase-field model. Computational Mechanics 46, 471–493 (2010) [37] Steinbach, I., Apel, M.: Multi phase field model for solid state transformation with elastic strain. Physica D 217, 153–160 (2006) [38] Ubachs, R., Schreurs, P., Geers, M.: A nonlocal diﬀuse interface model for microstructure evolution of tin–lead solder. Journal of the Mechanics and Physics of Solids 52, 1763–1792 (2004) [39] Ubachs, R., Schreurs, P., Geers, M.: Elasto-viscoplastic nonlocal damage modelling of thermal fatigue in anisotropic lead-free solder. Mechanics of Materials 39, 685–701 (2007) [40] Wang, Y., Chen, L.Q., Khachaturyan, A.: Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap. Acta Metallurgica et Materialia 41, 279–296 (1993)

Geometrically Nonlinear Continuum Thermomechanics Coupled to Diﬀusion: A Framework for Case II Diﬀusion Andrew T. McBride, Swantje Bargmann, and Paul Steinmann The authors would like to express their gratitude for the opportunity to contribute to this Springer volume on the occasion of the 60 th birthday of Professor Wolfgang Ehlers.

Abstract. This chapter introduces a geometrically nonlinear, continuum thermomechanical framework for case II diﬀusion: a type of non-Fickian diﬀusion characterized by the wave-like propagation of a low-molecular weight solvent in a polymeric solid. The key objective of this contribution is to derive the coupled system of governing equations describing case II diﬀusion from fundamental balance principles. A general form for the Helmholtz energy is proposed and the resulting constitutive laws are derived from logical, thermodynamically consistent argumentation. The chapter concludes by comparing the model developed here to various others in the literature. The approach adopted to derive the governing equations is not specific to case II diﬀusion, rather it encompasses a wide range of applications wherein heat conduction, species diﬀusion and finite inelastic eﬀects are coupled. The presentation is thus applicable to the generality of models for non-Fickian diﬀusion: an area of increasing research interest.

1 Introduction Fick’s law of diﬀusion [8] proposes a relationship between the diﬀusion flux vector and the gradient of the concentration field. It has been widely used to successfully model numerous physical processes involving diﬀusion. There are, however, several applications for which the classical model of Fickian diﬀusion does not suﬃce (see, for example, Aifantis [1] for an extensive overview). Case II diﬀusion is one such application. Andrew T. McBride · Paul Steinmann Chair of Applied Mechanics, University of Erlangen-Nuremberg, Egerlandstraße 5, 91058 Erlangen, Germany e-mail: {andrew.mcbride,paul.steinmann}@ltm.uni-erlangen.de Swantje Bargmann Institute of Mechanics, TU Dortmund University, Leonhard-Euler-Straße 5, 44227 Dortmund, Germany e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 89–107. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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Case II diﬀusion is the name given by Alfrey et al [2] to the physical process wherein a low molecular weight solvent diﬀuses within an initially glassy polymeric solid. In the vicinity of the glass transition temperature, the polymer undergoes a glass-to-rubber phase transition (see, for example, Vesely [26] for an overview of the process of liquid diﬀusion within solids). The glass-to-rubber transition occurs at a finite rate and thereby inhibits the solvent ingress. Thus, the resulting diﬀusion process is non-Fickian with the propagation of the solvent front wave-like in nature. Ahead of the sharp wave front the polymer is glassy and the solvent concentration low; behind the front the polymer has been plasticized and the solvent concentration is significantly increased. The molecular rearrangement of the polymeric solid during the diﬀusion process results in significant deformation at the macro-scale. Moreover, whereas the uptake of solvent is proportional to the square root of time in Fickian diﬀusion, it is linear in time for case II diﬀusion. Temperature also plays a role in the overall response of the system, but is generally neglected in the modeling literature. Case II diﬀusion occurs in many industrial and biological processes and is of importance in the pharmaceutical and automotive industries, for example. A typical system undergoing case II diﬀusion is poly(methyl methacrylate) (PMMA) and methanol [23], for example. One objective of this contribution is to derive the equations governing the response of a continuum solid undergoing finite, inelastic deformation coupled to species diﬀusion and heat conduction within a continuum mechanical setting. The equations are developed using fundamental balance principles and lead to thermodynamically consistent constitutive relations. The presentation draws on the seminal work by Govindjee and Simo [10] and extends these results to include thermal eﬀects. The nature of the coupling between heat, diﬀusion and deformation is described in detail. The framework presented here is general in nature; that is, it is applicable to problems wherein diﬀusion, deformation, and heat conduction are coupled. The specific choice of the Helmholtz energy allows this general framework to be specialized to describe case II diﬀusion. The key features of such a Helmholtz energy are presented. In summary, the key objectives and contributions of this work are: • to derive the equations governing the response of a thermomechanical solid undergoing finite, inelastic deformations with diﬀusion; • to explicitly derive the form of the constitutive relations using logical, thermodynamic arguments; • to clarify the nature of the coupling between heat conduction, species diﬀusion, and finite inelastic deformation; • to succinctly categorize and compare several models for case II diﬀusion (a detailed comparison can be found in Bargmann et al [3]). This chapter is organized as follows. The notation and certain key concepts are introduced in Sect. 2. The equations governing the response of the coupled

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system are derived from fundamental balance principles in Sect. 3. In particular, the conservation of solid and diﬀusing species mass, and the balance of linear and angular momentum are presented. Thereafter, the balance of energy and entropy are detailed. A thermodynamic framework is then utilized to determine the form of the thermodynamically consistent constitutive relations. The nature of the coupling between the various fields (displacement, temperature and chemical potential) is made clear. Specific choices for the Helmholtz energy that specialize the general theory of Sect. 3 for case II diﬀusion are presented in Sect. 4. We conclude in Sect. 5 by comparing the model presented here with other models in the literature and give some closing remarks.

2 Preliminaries: Notation and Key Concepts The purpose of this preliminary section is to summarize certain key concepts in nonlinear continuum mechanics and to introduce the notation adopted here. A detailed exposition on nonlinear continuum mechanics can be found in Truesdell and Noll [25], amongst others. Direct notation is employed throughout. The scalar product of two vectors a and b is denoted a · b. The scalar product of two second-order tensors A and B is denoted A : B. The action of a second-order tensor A on a vector b is represented as A · b, while the action of a fourth-order tensor A on a second-order tensor B is represented as A : B. The tensor product a ⊗ b of two vectors a and b is a second-order tensor defined by the relation [a ⊗ b] · c = [b · c]a for all vectors c. We denote by the open set B0 ⊂ R3 the reference (initial) placement of a continuum body, as depicted in Fig. 1, with material particles labeled X ∈ B0 . The surface of B0 , assumed smooth, is denoted as ∂B0 . The unit normal to ∂B0 is denoted N. Let T = [0, T ] ⊂ R+ denote the time domain. Here time simply provides a history parameter to order the sequence of events and quasi-static conditions are assumed henceforth. We denote by the map ϕ : B0 × T → R3 , a smooth motion of the reference placement for a time t ∈ T. The current placement of the body at time t associated with the motion ϕ is denoted Bt = ϕ(B0 (X), t), with particles designated as x = ϕ(X, t) ∈ Bt . Here and henceforth, the subscripts t and 0 shall designate spatial and material quantities, respectively, unless specified otherwise. The material velocity V(X, t) is defined by the conventional time derivative of the motion as V(X, t) :=

∂ϕ(X, t) . ∂t

A spatial description of the motion can be obtained from the material description by transforming the independent variables from the material coordinates to the spatial coordinates. The spatial velocity field, denoted v(x, t), is obtained from the material velocity field as v(ϕ (X, t) , t) = V(X, t) .

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Fig. 1 The motion ϕ of a continuum body from the initial (reference) configuration B0 to the current configuration Bt . The control regions B0 and Bt in the initial and deformed configurations, respectively, are also indicated.

The equations governing the coupled response of the system are obtained from the balances of solid and diﬀusing species mass, linear and angular momentum, and energy and entropy over a control region. The control region in the reference configuration B0 is denoted B0 with smooth boundary ∂B0 . These integral balance expressions are then localized at an arbitrary point in the bulk B0 . The unit normal to the surface ∂B0 is denoted M. The region B0 is mapped via the motion ϕ to the current configuration Bt as Bt = ϕ(B0 (X), t) ⊂ Bt . The smooth surface of Bt is denoted ∂Bt with outward unit normal m. The integral form of the balance equations typically include terms due to convective or physical fluxes, or both, of a quantity over the internal surface ∂B0 . The spatial velocity of the diﬀusing species is denoted v . The invertible linear tangent map F : T B0 → T Bt (that is, the deformation gradient) maps a line element dX ∈ T B0 in the reference configuration to a line element dx ∈ T Bt in the current configuration and is defined as the derivative of the motion with respect to the reference placement; that is, F(X, t) := Grad ϕ(X, t) , where Grad {•} := ∂ {•} /∂X is the gradient operator with respect to the reference placement. The notation T {•} denotes the tangent space to {•}. The action of the gradient operator with respect to the current placement is denoted grad {•} := ∂ {•} /∂x. The Jacobian determinant of the deformation gradient is denoted J(X, t) := det(F(X, t)) > 0. The inverse of the deformation gradient is denoted

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f := F−1 with an associated inverse Jacobian determinant j := det f = 1/J > 0. The Jacobian determinants are (invertible) volume maps that relate the spatial and material volume elements, dv and dV respectively, as dv = JdV

and

dV = jdv .

(1)

The right Cauchy-Green tensor, a measure of the deformation, is defined by C := Ft · F. Nanson’s formula relates oriented surface area elements d A ∈ T ∂B0 and da ∈ T ∂Bt as follows: n da = [cofF] · N dA

and

N dA = cof f · n da ,

(2)

where the cofactor of an arbitrary second-order tensor {•} is defined by: cof {•} := det {•} {•}−t . The governing equations derived in Sect. 3 are constructed from balances over a spatial control region Bt (that is, a material control region mapped by the deformation ϕ to the current configuration Bt ), as depicted in Fig. 1. Two forms of flux over the boundary of the spatial control region are accounted for: (1) convective flux due to the motion of the control region, and (2) physical flux due to the motion of the diﬀusing species. A key relation used when manipulating the integral balance equations is Reynolds’s transport theorem. Spatial scalar quantities {•} (x, t) are denoted {•}t . The material time derivative of an integral over a spatial control region Bt ⊂ Bt is given by Reynolds’s transport theorem as dt {•}t dv + {•}t v · m da . (3) Dt {•}t dv = Bt

Bt

∂Bt

Here Dt {•} and dt {•} denote the material and spatial time derivatives of a scalar {•}, respectively. Reynolds’s transport theorem equates the material rate of change of the integral of a spatial quantity {•}t in the bulk to the integral of the instantaneous rate of change of the quantity in the bulk and changes due to the convective flux of the quantity over the surface ∂Bt . The divergence theorem relates the integral of a vectorial quantity {•} over the boundary of a region to an integral over the region itself as follows div {•} dv = {•} · m da . (4) Bt

∂Bt

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3 Governing Equations The objective of this section is to derive the equations governing the response of an inelastic continuum body undergoing finite deformations coupled to thermal and diﬀusional eﬀects. The governing equations are systematically derived by considering the conservation of several key properties, namely solid mass, diﬀusing species mass, linear and angular momentum, (internal) energy and entropy, over a spatial control region. The Clausius-Duhem form of the second law of thermodynamics and the resulting local dissipation inequality provide the form for the constitutive relations. The restrictions that arise from the reduced dissipation inequality stipulate the thermodynamically consistent form of the remaining constitutive relations. The section concludes with the derivation of the temperature evolution equations.

3.1 Conservation of Solid Mass Consider the material and spatial control regions, denoted B0 and Bt respectively, shown in Fig. 1. The bulk densities of the solid in the reference and current configurations are denoted ρ0 [kg/m3 ] and ρt [kg/m3 ], respectively. The global statement of conservation of solid mass over the control region (i. e. equivalence of the solid mass of the control volumes in the undeformed and deformed configurations) reads ρ0 dV = ρt dv . (5) B0

Bt

Using Eq. (1)1 to change the limits of integration on the right-hand side of Eq. (5) to the reference configuration and applying the standard localization theorem renders the familiar local form for the conservation of solid mass [see e. g. 16] as ρt J = ρ0

or

ρt = jρ0

in B0 .

(6)

The localization process is valid for an arbitrary region B0 ⊂ B0 and hence Eq. (6) holds throughout the domain B0 .

3.2 Conservation of Diﬀusing Species Mass The mass of the diﬀusing species per unit volume of the (deformed) solid is denoted ρt [kg/m3 ]. The concentration of the diﬀusing species per unit volume of the (deformed) solid, denoted ct [mol/m3 ], is related to the diﬀusing species density ρt via the inverse of the diﬀusing species molar mass c [mol/kg] as ct = cρt

[mol/m3 ] = mol/kg · [kg/m3 ] .

Consider now a spatial control region Bt ⊂ Bt as shown in Fig. 1. The global statement of conservation of diﬀusing species mass over Bt reads

Geometrically Nonlinear Continuum Thermomechanics Coupled to Diﬀusion

Bt

dt ct dv = −

95

ct v · m da +

∂Bt

w dv ,

(7)

Bt

where v is the spatial velocity of the diﬀusing species and bulk sources of species generation are denoted w [mol/m3 s]. The quantity ct v = cρt v represents the physical species flux over the boundary ∂Bt . The negative sign in front of the first term on the right-hand side indicates that the flux of species into the domain causes an increase in the concentration within the domain. The preceding statement of conservation of diﬀusing species mass can be expressed, using Reynolds’s transport theorem (3), as Dt ct dv = − ct v − v · m da + w dv , (8) Bt

∂Bt

Bt

w

where w [mol/m2 s] is the (net) spatial diﬀusion flux vector. The Piola transform (pull-back) of the (net) diﬀusion flux vector in the spatial configuration to the corresponding quantity W in the reference configuration is W = Jw · f t

W = w · cofF .

or, equivalently,

(9)

The concentration of the diﬀusing species expressed in the current and reference configurations, ct and c0 , respectively, are related via a Piola transformation as c0 = Jct .

(10)

Transforming the limits of integration in the diﬀusing species mass balance (8) to the reference configuration using Eq. (1)1 and Nanson’s relation in Eq. (2)2 , and using the relations given in Eqs. (9)–(10) yields the material expression for the balance of diﬀusing species mass: [ct J] dV = Dt c0 dV Dt ct dv = Dt Bt

B0

=−

∂B0

c0

B0

[w · cofF] · M dA + W

B0

[wJ] dV ,

(11)

W

and using the divergence theorem (4)

Dt c0 dV = B0

[− Div W + W] dV . B0

Localizing the above relation to a point in the domain B0 yields the standard expression for the conservation of diﬀusing species concentration [see e. g. 10], which is simply the material expression of Fick’s second law:

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Dt c0 = − Div W + W

in B0 .

(12)

Remark 1. Fick’s second law together with a constitutive relation for the diﬀusion flux vector W form the basis for the vast majority of models for case II diﬀusion. The distinction between the material and spatial setting is neglected in most of these works as infinitesimal deformations are assumed, either implicitly or explicitly. The exception to this is the work of Govindjee and Simo [10] and the extensions thereof and elaborations thereupon in Vijalapura and Govindjee [27, 28].

3.3 Balance of Linear and Angular Momentum The Piola-Kirchhoﬀ stress tensor is denoted P. Inertial eﬀects are not considered for the sake of simplicity. The derivation of the equilibrium equation and the symmetry properties of the Cauchy stress tensor, denoted σ := jP · Ft , are standard [see e. g. 20] and further details omitted here for the sake of brevity. The derivation is based upon performing a balance of linear and angular momentum over a spatial control region Bt which yields: ⎫ ⎪ Div P + b0 = 0 ⎬ (13) in B0 , t t⎪ F· P = P·F ⎭ where b0 denotes the total volume force per unit reference volume. Remark 2. Coupled models for case II diﬀusion can be classified as weakly or strongly coupled. Strongly-coupled models explicitly account for the deformation of the polymeric solid using the equilibrium equation given in Eq. (13)1 , or more often its restriction to the case of infinitesimal deformations. The infinitesimal deformation model by Wu and Peppas [29] and the finite deformation model by Govindjee and Simo [10] are two examples of strongly-coupled frameworks. In weakly-coupled models, the deformation of the solid is not directly accounted for, but the rate-limiting influence of the solid state on the diﬀusion process is. Examples of weakly-coupled models include those by Thomas and Windle [22, 23, 24] and Cohen and White [4, 5].

3.4 Balance of Internal Energy The enthalpy of the diﬀusing species per unit volume of the (deformed) solid ϕt [N m/m3 ] that contributes to the internal energy is related to the diﬀusing species concentration in the bulk ct as

N m/m3 = [N m/mol] · mol/m3 , ϕt = ϕct where ϕ [N m/mol] is the enthalpy of the diﬀusing species per unit amount of diﬀusing species.

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Consider now a material control region B0 ⊂ B0 . The rate of change of enthalpy associated with the diﬀusing species is due to the net diﬀusing species flux (i. e. the net flux resulting from the physical and convective flux contributions) over the boundary ∂B0 and source terms in B0 . Recalling the definition of the spatial diffusion flux vector w := ct [v − v] in Eq. (8) and the pull-back thereof in Eq. (9), the contribution of the diﬀusing species to the rate of change of the internal energy follows directly from the balance of diﬀusing species mass given in Eq. (11) as Dt ϕ0 dV = − ϕW · M dA + ϕW dV , (14) B0

∂B0

B0

where ϕ0 = Jϕt .

(15)

Consider now the thermal contributions to the internal energy. The heat flux vector per unit current area in the bulk, denoted q [N m/m2 s], is related to the heat flux vector per unit reference area in the bulk Q via the Piola transform (pull-back) as Q = q · cofF . The material quantity Q [N m/m3 s] shall denote the heat sources. The global rate of change of the total internal energy u0 due to deformation, species diﬀusion and sources (see Eq. (14)), heat conduction and heat sources (i. e. the deformational, diﬀusional and thermal power) is therefore Dt u0 dV = P : Dt F + Q + ϕW ] dV − (16) Q + ϕW · M dA , B0

B0

Qeﬀ

∂B0

Qeﬀ

where Qeﬀ denotes the eﬀective heat source due to heat and species sources in the bulk, and Qeﬀ denotes the eﬀective heat flux vector. Localizing Eq. (16) at a point in the bulk B0 yields the local balance equation for internal energy, expressed in material quantities, as Dt u0 = P : Dt F + Qeﬀ − Div Qeﬀ

in B0 .

(17)

Remark 3. The energetic coupling between the heat conduction and species diffusion is clear from the eﬀective energetic quantities Qeﬀ := [Q + ϕW] and Qeﬀ := [Q + ϕW] in Eq. (17).

3.5 Balance of Entropy The entropy density of the diﬀusing species is denoted σt [N m/K m3 ] and is related to the diﬀusing species concentration ct via the entropy of the diﬀusing species per amount of diﬀusing species σ [N m/K mol] as

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σt = σct

N m/K m3 = [N m/K mol] · mol/m3 .

(18)

Consider now the control region B0 ⊂ B0 . Using the same arguments employed to derive the internal energy contribution due to diﬀusion and sources in Eq. (14), the rate of change of entropy associated with the diﬀusing species is given by Dt σ0 dV = − σW · M dA + σW dV , (19) B0

∂B0

B0

where σ0 = Jσt .

(20)

We denote by the material quantity s0 the total bulk entropy density. The second law of thermodynamics imposes the physical restriction that the rate of increase of entropy in a body is not less than the total entropy supplied to the body. We adopt the commonly made Clausius-Duhem assumption that the heat flux vector Q and the heat source Q are proportional to the entropy flux vector H and the entropy source H, respectively. Thus, H := θ−1 Q

H := θ−1 Q ,

and

(21)

where θ > 0 is the absolute temperature. We denote via the non-negative material quantity π0 the entropy production density in the bulk, and, furthermore, define the dissipation density by δ0 := π0 θ ≥ 0 .

(22)

The global balance of entropy (i. e. the entropy due to production, heat and diﬀusion) over a material control region B0 thus reads Dt s0 dV = π0 dV + [H + σW] dV − [H + σW] · M dA , (23) B0

B0

B0

Heﬀ

∂B0

Heﬀ

where Heﬀ denotes the eﬀective entropy source due to heat and species sources in the bulk, and Heﬀ denotes the eﬀective entropy flux vector. The local form of the conservation of entropy in the bulk (i. e. the ClausiusDuhem form) follows from Eq. (23) as Dt s0 = π0 + Heﬀ − Div Heﬀ

in B0 ,

(24)

and using the localized expression (12) for the conservation of species mass in the bulk yields Dt s0 = π0 + H − Div H + σDt c0 − W · Grad σ

in B0 .

(25)

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Multiplying both sides of the above equation by the bulk temperature θ, using the relations given in Eqs. (21)–(22) and exploiting the relationship θ Div H = [Div Q − H · Grad θ] yields θDt s0 = δ0 + Q + θσDt c0 − Div Q + H · Grad θ − θW · Grad σ .

(26)

Remark 4. The entropic coupling between the heat conduction and species diﬀusion is clear from the eﬀective entropic quantities Heﬀ := [H +σW] and Heﬀ := [H+σW] in Eq. (24).

3.6 Constitutive Relations As is customary in solid mechanics we choose to work with the Helmholtz energy ψ0 := u0 − θs0 and the chemical potential μ := ϕ − θσ, as defined via Legendre transforms [see e. g. 16]. The functional form of the Helmholtz energy density is chosen, generally, as ψ0 = ψ0 (C, c0 , θ, Ξ; X) ,

(27)

where Ξ denotes the set of material internal variables that describe the inelastic response of the solid. The argument of the Helmholtz energy density in the second slot denotes the parameterization in terms of position. This parameterization is required to account for inhomogeneity. Henceforth, for the sake of simplicity, we shall assume a homogeneous material response and furthermore, for the sake of brevity, omit the parameterization of the energy in terms of position from the notation. Note, that in order to satisfy the principle of material objectivity [25] the dependence of the Helmholtz energy ψ0 on the deformation gradient F is via the right Cauchy-Green tensor C. Using the local expressions for the conservation of diﬀusing species mass (12) and the conservation of energy (17) and entropy (26), one obtains from the definition of the Helmholtz energy, after some straightforward manipulations, the following relation: Dt u0 − θDt s0 = Dt ψ0 + s0 Dt θ = P : Dt F + μDt c0 − W · Grad μ − δ0 − Heﬀ · Grad θ ∂ψ0 ∂ψ0 ∂ψ0 ∂ψ0 = 2F · · Dt F + Dt θ + Dt c 0 + : D t Ξ + s 0 Dt θ . ∂C ∂θ ∂c0 ∂Ξ

(28)

Applying the standard Coleman-Noll procedure to Eq. (28) yields the constitutive relations for the Piola-Kirchhoﬀ stress P, the chemical potential μ and the entropy s0 as P = 2F ·

∂ψ0 , ∂C

μ=

∂ψ0 ∂c0

and

s0 = −

∂ψ0 . ∂θ

(29)

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Remark 5. The conjugate quantity to chemical potential μ is thus, clearly, the concentration c0 . That this is the case is not obvious in many models of case II diﬀusion that use the notion of ideal and non-ideal mixtures. In the Govindjee and Simo [10] model, the activity A (expressed as a function of the chemical potential μ), as opposed to the species concentration c0 , is viewed as the primary variable. This is possible as the actual expression for the term ∂ψ0 /∂c0 is invertible. Alternatively, a Legendre transformation of the Helmholtz energy as a function of the concentration to one that is a function of the chemical potential would also allow the chemical potential to be viewed as the primary variable. The reduced dissipation inequality follows from Eqs. (28)–(29) as δ0 = −W · Grad μ − Heﬀ · Grad θ −

∂ψ0 : Dt Ξ ≥ 0 . ∂Ξ

(30)

In order to automatically satisfy the reduced dissipation inequality (30), and thereby ensure thermodynamic consistency, we postulate constitutive relations of the form W ∝ − Grad μ

and

Heﬀ = [H + σW] ∝ − Grad θ .

(31)

Remark 6. The proposed structure for the eﬀective entropy flux vector Heﬀ illustrates the coupling between heat conduction and species diﬀusion. The enthalpy associated with the diﬀusing species gives rise to the additional term σW that would be otherwise absent in a purely thermomechanical model. The proposed structure for the diﬀusion flux vector W emphasizes that the natural quantity conjugate to the diﬀusion flux vector is the gradient of the chemical potential. 3.6.1

Forms for the Constitutive Relationships

A relationship for the diﬀusion flux vector W that satisfies the reduced dissipation inequality given by Eq. (31)1 is obtained from a classical Stokes-Einstein type argument [10] wherein the molecular velocity (here the spatial velocity of the diﬀusing species relative to the bulk) is assumed proportional to the driving force (i. e. the gradient of the chemical potential μ) via the, potentially concentration, temperature and deformation dependent, second-order, positive-definite mobility tensor B. Thus, v − v = −B(C, ct , θ) · grad μ , and using the definition of the spatial diﬀusion flux vector w given on the right-hand side of Eq. (8) to obtain w = ct v − v

= −ct B(C, ct , θ) · f t · Grad μ .

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The relation for the diﬀusing species flux vector in the reference configuration is thus W = −c0 B(C, c0 , θ) · C−1 · Grad μ . (32) Remark 7. Equation (32) is Fick’s first law in the setting of finite deformations and along with Fick’s second law in Eq. (12) they, or the infinitesimal restrictions thereof, form the basis for the vast majority of models for case II diﬀusion. Note, that even for a constant mobility tensor B there is a coupling between deformation and diﬀusion due to the presence of the term C−1 which is a measure of the deformation. A relationship of the form suggested by the reduced dissipation inequality given by Eq. (31)2 for the eﬀective entropy flux vector Heﬀ is given by a non-standard Fourier-Duhamel type law of the form Heﬀ = −K(C, c0 , θ) · C−1 · Grad θ ,

(33)

where K is the positive-definite, second-order conductivity-like tensor. Remark 8. Note that, as in the constitutive relation for the diﬀusing species flux vector in Eq. (32), even for a constant conductivity-like tensor K there is geometric coupling in Eq. (33) due to the presence of the term C−1 . Additional coupling arises as the non-standard eﬀective entropy flux vector Heﬀ = [H + σW] contains contributions due to thermal conduction and species diﬀusion. The polymer is assumed to behave as a viscoelastic solid. An evolution equation for the inelastic internal variables Ξ that satisfies the reduced dissipation inequality is required. By defining χ := −∂ψ0 /∂Ξ as the non-equilibrium stress conjugate to the history variable Ξ, a suitable form for the evolution of the history variable is [19]: Dt Ξ = F (C, c0 , θ, Ξ) : χ ,

(34)

where F is the positive-definite, fourth-order fluidity tensor. For further details on the development of constitutive relations for viscoelastic solids the reader is referred to Simo and Hughes [19]. Remark 9. The vast majority of models for case II diﬀusion describe the polymer as a viscoelastic solid where the relaxation time of the polymer depends on the concentration of the diﬀusing species.

3.7 Temperature Evolution Equation The final step in the development of the governing equations is to determine the evolution equation for the temperature. Using the localized expression for the entropy balance (24) and the reduced dissipation inequality (30), the equation for the evolution of the entropy becomes

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θDt s0 = −W · Grad μ + θHeﬀ − Div (θHeﬀ ) + χ : Dt Ξ , and using the constitutive relation for the entropy in Eq. (29)3 , ∂ψ0 ∂θ ∂ = −θ P : Dt F + μDt c0 − s0 Dt θ − χ : Dt Ξ ∂θ ∂2 ψ0 ∂ = −θ Dt θ − θ P : Dt F + μDt c0 − χ : Dt Ξ . ∂θ∂θ ∂θ

θDt s0 = −θDt

cv

=⇒ cv Dt θ = −W · Grad μ + θHeﬀ − Div (θHeﬀ ) + χ : Dt Ξ ∂ +θ P : Dt F + μDt c0 − χ : Dt Ξ , ∂θ

(35)

where cv denotes the specific heat capacity at constant deformation in the bulk. Remark 10. Even ignoring inelastic eﬀects, the form of Eq. (35) is non-standard due to the coupling of heat conduction and diﬀusion. The first term on the right-hand side of the equation arises purely from species diﬀusion. The second and third terms on the right-hand side represent the influence of eﬀective entropy sources (thermal and diﬀusive) and the eﬀective entropy flux vector on the evolution of the temperature. The term involving the thermal gradient of the stress power P : Dt F gives rise to the well known Gough-Joule eﬀect (that is, structural thermoelastic heating) which couples the temperature evolution and the deformation [see e. g. 11].

4 Key Features of the Helmholtz Energy Required to Reproduce Case II Diﬀusion The model of case II diﬀusion developed by Govindjee and Simo [10] arises from restricting the governing equations derived in Sect. 3 to the isothermal regime and selecting a specific form for the Helmholtz energy. The highly non-linear form of the Helmholtz energy, in general motivated from sound micro-mechanical arguments, allows the model to reproduce the key features of case II diﬀusion. It should be emphasized that prior to the specialization of the Helmholtz energy, the framework developed in Sect. 3 simply describes viscoelastic deformation coupled to thermal diﬀusion and species conduction. The objective of this section is to describe briefly certain key features of the Helmholtz energy function required to reproduce case II diﬀusion. A general form for the Helmholtz energy ψ0 is chosen as ψ0 (C, c0 , θ, Ξ) = ψeq (C, c0 , θ) + ψneq (C, c0 , θ, Ξ) + ψmix 0 (J, c0 , θ) , 0 0

(36)

Geometrically Nonlinear Continuum Thermomechanics Coupled to Diﬀusion eq

103

neq

where ψ0 and ψ0 are the parts of the Helmholtz energy that account for the equilibrium and non-equilibrium (viscoelastic) response, respectively, and ψmix is the 0 energy associated with the mixing of the solvent and the polymer. The form of the energy associated with the equilibrium response accounts for the thermalmechanical and thermal-diﬀusion coupling as well as purely diﬀusive eﬀects. For the sake of brevity, further details are omitted here and [10, 12] should be consulted for additional information. The energy associated with non-equilibrium eﬀects and mixing are discussed in additional detail.

4.1 Energy Associated with Viscoelastic Eﬀects Two forms for the energy associated with viscoelastic eﬀects are discussed. The model by Holzapfel and Simo [12] accounts for thermal-mechanical coupling while the model of Govindjee and Simo [10] accounts for diﬀusional-mechanical coupling. The Helmholtz energy proposed by Holzapfel and Simo is of the form eq

neq

ψHS 0 (C, θ, Ξ) = ψ0 (C, θ) + ψ0 (C, θ, Ξ)

(37)

where eq

eq:tm

ψ0 (C, θ) = ψ0

eq:t

(C, θ) + ψ0 (θ) ,

eq:tm

(38) eq:t

where ψ0 accounts for the thermomechanical coupling and ψ0 accounts for purely thermal eﬀects (see e. g. [14] for additional details). The non-equilibrium part of the Helmholtz energy proposed by Holzapfel and Simo [12] was developed for elastomers, here assumed to be composed of identical polymer chains, and is given by: eq ∂ β ψ (C, θ) ∞ 0 neq eq ψ0 (C, θ, Ξ) = ζ|Ξ|2 − 2 : Ξ + β∞ ψ0 (C, θ) , (39) ∂C where ζ ∈ R+ is a temperature dependent parameter and β∞ ∈ R+ is a nondimensional constitutive parameter associated with the temperature dependent relaxation time τ. The form of the energy in Eq. (39) is non-standard due to the presence of the derivative with respect to a measure of the deformation C in the second term on the right-hand side. The inelastic process is assumed here to be describable via a single internal variable Ξ. Here and henceforth, Ξ shall thus denote a single internal variable as opposed to a set of internal variables. The choice of the energy in Eq. (39) renders the fourth-order fluidity tensor F defined in Eq. (34) as F=

1 I, 2ζτ

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where I is the fourth-order identity tensor. The choice of the non-equilibrium energy in Eq. (39) coincides with an evolution equation for the internal variable χ of the form: ⎞ ⎛ ⎫ ⎪ ⎜⎜⎜ ∂ β∞ ψeq (C, θ) ⎟⎟⎟ ⎪ χ 0 cpl ⎪ ⎪ ⎟⎟⎠⎟ − χ ⎪ Dt χ + = Dt ⎜⎜⎝⎜2 ⎬ τ ∂C in B0 × T , (40) ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ χ|t=0 = χ0 where the thermal coupling term χcpl is defined by χcpl = 2Dt ζ Ξ .

(41)

The inelastic evolution is, thus, coupled to the temperature via the temperature dependent relaxation time τ and the tensorial thermal coupling term χcpl . The isothermal model of Govindjee and Simo accounts for the influence of the concentration of the diﬀusing species c0 on the viscoelastic response of the polymer via a concentration dependent relaxation time τ chosen to reflect the glass-to-rubber transition that occurs in case II diﬀusion.

4.2 Energy Associated with Mixing One of the major distinctions between the Helmholtz energy proposed by Govindjee and Simo [10] and that implied by the widely adopted model of Wu and Peppas [29] lies in the definition of the energy of mixing.1 Govindjee and Simo proposed an energy of mixing which extends the widely adopted Flory-Huggins model [9, 13] to the transient case and thereby permits independent deformation and concentration fields (i. e. perfect mixing is not assumed). Govindjee and Simo’s approach is based on the micromechanical argument that an energetically easily accessible space (referred to as free volume) exists in the polymer and, thus, full swelling cannot be achieved. The vast majority of other models for case II diﬀusion either adopt the Flory-Huggins model or impose the perfect mixing assumption in another form.

