Table Of ContentAdvanced Structured Materials
Holm Altenbach
Samuel Forest
Anton Krivtsov Editors
Generalized
Continua as Models
for Materials
With Multi-scale Effects or Under
Multi-field Actions
Advanced Structured Materials
Volume 22
Series Editors
Andreas Öchsner
Lucas F. M. da Silva
Holm Altenbach
For furthervolumes:
http://www.springer.com/series/8611
Holm Altenbach Samuel Forest
•
Anton Krivtsov
Editors
Generalized Continua
as Models for Materials
With Multi-scale Effects or Under
Multi-field Actions
123
Editors
HolmAltenbach AntonKrivtsov
Lehrstuhl fürTechnische Mechanik Department of Theoretical Mechanics
Otto-von-Guericke-Universität Magdeburg St.Petersburg StatePolytechnical
Magdeburg University
Germany St.Petersburg
Russia
Samuel Forest
Centre des Materiaux CNRSUMR 7633
Mines ParisTech
Evry Cedex
France
ISSN 1869-8433 ISSN 1869-8441 (electronic)
ISBN 978-3-642-36393-1 ISBN 978-3-642-36394-8 (eBook)
DOI 10.1007/978-3-642-36394-8
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013934985
(cid:2)Springer-VerlagBerlinHeidelberg2013
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Preface
TheMechanics ofGeneralizedContinua isanestablishedresearchtopic since the
end of the 1950s–early 1960s of the last century. The starting point of this
developmentwasthemonographoftheCosseratbrothers,publishedin19091,and
some previous works of such famous scientists like Lord Kelvin, Duhem, Helm-
holtz among others. All these contributions were focussed on the fact that in a
continuum one has to define translations and rotations independently (or in other
words,onehastoestablishforceandmomentactionsindependentlyasitwasdone
by Jakob Bernoulli and Euler). At the same time the continuum was not modeled
as an infinite number of continuously distributed points with properties like the
mass, but as an infinite number of continuously distributed infinitesimal small
bodies with properties like the mass.
Thereasonfortherevivalinthemidof1950softhelastcenturywasthatsome
effects of the mechanical behavior of solids and fluids could not be explained by
theavailableclassicalcontinuummodels.Examplesofthisaretheturbulenceofa
fluidorthebehaviorofsolidswithasignificantandverycomplexmicrostructure.
Since the suggested new models fulfill all requirements from Continuum Ther-
momechanics(thebalancelawswereformulatedandthegeneralrepresentationof
theconstitutiveequationsweregiven)thescientificcommunitywassatisfiedfora
while. At the same time real applicative developments were missed.
Indeed, for practical applications the developed models were not useful. The
reason for thiswas the gapbetween the formulatedconstitutive equationsand the
possibilitiestoidentifythematerialparameters.Asisoftenthecaseonehadmuch
more parameters compared to classical models, but no facilities to measure all
properties. In addition, computational progress and available machines in these
times were limited.
During the last ten years the situation has drastically changed. More and more
researches emerged, being kindled by the partly forgotten models. Now one has
available much more computational possibilities and very complex problems can
be simulated numerically. In addition, with the increased attention paid to a large
1 E.etF.Cosserat:Cosserat,F.:ThéoriedesCorpsDéformables,HermannEditeurs,Paris,1909
(Reprint,Gabay,Paris,2008).
v
vi Preface
number of materials with complex microstructure and a deeper understanding of
the meaning of the material parameters (scale effects) the identification becomes
much more well founded. Thus we have contributions describing the micro- and
macro-behavior,newexistenceanduniquenesstheorems,theformulationofmulti-
scaleproblems,etc.,andnowitistimetoponderagain2thestateofmatterandto
discuss 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 Seminar
Generalized Continua as Models for Materials With Multi-scale Effects or Under
MultifieldActions,whichheldattheLeucorea(LutherstadtWittenberg,Germany)
fromSeptember26upto30,2012.3SpecialthankstoAndreasKutschkewhotook
all duties connected with realization of the Seminar. In addition, we kindly
acknowledgeDr.ChristophBaumannandBenjaminFeuchter(SpringerPublisher)
for the support of the book project. Last but not least it should be mentioned that
the Seminar was sponsored by grants of the French National Center for Scientific
Research (CNRS), the German Research Foundation (DFG) AL341/41-1, and the
Russian Foundation for Basic Research 12-01-91260RFG.
Magdeburg, December 2012 Holm Altenbach
Paris Samuel Forest
St. Petersburg Anton Krivtsov
2 Thereweretwoproceedingswithinthelastyearswhichshouldbementionedhere:GérardA.
Maugin, Andrei V. Metrikine (Eds) Mechanics of Generalized Continua - One Hundred Years
After the Cosserats, Springer, 2010 (Advances in Mechanics and Mathematics, Vol. 21) and
HolmAltenbach,GérardMaugin,VladimirErofeev(Eds)MechanicsofGeneralizedContinua,
Springer,2011(AdvancedStructuredMaterials,Vol.7).
3 The First Trilateral French–German–Russian Seminar held also in Lutherstadt Wittenberg
(Germany)August9–11,2010.
