Advanced Structured Materials978-3-642-36394...Svyatoslav V. Derezin Faculty of Mathematics,...
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Advanced Structured Materials Volume 22 Series Editors Andreas Öchsner Lucas F. M. da Silva Holm Altenbach For further volumes: http://www.springer.com/series/8611
Transcript of Advanced Structured Materials978-3-642-36394...Svyatoslav V. Derezin Faculty of Mathematics,...
Advanced Structured Materials
Volume 22
Series Editors
Andreas ÖchsnerLucas F. M. da SilvaHolm Altenbach
For further volumes:http://www.springer.com/series/8611
Holm Altenbach • Samuel ForestAnton KrivtsovEditors
Generalized Continuaas Models for Materials
With Multi-scale Effects or UnderMulti-field Actions
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EditorsHolm AltenbachLehrstuhl für Technische MechanikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany
Samuel ForestCentre des Materiaux CNRS UMR 7633Mines ParisTechEvry CedexFrance
Anton KrivtsovDepartment of Theoretical MechanicsSt. Petersburg State Polytechnical
UniversitySt. PetersburgRussia
ISSN 1869-8433 ISSN 1869-8441 (electronic)ISBN 978-3-642-36393-1 ISBN 978-3-642-36394-8 (eBook)DOI 10.1007/978-3-642-36394-8Springer Heidelberg New York Dordrecht London
Library of Congress Control Number: 2013934985
� Springer-Verlag Berlin Heidelberg 2013This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed. Exempted from this legal reservation are briefexcerpts in connection with reviews or scholarly analysis or material supplied specifically for thepurpose of being entered and executed on a computer system, for exclusive use by the purchaser of thework. Duplication of this publication or parts thereof is permitted only under the provisions ofthe Copyright Law of the Publisher’s location, in its current version, and permission for use mustalways be obtained from Springer. Permissions for use may be obtained through RightsLink at theCopyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exemptfrom the relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.
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Preface
The Mechanics of Generalized Continua is an established research topic since theend of the 1950s–early 1960s of the last century. The starting point of thisdevelopment was the monograph of the Cosserat brothers, published in 19091 , andsome previous works of such famous scientists like Lord Kelvin, Duhem, Helm-holtz among others. All these contributions were focussed on the fact that in acontinuum one has to define translations and rotations independently (or in otherwords, one has to establish force and moment actions independently as it was doneby Jakob Bernoulli and Euler). At the same time the continuum was not modeledas an infinite number of continuously distributed points with properties like themass, but as an infinite number of continuously distributed infinitesimal smallbodies with properties like the mass.
The reason for the revival in the mid of 1950s of the last century was that someeffects of the mechanical behavior of solids and fluids could not be explained bythe available classical continuum models. Examples of this are the turbulence of afluid or the behavior of solids with a significant and very complex microstructure.Since the suggested new models fulfill all requirements from Continuum Ther-momechanics (the balance laws were formulated and the general representation ofthe constitutive equations were given) the scientific community was satisfied for awhile. At the same time real applicative developments were missed.
Indeed, for practical applications the developed models were not useful. Thereason for this was the gap between the formulated constitutive equations and thepossibilities to identify the material parameters. As is often the case one had muchmore parameters compared to classical models, but no facilities to measure allproperties. In addition, computational progress and available machines in thesetimes were limited.
During the last ten years the situation has drastically changed. More and moreresearches emerged, being kindled by the partly forgotten models. Now one hasavailable much more computational possibilities and very complex problems canbe simulated numerically. In addition, with the increased attention paid to a large
1 E. et F. Cosserat: Cosserat, F.: Théorie des Corps Déformables, Hermann Editeurs, Paris, 1909(Reprint, Gabay, Paris, 2008).
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number of materials with complex microstructure and a deeper understanding ofthe meaning of the material parameters (scale effects) the identification becomesmuch more well founded. Thus we have contributions describing the micro- andmacro-behavior, new existence and uniqueness theorems, the formulation of multi-scale problems, etc., and now it is time to ponder again2 the state of matter and todiscuss new trends and applications.
The main focus in this book will be directed on the following items:
• Modeling and simulation of materials with significant microstructure;• Generalized continua as a result of multi-scale models;• Multi-field actions on materials resulting in generalized material models; and• Comparison with discrete modeling approaches.