5 Discussion and Conclusions The equations governing the inelastic response of a continuum coupled to heat conduction and species diﬀusion have been derived from fundamental balance principles and are summarized in Table 1. The choice of the Helmholtz energy specializes this general structure to capture the key features of case II diﬀusion. With the exception of the model by Govindjee and Simo [10], the vast majority of the models in the literature are restricted to one space dimension and do not directly account for the deformation of the solid. The one-dimensional model by Wu 1

For a detailed comparison of the models of Govindjee and Simo [10] and Wu and Peppas [29], the reader is referred to Vijalapura and Govindjee [27].

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Table 1 Summary of the key relations governing coupled heat conduction and species diﬀusion within an inelastic solid. Conservation of: solid mass

ρ0 = Jρt

species mass

Dt c0 = − Div W + W

linear momentum

Div P + b0 = 0

angular momentum

F · P t = P · Ft

internal energy

Dt u0 = P : Dt F + Qeﬀ − Div Qeﬀ

entropy

Dt s0 = π0 + Heﬀ − Div Heﬀ

Constitutive relations Dissipation inequality

Temperature evolution

∂ψ0 ∂θ ∂ψ0 δ0 = −W · Grad μ − Heﬀ · Grad θ − : Dt Ξ ≥ 0 ∂Ξ =⇒ W ∝ − Grad μ and Heﬀ ∝ − Grad θ P = 2F ·

∂ψ0 , ∂C

μ=

∂ψ0 , ∂c0

s0 = −

cv Dt θ = −W · Grad μ + θHeﬀ − Div (θHeﬀ ) + χ : Dt Ξ ∂ +θ P : Dt F + μDt c0 − χ : Dt Ξ ∂θ

and Peppas [29] can be seen as a restriction of the model by Govindjee and Simo to infinitesimal deformations with the energy of mixing based on the Flory-Huggins approximation. Such fundamental restrictions seem excessive given that case II diffusion is a multi-dimensional process characterised by large deformations. Weakly-coupled models for case II diﬀusion take as their point of departure a simple model of stress-assisted diﬀusion due to Cottrell [6] wherein a term involving the gradient of the trace of the stress in the solid is added to Fick’s second law (32). The inelastic part of the stress is assumed to evolve according to a one-dimensional, infinitesimal, isothermal counterpart of the evolution equation for the inelastic internal stress (40) where the relaxation time is a function of the concentration. The thermodynamic consistency of such an approach is unclear. Weakly-coupled models are clearly unable to accurately describe actual three-dimensional applications undergoing case II type diﬀusion. The equations governing classical Fickian diﬀusion are parabolic. Hence, disturbances are propagated instantaneously throughout the domain. An infinite speed of disturbance propagation is physically unrealistic. This description is clearly inadequate when attempting to address the phenomena observed during case II diﬀusion where a sharp front propagates at a finite speed through the polymeric solid. An alternative framework to describe case II diﬀusion, termed hyperbolic diﬀusion, is motivated by the wave-like nature of the sharp front separating the glassy and swollen regions of the polymer. The key feature of hyperbolic models of diﬀusion

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is the introduction of an additional term into Ficks’s first law (32) involving the time derivative of the diﬀusion flux vector. This results in a hyperbolic model for diﬀusion when substituted into Ficks’s second law (12). The general framework for diﬀusion proposed by Aifantis [1] includes hyperbolic diﬀusion. Maxwell [17] originally included the time derivative term when developing the equations governing heat conduction from the classical kinetic theory of gases but subsequently discarded it as he viewed the contribution as insignificant. A hyperbolic diﬀusion model developed by Kalospiros et al [15] captures many of the key phenomena that characterize case II diﬀusion. The relative importance of the temporal term is also elucidated upon in the alternative Hamiltonian framework by El Afif and Grmela [7]. The extension of the framework presented here to include surface eﬀects, which play a role in case II diﬀusion, has been undertaken in McBride et al [18]. Furthermore, the framework has been recast in the illuminating configurational setting by Steinmann et al [21]. Acknowledgements. The financial support of the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), grant STE 544/39-1, is gratefully acknowledged. The first author thanks the National Research Foundation of South Africa for their support.

References [1] Aifantis, E.C.: On the problem of diﬀusion in solids. Acta Mechanica 37, 265–296 (1980) [2] Alfrey, T., Gurnee, E.F., Lloyd, W.G.: Diﬀusion in glassy polymers. Journal of Polymer Science Part C 12, 249–261 (1966) [3] Bargmann, S., McBride, A.T., Steinmann, P.: Models of solvent penetration in glassy polymers with an emphasis on case II diﬀusion. A comparative review. Applied Mechanics Reviews, doi:10.1115/1.4003955 (in press) [4] Cohen, D.S.: Theoretical models for diﬀusion in glassy polymers. Journal of Polymer Science: Polymer Physics Edition 2(22), 1001–1009 (1984) [5] Cohen, D.S., White, A.B.: Sharp fronts due to diﬀusion and stress at the glass transition in polymers. Journal of Polymer Science: Part B: Polymer Physics 27, 1731–1747 (1989) [6] Cottrell, A.H.: Eﬀect of solute atoms on the behavior of dislocations. In: Wills, H.H. (ed.) Report of a Conference on Strength of Solids, pp. 30–36. The Physical society, University of Bristol, London (1948) [7] El Afif, A., Grmela, M.: Non-Fickian mass transport in polymers. Journal of Rheology 46(3), 591–628 (2002) [8] Fick, A.: Über Diﬀusion. Poggendorﬀ’s Annalen der Physik und Chemie 94, 58–86 (1855) [9] Flory, P.J.: Thermodynamics of high polymer solutions. Journal of Chemical Physics 9(8), 66–661 (1941) [10] Govindjee, S., Simo, J.C.: Coupled stress–diﬀusion: Case II. Journal of the Mechanics and Physics of Solids 41(5), 863–887 (1993)

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[11] Holzapfel, G.A., Simo, J.C.: Entropy elasticity of isotropic rubber-like solids at finite strains. Computer Methods in Applied Mechanics and Engineering 132(1-2), 17–44 (1996) [12] Holzapfel, G.A., Simo, J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. International Journal of Solids and Structures 33(20-22), 3019–3034 (1996) [13] Huggins, M.L.: Solutions of long chain compounds. Journal of Chemical Physics 9, 440 (1941) [14] Javili, A., Steinmann, P.: On thermomechanical solids with boundary structures. International Journal of Solids and Structures 47(24), 3245–3253 (2010) [15] Kalospiros, N.S., Ocone, R., Astarita, G., Meldon, J.H.: Analysis of anomalous diﬀusion and relaxation in solid polymers. Industrial & Engineering Chemistry Research 30, 851–864 (1991) [16] Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice–Hall Inc., Englewood Cliﬀs (1969) [17] Maxwell, J.C.: On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London 157, 49–88 (1867) [18] McBride, A.T., Javili, A., Steinmann, P., Bargmann, S.: Geometrically nonlinear continuum thermomechanics with surface energies coupled to diﬀusion (2011) (review) [19] Simo, J.C., Hughes, T.J.R.: Computational inelasticity. In: Marsden, J.E., Wiggins, S., Sirovich, L. (eds.) Interdisciplinary Applied Mathematics, vol. 7. Springer, Heidelberg (1998) [20] Steinmann, P.: On boundary potential energies in deformational and configurational mechanics. Journal of the Mechanics and Physics of Solids 56, 772–800 (2008) [21] Steinmann, P., McBride, A., Bargmann, S., Javili, A.: A deformational and configurational framework for geometrically nonlinear continuum thermomechanics coupled to diﬀusion. International Journal of Non-Linear Mechanics, doi:10.1016/j.ijnonlinmec.2011.05.009 (in press) [22] Thomas, N.L., Windle, A.H.: A deformation model for case ii diﬀusion. Polymer 21(6), 613–619 (1980) [23] Thomas, N.L., Windle, A.H.: Diﬀusion mechanics of the system pmma-methanol. Polymer 22(5), 627–639 (1981) [24] Thomas, N.L., Windle, A.H.: A theory of case II diﬀusion. Polymer 23(4), 529–542 (1982) [25] Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 3rd edn. Springer, New York (2004) [26] Vesely, D.: Diﬀusion of liquids in polymers. International Materials Reviews 53(5), 299–315 (2008) [27] Vijalapura, P.K., Govindjee, S.: Numerical simulation of coupled-stress case II diﬀusion in one dimension. Journal of Polymer Science: Part B: Polymer Physics 41, 2091–2108 (2003) [28] Vijalapura, P.K., Govindjee, S.: An adaptive hybrid time-stepping scheme for highly non-linear strongly coupled problems. International Journal for Numerical Methods In Engineering 64, 819–848 (2005) [29] Wu, J.C., Peppas, N.A.: Modeling of penetrant diﬀusion in glassy polymers with an integral sorption Deborah number. Journal of Polymer Science: Part B: Polymer Physics 31, 1503–1518 (1993)

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics Marc-André Keip and Jörg Schröder

The authors would like to express their gratitude for the opportunity to contribute to this Springer volume on the occasion of the 60 th birthday of Professor Wolfgang Ehlers.

Abstract. The present contribution discusses a two-scale homogenization procedure for the continuum mechanical modeling of heterogeneous electro-mechanically coupled materials. The direct meso-macro formulation is implemented into an FE2 homogenization environment, which allows for the computation of a macroscopic boundary value problem in consideration of attached heterogeneous representative volume elements at each macroscopic point. The resulting homogenization approach is capable of computing the eﬀective elastic, piezoelectric, and dielectric properties of electro-mechanically coupled materials in consideration of arbitrary mesostructures.

1 Introduction Electro-mechanically coupled solids as piezo- and ferroelectric ceramics, from which the most prominent are the electroceramics Barium Titanate (BaTiO3 ) and Lead Zirconate Titanate (PZT), are in general polycrystalline materials and possess a heterogeneous micro- and mesostructure. The material behavior on each scale may strongly diﬀer from each other and therefore it has to be distinguished between the respective behavior on the micro-, meso-, and the macro-scale. Here, the microscopic scale may be considered as a spontaneous polarized crystal unit cell or a domain comprising homogeneously polarized unit cells, the mesoscopic scale may represent a grain containing a number of ferroelectric domains, and the macro-scale can be interpreted as the bulk ferroelectric material or a certain technical device, see Fig. 1. In general, the macroscopic response is strongly related to the composition of the micro- and mesostructure and dependent on the physical and geometrical Marc-André Keip · Jörg Schröder Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, Universitätsstraße 15, 45141 Essen, Germany e-mail: {marc-andre.keip,j.schroeder}@uni-due.de B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 109–128. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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Fig. 1 Macro-, meso-, and micro-scale of a ferroelectric material: schematic representation of a piezoelectric stack actuator (left), ferroelectric domains (middle), and ferroelectric unit cell of BaTiO3 (right).

constitution of the lower scales. Physically, the microscopic single crystal properties of BaTiO3 are well understood, see e. g. [59]. However, the overall macroscopic behavior of a bulk ferroelectric is also influenced by the specific geometry of the underlying micro- and mesostructure, which may exhibit certain inhomogeneities as for instance inclusions. Such materials can in general be referred to as multiphase materials. One class of multiphase solids that exhibits electro-mechanical coupling are piezocomposites, which are composed of a certain matrix material in combination with a diﬀerent material of the inclusions. Geometrically, such electroactive composites can be subdivided into specific types depending on the geometry of the matrix and the inclusions, see Fig. 2, where three typical kinds of two-phase piezocomposites are depicted.

a)

b)

c)

Fig. 2 Piezocomposites: a) 0-3 composite, b) 2-2 composite, and c) 1-3 composite.

One of the main characteristics of a composite is the connectivity of the phases it is composed of. It can be expressed by the number of dimensions in which the respective phase is continuously connected throughout a given volume. Each component of the composite can be connected in 0, 1, 2, or 3 dimensions. For example, the inclusions of the composite depicted in Fig. 2a) are not connected to each other and have therefore the connectivity 0. On the other hand, the matrix material is continuously distributed in three dimensions and has thus the connectivity 3. Supposed that this composite represents a composite composed of piezoelectric inclusions embedded in a polymeric matrix, the result is a 0-3 composite; an often used convention states that the first number refers to the more piezoactive material of the composite, see e. g. [52], where a comprehensive overview on piezocomposites is given.

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In each of the above mentioned cases, the prediction of the resulting eﬀective properties of the heterogeneous electro-mechanically coupled material is of great importance. In this connection it has in general to be distinguished between upper and lower bounds and appropriate estimates of the overall properties of the respective material. In case of mechanical properties, Voigt computed eﬀective elastic moduli assuming a constant strain field throughout the material [55]. On the other hand, the eﬀective elastic properties derived by Reuss are achieved assuming a constant stress field, see [37]. Later, Hill [18] showed that the properties computed by Voigt and Reuss are respectively upper and lower bounds of the elastic properties and are therefore referred to as Voigt and Reuss bounds. In the same contribution, Hill states that empirically the geometric and arithmetic mean of the Voigt and Reuss bounds serve as good approximations. Therefore, the latter one is also termed a Voigt-Reuss-Hill average, see [8]. Other estimates of such bounds are mainly based on the fundamental works [15–17], and later [13, 25, 34, 56, 57]. Analytical homogenization schemes that were developed for the computation of appropriate estimates of overall eﬀective material moduli are mainly based on the equivalent inclusion method based on Eshelby’s problem of a single ellipsoidal inclusion embedded in an infinite matrix [11]. It is the basis for e. g. the self-consistent method ([20], [5]), the Mori-Tanaka method ([33]), and the diﬀerential method ([35], [14]), compare [36]. General works on homogenization theory are [21, 22, 49] and [24]. The above mentioned methods have been applied for the prediction of mechanical as well as non-mechanical properties. In the following, a short literature review on homogenization of piezoelectric solids is given. In [6] exact results for the overall properties of piezoelectric composites consisting of transversely isotropic phases are proposed. Estimates for overall thermo-electro-elastic moduli of multiphase fibrous composites based on self-consistent and Mori-Tanaka methods are given in [7]. Eﬀective quantities of two-phase composites are evaluated in [10] and [9] using e. g. dilute, self-consistent, and Mori-Tanaka schemes. In this context we refer also to [1–4] and [12]. Universal bounds for eﬀective piezoelectric properties of heterogeneous materials are derived in [23], utilizing generalized Hashin-Shtrikman variational principles. As mentioned above, the overall properties of composites depend on the morphology of their mesostructure and the properties of the individual constituents. Therefore, it is possible to improve the performance of piezoelectric materials by means of topology optimization and homogenization techniques, see e. g. [45] and [46]. Utilizing a unit-cell method, [26] investigate the relation between eﬀective properties and the geometry of microvoids based on a three-dimensional finite element analysis. A multi-scale finite element modeling procedure for the macroscopic description of polycrystalline ferroelectrics is proposed by [53] and [54]; here a homogenization procedure based on asymptotic expansions of the displacements and the electric potential is utilized, for the mathematical background see e. g. [38]. An approximation of macroscopic polycrystals by discrete orientation distribution functions is discussed in [44]. This procedure is associated to the above mentioned Reuss- and Voigt-bounds. An algorithm for the description of heterogeneous coupled thermo-electro-magnetic continua is presented in [60].

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In the following a general direct homogenization procedure, which couples the macroscopic to the mesoscopic scale, is presented; in this context see also [28– 32, 39, 47, 48, 50, 51, 58, 61]. The procedure is as follows: 1. At each macroscopic point: Localize suitable macroscopic quantities (e. g. the strains and the electric field) to the mesoscale. To be more specific, apply constraint conditions or boundary conditions, e. g. driven by the macroscopic strains and the electric field, on a representative volume element, see e. g. [40] and [27]. 2. Next, solve the equations of balance of linear momentum and Gauß’s law on the mesoscale under the applied macroscopic loading conditions in order to obtain the dual mesoscopic quantities (e. g. the stresses and the electric displacements). 3. Next, perform a homogenization step, i. e. compute the average values of the dual quantities on the mesoscale. These macroscopic variables have to be transferred to the associated points of the macroscale. 4. Finally, solve the electro-mechanically coupled boundary value problem on the macroscale and proceed with step 1 until convergence is obtained on both scales. The numerical solution is based on separate finite element analyzes on each scale. The overall algorithmic moduli needed for the Newton-Raphson iteration scheme on the macroscale are eﬃciently computed during the standard solution procedure on the mesoscale.

2 Boundary Value Problems on the Macro- and the Mesoscale In the following we describe the electro-mechanically coupled boundary value problems (BVP) on both scales. The material behavior for the individual constituents on the mesoscale are modeled within a coordinate invariant formulation. Here we restrict ourselves to transversely isotropic material as presented in [41].

2.1 Macroscopic Electro-Mechanically Coupled BVP Let the body of interest on the macroscopic scale B ⊂ IR3 be parameterized in x. The macroscopic displacement field and the macroscopic electric potential are denoted as u and φ, respectively. The basic kinematic and electric variables are the linear strain tensor ε(x) := sym[∇u(x)] and the electric field vector E(x) := −∇ φ(x), where ∇ denotes the gradient operator w. r. t. x. The governing field equations for the quasi-static case are the balance of linear momentum and Gauß’s law div x [σ] + f = 0 and divx [ D] = q

in B ,

(1)

where div x denotes the divergence operator with respect to x, σ represents the symmetric Cauchy stress tensor, f is the given mechanical body force, D denotes the vector of electric displacements, and q is the given density of free charge carriers. The boundary conditions in terms of displacements and surface tractions t are u = ub

on ∂Bu

and

t = σ · n on ∂Bσ ,

(2)

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

113

and in terms of the electric potential and the electric surface charge Q we write φ = φb

on ∂Bφ

and

− Q = D · n on ∂BD ,

(3)

where n is a unit normal vector pointing outwards from the surface of the body, see also Fig. 3. t¯

Q

n ∂Bφ

x∈B

x∈B

∂Bu

∂BD

∂Bσ

Fig. 3 Boundary decomposition of ∂B into mechanical, i. e. ∂Bu ∪ ∂Bσ = ∂B with ∂Bu ∩ ∂Bσ = ∅, and electrical parts: ∂Bφ ∪ ∂BD = ∂B with ∂Bφ ∩ ∂BD = ∅.

In the following we do not postulate the existence of a thermodynamical potential on the macroscale. Instead, we attach a representative volume element (RVE) at each macroscopic point x, that delivers the constitutive response, see Fig. 4. RVE ε, σ

RVE E, D

u ∈ RVE

x

φ ∈ RVE

x

x ∈ RVE

x ∈ RVE

∂RVE

∂RVE

Fig. 4 Attached RVE at x, associated to the macroscopic mechanical and electrical quantities.

In order to link the macroscopic variables {ε, σ, E, D} with their microscopic counterparts {ε, σ, E, D}, we define in this two-scale approach the macroscopic variables in terms of some suitable surface integrals over the boundary of the RVE with volume V. It should be remarked, that a definition of macroscopic quantities in terms of surface integrals is necessary in general. Respective definitions of macroscopic values by means of simple volume averages could lead to physically unreasonable results and would not allow for reliable interpretations of simple experiments, see e. g. [39]. The macroscopic strains and stresses are given by 1 1 ε := sym[u ⊗ n] da and σ := sym[t ⊗ x] da , (4) V V ∂RVE

∂RVE

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where u and t are the displacement and traction vectors on the boundary of the RVE, respectively. Furthermore, the macroscopic electric field and electric displacements are defined by the surface integrals 1 1 E := −φ n da and D := −Q x da , (5) V V ∂RVE

∂RVE

governed by the electric potential φ and the electric charge density Q on ∂RVE.

2.2 Mesoscopic Electro-Mechanically Coupled BVP On the mesoscopic scale we consider a BVP defined on the RVE ⊂ IR3 , which is parameterized in the mesoscopic Cartesian coordinates x. The governing balance equations are the balance of linear momentum neglecting body forces, and the Gauß’s law neglecting the density of free charge carriers div[σ] = 0

and div[ D] = 0

in RVE .

(6)

The mesoscopic strains and electric field vector are given by ε := sym[∇u(x)] and E := −∇φ(x), where ∇ denotes the gradient operator and div the divergence operator with respect to x. In order to complete the description of the mesoscopic BVP we define some appropriate boundary conditions on ∂RVE or some constraint conditions in the whole RVE. Therefore, we apply a generalized macro-homogeneity condition 1 1 ˙ σ : ε˙ + D · E = σ : ε˙ dv + D · E˙ dv , (7) V V RVE

RVE

in this context see [19]. The generalized macro-homogeneity condition is fulfilled if P = 0 holds, where 1 1 ˙ . P := σ : ε˙ dv − σ : ε˙ + D · E˙ dv − D · E (8) V V RVE

RVE

The simplest assumption of the mesoscopic fields that automatically fulfills the condition P = 0 is achieved by setting σ = σ = const. or

ε˙ = ε˙ = const.

(9)

D = D = const. or

˙ = const. E˙ = E

(10)

and

for all points of the mesoscale. Equations (9)1,2 lead to the well-known Reuss- and Voigt-bounds, respectively. In the following we denote (9) and (10) as constraint conditions. More sophisticated expressions for suitable boundary conditions can be derived from the equivalent expression to (8)

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

115

1 1 ˙ ˙ P= (t − σ · n) · (u˙ − ε · x) da + (Q + D · n) (φ˙ + E · x)da, V V ∂RVE ∂RVE P1

(11)

P2

where we utilized the Gauß theorem, the balance of linear momentum (6), the Cauchy theorem t = σ · n, Gauß’s law and Q = −D · n. Evaluating P1 = 0 leads to the Neumann- or Dirichlet-boundary conditions for the mechanical part t = σ · n on ∂RVE

u˙ = ε˙ · x on ∂RVE .

or

(12)

Possible periodic boundary conditions, satisfying P1 = 0, are t + (x+ ) = −t − (x− )

+ (x+ ) = w − (x− ) and w

x± ∈ ∂RVE± ,

on

(13)

for an illustration see Fig. 5. + +

−

+

ε = ε + ∇ w

u ∈ RVE

n+

−

∂RVE

−

x ∈ RVE n−

+

u=ε·x+w

∂RVE

−

Fig. 5 Mesoscopic mechanical BVP: periodic boundary conditions on ∂RVE.

In analogy to the derivations of the mechanical boundary conditions we obtain possible boundary conditions for the electrical part by evaluating the expression P2 = 0. Possible Neumann- or Dirichlet-boundary conditions then appear as Q = −D · n on ∂RVE

or

˙ φ˙ = − E · x

on ∂RVE .

(14)

Periodic boundary conditions, satisfying P2 = 0, are given by the conditions Q+ (x+ ) = −Q− (x− )

and φ+ (x+ ) = φ− (x− ) on

x± ∈ ∂RVE± ,

(15)

for an illustration see Fig. 6. For a detailed derivation of the respective boundary conditions we refer to [40]. +

∂RVE

n−

x ∈ RVE −

∂RVE −

+

−

+

φ = −E · x + φ

E = E − ∇ φ n+

+ φ ∈ RVE −

Fig. 6 Mesoscopic electrical BVP: Periodic boundary conditions on ∂RVE.

116

M.-A. Keip and J. Schröder

3 Eﬀective Properties of Piezoelectric Materials The aim of this two-scale approach is to achieve quadratic convergence within the Newton-Raphson iteration scheme for the discretized BVP on the macroscale. For this we have to perform the consistent linearization of the macroscopic stresses and electric displacements with respect to the macroscopic strains and electric field. In detail, we need the macroscopic (overall) mechanical moduli C, piezoelectric moduli e, and dielectric moduli , which enter the incremental constitutive relations Δσ = C : Δε − eT ΔE , −Δ D = − e : Δε − ΔE .

(16)

Formally, we obtain the overall moduli by the partial derivatives of the volume averages of the mesoscopic stresses and electric displacements with respect to the macroscopic strains and electric field, i. e. ⎧ ⎫ ⎧ ⎫ ⎤ ⎡ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ ⎥⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎡ ∂ σ dv ∂ σ dv ⎪ ⎪ ⎢⎢⎢ ε ⎪ ⎡ ⎤ ⎤ E⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩RVE ⎭ ⎩RVE ⎭ ⎥⎥⎥⎥ ⎢⎢⎢ Δε ⎥⎥⎥ ⎢⎢⎢⎢ Δσ ⎥⎥⎥⎥ 1 ⎢⎢⎢⎢ ⎥ ⎥⎥⎥ . ⎧ ⎫ ⎧ ⎫ ⎥⎥⎥ ⎢⎢⎢ ⎢⎣⎢ ⎥⎦⎥ = ⎢⎢⎢ (17) ⎪ ⎪ ⎪ ⎪ V ⎢⎢⎢ ⎪ ⎪ ⎪ ⎪ ⎥⎥⎥⎥ ⎣ ΔE ⎦ −ΔD ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢⎢⎢ ⎬ ⎨ ⎬ ⎥⎥⎥ ⎢⎢⎢ −∂ε ⎨ D dv⎪ −∂ E ⎪ D dv⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎥ ⎪ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ ⎪ ⎪ ⎩RVE ⎭ ⎩RVE ⎭⎦ For the analysis of the mesoscopic BVP we additively split the mesoscopic strains and electric field into a constant part and a fluctuating part, i. e. ε = sym[∇u(x)] = ε + ε with ε dv = 0 (18) RVE

and with E = −∇φ = E + E

dv = 0. E

(19)

RVE

Exploiting these relations in (17) leads after application of the chain rule to ⎡ ⎤ ⎢⎢⎢ Δσ ⎥⎥⎥ 1 ⎢⎢⎢ ⎥⎥⎥ = ⎣ ⎦ V −ΔD

⎛ ⎡ ⎤ ⎜⎜⎜ ⎢⎢⎢ C −eT ⎥⎥⎥ ⎜⎜⎜ ⎢⎢⎢ ⎥⎥⎥ ⎜⎜⎝ ⎢⎢⎣ ⎥⎥⎦ dv + RVE −e − RVE

⎡ ⎤ ⎞ ⎥⎥ ⎟⎟ ⎡⎢ Δε ⎤⎥ ⎢⎢⎢ C : ∂ε ε −eT · ∂ E E ⎥⎥⎥ ⎥⎥⎥ ⎟⎟⎟ ⎢⎢⎢ ⎢⎢⎢ ⎥⎥⎦ . ⎥⎥⎦ dv⎟⎟⎟ ⎢⎢⎣ ⎢⎣ ⎠ ΔE −e : ∂ε ε − · ∂ E E

(20)

In order to compute the sensitivities of the fluctuation fields w. r. t. the macroscopic counterparts we first consider the coupled BVPs on the mesoscale. The weak forms of the balance of linear momentum and Gauß’s law are

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

Gu = −

divσ · δ w dv =

RVE

δ ε : σ dv −

RVE

δ w · (σ · n) da

(21)

∂RVE

Gext u

Gint u

and

117

divD δ φ dv =

Gφ = − RVE

· D dv − δE

RVE

δ φ( D · n) da,

(22)

∂RVE

Gext φ

Gint φ

respectively. The linearization, LinGu,φ = Gu,φ (•) + ΔG u,φ , of the weak forms yields for conservative loadings the linear increments int dv ΔGu = δ ε : ∂ε σ : Δ ε dv + δ ε : ∂ Eσ · Δ E (23) RVE

and ΔGint φ =

RVE

· ∂ε D : Δ δE ε dv +

RVE

· ∂E D · ΔE dv, δE

(24)

RVE

wherein we identify the tangent moduli C := ∂ε σ,

e = − {∂ E σ}T = ∂ε D,

= ∂ E D.

(25)

Finite Element Approximations: For the discretization of the weak forms we only have to account for the fluctuation fields of the displacements and electric potential. Thus, the approximation of the fluctuation fields, virtual fluctuation fields and incremental fluctuation fields appear as = w

n node

NuI duI ,

δ w=

I=1

and φ=

n node

n node

NuI δ duI ,

Δ w=

I=1

φ NφI dI ,

I=1

δ φ=

n node

n node

NuI Δ duI

(26)

φ NφI ΔdI ,

(27)

I=1

φ NφI δdI ,

I=1

Δ φ=

n node I=1

where Nu,φ denote the ansatz functions, du,φ the nodal degrees of freedom and nnode the number of nodes per element. Let Beu and Beφ characterize the B-Matrices associated to fluctuation strains and electric field. The finite element approximations of the actual, virtual and incremental strains are ε = Beu du,

δ ε = Beu δ du ,

Δ ε = Beu Δ du .

Analogously, the fields associated to the electric potential appear as = −Beφ E dφ ,

= Beφ δ δE dφ ,

= −Beφ Δ ΔE dφ .

118

M.-A. Keip and J. Schröder

Inserting the latter expressions into the linearized weak forms yields the discrete counterparts ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ nele ⎪ ⎪ ⎪ ⎪ ⎨ uT eT e u eT T e φ e⎬ δd ⎪ Bu CBu dv Δ d + Bu e Bφ dv Δ d + ru ⎪ =0 (28) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎪ V V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ keuu ke uφ

and

⎧ ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ n ele ⎪ ⎪ ⎪ ⎪ φT ⎨ eT e u eT e φ e⎬ δd ⎪ B eB dv Δ d − B B dv Δ d + r = 0, φ u φ φ φ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ e=1 ⎪ ⎪ V V ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ke ke φu

reu

(29)

φφ

reφ ;

with the right hand sides and nele denotes the number of finite elements. After the standard assembling procedure, ⎛ ⎞ ⎜⎜⎜ ⎟⎟ ⎡ u ⎤ ⎜⎜⎜⎜⎡ e ⎤ ⎡ u ⎤ ⎡ e ⎤ ⎟⎟⎟⎟⎟ e nele ⎢ δ d ⎥ ⎜⎢ k ⎢⎢⎢ ⎥⎥⎥ ⎜⎜⎜⎢⎢⎢ uu kuφ ⎥⎥⎥⎥ ⎢⎢⎢⎢ Δ d ⎥⎥⎥⎥ ⎢⎢⎢⎢ ru ⎥⎥⎥⎥ ⎟⎟⎟⎟ ⎢⎢⎢ ⎥ ⎜⎢ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ + ⎢⎢⎢ ⎥⎥⎥ ⎟⎟⎟ = 0, (30) ⎣ φ ⎥⎥⎦ ⎜⎜⎜⎜⎜⎢⎢⎣ e ⎣ e ⎦ ⎟⎟⎟ e ⎦ ⎣ φ ⎦ k k rφ ⎟⎟ e=1 δ d Δd φu φφ ⎜⎜⎜⎜ ⎟⎟⎠ ⎝ e e Δ De

K

R

we formally get the solution by inversion Δ D = −K −1 R,

nele

with

K=

A e=1

nele

Ke

and

R=

A R. e

(31)

e=1

Computation of Overall Moduli: Now we compute the sensitivities of the mesoscopic fluctuation fields w. r. t. their incremental macroscopic counterparts. Therefore we linearize the weak forms at an equilibrium state: RVE

RVE

δ ε : C : Δε + Δ ε dv −

RVE

· e : Δε + Δ δE ε dv +

dv = 0, δ ε : eT · ΔE + Δ E · · ΔE + Δ E dv = 0. δE

(32)

RVE

After inserting the finite element approximations of the displacements and electric potential we obtain

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

119

⎛ ⎧ ⎪ ⎜⎜⎜ ⎪ ⎜ ⎪ ⎨ uT ⎜⎜⎜⎜ eT e u ⎜ δ d B C dv Δε + BeT ⎪ ⎜ u u CB dv Δ d ⎪ ⎜⎜⎜ ⎪ ⎪ ⎜⎝ Be e=1 ⎪ ⎪ ⎩ Be

⎪ nele ⎪ ⎪

e luu

−

T BeT u e dv ΔE +

Be

T e φ BeT u e Bφ dv Δ d

Be

e luφ

and

⎞⎫ ⎟⎟⎟⎪ ⎪ ⎪ ⎪ ⎟⎟⎟⎟⎪ ⎪ ⎟⎟⎟⎬ =0 ⎪ ⎟⎟⎟⎪ ⎪ ⎪ ⎟⎠⎪ ⎪ ⎭

keuu

(33)

keuφ

⎛ ⎧ ⎪ ⎜⎜⎜ ⎪ ⎜ ⎪ ⎨ φT ⎜⎜⎜⎜ eT e u ⎜ δ d B · e dv Δε + BeT ⎪ ⎜⎜⎜ φ φ eBu dv Δ d ⎪ ⎪ ⎪ ⎜ ⎜ e=1 ⎪ ⎪ ⎝ Be ⎩ Be

⎪ nele ⎪ ⎪

e lφu

keφu

+

BeT φ dv ΔE −

Be

e φ BeT φ B dv Δ d

Be

e lφφ

keφφ

⎞⎫ ⎟⎟⎟⎪ ⎪ ⎪ ⎟⎟⎟⎪ ⎪ ⎬ ⎟⎟⎟⎪ ⎪ = 0. ⎟⎟⎟⎟⎪ ⎪ ⎪ ⎟⎠⎪ ⎪ ⎭

(34)

In a more compact notation we write nele

! e e δ duT luu Δε + keuu Δ du + luφ ΔE + keuφ Δ dφ = 0

(35)

! e e δ dφT lφu Δε + keφu Δ du + lφφ ΔE + keφφ Δ dφ = 0.

