Higher Gradient Materials and Related Generalized Continua

246 Pages·2019·4.078 MB·English

by  Holm Altenbach, Wolfgang H. Müller, Bilen Emek Abali

Advanced Structured Materials
Holm Altenbach
Wolfgang H. Müller
Bilen Emek Abali Editors
Higher Gradient
Materials
and Related
Generalized
Continua
Advanced Structured Materials
Volume 120
Series Editors
Andreas Öchsner, Faculty of Mechanical Engineering, Esslingen University of
Applied Sciences, Esslingen, Germany
Lucas F. M. da Silva, Department of Mechanical Engineering, Faculty of
Engineering, University of Porto, Porto, Portugal
Holm Altenbach, Faculty of Mechanical Engineering,
Otto-von-Guericke-UniversitätMagdeburg,Magdeburg,Sachsen-Anhalt,Germany
Common engineering materials reach in many applications their limits and new
developments are required to fulfil increasing demands on engineering materials.
The performance ofmaterials can beincreasedby combiningdifferent materials to
achieve better properties than a single constituent or by shaping the material or
constituents in a specific structure. The interaction between material and structure
mayariseondifferentlengthscales,suchasmicro-,meso-ormacroscale,andoffers
possible applications in quite diverse fields.
Thisbookseriesaddressesthefundamentalrelationshipbetweenmaterialsandtheir
structure on the overall properties (e.g. mechanical, thermal, chemical or magnetic
etc) and applications.
The topics of Advanced Structured Materials include but are not limited to
(cid:129) classical fibre-reinforced composites (e.g. glass, carbon or Aramid reinforced
plastics)
(cid:129) metal matrix composites (MMCs)
(cid:129) micro porous composites
(cid:129) micro channel materials
(cid:129) multilayered materials
(cid:129) cellular materials (e.g., metallic or polymer foams, sponges, hollow sphere
structures)
(cid:129) porous materials
(cid:129) truss structures
(cid:129) nanocomposite materials
(cid:129) biomaterials
(cid:129) nanoporous metals
(cid:129) concrete
(cid:129) coated materials
(cid:129) smart materials
Advanced Structured Materials is indexed in Google Scholar and Scopus.
More information about this series at http://www.springer.com/series/8611
ü
Holm Altenbach Wolfgang H. M ller
(cid:129) (cid:129)
Bilen Emek Abali
Editors
Higher Gradient Materials
and Related Generalized
Continua
123
Editors
HolmAltenbach WolfgangH.Müller
Institut für Mechanik Institut für Mechanik
Otto-von-Guericke-UniversitätMagdeburg Technische UniversitätBerlin
Magdeburg, Germany Berlin, Germany
Bilen EmekAbali
Institut für Mechanik
Technische UniversitätBerlin
Berlin, Germany
ISSN 1869-8433 ISSN 1869-8441 (electronic)
AdvancedStructured Materials
ISBN978-3-030-30405-8 ISBN978-3-030-30406-5 (eBook)
https://doi.org/10.1007/978-3-030-30406-5
©SpringerNatureSwitzerlandAG2019
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Preface
The idea for this volume of the Advanced Structured Materials Series was born
duringtwoseminars,namely“NewDevelopmentsinMicropolarTheory,”and“Ad-
vancedSeminar:GeneralizedContinuainEngineering—Theory,Experiments,and
Applications,”whichwereheldonNovember6-7,2017andSeptember3-5,2018,
respectively, both at the Technische Universität Berlin. The first seminar was or-
ganized by Wolfgang H. Müller (Berlin) and Elena Vilchevskaya (St. Petersburg)
and the second by Wolfgang H. Müller (Berlin) & Holm Altenbach (Magdeburg)
andattendedbymanyscientistsfromGermany,Russia,Italy,USA,Sweden,Geor-
gia, France, Estonia, and Finland. The organizers were assisted by B. Emek Abali
(Berlin).
GeneralizedContinuahaverecentlyseenaformidablerenaissance:theCosserat
brothersgaveafirstsummaryin1909.Theirideasstayeddormantforawhileand
werepickedupafterWorldWarIIbyEricksen&Truesdellresultinginacontinu-
ousstreamoftheoreticalpapersuntiltoday.Mostrecently,thetheorywascomple-
mentedbyapplicationsandembeddedinexperimentsfocusingonhowtodetermine
thevariousnewmaterialparametersrequiredformakingthetheoryapplicable.
