Applied Analysis of Composite Media: Analytical and Computational Approaches

369 Pages·2020·4.757 MB·English

by  Drygaś, Piotr, Gluzman, Simon, Mityushev, Vladimir, Nawalaniec, Wojciech

Applied Analysis of Composite
Media

Woodhead Publishing Series in Composites
Science and Engineering
Applied Analysis of
Composite Media
Analytical and Computational Results
for Materials Scientists and Engineers
Piotr Drygas´
Department of Mathematical Analysis
University of Rzeszow
Rzeszow, Poland
Simon Gluzman
Toronto, ON, Canada
Vladimir Mityushev
Institute of Computer Science
Pedagogical University of Cracow
Krakow, Poland
Wojciech Nawalaniec
Institute of Computer Science
Pedagogical University of Cracow
Krakow, Poland
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Contents
Biography xi
Preface xiii
Acknowledgments xvii
Nomenclature xix
1 Introductiontocomputationalmethodsandtheoryofcomposites 1
1.1 Traditionalapproaches 1
1.1.1 Self-consistentmethods 2
1.1.2 Seriesmethod 4
1.1.3 Gettingfromtraditionalapproachtoneoclassicalillustratedby
example 4
1.2 Mathematicalfoundationsofconstructivehomogenization 7
1.2.1 Differentapproachestoheterogeneityinmaterialsciencesand
instatisticalmechanics 7
1.2.2 Randomgeometryofcomposites 8
1.2.3 Themethodofstructuralsums 10
1.2.4 Randomhomogenizationandergodicity 13
1.3 Asymptoticmethods.Criticalindex 17
1.3.1 Methodofself-similarrenormalization 20
1.3.2 Criticalindex.Directmethods 27
1.3.3 Criticalindicesfromself-similarrootapproximants.Examples 37
1.3.4 Factorapproximantsandcriticalindex.Example 47
1.3.5 Additiveself-similarapproximantsandDLogadditive
recursiveapproximants 51
2 Algorithms,computationsandstructuralinformation 57
2.1 Computinginmoderncomputationalsciences 57
2.2 Symbolicrepresentationsofcoefficientsfortheeffective
conductivity 57
2.2.1 Algorithm1:ExpressA intermsofstructuralsums 58
q
2.2.2 Algorithm2:ReductionofdependentstructuralsumsinA 63
q
2.3 Algorithmsforcomputingstructuralsums 67
2.3.1 Simplifiedformulasfore ande 67
pp 2222
2.3.2 Calculationofmultidimensionaldiscretechainconvolutionsof
functions 68
2.3.3 Efficientcomputationofstructuralsums 70
2.3.4 Computingstructuralsumsofagivenorder 74
2.3.5 Convolutionsoffunctionsincaseofelasticcomposites 78
viii Contents
2.4 Symboliccalculationsforsystemsoffunctionalequations 80
2.4.1 Indefinitesymbolicsums 80
2.4.2 Symbolicrepresentationofsuccessiveapproximations 81
2.4.3 Applicationtothe3Deffectiveconductivity 81
2.5 Structuralinformationandstructuralfeaturesvector 82
2.5.1 Patternrecognitionandfeaturevectors 83
2.5.2 Fromcomputationalsciencetoinformationprocessing 83
2.5.3 Structuralinformation 84
2.5.4 Constructionofstructuralfeaturesvector 86
2.5.5 Structuralinformationinthree-dimensionalspace 88
2.6 Classificationandanalysisofrandomstructures 89
2.6.1 Circularinclusions 90
2.6.2 Non-circularinclusions 94
2.6.3 Irregularityofrandomstructuresanddatavisualization 95
2.7 ThePythonpackageforcomputingstructuralsumsandtheeffective
conductivity 101
3 Elasticityofhexagonalarrayofcylinders 103
3.1 Methodoffunctionalequationsforelasticproblems 103
3.1.1 Localfield.Analyticalapproximateformulas 110
3.1.2 Clusterapproachtoelasticityproblem 113
3.1.3 Rayleigh’sandMaxwell’sapproaches 118
3.2 Generalformulafortheeffectiveshearandbulkmoduli 120
3.3 Effectiveshearmodulusforperfectlyrigidinclusionsembeddedinto
matrix 123
3.3.1 ModifiedPadéapproximations 124
3.3.2 Additiveapproximants 125
3.3.3 Perfectlyrigidinclusions.DLogadditiveapproximants 127
3.3.4 Effectiveviscosityof2Dsuspension 130
3.4 Effectiveshearmodulusforsoftfibers 131