(36)

e=1

and

nele e=1

After the standard assembling procedure we obtain ⎡ u ⎤T ⎛⎡ ⎢⎢⎢ δ D ⎥⎥⎥ ⎜⎜⎜⎢⎢ Kuu ⎥⎥ ⎜⎜⎢⎢⎢ ⎢⎢⎢⎢ ⎣ φ ⎥⎥⎦ ⎜⎜⎝⎢⎣ Kφu δD

⎤ ⎡ ⎤⎞ ⎤⎡ Kuφ ⎥⎥ ⎢⎢⎢ Δ Du ⎥⎥⎥ ⎢⎢⎢ Luu Δε + Luφ ΔE ⎥⎥⎥⎟⎟⎟ ⎥⎥⎥ ⎢⎢ ⎥⎥⎥ + ⎢⎢⎢ ⎥⎥⎥⎟⎟⎟ = 0 ⎥⎦ ⎢⎣⎢ ⎥⎦ ⎢⎣ ⎥⎦⎟⎠ φ Kφφ Δ D Lφu Δε + Lφφ ΔE

(37)

with the abbreviations nele

Kuu =

A e=1

nele

keuu ,

Kuφ =

e=1

nele

Luu =

A e=1

A

nele

keuφ ,

Kφu =

nele

e luu ,

Luφ =

A e=1

A e=1

nele

keφu ,

Kφφ =

nele

e luφ ,

Lφu =

A e=1

Ak e=1

e φφ ,

(38)

nele

e lφu ,

Lφφ =

A e=1

e lφφ .

120

M.-A. Keip and J. Schröder

The solution, i. e. the incremental nodal fluctuations due to the incremental macroscopic strains and electric field, is formally given by ⎡ ⎤ ⎡ ⎢⎢⎢ Δ ⎢ Kuu Du ⎥⎥⎥ ⎥⎥⎥ = − ⎢⎢⎢⎢⎢ ⎢⎢⎢⎢ ⎥ ⎢⎣ ⎣ φ ⎦ Kφu ΔD

⎤−1 ⎡ ⎤ Kuφ ⎥⎥⎥ ⎢⎢⎢ Luu Δε + Luφ ΔE ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ . ⎥⎦ ⎢⎣ ⎥⎦ Kφφ Lφu Δε + Lφφ ΔE

(39)

Inserting the finite element approximations in (20) and exploiting the definitions of the L-Matrices (38) leads to ⎡ ⎤ ⎡ T T ⎤ ⎡ ⎡ ⎤ ⎤ ⎢⎢ C −eT ⎥⎥ ⎢⎢⎢ C −eT ⎥⎥⎥ ⎢L L ⎥ u 1 ⎢⎢⎢⎢⎢ uu φu ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ ∂ε Δ D ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥⎥⎥ ⎥⎥⎥⎥ = 1 ⎢⎢⎢⎢ (40) ⎥ ⎢ ⎥⎥ . ⎢⎢⎣ ⎥⎥⎦ dv + ⎢⎢⎢ ⎢⎢⎣ ⎥⎥⎦ V V ⎣ T T ⎥⎥⎦ ⎢⎣ ∂ Δ D φ ⎦ Luφ Lφφ −e − E RVE −e − Obviously, we have to compute the sensitivities of the nodal fluctuations w. r. t. the macroscopic strains and electric field. Utilizing (39), we obtain the derivatives of the incremental solutions ⎡ ⎤−1 ⎡ ⎤ ⎢⎢⎢ Kuu Kuφ ⎥⎥⎥ ⎢⎢⎢ ∂ε Δ Du ⎥⎥⎥ ⎢ ⎥⎥⎥ ⎥⎥⎥ = − ⎢⎢ ⎢⎢⎢⎢ ⎢⎢⎣ ⎥⎥⎦ ⎥⎦ ⎣ ∂ EΔ Dφ Kφu Kφφ

⎡ ⎤ ⎢⎢⎢ Luu Luφ ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ , Lφu Lφφ

which leads us to the final algorithmic expression for the macroscopic (overall) electro-mechanical tangent moduli ⎡ ⎤ ⎢⎢⎢ C −eT ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ 1 ⎥⎥⎥ = ⎢⎢⎢ ⎣ ⎦ V −e − RVE

⎡ ⎤ ⎢⎢⎢ C −eT ⎥⎥⎥ ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎣ ⎥⎥⎦ dv −e −

⎡ T T ⎤ ⎡ ⎢L L ⎥ 1 ⎢⎢⎢⎢⎢ uu φu ⎥⎥⎥⎥⎥ ⎢⎢⎢⎢⎢ Kuu − ⎢ ⎥ ⎢ V ⎢⎢⎣ T T ⎥⎥⎦ ⎢⎣ K φu Luφ Lφφ

⎤−1 Kuφ ⎥⎥⎥ ⎥⎥⎥ ⎦⎥ Kφφ

⎡ ⎤ ⎢⎢⎢ Luu Luφ ⎥⎥⎥ ⎥⎥⎥ ⎢⎢⎢⎢ ⎥⎥⎦ . ⎢⎣ Lφu Lφφ

The presented solution procedure is general in the sense that it can be applied to linear as well as to nonlinear problems. However, quadratic convergence on the macroscale can only be achieved, if an algorithmic consistent linearization of the nonlinear weak forms on the mesoscale is carried out.

4 Numerical Example In the following we analyze a two-dimensional BVP with attached heterogeneous representative volume elements. The eﬀective properties of the material will be computed and the distribution of some mechanical and electrical fields on the mesoscale will be analyzed. The material law that is used for the representation of the individual phases on the mesoscale will be discussed in the following.

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

121

4.1 Invariant Formulation and Material Parameters The material parameters that are applied in this contribution are taken from [59] and fitted to the underlying transversely isotropic material model with a least-squares approximation. We assume the existence of a quadratic electric enthalpy function from which the stresses and electric displacements are derived. In the present case of linear piezoelectric material behavior we obtain the general representation " # σ = C : ε − eT · E σi j = Ci jkl εkl − eki j Ek and (41) D= e:ε+ · E Di = eikl εkl + ik E k in direct and tensorial notation, respectively. Here C denotes the fourth-order elasticity tensor, e the third-order tensor of piezoelectric moduli, and the second-order tensor of dielectric moduli, see (25). Following [41] we use a coordinate-invariant representation in the framework of the invariant-theory and focus on transversely isotropic solids, where a (with ||a|| = 1) is the preferred direction of the transversely isotropic material. We obtain C = λ1 ⊗ 1 + 2μI + α3 [1 ⊗ m + m ⊗ 1] + 2α2 m ⊗ m + α1 Ξ,

(42)

where m = a ⊗ a denotes the second-order structural tensor, 1 is the second-order unity tensor, I denotes the fourth-order unity tensor, and Ξi jkl := [ai δ jk al + ak δil a j ]; δi j is the Kronecker Delta. The second-order tensor of dielectric moduli is given by = −2γ1 1 − 2γ2 m.

(43)

Finally, the third-order tensor of the piezoelectric moduli appears in the form e := −β1 a ⊗ 1 − β2 a ⊗ m − β3 e¯

(44)

with the abbreviation e¯ ki j := 12 [ai δk j + a j δki ]. Let the preferred direction a coincide with the x3 -axis, then we obtain the relations λ = C12 , α2 =

1 (C11 2

μ = 12 (C11 − C12 ),

+ C33 ) − 2C44 − C13 ,

β2 = e31 − e33 + 2e15 ,

β3 = −2e15

α1 = 2C44 + C12 − C11 , α3 = C13 − C12 , γ1 =

− 21 11 ,

β1 = −e31 ,

γ2 =

1 ( 2 11

(45)

− 33 )

between the parameters of the coordinate-independent and coordinate-dependent representation. The fitted material parameters are listed in Table 1. Table 1 Fitted material parameters of BaTiO3 [59]; [C] = GPa, [e] = C/m2 , [] = 10−9 C/Vm. C11

C12

C13

C33

C44

e31

e33

e15

11

33

222

108

111

151

30.5

-0.7

6.7

34.2

19.5

0.5

122

M.-A. Keip and J. Schröder

4.2 Investigation of the “Wolfgang Ehlers 60” Mesostructure As numerical example we consider an academic mesostructure that has been designed on the occasion of the 60th birthday of Professor Wolfgang Ehlers from the University of Stuttgart. It is given as a heterogeneous electro-mechanically coupled mesostructure composed of a piezoelectric matrix with piezoelectric inclusions, see Fig. 7. Herein, the preferred direction of the transversely isotropic matrix material is pointing upwards and the preferred direction of the inclusions points to the right.

Fig. 7 The WE60-mesostructure on the occasion of the 60th birthday of Wolfgang Ehlers.

In order to compute the eﬀective properties of the WE60-mesostructure by means of the FE2 -method, it is discretized with six-noded quadratic triangular finite elements and integrated into the proposed homogenization environment. For the computations, periodic boundary conditions along the boundary of the RVE are prescribed. Based on the material properties of the individual constituents, which were given in the previous paragraph, the macroscopic eﬀective moduli of the heterogeneous mesostructure can be computed. However, the accuracy of the results has to be ensured by means of a convergence study of the eﬀective coeﬃcients with respect to the mesh density of the mesostructure. The mesh discretizations that are utilized for the convergence study are depicted in Fig. 8.

Fig. 8 Discretized WE60-mesostructure with increasing mesh density, i. e. 346, 902, 2434, and 9292 quadratic triangular finite elements.

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

123

The results of the convergence study are depicted in Fig. 9. It can be seen that the coeﬃcients converge to a saturation value with increasing mesh density. For the computation of eﬀective moduli of three-dimensional RVEs see [42] and [43]. 11 [10−9 C/Vm]

C11 [GPa] 210.5

16

15.5 210 15 209.5 14.5

ele

209

a)

0

2500

5000

7500

10000

ele

14

b)

0

2500

5000

7500

10000

Fig. 9 Convergence of eﬀective moduli: a) mechanical moduli C11 b) dielectric moduli 11 .

In the following, some mechanical and electrical fields on the mesostructure will be analyzed. For the analysis we utilize the discretization with finest mesh density, i. e. the discretization with 9292 quadratic triangular finite elements, and attach it to a two-dimensional macroscopic BVP consisting of a statically determined squareshaped body with dimensions of 0.5 × 0.5 [mm2 ] that is loaded with an electric potential on its upper and lower edge as depicted in Fig. 10.

φ = −1 kV x φ = 1 kV

Fig. 10 Macroscopic electro-mechanical BVP with attached WE60-mesostructure.

In Fig. 11 a) the electric displacement in vertical direction D2 is depicted. Due to the perpendicular orientation of the preferred directions of the material of the matrix and the inclusion a heterogeneous distribution can be observed. In b) the distribution of the fluctuations of the vertical displacements w˜ 2 are shown. Here it should be remembered that periodic boundary conditions for both, the displacement field and the electric potential are applied. Therefore, the fluctuations of those fields on opposite boundaries of the representative volume element are identical. Finally, in c) the distribution of the electric field in vertical direction E2 is plotted. It becomes obvious that strong concentrations of the electric field occur at the boundary between the inclusion and the matrix.

124

M.-A. Keip and J. Schröder

a)

b)

c)

Fig. 11 Distributions of a) the vertical component of the electric displacement D2 [C/m2 ], b) the vertical component of the fluctuation of the displacement field w˜ 2 [mm], and c) the vertical component of the electric field E2 [kV/mm] (the strongly localized maximum value is E2 ≈ 18.2 kV/mm).

5 Conclusion In this contribution, we presented a general framework for the two-scale homogenization of heterogeneous piezoelectrics. The basis for this homogenization approach is a meso-macro transition procedure for electro-mechanically coupled materials. It was implemented into an FE2 -homogenization approach which allows for the computation of macroscopic boundary value problems in consideration of mesoscopic representative volume elements. It has been shown that the presented homogenization formulation is capable of computing eﬀective electro-mechanical moduli for composites with arbitrary inclusion geometries.

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References [1] Benveniste, Y.: Exact results in the micromechanics of fibrous piezoelectric composites exhibiting pyroelectricity. Proceedings of the Royal Society London A 441(1911), 59–81 (1993) [2] Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, Part I: Binary media: Local fields and eﬀective behavior. Journal of Applied Mechanics 60, 265–269 (1993) [3] Benveniste, Y.: Universal relations in piezoelectric composites with eigenstress and polarization fields, Part II: Multiphase mediaeﬀective behavior. Journal of Applied Mechanics 60, 270–275 (1993) [4] Benveniste, Y.: Piezoelectric inhomogeneity problems in anti-plane shear and in-plane electric fields – how to obtain the coupled fields from the uncoupled dielectric solution. Mechanics of Materials 25(1), 59–65 (1997) [5] Budiansky, B.: On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids 13, 223–227 (1965) [6] Chen, T.: Piezoelectric properties of multiphase fibrous composites: Some theoretical results. Journal of the Mechanics and Physics of Solids 41(11), 1781–1794 (1993) [7] Chen, T.: Micromechanical estimates of the overall thermoelectroelastic moduli of multiphase fibrous composites. International Journal of Solids and Structures 31(22), 3099– 3111 (1994) [8] Chung, D.H.: Elastic moduli of single crystal and polycrystalline MgO. Philosophical Magazine 8(89), 833–841 (1963) [9] Dunn, M.L., Taya, M.: An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society London A 443(1918), 265–287 (1993) [10] Dunn, M.L., Taya, M.: Micromechanics predictions of the eﬀective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures 30, 161–175 (1993) [11] Eshelby, J.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society London A 241, 376–396 (1957) [12] Fang, D.N., Jiang, B., Hwang, K.C.: A model for predicting eﬀective properties of piezocomposites with non-piezoelectric inclusions. Journal of Elasticity 62(2), 95–118 (2001) [13] Francfort, G.A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Archive for Rational Mechanics and Analysis 94, 307–334 (1986) [14] Hashin, Z.: The diﬀerential scheme and its application to cracked materials. Journal of the Mechanics and Physics of Solids 36(6), 719–734 (1988) [15] Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342 (1962) [16] Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids 10(4), 343–352 (1962) [17] Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids 11(2), 127–140 (1963) [18] Hill, R.: The elastic behaviour of a crystalline aggregate. Proceedings of the Royal Society London A 65(5), 349–354 (1952) [19] Hill, R.: Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372 (1963)

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[20] Hill, R.: A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13, 213–222 (1965) [21] Hill, R.: On constitutive macro-variables for heterogeneous solids at finite strain. Proceedings of the Royal Society London A 326(1565), 131–147 (1972) [22] Hill, R.: On the micro-to-macro transition in constitutive analyses of elastoplastic response at finite strain. Mathematical Proceedings of the Cambridge Philosophical Society 98, 579–590 (1985) [23] Hori, M., Nemat-Nasser, S.: Universal bounds for eﬀective piezoelectric moduli. Mechanics of Materials 30(1), 1–19 (1998) [24] Krawietz, A.: Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens. Springer, Berlin (1986) [25] Kröner, E.: Bounds for eﬀective elastic moduli of disordered materials. Journal of the Mechanics and Physics of Solids 25, 137–155 (1977) [26] Li, Z., Wang, C., Chen, C.: Eﬀective electromechanical properties of transversely isotropic piezoelectric ceramics with microvoids. Computational Materials Science 27(3), 381–392 (2003) [27] Lupascu, D.C., Schröder, J., Lynch, C.S., Kreher, W., Westram, I.: Mechanical properties of ferro-piezoceramics. In: Pardo, L., Ricote, J. (eds.) Multifunctional polycrystalline ferroelectric materials. Springer Series in Materials Science, vol. 140, pp. 485– 559. Springer, Heidelberg (2011) ISBN 978-90-481-2874-7 [28] Markovic, D., Niekamp, R., Ibrahimbegovic, A., Matthies, H., Taylor, R.: Multi-scale modeling of heterogeneous structures with inelastic constitutive behavior. International Journal for Computer-Aided Engineering and Software 22(5/6), 664–683 (2005) [29] Michel, J., Moulinec, H., Suquet, P.: Eﬀective properties of composite materials with periodic microstructure: a computational approach. Computer Methods in Applied Mechanics and Engineering 172, 109–143 (1999) [30] Miehe, C., Koch, A.: Computational micro-to-macro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4), 300–317 (2002) [31] Miehe, C., Schotte, J., Schröder, J.: Computational micro-macro transitions and overall moduli in the analysis of polycrystals at large strains. Computational Materials Science 16(1-4), 372–382 (1999) [32] Miehe, C., Schröder, J., Schotte, J.: Computational homogenization analysis in finite plasticity. Simulation of texture development in polycrystalline materials. Computer Methods in Applied Mechanics and Engineering 171, 387–418 (1999) [33] Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Mechanica 21, 571–574 (1973) [34] Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, London (1993) [35] Norris, A.N.: A diﬀerential scheme for the eﬀective moduli of composites. Mechanics of Materials 4(1), 1–16 (1985) [36] Qin, Q.H., Yang, Q.S.: Macro-Micro Theory on Multifield Coupling Behavior of Heterogeneous Materials. Higher Education Press, Springer, Bejing, Berlin (2008) [37] Reuss, A.: Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Zeitschrift für angewandte Mathematik und Mechanik 9(1), 49–58 (1929) [38] Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. Lecture Notes in Physics, vol. 127, pp. 46–293. Springer, Heidelberg (1980)

Effctive Electromechanical Properties of Heterogeneous Piezoelectrics

127

[39] Schröder, J.: Homogenisierungsmethoden der nichtlinearen Kontinuumsmechanik unter Beachtung von Instabilitäten. Bericht aus der Forschungsreihe des Instituts für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart (2000) [40] Schröder, J.: Derivation of the localization and homogenization conditions for electromechanically coupled problems. Computational Materials Science 46(3), 595–599 (2009) [41] Schröder, J., Gross, D.: Invariant formulation of the electromechanical enthalpy function of transversely isotropic piezoelectric materials. Archive of Applied Mechanics 73(8), 533–552 (2004) [42] Schröder, J., Keip, M.A.: A framework for the two-scale homogenization of electromechanically coupled boundary value problems. In: Kuczma, M., Wilmanski, K. (eds.) Computer Methods in Mechanics, Advanced Structured Materials, vol. 1, pp. 311–329. Springer, Heidelberg (2010) [43] Schröder, J., Keip, M.A.: Multiscale modeling of electro-mechanically coupled materials: Homogenization procedure and computation of overall moduli. In: Kuna, M., Ricoeur, A. (eds.) IUTAM Symposium on Multiscale Modelling of Fatigue, Damage and Fracture in Smart Materials. IUTAM Bookseries, vol. 24, pp. 265–276. Springer, Netherlands (2011) [44] Schröder, J., Romanowski, H., Kurzhöfer, I.: A computational meso-macro transition procedure for electro-mechanical coupled ceramics. In: Schröder, J., Lupascu, D., Balzani, D. (eds.) First Seminar on the Mechanics of Multifunctional Materials, Universität Duisburg-Essen, Bad Honnef, Germany (2007) [45] Silva, E., Fonseca, J., Kikuchi, N.: Optimal design of periodic piezocomposites. Computer Methods in Applied Mechanics and Engineering 159(1), 49–77 (1998) [46] Silva, E., Nishiwaki, S., Fonseca, J., Kikuchi, N.: Optimization methods applied to material and flextensional actuator design using the homogenization method. Computer Methods in Applied Mechanics and Engineering 172(1-4), 241–271 (1999) [47] Smit, R., Brekelmans, W., Meijer, H.: Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Computer Methods in Applied Mechanics and Engineering 155, 181–192 (1998) [48] Somer, D., de Souza Neto, E., Dettmer, W., Peric, D.: A sub-stepping scheme for multi-scale analysis of solids. Computer Methods in Applied Mechanics and Engineering 198(9-12), 1006–1016 (2009) [49] Suquet, P.M.: Elements of homogenization for inelastic solid mechanics. In: Suquet, P.M. (ed.) Homogenization Techniques for Composite Materials. Lecture Notes in Physics, vol. 272, pp. 193–278. Springer, Heidelberg (1986) [50] Terada, K., Kikuchi, N.: A class of general algorithms for multi-scale analyses of heterogeneous media. Computer Methods in Applied Mechanics and Engineering 190(40-41), 5427–5464 (2001) [51] Terada, K., Saiki, I., Matsui, K., Yamakawa, Y.: Two-scale kinematics and linearization for simultaneous two-scale analysis of periodic heterogeneous solids at finite strain. Computer Methods in Applied Mechanics and Engineering 192(31-32), 3531–3563 (2003) [52] Topolov, V.Y., Bowen, C.R.: Electromechanical properties in composites based on ferroelectrics. Springer, Heidelberg (2009) [53] Uetsuji, Y., Nakamura, Y., Ueda, S., Nakamachi, E.: Numerical investigation on ferroelectric properties of piezoelectric materials using a crystallographic homogenization method. Modelling and Simulation in Material Science and Engineering 317, S303– S317 (2004)

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[54] Uetsuji, Y., Horio, M., Tsuchiya, K.: Optimization of crystal microstructure in piezoelectric ceramics by multiscale finite element analysis. Acta Materialia 56(9), 1991– 2002 (2008) [55] Voigt, W.: Lehrbuch der Kristallphysik. Teubner (1910) [56] Walpole, L.J.: On bounds for the overall elastic moduli of inhomogeneous system. Journal of the Mechanics and Physics of Solids 14, 151–162 (1966) [57] Willis, J.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. Journal of the Mechanics and Physics of Solids 25, 185–202 (1977) [58] Xia, Z., Zhang, Y., Ellyin, F.: A unified periodical boundary conditions for representative volume elements of composites and applications. International Journal of Solids and Structures 40, 1907–1921 (2003) [59] Zgonik, M., Bernasconi, P., Duelli, M., Schlesser, R., Günter, P., Garrett, M.H., Rytz, D., Zhu, Y., Wu, X.: Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals. Physical Review B 50(9), 5941–5949 (1994) [60] Zohdi, T.: On the computation of the coupled thermo-electromagnetic response of continua with particulate microstructure. International Journal for Numerical Methods in Engineering 76(8), 1250–1279 (2008) [61] Zohdi, T., Wriggers, P.: Introduction to computational micromechanics. In: Pfeiﬀer, F., Wriggers, P. (eds.) LNACM, vol. 20, Springer, Heidelberg (2005)

Coupled Thermo- and Electrodynamics of Multiphasic Continua Bernd Markert

In honor of the 60 th birthday of my scientific mentor Wolfgang Ehlers.

Abstract. This chapter gives a compact overview of multi-phase continuum mechanics by recourse to the general concepts of mixture and porous media theories. Attention is focused on volumetrically coupled, multi-field formulations arising from the continuum mechanical treatment of multi-phase materials accounting for diﬀerent physical phenomena. For this purpose, starting with the basics of the macroscopic mixture approach, mixture kinematics, some aspects of electromagnetism and the stress concept are reviewed. Finally, the axiomatic conservation laws of thermodynamics and electrodynamics are fused and represented within the holistic framework of a general multiphasic theory.

1 Mixture and Porous Media Theories Thermoelasticity and incompressible viscous flow pose typical examples of coupled multi-field problems, where the diﬀerent physical fields (e. g. temperature and deformation or fluid velocity and pressure) are in immanent interaction within the same spatial domain. The interaction takes place at each material point of the continuum body which gives rise to a continuum mechanical description by a so-called volumetrically or materially coupled formulation [20]. Obviously, such coupled problems with the coupling inherent in the governing multi-field equations may emerge in solids as well as in fluids and a fortiori in multiphasic materials constituted of both. Therefore, it suggests itself to start from a more general theoretical framework that allows to address solids and fluids solely or interacting together, additionally having the possibility to account for thermal, chemical and electrical eﬀects in a modular fashion. However, it is beyond the scope of this work to start from scratch and derive all concepts, theories, or constitutive laws from a microscopic level. It Bernd Markert Institute of Applied Mechanics (Civil Engineering), University of Stuttgart, Pfaﬀenwaldring 7, 70569 Stuttgart, Germany e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 129–152. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

130

B. Markert

is rather envisaged to follow a macroscopic, mean-field approach on a phenomenological basis without getting lost in the ‘real’ physics of the processes and material behavior that are governed on the atomic, molecular, or some other lower scale.

1.1 The Macroscopic Mixture Approach In the framework of continuum physics, a mixture ϕ is understood as an idealized macroscopic model of superimposed continua of miscellaneous constituents ϕα without restriction to be solid, liquid, or gas. Here, the presentation is limited to solid-fluid aggregates with the solid-phase forming some interconnected porous matrix or skeleton, whereas the intrinsic fluid (pore fluid) itself may be a real mixture of sundry components ϕ β . Such fluid-saturated porous solids can be treated very elegantly by the Theory of Porous Media (TPM) [3, 9], where the material is represented as a de facto immiscible binary mixture of constituents ϕS (solid skeleton) and ϕF (pore-fluid mixture), which are assumed to be in a state of ideal disarrangement. Following this, the prescription of a real or a virtual averaging process over a representative elementary volume (REV) leads to a model ϕ of overlaid and interacting continua, viz. ϕ= ϕα = ϕS ∪ ϕF with ϕF = ϕβ . (1) α

β

In the arising macroscopic model, the incorporated physical quantities are then understood as the local averages of their microscopic representatives. In this regard, ϕS may be subdivided further in the sense of mixture theories, for instance, if fixed charges or other attached reactive species are to be considered, e. g. [11].

1.2 Volume Fractions, Saturation and Density In order to account for the local composition of the mixture, local volumetric ratios are introduced according to the concept of volume fractions. The volume V of the overall medium B results from the sum of the partial volumes V α of the constituent bodies Bα : V= dv = Vα with Vα = dv = dvα =: nα dv . (2) B

α

Bα

B

B

Consequently, the volume fractions nα of ϕα are defined as the local ratios of the partial volume elements dvα w. r. t. the volume element dv of the whole mixture ϕ: S dvα n : solidity , nα := , where (3) nF : porosity . dv As a further consequence of Eqs. (2), assuming fully saturated conditions, i. e., avoiding any vacant space in the porous medium, the saturation constraint yields

Coupled Thermo- and Electrodynamics of Multiphasic Continua

α

nα = nS + n F = 1

nF =

with

131

nβ ,

(4)

β

where n β is the volume fraction of the not further specified components ϕ β of ϕF . In presence of a pore-fluid mixture, it is moreover convenient to introduce the saturation of a fluid component ϕ β as the quotient of its volume fraction and the porosity: s β :=

nβ , nF

where

sβ = 1 .

(5)

β

Proceeding from the definition of the volume fractions (3), associated with each constituent ϕα is a material (realistic) density ραR defined as the local mass dmα of ϕα per unit of dvα and a partial density ρα , where the local mass element dmα is related to the bulk volume element dv. Moreover, the mixture density can be introduced as the sum of the partial densities ρα . In particular, they read ραR :=

dmα , dvα

ρα :=

dmα , dv

ρ :=

α

ρα ,

Moreover, one finds for the pore-fluid mixture ρFR = s β ρ βR and ρF = ρβ β

where ρα = nα ραR .

with ρ β = n β ρ βR .

(6)

(7)

β

According to (6), it is obvious that the property of material incompressibility of a constituent ϕα , defined by ραR = const., does not lead to macroscopic incompressibility as the partial density ρα , and thus, ρ can still change through changes in the volume fractions nα . Following the above, additional quantities can be introduced relevant for speβ cific fields of application. For example, the local mass fraction xαM (or likewise x M ), which is used in conserved phase-field models and chemical applications [4, 23], relates the constituent mass element to the bulk mass element: dmα xαM := , where xαM = 1 . (8) dm α Further chemical measures are the molar concentration cmβ given by the local amount of substance in moles dnmβ per volume of pore-fluid mixture dvF and the mole fracβ tion xm as the amount of a fluid component ϕ β in solution, cf. [10]: cmβ :=

dnmβ , dvF

dn β cβ xmβ := m β = m β , β dnm β cm

where

β

xmβ = 1 .

(9)

132

B. Markert χS (XS , t)

(t0 )

dVS

dv

PS

dVF

(t)

χF (XF , t)

PF

PS, PF

XS x XF

B0

B O

Fig. 1 Motion of a biphasic mixture.

2 Kinematical Relations 2.1 Mixture Kinematics Concerning the kinematics of mixtures, the idea of superimposed and interacting continua implies that, starting from diﬀerent reference positions Xα at time t0 , each constituent follows its individual Lagrangean motion function and has its own velocity and acceleration fields: x = χα (Xα , t) ,

xα =

dχα (Xα , t) , dt

xα =

d2 χα (Xα , t) . dt2

(10)

Following this, each spatial point x of the current configuration at time t is simultaneously occupied by material points Pα of both constituents (Fig. 1).1 Here, the presentation is restricted to standard Cauchy-Boltzmann continua without equipping the individual material points with more than the classical three translational degrees of freedom. The unique individual motion functions χα of Pα are assumed to be local diﬀeomorphisms such that unique inverse motions exist based on non-singular functional determinants (Jacobians) Jα : Xα = χ−1 α (x, t) ,

if

Jα := det

∂ χα (Xα , t) 0. ∂ Xα

(11)

Thus, using the inverse motion functions χ−1 α , the Eulerian (spatial) description of the velocity and the acceleration fields from (10) are given by 1

Concerning the notation, vectors and tensors are indicated by boldface small and capital letters except for the reference position vectors, which are traditionally denoted by capital letters defined as Xα := x0α with ( q )0α indicating an arbitrary initial state of the motion of ϕα at time t0 . This is also retained for line, area and volume elements, e. g., dVS := dv0S is the reference bulk volume element w. r. t. an initial (normally undeformed) solid state. The phase or species identifier α (and likewise β) is commonly used as subscript with kinematic quantities and as superscript otherwise.

Coupled Thermo- and Electrodynamics of Multiphasic Continua

xα = xα (x, t) ,

133

xα = xα (x, t) .

(12)

Furthermore, by use of the mixture density ρ from (6), the so-called mixture velocity x˙ describing the barycentric velocity of the overall medium and the diﬀusion velocity dα describing the relative velocity of ϕα w. r. t. ϕ can be introduced as x˙ =

1 α ρ xα , ρ α

dα = xα − x˙

with

α

ρα d α = 0 .

(13)

In analogy, one finds for the pore-fluid mixture

xF =

1 β ρ xβ , ρF β

dβF = xβ − xF

with

ρ β dβF = 0 ,

(14)

β

where dβF is the diﬀusion velocity of the components within the moving pore-fluid mixture. In the above equations, ( q )α (and likewise ( q )β ) indicates the material time derivative following the motion of ϕα (or ϕβ ) and ( q )˙ denotes the material time derivative following the barycentric motion of ϕ (mixture derivative). Suppose that Ψ and Ψ are arbitrary, steady and suﬃciently steadily diﬀerentiable scalar- and vector-valued field functions. Then, the respective derivatives are computed as Ψα

dα Ψ ∂ Ψ = + grad Ψ · xα , dt ∂t dΨ ∂ Ψ Ψ˙ = = + grad Ψ · x˙ , dt ∂t

=

dα Ψ ∂ Ψ = + (grad Ψ ) xα , dt ∂t dΨ ∂ Ψ ˙ = Ψ = + (grad Ψ ) x˙ . dt ∂t Ψα

=

(15)

Therein, the diﬀerential operator grad ( q ) := ∂ ( q )/∂x denotes the partial derivative w. r. t. the actual position vector x. In porous media theories, it is generally convenient to proceed from a Lagrangean description of the solid matrix via the solid displacement vector uS as the primary kinematic variable. In contrast, the pore-fluid flow is better expressed in a modified Eulerian setting via the seepage velocity wFS describing the fluid motion relative to the deforming skeleton. Moreover, relating wFS only to the fluid part of the mixture, the so-called filter or superficial velocity wF can be introduced. Thus, uS = x − XS ,

wFS = vF − vS

with vα := xα ,

wF = nF wFS .

(16)

The movement of the fluid components ϕ β may likewise be expressed w. r. t. the motion of the fluid mixture via dβF (14)2 or analogously to (16)2 w. r. t. the solid motion via wβS := vβ − vS = dβF + wFS . (17) Given (10)1 and (11)1 , the material deformation gradients and their inverses are defined by ∂x ∂ Xα Fα = =: Gradα x , F−1 = grad Xα , (18) α = ∂ Xα ∂x

134

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where the gradient operator Gradα ( q ) := ∂ ( q )/∂ Xα denotes the partial derivative w. r. t. the reference position vector Xα . Particularly, the solid deformation gradient FS can be expressed in terms of the solid displacement using (16)1 : FS =

∂x = GradS x = I + GradS uS . ∂ XS

(19)

By virtue of (11), the existence of uniquely invertible motions requires non-zero Jacobians Jα . Starting the deformation process from an undeformed state at time t0 , the initial condition Fα (t0 ) = I restricts the domain of det Fα to positive values, det Fα = Jα > 0 with

det Fα (t0 ) = 1 .

(20)

Moreover, the assumption of solid incompressibility (ρS R = const.) implies that volumetric compression is only possible until the pore space is completely closed (nF = 0). Hence, the lower limit of the finite volume dilatation eVS of the solid F matrix is predefined by the initial porosity n0S , viz. eVS =

dv − dVS F = det FS − 1 > −n0S dVS

(21)

directly yielding the stronger restriction det FS = JS > nS0S

with

F nS0S = 1 − n0S .

(22)

Therein, the transport property (25)3 of det FS describing finite volume changes is used and ( q )0S indicates initial values w. r. t. an undeformed solid state at time t0 .