DuringthelastdecadevariouscolloquiaheldinParis(2009)1,Wittenberg(2010,
2012)2,3,andMagdeburg(2015)4,aswellastheCISMCourse“GeneralizedCon-
tinua—from the Theory to the Engineering Applications” (Udine, 2011)5 helped
to promote this type of research. During this new “Advanced Seminar,” attention
1 Maugin,G.A.,Metrikine,A.V.(eds)MechanicsofGeneralizedContinua:OneHundredYears
AftertheCosserats.Springer,NewYork,2010
2Altenbach,H.,Maugin,G.A.,Erofeev,V.(eds)MechanicsofGeneralizedContinua.Advanced
StructuredMaterials,vol.7.Springer,Berlin,Heidelberg,2011
3Altenbach,H.,Forest,S.,Krivtsov,A.(eds)GeneralizedContinuaasModelsforMaterials.Ad-
vancedStructuredMaterials,vol.22.Springer,Berlin,Heidelberg
4 Altenbach, H., Forest, S. (eds) Generalized Continua as Models for Classical and Advanced
Materials.AdvancedStructuredMaterials,vol.42.Springer,Cham
5AltenbachH.,EremeyevV.A.(eds)GeneralizedContinuafromtheTheorytoEngineeringAp-
plications.CISMInternationalCentreforMechanicalSciences(CoursesandLectures),vol541.
Springer,Vienna,2013
v
vi Preface
waspaidonthemostrecentresearchitems,e.g.,newgeneralizedmodels,materials
withsignificantmicrostructure,multi-fieldloadingsoridentificationofconstitutive
equations.Finallyyetimportantly,acomparisonwithdiscretemodelingapproaches
andexperimentswasdiscussed.
Aseditors,weintendtothankallauthorsfortheircrucialcontributionsaswell
asallreviewersfortheirinvaluabletimeandeffort.WedelightedlyacknowledgeDr.
ChristophBaumann(SpringerPublisher)forinitiatingthebookproject.Inaddition,
wehavetothankDr.MayraCastro(SeniorEditorAppliedSciences;MaterialsSci-
ence; Materials Engineering; Nanotechnology and Nanomedicine) and Mr. Ashok
Arumairaj (Production Administrator) giving the final support. Last but not least,
the first editor has to acknowledge the Fundacja na rzecz Nauki Polskiej (Funda-
tionforPolishScience)allowingtofinalizethisbookatthePolitechnikaLubelska
(host: Prof. dr.hab.inz˙. Tomasz Sadowski, dr.h.c.) with the help of the Alexander
vonHumboldtPolishHonoraryResearchFelloship.
Berlin,Magdeburg HolmAltenbach
July2019 WolfgangH.Müller
BilenEmekAbali
Contents
1 AComputationalApproachforDeterminationofParametersin
GeneralizedMechanics....................................... 1
BilenEmekAbali,HuaYang,andPanayiotisPapadopoulos
1.1 Introduction.............................................. 1
1.2 HomogenizationBetweenMicro-andMacroscales ............. 4
1.3 DeterminationofParameters................................ 6
1.4 ComputationofOneSpecificCase........................... 8
1.5 AlgorithmforAllDeformationCases ........................ 9
1.6 Examples................................................ 10
1.7 Conclusion............................................... 13
References..................................................... 14
2 ExtensibleBeamModelsinLargeDeformationUnderDistributed
Loading:aNumericalStudyonMultiplicityofSolutions .......... 19
Francescodell’Isola,AlessandroDellaCorte,AntonioBattista,and
EmilioBarchiesi
2.1 Introduction.............................................. 20
2.2 TheModel............................................... 21
2.2.1 KinematicsandDeformationEnergy ................. 21
2.2.2 LagrangeMultipliersMethod ....................... 22
2.3 NumericalSimulations..................................... 23
2.3.1 NumericalMethods................................ 23
2.3.2 TheNumberofEquilibriumConfigurationswhenthe
LoadIncreases.................................... 25
2.3.3 EquilibriumConfigurations ......................... 26
2.3.4 ParametricStudyontheExtensionalStiffness.......... 29
2.4 Conclusions.............................................. 32
Appendix ...................................................... 36
References..................................................... 37
vii
viii Contents
3 OntheCharacterizationoftheNonlinearReducedMicromorphic
ContinuumwiththeLocalMaterialSymmetryGroup............. 43
VictorA.Eremeyev
3.1 Introduction.............................................. 43
3.2 MicromorphicContinua.................................... 44
3.3 LocalMaterialSymmetryGroup ............................ 46
3.4 RelaxedMicromorphicMediumasaMicromorphicSubfluid .... 47
3.5 Conclusions.............................................. 52
References..................................................... 52
4 Structural Modeling of Nonlinear Localized Strain Waves in
