Micromechanics with Mathematica

290 Pages·2016·19.355 MB·English

by  Nomura, Seiichi

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MICROMECHANICS
WITH MATHEMATICA
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MICROMECHANICS
WITH MATHEMATICA
SeiichiNomura
DepartmentofMechanicalandAerospaceEngineering
TheUniversityofTexasatArlington
Arlington,TX
USA
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Thiseditionfirstpublished2016
©2016JohnWiley&SonsLtd
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ISBN:9781119945031
Setin10/12pt,TimesLTStdbySPiGlobal,Chennai,India.
1 2016
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Contents
Preface ix
AbouttheCompanionWebsite xi
1 CoordinateTransformationandTensors 1
1.1 IndexNotation 1
1.1.1 SomeExamplesofIndexNotationin3-D 3
1.1.2 MathematicaImplementation 3
1.1.3 KroneckerDelta 6
1.1.4 PermutationSymbols 9
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1.1.5 ProductofMatrices 10
1.2 CoordinateTransformations(CartesianTensors) 11
1.3 DefinitionofTensors 13
1.3.1 TensorofRank0(Scalar) 13
1.3.2 TensorofRank1(Vector) 14
1.3.3 TensorofRank2 15
1.3.4 TensorofRank3 17
1.3.5 TensorofRank4 17
1.3.6 Differentiation 19
1.3.7 DifferentiationofCartesianTensors 20
1.4 InvarianceofTensorEquations 21
1.5 QuotientRule 22
1.6 Exercises 23
References 24
2 FieldEquations 25
2.1 ConceptofStress 25
2.1.1 PropertiesofStress 29
2.1.2 (Stress)BoundaryConditions 30
2.1.3 PrincipalStresses 31
2.1.4 StressDeviator 35
2.1.5 Mohr’sCircle 38
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vi Contents
2.2 Strain 40
2.2.1 ShearDeformation 47
2.3 CompatibilityCondition 49
2.4 ConstitutiveRelation,Isotropy,Anisotropy 50
2.4.1 Isotropy 52
2.4.2 ElasticModulus 54
2.4.3 Orthotropy 56
2.4.4 2-DOrthotropicMaterials 57
2.4.5 TransverseIsotropy 57
2.5 ConstitutiveRelationforFluids 58
2.5.1 ThermalEffect 58
2.6 DerivationofFieldEquations 59
2.6.1 DivergenceTheorem(GaussTheorem) 59
2.6.2 MaterialDerivative 60
2.6.3 EquationofContinuity 62
2.6.4 EquationofMotion 62
2.6.5 EquationofEnergy 63
2.6.6 IsotropicSolids 65
2.6.7 IsotropicFluids 65
2.6.8 ThermalEffects 66
2.7 GeneralCoordinateSystem 66
2.7.1 IntroductiontoTensorAnalysis 66
2.7.2 DefinitionofTensorsinCurvilinearSystems 68
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2.7.3 MetricTensor10,g 69
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2.7.4 CovariantDerivatives 70
2.7.5 Examples 73
2.7.6 VectorAnalysis 75
2.8 Exercises 77
References 80
3 InclusionsinInfiniteMedia 81
3.1 Eshelby’sSolutionforanEllipsoidalInclusionProblem 82
3.1.1 EigenstrainProblem 85
3.1.2 EshelbyTensorsforanEllipsoidalInclusion 87
3.1.3 Inhomogeneity(Inclusion)Problem 95
3.2 MultilayeredInclusions 104
3.2.1 Background 104
3.2.2 ImplementationofIndexManipulationinMathematica 105
3.2.3 GeneralFormulation 108
3.2.4 ExactSolutionforTwo-PhaseMaterials 116
3.2.5 ExactSolutionforThree-PhaseMaterials 123
3.2.6 ExactSolutionforFour-PhaseMaterials 132
3.2.7 ExactSolutionfor2-DMultiphaseMaterials 137
3.3 ThermalStress 137
3.3.1 ThermalStressDuetoHeatSource 138
3.3.2 ThermalStressDuetoHeatFlow 146
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3.4 Airy’sStressFunctionApproach 155
3.4.1 Airy’sStressFunction 156
3.4.2 MathematicaProgrammingofComplexVariables 161
3.4.3 MultiphaseInclusionProblemsUsingAiry’sStressFunction 163
3.5 EffectiveProperties 172
3.5.1 UpperandLowerBoundsofEffectiveProperties 173
3.5.2 Self-ConsistentApproximation 175
3.5.3 SourceCodeformicromech.m 178
3.6 Exercises 188
References 189
4 InclusionsinFiniteMatrix 191
4.1 GeneralApproachesforNumericallySolvingBoundaryValueProblems 192
4.1.1 MethodofWeightedResiduals 192
4.1.2 Rayleigh–RitzMethod 203
4.1.3 Sturm–LiouvilleSystem 205
4.2 Steady-StateHeatConductionEquations 213
4.2.1 DerivationofPermissibleFunctions 213
4.2.2 FindingTemperatureFieldUsingPermissibleFunctions 227
4.3 ElasticFieldswithBoundedBoundaries 232
4.4 NumericalExamples 238
4.4.1 HomogeneousMedium 238
4.4.2 SingleInclusion 240
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4.5 Exercises 251
References 252
AppendixA IntroductiontoMathematica 253
A.1 EssentialCommands/Statements 255
A.2 Equations 256
A.3 Differentiation/Integration 260
A.4 Matrices/Vectors/Tensors 260
A.5 Functions 262
A.6 Graphics 263
A.7 OtherUsefulFunctions 265
A.8 ProgramminginMathematica 267
A.8.1 ControlStatements 268
A.8.2 TensorManipulations 270
References 272
Index 273
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