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MICR OMECHANICS-B ASED PREDICTION OF
THERMOELASTIC PROPERTIES OF
HIGH ENERGY MATERIALS
by
Biswajit Banerjee
A researchproposalsubmittedto thefacultyofTheUniversityof Utah
in partialfulfillment of therequirementsfor thedegreeof
Doctorof Philosophy
Departmentof MechanicalEngineering
TheUniversityof Utah
January2002
ABSTRACT
High energy materialssuchaspolymerbondedexplosivesarecommonlyusedaspropellants.
Theseparticulatecompositescontainexplosive crystalssuspendedin a rubberybinder. However,
the explosive natureof thesematerialslimits the determinationof their mechanicalpropertiesby
experimentalmeans.Micromechanics-basedalternativesare,therefore,exploredin this research.
In particular, methodsfor the determinationof the effective thermoelasticpropertiesof polymer
bondedexplosivesareinvestigated.
Polymerbondedexplosivesaretwo-componentparticulatecompositeswith high volumefrac-
tions of particles(volumefraction � 90%) andhigh moduluscontrast(ratio of Young’s modulus
of particlesto binder of 5,000-10,000). Experimentallydeterminedelasticmoduli of one such
material,PBX 9501,areusedto validatethemicromechanicsmethodsexaminedin this research.
The literatureon micromechanicsis reviewed; rigorousboundson effective elasticpropertiesand
analyticalmethodsfor determiningeffectivepropertiesareinvestigatedin thecontext of PBX 9501.
Sincedetailednumericalsimulationsof PBXsarecomputationallyexpensive,simplenumerical
homogenizationtechniqueshave beensought. Two suchtechnqiuesexploredin this researchare
the generalizedmethodof cells and the recursive cells method. Effective propertiescalculated,
for PBX-like materials,using thesemethodshave beencomparedwith finite elementanalyses
andexperimentaldata. In addition,someshortcomingsof thesemethodshave beenidentifiedand
improvementssuggested.
CONTENTS
ABSTRACT �������������������������������������������������������������������������������������������������������������������LIST OF FIGURES ������������������������������������������������������������������������������������������������������� iv
LIST OF TABLES ��������������������������������������������������������������������������������������������������������� vii
CHAPTERS
1. INTR ODUCTION ��������������������������������������������������������������������������������������������������� 1
2. HIGH ENERGY COMPOSITES ��������������������������������������������������������������������������� 3
2.1 PBX 9501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Compositionof HMX in PBX 9501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.2 ElasticModuli of � -HMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.3 ThermalExpansionPropertiesof HMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Compositionof PBX 9501Binder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.5 ElasticPropertiesof PBX 9501Binder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.6 ThermalExpansionof PBX 9501Binder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.7 ManufacturingProcessfor PBX 9501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.8 ElasticPropertiesof PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Mock Propellants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3. MICR OMECHANICS OF COMPOSITES ������������������������������������������������������������� 16
3.1 RigorousBounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 Hashin-ShtrikmanBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.2 Third OrderBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 AnalyticalMethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.1 CompositeSpheresAssemblage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.2 Self-ConsistentSchemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.3 DifferentialEffective MediumApproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 NumericalApproximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.1 TheRepresentative VolumeElement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3.2 Finite DifferenceApproximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.3 Finite ElementApproximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3.3.1 RegularArraysin Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3.3.2 RandomDistributionsin Two Dimensions. . . . . . . . . . . . . . . . . . . . . . . 313.3.3.3 ApproximationsusingHomogenizationTheory . . . . . . . . . . . . . . . . . . . 333.3.3.4 ApproximationsusingStochasticFinite Elements. . . . . . . . . . . . . . . . . . 343.3.3.5 ThreeDimensionalApproximations . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3.4 DiscreteModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.5 Integral EquationBasedApproximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3.6 FourierTransformBasedApproximations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Methodof Cells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4. THE GENERALIZED METHOD OF CELLS ������������������������������������������������������� 41
4.1 AverageStrainRelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Stress-StrainRelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 Effective ThermoelasticProperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.4 Shear-CoupledMethodof Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5. THE RECURSIVE CELL METHOD ��������������������������������������������������������������������� 61
5.1 SubcellStiffnessMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.1.1 DisplacementBasedFour-NodedElement. . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.1.2 DisplacementBasedNine-NodedElement . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1.3 MixedDisplacement-PressureNine NodedElement . . . . . . . . . . . . . . . . . . . . 68
5.2 ModelingaBlock of Subcells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 BoundaryConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.3.1 Applicationof ConstraintEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.2 Applicationof SpecifiedDisplacements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.3 CalculatingVolumeAveragedStressesandStrains . . . . . . . . . . . . . . . . . . . . . 815.3.4 CalculatingEffective Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 CalculatingEffective Propertiesof theRVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6. VALID ATION OF GMC AND RCM ����������������������������������������������������������������������� 86
6.1 ComparisonsWith ExactRelations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 866.1.1 PhaseInterchangeIdentity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.1.2 MaterialsRigid in Shear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.1.3 TheCLM Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.1.4 SymmetricCompositeswith EqualBulk Modulus. . . . . . . . . . . . . . . . . . . . . . 986.1.5 Hill’ s Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1006.1.6 CommentsOn ComparisonsWith ExactSolutions . . . . . . . . . . . . . . . . . . . . . 101
6.2 ComparisonsWith NumericalResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.3 SpecialCases: StressBridging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.3.1 CornerBridging : X-ShapedMicrostructure. . . . . . . . . . . . . . . . . . . . . . . . . . 1066.3.2 EdgeBridging : FiveCases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3.2.1 ModelA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126.3.2.2 ModelB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.3.2.3 ModelC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146.3.2.4 ModelD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.3.2.5 ModelE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.4 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7. SIMULA TION OF PBX MICR OSTRUCTURES ��������������������������������������������������� 120
7.1 ManuallyGeneratedMicrostructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.1.1 FEM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217.1.2 GMC Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.1.2.1 Fifty PercentRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247.1.2.2 TheTwo-StepApproach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257.1.2.3 Effective Propertiesfrom GMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.1.3 RCM Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.2 RandomlyGeneratedMicrostructures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2.1 CircularParticles- PBX 9501Dry Blend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.2.1.1 FEM Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
ii
7.2.1.2 GMC Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.2.1.3 RCM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.2.2 CircularParticles- PBX 9501PressedPiece. . . . . . . . . . . . . . . . . . . . . . . . . . 1417.2.2.1 FEM Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.2.2.2 GMC Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.2.2.3 RCM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.2.3 SquareParticles- PressedPBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.2.3.1 FEM Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.2.3.2 GMC Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.2.3.3 RCM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8. PROPOSEDRESEARCH �������������������������������������������������������������������������������������� 157
8.1 CurrentStatusof Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1578.2 RemainingResearch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8.2.1 Improvementsto RCM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1598.2.2 FurtherFEM Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.2.3 Calculationsfor PBX 9501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
APPENDICES
A. PLANE STRAIN STIFFNESSAND COMPLIANCE MATRICES ������������������������� 161
REFERENCES ������������������������������������������������������������������������������������������������������������� 165
iii
LIST OF FIGURES
2.1 HMX particledistribution in thedry blend[8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Monoclinic structureof a -HMX crystal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 HMX particlesizesin PBX 9501beforeandafterprocessing.. . . . . . . . . . . . . . . . . . . 10
2.4 Young’s modulusvs. appliedstrainfor PBX 9501[21]at22� C andstrainrateof 0.001/s.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Young’s modulusvs. strainrateandtemperatureforglass/Estane(21%/70%by volume)mockpropellants.. . . . . . . . . . . . . . . . . . . . . . . . 14
2.6 Young’s modulusvs. strainrateandtemperatureforglass/Estane(44%/56%by volume)mockpropellants.. . . . . . . . . . . . . . . . . . . . . . . . 15
2.7 Young’s modulusvs. strainrateandtemperatureforglass/Estane(59%/41%by volume)mockpropellants.. . . . . . . . . . . . . . . . . . . . . . . . 15
3.1 Parameters�� and �� for thepenetrablespheremodel(* = ValuesComputedby Berryman[33].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Comparisonof boundson thebulk andshearmodulusof PBX 9501with experimentalvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Subcellsandnotationusedin GMC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1 Schematicof therecursive cell method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Fournodedelement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Ninenodedelement.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.4 A four subcellblock modeledwith four elements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 A four subcellblock modeledwith sixteenelements.. . . . . . . . . . . . . . . . . . . . . . . . . 75
5.6 Schematicof theeffectof auniformdisplacementappliedin the � direction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.7 Schematicof theeffectof auniformdisplacementappliedin the � direction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.8 Schematicof theeffectdisplacements,correspondingto apureshear, appliedat theboundarynodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.9 Schematicof theeffectdisplacements,correspondingto apureshear, appliedat thecornernodes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.10 Therecursive cellsmethodappliedto aRVEdiscretizedinto blocksof four subcells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.1 RVE for acheckerboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Validationof FEM, RCM andGMC usingthephaseinterchangeidentityfor acheckerboardcomposite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3 Variationof effective shearmoduliwith moduluscontrastfor acheckerboardcomposite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4 Ratioof effective shearmoduli predictedby FEM, RCM andGMC tothosepredictedby thephaseinterchangeidentity for acheckerboardcompositewith varyingmoduluscontrast.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.5 Convergenceof effective moduli predictedby finite elementanalyseswith increasein meshrefinementfor acheckerboardcompositewith shearmoduluscontrastof 25,000.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.6 RVE for asquarearrayof disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.7 Error in computationof ��� for aSquareArray of Disks. . . . . . . . . . . . . . . . . . . . . . . 104
6.8 Error in computationof ����� for asquarearrayof disks. . . . . . . . . . . . . . . . . . . . . . . . 105
6.9 Error in computationof ��� � for aSquareArray of Disks. . . . . . . . . . . . . . . . . . . . . . . 105
6.10 RVE usedfor cornerstressbridgingmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.11 Variationof � ���� with moduluscontrastfor ’X’-shapedmicrostructure.. . . . . . . . . . . . 108
6.12 Variationof ���� � with moduluscontrastfor ’X’-shapedmicrostructure.. . . . . . . . . . . . 109
6.13 Variationof � � � with moduluscontrastfor ’X’-shapedmicrostructure.. . . . . . . . . . . . 109
6.14 Comparisonof effective stiffnessmatrixfor cornerstressbridgingmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.15 Progressive stressbridgingmodelsA throughE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.16 Comparisonof normalizedeffective stiffnessesfor modelA. . . . . . . . . . . . . . . . . . . . 113
6.17 Comparisonof effective stiffnessesfor modelB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.18 Stressbridgingpathsfor ModelC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.19 Why RCM predictssquaresymmetryfor Model C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.20 Comparisonof effective stiffnessesfor ModelC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.21 Comparisonof effective stiffnessesfor ModelD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.22 Comparisonof effective stiffnessesfor ModelE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.1 Manuallygeneratedmicrostructuresfor PBXs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Effective stiffnessesfor thesix modelmicrostructuresfrom from detailedfinite ele-mentanalysesasaamultiple of thebinderstiffness.. . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.3 Applicationof fifty percentrule to amodelmicrostructure.. . . . . . . . . . . . . . . . . . . . . 125
7.4 Schematicof thetwo-stepGMC procedure.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 Ratiosof effective stiffnessescalculatedusingGMC (50%rule)andFEM. . . . . . . . . . 128
7.6 Ratiosof effective stiffnessescalculatedusingGMC (two-step)andFEM. . . . . . . . . . 129
7.7 Microstructureusedfor RCM calculationsfor model4. . . . . . . . . . . . . . . . . . . . . . . . 129
7.8 Ratiosof effective stiffnesscalculatedusingRCM andFEM. . . . . . . . . . . . . . . . . . . . 130v
7.9 Ratiosof effective stiffnesscalculatedusingFEM ( !#"#$&%'!#"#$ squareelements)andFEM (65,000triangularelements).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.10 Microstructureof PBX 9501[19]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.11 Microstructuresusingcircularparticlesbasedon thedry blendof PBX 9501. . . . . . . . 135
7.12 Approximatemicrostructureusedfor FEM andRCM calculationsonthe100particlemodelof PBX 9501basedon thedry blend. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.13 Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from FEM calculations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.14 Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from GMC calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.15 Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from RCM calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
7.16 Microstructuresusingcircularparticlesbasedon thepressedpiecesizedistribution of PBX 9501.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.17 Approximatemicrostructurefor the1000particlemodelof PBX 9501.. . . . . . . . . . . . 144
7.18 Effective stiffnessmatrix componentsfor microstructuresbasedon pressedPBX9501from FEM calculations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.19 Effective stiffnessmatrix componentsfor microstructuresbasedon pressedPBX9501from GMC calculations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.20 Effective stiffnessmatrix componentsfor microstructuresbasedon pressedPBX9501from RCM calculations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.21 Microstructuresusingsquareparticlesbasedon thepressedpiecesizedistribution of PBX 9501.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.22 Effective stiffnessmatrix componentsfrom FEM calculationsfor microstructurescontainingsquareparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
7.23 Microstructurefor the700particlemodelof PBX 9501usingsquare,alignedparticles.153
7.24 Effective stiffnessmatrix componentsfrom GMC calculationsfor microstructurescontainingsquareparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7.25 Effective stiffnessmatrix componentsfrom RCM calculationsfor microstructurescontainingsquareparticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
vi
LIST OF TABLES
2.1 Compositionsof commonPBX materials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Compositionof PBX 9501.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 HMX particlesizedistribution in PBX 9501[7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Differentphasesof HMX andtransitiontemperatures.. . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Componentsof thestiffnessmatrixof ( -HMX (GPa) [13, 14]. . . . . . . . . . . . . . . . . . . 6
2.6 Elasticpropertiesof ( -HMX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.7 Thermalexpansionpropertiesof ( -HMX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.8 Strain-rateandtemperaturedependentelasticmoduliof PBX 9501binder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.9 Elasticpropertiesof PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.10 Elasticpropertiesof sodaglass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.11 Young’s modulusof Estane5703atvarioustemperaturesandstrainrates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.12 Propertiesof sugar/bindermockpropellant[7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Elasticmoduli andCTEof PBX 9501andits componentsat roomtemperatureandlow strainrate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Voigt andReussboundsfor PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Hashin-Shtrikmanupperandlower boundsfor PBX 9501. . . . . . . . . . . . . . . . . . . . . . 20
3.4 Milton upperandlowerboundsfor PBX 9501.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Compositespheresassemblagepredictionfor PBX 9501. . . . . . . . . . . . . . . . . . . . . . . 24
3.6 Self consistentschemepredictionfor PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.7 Three-phasemodelpredictionfor PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Differentialschemepredictionsof effective properties.. . . . . . . . . . . . . . . . . . . . . . . . 27
6.1 Out-of-planepropertiesfor squarearrayof disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2 Componentsof effective stiffnessandcompliancematricesfor asquarearrayof disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Componentsof effective stiffnessandcompliancematricesfor acheckerboardcomposite. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4 Original andtranslatedtwo-dimensionalconstituentmodulifor checkingtheCLM condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Comparisonof effective moduli for theoriginal andthetranslatedcomposites.. . . . . . 98
6.6 Componentproperties,exacteffectivepropertiesandnumericallycomputedeffectivepropertiesfor two-componentsymmetriccompositewith equalcomponentbulk moduli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.7 Phasepropertiesusedfor testingHill’ s relationandtheexacteffective moduli of thecomposite.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.8 Numericallycomputedeffective propertiesfor asquarearrayof diskswith equalcomponentshearmoduli. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.9 Componentpropertiesusedby GreengardandHelsing[97]. . . . . . . . . . . . . . . . . . . . . 102
6.10 Comparisonof numericallycalculatedvaluesof two-dimensionalbulk andshearmoduli of squarearraysof disks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.11 Theelasticpropertiesof thecomponentsof the’X’ shapedmicrostructure.. . . . . . . . . 107
6.12 )�*+�+ , )�*+-, and ).*/�/ for X-shapedmicrostructurewith highestmoduluscontrast. . . . . . . 110
6.13 Materialsusedto testedgebridgingusingFEM, GMC andRCM.. . . . . . . . . . . . . . . . 112
6.14 Effective propertiesof ModelA from FEM, GMC andRCM. . . . . . . . . . . . . . . . . . . . 112
6.15 Effective propertiesof ModelB from FEM, GMC andRCM. . . . . . . . . . . . . . . . . . . . 114
6.16 Effective propertiesof ModelC from FEM, GMC andRCM. . . . . . . . . . . . . . . . . . . . 116
6.17 Effective propertiesof ModelD from FEM, GMC andRCM. . . . . . . . . . . . . . . . . . . . 117
6.18 Effective propertiesof ModelE from FEM, GMC andRCM. . . . . . . . . . . . . . . . . . . . 118
7.1 Experimentallydeterminedelasticmoduli of PBX 9501andits constituents[7]. . . . . . 121
7.2 Effective stiffnessfor thesix modelPBX 9501microstructuresfrom FEM calcula-tionsusing65,000six-nodedtriangleelements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.3 Effectivestiffnessfor thesix modelPBX 9501microstructuresfrom GMC calculations.127
7.4 Effectivestiffnessfor thesix modelPBX 9501microstructuresfrom RCM calculations.130
7.5 Effective stiffnessfor thesix modelPBX 9501microstructuresfrom FEM calcula-tionsusing 0#1#24350#1#2 squareelements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.6 Effective stiffnessfor the four modelPBX 9501microstructuresbasedon the dryblendof PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.7 Volumefractionsof particlesandmoduli of the“dirty” binderfor thefour pressedpiecebasedPBX microstructures.. . . . . . . . . . . . . . . . . . . . . . . . 144
7.8 Effectivestiffnessfor thefour modelPBX 9501microstructuresbasedonthepressedpieceof PBX 9501. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.9 Moduli of the“dirty” binderfor thethreePBX microstructureswith squareparticles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.10 Effective stiffnessfor thethreepressedPBX 9501modelmicrostructurescontainingsquareparticles.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
viii
CHAPTER 1
INTRODUCTION
High energy (HE) materialsare thosethat decomposerapidly and releaselarge amountsof
energy whenimpactedor ignited. Thesematerialsarecommonlyusedaspropellantsfor rockets.
In recentyears,the issueof safeguardingstockpilesof missilesin the United Stateshasgener-
atedrenewed interestin the mechanicalpropertiesof HE materials.Suchmaterialpropertiesare
essentialfor the predictionof the responseof containersfilled with HE materialsunderdifferent
circumstances.Mechanicalpropertiesof HE materialscanbedeterminedexperimentally. However,
thehazardsassociatedwith experimentsonthesematerials,aswell astheattendingcosts,make this
optionunattractive. As computationalcapabilitieshave grown andimprovednumericaltechniques
developed,numericaldeterminationof thepropertiesof HE materialshasbecomepossible.In this
research,we exploresomenumerical,micromechanics-basedmethodsfor thedeterminationof the
mechanicalpropertiesof HE materials.
Of thenumeroustypesof HE materialsthatexist, theonesthatareof interestin this research
arepolymerbondedexplosives(PBXs). Onereasonfor this interestis thatonesuchmaterial,PBX
9501,hasbeenextensively testedin variousNationalLaboratoriesin the United Statesandthus
providesabasisfor validatingnumericalcalculations.In addition,PBXsprovideuniquechallenges
for micromechanicalmodeling- thesematerialsare viscoelasticparticulatecomposites,contain
high volumefractionsof particles,andthe moduluscontrastbetweentheparticlesandthe binder
is extremelyhigh. For example,PBX 9501 containsabout92% by volumeof particlesand the
moduluscontrastbetweenparticlesand the binder, at room temperatureand low strain rates,is
around20,000.
Somesimplifying assumptionsaremadeaboutPBXsin this research.It is assumedthatPBXs
are two-componentparticulatecompositeswith the particlescompletelysurroundedby, andper-
fectly bondedto, the binder. The componentsof PBXs are assumedto be isotropic and linear
elastic,andonly thepredictionof elasticmoduli andcoefficientsof thermalexpansion(CTEs)of
PBXsis addressed.
A few PBX materialsandtheir compositionsareshown in Chapter2. SincePBX 9501is the
materialthatprovidesexperimentalvalidationof ourmicromechanicsmodels,thecompositionand
2
thermoelasticpropertiesof PBX 9501andit’s componentsarediscussedin detail in Chapter2. In
addition,two mockpropellantsthatdo not containexplosive crystalsarealsodiscussed.
Micromechanics-basedmethodsfor the determinationof effective propertiesof composites
arereviewed in Chapter3. Theseincluderigorousboundson the effective properties,analytical
solutionsand numericalmethods. The upperand lower boundson the effective elasticmoduli
of PBX 9501are found to be too far apartto be of practicaluse. Boundson the effective CTE
are, however, quite closeto eachother. Analytical solutionsfor simplified modelsare found to
underestimatethe effective elasticmoduli considerably. Hence,numericalmethodsare the only
viableapproachesfor thedeterminationof effective propertiesof PBXs.Thefinite elementmethod
(FEM) hasbeenchosento provide benchmarkcalculationsof effective propertiesin this research.
However, the computationalcost involved in detailedFEM calculationshas led us to consider
simplernumericalapproachesto modelPBXs.
Thegeneralizedmethodof cells(GMC) is asimpleapproachthathasbeenusedto computethe
effective propertiesof composites.A reformulationof this techniqueis discussedin Chapter4. It
hasbeendiscoveredthatGMC predictsinaccurateshearmodulianddoesnotcapturestressbridging
effectsadequately. An alternative GMC-basedapproachintendedto improve uponGMC is also
discussedin Chapter4.
A new techniquecalledtherecursivecell method(RCM) hasalsobeendevelopedto remedythe
drawbacksof GMC. Chapter5 discussesthe recursive cell methoddetail. Someimprovementsto
thismethodarealsosuggestedin thischapter.
Effective properties,computedusingGMC andRCM, arecomparedwith exactresultsandnu-
mericalsimulationsin Chapter6. It is observedthatbothmethodspredictrelatively accurateelastic
moduli directionsfor low volumefractionsof particlesandfor low moduluscontrasts.In addition,
GMC andRCM areusedto predicttheeffective propertiesof somespecialmicrostructures.Some
shortcomingsof thetwo techniquesareelucidatedby theresultsfrom thesevalidationexercises.
Proceduresof generatingmicrostructuresthat model PBXs are discussedin Chapter7. Mi-
crostructurescontainingcircularandsquareparticlesaregeneratedandtheeffective propertiesare
calculatedusing FEM. The effective propertiesof thesemicrostructures,calculatedusing GMC
and RCM, are comparedwith thosefrom FEM calculations. For thesemicrostructures,GMC
consistentlyunderestimatesthe effective propertieswhile the currentform of RCM consistently
overestimatestheeffective properties.Someimprovementsto RCM aresuggestedin thischapter.
Theremainingresearchproposedfor thePh.D.degreeis discussedin Chapter8. Theimprove-
mentsto RCM proposedin this chapterareexpectedto leadto considerableimprovementin the
ability to predicttheeffective propertiesof PBXs.
CHAPTER 2
HIGH ENERGY COMPOSITES
High energy materialsare usually compositescontainingtwo or more components.One of
the componentsis an explosive crystalwhile the othercomponentsact asa binder that provides
structuralsupportto thecrystals.Thepolymerbondedexplosives(PBXs)consideredin thisresearch
containa very high volumefractionof crystalsthatareconsiderablystiffer thanthebinder. Some
dataon thecompositionsof suchPBXs[1, 2, 3] areshown in Table2.1.
Table2.1. Compositionsof commonPBX materials.
BinderType PBX Explosive/Binder Weight(%) SourceFluoropolymer LX-10-1 HMX/V iton 95.5/4.5 [1](e.g.,Viton) PBX 9502 TATB/KEL-F-800 95/5 [1]
PBX 9010 RDX/KEL-F-3700 90/10 [2]PBX 9407 RDX/Exon-461 94/6 [2]PBX 9207 HMX/Exon-461 92/8 [2]
Polyeurethene PBX 9011 HMX/Estane5703F1 90/10 [2]EDC29 HMX/HTPB 95/5 [3]
Polyeurethene PBX 9404 HMX/NC+CEF(1:1) 94/6 [2](with PBX 9501 HMX/ 95/5 [2]Plasticizers) Estane5703+BDNPA-F(1:1)
EDC37 HMX/NC+K10(1:8) 91/9 [3]
2.1 PBX 9501The polymer-bondedexplosive of interestin this researchis PBX 9501. This material is a
compositeof crystalsof HMX (High Melting Explosive) anda bindercomposedof Estane5703
andBDNPA/F anda freeradicalinhibitor suchasdiphenylamineor Irgonox[4]. A moredetailed
compositionof PBX 9501is shown in Table2.2.
2.1.1 Compositionof HMX in PBX 9501
PBX 9501containsa mixtureof two differentsizedistributionsof HMX particlesbecausethe
smallerparticlesfit into the interstitial spacesbetweenthe larger particles. The mixture contains
4
Table2.2. Compositionof PBX 9501.
Component Chemical Weight VolumeComposition Fraction Fraction6
HMX 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane 0.949 0.927Estane5703 polybutyleneadipateand 0.025 0.039
4,4diphenylmethanediisocyanate-1,4-butanediolBDNPA/F bis-dinitropropylacetal-formal 0.025 0.033Irgonox 0.001 0.0Voids 0.0 0.01-0.02
a - Thevolumefractiondatahave beenobtainedfrom Dick et al. [5].
b - McAfeeet al. [6] cite volumefractionsof 0.912and0.088for HMX andbinderrespectively.
Class1 HMX (coarse)andClass2 HMX (fine) in a ratio of 3:1 by weight. Class1 HMX consists
of particlesprimarily between44 and300micronsin size. Thefiner gradeClass2 HMX alsohas
a few coarseparticles,but 75%of theparticlesarelessthan44 micronsin size[4]. SeveralHMX
particlesizedistributionsfor PBX 9501canbefoundin theliteraturethatdo notnecessarilymatch
oneanother. A goodapproximationthathasbeenlistedby Wetzel[7] is shown in Table2.3.A plot
of theparticledistributionsof thetwo gradesof HMX in PBX 9501obtainedfrom datageneratedby
Skidmoreet al. [8] is shown in Figure2.1. Theplot illustratesthebimodaldistribution of particles
in thedry blend.
Table2.3. HMX particlesizedistribution in PBX 9501[7].
Particle Class1 Class2Size(micron) HMX HMX8
44 3-13% at least75%874 14-26%8125 at least98%8149 40-60% 100%8297 84-96%
HMX crystalscanexist in threestablephases( 9 -HMX, : -HMX, and ; -HMX) dependingon
temperatureandpressure.Dataobtainedby Leiber[9] on thesephasesandtheir rangesof stability
areshown in Table2.4. The : -HMX phaseis dominantat or nearroom temperaturewhenlinear
elasticbehavior is expected.
The : -HMX crystalhasa monoclinicstructureasshown in Figure2.2. Theaxis < is the axis
of second-ordersymmetry(or equivalently the plane = - > is the planeof symmetry). At room
temperaturethe lattice parameters= , < and > are approximatelyin the ratio ?�@BADC.E#EFCHG@BA and
theangle: is approximatelyI#JLK (Bedrov etal. [10]).
5
1 10 100 10000
1
2
3
4
5
6
Vol
ume
Fra
ctio
n (%
)
Particle Diameter (microns)
Fine HMX (100%)Coarse HMX (100%)
Figure 2.1. HMX particledistribution in thedry blend[8].
Table2.4. Differentphasesof HMX andtransitiontemperatures.
Phase StableRegion Transitions( M C)N -HMX 103-162 ( N�OQP ) at 116M CP -HMX 20-103 ( P�OSR ) at 167-182M CR -HMX 162-melt ( N�OSR ) at 193-201M C
a
b
c β
a = b = c α = γ = 90 = βo
Figure 2.2. Monoclinic structureof a P -HMX crystal.
2.1.2 Elastic Moduli of T -HMX
Crystalsof P -HMX aremildly non-linearlyelasticat ambienttemperatures.As temperature
increases,voids develop in the crystalsthat may lead to degradationof elasticstiffnessprior to
6
melting.However, a linearelasticapproximationis adequatefor HMX below 40U C.
As mentionedin theprevioussection,crystalsof V -HMX aremonoclinicin structure.Following
Lekhnitskii [11], if the W axis(alsoreferredto asthe’ X ’ axisby Ting [12]) is theaxisof secondorder
symmetry, the elasticconstitutive relation for a HMX crystal is asshown in equation2.1 (Voigt
notation). YZZZZZZ[\^]�]\�_�_\�`�`\�_ `\^] `\^]-_acbbbbbbdfe
YZZZZZZ[g ]�] g ]-_ g ] ` h g ]-i hg ]-_ g _�_ g _ ` h g _�i hg ] ` g _ ` g `�` h g ` i hh h h gkj�j h gkjmlg ]-i g _�i g ` i h g i�i hh h h gkjml h gnl�l
acbbbbbbdYZZZZZZ[op]�]om_�_om`�`om_ `op] `op]-_acbbbbbbd (2.1)
In compactform, this relationcanbewritten asq esrutLinear elasticmoduli of V -HMX have beenestimatedusingexperiments(Zaug[13], Dick et
al. [5]) andby moleculardynamics(MD) simulations(Sewell et al. [14]). The dataobtainedby
Zaug[13] andSewell et al. [14] arethemostcomprehensive andareshown in Table2.5. Thedata
obtainedby Zaugwerecalculatedfrom measurementsof wave velocitiesthrougha singlecrystal
of V -HMX. The valuesof the 13 elasticcoefficientswerecalculatedat a temperatureof 107U C
usinga non-linearleastsquaressimplex fit of the experimentaldatausing the room temperature
valueof bulk modulus(12.5GPa) asa benchmark.Moleculardynamicssimulationsby Sewell et
al. [14] show resultsthatarecloseto thoseobtainedby Zaugandareshown insideroundbrackets
in Table2.5.
Table2.5. Componentsof thestiffnessmatrixof V -HMX (GPa) [13, 14].
rveYZZZZZZ[
19.8(18.7) 3.9(4.9) 12.5(7.7) 0 0.5(-1.7) 026.3(17.0) 6.5(7.3) 0 -1.4(3.0) 0
16.9(16.7) 0 0.1(0.2) 02.8(8.9) 0 2.9(2.4)
Symm. 6.4(9.3) 03.6(9.8)
acbbbbbbd(Numbersinsideroundbracketsshow valuesfrom MD simulations[14].)
Leiber[9] hascommentedthatthecouplingcoefficients(g ]-i , g _�i , and
gkjml) shown in Table2.5
have a significanteffect on the normalandshearstressesandstrainsandhenceisotropy is not a
goodapproximationfor V -HMX. However, the assumptionof isotropy providesa simpleway of
7
carryingout mesoscopicsimulationsof PBX materialsandhasbeenutilized in this investigation.
Variousestimatesof isotropicbulk modulus,shearmodulusandYoung’s modulusfor HMX are
shown in Table2.6. Thevaluesobtainedby Zaug[13] werecalculatedfrom ultrasonicsoundspeed
measurementsandhenceat high strain rates. The moleculardynamicssimulationsof Sewell et
al. [14] requirethe load to beappliedinstantaneouslyandthereforehigh strainratesareinvolved.
Theresultsobtainedby Dick etal. [5] arealsofrom highstrainrateimpacttests.SinceHMX is not
particularlysensitive to strainrateandwe assumethat thesepropertiescanbe usedfor low strain
ratesimulationsaswell.
Table2.6. Elasticpropertiesof w -HMX
Bulk Shear Young’s Poisson’s SourceModulus Modulus Modulus Ratio
(GPa) (GPa) (GPa)12.1 5.2 13.6 0.31 Zaug[13]10.2 7.3 17.7 0.21 Sewell et al. [14]14.3 5.8 15.3 0.32 Williams [15]
26.6 Dick etal. [5]
2.1.3 Thermal ExpansionPropertiesof HMX
Thecoefficientsof thermalexpansionof HMX crystalshave beenobtainedusingX-ray diffrac-
tion by Herrmann[16]. Thevaluesobtainedshow apronouncedanisotropy in the x latticedirection
comparedto the y and z directions. The angles { (betweenthe y and x lattice directions)and| (betweenthe y and z lattice directions)do not changesignificantlywith changingtemperature.
However, thereis a largechangein theanglew (betweenx and z ) of themonocliniclattice.Molec-
ular dynamics(MD) simulationsat room temperatureby Bedrov et al. [10] show resultssimilar
to thoseobtainedby Herrmann. Table2.7 shows the coefficients of thermalexpansionof HMX
obtainedexperimentallyby Herrmannandfrom moleculardynamicssimulationsby Bedrov etal.
2.1.4 Compositionof PBX 9501Binder
Thebinderin PBX 9501is essentiallyacombinationof Estane5703andaplasticizer(BDNPA-
F). A free radical inhibitor (Irgonox) is addedfor further stability of PBX 9501. The theoretical
maximumvolumeoccupiedby thebinderin PBX 9501is about8%of thetotal volume.
Estane5703is amorphousandthermoplasticwith a relatively low glasstransitiontemperature
(-31} C) anda meltingtemperatureof around105} C. It containssoft andhardsegmentsthatserve
to enhanceentanglementand lead to low temperatureflexibility, high temperaturestability and
8
Table2.7. Thermalexpansionpropertiesof ~ -HMX.
ThermalExpansion Experiments MD�������(/K) (Herrmann[16]) (Bedrov etal.[10])
LatticeParametersa -0.29 2.07
Linear/Angular b 11.60 7.2c 2.30 2.56~ 2.58
Volume 13.1 11.6
goodadhesive properties.Grayet al. [4] statethat theplasticizer(BDNPA/F) decreasesthebinder
strengthandstiffness.
Experimentaldataproducedby Grayetal. [4] show thatthemechanicalpropertiesof PBX 9501
areaffectedsignificantlyby theporosityof themix. Theporosityof PBX 9501is supposedlymostly
dueto cavitation in thebinderasthecompositerelaxesafter it hasbeenisostaticallypressed[17].
The voids thereforeoccupy a significantvolume fraction of the binder (20-50%)and affect the
mechanicalpropertiesof thebinderconsiderably. However, it theexperimentaldatain theliterature
areambiguousaboutwhatpercentagetheporosityof PBX 9501is dueto voidsin theHMX particles
or voidsin thebinder.
2.1.5 Elastic Propertiesof PBX 9501Binder
Theelasticpropertiesof thePBX 9501binderarequitesensitive to strainrateandtemperature.
Thishasledto experimentsonthebinderbeingcarriedoutatdifferentstrainratesandtemperatures.
Completebinderpropertiesarethereforeconsiderablymoredifficult to obtainfrom publishedex-
perimentaldatathanHMX properties.Few low strainratetestshave beencarriedout becauseof
the low stiffnessof the binder. High strain rate testsusing Hopkinsonbar type experimentsdo
not yield acceptableaccuracy in initial modulusvalues.Moleculardynamicssimulationshave not
beencarriedout on theconstituentsof thebinderbecauseEstane5703moleculesarecomplex and
containbothhardandsoft segments.
Numeroustestshave beencarriedout on the PBX 9501 binder by Dick et al. [5], Cady et
al. [18, 20], Grayet al. [4] andWetzel[7] at variousstrainratesandtemperatures.Datafrom these
sourceson the PBX 9501binderareshown in Table2.8. The datashow that at high strainrates
(keepingtemperatureconstant)theYoung’smodulusof thebinderis many timesgreaterthanat low
strainrates.Thishigherstiffnessathighstrainratesis becausethepolymerchainshave lesstimeto
flow. ThePoisson’s ratioof thebinderis closeto 0.5,ascanbeexpectedof rubbersandelastomers.
9
Table2.8. Strain-rateandtemperaturedependentelasticmoduliof PBX 9501binder.
Temperature Strain Young’s Poisson’s SourceRate Modulus Ratio
( � C) (strains/sec) (MPa)25 0.005 0.59 Wetzel[7]
0.008 0.73 Wetzel[7]0.034 0.81 Wetzel[7]0.049 0.82 Wetzel[7]2400 300 0.49 Dick etal. [5]
22 0.001 0.47 Cadyet al. [20]1 1.4 Cadyet al. [20]
2200 3.3 Cadyet al. [20]16 1700 22.5 Grayetal. [4]0 0.001 0.85 Cadyet al. [20]
1700 246 Grayetal. [4]2200 4 Cadyet al. [20]
-15 0.001 1.4 Cadyet al. [20]1 5.7 Cadyet al. [20]
1000 1600 Cadyet al. [20]-20 0.001 1.6 Cadyet al. [20]
1200 1600 Cadyet al. [20]1700 1333 Grayetal. [4]
-40 0.001 5.7 Cadyet al. [20]0.001 5.3 Grayetal. [4]1300 10000 Cadyet al. [20]
2.1.6 Thermal Expansionof PBX 9501Binder
Wetzel[7] citesthecoefficient of thermalexpansionof Estane5703to bebetween10 ��������� to
20 ����� ��� /K. Sincedataarenot availablefor thebinder, we shallassumethecoefficient of thermal
expansionof thebinderto bethesameasthatof Estane5703.
2.1.7 Manufacturing Processfor PBX 9501
The first stepin the manufacturingof samplesis to mix theconstituentsandto form molding
powdergranulesor prills of PBX 9501.Samplesarethenisostaticallycompresseduntil theporosity
is reducedto 1-2%. The theoreticalmaximumdensityfor the composite(1.860gm/cc) is used
to determinethe porosity. The processof isostaticpressingcauseslesssegregation of particles
awayfrom thepressingsurfacesthanunidirectionalcompression.Thepreparationof thematerialis
usuallycarriedoutatatemperatureof 90� C.Thesizedistributionof HMX particlesafterprocessing
is significantlydifferentfrom thatbeforeprocessing.Experimentsby Skidmoreet al. [8] show that
thecumulativevolumefractionof thefinersizedparticlesis dramaticallyhigherin thepressedPBX
10
9501comparedto thatof thedry blendof coarseHMX andfineHMX. Figure2.3showstheparticle
sizedistributionsof the dry blendof coarseandfine HMX particles,that of the molding powder
andthatof thepressedpiece.It is difficult to observe thebimodaldistribution of particlesin thedry
blendbecausethevolumefractionof finesis muchsmallerthanthatof thecoarsesizes.However,
the bimodaldistribution of particlesis clear in the plot pressedpiecesizedistribution. Pressing
considerablyincreasesthevolumefractionof smallersizedparticles.
10 100 10000
6
8
10
12
4
2
Particle Diameter (microns)
Vol
ume
Fra
ctio
n (%
)
Dry BlendMolding Powder
Pressed Piece
Figure 2.3. HMX particlesizesin PBX 9501beforeandafterprocessing.
Experimentsby Skidmoreetal. [19] haveshown thattheconsolidationof prills initially involves
little damageto thelargeHMX crystals.As porosityis decreased,thereis anincreasingincidence
of transgranularcrackingand twinning in the large HMX crystals. If porosity is decreasedto
lessthat1%,microcracksgrow acrosscrystalsdueto crystal-to-crystalcontactandintercrystalline
indentation.
2.1.8 Elastic Propertiesof PBX 9501
Theelasticmoduliof polymerbondedexplosivesarestronglyinfluencedby strainrateandtem-
perature[18] primarily becauseof thestrainrateandtemperaturedependentbehavior of thebinder.
It hasalsobeenobserved that thesecomposites(especiallyduring high-rateloading)continueto
strainafterthemaximumstresshasbeenachieved( [4], [18], [20], [21]). Therefore,sometimeand
historydependentbehavior is indicated.In general,thecompressive strengthsandelasticmodulus
of polymer-bondedexplosivesincreasewith decreasingtemperatureandincreasingstrainrate.The
11
above observationsarealsotruefor PBX 9501. In addition,somenon-linearityin thestress-strain
relationshipis indicatedfor PBX 9501in thesmallstraindomain.Dick et al. [5] have shown that
compressive stiffnessincreaseswith increasingvolumetricstrainfor smallstrainsprior to yielding.
Temperatureandstrainratemoduliof PBX 9501reportedby Wetzel[7] andobtainedfrom tests
carriedout by Wiegand[21], Dick et al. [5] andGray et al. [4] areshown in Table2.9. The data
show thesametrendsasthebinderbut higherstiffnessat room temperature.The high strainrate
Young’s modulusis around12 timesthelow strainrateYoung’s modulusat roomtemperature.
Table2.9. Elasticpropertiesof PBX 9501.