(cid:2)
3.4.1 Shearmodulusincriticalregimeforarbitraryν 134
3.4.2 DLogadditiveapproximantsforsoftfibers 138
3.5 Methodofcontrastexpansionforelasticincompressiblecomposites 140
3.5.1 MethodofSchwarz 140
3.5.2 Methodoffunctionalequations 143
3.5.3 Methodofsuccessiveapproximationsincontrastparameter 144
3.5.4 Secondorderapproximation 146
3.5.5 Thirdorderapproximation 148
3.5.6 Numericalsimulations 153
3.6 Practiceofasymptoticmethods 156
3.6.1 Glass-resincomposite 157
3.6.2 ComparisonwithnumericalFEMresults 163
3.6.3 Boron-aluminumcomposite 165
3.6.4 Boron-epoxycomposite 167
Contents ix
4 Randomelasticcompositeswithcircularinclusions 171
4.1 Introduction 171
4.2 Methodoffunctionalequationsforlocalfields 171
4.3 Averagedfieldsincompositesandeffectiveshearmodulus 176
4.4 Identicalcircularinclusions 180
4.5 Numericalexamples 183
4.5.1 Symmetriclocationofinclusionswithequalradii 183
4.5.2 Randomlocationofinclusionswithequalradii 185
4.5.3 Symmetriclocationofinclusionswithdifferentradii 187
4.5.4 Randomlocationofinclusionswithdifferentradii 189
4.6 Criticalindexfortheviscosityof2Drandomsuspension 190
4.7 Criticalbehaviorofrandomholes 192
5 Effectiveconductivityoffibrouscompositeswithcracksoninterface 195
5.1 Deterministicandrandomcracksoninterface 195
5.2 Boundaryvalueproblem 197
5.3 Maxwell’sapproach 201
6 Effectiveconductivityofarandomsuspensionofhighlyconducting
sphericalparticles 205
6.1 Introduction 205
6.2 Generalformulaforhighlyconductingspheres 207
6.2.1 Statementoftheproblem 207
6.2.2 Highconductinginclusions 209
6.3 ModifiedDirichletproblem 211
6.3.1 Finitenumberofinclusions 211
6.3.2 Computationofundeterminedconstants 216
6.4 Averagedconductivity 218
6.5 Effectiveconductivity 220
6.6 Numericalresultsforrandomcomposite 225
6.7 Foams 230
6.8 Slits,platelets 232
7 Permeabilityofporousmedia 237
7.1 Modelsofcriticalpermeability 237
7.2 Permeabilityofregulararrays 239
7.2.1 Transversepermeability.Squarearray 239
7.2.2 Transversepermeability.Hexagonalarray 241
7.2.3 Longitudinalpermeability.Squarearray 242
7.3 3Dperiodicarraysofsphericalobstacles 244
7.3.1 BCCandSClatticesofsphericalobstacles 244
7.3.2 FormulaforFCClattice 247
7.4 Permeabilityinwavy-walledchannels 249
7.4.1 Symmetricsinusoidaltwo-dimensionalchannel.Breakdownof
lubricationapproximation 252
x Contents
7.4.2 Symmetricsinusoidaltwo-dimensionalchannel.Breakdown
continued 254
7.4.3 Parallelsinusoidaltwo-dimensionalchannel.Novelcritical
index 257
7.4.4 Symmetricsinusoidalthree-dimensionalchannel.Two-fluid
model 260
8 Simplefluids,suspensionsandselectedrandomsystems 265
8.1 Compressibilityfactorofhard-sphereandhard-diskfluids.Index
function 265
8.2 “Sticky”rodsanddisks.MappingtoJanusswimmers 278
8.3 3Delasticity,orhigh-frequencyviscosity.Criticalindex 284
8.3.1 Modificationwithiteratedroots.Conditionimposedon
thresholds 286
8.3.2 Conditionimposedonthecriticalindex 287
8.3.3 Conditionsimposedontheamplitude 288
8.3.4 Minimalderivativecondition 289
8.4 Diffusioncoefficients 292
8.5 Non-localdiffusion 296
8.6 Sedimentation.Particlemobility 303
8.7 Polymercoilin2Dand3D 307
8.8 Factorapproximantsandcriticalindexfortheviscosityofa3D
suspension 311
Appendix 317
A.1 Equationsofviscousflowandelasticity 317
A.2 Eisenstein’sseries 319
A.3 Eisenstein–Natanzon–Filshtinskyseries 321
A.4 Clusterapproachanditslimitations 323
A.5 Mathematicalpseudo-language,transformationrulesequencesand
rule-basedprogramming 329
A.6 Implementationsinmathematica 329
A.6.1 ImplementationofcomputationsofcoefficientsA 329
q
A.6.2 Implementationofmirrortermsreduction 330
References 333
Index 349