2.2 Deformation and Strain Measures In the framework of a finite deformation theory of general fluid-saturated porous materials, where the constituents can exhibit intrinsic non-dissipative (elastic) as well as inelastic properties, it is common practice, particularly for the solid phase, to proceed from a local multiplicative split of the deformation gradient in Fα = Fel α Fα

(23)

in in into an elastic part Fel α and an inelastic part Fα . In principle, Fα may be used to describe any type of deformation-related dissipative process conceivable for the constituent ϕα , such as elastoplasticity [7] or viscoelasticity [19] among others. Basically, the multiplicative geometric concept can be phenomenologically introduced without insisting on somehow abstract and somewhat controversial interpretations. Instead, it can be understood as constitutive definition that will consequently yield a favorable structure of finite strain measures and their spatiotemporal derivatives. The theoretical description of the inelastic processes is commonly accomplished by introducing internal state variables associated with Fin α , which are nothing but additional fields that govern the inelastic evolution by some constitutive rate equations.

Coupled Thermo- and Electrodynamics of Multiphasic Continua

135

Strictly speaking, in combination with the momentum equation this constitutes a coupled multi-field problem. However, for the rest of this contribution, we leave intrinsic inelastic properties aside and rather concentrate on multi-phase and multiphysics problems, where the dependent variables are commonly treated as primal unknowns to be determined from a conservation principle and do not just represent history variables evolving with a rate equation. To continue, we stay with the material deformation gradient Fα , which is known to be a mixed-variant, two-field or two-point tensor with identity metric in terms of a curvilinear (natural) basis.2 This directly implies its special property of acting as a transport mechanism between configurations. To show this, let us apply Fα and F−1 α and their transposes as linear mappings to generic covariant vectors Ψ re f and Ψ act as well as contravariant vectors Ψ re f and Ψ act of the reference and current frame: Ψ

re f

Fα

Ψ act

F−1 α

Ψ

,

FTα −1

re f

FTα

Ψ act

.

(24)

The transformations (24)1 and (24)2 are commonly known as co- and contravariant push-forward (reference → actual) and pull-back (actual → reference) operations. Following this, the transport mechanisms of material line, oriented area and scalar volume elements of ϕα can be formulated: dx(k) = Fα dXα(k) ,

da(k) = cof Fα dAα(k) ,

dv = det Fα dVα .

(25)

Herein, dXα(k) := dx0α(k) and dx(k) as well as dAα(k) := da0α(k) and da(k) with k = 1, 2, 3 denote the covariant line and the contravariant area elements of the reference and the actual configuration. Moreover, dAα(k) = n0α(k) dAα(k) ,

da(k) = n(k) da(k)

|k

(26)

with dAα(k) := da0α(k) and da(k) as the scalar area elements, and n0α(k) and n(k) as the corresponding reference and actual outward-oriented unit surface normals. Next, the gradient operator obeys the following contravariant transport mechanisms: Gradα Ψ = Gradα Ψ

∂Ψ ∂ Xα

∂Ψ = ∂ Xα

FTα −1

∂Ψ = grad Ψ , ∂x

FTα 23

(I ⊗ FTα −1 ) T 23

(I ⊗ FTα ) T

∂Ψ = grad Ψ . ∂x

(27)

kl

Herein, ( q )T indicates an exchange of the kth and the lth basis vector included in the polyadic components of higher-order tensors. 2

Vectors and tensors can be expressed w. r. t. a co- or contravariant (local) basis in a curvilinear frame, i. e., admit a co- and a contravariant representation. Here, vectors and tensors are called co- or contravariant depending on the variant type of their component basis vectors [2].

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Then, by means of Fα as the basic kinematic quantity, the right and left CauchyGreen deformation tensors and their inverses are defined as usual: Cα = FTα Fα ,

Bα = Fα FTα ,

−1 T −1 C−1 , α = Fα Fα

T −1 −1 B−1 Fα . α = Fα

(28)

Given the definitions for the contravariant Green-Lagrangean and Almansian strains, 23 one obtains (FTα −1 ⊗ FTα −1 ) T 1 1 Eα = 2 (Cα − I) (I − B−1 (29) α ) = Aα . 2 23 (FTα ⊗ FTα ) T

Analogously, the covariant Karni-Reiner strain tensors are introduced by definition: 23

R

Kα =

1 2

(I − C−1 α )

(Fα ⊗ Fα ) T

1 2

23

−1 T (F−1 α ⊗ Fα )

(Bα − I) = Kα .

(30)

In order to formulate rate equations, time derivatives of the deformation and strain measures are required. To begin with, the material velocity gradient is given as (Fα )α =

d α ∂ x ∂ vα = = Gradα vα dt ∂ Xα ∂ Xα

(31)

and the corresponding spatial velocity gradient is defined by Lα := grad vα =

∂ vα ∂ v α ∂ Xα = = (Fα )α F−1 α . ∂x ∂ Xα ∂ x

(32)

By use of Lα , the material time derivatives of the infinitesimal line, area and volume elements given in (25) take the compact form (dx)α = Lα dx ,

(da)α = (Lα · I) I − LTα da ,

(dv)α = (Lα · I) dv .

(33)

The symmetric and skew-symmetric parts of Lα , commonly termed stretching or deformation rate tensor and spin or vorticity tensor, read Dα =

1 2

(Lα + LTα ) = DTα ,

Wα =

1 2

(Lα − LTα ) = −WTα .

(34)

In this regard, it can be shown that the deformation velocity tensor Dα admits a coand a contravariant representation in the frame of convected curvilinear coordinates, which by implication means that the material time derivative acts only on the timedependent metric coeﬃcients but not on the natural basis vectors. Thus, it holds T Dα = − 21 Fα (C−1 α )α F α =

1 2

FTα −1 (Cα )α F−1 α .

(35)

This motivates the introduction of a special class of objective time derivatives. In continuum mechanics, the most familiar is the convective or Oldroyd derivative, which applied to a tensor field is essentially its Lie derivative along the velocity vector. The procedure is to compute a pull-back to the reference configuration, take

Coupled Thermo- and Electrodynamics of Multiphasic Continua

137

the material time derivative and push-forward the result back to the actual configuration. One defines the lower (covariant) and upper (contravariant) derivatives via dα q T −1 FT = ( q ) − Lα ( q ) − ( q ) LT , ( q )α := Lvα ( q ) = Fα F−1 α ( ) Fα α α α dt (36)

−1 T −1 dα T q T q q q q q ( )α := Lvα ( ) = Fα F ( ) Fα Fα = ( )α + Lα ( ) + ( ) Lα . dt α In essence, the lower and upper Lie time derivatives are material time derivatives of the tensorial components while keeping the co- and contravariant tensorial bases fixed. Consequently, the Lie time derivative of a scalar field Ψ (x, t) coincides with the material time derivative, i. e., (Ψ )α = (Ψ )α = (Ψ )α . Then, applying the Lie time derivative to the Almansi and Karni-Reiner strains and accounting for the respective transport mechanisms and (35), it is easily concluded that 23

(Eα )α

=

1 2

(Cα )α

(FTα −1 ⊗ FTα −1 ) T (FTα

⊗

23

FTα ) T

(Aα )α = Dα , (37)

23

R

(Kα )α

=

− 12

(C−1 α )α

(Fα ⊗ Fα ) T (F−1 α

⊗

23

T F−1 α )

(Kα )α = Dα .

A further convective time derivative that accounts for volume changes is given by the flux derivative. It is of major importance for the treatment of the surface balances required in electrodynamics. It is easily derived by regarding the temporal change of a flux of a partial vector field Ψ α (x, t) through the material surface S of ϕ: dα dα α α Ψ · da = (Ψ · da) = (Ψ α )∗α · da . (38) dt S S dt S Then, with (33)2 and the identity Lα · I = div vα , one obtains the flux derivative as (Ψ α )∗α := (Ψ α )α + Ψ α div vα − (grad vα ) Ψ α

(39)

or in alternate representation3 (Ψ α )∗α =

∂ Ψα + vα div Ψ α − rot (vα × Ψ α ) . ∂t

In terms of mixture quantities, one finds analogously ⎧ ⎪ ˙ + Ψ div x˙ − (grad x˙ ) Ψ , Ψ ⎪ ⎪ ⎪ d ∗ ∗ ⎨ Ψ · da = Ψ · da with Ψ = ⎪ ⎪ ∂Ψ ⎪ dt S ⎪ S + x˙ div Ψ − rot (˙x × Ψ ) . ⎩ ∂t 3

(40)

(41)

This is obtained as follows: expand (Ψ α )α according to (15)2 , extend the equation by vα div Ψ α and exploit the rule for the rotation operator −rot (b × a) = a div b + (grad a) b − b div a − (grad b) a .

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3 Some Aspects of Electrodynamics Hitherto, we just introduced some basic quantities and notions of mixture and porous media mechanics, which would suﬃce to continue with the stress concept and the thermodynamical balance laws. However, we started with the objective to provide a more general framework of volume-coupled formulations, which also considers electromagnetic phenomena. Obviously, this makes it necessary to introduce additional fields and notions of electrodynamics before continuing with a universal representation of conservation principles.

3.1 Preliminaries Electrodynamics is a classical field theory dealing with electromagnetic phenomena, which in contrast to Newtonian mechanics is consistent with the theory of relativity. The modern form of electrodynamics is based on the four equations formulated in 1864 by and named after the Scottish physicist James Clerk Maxwell, from which all electromagnetic eﬀects can be deduced. The additional Lorentz-force equation formally establishes the link between electrodynamics and mechanics. In principle, Maxwell’s equations are valid in a vacuum as well as in matter of any kind. When dealing with electrodynamics in continuous media, which naturally implies a huge number of charged particles over several atomic length scales, it is sensible to proceed from Maxwell’s equations reformulated in terms of macroscopic fields. The macroscopic electromagnetic theory is based on a spatial averaging procedure, which guarantees that, on the one hand, atomic fluctuations are averaged out and, on the other hand, the interesting spatial dependencies on the macro scale remain unaﬀected. For an arbitrary microscopic field quantity am (x, t), the averaging can be carried out by [21]4 ⎧ ∂ am dm a m ⎪ ⎪ ⎪ = , ⎪ dt a = am := am fm dvm , where ⎨⎪⎪⎪ ∂ t (42) ⎪ ⎩ grad a = grad a . m m Herein, fm is a statistical distribution function of point-like particles of electric charge and dvm is the microscopic volume element (fluxion or phase-space element) such that fm dvm represents the probability of finding the particle system in dvm . Moreover, the averaging ( q ) commutes with the spatial and temporal derivatives, where dm ( q )/dt is a total time derivative in the sense of statistical mechanics. Following this, Table 1 introduces the macroscopic fields obtained through averaging from their atomic representatives.

4

In view of the common notation in continuum electrodynamics, the macroscopic vector fields are indicated by boldface small capitals. Note that all electromagnetic quantities and equations are given in the rationalized MKS (meter-kilogram-second) system of units.

Coupled Thermo- and Electrodynamics of Multiphasic Continua

139

Table 1 Macroscopic electromagnetic quantities. Symbol

Unit

Name

ρe

C/m3 V/m V s/m2 C/m2 A/m A/m2 C/m2 A/m

electric charge density electric field strength vector magnetic induction field vector (magnetic flux) electric displacement vector (charge potential) magnetic field strength vector (current potential) electric current density vector mean polarization field vector mean (Minkowski) magnetization field vector

e b d h j p m

In this context, it can further be shown that the following relationships hold [21]:

d = 0 e + p , h =

1 μ0

b − m,

(43)

where 0 and μ0 represent universal constants, namely the permittivity and the permeability of free space (vacuum).5 Moreover, p and m describe the neutral charge and current density distributions bound in the material, and thus, represent matter-dependent macroscopic quantities, which are related to the fields e and b via phenomenological material equations. For instance, proceeding from isotropic and homogeneous properties and further disregarding hysteretic eﬀects as they are observed for ferroelectric and ferromagnetic materials, the following approximate linear relations are valid:

p = 0 χe e , m = χm h

−→

d = e, b = μh.

(44)

Herein, χe and χm are the electric and magnetic susceptibilities, = 0 (1 + χe ) is the permittivity and μ = μ0 (1 + χm ) the permeability. A further celebrated phenomenological relation for isotropic conductors is given by Ohm’s law

j = σe

(45)

with σ in A/V m representing the electrical conductivity or specific conductance.

3.2 The Macroscopic Maxwell Equations The four macroscopic Maxwell equations are formulated in the averaged macroscopic fields of Table 1 and have a structure similar to their microscopic counterparts. It should however be noted that they are phenomenologically inspired rather than of a fundamental nature. In particular, they read:

5

In particular, 0 = 8.854187817... · 10−12 C/V m and μ0 = 4 π · 10−7 kg m/A2 s2 .

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B. Markert

div d = ρe

: Gauss’s law (Gauss’s flux theorem),

div b = 0

: Gauss’s law for magnetism,

(46) ∂b : Maxwell-Faraday equation (Faraday’s law of induction), ∂t ∂d rot h = j + : Maxwell-Ampère equation (corrected Ampère’s law). ∂t rot e = −

From the above equations, one easily derives the principle of charge conservation. To this end, take the divergence of Ampère’s law (46)4 , exploit the identity div rot h = 0 and substitute the Gauss’s law (46)1 to obtain the continuity equation div j = −

∂ ρe ∂t

: conservation of charge.

(47)

In order to formulate a general system of balance equations relative to inertial frames, Galilean invariance must be regarded as a fundamental requirement [24]. In fact, the thermodynamical balance laws in the sense of Newtonian mechanics are Galilean invariant as they admit the same expression in any co-moving frame {x, t} related to an inertial reference frame {x, t} by a Galilean transformation x = Q 0 x + v0 t + c 0 ,

t = t.

(48)

Herein, Q0 is a constant, proper orthogonal tensor (rotation tensor), v0 a constant, non-relativistic velocity and c0 some constant vector. Basically, the requirement of Galilean invariance imposes a specific dependence of the flux and supply terms on the velocity field. Now, verifying the macroscopic electromagnetic quantities in respect of (48), it can be shown that b, d, p and ρe fulfill the invariance requirement but the others do not [18]. Suppose x˙ to be the velocity of a moving continuum representing a Galilean frame according to (48). Then, with v0 = −˙x, one obtains the remaining electromagnetic quantities in a proper, co-moving reference frame as E

= e + x˙ × b

: electromotive intensity vector,

H

= h − x˙ × d

: magnetomotive intensity vector,

J

= j − ρe x˙

: conduction current density vector,

= m + x˙ × p

: Lorentz magnetization field vector.

M

(49)

Herein, in the conduction current (49)3 , ρe x˙ is commonly termed convection current density. To have Maxwell’s equations Galilean invariant, (46)3,4 and the charge conservation (47) are rewritten in terms of (49) and the flux derivative (41) yielding ∗

rot E = −b , ∗

rot H = J + d , div J = −˙ρe − ρe div x˙ .

(50)

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141

3.3 Fusion of Electrodynamics and Thermodynamics As long as our considerations are restricted to terrestrial problems, it is not a serious limitation to assume the involved velocity fields to be much smaller than the speed of light since macroscopic ponderable bodies cannot be accelerated to relativistic speeds. Thus, proceeding from non-relativistic mechanics, the fusion of the thermodynamic and electrodynamic balance laws can be carried out equivalently in three diﬀerent ways according to Box (51). Fusion approaches of electro- and thermodynamics [13, 18]: 1. Continuum-like approach: Consider the electromagnetic eﬀects as an action from the distance in analogy to gravitation. 2. Relativistic approach: Consider electromagnetic fields to exhibit inertia and contact actions. 3. Alternate approach: Assume electromagnetic fields to produce surface and volume forces but not inertia.

(51)

Here, we follow the continuum-like approach of linking the electromagnetic relations to the thermomechanical equations. To this end, it is necessary to introduce the respective electromagnetic force, couple and energy, which will later represent the supply and source terms in the combined set of general balance equations. Without presenting the derivation, they read (consult [13] for details): • Electromagnetic force density (electromagnetic body force): be = ρe e + j × b + (grad e) p + (grad b) m + div[(p × b) ⊗ x˙ ] + ∗

= ρe E + (J + p) × b + (grad E ) p + (grad b) M .

∂ (p × b) ∂t

(52)

• Electromagnetic momentum couple density: ce = p × e + m × b + x˙ × (p × b) = p × E + M × b .

(53)

• Total electromagnetic energy: ∂p ∂b εe = j · e + ·e−m· + div[˙x (p · e)] = x˙ · be + J · E + ρ E · π˙ − M · b˙ . (54) ∂t ∂t Therein, the second identities are the respective Galilean invariant forms w. r. t. a co-moving inertial frame, where π = p/ρ is the polarization per unit mass. It should be noted that the above only represents an incomplete summary of the basic quantities and equations of electrodynamics without giving a detailed interpretation of their mathematical meaning and phenomenological implications. Of course, there are excellent textbooks, which comprehensively elucidate the electrodynamic theory from the microscopic electromagnetism of point charges up to the electrodynamics of continua, see Jackson [16] among others. Of particular relevance for this contribution, are [13, 18, 21, 22].

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4 Balance Relations In this section, the fundamental balance laws of continuum thermodynamics and electrodynamics applied to mixtures are merged within an integrated framework. Therefore, the basic stress measures are introduced and the specific balances of the overall mixture as well as of the individual constituents are discussed in the context of the master balance principle. For a more detailed discussion of the continuum mechanics basics, the reader should consult respective subject books and the classical literature quoted therein. Concerning mixture theories, a comprehensive description of the thermodynamical balance principles can be found in [8, 9, 15]. Regarding the fusion of electrodynamics and thermodynamics, refer to the citations given in the previous section and, in the context of continuum mixtures, to [1, 17, 20].

4.1 Stress Concept and Dual Variables According to the intersection (free-body) principle, each partial body Bα ⊂ B of the mixture ϕ is subjected to an external mass-specific body (volume) force bα = bα (x, t) (e. g. gravitation) and an external traction (contact) force tα = tα (x, t, n) acting on parts or the whole of the boundary surface Sα ⊂ S with orientation n. Now, in consideration of electrodynamics, according to (52), additional volumespecific electromagnetic body forces bαe = bαe (x, t) are present. Then, the whole mixture body B = α Bα is exposed to the mechanical body force ρ b = α ρα bα , the electromagnetic body force be = α bαe and the total surface traction t = α tα α on the entire surface S = α S . In conclusion, one finds the resultant force vectors f and f α acting on the overall medium ϕ and on the constituents ϕα as f= ρ b dv + be dv + t da , f α = ρα bα dv + bαe dv + tα da. (55) B B S B B S α α fvm fve fa fvm fve faα Applying Cauchy’s theorem, one recovers the total Cauchy stress tensor of the mixture T = T(x, t) and the partial Cauchy stress tensor Tα = Tα (x, t) of ϕα from t(x, t, n) = T(x, t) n ,

tα (x, t, n) = Tα (x, t) n .

(56)

The total Cauchy stress may also be expressed by means of partial quantities, viz. (Tα − ρα dα ⊗ dα ) , (57) T= α

α

where ρ dα ⊗ dα represents an apparent stress comparable to the Reynolds stress in turbulent flows, which is caused by fluctuations of the entire velocity field due to the diﬀusion velocities dα superimposed on the averaged velocity of the whole mixture x˙ given in (13). The Cauchy stress as the true stress represents the current stress state at material points by describing the surface force acting on an area element in the actual

Coupled Thermo- and Electrodynamics of Multiphasic Continua

143

configuration. Proceeding from the partial incremental surface force on S, additional stress tensors are introduced

τα

dfaα = tα da = Tα da = Tα cof Fα dAα = Tα (det Fα ) FαT −1 dAα , Pα where in particular

(58)

Tα : Cauchy stress (true stress),

τα = (det Fα) Tα Pα = τα FαT −1

: Kirchhoﬀ stress (weighted Cauchy stress),

(59)

: 1st Piola-Kirchhoﬀ stress (nominal stress).

Moreover, it is of considerable assistance to formally introduce a further stress tensor operating on the reference configuration, viz. α −1 α T −1 : 2nd Piola-Kirchhoﬀ stress. Sα = F−1 α P = F α τ Fα

(60)

Note that the above stress measures represent covariant tensors, which is a direct consequence of the contravariant nature of the oriented area elements. Thus, in analogy to the covariant strain rates (37)2 , by use of the lower Oldroyd derivative (36)1 , convenient stress rates, which obey the same transport mechanisms as the corresponding stresses, can be introduced. In summary, we have 23

α

S

(Fα ⊗ Fα ) T

23

−1 T (F−1 α ⊗ Fα )

23

τ

α

,

(Sα )α

(Fα ⊗ Fα ) T

23

−1 T (F−1 α ⊗ Fα )

(τα )α .

(61)

The natural bond between certain stresses and strains is best described by the concept of dual variables stating that physically relevant scalar products between associated (conjugate) stress-strain pairs, such as the stress power, are invariant with any transport operation between configurations. It holds that, Sα · Eα = τα · Aα ,

(Sα )α · Eα = (τα )α · Aα ,

Sα · (Eα )α = τα · (Aα )α ,

(Sα )α · (Eα )α = (τα )α · (Aα )α .

(62)

Note that the scalar products in (62) including strain rates may also be expressed by means of the 1st Piola-Kirchhoﬀ stress. For example, assuming that τα = (τα )T (see Sect. 4.2), the volume-specific stress power can be rewritten as α τα · (Aα )α = τα · Dα = τα · Lα = (Pα FTα ) · [(Fα)α F−1 α ] = P · (Fα )α

(63)

with {Pα , (Fα )α } as a so-called work conjugate pair, which is obviously not consistent with all of the invariance conditions of dual variables in (62).

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B. Markert

4.2 Master Balance Principle for Mixtures According to Truesdell’s metaphysical principles given in Box (64), each constituent can be described by individual balance equations accounting for interactions with the other ϕα by additional production terms. The conservation equations of the whole mixture are then obtained as the sum of the balance equations of the constituents and must have the same form as the respective balance equations of a single-phase material. Due to the fact that the mathematical structure of the fundamental thermodynamic balance laws, namely mass, linear momentum, angular momentum, energy and entropy, is in principle identical, they can be expressed in the concise form of a master balance. The same applies to the basic electromagnetic relations in form of the Galilean invariant Maxwell equations (46)1,2 and (50). Truesdell’s metaphysical principles [25, p. 221]: 1. All properties of the mixture must be mathematical consequences of properties of the constituents. 2. So as to describe the motion of a constituent, we may in imagination isolate it from the rest of the mixture, provided we allow properly for the actions of the other constituents upon it. 3. The motion of a mixture is governed by the same equations as is a single body.

(64)

To begin with, let Ψ and Ψ be arbitrary scalar- and vector-valued densities of some physical quantity to be balanced. Then, proceeding from classical continuum mechanics of single-phase materials, the general volumetric balance relations of a mul tiphasic mixture ϕ = α ϕα take the global (integral) form: d Ψ dv = (φ · n) da + σ dv + Ψˆ dv , dt B S B B (65) d ˆ dv . Ψ dv = (Φ n) da + σ dv + Ψ dt B S B B Herein, φ · n and Φ n are the surface densities defined per unit current area representing the eﬄux of the physical quantity over the surface S of B, σ and σ are the volume densities describing the supply (external source) of the physical quantity, and Ψˆ and Ψˆ are the productions of the physical quantity due to couplings of ϕ with its environment. Analogously, a global surface balance is defined, d (Ψ · n) da = (φ · nt ) ds + (σ · n) da + (Ψˆ · n) da , (66) dt S L S S in which L is a closed material curve (line) bounding S and nt is the unit tangent vector along L. The other terms have a similar intuitive meaning as in the volume balance, that is, φ · nt is the flux of lines through S, and (σ · n) and (Ψˆ · n) are the supply and production of Ψ on S.

Coupled Thermo- and Electrodynamics of Multiphasic Continua

145

Supposing adequate continuity properties for all occurring fields, the equivalent local (diﬀerential) form of the master balance equations (65) results from diﬀerentiation of the left-hand sides and transformation of the surface integrals into volume integrals on the right-hand sides of (65): ˙ Ψ

+ Ψ div x˙ = div φ + σ + Ψˆ ,

˙ Ψ

+ Ψ div x˙ = div Φ + σ + Ψˆ .

(67)

From (66), one respectively obtains with the aid of the flux derivative (41) the local form of the surface balance ∗

Ψ

= rot φ + σ + Ψˆ .

(68)

Next, proceeding from the axiomatically introduced balance principles, namely of mass, linear momentum, angular momentum or moment of momentum, energy and entropy, as well as the macroscopic Maxwell equations, one identifies the quantities in Eqs. (65) and (66), as well as (67) and (68), respectively, according to Table 2. Table 2 Identified physical quantities of the volume and surface mixture master balances. Balance

Ψ, Ψ

φ, Φ

σ, σ

ˆ Ψˆ , Ψ

mass linear momentum angular momentum energy entropy

ρ ρ x˙ x × (ρ x˙ ) ρ ε + 12 x˙ · (ρ x˙ ) ρη

0

0 ρ b + be x × (ρ b + be ) + ce x˙ · (ρ b) + ρ r + εe ση

0

T x×T TT x˙ − q φη

charge Gauss’s law (elec.) Gauss’s law (magn.)

ρe 0 0

−J −d −b

0 ρe 0

0 0 0

Faraday’s law Ampère’s law

b

−E −H

0 J

0 0

−d

0 0

0 ηˆ

Therein, ρ x˙ is the momentum of the entire mixture and x×(ρ x˙ ) is the corresponding moment of momentum. Concerning the energy balance, ε is the internal energy, q is the heat influx vector and r is the external heat supply. The additional electromagnetic contributions are the electromagnetic body force, couple and energy be , ce and εe given through Eqs. (52)–(54). The separated middle three and last two lines represent the electromagnetic volume and surface conservation laws according to Maxwell’s equations given in Sect. 3.2. Moreover, η is the entropy, φη and ση are the eﬄux of entropy and the external entropy supply, and ηˆ ≥ 0 is the non-negative entropy production, which considers the irreversibility of the overall thermodynamic process. Furthermore, regarding the mixture as a closed system, all remaining production terms are equal to zero. Then, by inserting the respective quantities of Table 2 into the local master balances (67) and (68), respectively, one recovers the specific

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B. Markert

balance relations of continuum electro-magneto-thermomechanics as for singlephasic materials [13]. The complete set of equations is depicted in Box (69), where L = grad x˙ is the spatial velocity gradient of the mixture in analogy to (32). Specific conservation laws for the mixture: mass: linear momentum: angular momentum: energy: entropy:

ρ˙ + ρ div x˙ = 0 ρ x¨ = div T + ρ b + be 0 = I × T + ce −→ T TT ρ ε˙ = T · L − div q + ρ r + εe − x˙ · be ρ η˙ ≥ div φη + ση

charge: ρ˙e + ρe div x˙ = −div J 0 = div d − ρe 0 = div b

Gauss’s law (electric): Gauss’s law (magnetic):

∂ ρe ∂t

(69)

∂b ∂t ∂d −→ rot h = j + ∂t

∗

b = −rot E

Faraday’s law: Ampère’s law:

−→ div j = −

−→ rot e = −

∗

−d = −rot H + J

Moreover, regarding the total Cauchy stress tensor, which from (69)3 is obviously not symmetric for general electromagnetic materials, it is of considerable assistance to proceed from a constitutive split T = Tm + Te into a thermomechanical part Tm and an electromagnetic part Te , for both of which further material-dependent equations must be provided. Here, we introduce the electromagnetic stress tensor of the form [13, Sect. 3.6] ⎧ M 1 ⎪ ⎪ ⎨ Te = e ⊗ e + b ⊗ b − 2 (e · e + b · b) I , M Te = Te + Te with ⎪ (70) ⎪ ⎩ T e = p ⊗ E − b ⊗ M + ( M · b) I , which is decomposed into the symmetric Maxwell stress TeM = (TeM )T and another stress tensor Te depending on the polarization and magnetization. Then, it is easily proven that the electromagnetic momentum couple is related to the skew-symmetric part of Te , viz. 3

3

ce = −I × skw Te = E skw Te = E skw Te ,

(71)

3

where E is the rank-3 Ricci permutation tensor. In this context, it is also convenient to express the electromagnetic volume force be in some divergence form dependent on the electromagnetic stress field. This can be accomplished with the aid of the volume- and mass-specific electromagnetic momentum vectors

g=e×b

and

γ=

g ρ

(72)

Coupled Thermo- and Electrodynamics of Multiphasic Continua

147

describing the momentum carried by electromagnetic radiation, by which [21] be = div Te −

∂g = div (Te + x˙ ⊗ g) − ρ γ˙ . ∂t

(73)

Note that this gives rise to an alternate representation of the momentum balance when linking thermo- and electrodynamics as mentioned in Sect. 3.3.6 To continue with the derivation of the partial balance relations in consideration of electrodynamics, we proceed from the most general approach and split-up all the electromagnetic fields into parts to be associated with each constituent [17]. Thus, following Truesdell’s principles (64), the general balances of the individual constituents yield, in analogy to (65), the global form dα α ˆ α dv , Ψ dv = (φα · n) da + σα dv + Ψ dt B S B B (74) dα α α α α ˆ Ψ dv = (Φ n) da + σ dv + Ψ dv , dt B S B B and according to (67) the local form

(Ψ α )α + Ψ α div xα = div φα + σα + Ψˆ α ,

(75)

(Ψ α )α + Ψ α div xα = div Φα + σα + Ψˆ α .

In the same way, one adopts the global and local structure of the surface balance (66) and (68) to obtain d (Ψ α · n) da = (φα · nt ) ds + (σα · n) da + (Ψˆ α · n) da , dt S L S S (76) (Ψ α )∗α = rot φα + σα + Ψˆ α , where ( q )∗α represents the flux derivative according to (39) or (40), respectively. In the above equations, the partial quantities ( q )α have the same physical meaning as the quantities included in the master balances of the overall mixture. However, they have to satisfy additional conditions such that summation of a partial balance of all ϕα yields the respective fields of the multiphasic continuum ϕ. For the constituent volume balances (74), they read: α α Ψ = Ψ , φ·n= (φα − Ψ α dα ) · n , σ= σα , Ψˆ = Ψˆ , α

Ψ

6

=

α

Ψ α

α

,

Φn =

α

α

α

α

(Φ − Ψ ⊗ dα ) n , σ =

α

α

α

σ ,

ˆ Ψ

=

α

ˆ Ψ

α

.

(77)

Following the master balance approach, one may proceed from an equivalent representation of the momentum balance based on the identification Ψ = ρ x˙ + g, Φ = T + Te + x˙ ⊗ g and σ = ρ b.

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Respective constraints apply also to the partial surface balances (76) [17]: α Ψ = Ψ , φ · nt = (φα − Ψ α × dα ) · nt , α

(σ + Ψˆ ) · n =

α

α

(78)

(σα + Ψˆ α − dα div Ψ α ) · n .

In this regard, it should be noted that the supply σα and the production Ψˆ α of Ψ α on S can only hardly be distinguished. Hence, they are formally treated together. Next, the specific balance equations of the constituents are derived in analogy to those of single-phase materials except that one needs to account for the interaction mechanisms by additional production terms. Strictly speaking, the local partial continua represent open systems, which stay in thermo- and electrodynamic exchange with one another. In particular, one identifies the partial physical quantities according to the following table: Table 3 Identified physical quantities of the volume and surface constituent master balances. Balance

Ψα, Ψα

mass linear momentum angular momentum energy entropy

ρα ρα xα x × (ρα xα ) ραεα+ 12 xα · (ρα xα ) ρα ηα

charge ραe Gauss’s law (elec.) 0 Gauss’s law (magn.) 0 Faraday’s law Ampère’s law

bα −dα

φ α , Φα

σα , σα

ˆα Ψˆ α , Ψ

0

T x × Tα (Tα )T xα − qα φαη

0 ρα bα + bαe x × (ρα bα + bαe ) + cαe xα · (ρα bα ) + ραr α+ εαe σαη

ρˆα sˆα hˆ α

−J α −dα − bα

0 ραe 0

ρˆαe σ ˆα λˆα

−E α −Hα

0

ξˆ α Jˆ α

α

Jα

eˆ α ηˆ α

Herein, ρˆα is the mass production describing mass exchanges or phase transitions between a particular ϕα and the other constituents, sˆα is the total momentum production of ϕα and hˆ α is the total production of angular momentum. Moreover, eˆ α and ηˆ α represent the total energy and entropy productions of ϕα . Concerning the electromagnetic terms, ρˆαe is the production of electric charge (e. g. through reaction), σ ˆα α and Jˆ account for the fact that charge and current are not necessarily conserved for the individual ϕα , and λˆα and ξˆ α are additionally introduced productions associated with the magnetic flux and the electric field. In mixture theories, it is convenient to split the total production terms of the thermodynamical balances into direct parts and parts including productions of the preceding (lower) balances:

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ˆ α + x × (pˆ α + ρˆα xα ) , hˆ α = m

sˆα = pˆ α + ρˆα xα ,

eˆ α = εˆ α + pˆ α · xα + ρˆα (εα + 12 xα · xα ) ,

ηˆ α = ζˆα + ρˆα ηα .