Temperature Strain Young’s Modulus Poisson’s Source( � C) Rate Compressive Tensile Ratio
(strains/sec) (GPa) (GPa)55 2250 4.65 Grayet al. [4]40 2250 4.65 Grayet al. [4]27 0.001 0.96 Grayet al. [4]
0.011 1.02 Grayet al. [4]0.11 1.09 Grayet al. [4]
25 0.001 1.04 Wiegand[21]0.01 0.77 Dick et al. [5]0.05 1.013 7.3 0.35 Wetzel[7]0.44 1.15 Grayet al. [4]
17 2250 4.65 Grayet al. [4]0 2250 5.48 Grayet al. [4]
-20 2250 6.67 Grayet al. [4]-40 2250 12.9 Grayet al. [4]-55 2250 8.51 Grayet al. [4]
Ultimatecompressivestrengthsof PBX 9501havebeenfoundto bearound10- 15MPafor low
strainratetestsandaround50 - 90 MPa for high strainratetests[5]. Wiegand[21] hasshown that
after yielding, progressive damagedevelopsin PBX 9501andthe materialbecomesconsiderably
lessstiff asshown in Figure2.4.
2.2 Mock PropellantsVariousmock propellantshave beentestedto determinethe effectsof the binderon material
properties.A mock propellantmadeof monodispersed(650 � 50 microns)sphericalglassbeads
with Estaneasbinderhasbeentestedat theLos AlamosNationalLaboratory[18] at temperatures
rangingfrom -55� C to 25� C. Glassvolumefractionsof 21%,44%and59%(25%,50%and65%
by weight)wereusedin thetests.Low strainratecompressiontestsat 0.001,0.1and1 s�^� aswell
ashighstrainrateimpacttestsat3500s�^� wereconductedonthespecimens.Theglassbeadswere
12
0 0.02 0.04 0.06 0.08 0.1 0.12 0.140
0.2
0.4
0.6
0.8
1
1.2
1.4
Applied Strain
You
ng’s
Mod
ulus
(G
Pa)
Figure2.4. Young’s modulusvs. appliedstrainfor PBX 9501[21]at22� C andstrainrateof 0.001/s.
standardsodalime glasswith a densityof 2.5gm/cc.Elasticpropertiesof sodaglassareshown in
Table2.10.
Table2.10. Elasticpropertiesof sodaglass.
Young’s Poisson’s ShearModulus Ratio Modulus(MPa) (MPa)50,000 0.20 20,000
The Young’s modulusof Estane5703hasbeencalculatedat varioustemperaturesandstrain
rateson thebasisof thestress-straincurvesfrom experimentscarriedout by Cadyet al. [18]. The
PBX 9501binderis lessstiff thanEstane5703,but exhibits qualitatively similar temperatureand
straindependence.Table2.11shows theYoung’s modulusof Estane5703atdifferenttemperatures
andstrainrates.It canbeobserved from thesodaglassandEstane5703moduli that themodulus
contrastis around5-10timeslower thanthatfor PBX 9501for aglass-Estanecomposite.However,
for atestof themicromechanicstechniquesof interestin this investigation,thiscontrastis adequate.
13
Table2.11. Young’s modulusof Estane5703atvarioustemperaturesandstrainrates.
Temperature StrainRate Young’s Modulus( � C) (/sec) (MPa)23 0.001 3.3
1.0 6.72400 21
20 0.001 5.21.0 94
2400 2441.710 0.001 6
1.0 101.82400 2455.3
0 0.001 7.01.0 110.2
2400 2469-10 0.001 8.1
1.0 122.72400 2636.8
-20 0.001 9.31.0 136.5
2400 2816-30 0.001 266.7
1.0 877.32400 3356.2
-40 0.001 727.31.0 1621.2
2400 4000
Micromechanicstechniquescanbe usedto determinethe effective elasticpropertiesof glass-
Estanecompositesusingthepropertiesshown in Tables2.10and2.11.Thesecanthenbecompared
with experimentallydeterminedelasticpropertiesof mixturesof the two components(mock pro-
pellants).TheYoung’s moduli of thethreemockpropellantstestedby Cadyet al. [18] at different
temperaturesandstrainratesareshown in Figures2.5,2.6,and2.7.Thesedataprovideanadditional
checkof theaccuracy of micromechanicssimulationsof PBX-likematerials.
Mock propellantshave alsobeendesignedusingsugarinsteadof glassbeads.A sugarbased
PBX 9501mockhighenergy materialhasbeenstudiedby Wetzel[7]. Thecubicsugarcrystalstake
theplaceof HMX crystalswhile thebinderwaschosento have thesamecompositionasthebinder
for PBX 9501. Elasticproperties,densitiesandvolumefractionsof thesugarcrystals,the binder
andthe mock arelisted in Table2.12. The datalisted arefor a strainrateof 0.0336/sandroom
temperature.
14
10−4
10−2
100
102
104
100
101
102
103
104
105
Strain Rate (/s)
You
ng’s
Mod
ulus
(M
Pa)
20o C10o C0o C−10o C−20o C−30o C−40o C
Figure 2.5. Young’s modulusvs. strainrateandtemperatureforglass/Estane(21%/70%by volume)mockpropellants.
Table2.12. Propertiesof sugar/bindermockpropellant[7].
Property Sugar Binder MockVolumeFraction 0.96 0.04Density(gm/cc) 1.587 1.19 1.52TensileModulus(GPa) 740 0.81 741Poisson’s Ratio 0.2 0.5 0.38
Thebindercanbemodeledasaviscoelasticmaterial.Therefore,thePoisson’s ratioof themock
varieswith appliedstressandtime. However, we assumethat it is constantover therangeof strain
ratesandtemperaturesof interestin this research.
Wereview someof themethodsof determiningeffective thermoelasticpropertiesof composites
in Chapter3. Someof thesemethodsarealsousedto predicteffective elasticmoduli of PBX 9501
basedon thepropertiesof thecomponentsat roomtemperatureandlow strainrates.Thepredicted
valuesarecomparedwith experimentallydeterminedvaluesto determinethe efficacy of someof
thesemicromechanicsmethods.
15
10−4
10−2
100
102
104
100
101
102
103
104
Strain Rate (/s)
You
ng’s
Mod
ulus
(M
Pa)
20o C10o C0o C−10o C−20o C−30o C−40o C
Figure 2.6. Young’s modulusvs. strainrateandtemperatureforglass/Estane(44%/56%by volume)mockpropellants.
10−4
10−2
100
102
104
101
102
103
104
Strain Rate (/s)
You
ng’s
Mod
ulus
(M
Pa)
20o C10o C0o C−10o C−20o C−30o C−40o C
Figure 2.7. Young’s modulusvs. strainrateandtemperatureforglass/Estane(59%/41%by volume)mockpropellants.
CHAPTER 3
MICR OMECHANICS OF COMPOSITES
Thegoalof thisresearchis todeveloptoolsthatcanpredicttheeffective thermoelasticproperties
of PBX like materials.The term “micromechanics”describesa classof methodsfor determining
the effective materialpropertiesof compositesgiven the materialpropertiesof the constituents.
Governingequationsbasedonacontinuumapproximationareusedto solvetheproblemof effective
propertydeterminationin micromechanicsbasedmethods.
The materialpropertiesof interestin this researchare elasticpropertiesand coefficients of
thermalexpansionin thedomainof infinitesimalstrain.Wedo notdiscussmethodsof determining
theeffective thermalconductivity or effectivespecificheatsof PBX materials.This is becausethese
propertiesarerelatively closeto eachother for the componentsof PBX 9501. The high volume
fraction of the dispersedcomponentin PBXs as well as the high moduluscontrastbetweenthe
dispersedandthecontinuouscomponentsin thecompositeprovide themainchallenges.Accurate
yet computationallyinexpensive methodsthat canaddressthesechallengesaresought. The data
on thepropertiesof PBX 9501andits componentsthathave beenpresentedin Chapter2 provide
an excellentcheckof the accuracy of variousmicromechanicsmethodsin dealingwith PBX like
materials.In this chapter, we review somemicromechanicsmethodsanddiscusstheeffectiveness
of thesemethodsin thecontext of PBX 9501.
Excellent reviews of micromechanicsof compositematerialsare provided by Hashin [22],
Markov [23] andBuryachenko [24]. More detailedexpositionson the micromechanicsof com-
positescanbefoundin themonographsby Nemat-NasserandHori [25] andMilton [26].
Polymerbondedexplosivesform partof theclassof compositesknown asparticulatecompos-
ites.Theparticlesof thedispersedcomponentof thecompositearedistributedin threedimensions.
Hence,accuratemodelsof thesecompositesshouldbethree-dimensional.However, for simplicity,
weprimarily exploretwo-dimensionalmodelsin thisresearch.Micromechanicsmethodsthatapply
only to alignedfiber compositesare,therefore,alsoreviewedin thischapter.
Someboundson theeffective elasticpropertiesof particulatecompositesbasedon variational
principlesof the minimization of strain energy are discussedfirst. Theseupperand the lower
boundsarefoundto bequitedifferentfrom eachother. Analytical solutionsfor theeffective elastic
17
propertiesarediscussednext. Simplifiedmodelsof thecompositeareusedto obtainthesesolutions
and theseare found not to fare much better than the bounds. The final portion of this chapter
dealswith variousnumericaltechniquesthat have beenusedto solve the problemof determining
effective properties.Someof theadvantagesanddrawbacksof thesemethodsarealsodiscussedin
thecontext of PBX-like materials.
Table3.1 shows the elasticmoduli andthe coefficientsof thermalexpansion(CTE) of HMX,
Binder andPBX 9501at room temperatureandlow strainrate. Thesedataareusedto assessthe
predictionsof someof thetechniquesdiscussedin thischapter.
Table3.1. Elasticmoduli andCTE of PBX 9501andits componentsat roomtemperatureandlow strainrate.
Material Volume Young’s Poisson’s Bulk Shear CTEFraction Modulus Ratio Modulus Modulus
(%) (MPa) (MPa) (MPa) (10��� /K)HMX 92 15300 0.32 14300 5800 11.6Binder 8 0.7 0.49 11.7 0.23 20PBX 9501 1000 0.35 1111 370
3.1 RigorousBoundsThe mostelementaryrigorousboundson elasticmoduli are the Voigt (arithmeticmean)and
Reuss(harmonicmean)bounds[27]. In termsof isotropicbulk andshearmoduli, theseboundscan
beexpressedas ������ � ��� � � ���p��� � � Voigt Bounds (3.1)¡ � � � � ¡ � � ¡ � � � � ¡ � (3.2)
and, ¢� �£ � ¤ ¢�v¥ � �p�� � � � � ReussBounds (3.3)¢¡ �£ � ¤ ¢¡ ¥ � �p�¡ � � � ¡ (3.4)
18
where ¦p§�¨ thevolumefractionof theparticles,© §.¨ thebulk modulusof theparticles,ª §.¨ theshearmodulusof theparticles,¦L«¬¨ thevolumefractionof thebinder,© «¨ thebulk modulusof thebinder,ª «¨ theshearmodulusof thebinder,©�® ¨ theeffective bulk modulusof thecomposite,and,ª ® ¨ theeffective shearmodulusof thecomposite.
The subscript ¯ denotesthe upperboundwhile the subscript ° denotesthe lower boundon an
effective property.
Using the bulk andshearmoduli of the componentsshown in Table3.1 we cancalculatethe
Voigt and Reussboundson the effective moduli of the composite. Thesevaluesare shown in
Table3.2. TheVoigt andReussboundsshow that theactualelasticmoduli arecloserto theReuss
boundbut considerabledifferenceexistsbetweenthelower boundsandtheexperimentalvaluesof
compositemoduli.
Table3.2. Voigt andReussboundsfor PBX 9501.
Elastic Voigt Reuss ExperimentalModulus Bound Bound Modulus
(MPa) (MPa) (MPa)Bulk 13034 144 1111Shear 5332 3 370
3.1.1 Hashin-Shtrikman Bounds
Variationalprinciplesbasedon the conceptof a polarizationfield have beenusedby Hashin
andShtrikman[28] to obtainimprovedboundsontheeffectiveelasticmoduli thathavebeenshown
to be optimal for assemblagesof coatedspheres.For particulatecompositestheseboundscanbe
writtenas © ®±'²´³ ©�µ·¶D¸ ¦p§#¦L«�¹ © § ¶�© «mº�»¸½¼³ ©�µ^¾À¿ ª § Á Hashin-ShtrikmanUpperBounds (3.5)ª ® ± ²´³ ª µ�¶D ¦ § ¦L«�¹ ª § ¶ ª «mº�»Â ¼³ ª µ�¾FÃLÄ Á (3.6)
19
and, ÅÆ�ÇÈfÉËÊ ÅÆÍÌÏÎÑÐÓÒpÔ#ÒLÕÖ ÅÆ Ô Î ÅÆ Õ
×ÙØÐ ÚÛ ÅÆFÜÀÝ Þß Õ
à Hashin-ShtrikmanLowerBounds (3.7)
Åß ÇÈfÉ Ê Åß Ì ÎDÒpÔ#ÒLÕÖ Åß Ô Î Åß Õ
×ÙØÚÛ Åß Ü Ýfá à (3.8)
where,for any property â , we define ã â�ä É â Ô Ò Ô Ýfâ ÕmÒLÕ àÚã â�ä É â ÔåÒLÕ Ýfâ Õ�ÒpÔ à
and, æ É ß Ô4çè Æ Ô Ýfé ß ÔÆ Ô ÝFê ß Ôë àá É Åß Õ ç Þ Æ Õ Ý Ð ß Õè Æ Õ Ýfé ß Õëíì
Boundson the effective coefficient of thermalexpansion( î Ç ) of a two-componentisotropic
compositecanbecalculatedusingtheHashin-Shtrikmanbounds[29]. Theseboundsareî Ç ï É ã î·äðÝ ÐÓÒpÔåÒLÕ ß Õ�ñ Æ Õ Î Æ ÔÓòpñ î Õ Î î ÔÓòÞ Æ Ô Æ Õ Ý Ð ß Õã Æ ä Rosen-HashinUpperBound (3.9)î ÇÈ É ã î·äðÝ ÐÓÒpÔåÒLÕ ß Ô�ñ Æ Õ Î Æ ÔÓòpñ î Õ Î î ÔÓòÞ Æ Ô Æ Õ Ý Ð ß Ôã Æ ä Rosen-HashinLowerBound (3.10)
where î Ô.ó coefficient of thermalexpansionof theparticles,and,î Õó coefficient of thermalexpansionof thebinder.
For thecomponentsof PBX 9501listed in Table3.1 the Hashin-Shtrikmanboundshave been
calculatedand are shown in Table3.3. The datashow that only a very limited improvementis
obtainedover theVoigt-Reussboundsfor thebulk andshearmoduli. Theboundson thecoefficient
of thermalexpansionarewithin 1%of eachother.
20
Table3.3. Hashin-Shtrikmanupperandlower boundsfor PBX 9501.
Bulk Modulus ShearModulus ThermalExpansion(MPa) (MPa) ( ô�õ�ö�÷�ø /K)
UpperBound 11372 5257 12.2558LowerBound 148 11 11.6017Experiments 1111 370 -
3.1.2 Third Order Bounds
The boundsdiscussedso far have beenbasedonly on the volumefractionsof the component
materials. An improvementover the Hashin-Shtrikmanboundsis the applicationof threepoint
correlationfunctionsto incorporategeometricinformationinto thedeterminationof upperandlower
boundsof third order on the effective propertiesof the composite. A simplification of bounds
obtainedusingstatisticalmethodswith threepoint correlationfunctionsby Beran,Molyneux and
McCoy [30, 31] have beenprovidedby Milton [32]. TheMilton boundscanbeexpressedasù�úû'ü´ý ù�þ�ÿ���������� ù � ÿíù ���� ���ý ù�þ���� ý�� þ���� Milton UpperBounds (3.11)� ú û ü´ý�� þ ÿ�� � � ��� � � ÿ � ��� � �ý�� þ���� � (3.12)
and,
õù ú ü"! õù$# ÿ � �������&% õù � ÿ õù ��' � �( õù*) � � ( õ� ) � � Milton LowerBounds (3.13)
õ� ú ü ! õ� # ÿ ������� % õ� � ÿ õ� �+' �( õ� ) � ��, � (3.14)
where,for any property - , we define, ý - þ ü - � � � � - ����� ��ý - þ ü - ����� � - �.��� �ý - þ � ü - ��/+� � - � / � �ý - þ10 ü - �324� � - �+2�� �
21
and,
57698:<;�=?>�@BA 8=�C<DFE�G ;�HI>�;KJ�H E*L =M>NA 8HOC<DFE ;KJ�= E HI>+@BA 8HPC&Q;KR�= E*S HI> @ TU 6 8:<;1HI>+@V;W=M> D E*G ;�HI>�;KJ�H E*L =M>�;�HI> D E ;KJ�= E HI>+@X;�HI> Q;W= E�L HI>+@ YTheMilton boundsdependon two extrageometricparameters,Z+[ 6 8B\ Z�] and ^4[ 6 8B\ ^�] which
incorporatethethree-pointcorrelationfunctionsandhavebeenfoundto lie between0 and1. These
parameterscanbecalculatedusingthefollowing relations(following Berryman[33]):Z+[ 6 RL�_ [ _ ]O`badceNfIg `badceFh�fIikj elhe mon j eFhe mqp jsrt rvuxw�y n T p T�z�{n�p | @ y zx{ m z}T (3.15)^4[ 6 G Z+[L 8 E 8 G :~ _ [ _ ]�`dadceXf�g `dadcelh�f�i j elhe mon j elhe mqp j rt r u w y n T p T�zx{n�p |}� y zx{ m z (3.16)
where u w y n T p T�z�{ is the probability of a triangle (with two sides n and p and an includedangle��� p t r y zx{ ) having all threeverticeslie within particleswhenplacedrandomlyin thecomposite.The
terms | @ y zx{ and |}� y zx{ areLegendrepolynomialsof order2 and4 respectively andaregivenby| @ y zx{ 6 8L y J z @ \�8 {�T| � y zx{ 6 8S y J G z � \kJ�: z @ E J {�YFor compositeswith constituentsthathaveasmallmoduluscontrast,theMilton boundsareremark-
ably closeto eachother. However, this is not be true for large moduluscontrastcompositeslike
PBX 9501[32].
Onemethodof calculatingthevaluesof the parametersZ+[ and ^4[ is to convert digital images
of PBX 9501 into binary (black and white) imageswith black representingparticlesand white
representingbinderandthenfollowing theprocedureoutlinedby Berryman[33, 34, 35, 36]. For
PBX9501,thevolumefractionof theparticlesiscloseto92%.It isalsoobservedthatthereflectivity
of differentfacesof theHMX crystalson a SEM micrographvarieswidely. Hence,it is extremely
difficult to obtainabinaryimageof thePBX 9501microstructurein orderthatthetwo parametersbe
calculated.Instead,we canmake theassumptionthatthepenetrablespheresmodel(wherespheres
placedrandomlyin the RVE may overlap) is representative of high volume fraction particulate
compositesandusethe valuesof Z [ and ^ [ listed by Berryman[33]. Thesevaluesareplottedin
Figure3.1andcanbeobservedto increaselinearly with increasingvolumefraction.
22
0 0.2 0.4 0.6 0.8 10.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Volume Fraction
ζ p, ηp
ζp (Curve Fit)
ηp (Curve Fit)
ζp (Computed*)
ηp (Computed*)
Figure 3.1. Parameters�+� and �4� for thepenetrablespheremodel(* = ValuesComputedby Berryman[33].)
Linearextrapolationfrom thedatashown in Figure3.1 for a volumefractionof 0.92gives �+� =
0.956and �4� = 0.937.Usingthesevaluesto calculatetheMilton boundsgivesthevaluesshown in
Table3.4.
Table3.4. Milton upperandlowerboundsfor PBX 9501.
Elastic Upper Lower ExperimentalModulus Bound Bound Modulus
(MPa) (MPa) (MPa)Bulk 11306 224 1111Shear 4959 68 370
TheMilton boundsaredefinitelyanimprovementover theHashin-Shtrikmanbounds.However,
theupperandlower boundsarestill quitefar apartfrom eachotherandthereforeprovide no useful
engineeringapproximationon the effective elasticmoduli underconsideration.Figure3.2 shows
a comparisonof the Voigt-Reuss,Hashin-Shtrikmanand Milton boundson the bulk and shear
modulusof PBX 9501asratiosof theexperimentallydeterminedvaluesshown in Table3.1.
23
0
2
4
6
8
10
12
14
16
18
20
Bulk Modulus
Shear Modulus
Bou
nds/
Exp
erim
enta
l val
ues
Voigt/Reuss BoundsHashin−Shtrikman BoundsMilton Bounds
Figure3.2. Comparisonof boundson thebulk andshearmodulusof PBX 9501with experimentalvalues.
3.2 Analytical MethodsAnalyticalmethodsfor approximatingeffective elasticmoduli of randomcompositeshavebeen
developedby researcherssincethe early 1900s. Early developmentswere basedon dilute dis-
persionsof particlesin a continuousmatrix assumingthat therewere no particle-particleinter-
actions.More recentdevelopmentshave exploredconcentrateddispersionswhereparticle-particle
interactionsareallowed.For highvolumefractionparticulatecomposites,thecompositespheresas-
semblage,thethree-phasemodel,theself-consistentscheme,andthedifferentialeffective medium
approachareof interest.Eachof thesemethodsmakescertainsimplifying assumptionsaboutthe
microstructureof thecomposites.Thesemethodsarediscussedbriefly andthepredictedeffective
moduli for PBX 9501arecomparedwith theexperimentalvaluesshown in Table3.1.
3.2.1 CompositeSpheresAssemblage
Thecompositespheresassemblage(CSA)proposedby Hashin[37] idealizesaparticulatecom-
positeusing sphericalparticlescoatedwith a layer of binder. The volume of the compositeis
assumedto befilled completelywith varioussizesof thesecoatedspheres.Theratioof theradiusof
a sphericalparticleto thethicknessof its bindercoatingreflectsthevolumefractionof particlesin
24
thecomposite.Thesolutionof theprobleminvolvesplacingacoatedspherein theeffectivemedium
andapplyinga hydrostaticstressat theboundaryof thecoatedsphere.This approachleadsto an
expressionfor theeffective bulk modulusthatcanbewrittenas�?�<���O��� ����� ��� �O� � � ������������������ (3.17)
Theeffective coefficient of thermalexpansionfor anisotropiccompositeformedfrom isotropic
componentsis givenby [29]� �<��� �¡ � � � �O�£¢ � ��� � �� ��� ���+¤ ¢ �� � ��¥ ���¦&¤ � (3.18)
This equationrequiresonly the isotropicbulk modulusof thecompositeto calculatetheeffective
coefficient of thermalexpansion. For high concentrationsof particles,the shearmoduluscannot
befoundaccuratelyusingtheCSA modelthoughlow expressionsthatarevalid in thedilute limit
exist. The CSA predictionfor PBX 9501 is shown in Table3.5. This value matchesthe lower
boundpredictedby theHashin-Shtrikmanboundsandis considerablylowerthantheexperimentally
determinedbulk modulus.
Table3.5. Compositespheresassemblagepredictionfor PBX 9501.
CSA Predicted Experimental PredictedBulk Modulus Bulk Modulus ThermalExpansion
(MPa) (MPa) ( § �¨q©«ª /K)148 1111 12.2882
3.2.2 Self-ConsistentSchemes
Theeffective stiffnesstensorof a dilute distribution of particlesin a continuousbindercanbe
expressedas[23] ¬ � � ¬ �F� ���® ¬ �P� ¬ ��¯v°�± ¬ ��² ¬ ��¯��*³ K��� ¯ (3.19)
where � �I´ thevolumefractionof particles,¬ � ´ thestiffnesstensorof theparticles,¬ � ´ thestiffnesstensorof thebinder,¬ � ´ theeffective stiffnesstensor, and,± ´ thetensorthatrelatestheappliedstrainto thestrainin aparticle.
25
Whenthevolumefractionsof particlesis morethan5%, thebinderin thedilute approximationis
replacedwith a materialthat possessestheunknown effective elasticpropertiesof the composite.
Thus,theexpressionfor theeffective stiffnesstensoris changedtoµN¶¸·�µX¹Fº*»�¼®½Wµ�¼P¾MµX¹�¿vÀ�Á&½WµF¼xÂ.µV¶o¿�º�ÃP½K»�¼o¿�Ä(3.20)
The above equationcanbe solved for the effective stiffnessof the compositefor variousparticle
shapes.This procedureis calledthe “self-consistentscheme”,the “effective mediumapproxima-
tion” andalsothe“coherentpotentialapproximation”.
Varioustypesof “self-consistent”approximationsof effectivecompositepropertiescanbefound
in the literature. Someof theseapproximationshave beenfound to generateexcellent effective
propertiesat low concentrationsof thedispersedcomponent.However, athighconcentrationswhen
themoduluscontrastbetweenthecomponentsis large,thesemethodsdo not performwell. An ex-
cellentcomparisonof self-consistentmodelswith thecommonlyusedMori-Tanakaapproximation
hasbeenprovided by BerrymanandBerge [38]. A survey of thesemethodsanda critical review
hasalsobeengiven by Christensen[39]. In general,thesemethodsareunsuitablefor materials
with both a high volumefraction of particulatesaswell asa high moduluscontrastbetweenthe
constituentsasis seenin compositeslike PBX 9501.
For aparticulatecompositecontainingadispersionof elasticspheres,theself-consistentscheme
leadsto two equationsin Å ¶ and Æ ¶ whichhave to besolvediteratively. TheseareÅ ¶ · Å ¹Çº »�¼ Å ¶ ½ Å ¼�¾ Å ¹�¿Å ¶ ºÉÈ Ê Å ¶Ê Å ¶ º�Ë Æ ¶�Ì ½ Å ¼I¾ Å ¶ ¿  (3.21)
Æ ¶ · Æ ¹Çº »�¼ Æ ¶ ½ Æ ¼�¾ Æ ¹�¿Æ ¶ ºÉÈkÍ Å ¶ ºÏÎ4Ð Æ ¶Î4Ñ Å ¶ º�ÎÒ Æ ¶4Ì ½ Æ ¼I¾ Æ ¶ ¿ Ä (3.22)
For thecomponentsof PBX9501,theseequationsleadto theeffectivebulk andshearmoduliand
thecorrespondingeffective coefficient of thermalexpansionareshown in Table3.6. Thepredicted
valuesof themoduli areconsiderablyhigherthantheexperimentalvalues(about10 timesfor the
bulk modulusandabout13 timesfor theshearmodulus).However, thesevaluesarelower thanany
of theupperboundsdiscussedin theprevioussections.On theotherhand,thepredictedcoefficient
of thermalexpansionis higherthantheHashin-Rosenupperbound.
The three-phasemodeldevelopedby ChristensenandLo [40] is anotherexampleof a “self-
consistent”modelthathasbeenusedconsiderablyby engineers.In this model,a third outershell
of materialpossessingthe effective propertiesof the compositeis addedto the compositesphere
26
Table3.6. Self consistentschemepredictionfor PBX 9501.
Bulk Modulus ShearModulus ThermalExpansionPredicted Experimental Predicted Experimental Predicted
(MPa) (MPa) (MPa) (MPa) ( Ó<ÔÕqÖ«× /K)11,044 1,111 4,700 370 12.9420
assemblage.This model predictsthe samebulk modulusas the CSA model. In addition, the
effective shearmoduluscanbefoundaftersolvingaquadraticequationof theformØÚÙlÛIÜÛ�Ý�Þ&ßVà*á�â ÙlÛ�ÜÛ�ÝoÞãà�ä�å Õ (3.23)
whereØçæ�ØéèKê�ëíì Û ëíì Û�Ý ì�î�ëíì�î Ý.ï ìâ æ â èKê�ëqì Û ëíì Û�Ý ì�î�ëíì�î Ý�ï ì and,ä æ ä èKê�ëqì Û ëíì Û�Ý ì�î�ëíì�î Ý�ï�ð
The effective shearmodulusof PBX 9501 calculatedusing the three-phasemodel is shown in
Table3.7. This modelpredictsvaluesof effective shearmodulusthat are lower than the Milton
lowerboundsshown in Table3.4.
Table3.7. Three-phasemodelpredictionfor PBX 9501.
Predicted ExperimentalShearModulus ShearModulus
(MPa) (MPa)52 370
3.2.3 Differ ential Effective Medium Approach
The differentialeffective mediumapproachis anotherschemefor approximatingthe effective
propertiesof compositescontaininga continuouscomponent(binder)anda dispersedcomponent
(particles).This schemehasbeenutilized by variousresearchers,mostly for low volumefractions
of thedispersedcomponent[23, 38, 41].
Theideabehindthisapproachis thataninfinitesimalvolumefractionof theparticlematerialis
addedto thebinderandtheeffective propertiesarecalculatedusinga dilute approximation.Next,
the binder is replacedwith the effective materialandthe calculationis carriedout againwith an
infinitesimalvolumefractionof particles.This processis repeateduntil theactualvolumefraction
27
of particlesis reached.Mathematically, this approachcanbe representedby a coupledsystemof
linearfirst orderordinarydifferentialequationsof theformñ+ò&ó7ô�õ�ö3÷oø?ù÷ ô õûú ñ ø õ�ó ø ù ö&üýø?ùNþãÿ�������ùø õ þ�ÿ������ ù ��� (3.24)ñ+ò<ó7ô�õ�ö3÷�Iù÷ ô õ ú ñ � õIó � ù ö&ü��ùVþ���ùø õ þ�� ù � �where � ù ú ��ù� ü���ø ùVþ ����ùø ù þ���� ù ���Solving thesystemof equationsusinga fourth-orderRunge-Kuttaschemegivestheeffective bulk
andshearmodulishown in Table3.8.Theseresultsshow thatthedifferentialschemeunderestimates
the bulk and shearmoduli thoughthe valuesare inside both the Hashin-Shtrikmanand Milton
bounds.
Table3.8. Differentialschemepredictionsof effective properties.
Bulk Modulus ShearModulus ThermalExpansion(MPa) (MPa) ( � ��� /K)
Predicted 229 83 12.5218Experiments 1111 370 -
Severalotherapproximationschemesexist thatgenerateanalyticalequationsrelatingtheeffec-
tive elasticmoduli to theconstituentmoduli andvolumefractions. Detailsof theschemescanbe
foundin themonographby Milton [26] andthepaperby Buryachenko [24] andreferencestherein.
However, noneof theseschemesprovide sufficiently accurateestimatesof theeffective moduli of
compositeswith highvolumefractionsof particlesandhighmoduluscontrastsbetweentheparticles
andthebinder.
3.3 Numerical ApproximationsThe effective elasticmoduli of a compositecanbe determinedapproximatelyby solving the
governingequationsusingnumericalmethods.This processinvolves the determinationof a rep-
resentative volumeelement(RVE), theapproximationof themicrostructureof the composite,the
choiceof appropriateboundaryconditionsandthesolutionof theresultingboundaryvalueproblem.
Numericalsolutionof suchproblemsrequiresthe RVE to be discretizedso that the geometryis
adequatelyapproximated.Thestressesandstrainsthatsolve theproblemarethenaveragedover the
28
volume(V) of theRVE. Theeffectivestiffnesstensor������! #" of thecompositeis thencalculatedfrom
therelation $&%(' ���*) � ����+ ," $-%/. #" (3.25)
where
' ��� arethestressesand
. ��� arethestrains.
Theearliestnumericalapproximationsof effective elasticmoduli werecarriedout usingfinite
differenceschemesby AdamsandDoner[42, 43]. Theseweretwo-dimensionalapproximationsfor
regular arraysof fibersin a matrix. Randomlygeneratedmicrostructuresin two dimensionswere
simulatedsoonafterthesepreliminaryinvestigations[44]. With improvementin thecapabilitiesof
computersmany researchershave approachedthis problemusingfinite elementmethods[45, 46],
boundaryintegralmethods[47, 48, 49], Fouriertransformbasedmethods[50, 51] andsoon. Three-
dimensionalsimulationsarealsoincreasinglybeingcarriedout [52, 53]. Recently, researchershave
alsousedtheconceptsof homogenizationtheoryto solve theproblemof determinationof effective
propertiesof composites[54, 55]. Themethodof cells[56] is anothernumericaltechniquethathas
beenusedwith considerablesuccessfor fiber-matrix composites.
Somerecentdevelopmentson the determinationof an optimum RVE are discussedin this
section,followed by a review of the literatureon two-dimensionalsimulationsfor regular arrays
of fibers. Two-dimensionalsimulationsof randomlydistributed fibersarediscussednext. Finite
elementmethodsarethemostfrequentlyusedsolutiontechniquesfor thesestudies.Someboundary
integral andFourier transformbasedmethodarealso discussed.Three-dimensionalsimulations
have beenlessfrequentlycarriedout becauseof thehigh computationalcostsinvolved. A few of
thesesimulationsarealsoreviewed in this section.Finally, themethodof cellsandits application
to variousmicromechanicsproblemsis reviewed.
3.3.1 The Representative VolumeElement
Primaryto theuseof numericalapproximationsof theeffective propertiesof compositesis the
conceptof therepresentative volumeelement(RVE). Thisconceptis similar to thecrystallographic
unit cell which is thebuilding block in thestructureof crystals[57]. Squareor cubicRVEsareused
for most numericalapproximationsbecauseof the easeof numericallysolving boundaryvalues
problemswith thesegeometries.Thedifficultiesinvolvedin generatingstatisticalinformationabout
particledistributionsandconcentrationsleadsto difficulties in the rigorousdeterminationof RVE
sizes.Hence,for mostapplications,RVE sizeshave beenratherarbitrary.
Sab[58] hasshown that if an RVE exists for a randomcompositematerial,the homogenized
propertiesof the material can be calculatedby the simulation of one single realizationof the
medium.The“ergodic” hypothesis,whichassumesthattheensembleaverageis equalto thevolume
29
average,hasbeenusedto arrive at this conclusion.The ensembleaverageis the meanof a large
numberof realizationsof themicrostructure.Thevolumeaverage,on theotherhand,is theaverage
astheRVE volumebecomesinfinitely largecomparedto thevolumeof a particle. However, such
a realizationmay leadto a anextremelylargeRVE anda morepracticalapproachis to simulatea
large numberof differentrealizationson smallerRVEs so that boundson the effective properties
areobtained.
DruganandWillis [59] have shown, usingnon-localequationsfor theelasticresponse,that for
a randomdistribution of identical spheresthe RVE size is approximatelytwo spherediameters.
A three-dimensionalfinite elementstudy on the optimal size of the RVE hasbeencarriedout
by Gusev [52]. The compositeconsideredwascomposedof around26% by volumeof identical
spheresin a continuousmatrix. The moduluscontrastbetweenthe componentswas around20.
Thesimulationsby Gusev show thattheoptimumRVE sizeis around3-5 timesthespherediameter.
However, it is doubtfulif thesameconclusionscanbedrawn for polydispersecompositescontaining
spheresof many differentsizes.Two andthreedimensionalfinite elementanalyses(usingtriangular
andtetrahedralelements)of two phasecompositesmadeup of sphericalinclusionsin a continuous
matrix have beencarriedout by Bohm and Han [53]. The resultsshow that thoughrelatively
smallRVEs canbeusedfor determiningeffective elasticmoduli, elastic-plasticor othernonlinear
behaviors requiremuchlargerRVEs to beaccuratelypredicted.Thevalidity of theseconclusions
for compositeswith highmoduluscontrastcannotbeascertainedfrom thesenumericalstudies.
3.3.2 Finite Differ enceApproximations
Early numericalapproximationsof effective moduli were carriedout on unidirectionalfiber
compositeswith two-dimensionalapproximationsof the elasticfields. Regular arraysof circular
fibersweremodeledusingfinite differenceschemesfor transversenormalmoduli andlongitudinal
shearmoduli by AdamsandDoner[42, 43]. Thesesimulationswerecarriedout for fiber volume
fractionsup to 78%. Plotsweregeneratedfor the normalizedtransversenormalandlongitudinal
shearstiffnessesat variousmoduluscontrastsbetweenthefibersandthematrix. Theresultsshow
that beyond a moduluscontrastof around1000, the effective stiffnessbecomesconstantfor the
Poisson’s ratiosused.However, wehave foundthatthis is not truewhenoneof thecomponentsis a
rubberymaterial.Hexagonalrandompackingsof fiberswerestudiedby AdamsandTsai[44] using
finite differencesandfound to generatebetterapproximationsof actualfiber compositebehavior
thanregularsquarearrayor regularhexagonalarraypackings.
Finite differenceapproximationsweresoonreplacedby finite elementapproximationsas the
primary simulationtechnique.This wasprobablyduethe improved discretizationof particlege-
ometriesusingfinite elements.However, the finite differenceapproximationshave recentlybeen
30
usedby Ostoja-Starzewski et al. [60] to computethe effective propertiesof a randomcomposite
with circular inclusionsundergoing damage.The lattice-bondbasedstochasticmodelfor damage
growth is easily parallelizedwith finite differences. This methodhasalso beenparallelizedfor
gradedinterfacesbetweenparticlesandmatrixandshown to generatecloseboundson theeffective
thermalconductivity for particlevolumefractionsof about50%[61].
3.3.3 Finite ElementApproximations
Since actual microstructuresof compositesare difficult to obtain and simulate,most finite
elementsimulationsof themicromechanicsof compositeshave involvedsquareor hexagonalarrays
of fibersin two dimensions.With decreasingcomputationalcosts,complex two-dimensionaland
three-dimensionalproblemsin themicromechanicsof compositesarebeinginvestigatedwith finite
elements.
3.3.3.1 Regular Arrays in Two Dimensions
Finite elementapproximationsof theeffective behavior of regulararraysof unidirectionalcir-
cularfiberswerecarriedoutby AdamsandCrane[62] usingageneralizedplanestrainassumption.
The RVE was chosento containone fiber and was discretizedusing triangles. Eachnodewas
assignedfour degreesof freedom- two for thein-planedisplacementsandtwo for theout-of-plane
displacements.This formulationcanbeusedto determinethethree-dimensionalstateof stressfrom
a two-dimensionalsolution. It is not obvious, however, how the formulation can be usedfor a
randomdistribution of particlesastheboundaryconditionsbecomeconsiderablymorecomplex.
Standarddisplacementbasedfinite elementformulationsin two dimensionshave beenusedto
modelcircularfibersby ZhangandEvans[63] andto modelrectangularfibersby Shietal. [64]. The
studyby ZhangandEvansmodeleda RVE containinga circular fiber coatedby a annularmatrix
ring andvalidatedthe concentriccylinders model [65]. Thoughthe circular RVE leadsdirectly
to thepredictionof isotropicproperties(unlike a squareor rectangularRVE), this approachis not
applicableto highvolumefractionPBXssinceeachfiber is coateduniformly with matrixandthere
is no fiber-fiber contact. The useof rectangularfibersby Shi et al. to modelwhisker reinforced
compositesis a grosssimplificationof the actualmicrostructureandfiber geometry. Thoughthis
simplificationhastheadvantageof beingeasilydiscretized,thesharpcornersof theparticlesleadto
highstressconcentrationsandleadto numericalerrors.Weexplorebothcircularandsquareparticle
distributionsin Chapter7.
Displacementbasedfinite elementsolutionshave beenfound to predict effective properties
that overestimatethe actualproperties.On the otherhand,force basedsolutionsprovide a lower
boundon the actualproperties.However, force basedfinite elementmethodsarenot often used
31
becauseof thedifficultiesinvolvedin their formulation.Recently, Lukkassenetal. [66] haveshown
how homogenizationtheorycanbeusedto computetheeffective propertiesof unidirectionalfiber
compositesusingaforcebasedfinite elementapproximationof theeffectivepropertiesof aunit cell.
A recentfinite elementstudyhasbeencarriedouton regulararraysof fibersby Pecullanetal. [67].
This study hasfound that for a compositewith high moduluscontrastand a compliantmatrix,
the forcebasedeffective stiffnesstensoris moreaccurate.Forcebasedfinite elementmethodsare
not utilized in this researchbecauseof thedifficulties is formulationandimplementationof these
methods.
Interestingly, Pecullanet al. [67] have also observed that replacementof the smallestscale
microstructureby the equivalenthomogeneousmaterialdoesnot causelarge errorsin calculation
of the effective stiffnesstensors.This result is of interestin this researchbecausegenerationof
microstructuresoccupying more that 86% of the volume is difficult. Instead,we can generate
microstructuresthat occupy about86% of the volume and replacethe remainingvolume with a
“dirty” binder(abinderwith theeffectivepropertiesof amixtureof particlesandtheoriginalbinder)
without muchlossin accuracy.