Contents
Shells and Plates with Surface Effects . . . . . . . . . . . . . . . . . . . . . . . . 1
Holm Altenbach and Victor A. Eremeyev
Geometrical Picture of Third-Order Tensors . . . . . . . . . . . . . . . . . . . 17
Nicolas Auffray
Continuum Modelling of Shear-Coupled Grain
Boundary Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Stéphane Berbenni, Bhasker Paliwal and Mohammed Cherkaoui
Buckling of Nonlinearly Elastic Plates with Microstructure . . . . . . . . 65
Svyatoslav Derezin
Material Symmetry Group and Consistently Reduced
Constitutive Equations of the Elastic Cosserat Continuum . . . . . . . . . 77
Victor A. Eremeyev and Wojciech Pietraszkiewicz
Nonlinear Localized Strain Waves in a 2D Medium
with Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Vladimir I. Erofeev, Vladimir V. Kazhaev and Igor S. Pavlov
Cosserat Anisotropic Models of Trabecular Bone from
the Homogenization of the Trabecular Structure:
2D and 3D Frameworks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Ibrahim Goda, Mohamed Assidi and Jean-Francois Ganghoffer
Electro-Thermo-Elastic Simulation of Graphite Tools Used
in SPS Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Stefan Hartmann, Steffen Rothe and Nachum Frage
The Use of Moment Theory to Describe the Piezoelectric Effect
in Polar and Non-Polar Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Elena A. Ivanova and Yaroslav E. Kolpakov
vii
viii Contents
Description of Thermal and Micro-Structural Processes
in Generalized Continua: Zhilin’s Method
and its Modifications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Elena Ivanova and Elena Vilchevskaya
Hierarchical Architecture and Modeling of Bio-Inspired
Mechanically Adaptive Polymer Nanocomposites . . . . . . . . . . . . . . . . 199
Rasa Kazakevi(cid:2)ciu¯te_-Makovska and Holger Steeb
Justification of the Bending-Gradient Theory Through
Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Arthur Lebée and Karam Sab
Macroscopic Modeling of Size Effects in Foams Using
an Order-Parameter Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Bernd Lenhof, Alexander Geringer and Stefan Diebels
Measuring Material Coefficients of Higher Gradient Elasticity
by Using AFM Techniques and Raman-Spectroscopy. . . . . . . . . . . . . 255
Christian Liebold and Wolfgang H. Müller
On the Electrostatic Fields in Dielectric Continua
with Self-Similar Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Thomas M. Michelitsch and Gérard A. Maugin
Nonlinear Generalizations of the Born-Huang Model
and Their Continuum Limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Alexey V. Porubov, Eron L. Aero and Boris R. Andrievsky
Buckling of Inhomogeneous Circular Plate of Micropolar Material. . . 291
Denis N. Sheydakov
A Theory of Disclination and Dislocation Fields
for Grain Boundary Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
V. Taupin, L. Capolungo, C. Fressengeas, A. Das and M. Upadhyay
Material Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Pavel A. Zhilin, Holm Altenbach, Elena A. Ivanova
and Anton Krivtsov
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, Bolshoy
pr. 61, V.O., St. Petersburg, Russia 199178, e-mail: [email protected]
Mohamed Assidi Centre de Recherche Public Henri Tudor, 29, Avenue John
Kennedy, L-1885 Luxembourg, Luxembourg, e-mail: [email protected]
Nicolas Auffray Laboratoire Modélisation et Simulation MultiEchelle LMSME,
Université Paris-Est,MSME UMR 8208CNRS,5bdDescartes, 77454Marne-la-
Vallée, France, e-mail: [email protected]
Stéphane Berbenni Laboratoire d’Etudes des Microstructures et de Mécanique
des Matériaux, LEM3, University of Lorraine, UMR CNRS 7239, 57045 Metz,
France; Unité Mixte Internationale Georgia Tech Lorraine-CNRS, UMI CNRS
2958, 57070 Metz, France, e-mail: [email protected]
Laurent Capolungo Woodruff School of Mechanical Engineering, Georgia
Institute of Technology/CNRS, 57070 Metz Cedex, France, e-mail: laurent.capo-
[email protected]
Mohammed Cherkaoui Unité Mixte Internationale Georgia Tech Lorraine-
CNRS, UMI CNRS 2958, 57070 Metz, France; George W. Woodruff School of
Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-
0405, USA, e-mail: [email protected]
A.Das Laboratoire d’Etudedes Microstructures etdeMécaniquedes Matériaux,
Université de Lorraine/CNRS, Ile du Saulcy, 57045 Metz Cedex, France, e-mail:
[email protected]
ix
x Contributors
Svyatoslav V. Derezin Faculty of Mathematics, Mechanics and Computer
Science, Southern Federal University, Milchakova 8a, Rostov on Don, Russia
344090, 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 RASci
and South Federal University, Milchakova St. 8a, Rostov on Don, Russia
344090, e-mail: [email protected]; [email protected]
Vladimir I. Erofeev Nizhny Novgorod Mechanical Engineering, Research
Institute 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 of
the Negev, Beer-Sheva, Israel, e-mail: [email protected]
Claude Fressengeas 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]
Jean-François Ganghoffer LEMTA, Université de Lorraine, 2, Avenue de la
Forêt de Haye, 54504 Vandoeuvrelès-Nancy Cedex, France, e-mail: jean-franc-
[email protected]
Alexander Geringer Chair of Applied Mechanics, Saarland University, Campus
A4.2, D-66123 Saarbrücken, Germany, e-mail: [email protected]
IbrahimGoda LEMTA,UniversitédeLorraine,2,Avenuedela ForêtdeHaye,
54504 Vandoeuvrelès-Nancy Cedex, France, e-mail: [email protected]
nancy.fr
Stefan Hartmann InstituteofAppliedMechanics,ClausthalUniversityofTech-
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 in
MechanicalEngineering,RussianAcademyofSciences,Bolshoypr.V.O.,61,St.
Petersburg, Russia 199178, e-mail: [email protected]
Rasa Kazakevicˇiu¯te_-Makovska Mechanics-Continuum Mechanics, Ruhr Uni-
versity Bochum, Universitätsstr. 150, 44780 Bochum, Germany, e-mail: Rasa.
[email protected]
