Mechanics and Physics of Structured Media: Asymptotic and Integral Equations Methods of Leonid Filshtinsky.

528 Pages·2022·12.56 MB·English

by  Igor Andrianov, Simon Gluzman, Vladimir Mityushev

Mechanics and Physics of Structured
Media
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Mechanics and Physics
of Structured Media
Asymptotic and Integral Equations Methods of
Leonid Filshtinsky
Edited by
Igor Andrianov
RWTHAachenUniversity
Aachen,Germany
Simon Gluzman
Toronto,ON,Canada
Vladimir Mityushev
ResearchGroupMaterialica+
FacultyofComputerScienceandTelecommunications
CracowUniversityofTechnology
Kraków,Poland
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LeonidAnshelovichFilshtinsky
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Contents
Listofcontributors xvii
Acknowledgements xxi
1. L.A.Filshtinsky’scontributiontoAppliedMathematics
andMechanicsofSolids
VladimirMityushev,IgorAndrianov,andSimonGluzman
1.1 Introduction 1
1.1.1 Personalityandcareer 1
1.1.2 Lessonsofcollaboration(V.Mityushev) 3
1.1.3 Filshtinsky’scontributiontothetheoryofintegral
equations 5
1.2 Doubleperiodicarrayofcircularinclusions.Founders 7
1.2.1 Preliminaries 7
1.2.2 ContributionbyEisenstein 10
1.2.3 ContributionbyRayleigh 12
1.2.4 ContributionbyNatanzon 14
1.2.5 ContributionbyFilshtinsky 18
1.3 Synthesis.Retrospectiveviewfromtheyear2021 22
1.4 Filshtinsky’scontributiontothetheoryof
magneto-electro-elasticity 27
1.5 Filshtinsky’scontributiontothehomogenizationtheory 30
1.6 Filshtinsky’scontributiontothetheoryofshells 32
1.7 Decentandcreativeendeavor 33
Acknowledgment 34
References 34
2. Cracksintwo-dimensionalmagneto-electro-elastic
medium
DmytroNosov,LeonidFilshtinsky,andVladimirMityushev
2.1 Introduction 41
2.2 Boundary-valueproblemsforanunboundeddomain 42
2.3 Integralequationsforanunboundeddomain 45
2.4 Asymptoticsolutionattheendsofcracks 50
vii
viii Contents
2.5 Stressintensityfactors 52
AcrackinMMEplane 55
2.6 Numericalexample 56
2.7 Conclusion 60
References 60
3. Two-dimensionalequationsof
magneto-electro-elasticity
VladimirMityushev,DmytroNosov,andRyszardWojnar
3.1 Introduction 63
3.2 2Dequationsofmagneto-electro-elasticity 65
3.2.1 Linearequationsofmagneto-electro-elasticityand
potentials 66
3.2.2 Complexrepresentationoffieldvalues 67
3.3 Boundaryvalueproblem 70
3.4 Dielectrics 72
3.5 Circularhole 77
Numericalexample 80
3.6 MEEequationsandhomogenization 82
3.7 Homogenizationof2Dcompositesbydecompositionof
coupledfields 89
3.7.1 Straley-Milgromdecomposition 90
3.7.2 Rylkodecomposition 91
3.7.3 Example 94
3.8 Conclusion 95
References 96
4. Hashin-Shtrikmanassemblageofinhomogeneous
spheres
AndrejCherkaevandVladimirMityushev
4.1 Introduction 99
4.2 TheclassicHashin-Shtrikmanassemblage 100
4.3 HSA-typestructure 102
4.4 Conclusion 107
Acknowledgments 107
References 107
5. Inverseconductivityproblemforsphericalparticles
VladimirMityushev,NataliaRylko,ZhanatZhunussova,and
YeskendyrAshimov
5.1 Introduction 109
5.2 ModifiedDirichletproblem 110
5.2.1 Reductiontofunctionalequations 110
Contents ix
5.2.2 Explicitasymptoticformulas 114
5.3 Inverseboundaryvalueproblem 116
5.4 Discussionandconclusion 119
Acknowledgments 120
References 120
6. Compatibilityconditions:numberofindependent
equationsandboundaryconditions
IgorAndrianovandHeikoTopol
6.1 Introduction 123
6.2 GoverningrelationsandSouthwell’sparadox 126
6.3 Systemofninthorder 128
6.4 CounterexamplesproposedbyPobedryaandGeorgievskii 129
6.5 Variousformulationsofthelineartheoryofelasticityproblems
instresses 130
6.6 Otherapproximations 131
6.7 Generalization 132
6.8 Concludingremarks 133
Conflictofinterest 134
Acknowledgments 134
References 134
7. Criticalindexforconductivity,elasticity,
superconductivity.Resultsandmethods
SimonGluzman
7.1 Introduction 141
7.2 Criticalindexin2Dpercolation.Rootapproximants 144
7.2.1 Minimaldifferenceconditionaccordingtooriginal 145
7.2.2 Iteratedroots.Conditionsimposedonthresholds 145
7.2.3 Conditionsimposedonthecriticalindex 146
7.2.4 Conditionsimposedonamplitudes 147
7.2.5 Minimalderivative(sensitivity)condition 148
7.3 3DConductivityandelasticity 149
7.3.1 3Delasticity,orhigh-frequencyviscosity 150
7.4 Compressibilityfactorofhard-disksfluids 151
7.5 Sedimentationcoefficientofrigidspheres 152
7.6 Susceptibilityof2DIsingmodel 153
7.7 Susceptibilityofthree-dimensionalIsingmodel.Root
approximantsofhigherorders 155
7.7.1 Commentonunbiasedestimates.Iteratedroots 155
7.8 3DSuperconductivitycriticalindexofrandomcomposite 157
7.9 Effectiveconductivityofgraphene-typecomposites 159
7.10 Expansionfactorofthree-dimensionalpolymerchain 165
7.11 Concludingremarks 167