Recentfinite elementanalysesof the micromechanicsof two-dimensionalcompositeshave
focusedmostly on determiningthe effective inelastic response[68, 69, 70]. However, most of
theseapproachesusedisplacementbasedfinite elementmethodson regularpackingsof fibersand
do notattemptto solve theproblemsassociatedwith high volumefractions( 0 90%)of fibers.
3.3.3.2 RandomDistrib utions in Two Dimensions
Modelsusingregulararraysof fibersprovide reasonablygoodapproximationsof theeffective
elasticpropertiesof fiber composites.However, for particulatecomposites,this is not trueandthe
complex microstructurehasto betakeninto account.Thisimpliesthatthree-dimensionalmodelsare
required.Thehigh computationalcostinvolved in modelingparticulatecompositemicrostructures
in threedimensionshasledto thedevelopmentof two-dimensionaltechniquesthatperformwell for
someof thesecomposites.
Ramakrishnanet al. [71] have useda generalizedplanestrain approachto modelparticulate
metalmatrix composites.Thecompositesconsideredin thestudyhada maximumvolumefraction
of 40% of particles. Particlesof variousshapesand sizeswere randomlydistributed in a two-
dimensionalsquareRVE. The effective Young’s moduluswasdeterminedby the applicationof a
uniformunidirectionaldisplacement.Theeffectivebulk moduluswasdeterminedby applyingequal
displacementsin thethreeorthogonaldirections.TheeffectivePoisson’s ratiowasdeterminedfrom
the effective bulk and Young’s moduli. The effective coefficient of thermalexpansionwas also
determinedusingfinite elements.It is observed that the shapesof the particlesdo not have any
32
significanteffect on the effective elasticpropertieseven thoughmany particleshave sharpacute
anglesandthereforehigh stressconcentrations.Periodicboundaryconditionsarenot usedin the
approach.TheRVE sizeis alsochosenarbitrarily. Theuseof thebulk modulusto determinethe
Poisson’s ratioassumesthatthematerialis isotropic.However, thetwo-dimensionalapproximation
automaticallyimpliesthateachparticleextendscontinuouslyin theout-of-planedirectionandhence
makesthematerialanisotropic.Wedonotusethisapproachin thisresearchfor thesereasonsthough
theauthorscitegoodagreementwith experimentaldata.
Randomdistributions of particlesin two dimensionshave also beenstudiedby Theocariset
al. [72] in thecontext of determiningtheeffective Poisson’s ratio. Finite elementsimulationswere
carriedout on a unit cell. Periodicboundaryconditionsanduniform pre-stresseswereappliedto
theunit cell. Theeffective elasticpropertiesweredeterminedusinga strainenergy matchbetween
a cell simulatingthemicrostructureandanequivalenthomogeneouscell. This studyis of interest
to usbecausetheprocedureof determiningtheeffective moduli is well groundedin theoryanduses
theHill condition.It alsoshows thatthePoisson’s ratiosthatarecalculatedusingtwo-dimensional
modelsareactually two-dimensionalPoisson’s ratios that have an upperboundof 1.0 insteadof
the0.5 for the three-dimensionalcase.Thestudyby Theocariset al. [72] alsosuggeststhatsharp
cornersin particlesdo not have any significanteffect on theeffective elasticproperties- assuming
perfectinterfacialbonding.
JiaandPovirk [73] have useda subgridscalefinite elementmodelbasedtechniqueto calculate
theeffective moduli of a two phasecompositecontainingrandomlydistributedsquareinclusions.
A window of theRVE is chosenin thefirst stageof thecalculationsandmovedover theRVE. Cal-
culationsof theeffective propertiesarecarriedout at eachlocationof thewindow. Theseeffective
propertiesarethenassignedto a smallermeshfor fastercalculationsof the overall properties.It
is found that the error in the estimationis small for the componentpropertiesconsideredby Jia
andPovirk. This approachis similar in somerespectsto the two-stepgeneralizedmethodof cells
techniquediscussedin Chapter7.
Anothertechniquethathasbeenusedto determinetheeffective propertiesof two-dimensional
compositeswith complex microstructuresis the multiphasefinite elementmethod[74]. The ap-
proachis to assigndifferentmaterialpropertiesto differentGausspointsin afinite elementanalysis
of a complex microstructure.This approximationobviatesthe needto generatecomplex meshes
to describethe geometry. Mishnaevsky et al. [74] have usedthe methodto determinethe elastic
fieldsin metalmatrixcomposites.Microstructureshavebeenobtainedfor thesestudiesfrom digital
images.Thisapproachcanbeusedto modelmicrostructureswhereadjacentparticlesareveryclose
to oneanother. However, therobustnessof themethodis still not very well established.
33
Kwanet al. [75] have usedrandomlydistributedparticlesof arbitrarysizesandshapesto study
theeffectivebehavior of concrete.Thoughtheapproachis similar to mostdisplacementbasedfinite
elementapproaches,it is of interestthat the interfaceparticlesandthe matrix hasbeenmodeled
using zero-thicknessinterfaceelements.Theseinterfaceelementscanbe used,in this research,
both for themodelingof debondingandcracksaswell asfor a very thin layer of binderbetween
particles.
3.3.3.3 ApproximationsusingHomogenizationTheory
Themathematicaltheoryof homogenization[76] hasrecentlybecomeanestablishedapproach
for determiningeffective propertiesof periodiccomposites[77]. Thegoverningdifferentialequa-
tionswith rapidlyvaryingcoefficientsarereplacedby differentialequationswith constantor slowly
varying coefficients. Asymptoticexpansionsof the field variablesalongwith the assumptionof
periodicity lead to this transformedset of equations. The new set of equationsare called the
Y-periodic homogenizationproblembecausethe repeatingcell is called 1 in the notationused
in the theory. The Y-periodic homogenizationproblemcan be solved using finite elementsor
othertechniques.The assumptionof periodicity doesnot precludetheapplicationof this method
to particulatecomposites.We can always assumethat an RVE containingrandomlydistributed
particlesis repeatedperiodicallyin space.
HassaniandHinton [78] have usedfinite elementanalysesalongwith homogenizationtheory
to solve theeffective modulusproblemfor variousrankedlaminatesandfor cellularmaterialswith
rectangularholes. Incompatibleand hybrid finite elementshave also beenusedto solve the Y-
periodic homogenizationproblemfor fiber composites[79]. However, thesestudieshave used
microstructureswith regulararraysof fibers.
Ghoshandco-workers [54, 55] have usedhomogenizationtheoryalongwith the Voronoi cell
finite elementmethodto modelRVEs containingrandomdistributionsof particlesat volumefrac-
tions of up to 50%. In this approach,particle locationsare generatedwithin the RVE using a
randomprocess.A weightedVoronoi tessellationof theseparticlesis carriedout to generatea set
of Voronoicells.Eachcell is amulti-sidedpolygonandcontainsasingleparticle.Homogenization
theory is usedto model the effect of a single particle on the propertiesof a Voronoi cell. The
approachshows goodagreementwith detailedfinite elementanalysesof thesamemicrostructure.
However, for highparticlevolumefractions( 2 80%)theVoronoitessellationleadsto needleshaped
cells in two dimensions.If the methodis extendedto three-dimensions,not only is it difficult to
generateparticledistributionsthatfill morethan55%of thevolume,generationof weightedVoronoi
tessellationsbecomesconsiderablymoreinvolved. Moreover, theeffective propertiesobtainedby
finite elementanalysesdependstronglyon thechoiceof elementtypeandsizeascanbeobserved
34
from thesimulationsof randomlyorientedshortfibercompositesby CourageandSchreurs[80].
3.3.3.4 ApproximationsusingStochasticFinite Elements
In continuumdescriptionsof composites,the constitutive relationshipis only a function of
spatialposition.Stochasticdescriptionsof theconstitutive relationassumethat thestiffnesstensor
is a randomfield with continuousrealizations.In otherwords,anadditional“stochastic”variableis
addedthecontinuumdescription.Stochasticfinite elementanalysesattemptto solve this modified
problemusingfinite elementtechniques.Thesemethodsareapplicableto particulatecomposites
wheretheparticledistributionscanvary in a randommanner.
Ostoja-Starzewski [81] hasperformedstochasticfinite elementanalyseson two-dimensional
compositesreinforcedby randomlylocateddisks. Numeroussimulationshave beencarriedout
to obtain boundson the effective stiffnesstensor. Theseanalysesshow, for a given RVE size,
that thepredictedboundsdeviate from eachotherby a 0.5%asthemoduluscontrastbetweenthe
componentsreachesabout20. It is not clearfrom thedatawhatthedifferencebetweenthebounds
wouldbefor highermoduluscontrasts.
Huyseand Maes[82] have usedstochasticfinite elementanalyses(using a trussnetwork to
representaparticulatecomposite)to determinetheautocorrelationandcross-correlationcoefficients
betweenvarious elastic constants. Ostoja-Starzewski [83] has suggestedthat thesecorrelation
coefficientscouldbeeasilydeterminedfor particulatecomposites.This informationcouldbethen
be usedto generateboundson the effective elasticresponsewithout resortingto time consuming
numericalsimulationsof differentrealizationsof themicrostructure.
Theanalysesof HuyseandMaes[82] show thatforcebasedfinite elementformulationsprovide
betterestimatesof theeffectivepropertiesthanthedisplacementbasedmethod.Similarconclusions
canbedrawn form thestochasticfinite elementanalysescarriedoutby Kaminski andKleiber [84].
3.3.3.5 ThreeDimensionalApproximations
Most three-dimensionalfinite elementstudiesof themicromechanicsof compositeshave dealt
with periodic microstructures[46]. Tetrahedralelementsare the most commonlyusedin these
simulations[85] while someuse hexahedralelements[86]. The techniquesof computingthe
effective propertiesfor three-dimensionalproblemareessentiallythesameasthosediscussedfor
two-dimensionalproblems. However, three-dimensionalanalysesprovide someinsightsinto the
mechanicsof compositesthat arenot obvious in two-dimensionalstudies.Further, techniquesof
microstructureandmeshgenerationin threedimensionsareinstructive in thecontext of PBX-like
particulatecomposites.In addition,thecomputationalcostsof three-dimensionalstudiesleadto the
explorationof efficient implementationsof numericaltechniquesthatareof interestin thisresearch.
35
Variousapproacheshave beenusedto generatethree-dimensionalmicrostructuresfor simula-
tionsof particulatecomposites.Themostcommonlyusedmethodis randomsequentialplacement
of spheresin a RVE, alsocalledtheMonteCarlo approach.The threedimensionalfinite element
simulationsof Gusev [52] useda MonteCarloapproachto generaterealizationsof thedistribution
of identically sizedspheresinsidea cubic RVE. Only about30% of the volumeof the RVE was
filled in thestudy. Tetrahedralelementswereusedto discretizethegeometry. This approachis not
well suitedfor high volumefraction compositeslike PBX 9501. First, theMonteCarlo approach
of placingparticlesbecomeextremely inefficient beyond volume fractionsof 55-60%. Meshing
of close-packed particlesusingtetrahedralelementsleadsto extremelyskewedelementsandpoor
numericalperformance.
A morepracticableapproach,for PBX-like materials,is a digital imageprocessingbasedap-
proachadoptedby GarbozciandDay [87]. Themethodusesdigital imageprocessingtechniquesto
generatethree-dimensionalfinite elementmeshesfor complex microstructures.X-ray tomography
is usedto generatethree-dimensionalvoxelizedimages.Eachvoxel is thenmodeledasan eight-
nodedlinear displacementfinite element. Teradaet al. [88] have alsodevelopeddetaileddigital
imagebasedmodelsof compositesthat usetwo-dimensionalslicesto generatethreedimensional
microstructures.Hexahedralelementsaregenerateddirectly from theimagesfor thesemodelsalso.
Thoughdigital techniquesappearlucrative, advancedimageprocessingtechniquesarerequiredto
generatemicrostructuresfor PBX materials.This is becausethehigh volumefractionsandsimilar
densitiesandreflectivities of particlesandbindermake it difficult to identify thecomponentsof the
compositefrom images.
A questionthatarisesfor PBXsis whetherthebinder“wets” all theparticles.In otherwords,it
is of interestto know theamountof strainthat leadsto interfacialdebondingbetweentheparticles
andthebinder. A three-dimensionalfinite elementstudyof theeffect of interfaceson thestresses
in compositescontainingsphericalinclusionsarrangedin a cubic array hasbeencarriedout by
Dong and Wu [89]. The resultsindicate that the assumptionof perfectbondingusedin many
micromechanicsstudiesmay not be appropriatefor high concentrationsof particleseven when
smallstrainsareapplied.This is becauseveryhigh interfacialstressesaredevelopedis theparticles
areto remainbondedto thebinder.
Multigrid finite elementmethodsoftenprove to beconsiderablymorecomputationallyefficient
than standardfinite elementmethods. An implementationof a multigrid finite elementmethod
basedon uniform grids hasbeenusedby Zohdi and Wriggers [90] to solve three-dimensional
elasticityproblemsfor compositesreinforcedwith spheres.Variouserrorsin approximationhave
beenexploredfor volumefractionsof upto 50%of spheres.Resultsarecomparedto variousbounds
36
andcurvefits to thenumericalresultshavebeenpresented.It is doubtfulthattheresultingequations
canbeusedfor compositeswith otherconstituents.However, themultigrid methodcanbeusedto
computetheelasticfields for complex materialsbecauseof thehigh computationalefficiency that
canbeachievedby thismethods.
3.3.4 DiscreteModels
Discretemodels,e.g., springnetwork models,are receiving renewed attentionbecausesome
of the discretizationissuesinvolved in other numericalmodelsbecomemore tractable. Two-
dimensionaltriangularspringnetwork modelshave beenusedby Day et al. [91] to determinethe
effective elasticresponseof platescontainingrandomlylocatedholes.Thecomputationsshow that
thesemodelsgeneratequite accurateresults. Digital imagescan easily be resolved into spring
networks.Randomcompositescanthereforebeeasilymodeledusingthesetechniques.
Toi andKiyoshe[92] useda threedimensionaldiscretemodelconsistingof springsandrigid
crystalsto determinethe effective mechanicalpropertiesof polycrystalswith damage. The mi-
crostructureis generatedusinga three-dimensionalVoronoitessellationof asetof randomlygener-
atedpoints. This methodis of interestin this researchbecauseparticlesin PBXs arealmostrigid
comparedto thebinder. Thehighvolumefractionof particlesmakesPBXsappearlikepolycrystals.
However, the large variation in particle sizesin PBXs requiresthe use of a weightedVoronoi
tessellationto generatethemicrostructuresof interest.This processis extremelycomplex in three
dimensions.In addition,certainVoronoi cellshave to be filled with binderto accountfor the8%
of binderin PBX 9501,for example. This assignmentof binderto Voronoi cells will necessarily
be arbitraryandwill lead to pockets of binderas is observed in the squareparticledistributions
discussedin Chapter7.
3.3.5 Integral Equation BasedApproximations
Boundaryintegral basedmethodshave beenusedwith somesuccessfor determiningtheeffec-
tivemechanicalandthermalpropertiesof two-dimensionalcomposites(e.g.,RizzoandShippy [47],
AchenbachandZhu [48], Papathanasiouet al. [93], Helsing[94, 95]). Thecomputationsof Rizzo
and Shippy [47] for squareinclusionsavoided calculationsof stressesat the cornersingularity
regions. The calculationsof AchenbachandZhu [48] werecarriedout on singlecircular inclu-
sionsusingstandardboundaryelementtechniques.Similar methodshave beenusedto determine
the effective elasticmoduli of two-dimensionalcompositeswith low volumefractionsof circular
inclusionsby Papathanasiouetal. [93].
The interfaceintegral methodof Helsing[95, 96] hasbeenusedto generateaccurateeffective
elasticpropertiesof periodiccompositesin two dimensions.An Airy stressfunction basedcom-
37
plex variablerepresentationof the governingdifferentialequationis convertedinto the Sherman-
Lauricellatype integral equationin this technique.The integral equationis solved usinga matrix
free Nystrom algorithm[95]. The useof complex variablesleadsto the methodbeingapplicable
only to two-dimensionalproblemsin its currentform. If theparticlestoucheachotheror have high
moduluscontrast,convergenceis reportedto berelatively slow. Theimplementationof theNystrom
algorithmconsistsof severalstepsandis quiteinvolved.Thishasmadethismethodunattractive for
this research.
TheHelsingmethodhasbeenusedto determineaccurateeffective elasticmoduli of RVEscon-
taininglargenumbersof complex shapedinclusionsnearlyin contact[97]. Thistechniquepromises
to beoneof thebestavailablefor two-dimensionalanalysisof the low strainratemicromechanics
of composites.It is especiallysuitedfor problemsthatinvolve stresssingularities.
3.3.6 Fourier Transform BasedApproximations
Complex microstructureshave alsobeenstudiedby MoulinecandSuquet[50] usinga Fourier
transformbasednumericalapproachto solve theunit cell problem.This approachtakesadvanced
of theassumedperiodicityof theelasticfieldsandby reducingthegoverningdifferentialequations
to the Lippman-Schwingerequationform both in real spaceandFourier space. The solution is
thenobtainedusingan explicit algorithmthat alternatesbetweenthe real andthe Fourier spaces.
Discretizationof theproblemis carriedout usinga regulargrid of pixelsor voxelsgeneratedfrom
imagesof microstructures.Theadvantageof thismethodis thatspecialconsiderationis notrequired
for materialsthatarenearlyincompressible(asis neededto avoid elementlocking in finite element
approaches).However, for highmoduluscontrastbetweenthecomponents,therateof convergence
is slow. This problemhasbeenpartially solvedusinganacceleratedconvergencemethod[51, 98]
that convergesasthe squareroot of the moduluscontrast.This methodhasbeenappliedto two-
dimensionalcompositesbut caneasilybeextendedto three-dimensionalproblems.
The integral equationbasedmethodof Helsing and the Fourier transformbasedmethodof
MoulinecandSuquetappearto bethebestfor numericallystudyingthelinearelasticmicromechan-
icsof polymerbondedexplosives.TheFouriertransformbasedmethodis morelucrative becauseit
caneasilybeextendedto modelthree-dimensionalproblemsandinelasticmaterialbehavior.
In general,thenumericalsimulationof thethermomechanicalbehavior of particulatecomposites
requireslarge computationalresources.Sincesuchresourcesmay not alwaysbe available to an
engineer, we next exploresomesimplerapproximationsthatmaybeusedto generateengineering
estimatesfor the effective thermoelasticpropertiesof composites. In particular, we look at the
methodof cells[56].
38
3.4 Method of CellsThe methodof cells (MOC) [56, 99] hasbeenusedto model the micromechanicalbehavior
of different typesof compositeswith relative success.The advantageof this methodover other
numericaltechniquesis that the full set of effective elasticpropertiescan be calculatedin one
stepinsteadof solving a numberof boundaryvalueproblemswith differentboundaryconditions.
An averagingtechniquethatsatisfiessubcellcontinuityandequilibriumin anaveragesenseusing
integralsover subcellboundariesis usedby themethodof cells. Theproblemof discretizationis
alsominimizedbecausea regularrectangulargrid is used.This methodhasbeenshown to bemore
computationallyefficient thanfinite elementsfor modelingfibercomposites[99].
The original methodof cells was extendedby Paley and Aboudi [100] from using a single
subcell to representthe inclusionsto a more generalversionwith multiple subcells. This new
methodhasbeenreferredto astheGeneralizedMethodof Cells (GMC) [99]. Comparisonsof the
resultsfrom GMC with finite elementanalysesfor a boron/aluminumcompositewith a volume
fractionof 0.46of boronfibersshowed remarkableagreement[99]. In addition,GMC wasfound
to bemorecomputationallyefficient thanfinite elementanalysesfor squarearraysof fibersin two
dimensions[101]. FarfewerGMC subcellswerefoundto benecessarythanfinite elementsto arrive
at thesamedegreeof accuracy in thesolution. However, thecomputationalefficiency of GMC is
becomesworsethanthatof finiteelementsasthenumberof subcellsincreases.Thisisdueto thesize
of thematrix that is invertedis thesquare/cubeof thenumberof subcellsin two/threedimensions.
This leadsto large memoryrequirementsandlarge computationaltimeswhile modelingcomplex
microstructures.
RobertsonandMall [102] attemptedan improvementover GMC by extendingthe MOC ap-
proachto threedimensionswith the additionalrestrictionthat compositenormalstressesdo not
produceany shearstressesin thefiber or matrix. A setof closedform equationsweregivenfor the
effective elasticconstantsusingthecell modelthatshowedgoodagreementwith experimentaldata
from boron-aluminumcompositesfor fiber volumefractionsrangingfrom 40%to 70%. However,
themethoddoesnotallow for largegridsof subcells.Thismakesit attractive for modelingcomplex
microstructures.
Themethodof cellshasalsobeenextendedtosolvethethree-dimensionalproblemof short-fiber
compositesby Aboudi [103]. This formulationof GMC in threedimensionsleadsto a systemof
equationsof 3547698�: whereN is thenumberof subcellsin eachcoordinatedirection.Orozco[104]
haspartially solved this problemby identifying thesparsitycharacteristicsof thesystemof equa-
tionsandby usingtheHarwellBoeingsuiteof sparsesolvers.Thecomputationalefficiency of GMC
hasbeenfurtherimprovedafterreformulationby PinderaandBednarcyk [105, 106, 107, 108]. The
39
reformulationhastakenadvantageof thecontinuityof tractionsacrosssubcellsto obtaina system
of ;5<7=9>@? equationsin threedimensions,therebygreatlyimproving theefficiency of themethod.
Low et al. [109] have useda two-stephomogenizationschemeusing GMC to determinethe
effective propertiesof unidirectionalfiber compositeswith interphaseregions. The interphase
region is discretizedinto a numberof subcellsand the variation of elasticpropertiesalong the
interphaseis modeledby assigningdifferent valuesto different subcells. This subassemblyof
subcellsis thenhomogenizedusingGMC. Theinterfaceis thenrepresentedasa few homogeneous
cellsin thenext stepthatgeneratesthefinal homogeneouseffectivepropertiesusingGMC.A similar
concepthasbeenusedfor modelingPBX microstructuresin Chapter7.
GMC hasalso beenextendedto model interfacial debondingand the resultscomparedwith
finite elementsimulationsfor squarearraysof disks[110]. Displacementjumpsacrossinterfaces
aremodeledwith springsin this approach.An alternative approachusinga Gaussiandistribution
basedinterfacedebondingmodelhasbeendevelopedbyRobertsonandMall [111]. BoththeAboudi
modelandtheRobertsonandMall modelrequiresometrial anderrorto determinetheappropriate
modelparameters.Thesemodelshave beenappliedto metalmatrix composites[111, 112, 113]
but comparisonswith experimentaldatahave not beenprovided in mostcases.Theuseof spring-
like displacementjump factorshasalsobeenavoided in the study by Lissenden[114]. Instead,
interfacedebondingis describedby a cubic polynomialthat relatesthe interfacial tractionsto the
interfacial displacementsin a smoothfashion. Anotherapproachof modelinginterfacial damage
within the context of GMC hasbeento usean uniaxial constitutive law for the interfacial zone
andthento increasethe sizeof the zonewith progressive damage[115]. A Weibull distribution
basedprobability density function hasbeenusedto describethe effective interfacial debonding
strainin additionto theinterfaceconstitutive law in theprogressive damagemodelto obtainbetter
agreementwith experimentaldata[116]. Difficulties involved in the determinationof interfacial
propertiesmake thesemodelsdifficult to assessandvalidate.
Thereis a lack of couplingbetweenthenormalandshearstressesandstrainsin GMC. Bednar-
cyk andArnold [117] claimthatthis lackof couplingmakesfor an“ultra-efficient” micromechanics
model. However, our studieshave shown that this lack of couplingleadsto grossunderestimation
of shearmoduli. Recently, a few attemptshave beenmadeto rectify theshearcouplingproblem.
Williams andAboudi [118] have attemptedto solve the problemfor periodicarraysof fibersby
using a third order expansionfor the displacementinsteadof the first order expansionusedin
the original methodof cells [118]. However, this approachleadsto a large systemof equations
andtheefficienciesintroducedby PinderaandBednarcyk areno longerapplicable.An alternative
approachhasbeentakenby Ganetal. [119] to includenormal-shearcouplingin theGMC analysis.
40
The original GMC assumesthat thereis tractioncontinuity acrossall cell andsubcellinterfaces.
The modificationmadeby Ganet al. removes this constraintand insteadtries to satisfysubcell
equilibriumandcompatibility. Resultsobtainedby thenew methodshow a muchbetterprediction
of shearmoduli thantheoriginal GMC withoutmuchgreatercomputationalrequirements.
The generalizedmethodof cells hasbeenusedfor the calculationof effective propertiesof
polymerbondedexplosives in this research.The detailsof the methodareprovided in Chapter4
andsomeresultsusingGMC areprovidedin Chapters6 and 7.
CHAPTER 4
THE GENERALIZED METHOD OF CELLS
A simplifiedversionof thereformulatedthree-dimensionalGMC is describedin thischapter. It
is assumedthat a RVE exists for the compositeunderconsideration.Sincewe areinterestedin a
randomparticulatecomposite,we assumethat theRVE is cubicandthesubcellsareof equalsize
for simplicity in thefollowing derivation. A schematicof thediscretizationof theRVE alongwith
thenotationusedis shown in Figure4.1.
X,1
Y,2
Z,3
RVE
Subcell
α
β
γ
Figure 4.1. Subcellsandnotationusedin GMC.
For simplicity, we forego thederivationof theequationsfor effective plasticstrainsandtherep-
resentationof interfacialdebonding.It shouldbenoted,however, thatour implementationof GMC
includesthecapabilityof variablesubcellsizes,plasticstrains,andinterfacial debondingbetween
subcells.Wefollow thenotationusedby Aboudi [56] wherepossible.A differentform of theGMC
equationscanbefoundin thereportbyBednarcyk andPindera[107]. Thethree-dimensionalversion
of GMC canbe easily convertedinto the two-dimensionalversionby suppressingthe equations
relatingto thethird dimension.
42
In thisderivation,we assumea lineardisplacementfield for eachsubcellof theformA�BDC�EGFGHI JLK BDC�EGFGHI MONQPSRTN5U�RTN5VXWZY\[ B]C�HP_^ BDC�EGFGHI Y`[ BaEbHUdc BDC�EGFGHI Y`[ BeFGHVgf B]C�E@FGHI (4.1)
whereh representsthecoordinatedirectionandtakesthevalues’1’,’2’ or ’3’,MON P RTN U RTN V W is theglobalcoordinatesystemof theRVE,MO[ BDC�HP RT[ BaEbHU RT[ BeFGHV W is thecoordinatesystemlocal to asubcell MjilkZmnW+RA�BDC�EGFGHI opM AqB]C�E@FGHP R AqB]C�E@FGHU R A�BDC�EGFGHV W arethedisplacementsin asubcell MjilkZmZW+RK BDC�EGFGHI is thedisplacementat thecenterof asubcell MjilkZmnW+R^ BDC�EGFGHI is thelocal variationof displacementin the’1’ direction,c BDC�EGFGHI is thelocal variationof displacementin the’2’ direction,and,f BDC�EGFGHI is thelocal variationof displacementin the’3’ direction.
Williams andAboudi [118] have useda field containinghigherorder termsso that the shear
andnormaldisplacementscanbecoupled.However, that formulationleadsto a muchlargersetof
equations.Wedo notexploretheapproachof Williams andAboudi in this formulation.
4.1 AverageStrain RelationsThestraindisplacementequationsfor thesubcellaregivenbyr#BDC�EGFGHI�s Jutv Mjw I A BDC�EGFGHs Y�w s A BDC�EGFGHI W (4.2)
where w&PxJ ww�[ B]C�HP R w�UyJ ww�[ BaEbHU R w�VyJ ww�[ BaFGHV{zIf eachsubcell MjinkZmnW hasthesamedimensionsM v�| R v�| R v�| W thentheaveragestrainin thesubcellis
definedasavolumeaverageof thestrainfield over thesubcellas} r BDC�EGFGHI�s ~ J t���x�-��� r BDC�EGFGHI�s | ���(4.3)
where���
is thevolumeof thesubcell,and,t�����-� � | ��� o t��| V �\�� � �\�� � �\�� � | [ B]C�HP | [ BaEbHU | [ BaFGHV zTheaveragestrainin thesubcellcanthenbeobtainedin termsof thedisplacementfield variables.
For example,for thenormalstraincomponentin the r PTP direction,equation(4.3)becomes} r BDC�EGFGHPTP ~�J t�����-� ��� w P K BDC�EGFGHP Y ^ BDC�EGFGHP Y�w P [ BDE�HU�c B]C�E@FGHP Y�w P [ BeFGHV�f B]C�E@FGHP � | � � z (4.4)
43
We canobtainsimilar equationsfor theothernormalstraincomponents.For theshearstrains,we
getsimilarequationsin termsof thedisplacementfield variables.This is seenfrom theequationfor
theshearstrain �#��� shown below.� �#�]���@�G���� ��� ������&�� �¡G¢ �T£¤�D���G�G�� ¥�¦ �]���@�G�� ¥ ¢ �¨§n�a�b��d© �D���G�G�� ¥ ¢ �T§n�a�G�ª�« �D���G�G�� ¥¢ �#£¤�D���G�G�� ¥ ¢ �#§l�D�����¦ �]���@�G�� ¥\© �]���@�G�� ¥ ¢ �,§n�e�G�ª�« �]���@�G�� ¬® � ��¯ (4.5)
Now, £��]���@�G�� ° §n�D����±° §n�e�G�ª areindependentof §n�]���� . After somealgebraicmanipulation,equation(4.4)
canbewritten as � �#�]���@�G��T� ���²¦ �D���G�G�� ¯(4.6)
Similarly, usingthe fact that £��]���@�G�� ° §n�]����³° §l�e�G�ª areindependentof §n�]���� andthat £��]���@�G�ª ° §n�]����³° §l�a�b��areindependentof §n�a�G�ª , we get � �#�]���@�G��T� ���L© �D���G�G�� ° (4.7)� �#�]���@�G�ªTª ��� « �D���G�G�ª ¯
(4.8)
Theshearstrainequation(4.5)canbereducedto� � �D���G�G���� ���²¦ �]���@�G�� ¥\© �]���@�G�� ° (4.9)
Similarly, � ���D���G�G�� ª � �´© �D���G�G�ª ¥ « �]���@�G�� ° (4.10)� ���D���G�G�ª � ��� « �D���G�G�� ¥ ¦ �D���G�G�ª ¯(4.11)
For the casewherethe interfacesbetweensubcellsareperfectlybonded,the averagestrainin the
compositeRVE is givenby µ �T¶�·@¸ � �� ¹º � � � ���D���G�G�¶�· �®° (4.12)
where �� ¹º » �¼�½ ª ¹º��¾ � ¹º��¾ � ¹º�G¾ � °½is thelengthof asideof theRVE, and,¿is thenumberof subcellspersideof theRVE.
In the following derivation, tractionsareassumedcontinuousacrosssubcell interfaces. Dis-
placementsandtractionsin theRVE areassumedto beperiodic. In the“shearcoupled”versionof
themethodof cellspresentedby Ganet al. [119], thetractioncontinuityassumptionis replacedby
44
thesatisfactionof equilibriumandcompatibilityacrosssubcells,therebymakingit difficult to apply
interfacial jump conditionsto accountfor imperfectinterfaces.
Assumingdisplacementcontinuityacrossinterfaces,if À is theinterfacebetweentwo subcells,
then Áq¨ÃÄ�Å@ÆGÇÈ ÉÉÉ�Ê�Ë Áq Ä�Å@ÆGÇÈ ÉÉÉÌÊyÍ´Î-Ï (4.13)Á Â Ä ÃÅXÆGÇÈ ÉÉÉ�Ê�Ë Á  Ä�Å@ÆGÇÈ ÉÉÉÌÊyÍ´Î-Ï (4.14)Á  Ä�Å ÃÆ�ÇÈ ÉÉÉ�Ê�Ë Á  Ä�Å@ÆGÇÈ ÉÉÉÌÊyÍ´Î-Ð (4.15)
whereÑ Í�ÒÔÓ�Ó�Ó�Õ7ÖuË Ò@×+ÏØÙ ÍÛÚ ÙÝÜ Ò�Ï if Ò¤Þ Ù\ß Ö ;Ò�Ï if Ù Í´Ö ;(4.16)Øà Í Ú à Ü Ò�Ï if ÒáÞ à ß Ö ;Ò�Ï if
à Í´Ö ;(4.17)Øâ ÍãÚ â/Ü Ò�Ï if Ò�Þ âäß Ö ;Ò�Ï if â ÍåÖ .(4.18)
Writing theequations(4.13),(4.14,and(4.15)in termsof the local subcellcoordinatesystems,we
have Áq Ä�Å@ÆGÇÈ ÉÉÉÌæ�çÌè@éê`ëíì Ë Á�¨ÃÄbÅGÆGÇÈ ÉÉÉ æ�ç�îè@éê�ëðï�ì Í´Î-Ï (4.19)Á  Ä�Å@ÆGÇÈ ÉÉÉ æ�çÌñ#éò`ëíì Ë Á Â Ä ÃÅXÆGÇÈ ÉÉÉ æ�ç îñ,éò`ëðï�ì Í´Î-Ï (4.20)Á� Ä�ÅGÆGÇÈ ÉÉÉ æ�ç ó!éô ëíì Ë Á� Ä�Å ÃÆXÇÈ ÉÉÉ æ�ç7îó!éô ëðï�ì Í´Î-Ð (4.21)
Applying thesedisplacementcontinuityequationsonanaveragebasisover theinterfaceswe get,õ-öø÷áù Á  Ä�Å@ÆGÇÈ ÉÉÉ æ çÌè@éê`ëíì Ë Á ¨ÃÄbÅ@ÆGÇÈ ÉÉÉ æ ç�îè@éê`ëðï�ì�úüû�ý  Å�Çþ û�ý  ÆGÇÿ Í´Î-Ï (4.22)õ ö�÷ ù Á  Ä�ÅGÆGÇÈ ÉÉÉ æ�ç ñ�éò ëíì Ë Á Â Ä ÃÅGÆGÇÈ ÉÉÉ æ�ç îñ,éò ëðï�ì ú û ý  Ä�Ç� û�ý  ÆGÇÿ Í´Î-Ï (4.23)õ ö�÷ ù Á  Ä�Å@ÆGÇÈ ÉÉÉ æ�çÌó!éô ëíì Ë Á  Ä�Å ÃÆXÇÈ ÉÉÉ æ�çOîó,éô\ëðï�ì ú û�ý  Ä�Ç� û ý  Å�Çþ Í´Î-Ï (4.24)
whereõö�÷ û ý  ��������� �DÇÈ û�ý  ��������� �DÇ� � õ ìï�ì õ ìï�ì û ý  �����������aÇÈ û ý  ��������� �DÇ� Ð
45
Substitutingequation(4.1) into equations(4.22), (4.23), and(4.24) we get, after integrationand
somealgebra, ������������ ���! ���������� " �#�%$�&������ �'�! �%$�&������ (*),+ (4.25)� ���������� �'�.- �/�������� " � �/� $������ �0�.- ��� $�1���� (*),+ (4.26)�#�/�������� �'�.2 ���������� " �#�/���3$���� �'�.2 �����4$���� (*),5 (4.27)
Let 6 �/�&� betheinterfacebetweenthetwo subcells7�8:9<;:= and 7?>8@9<;:= . Theequationsrelatingadjacent
subcellscanbeexpressedin termsof a singlecoordinatesystemwith its origin at themid point of
the interfacebetweenthe subcells. The mappingbetweenthe subcellbasedcoordinatesand the
interfacebasedcoordinatescanbewrittenasA ���&�B ( A4CED F�GB "H�I+A �%$�.�B ( A C D F�GB �'�I+A �����J ( A4CED KEGJ "H�3+ (4.28)A � $���J ( A C D KEGJ �'�3+A � ���L ( A C D MNGL "H�3+AO�$�1�L ( A CED MNGL �'�35Evaluatingall quantitiesin equations(4.25),(4.26)and (4.27) at the interfacesusing equations
(4.28),we have,�#���������� "H� PRQQIS B ������������ "0 �/�������� TU" �#�%$�&������ "H� PVQQIS B ���%$�?������ "0 �%$�?������ TW(X),+ (4.29)� ���������� "H� PVQQIS J � ���������� "H- ���������� TY" � �/� $������ "H� PRQQ3S J � �/� $������ "Z- �/� $�1���� TW(X),+ (4.30)� ���������� "[� P QQ3S L � �/�������� "Z2 ���������� TY" � �/���3$���� "H� P QQ3S L � �����4$���� "Z2 �/���3$���� TW(X),5 (4.31)
Equations(4.29),(4.30),and(4.31)canbewrittenas\ �/�������� (*),+ (4.32)] �/�������� (*),+ (4.33)^ �/�������� (*),+ (4.34)
46
where_a`/b�c�d�ef gWh `/b�c�d�ef i�j `/b�c�d�ef kZh `%lb?c�d�ef i�j `%lb�c�d�ef m (4.35)n `/b�c�d�ef gWh `/b�c�d�ef i[o `/b�c�d�ef kZh `�b lc1d�ef i0o `�b lc1d�ef m (4.36)p `�b�c�d�ef gWh `�b�c�d�ef irq `�b�c�d�ef kZh `/b�c3ld?ef i�q `/b�c3ld?ef m (4.37)j `�b�c�d�ef gskutwvRxxIy{z h `�b�c�d�ef k}| `/b�c�d�ef ~ m (4.38)
o `/b�c�d�ef g�kut v xx3y�� h `�b�c�d�ef kZ� `/b�c�d�ef ~Hm (4.39)
q `/b�c�d�ef g�kut v xx3y�� h `/b�c�d�ef k�� `/b�c�d�ef ~[� (4.40)
Now, from equations(4.38),(4.39),and(4.40),sinceh `�b�c�d�ef is linearin y f and | `�b�c�d�ef m�� `�b�c�d�ef m and� `/b�c�d�ef areconstant,we have, xxIy{z j `/b�c�d�ef g*�,m (4.41)xx3y � o `/b�c�d�ef g*�,m (4.42)xx3y � q `/b�c�d�ef g*�,� (4.43)
Therefore,from equations(4.35),(4.36),(4.37)andequations(4.41),(4.42),and(4.43)we have,xx3y{z _ `/b�c�d�ef g xx3y{z h `�b�c�d�ef k xxIy{z h `%lb.c�d�ef g*�,m (4.44)xx3y � n `/b�c�d�ef g xx3y � h `�b�c�d�ef k xxIy � h `�b lc�d�ef g*�,m (4.45)xx3y�� p `/b�c�d�ef g xx3y�� h `�b�c�d�ef k xxIy�� h `�b�c3ld�ef g*�,� (4.46)
If we carryout asmoothingoperationwherethedisplacementat thecenterof eachsubcellis setto
beequalto theapplieddisplacement,thenall of theabove equationsinvolving thedisplacementsat
thesubcellcentersaresatisfied.Thus,we canassumeasolutionof theform� f gWh `�b�c�d�ef � (4.47)
Fromequations(4.35),(4.36),(4.37)and(4.47),wehave,
_ `/b�c�d�ef gXj `/b�c�d�ef i�j `%lb�c�d�ef m (4.48)n `/b�c�d�ef gYo `/b�c�d�ef i}o `%lb&c�d�ef m (4.49)p `/b�c�d�ef gXq `�b�c�d�ef i�q `%lb?c�d�ef � (4.50)
47
Usingequations(4.32-4.34)and(4.48-4.50),andsummingover thecoordinatedirections,we get���&�@���a� �������� � ���&�@�����@� �������� � �@�%��������� ���X� ���&�@�,�@� �������� �*�, (4.51)���?�@�,¡ � �������� � �����@� �£¢I� �������� � ¢3����?������ �¤�X� ���?�@� ¢I� �������� �*�, (4.52)�����@��¥ � �������� � �����@����¦ � �������� � ¦ ����?������ � �X� �����@�,¦ � �������� �*�,§ (4.53)
Substitutingequations(4.38-4.40)into equations(4.51-4.53)we have,���&�@�3¨ ��©wªV««I¬ �& � �������� ¨}®:� �������� ¯W�*�, (4.54)���?�@� ¨ ��©wªV««3¬�° � �������� ¨Z±�� �������� ¯W�*�, (4.55)�����@� ¨ ��©wª ««I¬�² � �������� ¨Z³u� �������� ¯W�*�,§ (4.56)
Therefore,pluggingequations(4.47)into equations(4.54-4.56),we have,���&�@� ��© ®:� �������� �X��´ ««3¬ �,µ � (4.57)�����@� ��© ±�� �������� �X��´ ««3¬�° µ � (4.58)�����@� ��© ³u� �������� �X��´ ««3¬�² µ � § (4.59)
We canshow, usingtheprecedingequations,that theaveragestrainsin theRVE canberepre-
sentedas ¶· �¹¸�º �¼»� ª ««3¬ ¸ µ � � ««3¬ � µ ¸ ¯H§ (4.60)
Let uscheckthis for ½ �¾¿� » . Usingequation(4.12)we have¶· ��� º �À»Á �� Á3ÂÄà · � ���������� Å §Substitutingequation(4.6) into theabove equationwe get¶�· ��� º � »Á �� Á  ®:� �������� § (4.61)
Now, if equation(4.57)is multipliedby Æ © ° with ½ � » andsummedover Ç and È we get
»Á �� Á4 ® � �������� � ««I¬ � µ � § (4.62)
48
Comparingequations(4.61)and(4.62),we have,É�Ê£Ë�ËÍÌÏÎ ÐÐ3Ñ Ë!Ò ËÔÓTherestof therelationsin equation(4.60)canbeshown to hold in asimilar way.