GeneralizedContinua........................................ 55
VladimirI.Erofeev,AnnaV.Leontyeva,AlexeyO.Malkhanov,and
IgorS.Pavlov
4.1 Introduction.............................................. 55
4.2 PrinciplesofStructuralModeling............................ 56
4.3 One-dimensional Model of a Nonlinear Gradient-elastic
Continuum. .............................................. 60
4.4 NonlinearStrainWaves .................................... 62
4.5 Conclusions.............................................. 66
References..................................................... 67
5 ADiffusionModelforStimulusPropagationinRemodelingBone
Tissues .................................................... 69
IvanGiorgio,UgoAndreaus,FarisAlzahrani,TasawarHayat,and
TomaszLekszycki
5.1 Introduction.............................................. 70
5.2 AcceptedAssumptionsandMainVariables.................... 72
5.3 PoromechanicalFormulation................................ 75
5.4 EvolutionaryEquationsforBoneRemodeling ................. 77
5.5 StimulusModelingWithoutTimeDelayandDiffusionPhenomena 79
5.6 AnImprovedVersionofStimulusModeling................... 80
5.7 NumericalSimulations..................................... 82
5.7.1 APhysiologicalCase .............................. 83
5.7.1.1 UniformTensionTest..................... 83
5.7.1.2 Non-uniformTensionTest ................. 85
5.7.2 SimulationofaHealingProcess ..................... 86
5.8 Conclusions.............................................. 87
References..................................................... 89
6 AC1IncompatibleModeElementFormulationforStrainGradient
Elasticity .................................................. 95
RainerGlüge
6.1 Introduction.............................................. 95
6.1.1 Outline .......................................... 96
6.1.2 Notation ......................................... 96
Contents ix
6.2 From Local Balance of Momentum to Minimization of the
ElasticPotential .......................................... 97
6.3 CiarletsElasticEnergy..................................... 99
6.3.1 StrainGradientExtension........................... 99
6.3.2 Stress-strainRelations..............................100
6.3.3 Materialparameters................................101
6.4 ElementFormulation ......................................101
6.4.1 Overview ........................................101
6.4.2 IncompatibleModeElementFormulation .............102
6.4.3 NumericalIntegration..............................105
6.4.4 TestingoftheImplementation .......................106
6.5 SingleForceIndentationSimulations.........................107
6.5.1 TheBoundaryConditions ..........................107
6.5.2 Meshing .........................................108
6.5.3 Results ..........................................108
6.5.3.1 TransitionBehaviorasα=0...∞ .........108
6.5.3.2 Singularity ..............................108
6.5.4 ACommentonPseudorigidBodies ..................110
6.6 SharpCornerSimulations ..................................111
6.6.1 TheBoundaryConditions ..........................111
6.6.2 Meshing .........................................112
6.6.3 Results ..........................................112
6.6.3.1 TransitionBehaviorasα→∞ .............112
6.6.3.2 ConvergenceonMeshRefinement ..........114
6.7 ImprovementoftheElementFormulation .....................116
6.8 ConvergenceStudy........................................116
6.9 Conclusion...............................................118
References.....................................................119
7 AComparisonofBoundaryElementMethodandFiniteElement
MethodDynamicSolutionsforPoroelasticColumn ............... 121
LeonidA.Igumnov,AleksandrA.Ipatov,AndreyN.Petrov,Svetlana
Yu.Litvinchuk,AronPfaff,andVictorA.Eremeyev
7.1 Introduction..............................................121
7.2 MathematicalModel.......................................123
7.2.1 us–p-formulationinLaplaceDomain .................124
i
7.2.2 us–p-formulationinTimeDomain ...................125
i
7.3 BoundaryIntegralEquationandBoundaryElementMethodology.125
7.4 LaplaceTransformInversion................................127
7.5 NumericalExample .......................................127
7.6 Conclusion...............................................130
Appendix ......................................................131
References.....................................................133