(79)

In Eqs. (79), pˆ α denotes the direct momentum production, which can be interpreted as the volume-specific local interaction force between ϕα and the other constituents ˆ α represents the direct moment of momentum production describing the of ϕ, and m angular momentum couplings between the constituents. Moreover, εˆ α and ζˆα are the direct terms of the energy and the entropy productions. As is usual in the framework of single-phase continua, one proceeds from an a priori constitutive assumption for the partial entropy eﬄux and the external entropy supply, viz. φαη = −

1 α q , Θα

σαη =

1 α α ρ r , Θα

(80)

where diﬀerent absolute Kelvin’s temperatures Θα > 0 allow for an individual temperature field for each constituent. Following this, the final form of the specific partial balance relations is obtained in analogy to those of the whole mixture (69): Specific constituent balance relations:

mass: (ρα )α + ρα div xα = ρˆα linear momentum: angular momentum: energy: entropy: charge: Gauss’s law (elec.): Gauss’s law (magn.): Faraday’s law: Ampère’s law:

ρα xα = div Tα + ρα bα + bαe + pˆ α ˆα 0 = I × Tα + cαe + m ρα (εα )α = Tα · Lα − div qα + ρα rα + + εαe − xα · bαe + εˆ α 1 1 ρα (ηα )α = div (− α qα ) + α ρα rα + ζˆα Θ Θ (ραe )α + ραe div xα = −div J α + ρˆαe 0 = −div dα + ραe + σ ˆα 0 = −div bα + λˆα

(81)

(bα )∗α = −rot E + ξˆ α −(dα )∗α = −rot Hα + J α + Jˆ α

According to the sum relations (77), the physical quantities of the mixture can be expressed by means of the respective partial quantities extended by a diﬀusion term in the case of the surface densities. As an example, recall the previously introduced total Cauchy stress tensor T in (57). Moreover, one finds the following restrictions on the partial production terms: ρˆα = 0 , sˆα = 0 , hˆ α = 0 , eˆ α = 0 , ηˆ α ≥ 0 . (82) α

α

α

α

α

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The partial Maxwell equations are recovered from Box (81), from which, through summation over all ϕα , the mixture relations must be obtained. In doing so, one finds the constraints for the electromagnetic productions consistent with (78): div jα = −

∂ ραe + ρˆαe ∂t

−→

div bα = λˆ α

−→

div dα = ραe + σ ˆα

−→

rot eα = − rot hα

α

∂b + ξˆ α − xα λˆ α ∂t

−→

∂ dα = jα + + Jˆ α + xα σˆ α −→ ∂t

α

α

α

ρˆαe = 0 , λˆ α = 0 , σ ˆα = 0,

(83)

ξˆ α − dα λˆ α = 0 , α

ˆ J

α

α

+ dα σ ˆα = 0.

It should again be noted that in this contribution only non-polar materials (Cauchy-Boltzmann continua) are considered, where the motion does not account for additional rotational (nor any other) degrees of freedom beside translation. Thus, proceeding from non-polar mixtures, evaluation of the mixture balance of angular momentum yields the symmetry of the thermomechanical part of the total Cauchy stress tensor Tm = TTm . However, since the electromagnetic phenomena in general induce an additional momentum couple ce (see Box (69)3 ) caused by electric and magnetic multi-pole moments, the combined stress tensor T = Tm + Te is not symmetric, recall (70). Now, regarding the partial moment of momentum balances of ˆ α must additionally the non-polar ϕα (81)3 , direct angular momentum couplings m be considered yielding together with the partial electromagnetic couples to nonsymmetric partial Cauchy stresses, Tα (Tα )T , where it is likewise assumed that Tα = Tαm + Tαe . Nevertheless, proceeding from intrinsically non-polar materials associated with symmetric thermomechanic Cauchy stresses on the micro scale, it can ˆ α ≡ 0 [9, 14]. Then, if be shown by homogenization that on the macroscopic level m one further proceeds from almost linear electromagnetic materials, for which Eqs. (44) are valid implying that skw Tαe ≡ 0 or equivalently cαe ≡ 0, also the electromagnetic stress tensor, and thus, the combined stress becomes symmetric. Consequently, the combined partial Kirchhoﬀ and 2nd Piola-Kirchhoﬀ stresses would also be symmetric, while the 1st Piola-Kirchhoﬀ stress is generally not (cf. (59) and (60)). For specific extensions of the balance principles for single- and multiphasic materials to micropolarity (Cosserat continua), the interested reader is referred to [5, 6, 9, 12] and the quotations therein.

5 Conclusion In summary, the present contribution provides a holistic approach for the macroscopic modeling of combined thermo- and electro-magnetomechanical phenomena

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in interacting multiphasic continua. Basically, it proceeds from a sensible combination of the specific local balance relations given in Boxes (69) and (81), which build a system of volume-coupled partial diﬀerential equations (PDEs) governing some region of interest. Accompanied by respective material-dependent constitutive equations, problem-specific assumptions, as well as boundary and initial conditions, this is what constitutes an initial-boundary-value problem in strong form. Consequently, one ends up with a volumetrically coupled formulation in the primal variables of choice, such as displacement, pressure, temperature, electric or magnetic potential among others, which can be made accessible to numerical solution. Logically, the involved PDEs by their very nature deal with continuous functions, and hence, have to be discretized in space and time for the computational treatment. The spatial semi-discretization, possibly using diﬀerent schemes for the diﬀerent physical fields, results in a system of ordinary diﬀerential equations (ODEs) or differential algebraic equations (DAEs)7 in time, which for real-scale problems may comprise millions of evolution equations that have to be integrated numerically. However, just as the choice of a specific volume-coupled formulation depends on the multi-field problem under study, the choice of a suitable solution strategy is by no means trivial. In essence, the appropriate solution approach (direct monolithic or iterative decoupled) is determined by the systems coupling characteristics (one-way or multi-way, linear or non-linear, weak or strong) as well as the physical properties and interdependencies controlled by the involved constitutive parameters, cf. [20]. At this point, we waive further problem-specific details, such as a comprehensive discussion of the entropy principle for mixtures paving the way to a particular constitutive setting and a tailored numerical solution strategy. Apparently, due to the manifold possibilities of physical processes involved, which essentially guide the sensible choice of the primary fields, it is virtually impossible to provide an allembracing presentation of the constitutive modeling and the numerical treatment as it is achieved for the conservation laws in the frame of the master balance principle.

References [1] Bennethum, L.S., Cushman, J.H.: Multicomponent, multiphase thermodynamics of swelling porous media with electroquasistatics: I. Macroscale field equations. Transport in Porous Media 47, 309–336 (2002) [2] de Boer, R.: Vektor- und Tensorrechnung für Ingenieure. Springer, Berlin (1982) [3] de Boer, R.: Theory of Porous Media. Springer, Berlin (2000) [4] Bowen, R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics, vol. III, pp. 1–127. Academic Press, New York (1976) [5] Diebels, S.: Mikropolare Zweiphasenmodelle: Formulierung auf der Basis der Theorie Poröser Medien. Habilitation, Bericht Nr. II-4 aus dem Institut für Mechanik (Bauwesen), Universität Stuttgart (2000) [6] Diebels, S., Ehlers, W.: On fundamental concepts of multiphase micropolar materials. Technische Mechanik 16, 77–88 (1996) 7

Algebraic equations typically occur when additional constraints or side conditions, such as material incompressibility, must be fulfilled, cf. [20].

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[7] Ehlers, W.: Toward finite theories of liquid-saturated elasto-plastic porous media. International Journal of Plasticity 7, 443–475 (1991) [8] Ehlers, W.: Grundlegende Konzepte in der Theorie Poröser Medien. Technische Mechanik 16, 63–76 (1996) [9] Ehlers, W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds.) Porous Media: Theory, Experiments and Numerical Applications, pp. 3–86. Springer, Berlin (2002) [10] Ehlers, W.: Challenges of porous media models in geo- and biomechanical engineering including electro-chemically active polymers and gels. International Journal of Advances in Engineering Sciences and Applied Mathematics 1, 1–24 (2009) [11] Ehlers, W., Karajan, N., Markert, B.: An extended biphasic model for charged hydrated tissues with application to the intervertebral disc. Biomechanics and Modeling in Mechanobiology 8, 233–251 (2009) [12] Eringen, A.C., Kafadar, C.B.: Nonlocal polar field theories. In: Eringen, A.C. (ed.) Continuum Physics, vol. IV, pp. 205–267. Academic Press, New York (1976) [13] Eringen, A.C., Maugin, G.A.: Electrodynamics of Continua I: Foundations and Solid Media. Springer, New-York (1990) [14] Hassanizadeh, S.M., Gray, W.G.: General conservation equations for multi-phasesystems: 2. mass, momenta, energy and entropy equations. Advances in Water Resources 2, 191–203 (1979) [15] Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004) [16] Jackson, J.D.: Classical Electrodynamics, 2nd edn. John Wiley & Sons, New York (1975) [17] Kelly, P.D.: A reacting continuum. International Journal of Engineering Science 2, 129–153 (1964) [18] Kovetz, A.: Electromagnetic Theory. Oxford University Press, Oxford (2000) [19] Markert, B.: A biphasic continuum approach for viscoelastic high-porosity foams: Comprehensive theory, numerics, and application. Archives of Computational Methods in Engineering 15, 371–446 (2008) [20] Markert, B.: Weak or Strong – On Coupled Problems in Continuum Mechanics. Professorial dissertation (Habilitation), Report No. II-20 of the Institute of Applied Mechanics (CE). Universität Stuttgart, Germany (2010) [21] Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids. North Holland - Elsevier, Amsterdam (1988) [22] Müller, I.: Thermodynamics. Pitman Publishing, Boston (1985) [23] Müller, I.: Thermodynamics of mixtures and phase field theory. International Journal of Solids and Structures 38, 1105–1113 (2001) [24] Ruggeri, T.: Galilean invariance and entropy principle for systems of balance laws. The structure of extended thermodynamics. Continuum Mechanics and Thermodynamics 1, 3–20 (1989) [25] Truesdell, C.: Thermodynamics of diﬀusion. In: Truesdell, C. (ed.) Rational Thermodynamics, 2nd edn. pp. 219–236. Springer, New York (1984)

Ice Formation in Porous Media Joachim Bluhm, Tim Ricken, and Moritz Bloßfeld

The authors feel honored and would like to express their gratitude towards the participation in this Springer volume on the occasion of the 60th birthday of Professor Wolfgang Ehlers.

Abstract. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this contribution, a macroscopic model based on the Theory of Porous Media (TPM) is presented toward the description of freezing and thawing processes in saturated porous media. Therefore, a quadruple model consisting of the constituents solid, ice, liquid and gas is used. Attention is paid to the description of capillary suction, liquid- and gas pressure on the surrounding surfaces, volume deformations due to ice formation, temperature distribution as well as influence of heat of fusion under thermal loading. For detection of energetic eﬀects regarding the control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange based on the local divergence of the heat flux is used. Numerical examples are presented to the applications of the model.

1 Introduction The behavior of liquid and gas filled porous media under cyclic thermal loading is strongly influenced by the phase transition as well as heat and mass transfer, e. g. drying of porous solids, the freezing of soils and concrete or geothermal investigations. In many branches of engineering, e. g. solid construction, material science and geotechnics, the coupled fluid-ice-solid behavior plays an important role in the prediction of solid frost resistance. Joachim Bluhm · Moritz Bloßfeld Institute for Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, 45117 Essen, Germany e-mail: {joachim.bluhm,moritz.blossfeld}@uni-due.de Tim Ricken Computational Mechanics, Faculty of Engineering, Department of Civil Engineering, University of Duisburg-Essen, 45117 Essen, Germany e-mail: [email protected] B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 153–174. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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Frost damage results from two diﬀerent mechanisms: (i) internal frost attack caused by the freezing of a liquid phase inside the material, (ii) surface scaling, normally caused by the freezing of weak salt solutions at the surface of the solid matrix, see [22, 33]. One basic common cause for both of them, namely the expansion of ice associated with the achievement of a critical degree of saturation in the pores has been expected in [17]. Due to the anomaly of water, a 9% increase of volume dilatation (relative to the volume of liquid) takes place during the phase change from liquid to ice, i. e., an equivalent water volume has to be fitted in the pores. If there is no void available, internal pressure acts on the surfaces of the pores and frost damage occurs at the so-called hydraulic pressure, compare [32], and exceeds the tensile strength. First laboratory tests of ice formation in porous media have been carried out in [1]. Regarding the qualitative development of frost heave in soils, the researchers in [29–31] have compared test results with results of their mathematical model by using the finite element method. Problems due to the volume dilatation and their resulting stresses during freezing further multiphase models have been developed. A one dimensional model for frost heave based on the balance equation of energy and the Richards equation, see [34], has been presented in [20]. A macroscopic ternary model consisting of liquid, crystal and solid analyzing the eﬀect of cooling rate and pore radius distribution during ice formation of water infiltrated porous materials below the freezing point has been discussed in [11]. The model is to be understood as an extension of both the standard theory of poroelasticity ([3]) and the energy approach of poromechanics ([10]). Furthermore, mass exchange between the phases has been considered in [11], but in comparison with the TPM note that the supply terms of mass do not influence the balance equation of energy. A four-phase unsaturated porous media with interaction of water, heat and deformation consisting of soil particles, ice, water and air under freezing and thawing conditions has been investigated in [28]. The main focus lies in the description of air in the unsaturated freezing soil. The aforementioned remarks illustrate the complexity of freezing and thawing processes in porous solids in consideration of diﬀerent constituents with diﬀerent physical properties and thermo-mechanical interaction between the phases on the one hand and the phase transition and transport mechanisms on the other hand. For the description of fluid saturated porous solids, the Theory of Porous Media (TPM) ([7, 8, 12–14]) has been established in recent years. Within the framework of the TPM a quadruple model consisting of the constituents solid (cement stone), liquid (freezable water), gas (air) and ice is considered. The influence of the pore solution (gel water and non freezable water, respectively) is not taken into account, i. e., the model is not able to cover the suction of water assigned to the gel water during thawing processes ([39]). With respect to the gas phase, partly liquid saturated porous materials are investigated. Consequently, capillary eﬀects, see [21, 37], are included in this model. Furthermore, the influence of the pore radius is neglected and only elastic deformations of solid and ice are taken into account. The presented quadruple model is to be understood as a fundamental description extending previously known results of freezing and thawing processes and capillary

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eﬀects in liquid and gas filled porous media. The aforementioned simplifications of the model were made for a more precise description of thermal eﬀects and capillary suction. Note that the phase transition acts between liquid and ice. As shown in [5, 6, 36], the fusion enthalpy in a material point can be described by controlling the divergence of the heat flux, especially during the thawing process. Thus, the ice formation is influenced by the local available energy. After discussion of the field equations, we derive restrictions for the constitutive relations and the dissipation mechanism of the simplified model. Furthermore, a calculation concept is presented. Their numerical solution has been treated within the framework of a standard Galerkin procedure, whereby the resulting weak formulations are implemented into FEAP (Finite Element Analysis Program). The usefulness of the presented model is demonstrated by a comparison of computationally and experimentally gained data of the CIF-Test (Capillary suction, Internal damage and Freeze-thaw test; [42]). The illustrated results show that the simplified model is capable of reproducing the experimental observations and describing capillary suction.

2 Basics Originally, the Theory of Porous Media (TPM) was designed for modeling of geotechnical problems within a macroscopic theory. It is based on the theory of mixtures extended by the concept of volume fractions, i. e., it is assumed that all constituents are statistically distributed over a control space and that the system is in ideal disorder. The diﬀerent phases of a saturated porous solid are identifiable given their volume fractions in an RVE (Representative Volume Element). Readers interested in the foundations of the TPM are referred to [7, 8, 12, 13, 15, 16]. The field equations for porous media are composed of the balance equations of the constituents taken from the mixture theory and the saturation condition. Excluding additional supply terms of moment of momentum, the balance equations for a saturated porous medium are given by the local statements of the balance equations of mass, momentum, moment of momentum and energy for each individual constituent: (ρα )α + ρα div xα = ρˆ α , div Tα + ρα ( b − xα ) = ρˆ α xα − pˆ α , ρα (εα )α − Tα · Dα − ρα rα + div qα = 1 = eˆ α − pˆ α · xα − ρˆ α ( εα − xα · xα ) . 2

Tα = (Tα )T , (1)

In these equations, ρα = nα ραR and Tα are the partial density and the partial Cauchy stress tensor of the constituent ϕα , where nα and ραR denote the volume fraction and the real density of the constituents. The vector b is the external acceleration, the velocity and the acceleration of the motion function χα are represented by the vectors

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xα and xα . The internal energy, the external heat supply and the influx of energy are denoted by εα , rα and qα . The quantities ρˆ α , pˆ α and eˆ α represent the local supply terms of mass, momentum and energy of ϕα arising out of all other constituents κ − 1 that occupy the same position as ϕα at time t. The tensor Dα = 1/2(Lα + LTα ) is the symmetric part of the spatial velocity gradient Lα = (Gradα xα )F−1 α = grad xα , where Fα = Gradα χα is the deformation gradient. In addition, “div” is the divergence operator and the symbol (. . . )α = ∂(. . . )/∂t + grad(. . . ) · xα (calculation specification for a scalar quantity) defines the material time derivative with respect to the trajectory of ϕα . For the supply terms of mass, momentum and energy the following restrictions are applied: κ α=1

ρˆ α = 0 ,

κ α=1

pˆ α = o ,

κ α=1

eˆ α = 0 .

(2)

Since the solid particles and all remaining κ − 1 constituents are assumed to occupy all the volume available, the porous solid is said to be saturated and κ α=1

nα =

κ ρα = 1. αR α=1 ρ

(3)

Equation (3) is the so-called saturation condition which restricts the motion of the individual phases. In order to gain restrictions for constitutive relations, the second law of thermodynamics (entropy principle) is a helpful tool. The local entropy inequality for the mixture with diﬀerent temperatures of the constituents is defined as κ α=1

[ ρα (ηα )α + ρˆ α ηα ] ≥

κ α=1

[

1 α α 1 ρ r − div ( α qα ) ] , Θα Θ

(4)

where ηα and Θα are the specific entropy and the absolute temperature of the constituent ϕα . Considering the balance equation of energy (1)4 , the local statement of the entropy inequality can be transferred into κ 1 1 { − ρα [ (ψα )α + (Θα )α ηα ] + Tα · Dα − α qα · grad Θα + α Θ Θ α=1 κ 1 + eˆ α − pˆ α · xα − ρˆ α ( ψα − xα · xα ) } + λ ( 1 − nα )S ≥ 0 , 2 α=1

(5)

see [13], where the Helmholtz free energy function ψα = εα − Θα ηα has been used. As an extension, the material time derivation of the saturation condition following the motion of solid together by using the concept of Lagrange multipliers has been introduced. The Lagrange multiplier λ is understood as a reaction force assigned to the saturation condition.

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3 Simplified Quadruple Model With regard to the description of freezing and thawing processes of liquid and gas saturated porous solid materials like concrete a quadruple model consisting of the constituents solid (cement stone), ice, liquid (freezable water) and gas (air) is presented (α = S, I, L, G). The following assumptions and simplifications, respectively, will be made: 1. The fluid flow is low during the process, therefore it seems reasonable to neglect dynamic eﬀects and the square of the velocities (xα = o, xα · xα = 0); 2. Internal energy terms are not taken into consideration (rα = 0); 3. Local temperatures of all constituents are equal (Θα = Θ); 4. Mass exchange only acts between the ice and liquid (ρˆ S = ρˆ G = 0, ρˆ I = −ρˆ L ); 5. Solid, ice and liquid are incompressible (ρβR = const., β = S, I, L); 6. Gas is compressible (ρGR const.); 7. Motions of ice and solid are identical except at an initial solid motion, i. e., χI = χS − χS0 (the initial solid motion χS0 is the accrued motion before the onset of the phase transition). The consideration of the initial solid motion is connected with a multiplicative decomposition of the deformation gradient of the solid phase: FS = FI FS0 ,

FS0 = Grad χS0 .

(6)

The here used order of the decomposition of FS has been introduced in [2, 23, 38], toward the modeling growth in biological tissues. The decomposition is connected with the existence of the right Cauchy-Green deformation tensors of solid, the initial part of solid motion and ice: CS = FTS FS ,

CS0 = FTS0 FS0 ,

ˆ I = F T FI . C I

(7)

The symbol (. .ˆ . ) signifies that the tensor is related to the intermediate configuration, in which the initial solid deformations are stored in the material. Furthermore, the decomposition (6)1 is accompanied with the representation forms JS = JI JS0 ,

(JS )S = JS ( DS · I ) = (JI )S JS0 ,

(JI )S = JS J−1 S0 ( DS · I )

(8)

of the Jacobian and the material time derivatives of the Jacobian of solid and ice. Note that as a result of load and temperature changes following a freeze-thaw cycle the intermediate configuration can change.

3.1 Field Equations Considering the aforementioned assumptions, the set of coupled field equations is given by the balance equations of mass,

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(nS )S + nS div xS = 0 , (nL )L + nL div xL = −

(nI )S + nI div xS =

ρˆ I , ρLR

ρˆ I , ρIR

(nG )G + nG div xG +

(9)

nG GR (ρ )G = 0 , ρGR

the balance equations of momentum for the mixture, for liquid and for gas, div T + ρ b = − ρˆ I wLS , div TL + ρL b = − ρˆ I xL − pˆ L ,

div TG + ρG b = − pˆ G ,

(10)

the balance equation of energy for the mixture, α α [ ρ (ψ )α + Θ ρα (ηα )α + ΘS ρα ηα ] + ρL ηL wLS · grad Θ + α

+ ρG ηG wGS · grad Θ − TSI · DS − TL · DL − TG · DG + div q =

(11)

= − p · wLS − p · wGS − ρˆ [ ψ − ψ + Θ ( η − η ) ] , ˆL

ˆG

I

I

L

I

L

and the material time derivative of the saturation condition along the trajectory of the solid, div( nL wLS + nG wGS + xS ) +

nG GR 1 1 (ρ )G − ρˆ I ( IR − LR ) = 0 , ρGR ρ ρ

(12)

where the expression εα = ψα + Θ ηα for the internal energy and the abbreviations α α α T = T , TSI = TS + TI , q = q , ρ = ρ , (13) α

α

α

have been used. The quantities wLS = xL −xS and wGS = xG −xS are the velocities of liquid and gas relative to solid. It can be summarized that the simplified quadruple model is described by 15 field equations for the 3D case, see (9) – (12). Note that the phase changes due to freezing and thawing processes or due to drying processes, like liquid or liquid parts turn into vapor are known as first-order transitions. For the modeling of drying processes in deformable porous bodies it is referred to e. g. [25–27, 43].

3.2 Constitutive Theory For the here discussed quadruple model consisting of three incompressible phases (solid, ice, liquid) and a compressible gas phase, the set of the 15 unknown field quantities reads: U = { χS = χI , χL , χG , Θ , nS , nI , nL , nG , ρGR } .

(14)

Considering that the acceleration of gravity b is given, the remaining quantities C = { TSI , TL , TG , q , ψα , ηα , pˆ L , pˆ G , ρˆ I }

(15)

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require constitutive assumptions or evolution equations. It is postulated that these variables can depend on the following set of process variables, where viscous eﬀects in the stresses are not considered: ˆ I , JL , nI , nL , nG , ρGR , P = { Θ , grad Θ , CS , C grad nL , grad nG , grad ρGR , wLS , wGS } .

(16)

To simplify the evaluation of the entropy inequality, for the Helmholtz free energy functions the following dependences are postulated: ψS = ψS ( Θ , CS ) ,

ˆ I , nI ) , ψI = ψI ( Θ , C

ψL = ψL ( Θ , nL , nG ) ,

ψG = ψG ( Θ , ρGR ) .

(17)

With the ansatz (17) it can be shown that the entropy inequality (5) is fulfilled for the following constitutive relations for the Cauchy stress tensors TSI = − nSI λ I + TSI E ,

TL = − nL λ I + TLE ,

TG = − nG λ I + TG E

(18)

with the abbreviation nSI = nS + nI and the eﬀective stresses ∂ψI ∂ψS T ∂ψI T S I I + 2 ρ F F + 2 ρ F F , S I ˆI I ∂CS S ∂nI ∂C L ∂ψL G L ∂ψ = − nL ρL L I , TG I. E = −n ρ G ∂n ∂n

I I TSI E = −n ρ

TLE

(19)

The Lagrange multiplier λ, i. e., the reaction force on the rate of the saturation condition, is constitutively determined by λ = (ρGR )2

∂ψG ∂ψL − ρL G . GR ∂ρ ∂n

(20)

The relations for the specific entropies read ηα = −∂ψα /∂Θ. The dissipations mechanism of (5) yields the evolution equations for the heat flux vector as well as for the supply terms of mass and momentum: q = − αΘ grad Θ − αwLS wLS − αwGS wGS , ρˆ I = − βIΨ IL ( Ψ I − Ψ L ) , pˆ L = λ grad nL − ρL pˆ G = ( λ + ρL

∂ψL grad nG + pˆ LE , ∂nG

∂ψL ) grad nG + pˆ G E , ∂nG

(21)

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where I L 1 1 I ∂ψ L L L ∂ψ ( λ + ρ ) , Ψ = ψ + ( λ + ρ ), ρIR ∂nI ρLR ∂nL L = − γΘ grad Θ − γwL LS wLS − γwL GS wGS ,

Ψ I = ψI + pˆ LE

(22)

G G G pˆ G E = − γΘ grad Θ − γwGS wGS − γwLS wLS

are the chemical potentials of ice and liquid and the eﬀective production terms of momentum of liquid and gas. The material parameters are restricted by αΘ ≥ 0 ,

βIΨ IL ≥ 0 ,

L αwLS + γΘ = 0,

where the quantities αΘ =

α

ααΘ ,

γwL LS ≥ 0 , γwGGS ≥ 0 ,

G αwGS + γΘ = 0,

αwLS =

α

ααwLS ,

γwL GS + γwGLS = 0 , αwGS =

α

ααwGS

(23)

(24)

are composed of the corresponding material parameters of all phases. For the Helmholtz free energy functions of the solid and ice phase the following ansatz are postulated: 1 1 S 1 [ λ ( log JS )2 − μS log JS + μS ( CS · I − 3 ) − S 2 ρ0S 2 Θ S S S S S − 3 αΘ k log JS ( Θ − Θ0S ) − ρ0S c ( Θ log S − Θ + ΘS0S ) ] , Θ0S 1 1 1 ˆI · I − 3) − ψI = IR [ λI ( log JI )2 − μI log JI + μI ( C 2 ρ0I 2 Θ I I − 3 αIΘ kI log JI ( Θ − ΘI0I ) − ρIR 0I c ( Θ log I − Θ + Θ0I ) − Θ0I ψS =

(25)

I I I I I I − ρIR 0I η0I ( Θ − Θ0I ) − 3 αnI k log JI ( n − n0I ) ] ,

Therein, the material parameters μγ , λγ for γ = {S , I} and kγ = 2/3μγ + λγ are the Lam´e constants and the bulk modulus of the corresponding constituents. The coeﬃcient of thermal expansion, the heat capacity and the coeﬃcient related to the volume fraction are denoted by ααΘ , cα and αInI , respectively. The quantity ηI0I is the reference value of the specific entropy at the phase transition to the ice phase. The diﬀerent reference temperatures of the constituents are given by Θα0α . The symbols (. . . )S0S = JS (. . . )S = const. and (. . . )I0S = JS (. . . )I const. denote quantities of solid and ice referring to the reference placement of solid at time t = 0. With (25) the following constitutive relations for the eﬀective stress tensors of solid/ice are determined, see (19)1 : TSI E =

1 { 2 μS KS + λS ( log JS ) I − 3 αSΘ kS ( Θ − ΘS0S ) I + JS + nI JS [ 2 μI KI + λI ( log JI ) I − 3 αIΘ kI ( Θ − ΘI0I ) I − − 3 αInI

k (n − I

I

nI0I

− n log JI ) I ] } , I

(26)

Ice Formation in Porous Media

161

where Kγ = 1/2 (Bγ − I) and Bγ = Fγ FTγ denote the Karni-Reiner strain tensor referred to the actual placement and the left Cauchy-Green deformation tensor. For the Helmholtz free energy functions of liquid and gas, the following ansatz are chosen: ψL =

nL0L 2 nL0L 1 1 L L L [ k ( log ) − 3 α k log ( Θ − ΘL0L ) − Θ 2 nL nL ρLR 0L Θ L LR L L L − ρLR 0L c ( Θ log L − Θ + Θ0L ) − ρ0L η0L ( Θ − Θ0L ) + Θ0L sL sL 1 1 L + kcap ( − dilog L − log ( 0L − sL0 ) log L + log( 1 − sL0 ) log 0L ) ] , s s s s

ψG = − cG ( Θ log

ρGR ρGR Θ G 0G − Θ + ΘG ( log 0G − 1). 0G ) + Θ R G GR ρ ρGR Θ0G

(27) GR Therein, ρLR and ρ denote the real densities of liquid and gas with respect to the 0L 0G corresponding reference placements, RG is the specific gas constant, kL is interpreted as a bulk modulus of the liquid phase and kLcap denotes the hydraulic pressure parameter. The quantity ηL0L is the reference value of the specific entropy at the phase transition to the liquid phase. The abbreviation sL =

nL nL = L SI 1−n n + nG

(28)

for liquid saturation is used and sL0 describes the limit of the so called eﬀective saturation, sLe = 1, see [21]. With (27) the constitutive equations TL = − nL pLR I ,

TG = − nG pGR I ,

(29)

for the stress of liquid and gas can be derived, where pLR = pGR + ρL

L ∂ψL L ∂ψ − ρ ∂nL ∂nG

= pGR + 3 αLΘ kL ( Θ − ΘL0L ) − kL log L − kcap [ log (

pGR = (ρGR )2

nL0L − nL

sL0 − sL0 ) − log ( 1 − sL0 ) ] , sL

(30)

ρGR ∂ψG 0G G GR = − Θ R ρ log 0G ∂ρGR ρGR

denote the realistic pressures of the phases. Considering (30)2 , the Lagrange multiplier λ, see (20), can be written as λ = pGR − ρL

sL0 ∂ψL GR L L = p − s k [ log ( − sL0 ) − log ( 1 − sL0 ) ] . cap ∂nG sL

(31)

162

J. Bluhm, T. Ricken, and M. Bloßfeld

Classically, the capillary pressure is introduced as the pressure diﬀerence between the non-wetting and the wetting fluid, e. g. see [9]. As aforementioned, the porous media consists of the solid and ice phase as well as of two fluid phases (liquid and gas). The gas can be identified as the non-wetting phase and the liquid as the wetting phase, see [19]. Postulating that the capillary pressure will only be influenced by the liquid saturation, it follows from (30): pC = pGR − pLR + pLR ( Θ ) − pLR ( nL ) L = kcap [ log (

(32)

sL0 − sL0 ) − log ( 1 − sL0 ) ] . sL

In comparison with the ansatz (32), Brooks & Corey and van Genuchten have formulated the following relations for the liquid saturation of multiphase systems, see [9, 18]: sL ( pC = pCBC ) = (

p BC λBC ) , pCBC

sL ( pC = pCvG ) = [ 1 + ( αvG pCvG )nvG ]−mvG . (33)

The corresponding capillary pressures read pCBC ( sL ) = p BC (sL )−1/λBC ,

pCvG ( sL ) =

1 [ (sL )−1/mvG − 1 ]1/nvG . αvG

(34)

An exponential ansatz for the capillary pressure and the liquid saturation, respectively has been used in [35]: L pCExp ( sL ) = kcap

− log sL ,

sL ( pCExp ) = exp[ − (

pCExp L kcap

)2 ] .

(35)

A similar ansatz has been discussed in [4]. The qualitatively diﬀerent shapes of the capillary-pressure-saturation relations are presented in Figs. 1 and 2.

Parameter Value

Unit

λBC

2.0

[−]

pBC nvG mvG

2.0 · 105 4.37 0.77

[N/m2 ] [−] [−]

αvG

0.37 · 10−5 [N/m2 ]

L kcap

5.0 · 105

[N/m2 ]

Fig. 1 Capillary-pressure-saturation relations without hysteretic according to (34) and (35)1 .

Ice Formation in Porous Media

163

Parameter plot 1 plot 2 plot 3 plot 4 Unit L kcap

1.0

sL0

0.95 0.95 0.95 0.90 [−]

1.5

2.0

1.0

[105 N/m2 ]

Fig. 2 Capillary-pressure-saturation relations without hysteretic according to (32).