In orderto relatethesubcellstrainsto thevolumeaveragedstrainsin theRVE, weapplyequation
(4.6) to equation(4.57)to get,for Õ Î�Ö,×ØÙ&Ú Ë�Û�ÜÞÝ<ß Ù�à�á�âË Î Û�ã ÐÐ3Ñ Ë!Ò Ë Ó
Substitutingfor Ý ß Ù�à�á�âËandusingequation(4.60)wehave,×ØÙ&Ú Ë Û�ÜYä Ê ß Ù�à�á�âË�Ë å Î Û�ã É�ÊEË�ËÍÌ¿Ó
(4.63)
Usingsimilar methodswe cangetthefull setof relationshipsbetweentheaverageRVE strains
andthesubcellstrains.Theseare ×ØÙ&Ú Ë Û�Ü ä Ê ß Ù�à�á�âË�Ë å Î Û�ã É�ÊEË�ËÍÌ¿æ(4.64)×Øà?Ú Ë Û�Ü ä Ê ß Ù�à�á�âç�ç å Î Û�ã É�Ê ç�ç Ì¿æ (4.65)×Øá�Ú Ë,Û�Ü ä Ê ß Ù�à�á�âè�è å Î Û�ã É�Ê è�è Ì¿æ (4.66)
and ×ØÙ&Ú Ë×Øà?Ú Ë&é Ü ç ä Ê ß Ù�à�á�âË ç å Î é ã ç É�Ê Ë ç Ìêæ (4.67)×Øà�Ú Ë×Øá�Ú Ë é Ü ç ä Ê ß Ù�à�á�âç�è å Î é ã ç É�Ê ç�è Ìêæ (4.68)×ØÙ&Ú Ë×Øá�Ú Ë é Ü ç ä Ê ß Ù�à�á�âË è å Î é ã ç É�Ê Ë è ÌêÓ (4.69)
The relationsbetweenthe averagestrainsin the RVE and the averagesubcellstrainscanbe
usedto generaterelationsbetweentheaverageRVE stressesandtheaverageRVE strainsusingthe
tractioncontinuitycondition. Ganet al. [119] diverge from thestandardGMC formulationat this
stageby usingsubcellequilibriumandcompatibilityequationsto arriveatthestress-strainrelations.
49
4.2 Stress-StrainRelationsLet theconstitutive equationbeof theform (in matrixnotation)ë,ì.í�î�ï�ð�ñóòwôöõ÷í�î�ï�ð�ñ4ëùø÷í/î�ï�ð�ñNòûú�ü�í�î�ï�ð�ñ�ýÞþ
(4.70)
where ëùì í�î�ï�ð�ñ òÿô���ë�� í�î�ï�ð�ñ��� ò���ë�� í/î�ï�ð�ñ� ò��ë�� í/î�ï�ð�ñ��� ò��� uë�� í�î�ï�ð�ñ�� ò�� uë�� í/î�ï�ð�ñ��� ò�� uë�� í/î�ï�ð�ñ�� ò������õ í�î�ï�ð�ñ ô
����������� í/î�ï�ð�ñ��� � í/î�ï�ð�ñ�� � í�î�ï�ð�ñ��� � � �� í/î�ï�ð�ñ�� � í/î�ï�ð�ñ� � í�î�ï�ð�ñ�� � � �� í/î�ï�ð�ñ��� � í/î�ï�ð�ñ�� � í�î�ï�ð�ñ��� � � �� � � � í/î�ï�ð�ñ��� � �� � � � � í/î�ï�ð�ñ��� �� � � � � � í�î�ï�ð�ñ���
�! "�
ë ø÷í�î�ï�ð�ñ ò ô ��ë�# í�î�ï�ð�ñ��� ò � ë�# í/î�ï�ð�ñ� ò � ë�# í/î�ï�ð�ñ��� ò � ë�# í/î�ï�ð�ñ�� ò � ë�# í/î�ï�ð�ñ��� ò � ë�# í�î�ï�ð�ñ�� ò�� �$�ü í�î�ï�ð�ñ ô��&% í�î�ï�ð�ñ��� ��% í/î�ï�ð�ñ� ��% í�î�ï�ð�ñ��� � � � � � � � �$'
Notethatweassumethatthematerialis atmostorthotropic.
Let usexpresstheequations(4.64-4.69)in termsof thesubcellstressesusingequation(4.70).
Thenwehave,()î+* �+, � í/î�ï�ð�ñ��� ë # í�î�ï�ð�ñ��� òûú � í/î�ï�ð�ñ�� ë # í�î�ï�ð�ñ� òÞú � í�î�ï�ð�ñ��� ë # í�î�ï�ð�ñ��� òûú-% í�î�ï�ð�ñ��� ýÞþ/.�ô10 2435� ���76 �(4.71)()ï8* � , � í/î�ï�ð�ñ�� ë # í�î�ï�ð�ñ��� ò ú � í/î�ï�ð�ñ� ë # í�î�ï�ð�ñ� ò ú � í�î�ï�ð�ñ�� ë # í�î�ï�ð�ñ��� ò ú-% í�î�ï�ð�ñ� ýÞþ . ô 0 2 35� �96 �(4.72)()ð�* � , � í/î�ï�ð�ñ��� ë # í�î�ï�ð�ñ��� òûú � í/î�ï�ð�ñ�� ë # í�î�ï�ð�ñ� òÞú � í�î�ï�ð�ñ��� ë # í�î�ï�ð�ñ��� òûú-% í�î�ï�ð�ñ��� ýÞþ/.�ô 0 2 35� ���96 �(4.73)
and
50:;<+=?> :;@ =?> ACB < @ED�FG�G H IKJ B < @�D�F>�L MON1P LQ LSR�T >�L9UWV (4.74):;@ =?> :;D =?> ACB < @ED�FX�X H I J B < @�D�FL�Y M N P LQ L R�T L�Y9UWV (4.75):;<+=?> :;D =?> ACB < @ED�FZ�Z H IKJ B < @�D�F>�Y MON P LQ LSR�T >�Y9UW[ (4.76)
From the assumptionof traction continuity normal to subcellsinterfaces,appliedin an average
sense,we have, I J B < @�D�F>�> M N I J B]\< @ED�F>�> M N_^ B @ED�F>�> VI�J B < @�D�FL�L MWN I�J B < \@`D�FL�L MON_^ B < D�FL�L V (4.77)I�J B < @�D�FY�Y MWN I�J B < @ \D`FY�Y MON_^ B < @8FY�Y Vwhere
B @ED�F>�> aretheof normalstressesin the’11’ direction,^ B < D�FL�L aretheof normalstressesin the’22’ direction,^ B < @8FY�Y aretheof normalstressesin the’33’ direction.
Similarly, for theshearstresses, I J B < @�D�F>�L M N I J B]\< @�D�F>�L M N_^ B @�D�F>�LI�J B < @�D�FL�> MWN I�J B < \@`D�FL�> MaN_^ B < D�FL�>I�J B < @�D�FL�Y MWN I�J B < \@`D�FL�Y MaN_^ B < D�FL�Y (4.78)I J B < @�D�FY�L M N I J B < @ \D`FY�L M N_^ B < @bFY�LI�J B < @�D�F>�Y MWN I�J B]\< @�D�F>�Y MaN_^ B @�D�F>�YI�J B < @�D�FY�> MWN I�J B < @ \D`FY�> MaN_^ B < @bFY�>where
B @ED�F>�L N_^ B < D�FL�> aretheof shearstressesin the’12’ direction,^ B < D�FL�Y N_^ B < @8FY�L aretheof shearstressesin the’23’ direction,^ B @ED�F>�Y N_^ B < @bFY�> aretheof shearstressesin the’13’ direction.
The symmetryof the shearstressesleadsto a reductionof one dimensionin the subcellstress
dependenciesfor thesheardirectionsasshown in equations(4.79-4.81).
51c�dfe�g�hi�j k cld!m`g�hj�i k c�dng�hi�j1o (4.79)c d!m`g�hj�p k c dqm8e8hp�j k c d!m+hj�pro (4.80)c dfe�g�hi�p k c d!m8ebhp�i k c d!e8hi�pts (4.81)
It is seenfrom theabove assumptionsthatwe canseparatethenormalcomponentsof theequations
from theshearcomponentsleadingto decouplingof theeffective normalandsheareffects.
Let usfirst look at thenormalcomponentsof thestress.Substitutingequations(4.71-4.73)into
equations(4.77)),wehave,uvm+w iKxzy d!m8e�g�hi�i cldfe�g�hi�i { y dqm8eEg�hi�j c�dqm`g�hj�j { y dqm8eEg�hi�p c�dqm8e8hp�p}|�k1~ ����� i�i���� uvm+w iK� dqm8eEg�hi�i � c o (4.82)uvebw i x y d!m8e�g�hj�i c dfe�g�hi�i { y dqm8eEg�hj�j c dqm`g�hj�j { y dqm8eEg�hj�p c dqm8e8hp�p | k ~ � ��� j�jE��� uve8w i � d!m8e�g�hj�j � c o (4.83)uvg�w i xzy d!m8e�g�hp�i c dfe�g�hi�i { y dqm8eEg�hp�j c dqm`g�hj�j { y dqm8eEg�hp�p c dqm8e8hp�p}|�k ~ � ��� p�p ��� uvg�w i � d!m8e�g�hp�p � c s (4.84)
Wecanrewrite equations(4.82-4.84)in theform� uvm+w i�y d!m8e�g�hi�i � c�dfe�g�hi�i { uvm+w iKy d!m8e�g�hi�j c�dqm`g�hj�j { uvm+w i�y d!m8e�g�hi�p cld!m8ebhp�p k~ � ��� i�i7��� � c uvm+w i�� dqm8eEg�hi�i o (4.85)
uvebw i y d!m8e�g�hj�i c dfe�g�hi�i {��� uvebw i y d!m8e�g�hj�j �� c d!m`g�hj�j�{ uve8w i y d!m8e�g�hj�p c dqm8e8hp�p k~ �4��� j�jE��� � c uvebw i � d!m8e�g�hj�j o (4.86)
uvg�w i y d!m8e�g�hp�i c dfe�g�hi�i { uvg�w i y d!m8e�g�hp�j c d!m`g�hj�j�{��� uvg�w i y dqm8eEg�hp�p �� c d!m8ebhp�p k~ ����� p�pE��� � c uvg�w iK� dqm8eEg�hp�p s (4.87)
Equation(4.85)canbeexpressedin matrix form as�q��� i ��� j ��� p���� k������ i�i�� {¡ l�5� j�jE� {¢ l�5� p�pE���¡� � c�£ i�i o (4.88)
52
where ¤$¥r¦!¤¨§§ ¤ª© §¬««« ¤¯®§¯°° °° ¤¨§© ¤ª©© ««« ¤¯®©W°° °° ¤¨§± ¤ª©± ««« ¤¯®±³²7´$µ (4.89)¶�·�¸ ¥º¹»»»¼½ · § ¾ ««« ¾¾ ½ · © ««« ¾...
.... . .
...¾ ¾ ««« ½ · ®¿!ÀÀÀÁ µ (4.90)
¶�·Ã ¥º¹»»»¼Äa· § ¾ ««« ¾¾ ÄÅ· © ««« ¾...
..... .
...¾ ¾ ««« ÄÅ· ®¿!ÀÀÀÁ µ (4.91)
¶�·ÇÆ ¥ ¹»»»¼È · §�§ È · §�© ««« È · §É®È · ©�§ È · ©�© ««« È · ©]®...
.... . .
...È · ®³§ È · ®Ê© ««« È · ®Ë®¿!ÀÀÀÁ µ (4.92)Ì ¥r¦ Ì § Ì © ««« Ì ® ² ´ µ and, (4.93)Í §�§ ¥r¦&Î · § Î · © ««« Î · ® ² ´ « (4.94)
Thecomponentsof thesub-matrices½ ·+Ï
,ÄÅ·+Ï
andÈ ·+ÏÑÐ
in thematrices¶�· § µ ¶�· © µ and
¶�· ± are
givenbelow. For matrices½ ·
½ ·+Ï ¥º¹»»»¼ÒÔÓ Ï § Õ ««« ÕÕ ÒÖÓ Ï© ««« Õ...
.... . .
...Õ Õ ««« ÒÖÓ Ï ®¿!ÀÀÀÁ (4.95)
whereÒÔÓ Ï× ¥ ®ØÙ+Ú §KÛËÜ Ù× ÏÞݧ�§ « (4.96)
For theÄÅ·
matrices, ÄÅ·ÇÏ ¥ß¹»»»¼à Ó Ï §�§ à Ó Ï §�© ««« à Ó Ï §É®à Ó Ï ©�§ à Ó Ï ©�© ««« à Ó Ï ©]®...
.... . .
...à Ó Ï ®³§ à Ó Ï ®Ê© ««« à Ó Ï ®Ë®¿!ÀÀÀÁ (4.97)
whereà Ó Ï á Ù ¥ ÛCÜ Ù
á Ïâݧ�© « (4.98)
For theÈ ·
matrices, È ·bÏÑÐ ¥º¹»»»¼ã8Ó ÏäЧ�§ ã8Ó ÏäЧ�© ««« ã8Ó ÏäЧɮã8Ó ÏäЩ�§ ã8Ó ÏäЩ�© ««« ã8Ó ÏäЩ]®...
.... . .
...ã8Ó ÏÑÐ®å§ ã8Ó ÏÑЮʩ ««« ã8Ó ÏÑЮC®¿!ÀÀÀÁ (4.99)
whereã8Ó ÏÑÐá�æ ¥1ç ÛCÜæfá ÏÞݧ ± µ if èé¥ëê ;Õ µ otherwise. « (4.100)
53
Thecomponentsof thesub-matricesof the ì matrix areìlí îËï�ð!ñ�ò î íâóî�î ñ�ò!ô íâóî�î õõõ ñ�òÞö íâóî�îø÷ (4.101)ì í ô ï ð!ñ ò î íâóô�ô ñ ò!ô íâóô�ô õõõ ñ òÞö íâóô�ô ÷ (4.102)ìlíùúï ð!ñ�ò î íâóù�ù ñ�ò!ô íâóù�ù õõõ ñ�òÞö íâóù�ù ÷ (4.103)
Thecomponentsof thematricesû í areû í ïýü7þ ÿ þ ÿ õõõ þ ÿ���� (4.104)
whereû í is a ����� matrix. Thecomponentsof thesub-matrices í are� í ï ð � ö��� î�� ò � î íâóî�î � ö��� î�� ò � ô íâóî�î õõõ � ö��� î�� ò � ö íâóî�î ÷ (4.105)
Similarly, equation(4.86)canbeexpressedin matrix form as,����� î ��� ô ��� ù�� ì�ï������ î�î! #"%$ ��� ô�ô &"(' �)� ù�ù* ,+.-0/ ô�ô +.1 ô�ô*2 ñ (4.106)
For thethird normaldirectiongivenby equation(4.87), wecangetamatrixrepresentationasshown
below. � �43 î �43 ô �43 ù0� ì�ï5�6��� î�î &"(' �7� ô�ô &"%8 ��� ù�ù ,+9- / ù�ù +.1 ù�ù 2 ñ (4.107)
Combiningthethreenormaldirectionequations,we get,:; � î � ô � ù��� î ��� ô �4� ù�43 î �43 ô ��3 ù <=>:; ì îì ôì ù <= ï :; û '' <= ��� î�î0 &" :; '$ ' <= ��� ô�ô &" :; ''8 <= �7� ù�ù? @+ 2 ñ :; 1lî�î1 ô�ô1 ù�ù <= (4.108)
Sincethenormalandshearstressesareuncoupled,we candealwith theshearresponsesepa-
rately from thenormalresponse.We canwrite theequationsrelatingthesubcellshearstressesto
theaverageshearstrainsin theRVE in matrix form as:; ��A ' '' ��B '' ' �4C <=>:; ì î ôì ô ùì î�ù<= ï :; û $'' <= �7� î ô D" :; '$E8' <= ��� ô ùF &" :; ''û 8 <= ��� î�ùF (4.109)
Thesub-matricesin theabove equationareexpandedasshown below.�5A ï :GGG;IH î J õõõ JJ H ô õõõ J...
..... .
...J J õõõ H ö<�KKK=
(4.110)
54
whereLNMPO QRS�TVU QRW TVUYX[Z S W M]\^I^ _ (4.111)
`�a Ocbdddef U g _?_?_ gg fih _?_?_ g...
.... . .
...g g _?_?_ f Qj�kkkl (4.112)
wheref S O QRW TVU QRM TVU X[Z S W M]\mIm _ (4.113)
`4n Ocbdddeo U g _?_?_ gg o h _?_?_ g...
.... . .
...g g _?_?_ o Qj�kkkl (4.114)
whereo W O QRS�TVU QRM TVU X[Z S W M]\pIp _ (4.115)q U h Osrut UU h t hU h _?_?_ t QU hwv!x (4.116)q hIy Osrut UhIy t hhIy _?_?_ t QhIywv x (4.117)q U y O r t UU y t hU y _?_?_ t QU y v x (4.118)
and, z|{ O {E} O z|} O�~�� h� h � h� h _?_?_ � h� h�� x (4.119)
wherethevectorhas� elements.
Wethushaveasetof equationsthatrelatethesubcellstressesto theaveragestrainsin theRVE.
We now have to developequationsthat relatetheaveragestressesin theRVE to averagestrainsin
theRVE usingequations(4.108)and(4.109).In otherwords,weneedanequationfor thecomposite
stress-strainequationsof theform, �)��� O5�P�,� �7�����.�i� ti� _ (4.120)
This equationcanbe solved for the componentsof theeffective stiffnessmatrix andthe effective
coefficientsof thermalexpansion.
55
4.3 EffectiveThermoelasticPropertiesFor thenormaldirections,giventheglobalaveragestrains ���?�I�0� , �����I�F� , �7���I�*� , andthetempera-
turechange�P� , we cancomputethenormalstresses.Fromequation(4.108)we cansolve for the��� variablesby invertingthe � matrix. Let � betheinverseof thematrix � . Then � is a square
matrixof dimension�¡ �¢£ ¥¤% ¦¢E >¤% �¢£ �§ . Expressedin matrix form,��¨c©ª �¬« � �¬« � �« ���®��c��®¯�°�4®±��4²&�c�4²±�°��²�� ³´Nµ·¶ �¹¸¨c©ª �º«±« �º«»® �¼«»²�%®&« �%®±® �½®±²�½²#« �½²�® �¾²�² ³´À¿(4.121)
where,�½Á� is of thesamedimensionsas �4Áàthoughtheelementsin eachsub-matrixmaychange.
Therefore,theequation(4.108)canbewrittenas©ªIÄ �Ä �Ä �³´ ¨ ©ª �º«±« �¼«Å® �¼«»²�%®&« �½®�® �½®±²�½²#« �¾²�® �¾²�² ³´ ©ªÇÆ ÈÈ ³´ �����I�0�#¤ ©ª �º«±« �º«»® �º«�²�%®&« �%®±® �%®¯²�½²#« �½²�® �½²±² ³´ ©ª ÈÉ È ³´ ���Ê�I�F�¯¤ (4.122)
©ª �º«±« �¼«Å® �¼«»²�%®&« �½®�® �½®±²�½²#« �¾²�® �¾²�² ³´ ©ª ÈÈË ³´ ���Ê�I�F��Ì.�P�Í©ª �º«±« �º«»® �º«�²�%®&« �%®±® �%®¯²�½²#« �½²�® �½²±² ³´ ©ª�Î �I�Î �I�Î �I�³´>¿
Thesematricescanbeexpandedout to getexpressionsin termsof thecomponentsof matrices�½Á Ïfor thestresses� µ �uÂ0¸Ð�Ð .
Theaveragecompositestressin termsof thesubcellstressesis givenby��Ñ �u ��¨°ÒÓ ÔÕ Ó±Ö�× Ñ µ�ØwÙ]Ú ¸�uÂ Û ¿ (4.123)
Usingequation(4.77)we thereforeget�¡Ñ&�I�0�@¨cÜ �Ý � ÔÕÙ�Þ � ÔÕÚ]Þ � � µ�ÙFÚ ¸�I� ß (4.124)
�¡Ñ±�I�F�@¨ Ü �Ý � ÔÕØ�Þ � ÔÕÚ]Þ � � µ�ØàÚ ¸�I� ß (4.125)
�¡Ñ±�I�F�@¨cÜ �Ý � ÔÕØ�Þ � ÔÕÙ�Þ � � µ�ØwÙ ¸�I� ¿(4.126)
WecanthenexpresstheaverageRVE stressesin termsof theaverageRVE strainsas�¡Ñ#�I�0�@¨�áãâ�I� �����I�0�&¤äáåâ�)� �)���I�*�#¤äáåâ�)� ���Ê�I�?�@Ì.�P�½�7áåâ�I�!æ â �I� ¤äáãâ�)�?æ â�I� ¤¼áãâ�)�*æ â�I� § ß (4.127)�¡Ñ��I�F�@¨�áãâ�0� �����I�0�&¤äáåâ�I� �)���I�*�#¤äáåâ�I� ���Ê�I�?�@Ì.�P�½�7áåâ�0�!æ â �I� ¤äáãâ�I�?æ â�I� ̽áãâ�I�*æ â�I� § ß (4.128)�¡Ñ �I� �@¨�á â�0� ��� �I� �&¤äá â�I� �)� �I� �#¤äá â�I� ��� �I� �@Ì.�P�½�7á â�0� æ â �I� ¤äá â�I� æ â�I� ¤¼á â�I� æ â�I� § ¿ (4.129)
wheretheeffective stiffnessterms á â areexpressedasimplesumsof termsof thematrices�¾Á Ï .
56
Thecoefficientsof thermalexpansioncanbefoundusingçè�é�êëIëé êìIìé�êíIí îï%ðcçèÊñãêëIëòñãêë ì ñãêë íñ êì ë ñ êìIì ñ êìIíñãêí ë ñãêíIì ñãêíIí îïNóë çèIô ëô ìô í îïÀõ
(4.130)
wheretheô#ö
termsarealsosumsover the componentsof thematrices÷½ø ù . Thesetermsinvolve
complex algebraicexpressionsandarenotpresentedhere.
The determinationof the effective shearstiffnessesis simplerbecauseof the lack of coupling
betweenthenormalandtheshearterms.Theexpressionsfor thesheartermsof theeffectivestiffness
matrixare ñ êúIú ð|û üþýÿ��� ë �� � õ (4.131)ñ ê��� ð û ü ýÿ��� ë � � õ (4.132)ñ ê� ð û ü°ýÿ� � ë �ñ � � (4.133)
Thus the completesetof effective stiffnesstermsis determined.Equations(4.111),(4.113)and
(4.115)show that� � ,
� andñ � arejust volumeaveragesof theshearcompliancesof thesubcell
materials.Therefore,theshearstiffnessespredictedby GMC areequalto theReuss(or harmonic)
bounds.Sinceharmonicboundsdo not representaccurateeffective shearmoduli,aswill beshown
in Chapters6 and 7, this featureis a shortcomingof GMC. The “shear-coupling” approaches
developedby Williams andAboudi [118] andGanetal. [119] attemptto alleviatethisproblem.
4.4 Shear-Coupled Method of CellsThe generalizedmethodof cells with shearcoupling as describedby Gan et al. [119] for
unidirectionalfiber compositesin two dimensionsis extendedto threedimensionsin this section.
We start with the relationsbetweenthe averageRVE strainsand the subcell strainsshown in
equations(4.64-4.69).Theseequations,which arebasedon thecontinuityof displacementsacross
subcells,canbewrittenas ��� ëIë�� ð üû ýÿ��� ë � ��� � � ���ëIë � õ(4.134)��� ìIì � ð üû ýÿ� � ë � ��� � � ���ìIì � õ(4.135)��� íIí � ð üû ýÿ��� ë � ��� � � ���íIí � õ(4.136)
57������� �"!$# �% �'&()�* � &(+ * �-, ��. ) +0/�1��� 243(4.137)���5��6 �"! # �% �7&(+ * � &(/ * � , � . ) +0/�1��6 2 3(4.138)������6 �"! # �% �'&()�* � &(/ * � , � . ) +�/�1��6 298(4.139)
In addition to continuity of displacements,GMC assumesthat tractionsare continuousacross
subcellboundaries.Theshear-coupledmethodof cellsdoesnot requiretractioncontinuityacross
subcells.Instead,subcellequilibriumandcompatibilityareenforcedin anaveragesense.
Theequationsof equilibriumare: �<;=���?> : ��;=���@> : 6�;=��6A!CB 3(4.140): � ; ��� > : � ; ��� > : 6 ; ��6 !CB 3(4.141): �<;=��6@> : ��;D��6@> : 6�;D6�6A!CB 8(4.142)
Theseequationsareapproximatedusingforwarddifferencesbetweenaveragesubcellstresses.Thus
thediscretizedequilibriumequationsare(assumingperiodicityof stressesin theRVE), ;?.5E) +0/�1��� 2GF , ;?. ) +0/�1��� 2 > , ;?. ) E+�/�1��� 2GF , ;?. ) +0/�1��� 2 > , ;?. ) + E/H1��6 2GF , ;?. ) +0/�1��6 2 !CB 3(4.143), ;?.5E) +0/�1��� 2GF , ;?. ) +0/�1��� 2 > , ;?. ) E+�/�1��� 2GF , ;?. ) +0/�1��� 2 > , ;?. ) + E/H1��6 2GF , ;?. ) +0/�1��6 2 !CB 3(4.144), ; .5E) +0/�1��6 2GF , ; . ) +0/�1��6 2 > , ; . ) E+�/�1��6 2GF , ; . ) +0/�1��6 2 > , ; . ) + E/H16�6 2GF , ; . ) +0/�16�6 2 !CB 8(4.145)
If weassumeaconstitutive equationof theformI !KJML 3wherethestiffnessmatrix
Jis orthotropic,we canexpressthe equilibriumequationsin termsof
theaveragesubcellstrainsasN .�E) +�/�1��� , � .�E) +�/�1��� 2GF N . ) +�/�1��� , � . ) +�/�1��� 2 > N .5E) +�/�1��� , � .5E) +0/�1��� 2OF N . ) +0/�1��� , � . ) +0/�1��� 2 >N .5E) +0/�1��6 , � .5E) +�/�16�6 2GF N . ) +�/�1��6 , � . ) +�/�16�6 2 > N . ) E+�/�1P�P , � . ) E+Q/�1��� 2RF N . ) +�/�1P�P , � . ) +�/�1��� 2 >N . ) + E/S1T�T , ��. ) + E/S1��6 2GF N . ) +�/�1T�T , ��. ) +�/�1��6 2 !CB(4.146)N . ) E+�/�1��� , � . ) E+Q/�1��� 2 F N . ) +�/�1��� , � . ) +�/�1��� 2 > N . ) E+�/�1��� , � . ) E+Q/�1��� 2 F N . ) +0/�1��� , � . ) +0/�1��� 2 >N . ) E+0/�1��6 , ��. ) E+Q/�16�6 2GF N . ) +�/�1��6 , ��. ) +�/�16�6 2 > N .5E) +�/�1P�P , ��.5E) +�/�1��� 2RF N . ) +�/�1P�P , ��. ) +�/�1��� 2 >N . ) + E/S1U�U , � . ) + E/S1��6 2GF N . ) +�/�1U�U , � . ) +�/�1��6 2 !CB(4.147)
58
and VXWZYS[D\]Q^_�` acb WZYS[D\]Q^_�_ dGe VXWZYS[ ]�^_�` acb WZYS[ ]�^_�_ dgf VMWZYS[h\]Q^i�` ajb WZYS[h\]Q^i�i dOe VMWkYS[ ]�^i�` acb WkYS[ ]�^i�i dlfV WkYS[D\]S^`�` a b WkYS[D\]S^`�` d e V WZYS[ ]�^`�` a b WZYS[ ]�^`�` d f V W5\YS[ ]�^m�m a b W5\YH[ ]�^_�` d e V WZYS[ ]�^m�m a b WZYS[ ]�^_�` d fVMWkY \[ ]�^n�n acb WkY \[ ]�^i�` dGe VXWZYS[ ]�^n�n acb WZYS[ ]�^i�` dOoCp (4.148)
Equations(4.146-4.148)form a systemof qGrts `equationsin thetermsof thesubcellstrains,out
of which qRrvuws ` eyx0z equationsareindependent.
Thecompatibilityequationsare { ii�i b _�_ f { i_�_ b i�i e}| { i_�i b _�i oCpg~ (4.149){ i`�` b i�i f { ii�i b `�` e}| { ii�` b i�` oCpg~ (4.150){ i_�_ b `�` f { i`�` b i�i e}| { i_�` b _�` oCpg~ (4.151)
{ i_�_ b i�` f { ii�` b _�_ e { i_�i b _�` e { i_�` b _�i oCpg~ (4.152){ ii�i b _�` f { i_�` b i�i e { ii�` b _�i e { i_�i b i�` oCpg~ (4.153){ i`�` b _�i f { i_�i b `�` e { i_�` b i�` e { ii�` b _�` oCpg~ (4.154)
where
{ i����� { i{D� � {D� �l�Theseequationsarediscretizedusingcentraldifferenceschemesof theform
{ i��{D� i o � ��� _�� � ev| � � � � f � ��� _�� �� i ~{ i �{D�-{D� o � ��� _�� � e � ��� _�� ��� _ e � � � � f � � � � � _� i �Thediscretizedcompatibilityequationsarea�b WZY \[ ]�^_�_ dOe}|�ajb WkYS[ ]�^_�_ dgf�a�b WZY��[ ]�^_�_ dgf�a�b W5\Y�[ ]�^i�i dRev|�ajb WZYS[ ]�^i�i dgf�a�b W �YH[ ]�^i�i d�e� a�b W5\YH[ ]�^_�i dOe4a�b W5\Y �[ ]�^_�i dRe4acb WZYS[ ]�^_�i dGe4ajb WZY �[ ]�^_�i dD�Me� a�b WZY \[ ]�^_�i dOe4a�b W �Y \[ ]�^_�i dRe4acb WZYS[ ]�^_�i dGe4ajb W �YS[ ]�^_�i dD��o�p�~ (4.155)
a�b WZYS[D\]Q^i�i dOe}|�ajb WkYS[ ]�^i�i dgf�a�b WZYS[ �]Q^i�i dgf�a�b WZY \[ ]�^`�` dRev|�ajb WZYS[ ]�^`�` dgf�a�b WZY �[ ]�^`�` d�e� a b WZY \[ ]�^i�` d e a b WkY \[ �]�^i�` d e a b WZYS[ ]�^i�` d e a b WZYS[ �]Q^i�` dD� e� a�b WZYS[D\]Q^i�` dOe4a�b WkY �[D\]�^i�` dRe4acb WZYS[ ]�^i�` dGe4ajb WZY �[ ]�^i�` dD��o�p�~ (4.156)
59�����5��H�� �¡¢�¢ £O¤}¥ �j��� �S�0 �¡¢�¢ £g¦ �����¨§�H�� �¡¢�¢ £g¦ ����� �S� � Q¡©�© £R¤v¥ �j��� �S�� �¡©�© £g¦ ����� �S� § Q¡©�© £�¤ª«��� �5��H�� �¡¬�¢ £O¤ ��� �5��H� § �¡¬�¢ £R¤ �c� � �S�� �¡¬�¢ £G¤ �j� � �S� § Q¡¬�¢ £DM¤ª«����� �S� � Q¡¬�¢ £O¤ ������§�H� � �¡¬�¢ £R¤ �c��� �S�� �¡¬�¢ £G¤ �j����§�S�� �¡¬�¢ £D�®�¯�° (4.157)�c� �5��H�� �¡©�¢ £G¤v¥ �c� � �S�� �¡©�¢ £M¦ �c� ��§�H�0 �¡©�¢ £�¦ ª±��� � � ��Q �¡¬�¬ £R¤ �c� � � �� § �¡¬�¬ £G¤ �c� � �S�0 �¡¬�¬ £G¤ ��� � �S� § S¡¬�¬ £cR¤ª²� � �5��H�� �¡¬�¢ £ ¤ � � �5�� §�� �¡¬�¢ £ ¤ � � � �S�0 �¡¬�¢ £ ¤ � � � � §�Q �¡¬�¢ £c ¤ª²�c��� �S� � S¡¬�© £R¤ �c����§�S� � Q¡¬�© £G¤ �c��� �S�0 �¡¬�© £G¤ ������§�H�� �¡¬�© £ct®C¯g° (4.158)� � � � ��Q �¡¬�¢ £ ¤v¥ � � � �S�� �¡¬�¢ £ ¦ � � � � §�� �¡¬�¢ £ ¦ ª±� � � �S� � S¡©�© £ ¤ � � ��§�H� � �¡©�© £ ¤ � � � �S�0 �¡©�© £ ¤ � � ��§�H�� �¡©�© £c ¤ª²�c��� � ��Q �¡¬�© £R¤ �c��� � �� § Q¡¬�© £G¤ �c��� �S�0 �¡¬�© £G¤ ����� �S� § S¡¬�© £cR¤ª²�c���5��H�� �¡©�¢ £R¤ �c���5��S� § Q¡©�¢ £G¤ �c��� �S�0 �¡©�¢ £G¤ ����� � §�Q �¡©�¢ £ct®C¯g° (4.159)� � � �S� � S¡¬�© £ ¤v¥ � � � �S�� �¡¬�© £ ¦ � � � �S� § Q¡¬�© £ ¦ ª±� � �5��H�� �¡¢�¢ £ ¤ � � �5�� §�� �¡¢�¢ £ ¤ � � � �S�0 �¡¢�¢ £ ¤ � � � � §�Q �¡¢�¢ £c ¤ª²�c� � �S� � S¡©�¢ £R¤ �c� ��§�S� � Q¡©�¢ £G¤ �c� � �S�0 �¡©�¢ £G¤ ��� ��§�H�� �¡©�¢ £cR¤ª²� ��� � ��Q �¡¬�¢ £ ¤ � ��� � �� § Q¡¬�¢ £ ¤ � ��� �S�0 �¡¬�¢ £ ¤ � ��� �S� § S¡¬�¢ £c ®C¯g° (4.160)
where ³´ ®¶µ ´ ¤y·S° if ¥R¸ ´ ¸y¹ ;¹t° if ´ ®º· ;³» ®¶µ » ¤¼·S° if ¥R¸ » ¸y¹ ;¹½° if» ®º· ;³¾ ®¶µ ¾ ¤y·S° if ¥X¸ ¾ ¸y¹ ;¹t° if ¾ ®º· .
The averageRVE strainequations(4.134-4.139),theequilibrium equations(4.146-4.148)and
thecompatibilityequations(4.155-4.160)canbecombinedinto a systemof equationsrelatingthe
subcellstrainsto theaverageRVE strains.Unlike theoriginalGMC formulation,thenormalandthe
shearstrainsarecoupledin this formulationthroughtheequilibriumandcompatibilityconditions.
Theaveragesubcellstrainscanbecalculatedfrom theappliedRVE strainsby invertingthesystem
of equations. The subcell stressescan be calculatedfrom the subcell strainsusing the subcell
constitutive equations.As in the original GMC formulation, the averageRVE stressescan then
berelatedto theaverageRVE strainto gettheeffective stress-strainresponse.
Thedrawbackof theshear-coupledapproachis thatamuchlargersystemof equationsis formed,
comparedto thereformulatedGMC discussedin thischapter. Hence,themethodis computationally
60
expensive. Any computationaladvantageover finite elementanalysisbasedapproachescould be
lostbecauseof thelargesizeof thematricesthathave to beinverted.
The requirementthat a large matrix be invertedto get the effective propertiesmakesthe gen-
eralizedmethodof cellsvery inefficient asthenumberof subcellsincreases.Whenmaterialssuch
asPBX 9501 aremodeled,the numberof subcellsneededto representa randomdistribution of
particlesnecessarilybecomeslarge. In such situations,the methodof cells basedapproaches
becomeinefficientandit maybepreferableto performsix differentfinite elementanalysesto getthe
effective propertiesratherthanonemethodof cellsbasedanalysis.This limitation, alongwith the
lackof shearcouplingin theoriginalmethodof cellshasledusto developanothermicromechanics
schemethat we call the “Recursive Cell Method (RCM)”. This methodis discussedin the next
chapter.
CHAPTER 5
THE RECURSIVE CELL METHOD
The original GMC techniquehasbeenfound to provide inadequateshearcoupling between
adjacentsubcells.In addition,theamountof computationaltime neededto calculatetheeffective
propertiesusingGMC increasesdramaticallyasthenumberof subcellsincreases.Therecursivecell
method(RCM),developedasanalternative toGMC,attemptsto resolvetheseproblemswithoutloss
in accuracy.
A schematicof the recursive cell methodis shown in Figure5.1. TheRVE is discretizedinto
subcellsas in GMC. However, insteadof calculatingeffective propertiesof the whole RVE in a
single step, the effective propertiesof small blocks of subcellsare determinedat a time. The
effective propertiesof theRVE arecalculatedby combiningtheeffective propertiesof blocksusing
arecursive process.Theeffective propertiesof eachblockof subcellsmaybedeterminedusingany
accuratenumericaltechnique.Weusea finite elementsbasedtechniquein this research.TheRCM
recursive schemehasbeenfound to reducethecomputationalcostandremedytheshear-coupling
problemof GMC.
Efficient recursionthroughthesubcellsrequiresthat thenumberof subcellsperblock, in each
stageof the recursion,be the same.The first stepin the recursive cells methodis, therefore,the
choiceof thenumberof subcellsto behomogenizedinto asingleblockfor thenext homogenization
stage.We have chosenblocksof four equalsizedsubcellsfor the computationsin this research.
Thereis, however, no upperlimit to the numberof subcellsneededto form a block. The only
constraintis thatif ¿ÁÀG¿ subcellsarehomogenizedinto ablock, thentheRVE hasto bediscretized
sothatthereareat least¿= subcellson eachsidewhereà is anintegergreaterthanzero.
We usea simplifiedfinite elementbasedapproachto homogenizeeachblock of subcells.The
planestrain assumptionis madein the two-dimensionalcalculationsperformedin this research.
To improve computationalefficiency, numericalintegrationis avoidedin our calculations.Instead,
explicit formsof thestrain-displacementandstress-strainrelationsareused.Theserelationsandthe
algebraleadingto themareshown in thefollowing sections.After theserelationsweredetermined,
a largenumberof validationrunswereperformedto determinetheappropriateboundaryconditions
to beusedin therecursivehomogenizationprocess.Theseboundaryconditionsarealsolistedin this
62
RVE - Level 1
RVE - Level 2Final RVE
RVE - Level 0
Figure5.1. Schematicof therecursive cell method.
chapter. TheRCM techniquehasbeendevelopedfor two-dimensionalproblemsso far. However,
extensionto threedimensionsis straightforward.
The recursive procedurecanbe usedwith techniquesother thanfinite elementssuchasfinite
differencesor integral equationbasedmethods.However, careshouldbetakensothat thecompu-
tationalefficiency of therecursive procedureis higherthanthatof explicit calculationsusingthese
othermethods.
5.1 SubcellStiffnessMatricesExplicit expressionsfor thestiffnessmatrix usinga displacementformulationhave beendevel-
opedfor a four-nodedsquareanda nine-nodedsquare.Thenine-nodedsquareelementis usedin
conjunctionwith a hybrid nine-nodeddisplacement/pressurebasedelementthat is usedto model
nearly incompressiblebehavior. The explicit form of the stiffnessmatrix eliminatesthe needfor
numericalintegrationsin thecalculations.