This book contains selected papers submitted to the Second Trilateral SeminarGeneralized Continua as Models for Materials With Multi-scale Effects or UnderMultifield Actions, which held at the Leucorea (Lutherstadt Wittenberg, Germany)from September 26 upto 30, 2012.3 Special thanks to Andreas Kutschke who tookall duties connected with realization of the Seminar. In addition, we kindlyacknowledge Dr. Christoph Baumann and Benjamin Feuchter (Springer Publisher)for the support of the book project. Last but not least it should be mentioned thatthe Seminar was sponsored by grants of the French National Center for ScientificResearch (CNRS), the German Research Foundation (DFG) AL341/41-1, and theRussian Foundation for Basic Research 12-01-91260RFG.
Magdeburg, December 2012 Holm AltenbachParis Samuel ForestSt. Petersburg Anton Krivtsov
2 There were two proceedings within the last years which should be mentioned here: Gérard A.Maugin, Andrei V. Metrikine (Eds) Mechanics of Generalized Continua - One Hundred YearsAfter the Cosserats, Springer, 2010 (Advances in Mechanics and Mathematics, Vol. 21) andHolm Altenbach, Gérard Maugin, Vladimir Erofeev (Eds) Mechanics of Generalized Continua,Springer, 2011 (Advanced Structured Materials, Vol. 7).3 The First Trilateral French–German–Russian Seminar held also in Lutherstadt Wittenberg(Germany) August 9–11, 2010.
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Contents
Shells and Plates with Surface Effects . . . . . . . . . . . . . . . . . . . . . . . . 1Holm Altenbach and Victor A. Eremeyev
Geometrical Picture of Third-Order Tensors . . . . . . . . . . . . . . . . . . . 17Nicolas Auffray
Continuum Modelling of Shear-Coupled GrainBoundary Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Stéphane Berbenni, Bhasker Paliwal and Mohammed Cherkaoui
Buckling of Nonlinearly Elastic Plates with Microstructure . . . . . . . . 65Svyatoslav Derezin
Material Symmetry Group and Consistently ReducedConstitutive Equations of the Elastic Cosserat Continuum . . . . . . . . . 77Victor A. Eremeyev and Wojciech Pietraszkiewicz
Nonlinear Localized Strain Waves in a 2D Mediumwith Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Vladimir I. Erofeev, Vladimir V. Kazhaev and Igor S. Pavlov
Cosserat Anisotropic Models of Trabecular Bone fromthe Homogenization of the Trabecular Structure:2D and 3D Frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111Ibrahim Goda, Mohamed Assidi and Jean-Francois Ganghoffer
Electro-Thermo-Elastic Simulation of Graphite Tools Usedin SPS Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143Stefan Hartmann, Steffen Rothe and Nachum Frage
The Use of Moment Theory to Describe the Piezoelectric Effectin Polar and Non-Polar Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Elena A. Ivanova and Yaroslav E. Kolpakov
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Description of Thermal and Micro-Structural Processesin Generalized Continua: Zhilin’s Methodand its Modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179Elena Ivanova and Elena Vilchevskaya
Hierarchical Architecture and Modeling of Bio-InspiredMechanically Adaptive Polymer Nanocomposites . . . . . . . . . . . . . . . . 199Rasa Kazakevi�ciut _e-Makovska and Holger Steeb
Justification of the Bending-Gradient Theory ThroughAsymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217Arthur Lebée and Karam Sab
Macroscopic Modeling of Size Effects in Foams Usingan Order-Parameter Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237Bernd Lenhof, Alexander Geringer and Stefan Diebels
Measuring Material Coefficients of Higher Gradient Elasticityby Using AFM Techniques and Raman-Spectroscopy . . . . . . . . . . . . . 255Christian Liebold and Wolfgang H. Müller
On the Electrostatic Fields in Dielectric Continuawith Self-Similar Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273Thomas M. Michelitsch and Gérard A. Maugin