Taking into consideration the afore postulated dependence of the Helmholtz free energy functions of the phases on selected process variables, see (17), the discussed constitutive relations and evolution equations as well as the material time derivative of the saturation condition (12), weighted with the Lagrange multiplier λ, the balance of energy for the mixture simplifies to Θ [ ρS (ηS )S + ρI (ηI )S + ρL (ηL )L + ρG (ηG )G ] + div q = = − pˆ LE · wLS − pˆ G ˆ I [ Ψ I − Ψ L + Θ ( ηI − ηL ) ] , E · wGS − ρ

(36)

where Ψ I = ψI +

I 1 I ∂ψ ( λ + ρ ), ρIR ∂nI

Ψ L = ψL +

L 1 L ∂ψ ( λ + ρ ) ρLR ∂nL

(37)

are the chemical potentials of ice and liquid. With the definitions of the specific enthalpies assigned to phase change of the constituents ice and liquid, sIp = Ψ I + Θ ηI = εI +

1 I p , ρI red

sLp = Ψ L + Θ ηL = εL +

1 L p , ρL red

(38)

where pIred = nI ( λ + ρI

∂ψI ), ∂nI

pLred = nL ( λ + ρL

∂ψL ∂ψG + ρG L ) L ∂n ∂n

(39)

denote the reduced hydraulic pressures, the balance of energy for the mixture can be transferred into Θ [ ρS (ηS )S + ρI (ηI )S + ρL (ηL )L + ρG (ηG )G ] + div q = = − pˆ LE · wLS − pˆ G ˆ I ( sIp − sLp ) . E · wGS − ρ

(40)

164

J. Bluhm, T. Ricken, and M. Bloßfeld

It is well-known, that the freezing point depends on the concentration of solutes dissolved in water and on pressure. The freezing point of water is 273.15 [K] under standard atmosphere pressure (p0 = 101325 [Pa] = 1.01325 [bar]). For Δp = 200 [bar] and Δp = 2000 [bar] the freezing point is depressed approximately by 2 [K] and 22 [K], respectively. In the considered examples of this contribution, the pressure of water and ice lies far below 200 [bar]. Therefore, the dependence of the freezing point on the pressure is not taken into account in the model. Thus, the freezing point can be defined as ΘI0I = ΘL0L = 273 [K] and the specific reference entropies for liquid and ice are ηL0L = 3.055 [kJ/kg K] and ηI0I = 1.832 [kJ/kg K], respectively. Furthermore, the influence of gel water, which has been investigated by employing the micro-ice-lens model in [40], is not considered. By using the relations for ψδ (δ = I , L), see (25)2 and (27)1 , in the restrictions δ ˆ I = I, which implies JI = 1, and nL = nL , the η = −∂ψδ /∂Θ, then at the positions C 0L δ functions of specific entropies η and specific enthalpies sδp for the non-deformed configurations of the phases can be written as Θ Θ ηI ˆ = cI log I + ηI0I , ηL L L = cL log L + ηL0L , CI = I n = n0L Θ0I Θ0L sIp ˆ = cI (Θ − ΘI0I ) + ηI0I ΘI0I , sLp L L = cL (Θ − ΘL0L ) + ηL0I ΘL0L , C =I n =n I

(41)

0L

where pδred is set to zero. The assumption pδred = 0 is based on the splitting pδred = p˜ δred − p0 , where the internal reduced hydraulic pressure p˜ δred is set to the standard atmosphere pressure p0 (˜pδred = p0 ). The functions of the specific entropies (41)1,2 and specific enthalpies (41)2,3 are plotted in Fig. 3 for the following values of ice and liquid: freezing point: ΘI0I = ΘL0L = 273 [K]; specific heat capacities: cI = 2.090 [kJ/kg K], cL = 4.183 [kJ/kg K]; specific reference entropies: ηL0L = 3.055 [kJ/kg K], ηI0I = 1.832 [kJ/kg K]. These functions illustrate the phase transition between ηI0I and ηL0L as well as between sIp0I and sLp0L . The diﬀerence fI = sLp0L −sIp0I = 834−500 = 334 [kJ/kg] is the so called heat of fusion of ice at the freezing point. The above specific entropies depend on the logarithm of Θ, whereas the specific enthalpies are linear functions. ηL0L

a)

sIp0I

Θα0α

ηα [kJ/kg K]

sLp0L

temperature Θ[K]

temperature Θ[K]

ηI0I

b)

Θα0α

sαp [kJ/kg]

Fig. 3 a) Specific entropy ηα (Θ) and b) Specific enthalpy sαp (Θ).

Ice Formation in Porous Media

165

Furthermore, note that the change of the pressure in order of ˜pδred = 20 [bar] leads to a change of the enthalpies of sLp = 2.0 [kJ/kg] and sIp = 2.198 [kJ/kg] for one component materials, which is negligibly small. In this model only mass exchange from liquid to ice and vice versa is considered. Thus, only the specific heat of fusion fI for the ice phase has to be taken into account, i. e., fS = fL = fG = 0 and fI 0. Due to the above remark, under the standard atmosphere pressure p0 , the latent heat of fusion for ice is considered as constant (fI = 334 [kJ/kg]), i. e., the influence of the deformations and the change of the pressure with respect to the enthalpies are neglected. Apart from the scalar quantities pˆ LE · wLS and pˆ G E · wGS , the balance equation of energy (40) for the quadruple porous medium has the structure as the balance of energy of a one-component material with an internal source represented by ρˆ I (sIp − sLp ). The investigation of a one dimensional freezing process of water and ice, see Fig. 4, where the phases water and ice are considered as incompressible one-component materials and not as a mixture, i. e., all supply terms are equal to zero and the interaction between the two phases is captured by the moving interface, yields the relation qI − qW (42) v = I W ρ ( s − sI ) for the velocity of the interface. Equation (42) shows that the driving force of the velocity of the interface is equal to the jump of the heat flux vectors divided by the corresponding specific enthalpies.

Fig. 4 Illustration of a one-dimensional freezing process of water and ice.

Referring to the statements of the one-dimensional freezing process of water and ice, the following ansatz for the prefactor βIΨ IL is postulated: βIΨ IL =

( ΨI

βI (div q)B . − Ψ L ) ( sIp − sLp )

(43)

Considering the restriction (21)2 the mass supply term reads ρˆ I = −

βI (div q)B , sIp − sLp

(44)

166

J. Bluhm, T. Ricken, and M. Bloßfeld

where the parameter βI is in the range 0 ≤ βI ≤ 1 and controls the contingent of internal energy which aﬀects the change of temperature or phase transition, see [36]. With respect to the numerical treatment, the symbol (. . . )B means that the quantity div q is acting only in the volume and cannot be transferred to the surface. A detailed derivation of the reformulation of the evolution equation for mass exchange between ice and liquid is given in [6].

4 Examples Toward the description of the capillarity suction during the freezing and thawing processes, the linearized weak forms of the balance equations of mass of ice, liquid and gas, momentum and energy of the mixture, all within the framework of a standard Galerkin procedure have been implemented in the finite element program FEAP. In these weak forms, the diﬀerence velocities have been expressed with help of the balance equations of momentum of liquid and gas in consideration of the relations for the supply terms. The boundary conditions of the problem can be formulated as Dirichlet or as Cauchy/Neumann constraints. The rectangular volume (2-D) is approximated by 8-node-elements of Taylor-Hood type, with quadratic shape functions for solid displacements and temperature as well as bi-linear shapefunctions for saturation pressure and volume fraction ice. In this section an application of the model is given. Specifically, the model is applied to a cuboid solid specimen with a cross-section of 15 cm2 and a height of 7.5 cm. The dimension of the specimen is taken from the CIF-Test (Capillary suction, Internal damage and Freeze-thaw test), see Fig. 5 a). During the CIF-Test, the bottom of the saturated solid is also cooled down and heated. The temperature load is given in Fig. 5 b). Due to lateral air layers such as thermal isolation, the heat transport is one dimensional. For a detailed description of the CIF-Test see [42].

temperature

20°C

0°C

-20°C

a)

b)

0h

4h

time

7h

11h 12h

Fig. 5 a) CIF-Test, see [41]; b) Temperature load CIF-Test.

Assuming that all unstressed surfaces are adiabatic to the environment, it is sufficient to analyze the middle surface of the cuboid, i. e., the ice formation can be modeled as a 2-D problem. Due to a realistic simulation of an adiabatic saturated cement specimen, the influence of vapor and air is considered during the process.

Ice Formation in Porous Media

167

4.1 Example 1: Capillary Suction during Freezing In the first example the specimen will be saturated within 5.5 hours. Afterwards, a freeze-thaw cycle of the following value and boundary conditions are chosen, see Fig. 6 a). The bottom of the specimen will be cooled in-line from 293 K to 253 K within 4 hours and afterward will be heated to 293 K within 4 hours, see Fig. 6 b).

a)

b) Fig. 6 a) System example 1; b) Temperature load example 1.

The material parameters of the initial configuration are given in Table 1. With respect to the detection of energy eﬀects during the phase transition for the mass exchange between ice and water and vice versa, the evolution equation based on the local balance of the heat flux is used, see (44). Furthermore, as previously mentioned for the numerical results discussed here it is postulated that the prefactor βI of the ansatz in the mass supply term of ice, see (44), is equal to 1 for the freezing as well as for the thawing process. Due to the numerical control for the time discretization in the Newmark method the time step t = 8.3 [min] is used. Table 1 Material parameters of the initial configuration. Parameter

Unit Concrete

Lam´e constant μα Lamé constant λ

α

compression modulus kα heat dilatation coeﬀ. ααΘ

(273 K)

specific heat capacity cα heat conduction coeﬃcient ααΘ specific heat of fusion fα real density ραR initial volume fraction n

N m2 N m2 N m2

1 K

J

g˛ K W mK

J˛ g˛ g˛

factor Darcy permeability γwFS

[–] Ns m4

Liquid water Gaseous air

12.5e+9

4.17e+9

–

–

8.33e+9

2.78e+9

–

–

16.67e+9

5.56e+9

20e+09

–

(0.9-1.2)e-6

5.1e-5

1.8e-4

–

900 (273-373 K) 2090 (273 K) 4190 (293 K)

720

1.1 (293 K)

2.2 (273 K)

0.58 (293 K)

0.02 (293 K)

2081 (1986 K)

334 (273 K)

–

14 (54 K)

2000

920

1000

1.204

0.5

0.0

0.1

0.4

–

–

0.9e+5

0.5e+5

m3 α

Solid ice

168

J. Bluhm, T. Ricken, and M. Bloßfeld

In the simulation the specimen is considered as jacketed. Thus, during the freezethaw cycle liquid water and gas cannot be sucked in or squeezed out to the lateral unstressed surfaces, i. e., the corresponding diﬀerence velocities wLS and wGS on the corresponding boundaries are equal to zero. The bottom of the specimen is assumed as water permeable (wLS o, wGS = o), whereas only the top of the specimen is assumed as gas permeable (wGS o, wLS = o). These boundary conditions, see also Fig. 6 a), are conformed with the CIF-Test. With respect to the capillary suction, depending on the initial volume fraction of solid, liquid and gas, see Table 1, Eq. (28) provides the initial liquid saturation sL = 0.2 [−]. Therefore, at the beginning of the simulation the initial fluid pressure pLR = −358 [Pa] is calculated by Eq. 0 (32), where the parameters for the limit of saturation sL0 = 0.9 [−] and the hydraulic L pressure parameter of liquid phase kcap = 100 [N/m2 ] are used. Because of the L definition of limit of the saturation with s0 = 0.9 [−], Eq. (28) yields the maximally occurring volume fraction of water nL = 0.45 [−] (water saturated). During the first part of the simulation, within 5.5 hours, the volume fraction of water increases (nL ≥ 0.1 [−], see Fig. 7 e)), because the existing capillary power leads to a capillary flow and more water gets sucked into the system, see Fig. 7 e) and Fig. 8 e). This eﬀect is called capillary suction. If the specimen is saturated (nL = 0.45 [−]) the internal pressure pLR is equal to zero, see Fig. 7 b) and Fig. 8 b). This situation describes the balance of power, that means the capillary power is equal to the weight of water column (pLR = ρL |b| h = 750 [Pa], |b| = 10[m/s2 ]). Afterwards in the second part of the simulation the bottom of the saturated specimen is cooled down and then it is heated again by the temperature load as previously defined, see Fig. 6 b). This process describes a freeze-thaw cycle from the CIF-Test, see Fig. 5 b). During the cooling phase in-between 293 − 273 K, a thermal propagation is observed in experimental measurement as well as in the simulation. At a temperature of Θ ≤ 273 K, the freezing of water in the pores begins and the phase change from liquid to ice occurs. While the volume fraction of water decreases, the one of ice increases. Bear in mind that in this example the volumetric deformation of the specimen is not illustrated. Figs. 7–9 a) to f) show the distributions of the temperature, the fluid pressure, the gas pressure, the volume fraction of solid (ice), the volume fraction of liquid (water) and the volume fraction of gas (air) after 0.7, 5.4 and 11.1 hours. Furthermore, in Fig. 10 a) to c) the volume fractions of solid, ice, liquid and gas are plotted over time with respect to the marked points P1 , P2 and P3 . These points are located at a height of 0.94, 1.94 and 2.94 cm above the specimen. The functions in Fig. 10 a) to c) show the significant change of volume fraction during capillary suction and the freezing process. During the saturation the volume fraction of liquid increases, that of gas decreases. At the beginning of the ice formation the volume fraction of ice increases, whereas gas decreases and liquid declines.

Ice Formation in Porous Media

169

temperature [K] 250

a)

259

268

time: 0.7 [h] 250

277

fluid pressure [Pa]

286

295

d)

0.10

0.20

time: 0.7 [h] 0

0.30

0.40

0.25

-438

-228

-19

-20

-14

-8

-2

4

10

0.075

0.075

0.05

0.05

0.05

0.025

0.025

b)

volume fraction solid (ice) [-] 0.00

-647

0.075

0 300

275

-1066 -857

gas pressure [Pa]

0

time: 0.7 [h] -1000

-500

0

0.025

c)

0

time: 0.7 [h]

volume fraction liquid (water) [-]

0.50

0.00

0.10

0.20

0.30

0.40

-15

0

15

volume fraction gas (air) [-]

0.50

0.00

0.10

0.20

0.30

0.40

0.50

0.075

0.075

0.075

0.05

0.05

0.05

0.025

0.025

0.025

0

0

0.5

e)

time: 0.7 [h] 0

0.25

0.5

f)

0

time: 0.7 [h] 0

0.25

0.5

Fig. 7 a) Temperature; b) Fluid pressure; c) Gas pressure; d) Volume fraction solid (ice); e) Volume fraction liquid (water); f) Volume fraction gas (air) after 0.7 hours. temperature [K] 250

a)

259

268

time: 5.4 [h] 250

277

fluid pressure [Pa]

286

295

d)

0.10

0.20

time: 5.4 [h] 0

0.30

0.40

0.25

-438

gas pressure [Pa]

-228

-19

-20

-14

-8

-2

4

10

0.075

0.075

0.05

0.05

0.05

0.025

0.025

0.025

b)

volume fraction solid (ice) [-] 0.00

-647

0.075

0 300

275

-1066 -857

0

time: 5.4 [h] -1000

-500

0

c)

0

volume fraction liquid (water) [-]

0.50

0.00

0.10

0.20

0.30

0.40

-15

time: 5.4 [h]

0

15

volume fraction gas (air) [-]

0.50

0.00

0.10

0.20

0.30

0.40

0.50

0.075

0.075

0.075

0.05

0.05

0.05

0.025

0.025

0.025

0

0

0.5

e)

time: 5.4 [h] 0

0.25

0.5

f)

0

time: 5.4 [h] 0

0.25

0.5

Fig. 8 a) Temperature; b) Fluid pressure; c) Gas pressure; d) Volume fraction solid (ice); e) Volume fraction liquid (water); f) Volume fraction gas (air) after 5.4 hours. temperature [K] 250

a)

259

268

time: 11.1 [h] 250

277

275

fluid pressure [Pa]

286

295

-1066 -857

d)

0.10

0.20

time: 11.1 [h] 0

0.30

0.25

-19

-20

-14

-8

-2

4

10 0.075

0.05

0.05

0.05

0.025

0.025

0.025

b)

0

time: 11.1 [h] -1000

-500

0

c)

volume fraction liquid (water) [-]

0.50

0.5

-228

0.075

0 300

0.40

-438

0.075

volume fraction solid (ice) [-] 0.00

-647

gas pressure [Pa]

0.00

0.10

0.20

0.30

0.40

0 -15

time: 11.1 [h]

0

15

volume fraction gas (air) [-]

0.50

0.00

0.10

0.20

0.30

0.40

0.50

0.075

0.075

0.075

0.05

0.05

0.05

0.025

0.025

0.025

0

0

e)

time: 11.1 [h] 0

0.25

0.5

f)

0

time: 11.1 [h] 0

0.25

0.5

Fig. 9 a) Temperature; b) Fluid pressure; c) Gas pressure; d) Volume fraction solid (ice); e) Volume fraction liquid (water); f) Volume fraction gas (air) after 11.1 hours.

170

J. Bluhm, T. Ricken, and M. Bloßfeld

Fig. 10 a) Volume fractions concerning marked point P3 ; b) Volume fraction liquid; c) Volume fraction gas concerning marked points P1 , P2 and P3 .

4.2 Example 2: Heat of Fusion during Phase Transition In the second example the comparison between the numerical results and experimental data is presented. For simplification a ternary model consisting of the phases solid, ice and liquid is used. These simplifications of the model were made for a more precise description of thermal eﬀects. Here, the bottom of the system is also cooled down and is then heated again by the temperature load as previously defined, see Fig. 5 b). Furthermore, the temperature load on the top of the specimen is considered by additional boundary value conditions. This temperature load is proportional to the load on the bottom, where on the top max. ΔΘ is set to 30 K instead of 40 K. The temperature development on the top is taken from the measurements in [24]. For this simulation of a freeze-thaw cycle the same material parameters, see Table 1, as in the above examples are used, except for the initial volume fractions of solid and water. In particular the initial volume fraction of water and solid is set to real values namely nL = 0.1[ − ] and nS = 0.9[ − ], respectively. Regarding the numerical control for the time discretization in the Newmark method, the time step t = 1.6 [min] is used. The numerical results clearly point out the influence of heat of fusion during freezing and thawing, see Fig. 11.

Fig. 11 Development of temperature over time concerning the marked points P1 , P2 and P3 .

Ice Formation in Porous Media

171

temperature Θ[o C]

Especially, the numerical simulation of the thawing process shows, that at the socalled melting point (Θ = 273 K), the feeding energy is used to melt the ice and is not accompanied with an increase of the local temperature. The underlying cause of this physical eﬀect is the heat of fusion during the phase change from ice to water. The temperature distribution of the measurements in [24], see Fig. 12, show similar temperature functions in comparison with numerical results, see Fig. 11. Hence, the increasing temperature of penetrating liquid near the freezing point results from the energy released during freezing. Note that the lower and upper temperature sensors are located at a height of 1 and 6 cm above the specimen, whereas the other sensors between are located in steps of 1 cm, see Fig. 5 a).

upper sensor (6 cm) lower sensor (1 cm)

time [hh:min]

Fig. 12 Development of temperature over time during the fourth freeze-thaw-cycle, see [24].

The comparison of the numerical results with the experimental data, see Figs. 11 and 12 show a good agreement. Thus, the model is capable of simulating the temperature distribution and energetic eﬀects during phase change.

5 Conclusion In this article, a quadruple macroscopic model has been presented for simulation of freeze-thaw cycles in consideration of capillary eﬀects in saturated porous media. In eﬀect, the model describes the following aspects of this process: expansion of ice during freezing, liquid- and gas pressure on the surrounding surfaces, capillary suction, temperature distribution and the influence of heat of fusion. The main focus lies in the description of the influence of the gas phase with respect to the compensation of volume expansion of water during freezing as well as in the description of capillary eﬀects before and during phase transition of water and ice and vice versa with respect to capillary suction under thermal loading. Therefore, a modified ansatz for the capillary pressure is discussed. For the detection of energetic eﬀects regarding the characterization and control of phase transition of water and ice, an evolution equation for the mass exchange, based on the local divergence of the heat flux divided by the diﬀerence of the specific enthalpies of ice and

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water, is used. Furthermore, ice formation in porous media results from a coupled heat and mass transport and is accompanied by the ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. Regarding the capillarity during the phase transition, significant change of the volume fractions of liquid and gas has been considered. The illustrated results of the examples show that the model is capable to predicting experimental observations. Beside the considered influence of the heat of fusion with respect to the development of the temperature, specifically, the capillary suction in porous media have been realized under thermal loading. Acknowledgements. This work has been supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under grant no. BL 417/6-2.

References [1] Aquirre-Punte, J., Philippe, A.: Quelques recherches eﬀectuées en france sur le problème de la congélation des sols. Revue Generale de Thermique 96, 1123–1141 (1969) [2] Ateshian, G., Ricken, T.: Multigenerational interstitial growth of biological tissues. In: Biomechanics and Modeling in Mechanobiology. Springer, Heidelberg (2010), doi:10.1007/s10237-010-0205-y [3] Biot, M.: General theory of three-dimensional consolidation. Journal of Applied Physics 12, 155–164 (1941) [4] Blome, K.P.: Ein Mehrpahsenmodell-Stoﬀmodell für Böden mit Übergang auf Interface-Gesetze. PhD thesis, Universität Stuttgart, Report No.: II-10. Institut für Mechanik (Bauwesen), Lehrstuhl II (2003) [5] Bluhm, J., Ricken, T., Bloßfeld, M.: Energetische Aspekte zum Gefrierverhalten von Wasser in porösen Strukturen. Proceedings in Applied Mathematics and Mechanics 8(10), 483–484 (2008) [6] Bluhm, J., Ricken, T., Bloßfeld, W.M.: Dynamic phase transition border under freezingthawing load in porous media – a multiphase description. Schröder, J. (ed.): Tech. rep., Report 47. Institute of Mechanics, University Duisburg-Essen, Germany (2009) [7] Bowen, R.: Incompressible porous media models by use of the theory of mixtures. International Journal of Engineering Science 18, 1129–1148 (1980) [8] Bowen, R.: Compressible porous media models by use of the theory of mixtures. International Journal of Engineering Science 20, 697–735 (1982) [9] Brooks, R., Corey, A.: Hydraulic propertiesof porous media. Tech. Rep. 3, Colorado State University, Fort Collins (1964) [10] Coussy, O.: Poromechanics. Wiley, New York (2004) [11] Coussy, O.: Poromechanics of freezing materials. Journal of the Mechanics and Physics of Solids 53, 1689–1718 (2005) [12] de Boer, R.: Theory of Porous Media – highlights in the historical development and current state. Springer, Berlin (2000) [13] Ehlers, W.: Poröse Medien – ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. In: Habilitation, Heft 47, Forschungsbericht aus dem Fachbereich Bauwesen. Universität-GH Essen, Essen (1989)

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[14] Ehlers, W.: Constitutive equations for granular materials in geomechanical context. In: Hutter, K. (ed.) Continuum Mechanics in Environmental Sciences and Geophysics. CISM Courses and Lectures, vol. 337, pp. 313–402. Springer, Wien (1993) [15] Ehlers, W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds.) Porous Media: Theory, Experiments and Numerical Applications, pp. 3–86. Springer, Berlin (2002) [16] Ehlers, W., Bluhm, J. (eds.): Porous media: Theory, experiments and numerical applications. Springer, Heidelberg (2002) [17] Fagerlund, G.: Internal frost attack – state of the art. In: Auberg, R., Setzer, M. (eds.) Frost Resistance of Concrete, RILEM Proceedings, vol. 34, pp. 321–338. E & FN Spon, London (1997) [18] van Genuchten, M.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Science Society of America Journal 44, 892–898 (1980) [19] Graf, T.: Multiphsic flow processes in deformable porous media uner consideration of fluid phas transition. PhD thesis, Universität Stuttgart, Report No.: II-17, Institut für Mechanik (Bauwesen), Lehrstuhl II (2008) [20] Guymon, G.L., Hromadka II, T.V., Berg, R.L.: A one dimensional frost heave model based upon simulation of simultaneous heat and water flux. Cold Regions Science and Technology 3, 253–262 (1980) [21] Helmig, R.: Multiphase flow and transport processes in the subsurface: a contribution to the modeling of hydrosystems. Springer, Heidelberg (1997) [22] Helmuth, R.: Investigations of the low temperature dynamic mechanical response of hardened cement paste. In: Tech. rep., Vol. 154. Dept. of Civil Engineering, Standfort University (1972) [23] Humphrey, J., Rajagopal, K.: A constrained mixture model for growth and remodelling of soft tissues. Mathematical Models and Methods in Applied Sciences 12, 407–430 (2002) [24] Kasparek, S.: Wärme- und Feuchtetransport in zementgebundenen Baustoﬀen während der Frostprüfung mit besonderer Beachtung des CIF-Testes. PhD thesis, Universität Duisburg-Essen (2005) [25] Kowalski, S.J.: Thermomechanics of drying processes. Springer, Heidelberg (2003) [26] Lewis, R., Strada, M., Comini, G.: Drying induced stresses in porous bodies. International Journal for Numerical Methods in Engineering 11, 1175–1184 (1977) [27] Lewis, R., Morgan, K., Thomas, H., Strada, M.: Drying – induced stresses in porous bodies – an elastoviscoplastic model. Computer Methods in Applied Mechanics and Engineering 20, 291–301 (1979) [28] Li, N., Chen, F., Xu, B., Swoboda, B.: Theoretical modeling framework for an unsaturated freezing soil. Cold Regions Science and Technology 54, 19–35 (2008) [29] Mikkola, M., Hartikainen, J.: Mathematical model of soil freezing and its numerical implementation. International Journal for Numerical Methods in Engineering 52, 543– 557 (2001) [30] Miller, R.: Freezing phenomena in soils. In: Hillel, D. (ed.) Applications of Soil Physics, pp. 254–299 (1980) [31] O’Neill, K., Miller, R.: Exploration of a rigid ice model of frost heave. Water Resources Research 21, 281–296 (1985) [32] Powers, T.: A working hypothesis for further studies of frost resistance of concrete. Journal of the American Concrete Institute 41, 245–272 (1945)

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[33] Powers, T., Brownyard, T.: Studies of the physical properties of hardened portland cement paste (4) – Part IV: The thermodynamics of adsorption of water on hardened paste. Journal of the American Concrete Institute 43, 549–602 (1946) [34] Richards, L.A.: Capillary conduction of liquids through porous mediums. Physics 1, 318–333 (1931) [35] Ricken, T.: Kappilarität in porösen Medien – theoretische Untersuchung und numerische Simulation. PhD thesis, Universität Duisburg-Essen, Fachbereich Bauwesen, Shaker Verlag, Aachen (2002) [36] Ricken, T., Bluhm, J.: Modeling fluid saturated porous media under frost attack. In: GAMM-Mitteilungen, vol. 33, pp. 40–56. WILEY-VCH Verlag, GmbH & Co, KGaA, Weinheim (2010) [37] Ricken, T., de Boer, R.: Multiphase flow in a capillary porous medium. Computational Materials Science 28, 704–713 (2003) [38] Rodriguez, E., Hoger, A., McCulloch, A.: Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics 27, 455–467 (1994) [39] Setzer, M.: Micro-ice-lens formation in porous solid. Journal of Colloid Interface Science 243, 193–201 (2001) [40] Setzer, M.: Development of the micro-ice-lens model. In: Auberg, R., Keck, H.J., Setzer, M. (eds.) Frost Resistance of Concrete, RILEM Proceedings, vol. 24, pp. 133–145. RILEM Publications, France (2002) [41] Setzer, M.: Modelling and testing the freeze-thaw attack by micro-ice-lens model and cdf/cif-test. In: Proceedings of the International Workshop on Microstructure and Durability to Predict Service Life of Concrete Structures, pp. 17–28. Hokkaido University, Sapporo (2004) [42] Setzer, M., Heine, P., Palecki, S., Auberg, R., Feldrappe, V., Siebel, E.: CIF-test – capillary suction, internal damage and freeze-thaw test: Reference method and alternative methods. Material and Structures 37, 743–753 (2004) [43] Whitaker, S.: Simultaneous heat, mass, and momentum transfer in porous media: A theory of drying. Advances in Heat Transfer 13, 119–203 (1977)

Optical Measurements for a Cold-Box Sand and Aspects of Direct and Inverse Problems for Micropolar Elasto-Plasticity Rolf Mahnken and Ismail Caylak

Abstract. This chapter is organized into two parts: The first part is concerned with the experimental determination of thermo-mechanical properties for a coldbox sand, which have a strong influence during the solidification in a sand casting process. To this end the uniaxial behavior for diﬀerent temperatures up to the casting temperature of an aluminium alloy is investigated. For the tests at room temperature an optical measurement equipment is used, which is ideally suited to measure shear bands. Furthermore, diﬀerent strain rates are applied to analyze rate dependent behavior. The second part of the chapter is concerned with elasto-plastic modeling for granular materials. Here particular attention is directed to the SD-eﬀect, by a generalized yield function for the non-polar part, and shear band development by a micropolar part for the yield function. In the outlook perspectives for error controlled adaptive strategies for solution of the direct equilibrium problem and the inverse parameter identification problem for micropolar elasto-plastic models are discussed.

1 Introduction Cold-box sand are used in forming and mold processes of high-quality and complex components for plant manufacturing, automotive and mechanical engineering. Characteristics and applications of the cold-box process and comparisons with the hot-box process are given in [22]. An investigation of the mechanical strength of cold-box mold components is presented in the work of Boenisch and Lotz [4]. From microscopic studies it is observed, that the binder consists mainly of irregular glued spherical particles [4]. The bending strength depends strongly on the variation of the solvent ratio. Boenisch and Lotz [4] verify an increasing bending Rolf Mahnken · Ismail Caylak Chair of Engineering Mechanics, University of Paderborn, Warburger Straße 100, 33098 Paderborn, Germany e-mail: {rolf.mahnken,ismail.caylak}@ltm.upb.de B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 175–196. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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strength with increasing solvent ratio of the binder for an open and hermetic 24 h stored sand. Causes for unexpected losses of strengths of cold-box cores are explained in [5]. Gas cured cold-box cores are strongly dependent on variation of the atmospheric moisture. A short and soft warming improves the moisture resistance. Wilhelm [39] investigated the eﬀect of alcohol-based coatings on mechanical strength of cold-box cores. Occupational medical and hygiene implications of different catalysing techniques are treated in Schmittner [34]. In [31] Rogers describes a multi-component gas phase model of reactive transport in mold and core sand. Concerning cost saving possibilities Werling considers in [38] aspects of reduced resin consumption and improved casting finish. Stress-induced quality defects like structural distortion and dimensional inaccuracy are very often the consequences. There exist many works for simulation of thermo-mechanical solidification of cast metals, see e. g. Celentano [8, 9, 11]. Commercial casting programs enable the simulation of diﬀerent casting processes. In the sand casting, however all mentioned works concentrate on the casting materials, neglecting the behavior of the cold box of a sand mold and core sand. Therefore, it becomes important to obtain an extensive thermo-mechanical characterization of the cold-box sand. As will become apparent in the chapter, experiments for a cold-box sand reveal certain characteristics, such as the SD-eﬀect, shear band development, rate dependency, triaxial dependency and thermo mechanical dependency. As for general granular materials, all tests can be separated into two phases. In the first phase the specimen behaves more or less homogeneous. Having passed a certain external load, in the second phase the specimen shifts to an inhomogeneous behavior. It is accompanied by localization, bifurcation and instability phenomena. As noted by Ehlers and Scholz in [15] “A careful investigation of these phenomena both on the microscopic and on the macroscopic level reveals that inhomogeneous deformations result in local concentrations of plastic strains in narrow zones forming shear bands in 2-dimensional applications like the biaxial test and shear domains in general 3-dimensional problems. Furthermore, it has been observed that with the onset of shear zones, there is a local switch of the material behavior from non-polar to micropolar. This behavior is due to the fact that the grains of granular material start to roll upon each other, thus initiating rotational degrees of freedom on the micro level corresponding to micropolar rotations on the macro level.” Concerning modeling aspects, in this chapter we will concentrate on the phenomena of the SD-eﬀect and shear band development. The SD-eﬀect (strength-diﬀerential eﬀect) shows the maximum strength of the material greater in uniaxial compression than in uniaxial tension. For its simulation the well known classical criterion of Drucker and Prager [12] is often used. It is characterized by a cone in the hydrostatic plane, unbounded in compression and allowing little strength in tension. Ehlers [14] proposes a single-surface yield function for modeling geomaterials such as soils and rocks. Special cases of the model are the Drucker and Prager [12] model and Greens ellipsoid [19]. Motivated by the work of Ehlers, Mahnken [24] developed a material model for the SD-eﬀect with pressure dependence of yielding by use of three basic invariants of a reduced stress tensor. As this model considers damage eﬀects, special cases are the

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Rousselier model [32] and the Gurson model [20]. A constitutive model for asymmetric eﬀects in rate dependent plasticity for polymers is given in [35]. For simulation of localization and shear bands, it is well known, that constitutive modeling within non-polar continuum mechanics leads to pathological meshdependency of FE-solutions. Therefore, several generalized continuum approaches have been developed, such as non-local formulations [1], gradient approaches [30], or micromorphic continua, see e. g. Erringen [17] and references therein. Subclasses are micropolar (Cosserat) continua developed in its origin by Cosserat and Cosserat [10], with extensions see e. g. in Mühlhaus [29] and Steinmann [29, 36], microstretch continua [17], and microstrain continua Forest and Sievert [18]. As noted by Ehlers and Scholz in [15] among others, micropolar continua can also be motivated by a homogenization procedure: Proceeding from homogenization techniques applied to representative elementary volumes (REV) of particle ensembles, the average of the local forces and moments of the microstructure results in macroscopic stresses and couple stresses, thus basically defining granular material as a micropolar continuum in the sense of the Cosserat brothers. Constitutive equations simulating both the SD-eﬀect and shear band development incorporate a relatively large set of material parameters, related to • the standard non-polar elasto-plastic continuum, • the micropolar elasto-plastic continuum and • the interaction between non-polar and micropolar elasto-plastic continuum. The first set of parameters can be obtained from homogeneous tests, measuring forces and displacements during the experiment and which are converted to stressstrain data. The second and third sets are more challenging, since they require determination of inhomogeneous data, which in this chapter are obtained with an optical system. An outline of this work is as follows: In Sect. 2 the material (composites of the cold-box sand) and the equipment for the tests are explained in detail. Furthermore a hydrostatic pressure cell for triaxial experiments is described. Section 3 concentrates on the uniaxial compression and tension tests at room temperature and considers the thermal and the thermo-mechanical characterization. In Sect. 5 the results of the triaxial experiments are illustrated. Section 6 summarizes the basic equations for modeling of micropolar solids. In particular, it presents a generalized class of yield functions for the non-polar part, and a possible extension for the polar part. In the outlook perspectives for error controlled adaptive strategies for solution of the direct equilibrium problem and the inverse parameter identification problem for micropolar elasto-plastic models are discussed.

Remarks on Notations Square brackets [•] are used throughout the chapter to denote ‘function of’ in order to distinguish from mathematical groupings with parenthesis (•).