63
5.1.1 DisplacementBasedFour-NodedElement
A schematicof thefour-nodedelementis shown in Figure5.2.Sincetheplanestrainassumption
is only valid for at mostanorthotropicmaterial[11], we assumethat thematerialconstitutingthe
elementis orthotropic.Thenodes1 through4 areorderedin acounterclockwisemanner.
X
Y
4 3
21
h
h
Figure 5.2. Fournodedelement.
Thedisplacementfunctionsfor thiselement,in isoparametricform, areÄ?ÅÇÆQÈ<ÉSÊ@Ë ÌÍ Î�ÏÑÐ=Ò Î Ä Î È ÓÔÅÇÆ�È<ÉSÊ@Ë ÌÍ Î�ÏÑÐÕÒ Î Ó ÎÕÖ (5.1)
where Ä Î and Ó Î arethedisplacementsin the Æ and É directionsat node × , respectively. Theshape
functionsÒ Î
aregivenby Ò Ð Ë Å¨ØAÙ Æ«Ê�ŨØ�ÙvÉSÊÚ È (5.2)ÒgÛ Ë Å¨ØÝÜÞÆ«Ê�ŨØ�ÙvÉSÊÚ È (5.3)Ògß Ë Å¨ØÝÜÞÆ«Ê�ŨØàÜáÉSÊÚ È (5.4)Ò Ì Ë Å¨ØAÙ Æ«Ê�ŨØàÜáÉSÊÚ Ö(5.5)
Thestrain-displacementrelationsare
â4Ë ãä�åÔæ5ç�çæ�è<èé ç�è-ê�ëì Ë ãííííííä ííííííåî Äîjïî ÓîDðî Äîjð Ü î ÓîDï ê ííííííëííííííì
Ö(5.6)
64
Theserelationscanbewritten in termsof thenodaldisplacements( ñhòôó�õSò ) asöø÷CùRú ó (5.7)
where úû÷ýü ñÿþ õ±þ$ñ � õ � ñ�� õ�� ñ�� õ�� � ó and, (5.8)ù'÷ ��� ������ � � ����� � � ��� � � ����� � �� � � � � � � � � � � � � � � � � �� � � ��� � � � � � ����� ��� � � ��� � � � � � ����� �����
(5.9)
Thestress-strainrelationsare !#"%$'&(&$�))* &()+#,- ÷/. !#"102&(&02)3)4 &()
+#,- ó (5.10)
where5 is theorthotropicstiffnessmatrix,.7÷ ���6 þ�þ 6 þ � �6 þ � 6 �2� �� � 6�727 ��8�(5.11)
For the specialcaseof isotropy ( 9 and : arethe Young’s modulusandPoisson’s ratios, respec-
tively), .º÷ 9; � � :�< ; � �>= :�<�� � � : : �: � � : �� � þ2? �2@� ��
(5.12)
Theelementstiffnessmatrix is givenbyA ÷CB þ?=þ B þ?=þ ù .gùEDGF(HGIKJ � J � ó (5.13)
whereI is theJacobianmatrix relatingthe;ML ó2NG< coordinatesystemto the
; � ó � < coordinatesystem.
Performingtheintegration,weget
A ÷�OOOOOOOOOO�=�P Q 9 R ��P �SQ T � R= 4 � R U �SQ � 4 R V=�P �SQ T R �WP Q= 4 � R V Q � 4=�P Q 9 RX NZY[Y � = 4 � R U=�P �SQ= 4
�]\\\\\\\\\\� (5.14)
whereP ÷ �^ ; 6 þ�þ �_6�727 <"ó Q ÷ �� ; 6 þ � �_6�727 <"ó 4t÷ �^ ; 6 �2� �_6�727 <"óR ÷ �� ; 6 þ � �`6�727 <"ó T ÷ �^ ; 6 þ�þ �`=a6�727 < ó 9 ÷ �^ ; 6�727��>=a6 þ�þb<"óU ÷ �^ ; 6 �2� �>=a6�727 < ó V ÷ �^ ; 6�727��c=a6 �2� < �
It maybenotedthatthereis nodependenceof thestiffnessmatrixon theelementsizeor location.
65
5.1.2 DisplacementBasedNine-NodedElement
A schematicof thenine-nodeddisplacementbasedelementis shown in Figure5.3.Thiselement
is usedin conjunctionwith the nine-nodeddisplacement/pressurebasedhybrid elementusedto
modelthenearlyincompressiblebinderof PBXs. The materialof theelementfollows the stress-
strainrelationsshown in equations(5.10),(5.11)and(5.12).In thiscase,thedisplacementfunctions
X
Y
4 3
21
h
h
5
6
7
8 9
Figure 5.3. Nine nodedelement.
for theelementare dfeMg hi�jlk mn oqpsrut o d o h v eMg hi�jlk mn oqpsrwt o v owx (5.15)
The shapefunctionsusedfor this elementare(usingan isoparametricformulationwhere
eMghi�j is
thelocal co-ordinatesystem[120])
66y{z�|~}����c������}����`�����(5.16)y{��| }����`� � ��}��������� � y{z� (5.17)y���| }����c� � ��}����_���� � y{z� (5.18)y{��| }����`� � ��}��������� � y z� (5.19)y{��| }����c� � ��}����`���� � y{z� (5.20)y��W| }����c�a��}��������� � y �� � y{�� � y{z� (5.21)y{��| }����E�a��}��������� � y �� � y{�� � y{z� (5.22)y � | }����E�a��}����>���� � y��� � y{�� � y{z� (5.23)y���| }����c�a��}����>���� � y �� � y �� � y z� (5.24)
The stiffnessmatrix for this elementis obtainedusingequations(5.6) and(5.13)andperforming
theintegrations.Theexplicit form of thestiffnessmatrixof thenine-nodedelementis shown in the
following page
67
�M�� �u�1�u�u�1�u�1�u���u�1�u�u�1�u�1�u�1�u���u�1�u�u�1�u�1�u�1�u�u ¡£¢¡£¤¥ ¢ ¦¡£§ ¦¡£¨ ¦¡£© ¦¥ ª¡ § ¦« ¢¬¢® ¨¡ ª® ©¡ ª« ¨ ¦¬¢ ¦¯ ¨¦¬¨
¡±°¡±§ ¦² ª ¦¡±© ¦¡´³ ¦¡ §² ¢ ¦¬µ¢¶ ¨¡ ª· ©¡ ª· ¨¬ ¢ ¦¶ ¢ ¦¬ ¨¦¸ ¨
¡£¢ ¦¡£¤ ¦¥ ª ¦¡£§ ¦¡ ¨¡ © ¦« ¢ ¦¬¢« ¨¬¢® © ¦¡ ª® ¨ ¦¡ ª ¦¯ ¨¬¨
¡±°¡±§² ¢¡±©¦¡ ³¬µ¢¶ ¨¦¬µ¢ ¦¶ ¢ ¦¡ ª· ¨ ¦¡ ª· ©¬µ¨ ¦¸ ¨
¡£¢¡£¤¥ ¢ ¦¡ §® ©¡ ª« ¨¦¬ ¢ ¦« ¢¬¢® ¨¡ ª¦¯ ¨¦¬¨
¡£°¡£§¦²µª¡ ª· ¨¬¢ ¦¶ ¢ ¦¬¢¶ ¨¡ ª· © ¦¬ ¨ ¦¸ ¨
¡£¢ ¦¡ ¤® © ¦¡ ª® ¨ ¦¡ ª ¦« ¢ ¦¬¢« ¨¬¢ ¦¯ ¨¬¨
¡ ° ¦¡ ª· ¨ ¦¡ ª· ©¬¢¶ ¨¦¬¢ ¦¶ ¢¬¨¦¸ ¨
¥ ¨¹¦¯ ¨ ¦¬ ¨ ¦¯ ¢¹ ¦¯ ¨¬µ¨® ¢¹
² © ¦¬¨ ¦¸ ¨¹ ¸ ¢¬ ¨ ¦¸ ¨¹¦· ª
¥ ©¹¦¯ ¨¬¨¯ ¢¹ ¦® ª¹
² ¨¬¨¦¸ ¨¹ ¦¸ ¢¹· ¢
¥ ¨¹¦¯ ¨ ¦¬¨® ¢¹
² © ¦¬µ¨ ¦¸ ¨¹ ¦· ª
º»¼¼¥½ ¾»
¥ ©¹¦® ª¹² ¨¹· ¢¿À ¯ ¨¹ ¿À ¸ ¨Á ÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuÂ�ÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuÂ1ÂuÂ�ÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuÂ1ÂuÂuÃ
(5.2
5)
68
wherethefollowing substitutionshave beenmade:Ä�Å�Æ ÇÈ�É'Ê±Ë Å2ÅÍÌ Ë�Î2Î�Ï ÄÑÐ�Æ ÇÈ�É�Ê±Ë Å2ÅlÒ Ë�Î2Î�Ï Ó Å�Æ È Ô Ê±Ë Å£ÐlÌ Ë�Î2Î�ÏÓ Ð�ÆÖÕ× Ê±Ë Å£ÐSÒ Ë�Î2Î�Ï Ø Å�Æ ÇÈ�É�Ê±Ë Ð2ÐlÌ Ë�Î2Î�Ï Ø Ð�Æ ÇÈ�É'Ê±Ë Ð2ÐSÒ Ë�Î2Î�ÏÙ Å�Æ ÕÈ�É Ê±Ë Å2ÅÍÌ È Ë�Î2ÎÚÏ Ù Ð�ÆÖÕ�ÛÈ�É Ê±Ë Å2ÅlÒ È Ë�Î2Î�Ï ÙÑÜ Æ ÕÈ�É Ê È Ë Å2ÅÍÌ Ë�Î2Î(ÏÙ�Ý ÆÖÕ�ÛÈ�É Ê È Ë Å2Å�Ò Ë�Î2Î�Ï Þ Å�Æ ÇÈ�É Ê È Ë Å2ÅÍÌ_ß Ë�Î2Î�Ï Þ Ð�Æ ÕÔ�à Ê È Ë Å2Å�Ò�ß Ë�Î2Î�ÏÞ Ü Æ ÇÈ�É Ê ß Ë Å2ÅÍÌ È Ë�Î2Î�Ï Þ Ý Æ ÕÔ�à Ê ß Ë Å2Å�Ò È Ë�Î2Î�Ï á Å�Æ ÕÈ�É Ê ß Ë Å2Å�Ò Õ�Û Ë�Î2Î�Ïá Ð�Æ ÕÈ�É Ê Õ�Û Ë Å2Å�Ò`ß Ë�Î2Î�Ï â Å�Æ ÕÈ�É Ê±Ë Ð2ÐlÌ È Ë�Î2Î�Ï â Ð�Æ Õ�ÛÈ�É Ê±Ë Ð2ÐSÒ È Ë�Î2Î�Ïâ Ü Æ ÕÈ�É Ê È Ë Ð2Ð�Ì Ë�Î2Î�Ï â Ý Æ Õ�ÛÈ�É Ê È Ë Ð2Ð�Ò Ë�Î2Î�Ï ã Å�Æ ÇÈ�É Ê ß Ë�Î2Î Ì È Ë Ð2Ð Ïã Ð�Æ ÕÔ�à Ê È Ë Ð2Ð�Ò`ß Ë�Î2Î�Ï ã Ü Æ ÇÈ�É�Ê È Ë�Î2Î Ì_ß Ë Ð2Ð Ï ã Ý Æ ÕÔ�à Ê ß Ë Ð2Ð�Ò È Ë�Î2Î�Ïä Å Æ ÕÈ�É'Ê ß Ë Ð2Ð Ò Õ�Û Ë Î2Î Ï ä Ð Æ ÕÈ�É�Ê Õ�Û Ë Ð2Ð Ò`ß Ë Î2Î Ï å Å Æ Ä�ÅÕ�Ûå Ð�Æ ß�Ä ÅÈ å Ü Æ Ó ÅÕ�Û å Ý Æ Ó ÅÈå�æ Æ Ô Ó ÅÕ�Û å�Î Æ Ó ÐÈ åÚç Æ Ø ÅÕ�Ûå�è Æ ß Ø ÅÈêéThe stiffnessmatrix is, like the four nodedelement,independentof the locationandsizeof the
element.
5.1.3 Mixed Displacement-Pressure Nine NodedElement
Thebindermaterialusedin PBXsis nearlyincompressible.This impliesthatthePoisson’s ratio
of thesematerialsis closeto 0.5andhencethebulk modulusis largecomparedto theshearmodulus.
Hence,thevolumetricstrainis smallandis equalto zeroin thelimit of incompressibility. Thestrain
is determinedfrom derivativesof displacements.In finite elementformulations,thederivativesof
displacementarelessaccuratelydeterminedthanthenodaldisplacements.Therefore,any error in
the predictedvolumetricstrainfor nearlyincompressiblematerialswill leadto large errorsin the
predictedstresses.Sincetheexternalloadsarebalancedby thestresses,this alsoimplies that the
predicteddisplacementswill be inaccurateunlessanextremelyfine meshis used.In practice,the
displacementspredictedby displacementbasedfinite elementsfor nearlyincompressiblematerials
aremuchsmallerthanthoseexpected[120]. Thisbehavior is calledelementlocking.
69
Theproblemof elementlockingcanbeavoidedby usinganonlinearmaterialmodelsuchasthe
Mooney-Rivlin rubbermodelfor thebinder. However, afinite elementrepresentationof thismodel
requiresthattheloadbeappliedin multiple steps.For isotropiclinearelasticmaterialsundergoing
small strains,a displacementandpressurebasedmixed formulationis adequate[120]. We usea
mixed formulationpresentedby Bathe[120] to modelthesubcellscontainingthebindermaterial.
Thebasisof theformulationis theWu-Hashizufunctionalform of theprincipleof virtual work.
The Wu-Hashizufunctional can be expressedas a sum of volumetric and deviatoric strain
energiesandequatedto theexternalvirtual work asë'ìîí�ï ð�ñóòZôöõ`ëGìî÷ ìùø òZôûúýü8þ(5.26)ñÿú�� � ø � þ(5.27)í ð ú í õ ÷ ì� � þ (5.28)
whereí ð
is thedeviatoric strainmatrix,ñis thedeviatoric stressmatrix,÷ ìis thevolumetricstrain,ø
is thehydrostaticpressure,üis theexternalvirtual work,�is thestressmatrix,íis thestrainmatrix,and,�is theKronecker delta.
In addition,thevolumetricstrainandthehydrostaticpressurearerelatedbyë ì�� ø� � ÷ ì��ø ò%ôûú �(5.29)
where�
is thebulk modulus,and,øis aweightingfunction.
Finite elementdisplacementandpressureinterpolationfunctionsfor theelementarechosenof the
form � ú ���� þ(5.30)ø ú ������� (5.31)
70
where � is theelementdisplacementvector,�� arethenodaldisplacementdegreesof freedom,�� aretheelementpressuredegreesof freedom.�arethedisplacementshapefunctions,and,���arethepressureshapefunctions.
Thevolumetricstrain ��� is givenby thesumof thestrainsin the two coordinatedirectionsandis
relatedto thedisplacementsby ���! "�$#%#'&(�$)*)+ -,/.,/0 &1,32,/465 (5.32)
where. and 2 arethedisplacementsin the 0 and 4 directions,respectively. Thestrain-displacement
relationsfor thedeviatoric straincomponentsare
798 :;;;;;;;<�$#%#>= �*�?�@)�)A= ���?B #%)�@C@CD= �*�?
EGFFFFFFFH :;;;;;;;;;;<I? ,/.,/0 =KJ? ,/2,/4I? ,/2,/4 = J? ,/.,/0,3.,/4 & ,/2,30=�J? ,/.,/0 =KJ? ,/2,/4
EGFFFFFFFFFFH L (5.33)
Thenwecanform thefollowing relationships7M8 N 8 �� 5 (5.34)���! NO� �� L (5.35)
Let therelationshipbetweenthedeviatoric stressandthedeviatoric strainbeP RQ 8 7 8 5 (5.36)
where Q 8 is thedeviatoric stiffnessmatrix. For an isotropic,nearlyincompressiblematerialwith
shearmodulusS , thedeviatoric stiffnessmatrix is
Q 8 :;;< I S T T TT I S T TT T S TT T T I SEGFFH L (5.37)
71
Application of the principle of virtual work leadsto a systemof equationsrelating the nodal
displacementsandelementpressuresto the externalappliedloads. This systemof equationscan
bewrittenas UVVW X3YOZ\[]'^ ] Z ]`_ba c XdYOZ\[Yegf _bac X3Y e [f Z Y _ha c XhY e [f eifj _ba>kGllmRn%op oqsrit nvu w ryx (5.38)
Wecanwrite thisequationin morecompactform asnvz|{}{ z|{ fz [ { f z f@f r n op oqsrit n u w ryx (5.39)
Staticallycondensingout thepressureterms,we have,~ z|{�{ c z|{ f z����f@f z [ { f�� op t u x (5.40)
Equation(5.40)is anequationthatcanbesolvedfor thedisplacements.Thepressuresdo not have
to bedeterminedexplicitly.
The nine nodeddisplacement/pressure elementwith threepressuredegreesof freedom(also
called a 9/3 u-p element)hasbeenproven to avoid elementlocking [120]. We have, therefore,
chosenthiselementfor theRCM calculationson subcellscontainingthebindermaterial.
The 9/3 u-p elementhasthe samegeometryandnodenumberingschemeas that of the nine
nodedelementshown in Figure5.3. The displacementinterpolationfunctionsarethoseshown in
equations(5.16-5.24).Thepressureinterpolationfunctionis chosento be�������$�d� t ������� � �������*� x (5.41)
Therefore,in termsof theisoparametriccoordinates�������9� , we have,eif t������ � � �}� x (5.42)
Thethreepressurerelateddegreesof freedomareinternalto theelement.
Thematricesz�{}{ , z|{ f and z f@f canbedeterminedby explicit integration. After performing
the integrationsandinsertingthe resultingmatricesinto equation (5.40)we get the explicit form
of thestiffnessmatrix for themixednine-nodeddisplacement-pressureelement.The form of this
stiffnessmatrix is shown in equation(5.43).
72
Thefollowing substitutionshave beenmadein equation(5.43).��� ¡9¢ £¤¦¥'§!¨ª©/«¬¤¦¥`(©/«¯® ° � ¥'§!¨ª© ®±d² � ¥}³´ ¥}µ�¶ °¨ ® ±�· � ¨ª¨ª³¥}¸ ¥}³ ° ® ±�¹ � ¥}³¨ ³ °º ®±�» � ³¥}¸ ¥ ¢ ³ °º ® ±�¼ � ¶9³º § ¨ªµ ° ® ±�½ � ¥}µª³¥}¸ § ¥}³ ° ®±¿¾ � ¥}³º § µ °º ® ±�À � ¥}³¨ § ¥ª¥ ¡ ° ® ±�Á � ¥}µª³ § ¥}µª³ °¨ ®± ²Ã �Ä ³ ¥}³ °¨ ® ± ²$² � ¥}³¨ ¸�¶ ° ® ± ²Ã· � ¥ ¢ ³ § ¡ ³ ° ®±d²Ã¹ � ³ (¸ ¢ °i® ±h²Å» � µ ¢ §!µª¨ °6® ±h²Ã¼ � ¥}³´(¨h¥�¶ °6®±d²Ã½ � ¥}³>§Æ¥ª¥ °i® ±h² ¾ � ¸ ¢ §Æ¥ª¥}¸ °6® ±h² À � ¥ ¢ ³Ç(¨ ¢ ³ °i®±d² Á � ¥}³´ °6® ±�·$ � ¥}¨ ¢ §Æ¥ª¥}¸ °i® ±�·È² � ¨ ¢ (³ ¡ ¨ °i®±É·$· � ¥}³ °6® ±�·$¹ � ³ ËÊTheabove relationsshow that thestiffnessmatrix dependsonly on thematerialpropertiesandnot
on the elementlocationandsize. After the stiffnessmatriceshave beencalculated,they canbe
assembledin theusualmanner.
73
Ì�ÍÎÏ Ð�ÐÑÐ�Ð�ÐÑÐ�ÐÑÐ�Ð�Ð�ÐÑÐ�Ð�ÐÑÐ�ÐÑÐ�ÐÑÐ�Ð�Ð�ÐÑÐ�Ð�ÐÑÐ�ÐÑÐ�ÐÑÐ�Ð�ÒÓÕÔÓÕÖÓÕ×ÓÙØÓÕÚÓÕÛÓÙÜ ÝÓÙØÓÕÞÓÕß ÝàÓ ÜÓ ÔáÓÕÔÔÓÕÖ×ÓÕÔÖÓ Ô×Ó ÔØ Ýâ ã
ÓÙÔ ÝÓÃØÓÃÜÓÙÛÓÙÚÓÃØÓÙ×ÓÙÔ×Ó ÔÖÓ Ö×Ó ÔÔÓÙÔáÝàÓÃÜÓ ßÓ Þ Ýâ ãÓ ÔØ
ÓÕÔ ÝÓÕÖÓÙÜÓÙØÓÕÚ ÝÓÕÛÓÕÞ ÝÓÕßÓ ÔÖ ÝÓÕÔ×ÓÕÔÔ ÝÓÕÖ× ÝàÓ Ü ÝÓ ÔáÓ ÔØâ ã
ÓÙÔ ÝÓÃØÓÙ× ÝÓÙÛÓÙÚ ÝÓ Ô×Ó ÔÖ ÝÓÙßÓ Þ ÝÓÙÔá ÝàÓÃÜ ÝÓ Ö×Ó ÔÔâ ãÓ ÔØ
ÓÕÔÓÕÖÓÕ×ÓÙØÓÕÔÔÓ Ö×Ó ÔÖÓ Ô×ÓÕÞÓ ß ÝàÓ ÜÓ ÔáÓ ÔØ Ýâ ã
ÓÕÔ ÝÓÙØÓÙÜÓÕÔá ÝàÓ ÜÓÕßÓ ÞÓÕÔ×ÓÕÔÖÓÕÖ×Ó ÔÔ Ýâ ãÓ ÔØ
ÓÕÔ ÝÓÕÖÓÕÔÔ ÝÓ Ö× ÝàÓ Ü ÝÓÕÔáÓÕÞ ÝÓ ßÓÕÔÖ ÝÓ Ô×Ó ÔØâ ã
ÓÕÔ ÝÓ Ôá ÝàÓ Ü ÝÓ Ö×Ó ÔÔ ÝÓÕÔ×ÓÕÔÖ ÝÓÕßÓ Þâ ãÓ ÔØ
ÓÙÔÚäÓ ÔØ ÝÓÙÖÖÓÙÔÛäÓÙÔØÓ ÖÖÓ ÔÜä
Ó ÔÞ Ýâ ãÓ ÔØäÓÕÔßâ ãÓ ÔØäÓ Öá
Ó ÔÞäÓÕÔØâ ãÓÕÔßäÓ Öáä
Ó ÔÚÓÕÖÖÓÕÔØäÓ ÔÛäÓ ÔÜ
ÓÕÔÚäÓÕÔØ ÝÓ ÖÖÓ ÔÜä
ÓÙÔÞ Ýâ ãÓ ÔØäÓ Öá
ÓÕÔÞäÓ Öáä
åæççèé êæ
Ó ÔÚäÓ ÔÜ Ó ÖÔä Ó ÖÔë ì�ìÑì�ì�ìÑì�ìÑì�ì�ì�ìÑì�ì�ìÑì�ìÑì�ìÑì�ì�ì�ìÑì�ì�ìÑì�ìÑì�ìÑì�ì�í(5
.43)
74
5.2 Modeling a Block of SubcellsThepresentimplementationof the recursive cellsmethodperformsfinite elementcalculations
on blocksof four subcellsata time. A schematicof sucha block is shown in Figure5.4.
î î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îî î î î î î î îï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï
ð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðð ð ð ð ð ð ð ðñ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ññ ñ ñ ñ ñ ñ ñ ñ
ò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òò ò ò ò ò ò ò òó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó óó ó ó ó ó ó ó ó
ô ô ô ô ôô ô ô ô ôô ô ô ô ôô ô ô ô ôô ô ô ô ôõ õ õ õ õõ õ õ õ õõ õ õ õ õõ õ õ õ õõ õ õ õ õ
1 2 3
45
6
78 9
1 2
3 4
X
Y
Figure5.4. A four subcellblockmodeledwith four elements.
Eachblock is periodically repeatedin spaceto form a compositematerial. If the effective
propertiesof this materialareto be determined,periodicboundaryconditionshave to be applied
to theblock to setup the requiredfinite elementanalyses.If four elementsareusedto modelthe
four subcellsin ablockasshown in Figure5.4,forcingdisplacementson theboundaryof theblock
to beperiodicleadsto forcesthatarenot periodicon theboundaries.This is becausetheelements
on oppositesidesof the boundarycanhave differentstiffnessandmay requiredifferentforcesto
achieve thesamedisplacement.Threetypesof displacementboundaryconditionsareappliedfor
thefinite elementanalyses- two normaldisplacementsin thetwo coordinatedirectionsandashear
displacementin theplane.Notethatalongtheboundarieswheredisplacementsarenotapplied,the
forcessumto zerothoughthey maynotbezeroatany of thenodeson thatboundary.
Both theforcesandthedisplacementscanbeforcedto beperiodicif sixteenfinite elementsare
usedto modelthefour subcellblock asshown in Figure5.5. In this case,theelementson opposite
sidesof theboundaryhavethesamestiffnessandthereforerequirethesameforceto achieveagiven
displacement.
Similar approachesareusedwith nine-nodedelements.It shouldbenotedthat thenine-noded
displacement-pressure elementsareusedonly for subcellscontainingthebinder. Oncenew material
75
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ø ø ø ø ø ø øø ø ø ø ø ø øø ø ø ø ø ø øø ø ø ø ø ø øø ø ø ø ø ø øø ø ø ø ø ø øù ù ù ù ù ù ùù ù ù ù ù ù ùù ù ù ù ù ù ùù ù ù ù ù ù ùù ù ù ù ù ù ùù ù ù ù ù ù ùú ú ú ú ú ú úú ú ú ú ú ú úú ú ú ú ú ú úú ú ú ú ú ú úú ú ú ú ú ú úú ú ú ú ú ú úû û û û û û ûû û û û û û ûû û û û û û ûû û û û û û ûû û û û û û ûû û û û û û û
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� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � �� � � �� � � �� � � �� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � � � � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �
� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �� � � � � � �
1 2 3 4 5
6 7 8 9 10
1112 13 14 15
16 17 18 19 20
2122 23 24 25
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
X
Y
Figure 5.5. A four subcellblock modeledwith sixteenelements.
propertieshavebeengeneratedfor ablockcontainingthebinder, thedisplacementbasednine-noded
elementis usedfor all furtherrecursions.
Thestiffnessmatrix for eachof theelementscanbecalculatedusingtheexplicit formsshown
in theprevioussection.Thesematricescanthenbeassembledby superpositionto form theglobal
stiffnessmatrix for this problem.Explicit formsof theglobalstiffnessmatrix for thefour-element
modelandthesixteen-elementmodelhave beenobtainedusingMaple6. Thesematricesbecome
smallenough,afterapplicationof boundaryconditions,thatexplicit solutionsfor thenodaldisplace-
mentscanbe obtained. However, suchexplicit forms areextremelycomplex for the nine noded
elementsandit is easierto calculatetheelementstiffnessesandto assemblethemnumerically. In
the presentimplementationof the recursive cell method,the global stiffnessmatrix is assembled
numerically. After the global stiffnessmatrix for a block of subcellshasbeendetermined,the
appropriateboundaryconditionsareappliedto obtainthedisplacementsolutionthat,in turn, leads
to theeffective elasticmoduli for theblock.
5.3 Boundary ConditionsThe finite elementprobleminvolves the solutionof a setof � linear simultaneousequations
relatingthedisplacements��� to theappliedforces ��� . Thissystemof equationscanbewrittenas
76���! #"%$'& �)( �+*-, & .0/2143517698;: (5.44)
Thestiffnessmatrix is singular, andthesetof equationscanonly besolvedupontheapplicationof
suitableboundaryconditions.Threesetsof boundaryconditionsareappliedon thefinite element
representationof thefour-subcellblocksothatthetwo-dimensionaleffectivepropertiesof theblock
canbecalculated.Theseare
1. auniform normaldisplacementin the < direction(’1’ direction),
2. auniform normaldisplacementin the = direction(’2’ direction),and,
3. asheardisplacementin the <>= -plane(’12’ plane).
A schematicof the four elementbasedmodel of a block of subcellsundergoing a normal
displacementin the < direction is shown in Figure5.6. The figure shows the original shapeand
the deformedshapeandthe correspondinglocationsof the nodes. A uniform displacement? is
appliedto nodes3, 6, 9 andnode1 is keptfixed. Nodes2 and3 arenot allowed to move in the =direction.Similarly, nodes4 and7 arenot allowedto move in the < direction.Nodes7,8and9 are
constrainedto moveanequalamountin the = direction.Thepairof nodes2 and8 areconstrainedso
thatthey move anequalamountin the < directionwhile nodes4 and6 areconstrainedsothat they
move anequalamountin the = direction. Theapplieddisplacement? andthefixeddisplacements
atnodes1, 2, 3, 4 and7 arecalledtheprescribeddisplacements.Theconstraineddisplacementsare
describedby constraintequations.Thealgebrausedto applyconstraintequationsandtheprescribed
displacementsis discussedin sections5.3.1and 5.3.2respectively. In equationform, theprescribed
displacementsfor thesituationshown in Figure5.6are( "@*-ACB DE"@*FACB DHGI*FAJB (LK * ? BD�KM*-ACB (LN *FACB (LO * ? B (QP *RASB(LT * ? :Theconstraintequationsfor this caseare(LU9VW( GM*-ACB D O V D N *-ACB D UXV D P *FACB D T9V D P *FA :
The effect of a uniform displacementappliedin the = directionon the positionsof the nodes
is shown in Figure5.7. Boundaryconditionssimilar to thosefor a displacementin the < direction
apply to this casetoo. Note that the constraintequationsare usedto satisfy periodicity of the
displacements.Theseconstraintsleadto stressstatesthatarenot purelyunidirectional.However,
77
4 5 6
1 2 3
7 8 9
1 2 3
4 5 6
7 8 9
X
Y
Figure 5.6. Schematicof theeffectof auniform displacementappliedin the Y direction.
4 5 6
1 2 3
7 8 9
1
X
Y
2 3
4
56
7 8 9
Figure 5.7. Schematicof theeffectof auniform displacementappliedin the Z direction.
for the materialsunderconsideration,the deviationsof the stressesfrom a unidirectionalstateof
stressaresmall.
The applicationof a puresheardisplacementis more problematic. Two schemeshave been
examinedfor this process.The first schemeinvolves prescribingdisplacementsthat correspond
78
to a pure shearat the boundarynodes. A schematicof this processis shown in Figure 5.8. In
this approach,node1 is fixed andnode9 is assigneddisplacementsof magnitude[]\9^_[)` in thea and b directions. Node3 is assigneda displacement[]\ in the a directionanda displacement[)` in the b direction. Similarly, node7 hasprescribeddisplacementsof [)` in the a directionand[]\ in the b direction. The nodeson the boundarythat arebetweenthe cornernodesareassigned
displacementssuchthattheboundariesremainstraightlines.Thevaluesof [ \ and [ ` arechosenso
that they correspondto a puresheardisplacement.Applicationof suchboundaryconditionsleads
to relatively highstressesin the a and b directionsanda relatively stiff response.
4 5 6
1 2 3
7 8 9
X
Y
4
79
2
5
8
6
3
1
Figure 5.8. Schematicof theeffect displacements,correspondingto apureshear, appliedat theboundarynodes.
An alternative to this approachof applicationof sheardisplacementboundaryconditionsis
shown in Figure5.9. In this case,thedisplacementsareprescribedonly at thecornernodeswhile
the othernodeson the boundaryareconstrainedso that they maintainperiodicity. Thusnodes2
and8 areconstrainedto have thesamedisplacementsin the a and b directionswith node8 being
allowed anadditionaldisplacementcorrespondingto thesheardisplacement.A similar constraint
equationrelatesthe displacementsat nodes4 and 6. This approachis usedfor the calculations
shown in Chapter6 and7. The normalstressesgeneratedusing this type of sheardisplacement
boundarycondition aremuch smallerthanwith the previous approach.However, when 9/3 u-p
elementsareusedunrealisticdisplacementsmay be obtainedat node5 which do not occurwhen
thefirst approachfor applyingsheardisplacementsis used.This issueis currentlybeingexplored.
Theprescribedsheardisplacementsfor theapproachshown in Figure5.9are
79
4 5 6
1 2 3
7 8 9
X
Y
1 23
4 56
7 89
Figure 5.9. Schematicof theeffect displacements,correspondingto apureshear, appliedat thecornernodes.
ced@fFgJh iEd@fFgCh cQjMf-k]d@h iHjMf-k)lMhcQmIf-k)l+h inmIf-k]d@h cQoMf-k]dqprk)lIh iHoMf-k]dqp7k)lIsThecorrespondingconstraintequationsarecQtXuvc>w+f-k]d9h iHtXuWi�wMf-k)l+h cQxXuWcLlIf-k)l+h iHxXuWiHlIf-k]d9s
Theapplicationof constraintequationsandprescribeddisplacementsto thefinite elementsys-
temof equationsshown in equation(5.44)is discussedin thefollowing sections.
5.3.1 Application of Constraint Equations
An equationthat relatesthe displacementsof two nodesis calleda constraintequation. For
example,for thecaseshown in Figure5.6aconstraintequationiscLl9uWcQxMfFgwhere,c l is thedisplacementin the y directionat node2, and,cLx is thedisplacementin the y directionat node8.
In this case,cLl is theprimedegreeof freedomsinceit hasa coefficient of +1. Therecanbemany
suchconstraintequations.In generalform, theseconstraintequationscanbewritten as,z{|!} d%~ | c | f ~)� (5.45)
80
where ���9��� whentheprimedegreeof freedomis �>� . We do not have to divide the ��� valuesby��� to getto this form of theconstraintequation.This is becausetheconstraintequationsapplicable
for the threesetsof boundaryconditionsusedin the RCM calculationsautomaticallysatisfy the
requirementthat �!�#��� . UsingtheLagrangemultiplier technique,theoriginal setof equationscan
thenbereducedby oneto getasetof equationsof theform :���!�#� ����� �I� ��� �'� ��� � �]� ������� � �0� � �����E�E�M�_� � � �)� ��� ��� � � ������� � ��� � ��� � �¢¡�R£¤� (5.46)
Repeatedapplicationof thisapproachfor eachof theconstraintequationsgivesusasetof equations
with the redundantdegreesof freedomremoved. If thereare ¥�¦ constraintequations,the reduced
systemof equationscanbewritten as�>§>�©¨����#� �'� � ���+�-� � � �Cª4«5ª7¥ � ¥¬¦�¯® (5.47)
5.3.2 Application of SpecifiedDisplacements
Thesetof equationsremainingafter theconstraintequationshave beenappliedandtheredun-
dantdisplacementsremovedfrom theequation,canbewritten in matrix form as°¢± �F² (5.48)
If we decomposethe matrix°
into parts that are relatedto the specified(subscript ³ ) and the
unspecified(subscript� ) displacements,andthe forcevectorinto theapplied(subscript ) andthe
reaction(subscriptµ ) forces,we have,¶ ° ¦�¦ ° ¦¸·°º¹¦¸· ° ·¤·¤»½¼ ± ¦± ·�¾ � ¼ ²À¿¦² ¿· ¾ � ¼ ²ÂÁ¦² Á· ¾ (5.49)
Thespecifieddisplacements± · areknown. Hence,thematrixequation° ¹ ¦¸· ± ¦Ã� ° ·Ä· ± ·��-² ¿· �r² Á· (5.50)
is redundant.Therefore,weonly needequations° ¦¸¦ ± ¦Ã� ° ¦¸· ± ·��F² ¿¦ ��² Á¦ (5.51)
to determinethe unknown displacements± ¦ . Now, the reactionsat thepointswhereno displace-
mentsarespecifiedarezero,i.e., ² Á¦ �-Å (5.52)
Therefore, ° ¦¸¦ ± ¦ �F² ¿¦ � ° ¦¸· ± · (5.53)
After theconstraintequationsandtheprescribeddisplacementsareapplied,theunknown nodal
displacementscannow beobtainedfrom thereducedsystemof equations.It shouldbenotedthat
81
therearestill someunknown nodal forcesin the expressionsfor the force vectorbecauseof the
constraintequations.Thesecanbesetto zeroif weassumethattheaverageforcesarezero.
For the four elementmodel subjectedto a uniform normal displacementin the Æ direction
(shown in Figure5.6),we setÇ�ÈÀÉËÊrÇ�ÈÂÌ9Í-ÎCÏ Ç�ÐÒÑ;ÊrÇ�ÐÒÓXÍFÎCÏ Ç�ÈÀÔ@ÍFÎJÏÇ�Ð Ô Í-ÎCÏ Ç�ÐÕ;ÊrÇ�Ð Ì ÊrÇ�ÐÒÖXÍFÎC×where
Ç Èand
Ç Ðarethenodalforcesin the Æ andØ directionsrespectively. Thesubscripts2,4,5,6,7,8
and9 refer to nodesat which the forcesareapplied. Similar equationsareusedwhena uniform
normaldisplacementis appliedin the Ø direction.
For thefour elementmodelsubjectedto displacementsthatcorrespondto a pureshear(shown
in Figure5.9),we againassumethattheconstrainednodalforcesaverageto zero,i.e.,Ç ÈÀÉ ÊrÇ ÈÂÌ ÍFÎJÏ Ç ÐÉ ÊrÇ ÐÒÌ Í-ÎCÏ Ç ÈÂÑ ÊrÇ ÈÂÓ ÍFÎJÏ Ç ÐÒÑ ÊrÇ ÐÒÓ ÍFÎJÏÇ�ÈÀÔ9ÍFÎJÏ Ç�ÐÔ@Í-ÎC×Oncetheunknown forceshavebeenremovedusingtheaboveprocedure,thesystemof equations
can be solved for the unknown displacements.We use Gaussianelimination to solve for the
displacements.This is in order to eliminateany problemsdueto ill conditioningof the stiffness
matrixwhichmayoccurbecauseof thelargemoduluscontrastbetweentheparticlesandthebinder
in PBX materials.
5.3.3 Calculating VolumeAveragedStressesand Strains
Theeffective stiffnessmatrix ÙSÚ of ablock of subcellscanbeobtainedfrom therelationÛ�ÜMÝ Í Ù Ú Û�ÞHÝ (5.54)
whereÛ�ÜMÝ
is thevolumeaveragedstressin theblock,and,Û�ÞHÝis thevolumeaveragedstrainin theblock.
A block is modeledusingeitherfour or sixteenelementsasshown in Figures5.4 and5.5. Since
the elementsareall the samesize,the volumeaveragedstressor strainin a block is equalto the
ensembleaverageof theaveragestressesor strainsin eachelement.
82
Thevolumeaveragedstrainandstressin anelementaregivenbyß�àHá¯âäãåçæéè àXê åìë(5.55)ß�íIá¯â ãåçæ è íîê åðï(5.56)
For thefour nodedelement,usingthestrain-displacementrelationsandintegratingover theelement
we candeterminethevolumeaveragedstrainsin anelement.Theseexplicit expressionsfor these
are ß�ñÒòóòHá¬â ãôHõ¢ö0÷Iøeù#úûøLü¬ú�øLý9÷Wø>þ]ÿ ë (5.57)ß�ñ�����á¬â ãôHõ¢ö0÷��©ù¯÷��Hü¬ú��Hý¬ú���þ]ÿ ë (5.58)ß�ñ�ò���á¬â ãôHõ ö0÷Iøeù¯÷WøLü¬ú�øLý¯úûø>þI÷�©ùqú��ü¯ú��HýX÷���þÀÿ ï (5.59)
Theaveragestressescanbeobtainedsimilarly from thestress-strainrelations.Theexpressionsfor
theaverageelementstressesareß�� òóò á¯â ãôHõ ö�� ü ú�� ý ÿ)ö�� ý ÷�� ù ÿÃú-ö�� ù ÷�� ý ÿ)ö ø ü ÷Wø þ ÿ�úö��@üX÷��@ýÂÿ)ö���þ+÷��HüóÿÃú-ö��Mùqú��@ýÂÿ)ö øLýX÷WøÃù�ÿ�� ë (5.60)ß�������á¯â ãôHõ ö����¬ú����Âÿ)ö���ýM÷��EùÿÃú-ö��@üX÷����Àÿ)ö øLüX÷WøLþ]ÿ�úö����X÷����Âÿ)ö���þ+÷��HüóÿÃú-ö��@ü¬ú����Âÿ)ö øLýX÷WøÃù�ÿ�� ë (5.61)ß��%ò���á¯â ãôHõ ö����¬ú����Âÿ)ö���ýM÷��EùÿÃú-ö��@ýX÷����Àÿ)ö øLüX÷WøLþ]ÿ�úö����X÷����Âÿ)ö���þ+÷��HüóÿÃú-ö��@ý¬ú����Âÿ)ö øLýX÷WøÃù�ÿ�� ï (5.62)
For aRVE composedof many suchelements,anarithmeticaverageof theelementaveragestresses
andstrainscanbetakento calculatethevolumeaverageover theRVE. Similar expressionsfor the
averagestressesandstrainscanbeobtainedfor theninenodedelements.