Nonlinear Generalizations of the Born-Huang Modeland Their Continuum Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Alexey V. Porubov, Eron L. Aero and Boris R. Andrievsky
Buckling of Inhomogeneous Circular Plate of Micropolar Material . . . 291Denis N. Sheydakov
A Theory of Disclination and Dislocation Fieldsfor Grain Boundary Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303V. Taupin, L. Capolungo, C. Fressengeas, A. Das and M. Upadhyay
Material Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321Pavel A. Zhilin, Holm Altenbach, Elena A. Ivanovaand Anton Krivtsov
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Contributors
Eron L. Aero Institute of Problems in Mechanical Engineering, Bolshoy pr. 61,V.O., St. Petersburg, Russia 199178, e-mail: [email protected]
Holm Altenbach Lehrstuhl für Technische Mechanik, Fakultät für Maschinen-bau, Institut für Mechanik, Otto-von-Guericke-Universtät Magdeburg, Universi-tätsplatz 2, 39106 Magdeburg, Germany, e-mail: [email protected]
Boris R. Andrievsky Institute of Problems in Mechanical Engineering, Bolshoypr. 61, V.O., St. Petersburg, Russia 199178, e-mail: [email protected]
Mohamed Assidi Centre de Recherche Public Henri Tudor, 29, Avenue JohnKennedy, L-1885 Luxembourg, Luxembourg, e-mail: [email protected]
Nicolas Auffray Laboratoire Modélisation et Simulation MultiEchelle LMSME,Université Paris-Est, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France, e-mail: [email protected]
Stéphane Berbenni Laboratoire d’Etudes des Microstructures et de Mécaniquedes Matériaux, LEM3, University of Lorraine, UMR CNRS 7239, 57045 Metz,France; Unité Mixte Internationale Georgia Tech Lorraine-CNRS, UMI CNRS2958, 57070 Metz, France, e-mail: [email protected]
Laurent Capolungo Woodruff School of Mechanical Engineering, GeorgiaInstitute of Technology/CNRS, 57070 Metz Cedex, France, e-mail: [email protected]
Mohammed Cherkaoui Unité Mixte Internationale Georgia Tech Lorraine-CNRS, UMI CNRS 2958, 57070 Metz, France; George W. Woodruff School ofMechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA, e-mail: [email protected]
A. Das Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux,Université de Lorraine/CNRS, Ile du Saulcy, 57045 Metz Cedex, France, e-mail:[email protected]
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Svyatoslav V. Derezin Faculty of Mathematics, Mechanics and ComputerScience, Southern Federal University, Milchakova 8a, Rostov on Don, Russia344090, e-mail: [email protected]
Stefan Diebels Chair of Applied Mechanics, Saarland University, Campus A4.2,D-66123 Saarbrücken, Germany, e-mail: [email protected]
Victor A. Eremeyev Lehrstuhl für Technische Mechanik, Fakultät für Maschi-nenbau, Institut für Mechanik, Otto-von-Guericke-Universtät Magdeburg, Uni-versitätsplatz 2, 39106 Magdeburg, Germany; South Scientific Center of RASciand South Federal University, Milchakova St. 8a, Rostov on Don, Russia344090, e-mail: [email protected]; [email protected]
Vladimir I. Erofeev Nizhny Novgorod Mechanical Engineering, ResearchInstitute of Russian Academy of Sciences, IMASH RAN, 85, Belinskogo st.,Nizhny Novgorod, Russia 603024, e-mail: [email protected]
Nachum Frage Department of Materials Engineering, Ben-Gurion University ofthe Negev, Beer-Sheva, Israel, e-mail: [email protected]
Claude Fressengeas Laboratoire d’Etude des Microstructures et de Mécaniquedes Matériaux, Université de Lorraine/CNRS, Ile du Saulcy, 57045 Metz Cedex,France, e-mail: [email protected]
Jean-François Ganghoffer LEMTA, Université de Lorraine, 2, Avenue de laForêt de Haye, 54504 Vandoeuvrelès-Nancy Cedex, France, e-mail: [email protected]
Alexander Geringer Chair of Applied Mechanics, Saarland University, CampusA4.2, D-66123 Saarbrücken, Germany, e-mail: [email protected]
Ibrahim Goda LEMTA, Université de Lorraine, 2, Avenue de la Forêt de Haye,54504 Vandoeuvrelès-Nancy Cedex, France, e-mail: [email protected]
Stefan Hartmann Institute of Applied Mechanics, Clausthal University of Tech-nology, Adolph-Roemer-Str. 2a, 38678 Clausthal-Zellerfeld, Germany, e-mail:[email protected]