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2 Specimens and Testing Equipment 2.1 Materials and Specimen Preparation The cold-box sand used in this study was produced at the Giesserei-Institute at the RWTH Aachen University, Germany. The composite is given in Table 1. For preparation 4000 gram of the glass sand is mixed with 64 gram of a binder, 32 gram phenolic resin and 32 gram polyisocyanat, in total 1.6 percent of the whole mass. The core box, which is one of various specimen negative forms, is illustrated in Fig. 1. Two diﬀerent specimens are shown in Fig. 2 a) and b). Specimen a) is used to evaluate the eﬀect of hydrostatic pressure and is designed to fit into a hydrostaticpressure cell unit. Specimen b) of Fig. 2 is designed to evaluate the eﬀect of the uniaxial mechanical and thermo-mechanical material behavior. Preliminary tests show, that the reduced inner radii in Figure 2 b) are necessary to obtain a uniaxial tension stress. Otherwise the specimen would be damaged due to the radial pressure of the clamping jaw. Table 1 Composition of the cold-box sand.

Cold box

Glass sand

Phenolic resin binder

Polyisocyanat binder

4000 gram

32 gram

32 gram

Fig. 1 Core box of the cold-box sand.

2.2 Experimental Equipment In order to characterize the uniaxial mechanical and thermo-mechanical behavior of the core sand the uniaxial testing machine in Fig. 3 a) is used, which enables for tension and compression loading, respectively. The maximum force is 10 kN, and the

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Fig. 2 Sand specimens: a) Cylindrical specimen for triaxial tests; b) Geometry and specimen with reduced inner radius for uniaxial tests.

displacement range is ±50 mm. The clamping jaws of Fig. 3 a) are special designs for the specimen illustrated in Fig. 2 b). For characterization of hydrostatic pressure dependence compression tests with variation of radial pressure are conducted in the pressure cell of the triaxial testing machine depicted in Fig. 3 b). The sand specimen in Fig. 2 a) is used for triaxial experiments. This unit is also used for testing at atmospheric pressure for comparison with the uniaxial testings.

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Fig. 3 Experimental equipment: a) Testing machine for uniaxial characterization; b) Testing machine for triaxial characterization.

Furthermore, an optical measurement system is used, which enables to analyze nonhomogeneous 3D deformations in the whole region of the sand specimen. A random or regular structure is applied to the object surface such that it deforms with the object. Synchronized stereo images of the pattern are recorded at diﬀerent load stages using CCD cameras. The first image processing steps define the macroimage facets. These facets are tracked in each successive image with sub-pixel accuracy. 3D coordinates, 3D displacements and the plane strain tensor are calculated

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automatically using photogrammetric evaluation procedures. The results as 3D visualization (contour plots), section diagrams, time based analysis or analog values are displayed in reports or exported in standard file formats. In this way the system is suited for precise material characterization and for validation of simulation tools. For the thermo-mechanical characterization two diﬀerent heating strategies are compared: heating with an inductor and heating in an oven. Both variants are illustrated in Fig. 4 a) and in Fig. 4 b). Since the cold-box sand is nonmagnetic a stainless steal heat exchanger is used, which enables the application of thermal loading. Five diﬀerent ring geometries are used for comparison and are summarized in Fig. 5. An important criterion for the optimal choice is, that the temperature in the core of the sand should reach the reference temperature as fast as possible. To this end the gap between the sand specimen and the ring as well as the thickness of the ring is varied. Ring A has a thickness of 5 mm and a gap of 10 mm between the sand and the ring, whereas the gap for ring B is only 1 mm without varying the thickness. The thickness for ring C is reduced to 3 mm. While ring B and C are limited to the reduced radius, ring D also includes the outer radius. Furthermore, ring D has a thickness of 7 mm. Ring E is similar to ring C, where the thickness in the region of the reduced radius is chosen to 5 mm.

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Fig. 4 Diﬀerent heating methods: a) Heating with an induction coil; b) Heating in an oven.

3 Uniaxial Compression and Tension Tests The main aspects in the uniaxial compression and tension tests at room temperature are the SD-eﬀect, the application of an optical method, the rate dependency and the influence of the storage time for the cold-box sand.

3.1 SD-Eﬀect and Optical Measurements In Figs. 6 and 7 the results for compression and tension tests are compared. Figures 6 a) and 7 a) depict the force versus the vertical displacement for the compression and tension test, respectively. The maximum force in compression is approximately 4000 N, whereas the maximum force in tension reaches only approximately 1300 N. Thus the material shows an SD-eﬀect, where the strength in compression is higher

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E Fig. 5 Five diﬀerent ring geometries for the heat exchanger.

than in tension. The corresponding contour plots of the optical measurements are illustrated in Figs. 6 b) and 7 b). The numbers from 1-8 are related to the load steps in Figs. 6 a) and 7 a). In Fig. 6 b) at load step 5 a shear band develops until a macroscopic crack becomes visible at load step 8. Furthermore, Fig. 6 b) shows that

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Fig. 6 Compression test: a) Force versus displacement including the number of load steps; b) Contour plots of the optical measurement in vertical direction for diﬀerent load steps.

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b)

Fig. 7 Tension test: a) Force versus displacement including the number of load steps; b) Contour plots of the optical measurement in vertical direction for diﬀerent load steps.

the crack direction in compression attains an angle of approximately 45◦ , whereas in Figure 7 b) the angle for the crack direction in tension is 0◦ . Additionally, in Fig. 8 a) a contour plot for the displacement in horizontal direction for the compression test is shown, whereas the corresponding eﬀective strain contours are depicted in Fig. 8 b).

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Fig. 8 Compression test: a) Contour plot for the displacement in horizontal direction; b) Contour plot for the eﬀective strain.

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3.2 Rate Dependency and Reproducibility Three diﬀerent rates, a) 0.00025 mm/sec, b) 0.001 mm/sec and c) 0.01 mm/sec, are applied and the corresponding force versus displacement curves are illustrated in Fig. 9 a)-c). In all three cases two experiments are performed for reproducibility. Furthermore, Fig. 9 d) summarizes the results of Fig. 9 a)-c). It is evident, that with increasing strain rate the forces increase, thus illustrating a clear rate dependence for the cold-box sand.

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Fig. 9 Reproducibility and the eﬀect of rate dependency.

3.3 Influence of Storage Time The storage time of the sand is also varied, since it influences the material characteristic of the binder in the cold-box. The influence is shown in Fig. 10, where it should be mentioned, that the sand has always been stored under the same conditions, like constant temperature and hermetic storage. The maximum strength increases with the storage time of the sand, because the binder in the cold-box sand cured. An almost linear dependency is observed.

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Fig. 10 Maximum strength with varying storage time.

4 Thermo-Mechanical Characterization 4.1 Heat Exchanger Variation for Thermal Loading

Temperature [°C]

An important prerequisite for thermo-mechanical characterization is the achievement of a homogeneous temperature field, as explained in Sect. 2.2. In order to obtain an optimal experimental setup for this challenge, diﬀerent ring geometries are used for the heat exchanger to warm up the sand with an inductor. All diﬀerent setups A-E are illustrated in Fig. 5. Figure 11 illustrates the comparison between the reference temperature and the temperature in the core of the cold-box sand for all heat exchanger geometries A-E versus time. The top curve represents the ring temperature and can be regarded as the reference curve. The geometry A provides

225 200 175 150 125 100 75 50 25 0

Ring temperature Sand temperature, Sand temperature, Sand temperature, Sand temperature, Sand temperature,

0

250

500

ring C ring B ring D ring E ring A

750 1000 1250 1500 1750 2000

Time [s]

Fig. 11 Temperature in the cold-box sand: Comparison between reference temperature and diﬀerent heat exchanger geometries.

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unsatisfactory results, because the air gap between the exchanger and the cold-box sand is too large. It is obviously that ring C shows the best results compared with all other ring types, due to reduced thickness to 3 mm in the region of the reduced radius. Further reproduced tests verify the curves illustrated in Fig. 11. In Fig. 12, also the results of the heat exchanger ring C with a cold box heated in an oven are compared, where ring C shows better results. From now on all thermal and thermo-mechanical investigations are obtained for ring C. The temperature in the cold-box sand versus time up to 600◦ C for the heat exchanger ring C is depicted in Fig. 13.

Temperature [°C]

225 200 175 150 125 100 75 50 25 0

Ring temperature Sand temperature, ring C Sand temperature, oven

0

250

500

750 1000 1250 1500 1750 2000

Time [s] Fig. 12 Temperature in the cold-box sand: Comparison between Ring C and oven.

Fig. 13 Temperature in the cold-box sand up to 600◦ C.

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4.2 Mechanical Loading for Diﬀerent Isothermal Conditions The last task of the uniaxial characterization is the mechanical investigation of the cold-box sand due to diﬀerent isothermal loadings up to 600◦ C, which is a typical casting temperature for aluminium alloys. The heating of the specimen and the ensuing mechanical loading was done in two steps. In the first step the cold-box sand is heated up to the appropriate temperature, as explained in Sect. 4.1. In a second step a mechanical loading is applied. Cooled clamping jaws are used in order to prevent the heating-up of the testing machine.

a)

b)

c)

Fig. 14 Damaged states for a) 20◦ C, b) 200◦ C and c) 500◦ C.

The qualitative damage states for 20◦ C, 200◦ C and 500◦ C are summarized in Fig. 14 a), b) and c). Figure 15 shows the maximum strength dependence in compression with respect to diﬀerent thermal loading. It can be seen, that up to 200◦C the maximum strength decreases with increasing temperature. For higher values the maximum strength shows a small increase with increasing temperature. This may depend on the chemical process during the heating and the cooling phase. A further improvement of the heating procedure could eliminate these eﬀects, where the goal is to reach the reference temperature faster than with heat exchanger ring C.

Fig. 15 Maximum strength in compression up to 600◦ C.

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5 Triaxial Characterization Additional, triaxial tests of the cold-box sand were done at the laboratory of the Geotechnical Institute at the RWTH Aachen University, Germany. For these tests the solid core sand cylinder of Fig. 2 a) and the testing machine of Fig. 3 b) are used. The schematic of loading is illustrated in Fig. 16 a), where the laboratory configuration is shown in Fig. 16 b). Tests at atmospheric pressure verify the results obtained at our laboratory (LTM, University of Paderborn in Germany). Three different radial pressures are applied (p = 1, p = 5 and p = 10 bar) to analyze the hydrostatic pressure dependence. Figure 17 a) illustrates that the resulting eﬀective shear stress (σ1 − σ3 )/2 of Mohr’s circle increases with the radial pressure. Figure 17 b) shows a linear dependence between the eﬀective shear stress (σ1 − σ3 )/2 on the horizontal and the total stress (σ1 + σ3 )/2 on the vertical axis. Based on these results we intend to simulate the hydrostatic pressure dependence, according to the approaches in [14] and [24], in future work.

a)

b)

Fig. 16 Configuration of triaxial tests: a) Schematic of loading; b) Laboratory setup.

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ε(%)

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Fig. 17 Triaxial characterization for three diﬀerent pressures: a) Eﬀective shear stress versus strain; b) Eﬀective shear stress versus total stress.

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6 Modeling of Micropolar Continua The basic setting of granular materials in the framework continuum mechanics can be embedded in the Theory of Porous Media, explained fundamentally in Biot [3], Bowen [7], de Boer [6] and Ehlers [13]. In this section we summarize only some basic equations for modeling granular materials for a geometrically linear theory. Furthermore, the aspects of thermo-mechanical coupling will be left out.

6.1 Basic Equations In addition to the translational degrees of freedom of standard continua, summarized in the displacement vector u, for non-polar continua a supplementary and independent rotational field is introduced, labeled henceforth as ϕ. ¯ Macroscopically it describes the change of the particle orientation. Taking the gradients with respect to the spatial coordinates, both fields render the Cosserat and curvature tensors 3

1. ε = Gradu+ E ϕ¯ , 2. κ = Gradϕ¯ .

(1) 3

Here “Grad(·)” denotes the partial derivative of (·) and E is the Ricci permutation tensor. The balance equations in local form are 1. Divσ + ρ0 g = 0, 2. Divμ + ρ0 c + 1 × σ = 0 ,

(2)

where σ is the Cauchy stress tensor, ρ0 the density, g the gravitation, 0 is the nullvector, Div corresponds to Grad, μ is the couple stress, c the body couple, and 1 is the second order unity tensor. A main consequence of (2) is the unsymmetry of the Cauchy stress tensor, i. e. σ σT . The basis for the constitutive equations of micropolar plasticity are the additive splits for the Cosserat and curvature tensors: 1. ε = εel + ε pl , 2. κ = κel + κ pl .

(3)

Here εel , κel and ε pl , κ pl are the elastic and the plastic parts of the corresponding quantities in Eq. (1), respectively. In order to relate the elastic quantities of (3) to the stress like quantities in (2) a free energy function Ψ consisting of a nonpolar part Ψ npo and a micropolar part Ψ pol is introduced as 1. ρ0 Ψ 2. ρ0 Ψ npo 3. ρ0 Ψ pol

= ρ0 Ψ npo + ρ0 Ψ pol , where 1 2 = μεelsym : εelsym + λeVel , 2 = μc εelskw : εelskw + μc (lc )2 κel : κel .

(4)

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Here μ and λ are the Lamé constants, eVel = 1 : εel is the volumetric part of the elastic strain, μc is an additional parameter governing the influence of the skew symmetric part of the elastic Cosserat strain, and lc can be interpreted as an intrinsic length scale. Taking the derivative of ρ0 Ψ with respect to εel and κel yields the Cauchy stress and the couple stress tensor: ∂Ψ sym = 2μεel + 2μc εelskw + λeVel 1 , ∂εel ∂Ψ μ = ρ0 = 2μc (lc )2 κel . ∂κel σ = ρ0

(5)

We would like to point out, that the above simple structure of equations does not consider more advanced eﬀects, e. g. interaction between non-polar and micropolar eﬀects, constraints for maximal volumetric compression, see e. g. [15], or thermomechanical coupling.

6.2 Yield Function and Plastic Potential Before introducing a generalized class for micropolar continua we summarize some basic requirements for its formulation: 1. In the non-polar part the model should account for the SD-eﬀect, i. e. diﬀerent behavior in tension and compression. 2. The model should account for a non-associated flow rule. 3. In the micropolar part the model should account for shear band development without pathological mesh-dependency of the FE-solutions. 4. The model should be computationally eﬀective. This includes e. g. the requirement for a smooth single surface model and the avoidance of high power terms, which may lead to diﬃculties in the local iteration scheme. 5. The number of independent material parameters should be as low as possible. 6.2.1

A Generalized Class of Yield Functions for the Non-polar Part

For a non-polar elastic/plastic continuum a yield criterion is formulated in terms of the symmetric Cauchy stress tensor σ sym = σ symT as Φnpo [σ sym ] = 0. For the case of sym sym isotropy the yield function reduces to the form Φnpo = Φnpo [I1 , II d , III d ], where the three basic invariants are determined according to sym

I1 = σ sym : 1 , II d sym

=

1 sym 2 1 sym sym (σ ) : 1 , III d = (σdev )3 : 1 , 2 dev 3

(6)

and where σdev denotes the deviatoric part of σ sym. Now the following more general yield function is considered, see Mahnken [24, 25],

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sym

sym

1. Φnpo [I1 , II d , III d ] = 3II d − HN [h] D[h] , N 2. HN [h] = ai [h] I1 i ,

where

i=0

3.

D[h] = (1 + γ[h]ξ)−m ,

4.

h = [ev , f, ....] .

(7)

Here the ratio ξ=

√ sym 27 III d 2 (II dsym )3/2

(8)

detects the diﬀerent stress modes, i. e. tension ξ = 1, compression ξ = −1, shear ξ = 0. As noted in [13], the parameters γ and m in (7.3) can be merged using the convexity limit of the yield criterion √ 2 3 m= + . (9) 9 3γ Additionally, in a general case both functions H[h] and D[h] in (7) are dependent on a process vector h, which collectively summarizes the plastic strain ev , a softening parameter f and possible further process variables. This formulation generalizes the idea of Ehlers [14], where the coeﬃcients a1 , ..., a4 are kept as constant material parameters. Moreover, as shown in [25] the following special cases are obtained from (7) von Mises: Drucker Prager [12]: Green [19]: Ehlers [14]: Gurson [20]: Rousselier [32]: Lemaitre [23]: 6.2.2

N N N N N N N

= 0, = 2, = 2, = 4, = ∞, = ∞, = 0,

D = 1, D = 1, D = 1, D 1, D = 1, D = 1, D = 1,

a0 = Y 2 [ev ] , a0 = Y 2 [ev ], a1 , a2 0 , a0 [ev , f ], a1 = 0, a2 [ev , f ] , a0 , ..., a4 , γ are constants , ai [ev , f ], i = 0...., ∞ , ai [ev , f ], i = 0...., ∞ , a0 = Y 2 [ev ](1 − f )2 .

(10)

Extension of the Yield Function by a Micropolar Part

The yield function of the previous section 6.2.1 has been formulated for non-polar materials with symmetric stresses σ sym . Therefore, for micropolar materials it has to be extended by further terms considering the skew-symmetric stress σ skw and the couple stress μ. Several possibilities can be envisaged to extend the general structure in (7), where e. g. invariants formulated in terms of σ sym , σ skw , μ could be defined. However, this in turn would also extend the number of related material parameters drastically. In order to keep the formulation simple, in addition to (6) the following invariants will be introduced.

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II dskw =

191

1 skw 2 (σ ) : 1 , 2 dev

II μ = μ · μ .

(11)

Consequently, a simple ansatz for a micropolar part of the yield function is of the form Φ pol = kσ II dskw + kμ II μ ,

(12)

and where kσ and kμ are additional material parameters. 6.2.3

Combined Yield Function and Plastic Potential

With contributions from the non-polar part (7) and the micropolar part (12) a combined yield function takes the form Φ = Φnpo + Φ pol .

(13)

In most cases frictional materials show non-associated plasticity properties. This can be taken into account by formulating plastic potentials Φnpo∗ , Φ pol∗ , where both functions have the same mathematical structure as in (7) and in (12), however, only the related material parameters are diﬀerent. Without going into details, in this way a generalized plastic potential takes the form Φ∗ = Φnpo∗ + Φ pol∗ . 6.2.4

(14)

Evolution Equations

Having defined the plastic potential the evolution of the plastic strain and the curvature are obtained as ∗

∂Φ ε˙pl = λ˙ , ∂σ

∗

∂Φ κ˙pl = λ˙ , ∂μ

(15)

where λ˙ is a plastic multiplier satisfying the Kuhn-Tucker conditions Φ ≤ 0,

λ˙ ≥ 0 ,

˙ = 0. λΦ

(16)

Additionally, it becomes necessary to formulate evolution equations for the process variables h in (7.4), which however, will not be elaborated here.

7 Direct and Inverse Problems for Micropolar Solids 7.1 Direct Problem: Weak Formulation The basis for solution of the basic equations in Section 6 is the transfer of the strong form balance equations (2) into a weak formulation. With test functions δu for the translation field and δϕ¯ for the rotation field it is written as

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g1 [u, ϕ] ¯ = g2 [u, ϕ] ¯ =

Ω

σ : Grad(δu) dv −

μ : Grad(δϕ) ¯ dv − Ω − (1 × σ) · δϕ¯ = 0 ,

Ω

Ω

ρ0 g · δudv − ρ0 c · δϕdv ¯ −

Γσ

(σ · n) · δuda = 0 ,

Γμ

(μ · n) · δϕda ¯

(17)

Ω

where Ω is the spatial domain, and Γ its boundary, n denotes the outward-oriented unit surface normal of the Neumann boundaries Γσ and Γμ . We can summarize the two field form as ¯ g1 [u, ϕ] = 0. (18) g[u, ϕ] ¯ = g2 [u, ϕ] ¯ Outlook and perspectives: For numerical solution of the two-field weak form (18) a discretization, both in in time and in space becomes necessary. To this end, the Galerkin method, typically used for discretization in space, can also be used for discretization in time. Then, cG(s)dG(r) describes a method with conforming (continuous) discretization in space of order s and discontinuous discretization in time of order r [16]. The formulation of the corresponding cG(s)cG(r) discretization is more involved since one has to ensure the global continuity of functions in the trial space. Typically, the numerical solution is accompanied by numerical errors, which should be controlled by an adaptive strategy. Adaptive simulations are based on computable a priori error estimates, where the energy norm is a common choice. However, in engineering applications very often the control of local goal quantities Q is of interest, such as e. g. (measured) displacements or stresses. For a one field direct problem in [33] an a priori error estimation is of the form Q(u) − Q(uhk ) ≈ “Residual” × “Local Error” ≈ ηk + ηh , (19) where uhk is the solution of the discretized problem in space and in time. An important ingredient for derivation of the above error estimator is the introduction of a Lagrangian and the solution of a dual problem, in addition to the primal problem. In this way, the term “Residual” comprises primal and dual residuals and “Local Error” comprises local errors for primal and dual problems, respectively. Furthermore, ηk and ηh estimate the temporal and spatial discretization errors, which are computed as cell-wise (element-, patch-, node-wise) contributions. To the authors knowledge, adaptive methods have not been applied for direct problems of micropolar solids and therefore constitute a challenging field in future research.

7.2 Inverse Problem: Constrained Least Squares Problem Confining ourselves to parameter identification as an inverse problem, three classes of material parameters for micropolar continua have to be identified:

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1. Macroscopic material parameters, occurring in the nonpolar parts of the free energy function Ψ npo in (4), the yield function Φnpo in (7) and the plastic potential Φnpo∗ in (14), 2. microscopic material parameters, occurring in the micropolar parts of the free energy function Ψ pol in (4), the yield function Φ pol in (7) and the plastic potential Φ pol∗ in (14), 3. scale transition parameters, which however have not been introduced in the model equations of this chapter. For identification of a vector of material parameters p, in general, one should strive for simple experiments on the basis of experimental data with homogeneous behavior for the state variables within the specimen. Concerning granular media modeled as a micropolar continuum with state variables (u, ϕ) ¯ this ideal situation is diﬃcult to achieve, since no procedure is known so far to measure the kinematic variables ϕ. ¯ Instead, localization phenomena in certain zones are activated, such that parameter identification for inhomogeneous behavior on the basis of experimental data u¯ and least-squares minimization ¯ + ||p − p|| ¯ → min J(u(p), ϕ(p) ¯ = ||C(u, ϕ) ¯ − u||

(20)

becomes necessary, Here C(·) represents an observation operator, mapping the configuration trajectory (u, ϕ) ¯ to points of the observation space. Note, that the observation space does not contain experimental data for the rotation field ϕ, ¯ which constitutes a main diﬃculty for this kind of inverse problems. The inhomogeneous field u is approximated by the finite element method on the basis of the weak form (18), see e. g. [15] and [26–28] on related subjects. Simultaneously the basic independent variables u, ϕ¯ have to satisfy the weak forms at all time steps. This in turn motivates the formulation of a Lagrangian N L[{n u,n ϕ} ¯ n=1 , p] = J[p] +

N

g λ,

n Tn

(21)

n=1

where n λ is a Lagrange multiplier and where the upper index n = 1, ..., N denotes the time step. Concerning a generalized solution strategy for the above coupled problem we refer to [28]. Outlook and perspectives: Considering hyperbolic one field problems goal orientated a posteriori estimates for parameter identification with adaptive finite element methods are derived in [2, 21] and are of the form "Residuals” × “Local Errors” ≈ ηk + ηh . (22) Q(p) − Q(pkh ) = Without going into details, here Q is a goal quantity of interest, phk is the solution for the material parameters as a result of the inverse discretized problem in space and in time. The corresponding “Residuals” are a state equation, an adjoint state equation and a gradient equation. A simpler error estimator in [37] uses only “Residuals” for a state equation and an adjoint state equation and - more important from the

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practical point of view - avoids second order derivatives of the state equation and the observation operator. We also remark the analogous structure to the estimate for the direct problem in Eq. (19). To the authors knowledge, adaptive methods have not been applied for inverse problems of micropolar solids and therefore constitute a challenging field in future research.

8 Summary and Outlook In this contribution, a uniaxial characterization of the cold-box sand at room temperature and the thermo-mechanical characterization up to the casting temperature of an aluminium alloy has been provided. The experimental results at room temperature show a strength-diﬀerential eﬀect in tension and compression, with higher maximum strength in compression than in tension. Furthermore, the material behaves rate dependent, where the maximum strength increases with increasing strain rate. The maximum strength increases almost linearly with storage time of the cold-box sand. The results of optical measurements give comprehensive information about the inhomogeneous behavior within the specimen. In particular the development of shear-band could be observed. With a heat inductor and heat exchanger an eﬀective heating procedure is developed to reach the reference temperature of the ring as fast as possible. Heat exchanger Ring C provide the best results. During the mechanical investigation of the cold-box sand due to diﬀerent isothermal loadings up to 600◦C the maximum strength decreases with increasing temperature. Triaxial tests show a hydrostatic pressure dependence of the cold-box sand, where the eﬀective shear stress increases almost linearly with increasing radial pressure. Aspects of future works are: • A further improvement of the heating procedure to reach the reference temperature faster than heat exchanger ring C. • Constitutive modeling of the cold-box sand in order to simulate the solidification, which is part of a sand casting process. • Parameter identification for the non-polar and the polar parts proposed in Sect. 6 and moreover transition scale parameters. In particular the latter two sets are of much interest: Due to the solution of a boundary value problem at each step of the optimization procedure, the total computation can become very time consuming. Therefore, in order to optimize the finite element calculations by still preserving accuracy, error controlled adaptive finite element procedures should be applied for this challenging part of parameter identification. Acknowledgements. Parts of the research work in this chapter are obtained under grant MA 1979/8-1 within the joint research project: “Thermo-mechanical modeling and characterization of the solid-liquid interactions in casting processes”. The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG) for its financial support.

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References [1] Bazant, Z.: Why continuum damage is nonlocal: Micromechanic arguments. Journal of Engineering Mechanics 117, 397–415 (1988) [2] Becker, R., Vexler, B.: A posteriori error estimation for finite element discretization of parameter identification problems. Numerische Mathematik 96, 435–459 (2004) [3] Biot, M.A.: Theory of propagation of elastic waves in a fluid-saturated porous solid, I. Low frequency range. Journal of the Acoustical Society of America 28, 168–178 (1956) [4] Boenisch, D., Lotz, W.: Mechanical strength of cold-box mold components. Giesserei 71(5), 187–196 (1984) [5] Boenisch, D., Lotz, W.: Causes for unexpected losses of strengths of cold box cores. Giesserei 72(4), 83–88 (1985) [6] de Boer, R.: Theory of porous media. Springer, Berlin (2000) [7] Bowen, R.M., Mahnken, R.: Incompressible porous media models by use of the theory of mixtures. International Journal of Engineering Sciences 18, 1129–1148 (1980) [8] Celentano, D., Oller, S.: Finite element model for thermomechanical analysis in casting processes. Journal De Physique 3, 1171–1180 (1993) [9] Celentano, D., Oller, S.: A coupled thermomechanical model for the solidification of cast metals. International Journal of Solids and Structures 33, 647–673 (1996) [10] Cosserat, E., Cosserat, F.: Théorie des corps déformables. A. Hermann et fils, Paris (1909); Theory of deformable bodies, NASA TT F-11 561(1968) [11] Cruchaga, M., Celentano, D.J., Lewis, R.W.: Modeling fluid-solid thermomechanical interactions in casting processes. International Journal of Numerical Methods for Heat and Fluid Flow 14, 167–186 (2004) [12] Drucker, D.C., Prager, W.: Soil mechanics and plastic analysis of limit design. Quarterly Journal of Applied Mathematics 10, 157–165 (1952) [13] Ehlers, W.: Constitutive equations for granular materials in geomechanical context. In: Hutter, K. (ed.) Continuum Mechanics in Environmental Sciences and Geophysics, CISM Courses and Lectures, vol. 337, pp. 313–402. Springer, Wien (1993) [14] Ehlers, W.: A single-surface yield function for geomaterials. Archive of Applied Mechanics 65, 246–259 (1995) [15] Ehlers, W., Scholz, B.: An inverse algorithm for the identification and the sensitivity analysis of the parameters governing micropolar elasto-plastic granular material. Archive of Applied Mechanics 77, 911–931 (2007) [16] Eriksson, K., Johnson, C., Logg, A.: Adaptive computational methods for parabolic problems. In: Stein, E., de Borst, R., Hughes, J., Wiley, J. (eds.) Encyclopedia of Computational Mechanics, ch.24, Sons, Ltd (2004) [17] Eringen, A.C.: Microcontinuum field theories. In: I. Foundations and Solids, vol. I. Springer, NewYork (1999) [18] Forest, S., Sievert, R.: Nonlinear microstrain theories. International Journal of Solids and Structures 43, 7224–7245 (2006) [19] Green, R.J.: A plasticity theory for porous solids. International Journal of Mechanical Sciences 14, 215–224 (1972) [20] Gurson, A.L.: Continuum theory of ductile rupture by void nucleation and growth – I. Yield criteria and flow rules for porous ductile media. Journal of Engineering Materials and Technology 99, 2–15 (1977) [21] Johansson, H., Runesson, K.: Parameter identification in constitutive models via optimization with a posteriori error control. International Journal for Numerical Methods in Engineering 62, 1315–1340 (2005)

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[22] Kogler, H.: Characteristics and application of the cold box process and comparisons with the hot box process. Giesserei 64(5), 95–100 (1977) [23] Lemaitre, J.: Coupled elasto-plasticity and damage constitutive equations. Computer Methods in Applied Mechanics and Engineering 51, 31–49 (1985) [24] Mahnken, R.: Strength diﬀerence in compression and tension and pressure dependence of yielding in elasto-plasticity. Computer Methods in Applied Mechanics and Engineering 18, 801–831 (2002) [25] Mahnken, R.: Theoretical, numerical and identification aspects of a new model class for ductile damage. International Journal of Plasticity 18, 801–831 (2002) [26] Mahnken, R.: Identification of material parameters for constitutive equations. In: Stein, E., de Borst, R., Hughes, J.R., Wiley, J. (eds.) Encyclopedia of Computational Mechanics. Sons, Ltd. (2004) [27] Mahnken, R., Kuhl, E.: A finite element algorithm for parameter identification of gradient enhanced damage models. European Journal of Mechanics - A/Solids 18, 819–835 (1999) [28] Mahnken, R., Steinmann, P.: A finite element algorithm for parameter identification of material models for fluid-saturated porous media. International Journal for Numerical and Analytical Methods in Geomechanics 25, 415–434 (2001) [29] Mühlhaus, H.B.: Application of cosserat-theory in numerical solutions of limit point load problems. Archive of Applied Mechanics 59, 124–137 (1989) [30] Mühlhaus, H.B., Aifantis, E.C.: A variational principle for gradient plasticity. International Journal of Solids and Structures 28, 845–858 (1991) [31] Rogers, C.: Numerical mold and core sand simulation. Modeling of Casting, Welding and Advanced Solidification, 625–632 (2003) [32] Rousselier, G.: Ductile fracture models and their potential in local approach of fracture. Nuclear Engineering and Design 105, 97–111 (1987) [33] Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space-time finite element discretizations of parabolic equations. SIAM Journal on Scientific Computing 30, 369– 393 (2008) [34] Schmittner, H.: Occupational medical and hygiene implications of diﬀerent catalysing techniques. Giesserei 71(23), 895–902 (1984) [35] Shaban, A., Mahnken, R., Wilke, L., Potente, H., Ridder, H.: Simulation of rate dependent plasticity for polymers with asymmetric eﬀects. International Journal of Solids and Structures 44, 6148–6162 (2007) [36] Steinmann, P.: A micropolar theory of finite deformation and finite rotation multiplicative elasto-plasticity. International Journal of Solids and Structures 31, 1063–1084 (1994) [37] Vexler, B.: Adaptive finite element methods for parameter identification problems. In: für Angewandte Mathematik I, Dissertation. Universität, Heidelberg (2004) [38] Werling, J.: Does your core sand measure up? Modern Casting 93(2), 37–39 (2003) [39] Wilhelm, C.: Eﬀect of alcohol-based coatings on mechanical strength of cold-box cores. Giesserei 73(11), 317–321 (1986)

Model Reduction for Complex Continua – At the Example of Modeling Soft Tissue in the Nasal Area Annika Radermacher and Stefanie Reese

Dedicated to our colleague Wolfgang Ehlers on the occasion of his 60 th birthday, with many thanks for all the support and advice.

Abstract. Numerical simulation plays an important role in research fields with increasing complexity. Among these are e. g. extended continua, biomechanics, production technology, medical technology and many more. Modern simulation tools often provide both, realistic results and a close link to the physics of the problem. However, increased accuracy is paid by very high computational eﬀort which makes realtime simulation impossible. Nevertheless the latter is urgently needed in important application fields. A good example are surgery training and on-line support during minimally invasive real surgeries. Here, a numerical model can be well used to give otherwise unaccessible information about the stress and strain state in the biological material. The development of such a real-time computation method requires to extend existing model reduction concepts to non-linear solid mechanics based on complex continua. This paper discusses the eﬀectiveness of two singular value decomposition based model reduction methods in this context. Currently, the modal basis reduction method as well as the proper orthogonal decomposition method are widely used for solving linear problems. We will extend these approaches to nonlinear elasticity including large deformations. The performance of the extended concepts is first investigated for the simple geometry of a cantilever beam and then for the very complex system of a human nose. Comparing the results puts us into the position to discuss the suitability of the two model reduction methods for non-linear solid mechanics, especially from the point of view of biomechanics.