5.3.4 Calculating Effective Properties
Theeffective propertiesarerelatedto thevolumeaveragestressesandstrainsby thefollowing
relation: ���� ß�� òóò áß������Háß��)ò���á �!" â$#% �'&ù�ù �(&ù¸ü �'&ù��� &ù¸ü � &ü�ü � &ü)��'&ù�� �(&ü)� �'&�)�
*+ ���� ß�ñ òóò áß�ñ����Háß-,Eò���á �!" (5.63)
To solve for thesix componentsof thestiffnessmatrix,we needsix independentequationsrelating
thevolumeaveragedstressesto thevolumeaveragedstrains.
83
For thecasewhereanormaldisplacementis appliedin the . direction,we have/102(354'60)087 02:9�4(60�;<7 0=>9�4(60�?�7 02�= (5.64)/ 0= 354 60�; 7 02 9�4 6;); 7 0= 9�4 6;)? 7 02�= (5.65)/ 02�= 354 60�? 7 02 9�4 6;)? 7 0= 9�4 6;)? 7 02�= (5.66)
where / 02 3A@�B 2�21CED / 0= 3F@�B =8=GCED / 02�= 3F@�H 2�=ICED 7 02 3F@�J 2�2KCED 7 0= 3F@�J =�=IC , and,7 02�= 3L@�M 2�= CONWhenanormaldisplacementin appliedin the P direction,we have/ ;2 354 60)0 7 ;2 9�4 60�; 7 ;= 9�4 60�? 7 ;2�= (5.67)/ ;= 354 60�; 7 ;2 9�4 6;); 7 ;= 9�4 6;)? 7 ;2�= (5.68)/ ;2�= 354 60�? 7 ;2 9�4 6;)? 7 ;= 9�4 6;)? 7 ;2�= (5.69)
where / ;2 3A@�B 2�2 CED / ;= 3F@�B =8= CED / ;2�= 3F@�H 2�= CED 7 ;2 3F@�J 2�2 CED 7 ;= 3F@�J =�= C , and,7 ;2�= 3L@�M 2�=GCONFor asheardisplacementin the .QP -plane,wehave,/SR2(354'60)087 R2:9�4(60�;<7 R=>9�4(60�?�7 R2�= (5.70)/ R= 354 60�; 7 R2 9�4 6;); 7 R= 9�4 6;)? 7 R2�= (5.71)/ R2�= 354 60�? 7 R2 9�4 6;)? 7 R= 9�4 6;)? 7 R2�= (5.72)
where / R2 3A@�B 2�21CED / R= 3F@�B =8=GCED / R2�= 3F@�H 2�=ICED 7 R2 3F@�J 2�2KCED 7 R= 3F@�J =�=IC , and,7 R2�=T3L@�M 2�=GCONThesenineequationsmayalwaysbe independent,especiallywhena block possessessquaresym-
metry. However, the following combinationof theseequationsalways leadsto six independent
equationsin thesix unknown effective stiffnessmatrix termsUVVVVVVW VVVVVVX/ 02/ 0= 9 / ;2/ 02�= 9 / R2/ ;=/ ;2�=�9 / R=/ R2�=
Y�VVVVVVZVVVVVV[3\]]]]]]^7 02 7 0= 7 02�= _ _ _7 ;2 7 02 9 7 ;= 7 ;2�= 7 0= 7 02�= _7 R2 7 R= 7 02 9 7 R2�= _ 7 0= 7 02�=_ 7 ;2 _ 7 ;= 7 ;2�= __ 7 R2 7 ;2 7 R= 7 ;= 9 7 R2�= 7 ;2�=_ _ 7 R2 _ 7 R= 7 R2�=
`baaaaaacUVVVVVVW VVVVVVX4 60)04 60�;4 60�?4 6;);4 6;)?4 6?)?
Y�VVVVVVZVVVVVV[ (5.73)
Thisequationcanbesolvedto determinetheeffective stiffnessmatrixof ablock of subcells.
84
5.4 Calculating EffectivePropertiesof the RVEA sampleensembleof particlesin a matrix is shown in Figure5.10. This ensembleis divided
into a grid of subcellsin sucha way thateachsubcellof thegrid is composedof only onematerial
andthenumberof divisionspersideof theensembleis anintegerfactorof 2.
d d d d d d d d d d d d d d d d d de e e e e e e e e e e e e e e e e effffffffffffffffffff
ggggggggggggggggggggh h h h h h h h h h h h h h h h h h h h hi i i i i i i i i i i i i i i i i i i i i
jjjjjjjjjjjjjjjjjjjj
kkkkkkkkkkkkkkkkkkkk
l l ll l ll l lm m mm m mm m m
n n nn n nn n no o oo o oo o op p pp p pp p pq q qq q qq q q
r r rr r rr r rs s ss s ss s st t tt t tt t tu u uu u uu u u
v v v vv v v vv v v vw w w ww w w ww w w wx x xx x xx x xy y yy y yy y y
z z z zz z z zz z z z{ { { {{ { { {{ { { {
| | || | || | |} } }} } }} } }~ ~ ~~ ~ ~~ ~ ~� � �� � �� � �
� � �� � �� � �� � �� � �� � �� � � �� � � �� � � �� � � �� � � �� � � � � � �� � �� � �
� � �� � �� � �
� � � �� � � �� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �� � � �� � � �� � �� � �� � �� � �� � �� � �
� � �� � �� � �� � �� � �� � �� � � �� � � �� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �� � � �� � � �� � �� � �� � �� � �� � �� � �
� � � �� � � �� � � �� � � �� � � �� � � �
��������������������
��������������������
� � � � � � � � � � � � � � � � � � � � �� � � � � � � � � � � � � � � � � � � � �
� � �� � �� � �� � �First Iteration
Second Iteration
Third Iteration
Particles
Binder2
3
1 1 1 1
1
1
1111
1
1 1 1
1 1
2 2
22
3
1
X
Y
Figure 5.10. Therecursive cellsmethodappliedto aRVEdiscretizedinto blocksof four subcells.
The first iterationis carriedout with the four cells aroundthe nodesmarked 1. This leadsto
homogenizedcells thatareusedin theseconditerationwith thecompositecellsaroundthenodes
marked 2. The final iterationshown in thefigure is for the four compositecells that make up the
RVE at thisstage.Thefinite elementprocedureoutlinedabove is usedfor eachfour cell ensemble.
Thisapproachhasbeenimplementedusingaquadtree-baseddatastructure[121]. Finiteelement
analysesareusedto calculatetheeffective moduli at the lowestnodesof thestructure.Thevalues
obtainedat thesenodesof the quadtreeare assignedto the next higher level and the effective
85
propertiesarecalculatedat this level. This processis repeatedrecursively until thefinal effective
propertiesof theRVE areobtained.
Theeffective propertiesthathave beendeterminedusingRCM in Chapters6 and7 have been
calculatedusingfour-nodedelementswith sixteenelementsbeingusedto representablock of four
subcells.Theuseof four-nodedelementsleadsto a stiffer responsethanthatobserved with nine-
nodedelements.In addition,since9/3u-pelementsarenotusedto modelthebinder, someelement
lockingmaybeexpected.
CHAPTER 6
VALID ATION OF GMC AND RCM
In the first sectionof this chapter, exact relationsare usedto calculatethe effective elastic
propertiesof a few two-componentcomposites. Theseeffective propertiesare comparedwith
predictionsfrom detailedfinite elementanalyses.The effective propertiesfor thesecomposites
arealsocalculatedusingGMC andRCM andcomparedwith theexactsolutionswherepossible.
Thesecondsectionof this chapterdealswith thepredictionof theeffective elasticpropertiesof
microstructuresandcomponentmaterialsfor which no exact solutionsexist. We usetheeffective
propertiespredictedby detailedfinite elementanalysesasa benchmarkfor evaluatingGMC and
RCM asappliedto thesemicrostructures.Microstructuresfor which bothGMC andRCM perform
well arediscussedfollowedby somespecialmicrostructures.
In what follows, the detailedfinite elementanalyseshave beencarriedout with ANSYS 5.6
usingfour-, six-or eight-nodeddisplacementbasedfinite elements.Periodicdisplacementboundary
conditionshavebeenapplied.TheGMC resultshavebeencalculatedusingthetechniquediscussed
in Chapter4, without couplingof thenormalandtheshearbehaviors. TheRCM calculationshave
beenperformedusingblocksof four subcellsmodeledwith sixteenfour-nodedelements.
6.1 ComparisonsWith Exact RelationsExactrelationsfor theeffectiveelasticpropertiesof two-componentcompositescanbeclassified
into threetypes.Thefirst typeconsistsof relationsthathavebeendeterminedfrom thesimilarity of
thetwo-dimensionalstressandstrainfieldsfor certaintypesof materials.Theseexactrelationsare
calledduality relations[122]. Thesecondtypeof exactrelations,calledtranslation-basedrelations,
statethat if a constantquantityis addedto theelasticmoduli of thecomponentmaterialsthenthe
effectiveelasticmoduliarealso“translated”by thesameamount.Microstructureindependentexact
relations,valid for specialcombinationsof theelasticpropertiesof thecomponents,form thethird
category [123].
Many of theseexact relationsrequireeithersomeform of rigidity or incompressibilityin the
phasesof the composite. Since neither GMC nor RCM can deal with purely rigid or purely
87
incompressiblebehavior, we have to assumea suitably high value of the modulusor Poisson’s
ratio to approximatetherequirementsfor theexactrelationsto hold.
6.1.1 PhaseInter changeIdentity
A duality-basedexactrelationis thephaseinterchangeidentity [26] for theeffectiveshearmod-
ulusof asymmetrictwo-dimensionaltwo-componentisotropiccomposite.A symmetriccomposite
is invariantwith respectto interchangeof the components.The phaseinterchangeidentity states
thattheeffective shearmodulus( �T� ) of suchacompositeis givenby
� ���F� �¡ ¢�¤£G¥ (6.1)
where � and � £ aretheshearmoduli of thetwo components.
Thelinearelasticconstitutive relationshipfor a two-dimensionalisotropicmaterialcanbewrit-
tenas ¦§©¨ ) ¨ £)£ª �£
«¬ �¦§)¯® � ±° � ²³° � ´® � ²² ² �
«¬ ¦§<µ ) µ £)£¶ �£
«¬(6.2)
where¨ ) , ¨ £)£ and ª �£ arethestresses,
µ ) , µ £)£ and ¶ �£ arethestrains,and,
and � arethetwo-dimensionalbulk andshearmoduli, respectively.
Foramaterialwith squaresymmetry, theshearmodulusisnotthesameall directionsandtheslightly
modifiedconstitutive equationis writtenas¦§ ¨ ) ¨ £)£ª �£
«¬ �¦§ ·®�¸ ±°¸ ²¹°�¸ ´®¸ ²² ² ¸ £
«¬ ¦§ µ ) µ £)£¶ �£
«¬(6.3)
where¸ is theshearmodulusfor shearappliedalongthediagonalsof thesquare,and,
¸ £ is theshearmodulusfor shearappliedalongtheedgesof thesquare.
A checkerboard,asshown in Figure6.1, is an exampleof a symmetriccomposite.However,
a checkerboardexhibits squaresymmetryinsteadof isotropy, i.e., ¸ and ¸ £ aredifferent. Since
the phaseinterchangerelation is valid only when the compositeis isotropic,we choosethe two
componentswith low moduluscontrastandcomparetheshearmoduli predictedby finite elements,
RCM andGMC.
88
Figure 6.1. RVE for acheckerboard.
The two materialsthat form the checkerboardcompositeswere assignedthe sameYoung’s
modulusof 15,300MPa(theYoung’smodulusof HMX). ThePoisson’s ratioof thefirst component
wasfixedat 0.32while thatof thesecondcomponentwasvariedfrom 0.1to 0.49.
The exact effective shearmodulusfor the checkerboardhasbeenplotted as a solid line in
Figure6.2. The two effective shearmoduli, º¼» and º¾½ , calculatedusingfinite elements(FEM),
RCM and GMC, have beenplotted as points in the figure. The resultsshow that all the three
methodsperformwell (themaximumerroris 0.1%)in predictingtheeffective shearmoduluswhen
themoduluscontrastis small, i.e., whenthecompositeis nearlyisotropic. It canalsobeobserved
thatthevaluesof º » and º ½ arewithin 1%of eachotherfor thechosencomponentmoduli.
Anothersetof numericalcalculationshasbeenperformedon thecheckerboardmicrostructure
to observe the effect of increasingmoduluscontrast. In this case,the first componentof the
checkerboardwasassignedaYoung’smodulusof 15,300MPaandaPoisson’s ratioof 0.32.For the
secondcomponent,thePoisson’s ratio wasfixedat 0.49andtheYoung’s moduluswasvariedfrom
0.7MPato 7000MPa.
Whenthemoduluscontrastbetweenthecomponentsof thecheckerboardincreases,thematerial
canno longerbeconsideredisotropicandthevaluesof º¼» and º¾½ areconsiderablydifferentfrom
theeffective shearmodulus ¿'À predictedby thephaseinterchangeidentity. This canbeobserved
from theplot of the effective º » and º ½ versusthe ratio of ¿ » and ¿ ½ shown in Figure6.3. The
exacteffective shearmoduli for isotropic,symmetriccompositesof thetwo componentshave been
plottedwith a solid line in thefigure. The correspondingvaluesof º » and º ½ predictedby FEM,
89
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.55400
5500
5600
5700
5800
5900
6000
6100
6200
6300
6400
Poisson’s ratio of the second component
µ1 and
µ2
Exactµ1 (FEM)µ1 (RCM)µ1 (GMC)µ2 (FEM)µ2 (RCM)
Figure6.2. Validationof FEM, RCM andGMC usingthephaseinterchangeidentityfor acheckerboardcomposite.
RCM andGMC areshown aspointson the plot (note that GMC predictsthe samevaluesof Á¼Âand Á¾Ã for materialswith squaresymmetry). The FEM-basedpredictionsof Á  (circles)and Á¾Ã(diamonds)show thatthesearetheclosestto theexactresults.Theratioof Á à to Á¼Â increasesasthe
moduluscontrastthetwo componentsincreases.For relatively low moduluscontrast,theeffective
shearmoduluspredictedby the phaseinterchangeidentity is approximatelyequalto the meanofÁ  and Á¾Ã predictedby FEM. The effective Á  and Á¾Ã predictedby RCM arehigher than those
predictedby FEM while thosepredictedby GMC arelower.
The ratio of Á  and Á¾Ã to the ÄTÅ predictedusing the phaseinterchangeidentity is shown in
Figure6.4. It canbe observed that the FEM computationsproducegood approximationsto the
exact resultsfor shearmoduluscontrastsof up to 500. For shearmoduluscontrastsabove 500,
isotropy is no longeran adequateassumptionand the FEM resultsdiverge from thosepredicted
by thephaseinterchangeidentity. Theplot alsoshows that theRCM predictionsof Á¼Â and Á à are
consistentlyhigherthanthosepredictedby FEM while theGMC predictionsareconsistentlylower.
TheRCM resultsarecloserto theFEM resultsthanaretheGMC predictions.
The above resultsshow that finite elementanalysesmay provide a benchmarkfor evaluating
the RCM and GMC techniqueswhen exact relationsare not available. However, we have to
90
100
101
102
103
104
105
−500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Ratio of the shear moduli of the two components
µ1 and
µ2
Exactµ1 (FEM)µ1 (RCM)µ1 (GMC)µ2 (FEM)µ2 (RCM)
Figure 6.3. Variationof effective shearmoduli with moduluscontrastfor acheckerboardcomposite.
ascertainwhetherthe finite elementresultsthat we comparethe RCM andGMC resultsagainst
haveconvergedto asteadysolution.Thecheckerboardmaterialprovidesanextremecaseto testthe
convergenceof theFEM solutionbecausethecornersingularitiesleadto high stresses.Therefore,
veryhighmeshrefinementis requiredto minimizetheeffectof highcornerstressesontheeffective
moduli. In thisresearchweperformfinite elementanalyseson ÆÈÇÈÉËÊÌÆÈÇÈÉ squaregrids.TheeffectiveͼΠand Í¾Ï of a checkerboardwith a shearmoduluscontrastof about25,000(correspondingto the
highestmoduluscontrastshown in Figure6.4) have beencalculatedusingvariouslevels of mesh
refinementandplottedin Figure6.5. Theseplotsshow that theeffective ͼΠconvergesto a steady
valuewhenabout Ð<ÆÈÑÒÊÓÐ<ÆÈÑ elementsareusedwhile Í Ï convergesto asteadyvaluewhen ÆÈÇÈÉÒʤÆÈÇÈÉelementsareusedto discretizetheRVE. Thehighernumberof elementsin theplot correspondsto
a grid of ÔÈÇGÕÖÊ�ÔÈÇGÕ elements.Hence,our choiceof ÆÈÇÈÉÖÊ�ÆÈÇÈÉ elementsis justified andcanbe
expectedto generateeffective elasticmoduli thatcanbeusedasbenchmarks.
The checkerboardmicrostructurealso hasanothersignificancewith respectto the recursive
methodof cells. For the RCM procedurethat usesblocksof Æ×Ê�Æ subcells,oneof the possible
microstructuresis a checkerboard.The convergenceplot shown in Figure6.5 suggeststhat using
sixteenfour-nodedelementsto model the block may lead to an overestimationof the effective
91
100
101
102
103
104
105
10−2
10−1
100
101
102
Ratio of the shear moduli of the two components
Rat
io o
f µ1 a
nd µ2 to
G*
µ1 (FEM)µ1 (RCM)µ1 (GMC)µ2 (FEM)µ2 (RCM)
Figure 6.4. Ratioof effective shearmoduli predictedby FEM, RCM andGMC tothosepredictedby thephaseinterchangeidentity for acheckerboardcomposite
with varyingmoduluscontrast.
properties. It may be preferableto modeleachsubcellusingmore than16 elementsto obtaina
betterapproximationfor theeffective elasticproperties.
6.1.2 Materials Rigid in Shear
The stress-strainresponseof two-dimensionalcompositesthat arerigid with respectto shear
canberepresentedby ØÙ<ÚÜÛ)ÛÚ�Ý)ÝÞ Û�Ý
ßà�á ØÙ�âãÛ)ÛäâãÛ�ÝäåâãÛ�ÝæâçÝ)Ýäåå å å
ßà ØÙ�èéÛ)ÛèêÝ)Ýë Û�Ý
ßàíì(6.4)
whereèéÛ)Û , èQÝ)Ý and ë Û�Ý arethestresses,
ÚÜÛ)Û , Ú�Ý)Ý and Þ Û�Ý arethestrains,and,
âïîñð arethecomponentsof thecompliancematrix.
Two duality-basedrelationsthatarevalid for two-componentscompositescomposedof suchmate-
rialsare[122] :
92
100
101
102
103
104
105
106
0
500
1000
1500
2000
2500
3000
Number of elements
µ1 and
µ2
µ1
µ2
Figure 6.5. Convergenceof effective moduli predictedby finite elementanalyseswith increasein meshrefinementfor a checkerboardcomposite
with shearmoduluscontrastof 25,000.
1. If òãó)ó8òçô)ôéõíö�òãó�ô<÷ ô:øúù for eachphase(whereù is aconstant),thentheeffectivecompliance
tensoralsosatisfiesthesamerelationship,i.e., òüûó)ó òýûô)ô õ�ö�òýûó�ô ÷ ô øúù . This relationis truefor
all microstructures.
2. If thecompliancetensorsof thetwo phasesareof theform þ ó øúÿ ó�� and þ ô øúÿ ô�� where� is a constantmatrix, then the effective compliancetensorof a checkerboardof the two
phasessatisfiestherelation òüûó)ó òýûô)ô õ ö�òýûó�ô ÷ ô øúÿ ó ÿ ô ö�� ó)ó � ô)ô õ ö�� ó�ô ÷ ô ÷ .Thefinite elementanalysesperformedin this researcharetwo-dimensionalandbasedon theplane
strainassumption.Theeffective compliancematrixcannotbedetermineddirectly from planestrain
computations(AppendixA). Therefore,an approximatecompliancematrix is calculatedfor the
finite elementand RCM validationsusing the methoddiscussedin Appendix A. The effective
compliancematrix canbecalculateddirectlyby GMC.
Numericalexperimentsusing a squarearray of disks occupying an areafraction of 70% (as
shown in Figure 6.6) have beencarried out to check if the first of the above relationscan be
reproducedby finite elementanalyses,GMC andRCM. The þ matricesthat have beenusedfor
93
the disks (superscript� ) and the matrix (superscript� ), and the correspondingvaluesof � are
shown below. Thesematriceshave beenchosenso that the valueof � is constantandtherefore
shouldbeequalto thatfor theeffective compliancematrix.��� � ������� ������� �������� ������� �� � ��������� ���� � ��� ���! "����# �and �%$�� � ��� �'& �(� ��)���*+�(��� ���)���*+�(��� ��� �'& �(� �� � ��������� ���� � ��� ���! ,����#��The shearmodulusfor both materialsis ������� - around ����- times the Young’s modulus. Higher
valuesof shearmodulushavebeentestedandfoundnot to affect theeffective stiffnessmatrix terms
significantly.
Figure 6.6. RVE for asquarearrayof disks.
We require the out-of-planeYoung’s modulusand Poisson’s ratio to calculatethe effective
compliancematrix of the composite(asexplainedin AppendixA). Theseproperties,calculated
using the rule of mixtures(ROM) andfrom the effective compliancematrix predictedby GMC,
are shown in Table6.1. For the ROM calculations,the valuesof Young’s modulusfor the two
componentsare �������. /���+021 and� ��� & /���+043 . The correspondingPoisson’s ratiosare ���(��� and��� &5� .
94
Table6.1. Out-of-planepropertiesfor squarearrayof disks.
ROM Based GMC Based6�7 8�7 6�7 8�7( 9;:�<+=4> ) ( 9;:�<+=4> )
9.741 0.357 9.800 0.365
The compliancematricescalculatedusing finite elements(123,000elements),GMC (4,100
subcells)andRCM (4,100subcells)areshown in Table6.2.Thecorrespondingvaluesof ? arealso
shown in thetable. Thecompliancematriceshave beencalculatedusingboththeROM basedand
theGMC basedout-of-planeproperties.Sincethemoduluscontrastbetweenthe two components
of the compositeis small, the calculatedeffective propertiesareexpectedaccurate.However, the
resultsin Table6.2show thatboththedetailedfinite elementcalculationsandtheRCM calculations
leadto around10%error in theestimationof ? . On theotherhand,theGMC calculationsleadto
anerrorof only about4.5%.
Table6.2. Componentsof effective stiffnessandcompliancematricesfor asquarearrayof disks.@BACDC @BACFE
( 9;:�<+= 7 ) ( 9;:�<+= 7 )FEM 1.95 1.23GMC 1.77 1.05RCM 1.98 1.26G ACDC G ACFE ? Error
( 9�:�< E ) ( 9;:�< E ) ( 9�:�<�H ) (%)GMC 10.08 -3.82 8.69 -4.5ROM-BasedFEM 9.81 -4.05 7.98 -12.3GMC-BasedFEM 9.86 -4.00 8.12 -10.8ROM-BasedRCM 9.78 -4.07 7.91 -13.1GMC-BasedRCM 9.83 -4.02 8.04 -11.6
Theabove resultsimply thatthefinite elementanalysesandtheRCM calculationsoverestimate
theeffectivepropertiesof thesquarearrayof disks.Thismaybebecausetherigidity of thematerial
in shearis not well approximatedby the finite elementcalculations. Higher valuesof the shear
modulusof thecomponents,e.g., IKJL:�<NM , leadto essentiallythesameeffective stiffnessmatrix
components@BACDC and
@BACFE . Another reasonfor the error could be that the effective compliance
matrix terms,for the’3’ direction,arenotapproximatedwell by theruleof mixturesor by theGMC
calculations.Thiscanbeverifiedby carryingout three-dimensionalcalculationsfor thiscomposite.
95
Theslight differencebetweentherule of mixtures(ROM) basedcalculationsandtheGMC based
calculationsis probablydueto machineprecisionbecausethevaluesof OQPRFS and OQPSDS calculatedby
GMC for two-dimensionalproblemsis essentiallya ruleof mixturescalculation.
Theseconddualityrelationfor materialsthatarerigid in shearrequirestheuseof acheckerboard
geometryasshown in Figure6.1.Thestresssingularitiesat thecornercontactsof thecheckerboard
leadsto relatively largeerrorsin thecomputationof theeffective propertiesaswasobservedfor the
phaseinterchangeidentity discussedbefore. However, it is interestingto observe how well finite
elements,GMc andRCM performin calculatingthis result.To testthesecondduality relation,we
chooseT R�UWV�X�X , TZY UWV�X�X�X and, [ U]\ V�X ^�_^`_ V�XZacbThenthecompliancematricesfor thetwo componentsof thecompositeared R UWV�X�X \ V�X ^�_^�_ V�Xea and
d Y UfV�X�X�X \ V�X ^`_^�_ V�Xea bThe duality relationrequiresthat the effective compliancematrix of the checkerboardcomposite
shouldbesuchthat g�h�ikj d Pml U O PRDR O PYDY ^ j O PR Y l Y U�n b V�Xpo,V�X�q bFor theconstituentmaterialpropertiesthevaluesof r S and s S are t bvu owV�X+x4y and X b _ respec-
tively. The samevaluesareobtainedusing the rule of mixturesand the O PRFS and O PSDS calculated
usingGMC. Finiteelementanalysesusingaround123,000elementshavebeenusedto calculatethe
effective stiffnessmatrix for thecheckerboard.TheGMC andRCM calculationshave usedaround
4,100elementsto determinetheeffectivestiffnessmatrix. Theeffectivecompliancematrixhasbeen
determinedusingthemethoddescribedin AppendixA for thefinite elementandRCM calculations
anddetermineddirectlyusingGMC.
Theresultsfrom thesethreemethodsaretabulatedin Table6.3. Interestingly, thefinite element
calculationsleadto quiteanaccurateeffective compliancematrix andthedeviation from theexact
result is only around7%. The GMC calculationsoverestimatethe compliancematrix and the
determinantof the compliancematrix is around1.5 times higher than the exact result. On the
otherhand,thoughtheRCM calculationsunderestimatethecompliancematrix, they leadto avalue
of thedeterminantthatis closerto theexactvaluethantheGMC results.
The higher valuesof effective stiffness,predictedby RCM, are due to eachblock of z o zsubcellsbeingmodeledusing16 elements.As canbeseenfrom Figure6.5 theeffective stiffness
predictedusing sixteenelementsis considerablyhigher than that predictedusinga more refined
mesh.Therefore,a way of improving theperformanceof RCM would beto usemoreelementsto
modelablock of subcells.
96
Table6.3. Componentsof effective stiffnessandcompliancematricesfor acheckerboardcomposite.{}|~D~ {}|~F�( �;���+�4� ) ( �;���+�4� )
FEM 4.86 2.96GMC 2.45 1.05RCM 7.40 3.17�Q|~D~ ��|~F� �����k��� |m� Error
( �;����� ) ( �;����� ) ( ������� ) (%)FEM 3.44 -1.84 8.48 -6.8GMC 5.17 -1.98 22.80 150.5RCM 1.82 -0.54 3.01 -66.9
6.1.3 The CLM Theorem
The Cherkaev, Lurie and Milton (CLM) theoremis a well known “translation” basedexact
relation for two-componentplanarcomposites(Milton [26] and referencestherein). For a two-
dimensionaltwo-componentisotropiccomposite,this theoremcanbestatedasfollows.
Let theisotropicbulk moduli of thecomponentsbe � ~ and � � . Let theshearmoduliof thetwo
componentsbe � ~ and � � . Theeffective bulk andshearmodulusof a two-dimensionalcomposite
madeof thesetwo componentsare � | and � | respectively.
Let usnow createtwo new materialsthatare“translated”from theoriginalcomponentmaterials
by aconstantamount� . Thatis, let thebulk andshearmoduliof thetranslatedcomponentmaterials
begivenby �� ~ (translated) � �� ~�� ����� � (translated) � �� � � ����� ~ (translated) � �� ~� ����� � (translated) � �� ��� ���The CLM theoremstatesthat the effective bulk andshearmoduli of a two-dimensionalcom-
positeof thetwo translatedmaterials,having thesamemicrostructureastheoriginalcomposite,are
givenby
97��"�(translated) � �� ���¡ and, (6.5)�¢ �(translated) � �¢ ��£¤ ¦¥
Therequirementof isotropy canbesatisfiedapproximatelyfor numericalexperimentsby choos-
ing componentmaterialpropertiesthatarevery closeto eachother. Sinceour goal is to determine
how well GMC andRCM performfor high moduluscontrast,choosingmaterialswith smallmod-
ulus contrastis not adequate.Anotheralternative is to choosea RVE that representsa hexagonal
packingof disks. However, suchan RVE is necessarilyrectangularandcannotbe modeledusing
RCM in its currentform. It shouldbenotedthatRCM caneasilybemodifiedto dealwith elements
thatarenot squareandhenceto modelrectangularregions.
Anotherproblemin theapplicationof theCLM theoremis thatthevalueof hasto besmallif
thedifferencebetweentheoriginalandthetranslatedmoduli is largeandviceversa.If thevalueof is small,floatingpointerrorscanaccumulateandexceedthevalueof . On theotherhand,if is
large,theoriginalandthetranslatedmoduli arevery closeto eachotherandthedifferencebetween
thetwo canbelost becauseof errorsin precision.Hence,thenumbershave to bechosencarefully
keepingin mind thelimits on thevalueof thePoisson’s ratio.
However, it is interestingto observethedifferencesbetweentheeffectivemodulipredictedfinite
elements,GMC andRCM beforeandaftera translation.We,therefore,testthetranslationideaona
squarearrayof disksoccupying avolumefractionof 70%asshown in Figure6.6.ThisRVE exhibits
squaresymmetry, i.e., theshearmoduli §©¨ and §¦ª shown in equation(6.3)arenotequal.Wecannot
calculateauniquevalueof theeffectiveshearmodulusfor thisRVE. Instead,wecalculatethevalue
of the effective translatedshearmodulusfrom equation(6.5) by first setting¢«�
equalto §©¨ and
thento §¦ª . We thencomparethese“exact” valueswith the §©¨ and §¦ª valuespredictedusingfinite
elementanalyses,GMC andRCM.
The original set of elasticmoduli for the RVE is chosento reflect the elasticmoduli of the
constituentsof PBXs. Thesemoduli are then translatedby a constant �¬ ¥ ¬�¬ � . The original
andthetranslatedconstituenttwo-dimensionalmoduli areshown in Table6.4(phase’p’ represents
the particlesand phase’b’ representsthe binder). The three-dimensionalmoduli can easily be
calculatedfrom the two-dimensionalmoduli shown in Table6.4 usingthe relationsgiven by Jun
andJasiuk[41].
Theeffective bulk andshearmoduli of theoriginalandthetranslatedmaterialhave beencalcu-
latedusingfinite elements(123,000elements),GMC (4,100subcells)andRCM (4,100subcells).
Theseareshown in Table6.5. Thevaluesof err shown in thetablehave beencalculatedusingthe
equations
98
Table6.4. Originalandtranslatedtwo-dimensionalconstituentmodulifor checkingtheCLM condition.®!¯ °±¯ ®p² °³² ®}¯5´'®µ² °±¯¶´�°³²·¹¸;º�»�¼k½ ·¹¸;º�»�¼k½ ·¹¸;º�»�¼'½ ·¹¸;º�»�¾k½
Original 9.60 4.80 10.07 0.20 0.95 2.38Translated 240.0 3.24 10.17 0.20 23.5 1.61
¿err À · ¿ ´ »�Á�»�»�ºc¤ºk½±¸,º�»�»�Ã
where,¿ À º ´'®"Äoriginal
¤º ´'®"Ätranslated
Ãor,¿ À º ´ÆÅ%ÇÉÈ
translated¤º ´ÆÅ%ÇÉÈ
originalÁ
Table6.5. Comparisonof effective moduli for theoriginalandthetranslatedcomposites.® Ä Å©Ê È Å ¼ ÈOrig. Trans.
¿err Orig. Trans.
¿err Orig. Trans.
¿err
(¸;º�» Ê
) (¸;º�» Ê
) (%) (¸;º�» Ê
) (¸;º�» Ê
) (%) (¸;º�»+Ë Ê
) (¸;º�»+Ë Ê
) (%)FEM 3.64 3.78 -0.8 1.01 1.00 -3.1 9.14 9.13 22GMC 3.40 3.51 -1.3 0.382 0.380 5.3 6.69 6.68 30RCM 4.25 4.45 6.1 2.98 2.90 -6.9 13.09 13.13 -292
Even thoughthe moduluscontrastbetweenthe two componentsof the compositeis high, the
effective propertiespredictedby FEM, GMC andRCM arecloseto eachotherin magnitude.It is
alsointerestingto observe that thecalculatedvaluesof¿
closelyapproximatethe exact valuefor
an isotropiccompositefor all the threemethodseven thoughthevalueis small (¿ À »�Á�»�»�º
). The
detailedfinite elementanalysesproducethemostaccurateresultsfor this problem.If we compare
the finite elementanalysisbasedeffective moduli with thosecalculatedusingGMC, we observe
that the moduli areslightly underestimatedby GMC. On the otherhand,RCM overestimatesthe
effectivepropertiesslightly. If wecomparethe¿
valuesproducedby thethreemethods,weobserve
thatGMC producesvaluesof¿
thatarecloserto 0.001thanRCM does.
6.1.4 Symmetric Compositeswith Equal Bulk Modulus
Thetranslationprocedurecanalsobeusedto generateanexactsolutionfor theeffective shear
modulusof two-dimensionalsymmetrictwo-componentcompositeswith bothcomponentshaving
thesamebulk modulus[26]. This relationis
99Ì"Í;ÎÏÌ ÎÐÌÒÑ,ÎÐÌpÓÔ Í;Î ÌÕ}Ö�× ØÙÙÚ Û Ö�× ÌÔ ÑÝÜ Û Ö�× ÌÔ Ó�Ü (6.6)
We testthis relationon thecheckerboardmodelshown in Figure6.1usingthecomponentmaterial
propertiesgiven in Table6.6. The exact effective propertiesfor the composite,calculatedusing
equation(6.6),arealsogivenin thetable.Thevaluesof theeffective moduli calculatedusingfinite
elements(FEM), GMC andRCM arealsoshown in Table6.6.
Table6.6. Componentproperties,exacteffective propertiesandnumericallycomputedeffectivepropertiesfor two-componentsymmetriccomposite
with equalcomponentbulk moduli.Þ ß ÌµÓáà Ôâ¹ã Ö�ä ÓÆå â¹ã Ö�ä�æ å â¹ã Ö�ä ÓkåComponent1 25.00 0.25 2.0 10.0Component2 1.19 0.49 2.0 0.4Composite 5.12 0.46 2.0 1.76Ì Í ç Ñéè
Diff.ç ÓÝè
Diff. 0.5(ç Ñéè × ç ÓÝè ) Diff.â¹ã Ö�ä�æ å â¹ã Ö�ä Ó'å %
â¹ã Ö�ä Ókå %â¹ã Ö�ä Ókå %
FEM 2.0 1.29 -26.8 2.54 44.4 1.91 8.8GMC 2.0 0.77 -56.3 0.77 -56.3 0.77 -56.3RCM 2.0 2.96 68.0 4.41 150.9 3.68 109.5
Theseresultsshow that theeffective two-dimensionalbulk modulusis calculatedcorrectlyby
all thethreemethods.However, theshearmodulicalculatedfor thecheckerboardmicrostructureare
quitedifferentfrom theexactresult.Hence,thisexactresultdoesnotappearadequatefor examining
thevalidity of theapproaches.Thefinite elementcalculationshave beencarriedout using123,000
elementsandhenceareexpectedto be quite accurate.In addition, if we examinethe averageof
the two shearmoduli calculatedusingFEM, we get a valuequite closeto the exact valuefor an
isotropiccomposite.This alsosuggeststhatthefinite elementcalculationsareaccurate.TheGMC
andRCM calculations,ashasbeenobserved before,predict lower andhighervaluesof theshear
moduli that thefinite elementcalculations,respectively. TheRCM resultsarehigherbecauseonly
sixteenelementsareusedto calculatetheeffective properties.We would get resultscloserto the
finite elementresultsif moreelementswereusedto calculatetheeffective propertiesin RCM.
100
6.1.5 Hill’ sEquation
Hill’ s equation[124] is anexactrelationthatis independentof microstructure.This equationis
valid for compositescomposedof isotropiccomponentsthat have thesameshearmodulus.For a
two-dimensionaltwo-componentcomposite,thisequationcanbewrittenasêë"ì�íïîñð ò¹óë ó í¤î í ò�ôë ô í¤î (6.7)
where,ë ìöõ ë ó õ ë ô arethetwo-dimensionalbulk moduli of thecomposites,particles
andbinder, respectively,î ð î ì ð î ó ð î ô is theshearmodulusof thecompositeandits components,and,ò¹ó and ò�ô arethevolumefractionsof theparticlesandthebinder.
We attemptto verify this relationshipusing the RVE containingan arrayof disksoccupying
70%of thevolumeasshown in Figure6.6. Table6.7shows thepropertiesof the two components
usedto comparethe predictionsof finite elements,GMC andRCM with the exact valueof bulk
moduluspredictedby Hill’ sequation.
Table6.7. Phasepropertiesusedfor testingHill’ s relationandtheexacteffective moduli of thecomposite.
Vol. ÷ ø î ëµùáúFrac. û¹ü ê�ý�þÆÿ û¹ü ê�ý�þÆÿ û¹ü ê�ý�þkÿ
Disks 0.7 3.00 0.25 1.20 2.40Binder 0.3 3.58 0.49 1.20 60.00Composite 1.0 3.22 0.34 1.20 3.82
Since the moduluscontrastis small, we expect the squarearray of disks to exhibit nearly
isotropic behavior. Therefore,we expect the predictionsof finite elements,GMC and RCM to
becloseto theexactvaluesof theeffective propertiesof thecomposite.It shouldbenotedthat the
materialschosenarenotquiterepresentative of PBX materials.
The numericallycalculatedvaluesof the effective two-dimensionalbulk andshearmoduli of
the compositeareshown in Table6.8. The finite elementcalculationshave usedaround123,000
elements,theGMC calculationshave usedaround4,100subcells,andtheRCM calculationshave
usedaround65,000subcells.
Theeffective shearmoduli predictedby all thethreemethodsareexact. In caseof theeffective
bulk moduli, theRCM predictionsarethemostaccuratefollowed by GMC andthefinite element
basedcalculations.Thefiniteelementbasedcalculationsoverestimatetheeffectivetwo-dimensional
101
Table6.8. Numericallycomputedeffective propertiesfor asquarearrayof diskswith equalcomponentshearmoduli.
���% diff ��� � ��� �
( � ��� ) ( � ��� ) ( � ��� )FEM 3.98 4.4 1.20 1.20GMC 3.66 -4.2 1.20 1.20RCM 3.92 2.7 1.20 1.20
bulk modulusby around4.4%while GMC underestimatesthebulk modulusby around4.2%.This
testappearsto show that theaccuracy we shouldexpectfrom the threemethodsof calculatingthe
effective propertiesis around� 5%.
6.1.6 CommentsOn ComparisonsWith Exact Solutions
The above comparisonswith exact solutionsshow that noneof the exact relationsprovide a
definitive testof theaccuracy of thefinite elementcalculationsof theeffective propertiesfor mate-
rialssuchasPBX 9501.However, weobserve thatthefinite elementcalculationsarequiteaccurate
for materialswith low moduluscontrasts,evenfor extremegeometrieslike thecheckerboardmodel.