Elena A. Ivanova St. Petersburg State Polytechnical University (SPbSPU),Politekhnicheskaya 29, St. Petersburg, Russia 195251; Institute for Problems inMechanical Engineering, Russian Academy of Sciences, Bolshoy pr. V.O., 61, St.Petersburg, Russia 199178, e-mail: [email protected]
Rasa Kazakeviciut _e-Makovska Mechanics-Continuum Mechanics, Ruhr Uni-versity Bochum, Universitätsstr. 150, 44780 Bochum, Germany, e-mail: [email protected]
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Vladimir V. Kazhaev Nizhny Novgorod Mechanical Engineering, ResearchInstitute of Russian Academy of Sciences, IMASH RAN 85, Belinskogo st.,Nizhny Novgorod, Russia 603024
Yaroslav E. Kolpakov Morion Inc., Kima ave. 13a, St. Petersburg, Russia199155, e-mail: [email protected]
Anton Krivtsov Institute for Problems in Mechanical Engineering (IPME RAS),Bolshoy pr. V.O., 61, St. Petersburg, Russia 199178, ; St. Petersburg State Poly-technical University (SPbSPU), Politekhnicheskaya 29, St. Petersburg, Russia195251, e-mail: [email protected]
Arthur Lebée Laboratoire Navier (ENPC-IFSTTAR-CNRS), Ecole des PontsParis Tech, Université Paris-Est, 6-8 avenue Blaise Pascal, 77420 Champs-sur-Marne, France, e-mail: [email protected]
Bernd Lenhof Chair of Applied Mechanics, Saarland University, Campus A4.2,D-66123 Saarbrücken, Germany, e-mail: [email protected]
Christian Liebold Lehrstuhl für Kontinuumsmechanik und Materialtheorie,Institut für Mechanik, TU Berlin, Einsteinufer 5, 10587 Berlin, Germany, e-mail:[email protected]
Gérard A. Maugin Institut Jean le Rond d’Alembert, CNRS UMR 7190,Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris Cedex 05,France, e-mail: [email protected]
Thomas M. Michelitsch Institut Jean le Rond d’Alembert, CNRS UMR 7190,Université Pierre et Marie Curie (Paris 6), 4 Place Jussieu, 75252 Paris Cedex 05,France, e-mail: [email protected]
Wolfgang H. Müller Lehrstuhl für Kontinuumsmechanik und Materialtheorie,Institut für Mechanik, TU Berlin, Einsteinufer 5, 10587 Berlin, Germany, e-mail:[email protected]
Bhasker Paliwal Unité Mixte Internationale Georgia Tech Lorraine-CNRS, UMICNRS 2958, 57070 Metz, France; George W. Woodruff School of MechanicalEngineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA,e-mail: [email protected]
Igor S. Pavlov Nizhny Novgorod Mechanical Engineering, Research Institute ofRussian Academy of Sciences, IMASH RAN, 85, Belinskogo st., Nizhny Nov-gorod, Russia 603024, e-mail: [email protected]
Wojciech Pietraszkiewicz Institute of Fluid-Flow Machinery, PASci, ul., Gen.J. Fiszera 14, 80-952 Gdansk, Poland, e-mail: [email protected]
Alexey V. Porubov Institute for Problems in Mechanical Engineering (IPMERAS), Bolshoy pr. V.O., 61, St. Petersburg, Russia 199178, e-mail:[email protected]
Contributors xi
Steffen Rothe Institute of Applied Mechanics, Clausthal University of Technol-ogy, Clausthal-Zellerfeld, Germany, e-mail: [email protected]
Karam Sab Laboratoire Navier (ENPC-IFSTTAR-CNRS), Ecole des Ponts ParisTech, Université Paris-Est, 6-8 avenue Blaise Pascal, 77420 Champs-sur-Marne,France, e-mail: [email protected]
Denis Sheydakov South Scientific Center of Russian Academy of Sciences,Chekhova Ave. 41, Rostov on Don, Russia 344006, e-mail: [email protected]
Holger Steeb Mechanics-Continuum Mechanics, Ruhr University Bochum,Universitätsstr. 150, 44780 Bochum, Germany, e-mail: [email protected]
Vincent Taupin Laboratoire d’Etude des Microstructures et de Mécanique desMatériaux, Université de Lorraine/CNRS, Ile du Saulcy, 57045 Metz Cedex,France, e-mail: [email protected]
Manas Upadhyay Woodruff School of Mechanical Engineering, Georgia Insti-tute of Technology/CNRS, 57070 Metz Cedex, France, e-mail: [email protected]
Elena Vilchevskaya Institute for Problems in Mechanical Engineering,Russian Academy of Sciences, Bolshoy pr. 61 V.O., St. Petersburg, Russia199178, e-mail: [email protected]
xii Contributors
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