1 Indroduction The constantly rising requirements for biomechanical simulations yield numerical models with an increasing number of degrees-of-freedom. Together with additional Annika Radermacher · Stefanie Reese Institute of Applied Mechanics, RWTH Aachen University, Mies-van-der-Rohe-Straße 1, 52074 Aachen, Germany e-mail: {annika.radermacher,stefanie.reese}@rwth-aachen.de B. Markert (Ed.): Advances in Extended & Multifield Theories for Continua, LNACM 59, pp. 197–217. c Springer-Verlag Berlin Heidelberg 2011 springerlink.com

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non-linearity computational costs constantly increase. Simulations which serve for surgery training or on-line support of real surgeries, for example the functional endoscopic sinus surgery, should be carried out as quickly as possible. The ultimate goal is certainly a realistic analysis of the biomechanical problem in realtime. To achieve this and to minimize the computational costs model reduction is needed. Model reduction concepts for non-linear solid mechanics are hardly available in the literature. In fact, to the knowledge of the authors, the step to reduce a standard continuum made of non-linear elastic material has not yet been accomplished in the literature. Note that this is the simplest model which can be used to mimic the behavior of biological material. It is not suitable to display the realistic behavior very well. This is due to several reasons. First of all, most biological materials show diﬀerent types of inelasticity, frequently viscous eﬀects. Secondly, many biological structures have to be considered as composites, e. g. comprising diﬀerently oriented fiber bundles embedded into an isotropic matrix. Finally, certain parts of the human body, for instance articular cartilage, show diﬀerent behavior in dependence of the water content, the solid skeleton and other properties of such mixtures. In summary, the more complex the modeling at the continuum level, the more computational eﬀort has to be invested. This holds especially for extended continua. Thus, just in these cases model reduction is needed to reduce the numerical eﬀort to an acceptable level, preferably to realtime. The present paper is devoted to the first step, i. e. to the model reduction of non-linearly elastic structure. The treatment of extended continua can be performed along the same line as long as all degrees-of-freedom are treated in the same way. It is an important point of future research whether the additional degrees-of-freedom of e. g. Cosserat continua require specifically designed reduction strategies. Analogously we may pose the question whether diﬀerent field quantities (e. g. the displacement, the velocity and the pressure in a porous medium) call for diﬀerent degrees and methods of model reduction. However, for the time being, these highly challenging tasks have to postponed until a satisfactory level to model reduction in s¨tandard¨non-linear solid mechanics is reached. In the area of medicine most training simulations for surgeries are based on spring mass or tensor mass systems. In the spring mass model, point masses are connected with springs. The main problem of this method is to determine the correct spring parameters, as these are not directly related to physical material properties [14, 17]. Lloyd et al [13, 14] determine the spring parameters by comparing the matrices of the spring mass model with the corresponding finite element matrices. Kühnapfel et al [12] apply the spring mass method to compute non-linear material behavior by approaching the non-linear behavior by a third degree polynomial curve. The tensor mass model, on the other hand, is based on a tetrahedral mesh with barycentric shape functions. The stiﬀness matrix is parameterized in such a way that it consists of two terms, each one depends on one material parameter (i. e. [6, 7, 17, 21]). Cotin et al [7] use the tensor mass method to simulate cutting and tearing during a surgery. Non-linear material behavior is considered by Schwartz et al [21]. The

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non-linearity is contained in non-linear corrective functions preceding the terms in the parameterized form of the stiﬀness matrix. Other examples of tensor mass models for surgery simulations can be found in [18] and [26]. In other research fields like turbulence modeling, fluid dynamics, image processing or signal analysis many diﬀerent approaches to model reduction exist where the system of equations is reduced. We can diﬀerentiate between two classes of model reduction: the singular value decomposition (SVD)-based methods and the Krylovbased methods [1, 9]. The group of SVD-based methods projects the equations on a smaller-dimensional subspace. There is a variety of choices for the subspace. In the modal basis reduction method we choose the modal eigenforms by solving the eigenvalue problem to build the subspace. The projection yields a sparsely populated system of equations [8, 16, 20]. Another possibility is the proper orthogonal decomposition (POD) method. Here the subspace is built by solving the eigenvalue problem of the correlation matrix of an ensemble of snapshots (i. e. [5, 15]). Finally the balanced truncation method should be mentioned. This method is developed for first order state space problems. The subspace needed for the reduction is determined by a singular value decomposition of the controllability and observability Gramians (i. e. [1, 4, 25]). The second group of methods, the Krylov-based methods, uses moment matching of the transfer functions. There exist two diﬀerent algorithms for this purpose: the Lanczos and the Arnoldi procedure [1]. All concepts mentioned above were developed and are widely used for solving linear problems. There exist only a few strategies to expand some of these methods to non-linear systems. Remke and Rothert [19] use modal basis reduction for geometrically non-linear problems. Instead of solving the eigenvalue problem in each step, they approximate the eigenvectors by Ritz and displacement vectors. However, the method is only applied on truss systems. Another example for the extension to non-linear problems can be found in [11, 24] or [23]. They use the proper orthogonal decomposition method to solve the Burgers equation, the Helmholtz equation or signal transfer problems. The Krylov-based methods are recognized as being not useful for non-linear problems (i. e. [25]). A general method for model reduction in non-linear structural mechanics is not available yet. There are only a few papers which discuss model reduction with special emphasis on non-linear structural mechanics. For example Herkt et al [10] developed a method based on POD combined with a look up table for solving a complex non-linear structural problem. In the present paper we extend the modal basis reduction and the proper orthogonal decomposition method to non-linear elasticity including large deformations. The methods are tested for diﬀerent soft tissue materials, from nearly linear to strongly non-linear behavior. The performance of the extended concepts is first investigated for a cantilever beam and then for the very complex system of a human nose.

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2 Model Reduction for Non-linear Structural Mechanics The paper focuses on SVD-based model reduction methods which project the system onto a subspace of smaller dimension. In Sect. 2.1 we will derive the procedure of model reduction for non-linear structural mechanics. For this purpose two methods are used, the modal basis reduction method and the proper orthogonal decomposition method. In Sect. 2.2 a relative error is defined which permits a detailed comparison of the two strategies.

2.1 SVD-Based Reduction The three basic equations of mechanics (kinematics, the balance of linear momentum and the constitutive law) lead in case of a displacement-based finite element formulation to the non-linear vector equation G(U) = R(U) − Fext = 0 ,

(1)

where the n × 1 vector U represents the vector of the unknown nodal displacements at the global finite element level, Fext the global load vector and R(U) the residual force vector. In the present work, mass eﬀects are neglected. The first step in reducing Eq. (1) is to approximate the variable U by ¯ = Φ Ured , U

(2)

where the n×m matrix Φ is the chosen subspace and the m×1 vector Ured represents the reduced vector of unknowns. The original system has n degrees-of-freedom with n ≥ m. The subspace matrix Φ consists of m linearly independent basis vectors Vi (i = 1, ..., m) such that Φ = [V1 ; ...; Vm ] holds. Inserting (2) into Eq. (1) yields G(Φ Ured ) = R(Φ Ured ) − Fext = 0 .

(3)

Equation (3) contains n scalar equations for m ≤ n unknowns Ured,i (i = 1, ..., m) and is as such overdetermined. To solve this problem we use the least square approach. We multiply the system by the transpose of the subspace Φ and obtain the normal form ΦT G(Φ Ured ) = ΦT R(Φ Ured ) − ΦT Fext = 0 . (4) To solve the reduced system (4) the standard Newton-Raphson method is used. The Taylor expansion of Eq. (3) multiplied by the subspace yields T i+1 T i T ∂G Φ G(Φ Ured ) = Φ G(Φ Ured ) + Φ ΔUi+1 (5) red , ∂Ured Uired i+1 i+1 i where the increment ΔUi+1 red is given by ΔUred = Ured − Ured . During one NewtonRaphson iteration procedure the subspace matrix Φ is held constant. The subspace

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is only updated in the load steps. The partial derivative in Eq. (5) includes the tangential stiﬀness matrix KT = ∂G , which is known from the non-linear finite element ∂U method: ∂G ∂G ∂U = = KT Φ . (6) ∂Ured i ∂U ∂Ured i Ured

Ured

Finally we arrive at the m-dimensional equation system T ΦT KT Φ ΔUi+1 G. red = − Φ Gred KT,red

(7)

It remains to properly choose the subspace matrix Φ. In the next section we will discuss two possibilities to achieve this: the modal basis reduction and the proper orthogonal decomposition. 2.1.1

Modal Basis Reduction Method

The idea of modal basis reduction is to build the subspace matrix Φ by means of m eigenvectors of the eigenvalue problem (i. e. [16, 19]) (KT − λi I) Vi = 0 .

(8)

If m = n holds, the projection corresponds to a coordinate transformation and the solutions of Eq. (1) and (4) are identical. In most cases only a few modal parameters (pairs of eigenvalues and its corresponding eigenvectors) dominate the system behavior. So by taking a subspace built upon these eigenvectors the main eﬀects of the system behavior can be represented. In most cases the smaller eigenfrequencies, i. e. the smaller eigenvalues, and its corresponding eigenvectors are relevant. In general the tangential stiﬀness matrix changes significantly during each load step and it is necessary to solve the eigenvalue problem several times. This is a major drawback of the method since solving an additional eigenvalue problem increases the computational cost immensely. A possible way to reduce the computational eﬀort is to use a modified modal basis reduction (MORNL1), where the eigenvalue problem is solved only after the global residual |G| exceeds a user-defined tolerance rglobal . In Sect. 3 both the standard modal basis reduction method (MORNL) with solving the eigenvalue problem in each load step and the modified version (MORNL1) are implemented and compared. 2.1.2

Proper Orthogonal Decomposition Method

The proper orthogonal decomposition method uses a certain data set, an assembly of so-called snapshots, to build the subspace matrix Φ. See [2, 3, 5, 15, 16] for a more detailed overview.

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Snapshots are solution vectors of the problem one wishes to compute. To calculate the snapshots an analysis of the full system is needed beforehand. During this first analysis, called precomputation, the solution vector U j is saved at the first l time or load steps into a n × l snapshot matrix: D = [U1 , ..., Ul ] .

(9)

The proper orthogonal decomposition method searches for the optimum subspace matrix Φ to reduce the system with the help of the snapshot data set (m ≤ l). The optimum basis vectors are defined by maximizing the projection of the snapshots U j on the unknown basis vectors Vi (see [15]). This variational problem yields an eigenvalue problem (R − λi I) Vi = 0 (10) of the correlation matrix defined by R=

1 (D DT ) . l

(11)

The subspace matrix Φ includes m eigenvectors Vi computed from (10). There are diﬀerent ways to determine the subspace matrix. The first is to compute the singular value decomposition of the snapshot matrix: D = A Σ BT .

(12)

The subspace matrix Φ includes m columns of A. The same subspace matrix can be obtained by solving the eigenvalue problem of the correlation matrix R (Eq. (11)). A major drawback of both procedures is that one has to solve a large n × n problem. To circumvent the high computational cost Breuer and Sirovich [3] work with a new ˆ (dimension l × l) according to correlation matrix R ˆ = 1 (DT D) . R l

(13)

The m eigenvectors Vi to be included into the subspace matrix are defined by Vi =

D VR,i ˆ |D VR,i ˆ |

,

(14)

where VR,i are the eigenvectors of the eigenvalue problem of the correlation ˆ ˆ matrix R. The crucial question at this point is how well the POD subspace can represent the non-linear eﬀects. To investigate this open issue the example systems in Sect. 3 are reduced by means of diﬀerent subspace matrices.

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2.2 Error Definition For the purpose of comparing the presented methods, an average relative error of the displacements about the N load steps is defined as =

k k 1 N |U − Uorg | Σk=1 , N |Ukorg |

(15)

where Uk is the solution of the reduced system at the load step k (U = Φ Ured ), Ukorg is the solution of the original system and | • | is the L2 norm. In order to compute the stiﬀness matrix, the residual vector and the force vector the finite element solver FEAP developed by Taylor1 is used. The model reduction methods are implemented in Matlab2 and combined with FEAP by using the interface matfeap3 .

3 Biomechanical Structural Applications The two methods presented in Sect. 2 are validated by means of two examples. The first one is a simple beam structure. The second concerns the biomechanical analysis of a realistic nose geometry.

3.1 Examples As first example a cantilever beam (Fig. 1) is chosen. The beam’s dimensions are 5 m × 1 m × 1 m. It is meshed by mixed (Hu-Washizu variational principle) eightnode hexahedra. At the boundary x = 0 m all degrees-of-freedom are fixed, at the boundary x = 5 m a vertical load is applied (see Fig. 1). The load increases linearly in 100 load steps until a value of 80 N is reached. The biomaterial is modeled as being non-linearly elastic. The Neo-Hookean strain energy function is chosen. The material parameters, Young’s modulus (E) and Poisson’s ratio (ν), are chosen according to [27] and [22]. The material behavior varies from being stiﬀ (E = 80 kPa, ν = 0.45) to soft (E = 20 kPa, ν = 0.45). Altogether four diﬀerent materials are analyzed (see Table 1). The analysis is geometrically and materially non-linear. We use the vertical displacement in point P to compare the results. Figure 2 shows the force-displacement relation at point P for the investigated materials. The model of the human nose is illustrated in Fig. 3. The geometry is built up from CT images and meshed by four-node tetrahedra which use the mixed finite element technology, too. The model is fixed on the back side (Fig. 3). We simulate a point load on the nasal septum which increases linearly in 10 load steps until 0.05 N. We work with the same materials as before. 1

2

3

Department of Civil and Environmental Engineering, University of California at Berkeley, USA, www.ce.berkeley.edu/projects/feap A numerical computing environment and fourth-generation programming language, developed by MathWorks, www.mathworks.de www.cs.cornell.edu/ bindel/cims/matfeap

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Fig. 1 Cantilever beam - geometry and boundary conditions. Table 1 Overview about investigated materials. Number

Young’s modulus

Poisson’s ratio

material 1 material 2 material 3 material 4

80 kPa 60 kPa 40 kPa 20 kPa

0.45 0.45 0.45 0.45

force fz at point P [MN]

0

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

-5e-07

-1e-06

-1.5e-06

-2e-06

-2.5e-06 -1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

displacement uz at point P [m]

Fig. 2 Investigated material behavior.

0

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To compute the original solution (Uorg in Eq. (15)) a full finite element analysis of both models is conducted.

Fig. 3 Nose model - geometry and boundary conditions.

3.2 Study of Convergence The study of convergence is needed to determine a mesh which leads to a suﬃciently accurate solution. Therefore the mesh for the cantilever beam has been refined from two elements over length and one over height to 256 elements over length and 16 over height. The load is chosen such that the number of elements over the thickness has no influence on the results. Accordingly only one element over the thickness is used. Figure 4 shows the study of convergence for the cantilever beam with material one. It is seen that the refinement in x-direction leads to a softening of the behavior whereas the refinement in z-direction lets the displacement in point P decrease. Furthermore, a mesh-independent solution is already reached for a mesh of 64 × 1 × 8 elements. We choose this number of elements for the following analysis.

3.3 Study of Parameters 3.3.1

Modal Basis Reduction Method

The quality of the reduction is judged by looking at the error (15). The error of the modal basis reduction method (MORNL) for the cantilever beam is shown in the upper part of Fig. 5. As expected the error grows with smaller order of reduction (m). The cantilever beam has a total number of 3456 degrees-of-freedom. Setting the order of reduction to 3456 the original solution is obtained. The error increases with increasing material non-linearity. Using the same time increment, a stronger

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displacement uz at point P [m]

0

elements over height: 1 elements over height: 2 elements over height: 4 elements over height: 8 elements over height: 16

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 0

50

100

150

200

250

300

number of elements over the length [-]

displacement uz at point P [m]

0 -0.2 -0.4 -0.6 elements over length: 2 elements over length: 4 elements over length: 8 elements over length: 16 elements over length: 32 elements over length: 64 elements over length: 128 elements over length: 256

-0.8 -1 -1.2 0

2

4

6

8

10

12

14

16

number of elements over the height [-]

Fig. 4 Study of convergence: refinement over the length (top) and refinement over the height (bottom).

non-linearity leads to a larger change in the displacement vector during one Newton iteration (one time step). The at the beginning of each time step specified subspace does not consider this change. For this reason, the curves for the diﬀerent materials come closer to each other if the time increment is chosen according to the nonlinearity of the material (see lower part in Fig. 5). This is obviously a critical disadvantage of the modal basis reduction method. To achieve good results with increasing non-linearity, the time increment has to be chosen suﬃciently small. This results into a direct increase of the computational cost in addition to the overhead due to the solution of the eigenvalue problems. Furthermore it is not known which time increment yields suﬃciently accurate results.

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0.04

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.035 0.03

ε [-]

0.025 0.02 0.015 0.01 0.005 0 5

10

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500 1000

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order of reduction [-] (original model 3456 DOF) MORNL 0.016

E=80 kPa; ν=0.45; dt=0.10 s E=60 kPa; ν=0.45; dt=0.08 s E=40 kPa; ν=0.45; dt=0.05 s E=20 kPa; ν=0.45; dt=0.03 s

0.014 0.012

ε [-]

0.01 0.008 0.006 0.004 0.002 0 5

10

50

100

500 1000

3500

order of reduction [-] (original model 3456 DOF)

Fig. 5 Development of with fixed (top) and varied dt (bottom) with MORNL (diﬀerent ranges).

To reduce the computational cost of the modal basis reduction method, the modified method MORNL1 (see Sect. 2.1.1) is used to analyze the first example. Figure 6 shows the error development for MORNL1. The analysis is performed for diﬀerent criteria rglobal (see Sect. 2.1.1). It turns out that the accuracy of the method depends very much on the number of the solved eigenvalue problems. Choosing a tolerance rglobal = 10−4 leads to an error which is about 100 times higher than the one of the method MORNL. Despite the larger error, the reduction of the computational cost is comparatively low. Instead of 100 eigenvalue problems we have to solve about 75 eigenvalue problems in the case of rglobal = 10−4 .

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ε [-]

0.5 0.4 0.3 0.2 E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.1 0 5

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order of reduction [-] (original model 3456 DOF) -4

MORNL1, rglobal=10 , number of EVP: 75 0.25

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.2

ε [-]

0.15

0.1

0.05

0 5

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order of reduction [-] (original model 3456 DOF) MORNL1, rglobal=10-6, number of EVP: 99 0.04

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.035 0.03

ε [-]

0.025 0.02 0.015 0.01 0.005 0 5

10

50

100

500 1000

3000

order of reduction [-] (original model 3456 DOF)

Fig. 6 Development of for diﬀerent values of rglobal with MORNL1 (diﬀerent ranges).

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To investigate the influence of the shape we vary the length of the cantilever beam from 1 m to 5 m. Figure 7 shows that the error increases with shorter lengths. The shorter beam’s behavior is not dominated by deflection. Therefore the eigenvectors do not well represent the general deformation behavior of the beam. MORNL 0.09

l=5 m l=2 m l=1 m

0.08 0.07

ε [-]

0.06 0.05 0.04 0.03 0.02 0.01 0 5

10

50

100

500

1500

order of reduction [-] (original model 3456 DOF)

Fig. 7 Development of for diﬀerent lengths of the cantilever beam (MORNL).

In summary we can say that it is possible to reduce this model from 3456 degreesof-freedom to 50 with an error of only several per mill. We have detected an influence of the time increment and of the geometry. Stronger non-linearity requires to work with smaller time increments if one aims at the same accuracy as expected for the model reduction of weakly non-linear systems. With compacter shapes we need more eigenvectors to reach suﬃciently accurate results. The attempt to reduce the increased computational cost by using the modified method (MORNL1) has reduced the accuracy too much to be useful. 3.3.2

Proper Orthogonal Decomposition Method

At first we compare diﬀerent ways to compute the proper orthogonal decomposition subspaces as mentioned in Sect. 2.1.2. Four diﬀerent subspaces, computed from the same given snapshots, will be used for this purpose. The method where the subspace is determined by means of a singular value decomposition of the snapshot matrix is called SVD. Further, a modified SVD method (SVDmM) in which a singular value decomposition of the centering snapshot matrix is computed, is investigated. Finally, we use the method of Sirovich, explained in Sect. 2.1.2, on the one hand with centering the snapshots by subtraction the average value of each snapshot (POD) and on the other hand without centering (PODoM). Figure 8 shows the development of the error for the diﬀerent subspaces mentioned above. The plot shows the dependence on the number of snapshots. The

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results in Fig. 8 diﬀer only for small snapshot numbers (up to 50). The variant SVDmM gives the best results. The drawback of this method is the necessity to solve a singular value decomposition of the full system. It is therefore more expensive than the POD variant. For the following discussion we will concentrate on the POD method, as it provides better results as the PODoM and has lower computational cost as the SVDmM variant. material 4 0.006

POD PODoM SVD SVDmM

0.005

ε [-]

0.004 0.003 0.002 0.001 0 5

25

50

100

number of snapshots [-]

Fig. 8 Development of of diﬀerent subspace bases (material 4).

Figure 9 shows the error evolution for the POD method in the context of the cantilever beam made of the diﬀerent materials. From the upper part of Fig. 9 it is obvious that one snapshot is not enough to compute the model with suﬃcient accuracy. The lower part of Fig. 9 shows the error development in the range of 5 to 100 snapshots for a better overview. As expected the error decreases with an increasing number of snapshots. Furthermore Fig. 9 shows a dependence of the error on the degree of non-linearity. Figure 10 detect a drawback of the POD method. The subspace determined by the given snapshots can only represent a specified behavior. One has to choose the snapshots well, because for geometrically as well as materially non-linear problems the deformation changes significantly over time. If we compute the subspace with only one snapshot (solution at the first time step of the precomputation), we can only represent the first steps. Figure 10 shows a comparison between material 1 (top) and material 4 (bottom). In both cases the displacement curve from the POD analysis approaches the original curves with an increasing number of snapshots. In case of the non-linear material 4 we need more snapshots to minimize the diﬀerence between the approximated and the original displacement curve. In the following, the influence of the size of the time increment of the precomputation (0.03 s, 0.10 s, 0.33 s) is studied. The subspaces are again computed using

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211 POD

0.7

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.6

ε [-]

0.5 0.4 0.3 0.2 0.1 0 1

5

25

number of snapshots [-] POD 0.0016

E=80 kPa; ν=0.45 E=60 kPa; ν=0.45 E=40 kPa; ν=0.45 E=20 kPa; ν=0.45

0.0014 0.0012

ε [-]

0.001 0.0008 0.0006 0.0004 0.0002 0 25

50

80

100

number of snapshots [-]

Fig. 9 Development of for diﬀerent materials (POD) until 25 snapshots (top) and from 25 to 100 snapshots (bottom).

diﬀerent numbers of snapshots which depends on the total time of the precomputation. The model reduction analysis remains unchanged with 100 load steps with a time step size of 0.1 s. Figure 11 shows that compared to the number of snapshots the time step size of the precomputation has very little influence on the accuracy of the model reduction analysis. The direct comparison of the two materials 1 and 4 in Fig. 11 shows again the influence of non-linearity. The nearly linear behavior of material 1 requires a much smaller number of snapshots for a good result. In summary we find that the first example can be analyzed successfully by using the POD method. For acceptable errors it is important that the snapshots are a representation of the complete period of investigation, especially for non-linear problems. We have also seen that it is possible to carry out the precomputation, necessary to

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displacement uz at point P [m]

0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35

Uorg UPOD; l= 1 UPOD; l= 5 UPOD; l= 25 UPOD; l=100

-0.4 -0.45 -0.5 0

20

40

60

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number of loadsteps (dt=0.1 s) [-] material 4, number of snapshots l

displacement uz at point P [m]

0 -0.2 -0.4 -0.6 -0.8 -1 -1.2 Uorg UPOD; l= 1 UPOD; l= 5 UPOD; l= 25 UPOD; l=100

-1.4 -1.6 -1.8 0

20

40

60

80

100

number of loadsteps (dt=0.1 s) [-]

Fig. 10 Development of the displacement of the beam with material 1 (top) and material 4 (bottom) at point P over the time (POD).

determine the snapshots, with much larger time increments than needed for the POD simulation. This reduces the dimension of the snapshot matrix. Comparing the POD method (Fig. 9) with the modal basis reduction method (Fig. 5) the error decreases more rapidly with the order of reduction/number of snapshots. On the other hand, in this first and simple example the use of the MORNL method leads to a smaller error than the POD.

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POD, material 1 2e-05

dt=0.33 s dt=0.10 s dt=0.03 s

ε [-]

1.5e-05

1e-05

5e-06

0 0

1

2

3

4

5

6

7

8

9

10

total time of precomputation [s] POD, material 4 0.006

dt=0.33 s dt=0.10 s dt=0.03 s

0.005

ε [-]

0.004 0.003 0.002 0.001 0 0

2

4

6

8

10

total time of precomputation [s]

Fig. 11 Influence of precomputation’s time increment (material 1 (top) and 4 (bottom)).

3.4 Human Nose Model After discussing the two reduction methods for a rather simple geometry, we will investigate the methods in the context of a more complex geometry, a human nose (Fig. 3). 3.4.1

Model Basis Reduction Method

Figure 12 shows the error at the loaded point of the nose, determined by using the modal basis reduction method (MORNL). The original nose system has 35961 degrees-of-freedom. The error for a reduced system with 2000 degrees-of-freedom

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E=80 kPa; ν=0.45 E=20 kPa; ν=0.45

0.3

ε [-]

0.25

0.2

0.15

0.1 50

500

1000

2000

order of reduction [-] (original model 35961 DOF)

Fig. 12 Development of for the nose model (MORNL). POD, nose geometry 0.012

E=80 kPa; ν=0.45 E=20 kPa; ν=0.45

0.01

ε [-]

0.008 0.006 0.004 0.002 0 1

2

3

4

5

6

7

number of snapshots [-]

Fig. 13 Development of for the nose model (POD).

is about 10 %. Obviously the modal basis reduction is not useful for this compact geometry. The eigenvectors do not represent the overall behavior of the structure. 3.4.2

Proper Orthogonal Decomposition Method

Figure 13 shows the error evolution at the loaded point of the nose, determined by using the proper orthogonal decomposition method (POD). Even a simulation with a low number of snapshots (up to three) yields a suﬃciently accurate result in the

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case of the human nose model. In contrast to the modal basis reduction method the geometry does not have a significant influence when using the proper orthogonal decomposition method. The eigenvectors of the correlation matrix included in the subspace matrix Φ represent the general behavior of the nose system very well. This is because the snapshots are chosen to cover the most important eﬀects of the nose deformation.

4 Conclusion In this paper we have presented two model reduction methods. Both belong to the class of singular value decomposition based methods and were originally developed to solve linear problems. This paper shows that it is possible to use these methods in the context of non-linear biomechanics. The application of the developed methods on a simple geometry of a cantilever beam shows suﬃciently accurate results for the displacement field. By applying the methods on the more complex geometry of a human nose the proper orthogonal decomposition method yields significantly better results than the modal basis reduction method. In particular this concerns computational cost and the relative error of displacement. The application to a complex biomechanical structure, a human nose, permits a first judgment of accuracy and computational cost. The performance of the POD is very promising. However, it is important to build the subspace matrix on snapshots which are representative for the displacement during the surgery. This certainly requires to devote more time and eﬀort into the precomputation. The reduced system can then be very small. Acknowledgements. The CT images used to build up the geometry of the human nose have been provided by the HNO Klinik Bonn (Germany) and the Institute of Robotics and Process Control, Technical University of Braunschweig (Germany).

References [1] Antoulas, A., Sorensen, D.: Approximation of large-scale dynamical systems: An overview. International Journal of Applied Mathematics and Computer Science 11, 1093–1121 (2001) [2] Aquino, W., Brigham, J.C., Earls, C.J., Sukumar, N.: Generalized finite element method using proper orthogonal decomposition. International Journal for Numerical Methods in Engineering 79, 1097–1207 (2009) [3] Breuer, K.S., Sirovich, L.: The use of the Karhunen-Loéve procedure for the calculation of linear eigenfunctions. Journal of Computational Physics 96, 277–296 (1991) [4] Chahlaoui, Y., Lemonnier, D., Vandendorpe, A., Dooren, P.V.: Second-order balanced truncation. Linear Algebra and its Applications 415, 373–384 (2004) [5] Chatterjee, A.: An introduction to the proper orthogonal decomposition. Current Science 78, 808–817 (2000)

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[6] Cotin, S., Delingette, H., Ayache, N.: Real-time elastic deformations of soft tissues for surgery simulation. IEEE Transactions on Visualization and Computer Graphics 5, 62–73 (1999) [7] Cotin, S., Delingette, H., Ayache, N.: A hybrid elastic model for real-time cutting, deformations, and force feedback for surgery training and simulation. The Visual Computer 16, 437–452 (2000) [8] Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA Journal 6, 1313–1319 (1968) [9] Gugercin, S.: An iterative svd-krylov based method for model reduction of large-scale dynamical systems. Linear Algebra and its Applications 428, 1964–1986 (2008) [10] Herkt, S., Dreßler, K., Pinnau, R.: Model reduction of nonlinear problems in structural mechanics. Berichte des Fraunhofer ITWM, Kaiserslautern 175, 1–23 (2009) [11] Hinze, M., Volkwein, S.: Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: Error estimates and suboptimal control. In: Barth, T.J., Griebel, M., Keyes, D.E., Nieminen, R.M., Roose, D., Schlick, T., Benner, P., Sorensen, D.C., Mehrmann, V. (eds.) Dimension Reduction of Large-Scale Systems. LNCS, vol. 45, pp. 261–306. Springer, Berlin (2005) [12] Kühnapfel, U., Cakmak, H.K., Maaß, H.: Endoscopic surgery training using virtual reality and deformable tissue simulation. Computer & Graphics 24, 671–682 (2000) [13] Lloyd, B.A., Székely, G., Harders, M.: Identification of spring parameters for deformable object simulation. IEEE Transactions on Visualization and Computer Graphics 13, 1081–1094 (2007) [14] Lloyd, B.A., Kirac, S., Székely, G., Harders, M.: Identification of dynamic mass spring parameters for deformable body simulation. The Eurographics Association (2008) [15] Lumley, J.L., Holmes, P., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996) [16] Meyer, M., Matthies, H.G.: Eﬃcient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods. Computational Mechanics 31, 179–191 (2003) [17] Mollemans, W., Schutyser, F., Nadjmi, N., Suetens, P.: Very fast soft tissue predictions with mass tensor model for maxillofacial surgery planning systems. International Congress Series , vol. 1281, pp. 491–496 (2005) [18] Picinbono, G., Delingette, H., Ayache, N.: Nonlinear and anisotropic elastic soft tissue models for medical simulation. In: Proceedings of IEEE International Conference on Robotics and Automation, ICRA, vol. 2, pp. 1370–1375 (2001) [19] Remke, J., Rothert, H.: Eine modale Reduktionsmethode zur geometrisch nichtlinearen statischen und dynamischen Finite-Element-Berechnung. Archive of Applied Mechanics 63, 101–115 (1993) [20] Rickelt-Rolf, C.: Modellreduktion und Substrukturtechnik zur eﬃzienten Simulation dynamischer, teilgeschädigter Systeme. PhD thesis, Technische Universität CaroloWilhelmina zu Braunschweig (2009) [21] Schwartz, J.M., Denninger, M., Rancourt, D., Moisan, C., Laurendeau, D.: Modelling liver tissue properties using a non-linear visco-elastic model for surgery simulation. Medical Image Analysis 9, 103–112 (2005) [22] Su, J., Zou, H., Guo, T.: The study of mechanical properties on soft tissue of human forearm in vivo. In: 3rd International Conference on Bioinformatics and Biomedical Engineering, ICBBE 2009, pp. 1–4 (2009)

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[23] Verhoeven, A., Striebel, M., Rommes, J., Maten, E.J.W., Bechtold, T.: Proper orthogonal decomposition model order reduction of nonlinear ic models. In: Bock, H.G., Hoog, F., Friedman, A., Gupta, A., Neunzert, H., Pulleyblank, W.R., Rusten, T., Santosa, F., Tornberg, A.K., Capasso, V., Mattheij, R., Neunzert, H., Scherzer, O., Fitt, A.D., Norbury, J., Ockendon, H., Wilson, E. (eds.) Progress in Industrial Mathematics at ECMI 2008. Mathematics in Industry, vol. 15, pp. 441–446. Springer, Berlin (2010) [24] Volkwein, S., Hepberger, A.: Impedance identification by pod model reduction techniques at-Automatisierungstechnik, pp. 437–446 (2008) [25] Willcox, K., Peraire, J.: Balanced model reduction via the proper orthogonal decomposition. AIAA Journal 40(11), 2323–2330 (2002) [26] Xu, S., Liu, X., Zhang, H., Hu, L.: A nonlinear viscoelastic tensor-mass visual model for surgery simulation. IEEE Transactions on Instrumentation and Measurement 60, 14–20 (2011) [27] Zheng, Y.P., Choi, A.P.C., Ling, H.Y., Huang, Y.P.: Simultaneous estimation of poisson’s ratio and young’s modulus using a single indentation: A finite element study. Measurement Science and Technology 20, 045, 706 (2009)

Author Index

Ammar, Kais 68 Appolaire, Benoît 68 Bargmann, Swantje 88 Bloßfeld, Moritz 152 Bluhm, Joachim 152 Caylak, Ismail

174

Diebels, Stefan 19 Dyskin, Arcady V. 47 Forest, Samuel Keip, Marc-André

68 108

Mahnken, Rolf 174 Markert, Bernd 128 McBride, Andrew T. 88 Mühlhaus, Hans-Bernd 47 Pasternak, Elena

47

Radermacher, Annika Reese, Stefanie 196 Ricken, Tim 152 Scharding, Daniel 19 Schröder, Jörg 108 Steinmann, Paul 88 Svendsen, Bob 1

196