In addition,we observe that both GMC andRCM performwell for materialswith low modulus
contrast. Detailedfinite elementcalculationsof the effective propertiesof compositeswith high
moduluscontrastscanbepresumedto bequiteaccuratebecauseof thehigh level of discretization
andfrom comparisonswith someof the exact solutions. If we comparethe finite elementbased
solutionswith RCM andGMC, themostcommonfinding is that theRCM resultsarecloserto the
finite elementsolutionsthanthe GMC resultsthoughRCM overestimatesthe effective properties
while GMC underestimatestheseproperties.
Improved estimatesareobtainedfor the effective elasticpropertieswhen the amountof dis-
cretizationof theRVE is increased.This is true for all the threemethods- exceptfor the special
caseof the checkerboardmicrostructure. For the checkerboardmicrostructure,for any amount
of discretization,the presentimplementationof RCM predictsthe samevaluesas that of a RVE
discretizedinto ���� subcells,for obviousreasons.
6.2 ComparisonsWith Numerical ResultsIn this section,we comparethe predictionsof effective elasticpropertiesusingfinite element
analyses,GMC and RCM with accuratenumericalresultsobtainedby other researchers.These
testsserve thepurposeof furtherconfirmingtheaccuracy of finite elementprocedureusedin this
research.In addition,weobtainabetterestimateof theaccuracy of GMC andRCM with respectto
thedetailedfinite elementanalysisbasedcalculations.
102
The current limit on the amountof discretizationpossiblefor the finite elementanalysesis
around������������� elementsfor a squareRVE, thatfor GMC is around� ������� ��� subcellsandthat
for RCM is around� ��� �!�"� ��� � subcells.Theselimits applyfor a four processorSunUltra-80with
4 Gbof mainmemoryin theabsenceof otherusers.In this research,for reasonsthathavemostlyto
dowith availability of computationalresources,wehaveused����#��$����# elementsfor finite element
analyses,%����&%�� elementsfor GMC and ����#��'����# subcellsfor RCM for thecomparisonsof the
threemethods.
Highly accurateestimatesof theeffectivepropertiesof squarearraysof diskshavebeenobtained
by GreengardandHelsing[97] usinganintegral equationbasedmethod.Weutilize theseresultsas
abenchmarkagainstwhich theaccuracy of thethreemethodsexploredin this research.
Squarearraysof diskssimilar to that shown in Figure6.6 aremodeledusingfinite elements,
GMC andRCM. Volumefractionsof thedisksarevariedfrom 10%to 70%. Thetwo-dimensional
effective propertiesdeterminedusingthesemethodshave beencomparedwith thosedeterminedby
GreengardandHelsing[97].
Thematerialpropertiesof thedisksandthebinder, usedby GreengardandHelsing,areshown
in Table6.9.
Table6.9. Componentpropertiesusedby GreengardandHelsing[97].( ) *,+.- /
0 ��� �+21 0 ��� �
+21 0 ��� �+31
Disks 3.24 0.20 2.25 1.35Binder 0.03 0.35 0.03 0.01Contrast 1.20 0.68 1.35
Thevaluesof theeffective two-dimensionalbulk andshearmoduli have beencalculatedusing
finite elements(FEM),GMC andRCM. Wehaveassumedthattheresultsof GreengradandHelsing
(G&H) arethemostaccurateandhavecalculatedthepercentageerrorsin theestimationof effective
propertiesby FEM, GMC, and RCM with respectto the G&H results. Table 6.10 shows the
resultsof GreengardandHelsingalongwith thosefrom the FEM, GMC, andRCM calculations
andtherelative errorsin approximationof the two-dimensionalbulk andshearmoduli. TheFEM
calculationswereperformedusing �5476.����� nodesand � �86.����� six nodedtriangularelements.The
GMC calculationswere carriedout using %����9%�� 0 476:� ���1
subcellsand the RCM modelsused
����#;�<����# 0 #��76=����#1
subcells.
The datain Table6.10show that all thecomputedeffective properties(from FEM, GMC and
RCM) are quite close to eachother when comparedto the moduluscontrastbetweenthe two
103
Table6.10. Comparisonof numericallycalculatedvaluesof two-dimensionalbulk andshearmoduli of squarearraysof disks.
Vol. >�?Frac. G&H FEM % Diff. GMC % Diff. RCM % Diff.0.1 3.8 3.8 -0.2 3.8 -1.3 3.9 1.50.2 4.4 4.4 1.0 4.3 -2.8 4.8 10.20.3 5.1 5.2 0.7 5.0 -2.7 5.4 5.80.4 6.1 6.2 1.1 6.0 -2.3 6.5 6.00.5 7.6 7.7 1.2 7.5 -1.0 8.0 5.20.6 9.9 10.0 0.5 10.1 1.2 10.4 4.70.7 15.4 15.8 2.5 14.8 -4.1 16.5 6.9@�ACB
G&H FEM % Diff. GMC % Diff. RCM % Diff.0.1 1.2 1.2 -0.3 1.2 -1.7 1.5 21.30.2 1.5 1.5 1.2 1.4 -3.3 2.2 45.60.3 1.9 1.9 0.8 1.9 -3.1 2.9 47.00.4 2.6 2.6 0.6 2.5 -3.8 4.1 55.20.5 3.8 3.8 0.7 3.6 -4.4 5.5 44.90.6 5.9 5.8 -1.1 5.7 -4.3 7.8 31.40.7 10.9 10.9 0.1 9.6 -11.9 13.2 21.3@�D=B
G&H FEM % Diff. GMC % Diff. RCM % Diff.0.1 1.2 1.2 3.0 1.1 -4.1 1.2 -1.00.2 1.3 1.4 4.5 1.2 -7.0 1.5 10.50.3 1.5 1.6 3.3 1.4 -7.6 1.6 5.30.4 1.8 1.9 3.3 1.7 -7.8 1.9 2.90.5 2.2 2.2 2.2 2.0 -8.0 2.4 10.80.6 2.8 2.8 1.1 2.5 -10.8 3.2 13.60.7 4.3 4.4 3.3 3.2 -24.2 5.2 22.0
componentsof the composite. Therefore,thesemethodsare performingquite well. Moreover,
theFEM resultsmatchvery well with the resultsof GreengardandHelsingthoughtheGMC and
RCM resultsdiffer slightly from theaccuratecalculations.This shows that theapproachwe have
takento calculatethevaluesof theeffective propertiesfrom FEM basedresultsis accurate.It also
shows thattheGMC andRCM approachesarequiteaccuratethoughthey couldbeimproved.
Figure 6.7 shows plots of the error in the two-dimensionalbulk moduli calculatedby FEM,
GMC andRCM relative to thosecalculatedby GreengardandHelsing. This plot shows that the
error in estimationof effective propertiesis independentof volumefraction. The fluctuationsin
the error canprobablybe attributedto errorsin discretization.It is observed that GMC performs
slightly betterthanRCM in thiscase.
Therelativeerrorin theestimationof theshearmodulus@�A ? is shown in Figure6.8. In thiscase,
104
0 10 20 30 40 50 60 70 80−20
−15
−10
−5
0
5
10
15
20
Volume Fraction (%)
Err
or in
K* (
%)
GMCRCMFEM
Figure6.7. Error in computationof E�F for aSquareArray of Disks.
theerror in theRCM estimatesis larger thanthatof theGMC estimates.This is becausethevalue
of GHFIKJ is usuallyoverestimatedby RCM. Theoverestimationcanbereducedif eachblock in RCM
is discretizedinto morethanfour elementsthusleadingto a lessstiff response.
Figure6.9shows therelative errorin theestimationof theshearmodulusL J F . GMC underesti-
matesthisshearmodulusbecauseit predictsamodulusthatrepresentstheReusslowerboundonthe
shearmodulus.TheFEM predictionsareabout1%to 5%higherthanthosepredictedby Greengard
andHelsing. This is probablybecausethedisplacementbasedfinite elementcalculationsproduce
a responsethat is stiffer thanactual. Interestingly, if theboundariesof theRVE areconstrainedto
remainstraightlines in shear, an even higherstiffnessis predictedbecauserelatively high normal
stressesaregenerated.TheRCM predictionsarecloserto theFEM predictionsandusuallyhigher
thantheFEM predictions.Thereasonfor this is, onceagain,theminimaldiscretizationthatis used
for theRCM calculations.
On average,GMC is performsbetterthanRCM for the squarearraysof disksfor the chosen
componentmaterialproperties.The FEM calculationsshow excellentagreementwith the highly
accurateresultsof GreengardandHelsing.Therefore,we usedetailedFEM calculationsasbench-
marksfor estimatingtheaccuracy of predictionsfrom GMC andRCM in futurevalidationchecks.
105
0 10 20 30 40 50 60 70 80−60
−40
−20
0
20
40
60
Volume Fraction (%)
Err
or in
µ1* (
%)
GMCRCMFEM
Figure 6.8. Error in computationof M�NCO for asquarearrayof disks.
0 10 20 30 40 50 60 70 80−30
−20
−10
0
10
20
30
Volume Fraction (%)
Err
or in
µ2* (
%)
GMCRCMFEM
Figure 6.9. Error in computationof M�P.Q for aSquareArray of Disks.
106
6.3 SpecialCases: StressBridgingComparisonsof effective propertiespredictedusingGMC andRCM with exact relationsand
othernumericallydeterminedresultsshow that thesemethodsperformquitewell for low modulus
contrastmaterialsevenfor extrememicrostructureslikethecheckerboardmodel.Fromourresearch,
somemicrostructureshave beendiscoveredfor which GMC performslessthanadequatelywhile
thereareothermicrostructuresfor which RCM doesnot performwell. Someof thesemicrostruc-
turesarediscussedin thissection.Effectivepropertiesarecalculatedfor thesemicrostructuresusing
GMC andRCM. Comparisonsaremadewith thecorrespondingpropertiespredictedusingdetailed
finite elementanalysesand,wherepossible,reasonsfor thepoorperformanceof GMC or RCM are
discussed.
Thesquarearrayof disksrepresentsa situationin which thereis no contactbetweenparticles.
Sincethe PBXs of interestto this researchcontainmorethan90% particlesby volumethereare
boundto be particlesthat areeithervery closeto eachotheror in contact. To checkthe efficacy
of GMC and RCM when particlesare in contact,we have simulateda numberof “bridging”
models. The “bridging” is dueto contactbetweenparticlesthat leadsto preferentialstresspaths
or stressbridging. This problemwasfirst recognizedwhencalculatingthe effective moduli of a
randomdistribution of particles(with somecontactbetweenparticles).It wasobserved thatGMC
consistentlygeneratedlow valuesof the effective stiffnessmatrix terms. This, in turn, led to the
developmentof the RCM modelasan alternative to GMC. We discusssomeof these“bridging”
modelsand the effective elasticpropertiescalculatedusing GMC, RCM and finite elementsfor
thesemodels.
6.3.1 Corner Bridging : X-ShapedMicr ostructure
Thecheckerboardmodelshown in Figure6.1hassomestressbridgingthroughcornercontacts.
However, becauseof therelatively smallmoduluscontrastbetweenthephasesthatwereusedto test
theexactrelations,thedifferencebetweentheeffective moduli predictedby GMC, RCM andfinite
elementswasnot very large. Whenthemoduluscontrastis increasedto matchthatof PBX 9501
at room temperaturethe differencesbetweenthe resultsproducedby the threemethodsbecome
pronounced.We canobserve this for theRVE shown in Figure6.10. In this RVE theparticlesare
square,arrangedin theform of an’X’ andoccupy avolumefractionof 25%.Theparticlestransfer
stressthroughcornercontacts.Thematerialpropertiesusedarethosefor HMX andbinderasshown
in Table3.1.
Thesharpcornersin theparticlesleadto stresssingularities.We assumethat thehigh stresses
are averagedout during the calculationof effective properties(note that this is also one of the
assumptionsof the RCM technique). For the ’X’ shapedmicrostructureshown in Figure 6.10,
107
R R RR R RR R RS S SS S SS S S
T T TT T TT T TU U UU U UU U U
V V VV V VV V VW W WW W WW W W
X X XX X XX X XY Y YY Y YY Y Y
Z Z ZZ Z ZZ Z Z[ [ [[ [ [[ [ [
\ \ \\ \ \\ \ \] ] ]] ] ]] ] ]
^ ^ ^^ ^ ^^ ^ ^_ _ __ _ __ _ _
` ` ` ` ` `` ` ` ` ` `` ` ` ` ` `` ` ` ` ` `` ` ` ` ` `` ` ` ` ` `
a a a a a aa a a a a aa a a a a aa a a a a aa a a a a aa a a a a a
b b bb b bb b bc c cc c cc c c
d d dd d dd d de e ee e ee e e
f f ff f ff f fg g gg g gg g g
h h hh h hh h hi i ii i ii i i
j j jj j jj j jk k kk k kk k k
X,1
Y,2
Figure6.10. RVE usedfor cornerstressbridgingmodel.
we first observe the variation in the effective elasticpropertieswith increasein moduluscontrast
betweenthe particlesand the binder. The five setsof materialsthat are explored are shown in
Table6.11.
Table6.11. Theelasticpropertiesof thecomponentsof the’X’ shapedmicrostructure.
Model lnm opm q m rsr q m rKt q musuvpw�x y�z�{ vpw�x y�z�{ vpw�x y�z2{ vpw�x y�|3{(MPa) (MPa) (MPa) (MPa)
All 1.53 0.32 2.19 1.03 5.79l~} o�} q }rsr q }rKt q }usuvpw�x y r { vpw�x y | { vpw�x y | { vpw�x y r {
(MPa) (MPa) (MPa) (MPa)a 0.07 0.49 0.012 0.011 0.023b 0.7 0.49 0.12 0.11 0.23c 7 0.49 1.2 1.15 2.35d 70 0.49 11.98 11.5 23.49e 700 0.49 119.8 115.1 234.9
Model q m rsr�� q }rsr q m rKt3� q }rKt q musu � q }usuvpw�x y r { vpw�x y t {a 182.75 895.13 246.70b 18.28 89.51 24.67c 1.83 8.95 2.47d 0.18 0.89 0.25e 0.02 0.09 0.02
108
Figure 6.11 shows the variation of the effective ����s� , normalizedwith respectto the binder
properties,with increasingmoduluscontrastbetweenthe particlesand the binder. The effective
propertieshave beencalculatedusingfinite elements(FEM), GMC andRCM. It canbe observed
that theFEM, GMC andRCM estimatesarecloseto eachotherfor moduluscontrastsbelow 200.
For highermoduluscontrasts,theratio of theeffective moduluspredictedby GMC to themodulus
of the binderremainsrelatively constantandmuchlower thanthat predictedby FEM andRCM.
Thisshows thatGMC doesnotdealasaccuratelywith cornercontactsasFEM or RCM.
0.1 1 10 100 1000 4000−10
0
10
20
30
40
50
60
70
80
90
100
C* 11
/Cb 11
Cp11
/Cb11
FEMGMCRCM
Figure 6.11. Variationof � ��s� with moduluscontrastfor ’X’-shapedmicrostructure.
Similarplotsareshown in Figure6.12for theeffectiveproperty� ��K� . For thisproperty, it is again
observedthattheRCM predictionsarecloserto theFEM predictionsthantheGMC predictions.At
relatively low moduluscontrasts,all the threemethodspredictapproximatelythe samevaluesof
�H��K� . GMC is againfoundto predictlow valuesof ����K� at highmoduluscontrasts,implying thatthe
cornercontactsbetweenparticlesarenotaccuratelytakeninto account.
Figure6.13shows thevaluesof ����s� calculatedusingFEM, GMC andRCM. Onceagain,GMC
is foundto predictvaluesthatarequitelow comparedto thosepredictedby FEM.
The componentsof PBX 9501have the materialpropertiesthat correspondto thoseof model���5�
shown in Table6.11andhave the highestmoduluscontrast.The valuesof ����s� , �H��K� and �H��s�predictedfor the X-shapedmicrostructurefor this caseareshown in Table6.12. The ratio of the
109
0.06 1 10 100100 1000−5
0
5
10
15
20
25
30
35
40
45
50
C* 12
/Cb 12
Cp12
/Cb12
FEMGMCRCM
Figure 6.12. Variationof �H��K� with moduluscontrastfor ’X’-shapedmicrostructure.
1.5 10 100 1000 100000
500
1000
1500
2000
2500
3000
3500
C* 66
/Cb 66
Cp66
/Cb66
FEMGMCRCM
Figure 6.13. Variationof � ��s� with moduluscontrastfor ’X’-shapedmicrostructure.
110
FEM resultsto theGMC resultsshowsthelargeerrorin theGMC predictionscomparedto theRCM
predictions.Therefore,the currentversionof RCM is a definiteimprovementover GMC for this
microstructure.
Table6.12. ����s� , ����K� and �H��s� for X-shapedmicrostructurewith highestmoduluscontrast.
FEM GMC FEM/GMC RCM RCM/FEM�p��� � �3� �p��� � �2� �p��� � �3�� ��s� 3.50 0.16 21 11.33 3� ��K� 3.49 0.16 22 5.33 1.5����s� 4.00 0.003 1245 7.43 1.8
A comparisonof theeffective stiffnessmatrix terms � ��s� , � ��K� and � ��s� calculatedusingGMC,
RCM andFEM is alsoshown in Figure6.14.Thefigureshows thattheeffective moduli calculated
by GMC areconsiderablylower thanthosecalculatedusingfinite elements.Themoduli calculated
usingRCM areconsistentlyhigherthanthefinite elementresults.WeconcludethatGMC doesnot
“see” thecontactpointsbetweenparticles,especiallywheresheareffectsareconcerned.
0
500
1000
1500
C*11
C*12
C*66
C* 11
, C* 12
and
C* 66
GMCRCMFEM
Figure 6.14. Comparisonof effective stiffnessmatrixfor cornerstressbridgingmodel.
111
6.3.2 EdgeBridging : FiveCases
Furtherevaluationof GMC andRCM is performedusingmodelsA throughE shown in Fig-
ure6.15.Theobjective of thisstudyis to explorethebehavior of GMC andRCM asstressbridging
is increasedprogressively from cornerbridging to partial edgebridging followed by continuous
stressbridging.
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¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯
° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °
± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±
² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²² ² ² ²
Model A Model B
Model C Model D
Model E
X,1
Y,2
Figure6.15. Progressive stressbridgingmodelsA throughE.
ModelA containsasquareparticlethatoccupies25%of thevolumeandis centeredin theRVE
anddoesnot have any stressbridging. Model B hasthreeparticlesthat touchat two cornersof the
centralparticle. In modelC theamountof contactis increaseduntil thereis a singleline of stress
112
bridging along the centerof the RVE. This microstructurehasbeenchosensuchthat RCM will
predictsquaresymmetrywhena ³µ´�³ subcellbasedblock is used,eventhoughthemicrostructure
doesnothave squaresymmetry. Model D extendstheline of bridgingto anareaof bridgingin one
directionandmodelE extendsthe bridging to both directions. In what follows, the ’1’ direction
correspondsto the ¶ axisshown in Figure6.15andthe’2’ directioncorrespondsto the · axis.
The materialpropertiesof the constituentsof PBX 9501 at room temperature,as shown in
Table6.13,areusedfor thesetests.GMC simulationsof theRVEswerecarriedoutwith ¸ ¹�¹�´&¸ ¹�¹subcells,RCM simulationsused³�º�»~´µ³�º�» subcellsandfinite elementcalculationswerecarriedout
usingaround ¹½¼.¹�¹�¹ eightnodedquadrilateralelements.
Table6.13. Materialsusedto testedgebridgingusingFEM, GMC andRCM.¾ ¿ À�ÁsÁ À�ÁK ÀÄÃsÃ
(MPa) (MPa) (MPa) (MPa)Particle 15300 0.32 21894 10303 5795Binder 0.7 0.49 11.98 11.51 0.23
6.3.2.1 Model A
The effective propertiesof Model A, calculatedusing FEM, GMC and RCM, are shown in
Table6.14.Thedifferencesshown in thetablearepercentagesof thevaluescomputedusingFEM.
SinceModel A exhibits squaresymmetry, i.e.,ÀHÅÁsÁÇÆ À�ÅÂsÂ
, thevaluesofÀ�ÅÂsÂ
arenot shown in the
table.Thevaluesof theeffective propertiesfrom thethreetechniquesarequitecloseto eachother,
especiallywhenwe considerthe large moduluscontrastbetweenthe particleandthe binder. The
percentagedifferenceswith respectto theFEM resultsshow thatGMC is in betteragreementwith
FEM for thequantitiesÀ ÅÁsÁ
andÀ ÅÁKÂ
while RCM is in betteragreementforÀ ÅÃsÃ
.
Table6.14. Effective propertiesof ModelA from FEM, GMC andRCM.
FEM GMC Diff. RCM Diff.(MPa) (MPa) (%) (MPa) (%)À ÅÁsÁ16.4 16.2 -1.5 18.9 15À ÅÁKÂ15.1 15.2 0.9 13.3 -12ÀHÅÃsÃ0.38 0.31 -18 0.37 -3
Figure6.16shows acomparisonof theeffective stiffnessmatrixcomponentsÀHÅÁsÁ
,ÀHÅÂsÂ
,À�ÅÁKÂ
andÀ ÅÃsÃ
for modelA, normalizedwith respectto thebinderproperties.Theresultsshow that,for model
A, the binderdominatesthe effective elasticresponse.The plots in Figure6.16 further illustrate
113
thattheeffective moduli calculatedfor modelA usingGMC andRCM arequiteaccurate.It is also
observedthatthevaluesof ÈHÉÊsÊ and ÈHÉËsË areidentical.
0
0.5
1
1.5
2
C*11
C*22
C*12
C*66
Rat
io o
f C* to
Cbi
nder
GMCRCMFEM
Figure 6.16. Comparisonof normalizedeffective stiffnessesfor modelA.
6.3.2.2 Model B
The effective stiffnessmatricesfor the corner-bridging model (Model B) are shown in Ta-
ble 6.15. This modelalsoexhibits squaresymmetry. The stressbridgealongthe diagonalleads
to muchhigherstiffnessthanwouldoccurfor asingleparticleoccupying thesamevolumefraction.
This modelis similar to hecheckerboardmodelin somerespectsandexhibits similar trends.The
GMC calculationspredictvaluesof ÈHÉÊsÊ and ÈHÉÊKË that are lower by a factorof 18 thanthe FEM
results.Thevalueof È ÉÌsÌ calculatedusingFEM arearound1,400timesthatcalculatedusingGMC.
SincetheFEM calculationsarequite accurate,it is clearthat theaveragingprocessusedin GMC
is not adequateto capturetheeffectsof cornerstressbridging. On theotherhand,thevalueof È ÉÊsÊpredictedby RCM is around3 timesthatpredictedby FEM. Therefore,RCM predictsvaluesthat
arequitecloseto thosepredictedby FEM thoughthereis roomfor improvement.TheRCM based
valuesof È ÉÊKË and È ÉÌsÌ areeven closerto the FEM basedvalues- around1.5 timeshigher. Thus,
RCM doesquitewell in improving uponthedrawbackin GMC relatingto cornerbridging.A plot of
theeffective stiffnessmatrix terms,calculatedfor Model B usingFEM, GMC andRCM, is shown
in Figure6.17.
114
Table6.15. Effective propertiesof ModelB from FEM, GMC andRCM.
FEM GMC FEM/GMC RCM RCM/FEMÍpÎ�Ï Ð�Ñ3Ò ÍpÎ�Ï Ð�Ñ2Ò ÍpÎ�Ï Ð�Ñ3Ò(MPa) (MPa) (MPa)Ó�ÔÕsÕ 3.4 0.2 17 11.4 3.4Ó�ÔÕ Ñ 3.4 0.2 18 5.3 1.6Ó�ÔÖsÖ 5.4 0.004 1429 7.4 1.4
0
200
400
600
800
1000
1200
1400
C*11
C*22
C*12
C*66
C* 11
, C* 22
, C* 12
and
C* 66
GMCRCMFEM
Figure6.17. Comparisonof effective stiffnessesfor modelB.
6.3.2.3 Model C
ModelC hasacontinuouspaththroughparticlesalongthe × -axis(the’1’ direction)andanother
continuousparticlepathalongonediagonal.Thestressbridgepathalongthe’1’ directionimpacts
mostly the normalcomponentsof stiffnesswhile the bridgealongthe diagonalimpactsthe shear
stiffness.Thesepathsareshown by dashedlines(for normalstressbridge)andby dottedlines(for
shearstressbridge)in Figure6.18.Thestressbridgein the’1’ directionleadsto avalueofÓ�ÔÕsÕ that
is considerablyhigherthanÓ ÔÑsÑ .
This modelmicrostructurehasbeenchosento illustratewhy RCM mayfail to predictaccurate
effectivepropertiesfor certainmicrostructures.Theprocedureusedby RCM to predicttheeffective
propertiesof Model C is shown in Figure 6.19. The schematicshows that in the last recursion
performedby RCM, the four subcellsappearto belongto a materialwith squaresymmetryeven
thoughthis is not the case. This is why, for this type of microstructure,RCM predictseffective
115
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Ü Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü ÜÜ Ü Ü Ü Ü Ü Ü Ü Ü Ü
Ý Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý ÝÝ Ý Ý Ý Ý Ý Ý Ý Ý Ý
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ß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ßß ß ß ß ß à à à à àà à à à àà à à à àà à à à àà à à à àà à à à àà à à à àà à à à àà à à à àà à à à à
á á á á áá á á á áá á á á áá á á á áá á á á áá á á á áá á á á áá á á á áá á á á áá á á á á
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ä ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ää ä ä ä ä
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é é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é éé é é é é é é é é é
ê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê êê ê ê ê ê
ë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ëë ë ë ë ë ì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ìì ì ì ì ì
í í í í íí í í í íí í í í íí í í í íí í í í íí í í í íí í í í íí í í í íí í í í íí í í í í
î î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î îî î î î î î î î î î
ï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ïï ï ï ï ï ï ï ï ï ï
X,1
Y,2
Figure 6.18. Stressbridgingpathsfor ModelC.
propertiesthat display squaresymmetry. The problemcan be alleviated to someextent if the
discretizationof theRVE is carriedout in amannerthatthesymmetryis brokenduringrecursion.
Table6.16showstheeffectivepropertiesof ModelC thathavebeencalculatedusingFEM,GMC
andRCM.CalculationsusingGMC show thattheeffectof thestressbridgingis notreflectedby this
methodfor thismodelRVE andlow valuesof theeffective stiffnesstensorarepredicted.TheRCM
calculationscapturethe diagonalsheareffect quite accuratelybut underestimatethe stressbridge
in the ’1’ directionandoverestimatethebridgein the ’2’ directionbecausethepredictedeffective
stiffnessmatrix is squaresymmetric. The correctanisotropy is displayedby the finite element
calculationsthat result in a ðHñòsò of around ó�ô�ô�ô anda ð�ñõsõ of around ö ô�ô�ô . A comparisonof the
propertiespredictedby GMC, RCM andfinite elementsfor modelC is alsoshown in Figure6.20.
6.3.2.4 Model D
ModelD is aRVE with anareaof stressbridgingin the’1’ directionandadiagonalstressbridge.
In thiscase,all threetechniques(GMC,RCM andfinite elements)capturethestressbridgein the’1’
directionadequately. However, GMC againfails to predictthestiffeningeffect in the ’2’ direction
wherea directstresspathdoesnot exist. Theshearbridgeis alsoignoredby GMC. RCM, on the
otherhand,performsquitewell in predictingthecomponentseffective stiffnessmatrix. Thevalues
116
X,1
Y,2
÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷
ø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø øø ø ø ø ø ù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ùù ù ù ù ù
ú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú úú ú ú ú ú
û û û û û ûû û û û û ûû û û û û ûû û û û û ûû û û û û ûü ü ü ü üü ü ü ü üü ü ü ü üü ü ü ü üü ü ü ü ü
ý ý ý ý ýý ý ý ý ýý ý ý ý ýý ý ý ý ýý ý ý ý ýþ þ þ þ þþ þ þ þ þþ þ þ þ þþ þ þ þ þþ þ þ þ þ
ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � � �� � � � � �� � � � � �� � � � � �� � � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �
� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �
� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �� � � � � � � � � � �
� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �
� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �
� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �� � � � � � � � � �Square Symmetry
Figure 6.19. Why RCM predictssquaresymmetryfor ModelC.
Table6.16. Effective propertiesof ModelC from FEM, GMC andRCM.
FEM GMC FEM/GMC RCM RCM/FEM����������� ��� �����!� �����������(MPa) (MPa) (MPa)"$#%&% 4.1 0.025 164 2.2 0.5" #'&' 0.9 0.024 37 2.2 2.5"$#%(' 1.5 0.023 64 0.8 0.5" #)&) 1.1 0.0004 2500 1.0 0.9
of effective stiffnesscalculatedusingFEM, GMC andRCM areshown in Table6.17.Thevalueof"*#%&% predictedby GMC is around1.05timeslower thanthatpredictedby FEM while thatpredicted
by RCM is around1.05timesthatpredictedby FEM.Thevalueof" #'&' predictedby GMC is around
1/40th thatpredictedby FEM while RCM predictsa valuethatis around1.6 timestheFEM based
prediction. The valueof" #%(' predictedby RCM is againquite closeto that predictedby FEM -
around1.4timesof theFEM value.On theotherhand,GMC predictsavaluethatis around1/20th
of theFEM prediction.Finally, theshearstiffnessterm" #)&) predictedby RCM is around0.9 times
theFEM valuewhile thatpredictedby GMC is about1/1900th of theFEM value.Therefore,RCM
117
0
1000
2000
3000
4000
5000
C*11
C*22
C*12
C*66
C* 11
, C* 22
, C* 12
and
C* 66
GMCRCMFEM
Figure6.20. Comparisonof effective stiffnessesfor ModelC.
performswell for this microstructureandtheproblemsassociatedwith Model C arenot observed
for Model D. GMC still fails to provide adequatepredictionsof theeffective elasticmoduli. This
testalsoshows thata continuousareaof stressbridgingacrosstheRVE mayberequiredfor GMC
to capturetheeffectsof stressbridging.This issueis exploredin Model E. Thevaluesof +$,-&- , +*,.&. ,+ ,-(. and + ,/&/ for modelD arealsoshown in Figure6.21.
Table6.17. Effective propertiesof ModelD from FEM, GMC andRCM.
FEM GMC FEM/GMC RCM RCM/FEM0�1�2�3�4�5 0�1 2�3�4!5 0�1�2�3�4�5(MPa) (MPa) (MPa)
+ ,-&- 9.0 8.5 1.05 9.4 1.04+ ,.&. 1.4 0.03 42 2.2 1.6+$,-(. 0.5 0.02 23 0.7 1.4+ ,/&/ 1.2 0.0006 1887 1.1 0.9
6.3.2.5 Model E
For Model E therearetwo direct stressbridgesin the ’1’ and’2’ directionsandtwo diagonal
stressbridges. The effective propertiesfor this microstructure,calculatedusingFEM, GMC and
RCM, areshown in Table6.18.Both GMC andRCM predicttheeffective stiffnessterms +*,-&- , +*,.&.and + ,-(. quite accurately. This shows that GMC requirescontinuousstresspathsin the direction
118
0
2000
4000
6000
8000
10000 C*11
C*22
C*12
C*66
C* 11
, C* 22
, C* 12
and
C* 66
GMCRCMFEM
Figure 6.21. Comparisonof effective stiffnessesfor ModelD.
of the coordinateaxes to be able to predict the effect of stressbridging. Onceagain,the shear
stiffnesspredictedby GMC is just a volumefractionweightedaverageof theshearmoduli of the
particlesandthebinder. Therefore,theextrastiffeningof theeffectiveshearresponsebecauseof the
diagonalstressbridgesis overlooked by GMC. A plot theeffective stiffnesscomponentspredicted
by thethreemethodsis shown in Figure6.22.
Table6.18. Effective propertiesof ModelE from FEM, GMC andRCM.
FEM 6�7�8�9�:<; GMC 6�7�8�9�:�; FEM/GMC RCM 6�7 8�9�:�; RCM/FEM(MPa) (MPa) (MPa)=*>?&? 10.0 9.0 1.1 11.0 1.1= >?(@ 2.9 2.1 1.3 3.5 1.2=*>A&A 1.8 0.0009 1915 2.5 1.4
6.4 SummaryExact relationsfor the effective propertiesof two-componentcompositeshave beenexplored
using detailedfinite elementanalyses,GMC and RCM. We observe that detailedfinite element
analysespredictquiteaccurateeffective properties,eventhoughmostof theexact relationscanbe
verifiedonly approximatelyusingour approach.For moduluscontrastsof lessthan500,GMC and
RCM arealsoquiteaccuratein predictingeffective elasticmoduli.
119
0
2000
4000
6000
8000
10000
12000 C*11
C*22
C*12
C*66
C* 11
, C* 22
, C* 12
and
C* 66
GMCRCMFEM
Figure 6.22. Comparisonof effective stiffnessesfor ModelE.
Comparisonswith numericalexperimentsconductedby otherresearchersalsoshow thatall the
threetechniquesarequiteaccuratefor thematerialsconsidered.In general,wefind thatRCM tends
to overestimatetheeffective stiffnessslightly. Thediscretizationusedfor theblocksat the lowest
levelsof recursioncouldbe increasedto obtaina lessstiff response.GMC tendsto underestimate
the effective stiffness,especiallyin the presenceof stressbridging. The shearstiffnessis also
underestimateby GMC whenstressbridgingis present.Couplingof thenormal-shearstress-strain
responsecouldalleviate this problem.For RVEs with stressbridging,RCM is foundto predictthe
effectivestiffnessbetterthanGMC andtheproblemsassociatedwith lackof normal-shearcoupling
aredealtwith successfully.
In thefollowing chapter, weexploremethodsof modelingmicrostructuresthatarerepresentative
of PBX materials,specificallyPBX 9501. We apply GMC and RCM to calculatethe effective
propertiesof thesemicrostructures.The effective propertiesthusobtainedarecomparedto finite
elementanalysisbasedsolutionsandto experimentaldata.
CHAPTER 7
SIMULA TION OF PBX MICR OSTRUCTURES
Early simulationsof themicromechanicalresponseof PBX 9501usingGMC involved theuse
of a RVE subdivided into 25 subcellsin two dimensions(125 subcellsin threedimensions)[15].
The determinationof theexact geometryof theRVE andthe assignmentof constituentphasesto
subcellsin thisRVE wasbasedon trial anderroruntil theelasticpropertieswerein agreementwith
a particularsetof experimentaldata.However, any changein theconstituentmaterialsandvolume
fractionsrequiredthisprocessto berepeateduntil amatchwasfound.Thisapproachdefeatspartof
thepurposeof micromechanicalmodelingasit assumesknowledgeof experimentallydetermined
propertiesof thecompositePBX.
Most micromechanicalcalculationsfor PBXshave beencarriedout usingsubgridmodelsthat
usehighly simplifiedmodelsof themicrostructure(for example,sphericalgrainscoatedwith binder
or sphericalvoids in an effective PBX material[125]). This is becausecomplicatedphysicaland
chemicalphenomenaareusually involved andcomplex geometriesarenot only computationally
expensive to modelbut alsoarenot necessarilyaccuraterepresentationsof the actualmicrostruc-
ture. Closedform solutionsfrom thesesimplemodelshave beenusedto provide propertiesto the
macroscopicsimulations.
More detailedcalculationshave usedmicrostructurescontainingorderedarraysof circles or
polygonsin two or threedimensionsto modelPBXs [126, 127]. Thesemodelsdo not reflectthe
microstructureof PBXsandhencehave limited usefor predictingthermoelasticproperties.Better
two-dimensionalapproximationsof themicrostructurehavebeenconstructedfrom digital imagesof
thematerialandusedby BensonandConley [128] to studysomeaspectsof themicromechanicsof
PBXs.However, suchmicrostructuresareextremelydifficult to generateandrequirestate-of-the-art
imageprocessingtechniquesandexcellentimagesto accuratelycapturedetailsof thegeometryof
PBXs. Thereis alsotheproblemof creatinga threedimensionalimagebasedon two-dimensional
datathatrequiresalargenumberof crosssections.MorerecentlyBaer[129] hasusedacombination
of MonteCarloandmoleculardynamicstechniquesto generatethree-dimensionalmicrostructures
thatmodelPBXs.Microstructurescontainingspheresandorientedcubeshavebeengeneratedusing
thesetechniquesandtheseappearto mimic PBX microstructureswell. However, thegenerationof
121
a singlerealizationof thesemicrostructuresis very time consumingandoftenleadsto a maximum
packingfractionof at themost70%.Periodicityis alsoextremelydifficult to maintainin theRVEs
generatedby this method.Themethodusedfor simulatingmostof thesemicrostructureshasbeen
finite elementanalysis.
The approachwe have taken in this researchis to apply our techniques(GMC andRCM) to
both manuallygeneratedmicrostructuresandrandomlygeneratedmicrostructures.The manually
generatedmicrostructuresarediscussedfirst andresultsareshown for a few microstructures.The
procedurewe usefor automaticallygeneratedrandommicrostructuresis discussednext, followed
by a few resultsfor thesemicrostructuresusingGMC andRCM.
The materialpropertiesusedfor the particlesand the binder in thesesimulationsare shown
in Table7.1. For the binder, we have usedthe Young’s modulusat 25B C and0.049/sstrainrate
determinedby Wetzel[7]. Theexperimentallydeterminedvaluesof thestiffnessmatrix termsfor
PBX 9501,at 25B C and0.05/sstrainrate,arealsogivenin Table7.1. Theeffective elasticmoduli
calculatedusingfinite elements(FEM), GMC andRCM have beencomparedwith the valuesfor
PBX 9501shown in Table7.1.
Table7.1. Experimentallydeterminedelasticmoduli of PBX 9501andits constituents[7].
Material C D E$FG&G = E*FH&H E*FG(H E*FI&I(MPa) (MPa) (MPa) (MPa)
Particles 15300 0.32 21894 10303 5795Binder 0.7 0.49 11.97 11.51 0.235PBX 9501 1013 0.35 1626 875 375
7.1 Manually GeneratedMicr ostructuresSix two-dimensionalmicrostructuresrepresentingPBXshavebeencreatedmanuallyto observe
theeffect of particledistribution of theeffective elasticmoduli. Circularparticlesof varioussizes
have beenusedto fill a squareRVE. SincePBXsaretypically a mixtureof coarseandfine grains
with thefiner grainsforming filler betweencoarsergrains,mostof themodelscontainoneor a few
large particlessurroundedby smallerparticles.Themodelsareshown in Figure7.1. Thevolume
fractionsof particlesin eachof thesemodelsis around90J 0.5%. All the modelspossesssquare
symmetry.
7.1.1 FEM Calculations
Detailedfinite elementanalyseshavebeenperformedto determineaccurateeffective properties
of eachof themodelsshown in Figure7.1.Around65,000six nodedtriangularelementshavebeen
122
Model 1 Model 2
Model 3 Model 4
Model 5 Model 6
Figure7.1. Manuallygeneratedmicrostructuresfor PBXs.
usedto discretizeeachRVE andthegeometryof theparticleshasbeenpreserved. To alleviateany
elementshapeproblemsthat could occurbecauseof the closeproximity of the circular particles,
no particlesare allowed to be in contactwith eachother in the models. In addition, periodic
displacementboundaryconditionshave beenappliedto determinethestressandstrainfields. On
the basisof the resultsfrom Chapter6, we assumethat the effective propertiescalculatedusing
FEM arecloseto theactualvaluesfor themicrostructuresandcomponentmaterialsused.
The computedvaluesof the effective stiffnessmatrix terms K$LM&M , K*LM(N and K*LO&O are shown in
Table7.2. Thevaluesof K LM&M for thesix modelsvary from 175MPato 240MPawith ameanof 192
MPa, K*LM(N variesbetween75 MPa and114MPa with a meanof 91 MPa, and K*LO&O hasa rangeof 8
MPato 38MPawith ameanof around20MPa. Thestandarddeviationof K LM&M is 17%of themean,
123
that for P*QR(S is 13%of themean,andfor P$QT&T it is 59%of themean.Therefore,theshearstiffness
shows thelargestvariability for thesix models.On thewhole,thesix RVEs predictapproximately
thesameeffective propertieswhencomparedwith themoduluscontrastbetweentheparticlesand
thebinder.
Table7.2. Effective stiffnessfor thesix modelPBX 9501microstructuresfrom FEM calculationsusing65,000six-nodedtriangleelements.
P*QR&R FEM/ P*QR(S FEM/ P$QT&T FEM/U�V�W�X S�YExpt.
U�V W�X R�YExpt.
U�V�W�X R�YExpt.
(MPa) (%) (MPa) (%) (MPa) (%)Model1 1.77 11 9.02 10 1.11 3Model2 1.81 11 8.64 10 1.16 3Model3 1.86 11 8.76 10 1.54 4Model4 1.43 9 11.4 13 3.26 9Model5 2.37 14 9.42 11 3.83 10Model6 2.29 14 7.59 9 0.85 2Mean 1.92 11 9.14 10 1.96 5Std.Dev. 0.32 1.15 1.15
Theeffective propertiesfor thesix modelsfrom FEM calculationsarealsoshown in Figure7.2.
From the Figures7.2 and 7.1 it canbe seenthat the ratio P QR&R[Z P QR(S increaseswhen,at the edges
of theRVE, theamountof bindermaterialdecreases.Increasein this ratio is accompaniedby an
increasein P*QR&R andadecreasein P$QR(S . Thevalueof theshearmodulusP$QT&T increasesastheamount
of binderalongthediagonalof theRVE decreases.
Models1 through3 have a singlelarge particleandmany smallerparticlesandshow approxi-
matelythesameeffective behavior. Model4 hasasmallerratio betweentheradiusof thelargestto
thesmallestparticlesandgenerateslower effective P*QR&R thantheaverageandhighereffective P*QR(Sand P QT&T thanthe average.Models5 and6 which have larger lengthsof the boundarycontaining
particlesshow a stiffer P*QR&R thanthe othermodelseven thoughthe volumefractionsoccupiedby
particlesis slightly lower thanin theothercases.
Theeffective stiffnessfrom FEM calculationsis alsoshown asa percentageof theexperimen-
tally determinedstiffnessof PBX 9501 in Table 7.2. It is observed that the FEM calculations
underestimatethe stiffnessby an order of magnitude. The reasonsfor this could be that more
stressbridging is involved in real microstructuresleadingto increasedstiffness. Another reason
couldbethattwo-dimensionalcalculationsarenotaccurateenough.Intuitively, however, it appears
thatthree-dimensionalcalculationsshouldproducea lowereffectivestiffnessthantwo-dimensional
calculations. It is alsopossiblethat the materialpropertiesusedfor the binderarenot accurate.
124
0
20
40
60
80
100
120
140
160
180
1 2 3
4
5 6
1 2 34
56
1 23
45
6
C*11
C*12
C*66
C* (
FE
M)
/ C (
Bin
der)
Figure7.2. Effective stiffnessesfor thesix modelmicrostructuresfrom from detailedfiniteelementanalysesasaamultipleof thebinderstiffness.
However, from thedatashown in Table2.8it canbeseenthatthemodulusof thebinder, determined
by variousresearchers,doesnotdiffer significantlyfrom thevalueusedin thesecalculations.Hence,
the mostprobablereasonfor the low valuesof the effective stiffnessis that stressbridging is not
consideredin thechosenmicrostructures.
7.1.2 GMC Calculations
Two approacheshave beenusedto determinethe effective propertiesof the six modelsusing
GMC. Thefirst appliesa 50%rule to determineinitial subcellpropertieswhile thesecondapplies
of aninitial GMC stepto determinethese.A full GMC calculationis thenperformed.
7.1.2.1 Fifty Percent Rule
In thiscase,subcellsareassignedparticlematerialpropertiesif particlesoccupy morethan50%
of thesubcell.Binderpropertiesareassignedotherwise.Theparticlesarenot resolved well when
a squaregrid is overlaid on the microstructurein this manner. Stressbridging pathsarecreated
wheretherearenonein theactualmicrostructure.This leadsto thepredictionof higherthanactual
stiffnessvalues.Figure7.3shows theeffectof applicationof the50%ruleon themicrostructureof
oneof themodels.TheRVE hasbeendiscretizedinto \�]�]_^`\�]�] subcells.
125
Figure 7.3. Applicationof fifty percentrule to amodelmicrostructure.
7.1.2.2 The Two-StepApproach
Thelargesizeof thematrix to beinvertedin GMC limits thenumberof subcellsthatcanbeused
to discretizeaRVE. To improve theaccuracy of GMC evenwhenthenumberof subcellsis limited,
a two-stepapproachhasbeendevised [130] to determinethe effective propertiesof the RVE. A
schematicof thetwo-stepGMC procedureis shown in Figure7.4.
In this approach,the RVE is subdivided into a numberof subcells. The cumulative volume
fraction of particlesthat fall in eachsubcell is calculated. Eachsubcell is now assumedto be
a compositecontaininga squarearray of particles,occupying the calculatedcumulative subcell
volumefraction,in acontinuousbinder. Theeffective propertiesof eachsubcellarenext calculated
usingtheoriginal methodof cellsprocedure[56]. This procedureutilizesonesubcellto represent
theparticlesandthreesubcellsto representthebinderin a two-dimensionalcalculationusingfour
subcells.Mostsubcellsareeithercompletelyfilled with particleor binderandhenceonly a limited
numberof intermediatematerialsareproducedthathomogenizethepropertiesof areasthatcontain
bothparticlesandbinder.
Thesecondstepof thecalculationinvolvescalculatingtheeffective propertiesof the full RVE
basedon thepropertiescalculatedfor eachsubcellin thefirst step. Thegivesusa betterestimate
of the effective moduli without the full effect of stressbridging artifactscausedby discretization
errors.
126
GMC DiscretizationRVE
Homogenized RVE First Homogenization Step
Figure 7.4. Schematicof thetwo-stepGMC procedure.
127
7.1.2.3 Effective Propertiesfr om GMC
Theeffectivepropertiesof thesixmicrostructureshavebeencalculatedusingthetwoapproaches
discussedabove. Theseareshown in Table7.3.
On the average,the GMC calculationsbasedon the 50% rule predict valuesof a*bc&c that are
around2.5timestheFEM basedvalues.Surprisingly, thepredictionsof a bc(d areverycloseto those
predictedby FEM calculations,for all themodels.Thevaluesof a*be&e from GMC are,asexpected,
around10%of theFEM values.
Interestingly, thevaluesof a bc&c from the two-stepGMC calculationsshow someimprovement
over the50% rule basedGMC results.For models3 and4, thevaluesof a*bc&c and a$bc(d arewithin
5% of theFEM values.For models5 and6, thesearearound1.5 timesthefinite elementresults.
However, for models1 and2 thevalueof a*bc&c is still around2.5timesthatof theFEM basedvalues.
Thoughthereis someincreasein thevaluesof a be&e calculatedby GMC, thesearegenerallyaround
onefifth of theonescalculatedusingfinite elements.
Table7.3. Effective stiffnessfor thesix modelPBX 9501microstructuresfrom GMC calculations.
a*bc&c GMC/ a*bc(d GMC/ a$be&e GMC/f�g�h�i d�jFEM
f�g h�i d!jFEM FEM
(MPa) (MPa) (MPa)Fifty PercentRuleApproach
Model1 8.14 4.6 1.19 1.3 2.42 0.2Model2 8.07 4.5 1.12 1.3 2.28 0.2Model3 8.15 4.4 1.08 1.2 2.31 0.2Model4 1.16 0.9 1.12 1.0 2.35 0.1Model5 1.32 0.6 1.00 1.1 2.26 0.1Model6 1.32 0.6 0.93 1.2 2.14 0.3Mean 4.71 2.5 1.07 1.2 2.29 0.1Std.Dev. 3.41 0.08 0.01
Two-StepApproachModel1 4.79 2.7 1.03 1.1 2.32 0.2Model2 4.77 2.6 1.03 1.2 2.31 0.2Model3 1.93 1.0 0.89 1.0 2.22 0.1Model4 1.42 1.0 1.24 1.1 2.62 0.1Model5 3.23 1.4 1.04 1.3 2.48 0.1Model6 3.34 1.5 1.00 1.1 2.46 0.3Mean 3.25 1.7 1.04 1.1 2.40 0.1Std.Dev. 1.28 0.10 0.13
The valuesof a bkmlonqp a brms l for the six models,usingthe 50% rule approach,areshown in
Figure7.5. Fromthefigure,we canobserve that theGMC modeloverestimatesa$bc&c in models1,
2 and3. This is becauseerrorsin discretizationin thesethreemodelslead to continuousstress
128
bridgingpathsacrosstheRVE. Model 4 predictst*uv&v quiteaccuratelybecausethegeometryof the
particlesis betterrepresentedthanin models1 through3. Models5 and6 slightly underestimate
thevalueof t*uv&v . However, thereasonfor this is notobvious.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
1 2 3
45 6
1 2 3 4 5 6
1 2 3 4 5 6
C*11
C*12
C*66
C* (
GM
C −
50%
Rul
e) /
C* (F
EM
)
Figure 7.5. Ratiosof effective stiffnessescalculatedusingGMC (50%rule)andFEM.
The ratio of the effective stiffnessescalculatedby the two-stepGMC approachto thosefrom
FEM calculations,areshown in Figure7.6. Comparisonwith theresultsshown in Figure7.5shows
that the two-stepGMC calculationis an improvementover the50%rule basedGMC. Hence,this
approachis usedin all furtherGMC basedcalculations.It is alsoobservedthattheshearmoduliare
underpredictedby bothGMC approaches.Oneway of rectifying this problemwould beto usethe
shearcoupledmethodof cellsdiscussedin Chapter4.
7.1.3 RCM Calculations
Next, theeffectiveelasticmoduliwerecalculatedfor thesix modelRVEsusingRCM. TheRVE
wasdiscretizedinto w�x�y{z|w�x�y equalsizedsquares.Eachsubcellof theRVE wasassignedmaterial
propertiesusingthe50%rule. An exampleof theresultingparticledistribution for model4 is shown
in Figure7.7.Thisfigureshows thatthemicrostructureis approximatedquiteaccuratelythoughthe
edgesof theparticlesarenot smooth.
Theeffectivestiffnessfor thesix microstructures,calculatedusingRCM,areshown in Table7.4.
Ratiosof RCM to FEM resultsarealsoshown in thetable.It canbeobservedthatthevaluesof t*uv&v
129
0
0.5
1
1.5
2
2.5
1 2
3 45 6
1 23 4 5
6
1 2 3 4 56
C*11
C*12
C*66
C* (
GM
C −
Tw
o−S
tep)
/ C* (
FE
M)
Figure7.6. Ratiosof effective stiffnessescalculatedusingGMC (two-step)andFEM.
Figure7.7. Microstructureusedfor RCM calculationsfor model4.
130
predictedby RCM areabout35timesthosepredictedby FEM. For }*~�(� this ratio is about20andfor
} ~�&� theratio is around83. Theseratiosarealsoshown in Figure7.8. Interestingly, it is now model
4 thatshows thehighestnormalstiffnesswhereasFEM calculationspredictthat this modelshould
have thelowestnormalstiffnessof thesix.
Table7.4. Effective stiffnessfor thesix modelPBX 9501microstructuresfrom RCM calculations.
}$~�&� RCM/ }$~�(� RCM/ }*~�&� RCM/�����������FEM
�����������FEM
�����������FEM
(MPa) (MPa) (MPa)Model1 6.95 39 1.44 16 1.12 101Model2 5.45 30 0.84 10 0.78 67Model3 6.10 33 1.28 15 1.26 82Model4 9.20 64 2.66 23 1.97 61Model5 7.45 31 2.55 27 2.56 67Model6 6.88 30 2.11 28 2.04 242Mean 7.01 36 1.81 20 1.62 83Std.Dev. 1.18 0.67 0.61
0
50
100
150
200
250
1 2 34
5 61 2 3 4 5 6
1
23
4 5
6
C*11
C*12
C*66
C* (
RC
M)
/ C* (
FE
M)
Figure7.8. Ratiosof effective stiffnesscalculatedusingRCM andFEM.
The questionthat arisesat this point is why doesRCM predicteffective propertiesfor these
microstructuresthatare30 to 80 timeshigherthanthosepredictedby FEM. Thebestestimatesthat
131
we canexpectto obtainwith RCM arethosefrom a setof finite elementanalyseson approximate
microstructuressuchas the one shown in Figure 7.7. Thesemicrostructuresare basedon the
50% rule of assigningelementmaterialproperties. The effective propertiesfor the six models,
determinedby finite elementanalyseson microstructuresdiscretizedinto ������������� four-noded
squareelements,areshown in Table7.5. Theratio of thesevalueswith thosecalculatedby RCM
arealsoshown in thetable. We canobserve that thestandarddeviation of thevaluespredictedfor
thesix modelsis quitelargecomparedto thosefor theFEM calculationsusingtriangularelements
asshown in Table7.2.This impliesthatdiscretizationerrorscanhaveconsiderableinfluenceonthe
calculationof effective propertiesof highvolumefractionparticulatecomposites.
Table7.5. Effective stiffnessfor thesix modelPBX 9501microstructuresfrom FEM calculationsusing �����o������� squareelements.
�*��&� FEM/ RCM/�*��(� FEM/ RCM/
�*��&� FEM/ RCM/� �������!� Expt. FEM� ����� � � Expt. FEM
� ����� � � Expt. FEM(MPa) (MPa) (MPa)
Model1 1.43 0.9 5 1.90 0.2 8 5.24 0.14 21Model2 1.46 0.9 4 1.59 0.2 5 3.83 0.10 20Model3 0.21 0.1 29 0.79 0.1 16 1.90 0.05 66Model4 0.30 0.2 31 1.41 0.2 19 5.30 0.14 37Model5 1.09 0.7 7 8.66 1.0 3 73.2 1.95 3Model6 0.27 0.2 28 0.74 0.1 28 1.13 0.03 181Mean 0.79 0.5 9 2.51 0.3 7 15.1 0.40 11Std.Dev. 0.55 2.78 26.0
Figure7.9 shows the ratio of the effective stiffnessfrom FEM calculationsusing �������������squareelementsto thoseusing65,000six-nodedtriangularelements.Thefigureshows thatusing
squareelementsto discretizethe six modelsleadsto valuesof� ��&� that are,on average,around
12 times thoseobtainedfrom a FEM model that capturesthe particleboundarieswell. For�*��(�
this ratio is around10. The variationof the ratio for the six models,for� ��&� , is larger than for
�*��&� and�$��(� andvaluesrangefrom 3 to 10 with a medianof around5. This shows that errorsin
discretizationof particleboundariescanhave a disproportionateeffect of the effective properties
of high moduluscontrastcomposites.A solutionto this sproblemwould be to computeeffective
propertiesof elementsattheedgesof particlesusingamethodsimilar to thefirst stepin thetwo-step
GMC approach.Weproposeto explorethispossibilityin this research.
Table 7.5 shows that the effective stiffnessespredictedby RCM are still many times those
predictedby FEM using ������������� squareelements. For� ��&� thesevaluesrangefrom 4 times
to 31 timesthe FEM values,for� ��(� the rangeis from 3 timesto 28 timesthe FEM values,and
132
0
2
4
6
8
10
12
14
16
1 2 3
4
5 6
1 2 3
4
5
6
1 23
45
6
C*11
C*12
C*66
C* (
FE
M −
256
x256
) / C
* (F
EM
)
Figure 7.9. Ratiosof effective stiffnesscalculatedusingFEM ( �����o������� squareelements)andFEM (65,000triangularelements).
for $¡¢&¢ therangeis evenhigher, from 3 timesto 181timestheFEM values.Thesedifferencesbe
attributed to discretizationerrorsalone. We proposeto explore the reasonsfor the errorsand to
suggestwaysof improving theRCM approximationsin this research.
Preliminaryinvestigationsof thestepsin theRCM calculationssuggestthe following reasons
for theerrors:
1. At theeachlevel of recursion,ablockof 4 subcellsis homogenized.Theeffective stiffnesses
of theseblocksfrom RCM calculationsarehigherthanactualbecausea limited numberof
elementsis usedto discretizetheblocks.
2. The errorsaddat eachlevel of recursionbecauseaccuratesolutionsarenot obtained. In-
vestigationshave showed that at the level of recursionat which sixteensubcellshave been
homogenizedinto blocks,theerrorsin estimationarequitesmall.However, evenat this level,
thereareblockswhichoverestimatetheeffectivenormalmoduliby 4 to 10times.In addition,
theshearmoduli canbegrosslyoverestimatedor underestimateddependingonthegeometry.
The approximationsfrom RCM can be improved by increasingthe numberof subcellsin a
block andthenumberof elementsusedto modela block. This would leadto a betterestimateof
thepropertiesof a block. In addition,thenumberof recursionsshouldbeminimizedsothaterrors
accumulateonly minimally.
133
Thestiffnesspredictedby RCM canbereducedto someextentby usingtheninenodedformu-
lationdiscussedin Chapter5. Anotherwayto dealwith theproblemwouldbeto use£¥¤¦£ , §¨¤©§ orª<« ¤ ª<« subcellsfor eachblock in therecursivecalculation.Weproposeto exploretheseapproaches
in theremainderof this research.
7.2 Randomly GeneratedMicr ostructuresThemicrostructureof PBX 9501is shown in Figure7.10. Theparticlesareirregularly shaped
andareof a large numberof sizes.Thevolumefractionof particlesin PBX 9501is around92%.
However, if the imageis manuallyassignedmaterialsthehighestvolumefraction that is obtained
is around70%. Therefore,a digital imageis not the bestpossiblesourcefor the generationof
microstructures.Instead,we try to simplify the shapeof the particlesandautomaticallygenerate
particledistributionsthataremodelsof theactualmicrostructure.
Figure7.10. Microstructureof PBX 9501[19].
The preferredmethodfor generatingclosepacked microstructuresfrom a set of particlesis
to useMonteCarlobasedmoleculardynamicstechniques[131] or Newtonianmotionbasedtech-
niques[132]. In thismethod,adistributionof particlesis allocatedto thegrid pointsof arectangular
latticeusinga randomplacementmethod.Moleculardynamicssimulationsarethencarriedout on
134
thesystemof particlesto reachthepackingfraction that correspondsto equilibrium. A weighted
Voronoi tessellation[133] is thencarriedout on the particleswith the weightsdeterminedby the
sizesof theparticles.Theparticlesarenext movedtowardsthecenterof thepackingvolumewhile
maintainingthat they remaininsidetheir respective Voronoi cells. This processis repeateduntil
all the particlesare as tightly packed as possible. For our purposes,periodicity of the particles
at the boundarieshasto be maintained.This is usuallydoneby specifyingextra particlesat the
boundariesthatmovein andoutof thevolume.Thisprocess,with somemodifications,hasbeenthe
only efficientmethodof generatingclosepackedsystemsof particlesin threedimensions.However,
it is difficult to getpackingfractionsof morethan70-75%whenusingsphericalparticles.It is also
quitetimeconsumingto generatetight packing.
Sincemostof our studieshave beenin two dimensionswe usea fastermethodfor generating
particlepackings- randomsequentialpacking.The largestparticlesarefirst placedin thevolume
followed by progressively smallerparticles. If thereis any overlap betweena new particle and
theexisting set,thenew particleis moved to a new position. If a particlecannotbeplacedin the
volumeaftera certainnumberof iterations,thenext lower sizedparticleis chosenandtheprocess
is continueduntil therequiredvolumefraction is achieved. Thoughthis methoddoesnot preserve
theparticlesizedistributionsasaccuratelyastheMonteCarlobasedmoleculardynamicsmethods,
it is much fasterand canbe usedto generatehigh packingfractionsin two dimensionswithout
particlelocking. In threedimensions,this methodis highly inefficient andpackingsabove 60%are
extremelytimeconsumingto achieve.
7.2.1 Cir cular Particles - PBX 9501Dry Blend
Thedry blendof PBX 9501hastheparticlesizedistribution shown in Table2.3.Thecoarseand
thefine particlesareblendedin a ratio of 1:3 by weightandcompactedto generatethedistribution
of particlesizesshown for thepressedpiecein Figure2.3. In thissection,circularparticlesareused
to approximatethemicrostructureof PBX 9501.
Four microstructuresbasedon theparticlesizedistribution of thedry blendareshown in Fig-
ure 7.11. The numberof particlesusedfor the four microstructuresare100, 200, 300 and 400
respectively. The particlesoccupy a volume fraction of about 85-86%. The RVE widths are
650 ¬ m, 940 ¬ m, 1130 ¬ m and 1325 ¬ m, respectively. The remainingvolume of particlesis
assumedto beoccupiedby finesthatarewell separatedin sizefrom thesmallestparticlesusedin
themicrostructure.However, microstructurescontaininggreaterthan86%by volumeof particles
areextremelytimeconsumingto generatefrom theparticlesizedistribution of thedry blend.
The materialpropertiesof the constituentsare given in Table 7.1. Sinceabout92% of the
total volumeis occupiedby particlesin PBX 9501andthe samplemicrostructuresarefilled only
135
100 Particles 200 Particles
300 Particles 400 Particles
Figure 7.11. Microstructuresusingcircularparticlesbasedon thedry blendof PBX 9501.
136
up to 86%, we assumethat the binder is “dirty”, that is, it containsaround36% of particlesby
volume. The effective elasticpropertiesof the “dirty” binderarecalculatedusingthe differential
effective mediumapproximationfor avolumefractionof 36%of particles.For thematerialsshown
in Table7.1,thepropertiesof the“dirty” binderare ¯®±°³²µ´·¶¹¸<º and »_®±´¹²½¼¾¶¹¸<¿ . Theseproperties
areusedinsteadof theusualbinderpropertiesin theGMC, RCM andFEM calculations.
For theFEM calculations,theRVEs werediscretizedinto °�À�º|Á°�À�º squareelementsandthe
50% rule was applied to assignmaterialsto elements. This was requiredbecauseof the close
proximity of most particles. The RCM calculationsusedthe samediscretizationas the FEM
calculations.Figure7.12showstheapproximatemicrostructure,for the100particlemodel,usedfor
theFEM andRCM calculations.TheGMC calculationswerecarriedout with ¸�´�´ÃÁĸ�´�´ subcells
usingthetwo-stepprocessdiscussedin theprevioussection.
Figure7.12. Approximatemicrostructureusedfor FEM andRCM calculationson the100particlemodelof PBX 9501basedon thedry blend.
Theeffective propertiesof thefour models,calculatedusingFEM, GMC, andRCM, areshown
in Table7.6. On theaverage,thefour microstructuresexhibit squaresymmetry. TheFEM calcula-
tionsoverestimatetheeffectivepropertiesof PBX 9501by 1.2to 3 times.Thevaluesdeterminedby
RCM arearound2 timestheFEM valuesfor thenormaldirectionsandaround0.6 timestheFEM
137
calculationsfor shear. GMC, asseenbefore,predictsvaluesmuchlower thanthosefrom FEM.This
is probablybecausestressbridgesarenotcapturedaccuratelyby GMC.
Table7.6. Effective stiffnessfor thefour modelPBX 9501microstructuresbasedon thedry blendof PBX 9501.
FEM CalculationsRVE No. of Å*ÆÇ&Ç FEM/ Å*ÆÈ&È FEM/ Å$ÆÇ(È FEM/ Å*ÆÉ&É FEM/
Parts. Ê�Ë�Ì�Í�Î�Ï Expt. Ê�Ë Ì�Í�Î!Ï Expt. Ê�Ë�Ì�Í�Î�Ï Expt. Ê�Ë�Ì�Í�Î�Ï Expt.(MPa) (MPa) (MPa) (MPa)
1 100 2.41 1.5 2.12 1.3 0.66 0.8 0.79 2.12 200 3.63 2.2 1.65 1.0 0.65 0.7 0.75 2.03 300 2.54 1.6 3.39 2.1 1.15 1.3 1.32 3.54 400 5.28 3.2 5.13 3.2 1.73 2.0 1.70 4.5Mean 3.46 2.1 3.07 1.9 1.05 1.2 1.14 3.0
GMC CalculationsRVE No. of Å*ÆÇ&Ç GMC/ Å*ÆÈ&È GMC/ Å$ÆÇ(È GMC/ Å*ÆÉ&É GMC/
Parts. Ê�Ë�Ì�Í È Ï FEM Ê�Ë Ì�Í È Ï FEM Ê�Ë�Ì�Í È Ï FEM FEM(MPa) (MPa) (MPa) (MPa)
1 100 1.52 0.06 1.48 0.07 1.22 0.2 4.9 0.0062 200 1.46 0.04 1.44 0.09 1.22 0.2 4.9 0.0073 300 1.49 0.06 1.48 0.04 1.25 0.1 5.0 0.0044 400 1.44 0.03 1.46 0.03 1.20 0.1 4.8 0.003Mean 1.48 0.04 1.47 0.05 1.22 0.12 4.9 0.004
RCM CalculationsRVE No. of Å ÆÇ&Ç GMC/ Å ÆÈ&È GMC/ Å ÆÇ(È GMC/ Å ÆÉ&É GMC/
Parts. Ê�Ë�Ì�Í Î Ï FEM Ê�Ë Ì�Í Î Ï FEM Ê�Ë�Ì�Í Î Ï FEM Ê�Ë�Ì�Í Î Ï FEM(MPa) (MPa) (MPa) (MPa)
1 100 6.46 2.7 6.56 3.1 1.31 2.0 0.54 0.72 200 7.84 2.2 7.28 4.4 1.68 2.6 0.49 0.73 300 7.56 3.0 7.83 2.3 1.77 1.5 0.69 0.54 400 8.85 1.7 8.79 1.7 2.26 1.3 1.09 0.6Mean 7.68 2.2 7.62 2.5 1.75 1.7 0.70 0.6
7.2.1.1 FEM Calculations
Plotsof thecomponentsof theeffectivestiffnessmatrix for thefour microstructures,from FEM
calculations,areshown in Figure7.13.Theratiosof thesecomponentsto experimentaldataonPBX
9501arealsoshown in thefigure.
As mentionedearlier, aregulargrid is usedfor theFEM calculationsbecausethemeshgenerator
createspoorelementswhenwe attemptto discretizetheparticleboundariesaccurately. It hasbeen
138
0
1000
2000
3000
4000
5000
1
2
3
4
12
3
4
1 23
4
1 23
4
C*11
C*22
C*12
C*66
C* (
FE
M)
(MP
a)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
1
2
3
4
12
3
4
1 23
4 1 2
3
4
C*11
C*22
C*12
C*66
C* (
FE
M)
/ C* (
PB
X 9
501)
Figure7.13. Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from FEM calculations.
139
observed,for thesix manuallygeneratedmicrostructuresdiscussedpreviously, thattheregulargrid
leadsto a stiffer responsebecausesomecontactartifactsaregeneratedin the process.The FEM
calculationson thefour microstructuresrepresentative of thedry blendof PBX 9501are,therefore,
expectedto predicthigherthanactualvaluesof effective stiffness.
Fromtheplotsof theeffectivestiffnesspredictedby FEM in Figure7.13,it canbeobservedthat
theeffectivestiffnessincreases,onaverage,asthenumberof particlesin theRVE increases.This is
becausethegrid sizeis thesamefor thefour microstructuresandartificial contactsincreaseasthe
numberof particlesincreases.
For a particulatecomposite,anacceptableRVE is onethatexhibits isotropy or, at least,square
symmetryin two dimensions.The FEM calculationsshow that the RVE containing400 particles
is the closestto exhibiting squaresymmetry. Therefore,the 400 particle microstructurecan be
consideredmost representative. However, discretizationerrorsare, presumably, the largest for
this microstructure.The optimummicrostructurewould thereforebe onethat canbe discretized
accuratelyaswell asonethatexhibitsbehavior closeto squaresymmetry. If we take theaverageofÐ*ÑÒ&Ò and
Ð*ÑÓ&Ó andassignthisaveragevalueto bothÐ*ÑÒ&Ò and
Ð$ÑÓ&Ó beforecalculatingÔ Ñ , Õ Ò×Ö and Õ ÓØÖ ;we find thatthefour modelsareall remarkablycloseto beingisotropic.
TheFEM calculationspredictvaluesofÐ ÑÒ&Ò and
Ð ÑÓ&Ó thatareabout1 to 3 timestheexperimental
valuesfor PBX9501.Theshearstiffnessespredictedby FEM arearound2 to5 timestheexperimen-
tal valueswhile theÐ ÑÒ(Ó valuesarearound1 to 2 timestheexperimentalvalues.This is encouraging
becausetwo-dimensionalmodelsof particulatecompositesareexpectedto predicthigherstiffnesses
thanthree-dimensionalmodelsandtheexperimentaldataarefor three-dimensionalmaterials.
The experimentallydeterminedpropertiesof PBX 9501 are approximatevalues,as are the
materialpropertiesof the components.Therefore,therewill alwaysbe somedifferencebetween
numericalresultsandexperimentaldata.Thefinite elementsolutionsprovide thebestapproxima-
tionsfor thegivenmicrostructuresandmaterialproperties.
7.2.1.2 GMC Calculations
Figure7.14shows thecomponentsof theeffective stiffnessmatrix calculatedusingGMC and
the ratio of thesevaluesto thosecalculatedusingFEM. It canbe seenthat quite low valuesare
predictedby GMC for all thestiffnessterms.
The effective compliancespredictedby GMC are, however, relatively uniform for the four
microstructures.The valuesofÐ ÑÒ&Ò and
Ð ÑÓ&Ó arecloseto 150 MPa,Ð ÑÒ(Ó is around120 MPa and
Ð ÑÙ&Ù is around5 MPa. Inadequateaccountingfor stressbridgingcausesGMC to predictvaluesofÐ ÑÒ&Ò and
Ð ÑÓ&Ó thatareabout0.05timestheFEM results.
140
0
20
40
60
80
100
120
140
160
1 2 3 4 1 2 3 4
1 2 3 4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
GM
C)
(MP
a)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
12
3
4
12
34
1 2
3
4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
GM
C)
/ C* (
FE
M)
Figure7.14. Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from GMC calculations.
141
As expected,thevaluesof Ú*ÛÜ&Ü predictedby GMC arealsoquitelow comparedto theFEM pre-
dictions.Thereasonis thelack of couplingbetweenthetensileandshearbehaviors. Improvement
canbeobtainedby usingtheshearcoupledGMC formulationdiscussedin Chapter4. However, the
considerablylarger matricesgeneratedby the shearcoupledmethodof cells detractconsiderably
from theimprovementof efficiency over FEM thatis soughtby thismethod.
7.2.1.3 RCM Calculations
Theeffective stiffnessesfrom RCM calculationsandthecorrespondingratioswith valuesfrom
FEM calculationsareshown in Figure7.15.RCM appearsto performmuchbetterin this casethan
for themanuallydesignedmicrostructuresdiscussedbefore.
Valuesof Ú ÛÝ&Ý predictedby RCM, for thefour dry blendbasedmodels,vary from 6500MPa to
8500MPaor around1.7to 3 timestheFEM values.Similarvaluesareobtainedfor Ú*ÛÞ&Þ . Thevalues
of Ú*ÛÝ(Þ rangefrom about1500MPa to 2500MPa, or about1.5 to 2.5 timestheFEM values.The
shearmodulus Ú ÛÜ&Ü variesfrom 500MPa to 1000MPa andit generallyaround0.6 timestheFEM
values(andthereforecloserto theexperimentaldata).
The reasonsfor the higherstiffnesspredictedby RCM have beendiscussedfor the manually
generatedmicrostructuresandthesamereasonshold for thefour dry blendbasedmicrostructures.
Thepredictionscanbe improved by finer discretizationat eachlevel of recursionandusingfewer
recursionssothaterrorsdonotaccumulate.
It is alsoobservedthatthepredictedstiffnessincreaseswith increasingnumberof particlesin a
RVE. This is causedby artificial contactscreateddueto errorsin discretization.Theseerrorscan
be reducedto someextent if, insteadof usingthe 50% rule for assigningmaterialsto subcells,a
two-stepprocedureis utilized,similar to theoneusedfor theGMC calculations.Theeasiestwayof
implementingthis procedurewould beto calculatethevolumefractionof particlesin eachsubcell
andthento usetheeffective mediumapproximation(alsousedfor the ’dirty’ binder)to calculate
theeffective propertiesof subcellsthatarenotall particleor binder.
7.2.2 Cir cular Particles - PBX 9501PressedPiece
Fourparticledistributionsgeneratedonthebasisof theparticlesizedistribution in pressedPBX
9501areshown in Figure7.16.Thepressingprocessleadsto particlebreakageandhencethelarger
volumefractionof smallersizedparticles.A few largersizedparticlesremainandthis is reflectedin
thegeneratedmicrostructurescontaining100,200,500,and1000particles.Thesizesof theRVEs
aresmallerthanthosefor the dry blend,for the samenumberof particles. In this casethe RVE
widthsare360 ß m, 420 ß m, 535 ß m, and680 ß m, respectively. Thus,the1000particlebasedRVE
for thepressedpiecehasdimensionssimilar to the100particlebasedRVE for thedry blendshown
142
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
1
2 3
4
12
34
1 2 3 4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
RC
M)
(MP
a)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
12
3
4
1
2
34
12
3 4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
RC
M)
/ C* (
FE
M)
Figure7.15. Effective stiffnessmatrix componentsfor microstructuresbasedon thedry blendofPBX 9501from RCM calculations.
143
in Figure7.11. Thesizeof theRVE thatcanbeadequatelydiscretizedis thereforesmallerfor the
pressedpiece.
100 Particles 200 Particles
500 Particles 1000 Particles
Figure 7.16. Microstructuresusingcircularparticlesbasedon thepressedpiecesizedistribution of PBX 9501.
The target volume fraction of 91-92%is not attainedin any of the distributions. A “dirty”
binder, whosepropertiesarecalculatedusingthedifferentialeffective mediumapproach,is usedin
theeffective stiffnesscalculations.Theeffective moduli of the “dirty” binderfor the four models
areshown in Table7.7.
EachRVE wasdiscretizedinto à�á�â�ã`à�á�â elementsfor the FEM andRCM calculations,and
into ä�å�åæãÄä�å�å subcellsfor theGMC calculations.Theapproximatemicrostructureusedby FEM
andRCM for the1000particlemodelis shown in Figure7.17.
Theapproximatemicrostructuresfor FEM andRCM calculationswerebasedon the50%rule,
with the original binder being replacedby the “dirty” binder. The GMC calculationsusedthe
144
Table7.7. Volumefractionsof particlesandmoduli of the“dirty” binderfor thefour pressedpiecebasedPBX microstructures.
Model No. of ç�è ç�è Propertiesof “dirty” binderParticles in binder E (MPa) é
1 100 0.89 0.28 1.583 0.4842 200 0.87 0.37 2.1395 0.4813 500 0.86 0.43 2.713 0.4784 1000 0.855 0.45 2.952 0.477
Figure7.17. Approximatemicrostructurefor the1000particlemodelof PBX 9501.
two-stepapproachdiscussedpreviously. The effective propertiescalculatedfor the four models
usingFEM, GMC andRCM areshown in Table7.8.
7.2.2.1 FEM Calculations
Plotsof thecomponentsof theeffective stiffnessmatrix for the four pressedpiece(PP)based
microstructures,from FEM calculations,areshown in Figure7.18.Theratiosof thesecomponents
to experimentaldatafor PBX 9501arealsoshown in thefigure.
Comparisonof theeffective properties,for thePP-basedmicrostructures,with thosefor thedry
blend(DB) basedmicrostructures(shown in Figure7.13)shows that thePP-basedmodelspredict
a stiffer responsethan the DB-basedmodels. In addition, the PP-basedmodelsshow increasing
effective moduliwith increasingnumberof particles.
145
0
1000
2000
3000
4000
5000
6000
7000
8000
12
34
12
3
4
1 23
4
1 23
4
C*11
C*22
C*12
C*66
C* (
FE
M)
(MP
a)
0
1
2
3
4
5
6
7
12
34
12
34
1 2
34
1 2
3
4
C*11
C*22
C*12
C*66
C* (
FE
M)
/ C* (
PB
X 9
501)
Figure7.18. Effective stiffnessmatrix componentsfor microstructuresbasedonpressedPBX9501from FEM calculations.
146
Table7.8. Effective stiffnessfor thefour modelPBX 9501microstructuresbasedon thepressedpieceof PBX 9501.
FEM CalculationsRVE No. of ê*ëì&ì FEM/ ê*ëí&í FEM/ ê$ëì(í FEM/ ê*ëî&î FEM/
Parts. ï�ð�ñ�ò�ó�ô Expt. ï�ð ñ�ò�ó!ô Expt. ï�ð�ñ�ò�ó�ô Expt. ï�ð�ñ�ò�ó�ô Expt.(MPa) (MPa) (MPa) (MPa)
1 100 3.18 2.0 3.59 2.2 1.00 1.1 1.26 3.42 200 3.90 2.4 2.69 1.7 1.05 1.2 1.23 3.33 500 6.31 3.9 6.23 3.8 2.07 2.4 2.08 5.54 1000 7.36 4.5 7.59 4.7 2.57 2.9 2.54 6.8Mean 5.19 3.2 5.03 3.1 1.67 1.9 1.78 4.7
GMC CalculationsRVE No. of ê*ëì&ì GMC/ ê*ëí&í GMC/ ê$ëì(í GMC/ ê*ëî&î GMC/
Parts. ï�ð�ñ�ò í ô FEM ï�ð ñ�ò í ô FEM ï�ð�ñ�ò í ô FEM FEM(MPa) (MPa) (MPa) (MPa)
1 100 1.80 0.06 1.88 0.05 1.31 0.13 4.77 0.0042 200 1.70 0.04 1.90 0.07 1.32 0.13 5.71 0.0053 500 1.81 0.03 1.81 0.03 1.33 0.06 6.57 0.0034 1000 1.82 0.02 1.86 0.02 1.29 0.05 6.91 0.003Mean 1.78 0.03 1.86 0.04 1.31 0.08 5.99 0.003
RCM CalculationsRVE No. of ê*ëì&ì GMC/ ê*ëí&í GMC/ ê$ëì(í GMC/ ê*ëî&î GMC/
Parts. ï�ð�ñ�ò ó ô FEM ï�ð ñ�ò ó ô FEM ï�ð�ñ�ò ó ô FEM ï�ð�ñ�ò ó ô FEM(MPa) (MPa) (MPa) (MPa)
1 100 9.22 2.9 9.34 2.6 2.42 2.4 0.91 0.72 200 7.05 1.8 8.31 3.1 1.69 1.6 0.66 0.53 500 9.93 1.6 9.91 1.6 2.76 1.3 1.20 0.64 1000 10.3 1.4 10.2 1.3 2.94 1.1 1.41 0.6Mean 9.11 1.8 9.44 1.9 2.46 1.5 1.04 0.6
For the100and200particlePP-basedmodels,thesinglelargeparticlecontributesconsiderably
to thestiffer response.For the500and1000particlePP-basedmodels,errorsin thediscretization
of particle boundarieslead to artificial stressbridging pathsandhencea stiffer responseis pre-
dicted. The larger RVEs aremorerepresentative of PBX 9501,andit is believed that improving
thediscretizationof thesemodelswill provide a betterestimateof theeffective propertiesof PBX
9501. It shouldalsobe notedthat thesizedistribution of the pressedpieceof PBX 9501is more
representative of the actualmicrostructurethan the dry blend size distribution. Therefore,we
proposeto obtain improved estimatesof the effective propertiesof the PP-basedmicrostructures
usingFEM calculationson afiner regulargrid.
Thevaluesof ê*ëì&ì and ê*ëí&í for the500and1000particlemodelsarequitecloseto eachother. It
147
hasalsobeenfoundthatthevaluesof õ÷ö×ø and õúùØø for thesemodelsarealmostequal.This implies
thatthesePP-basedmicrostructuresexhibit approximatelyisotropicbehavior.
The FEM calculationson the PP-basedmodelspredictvaluesof û*üö&ö areare2 to 5 timesthe
experimentalvaluesfor PBX 9501.Thevaluesof û üý&ý are3 to 7 timestheexperimentalvalues.In
additionto discretizationerrors,thesediscrepanciesmaybecausedby thefactthatour calculations
aretwo-dimensionalandhencestiffer. Therecould alsobe voids,cracksandinterfacial debonds,
whichwe donot considerin thesecalculations,thatreducethestiffnessof PBX 9501.
7.2.2.2 GMC Calculations
Figure7.19shows thecomponentsof theeffective stiffnessmatrix calculatedfor thePP-based
microstructuresusingGMC.
0
20
40
60
80
100
120
140
160
180
200
1 23 4 1 2 3 4
1 2 3 4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
GM
C)
(MP
a)
Figure7.19. Effective stiffnessmatrix componentsfor microstructuresbasedonpressedPBX9501from GMC calculations.
The GMC basedcalculationsgive valuesof û*üö&ö and û*üù&ù of around180 MPa while û*üö(ù is
around130 MPa. Thesevaluesareslightly higherthanthoseobtainedfrom the dry blendbased
microstructures.Thevaluesof û*üý&ý areagainquitelow - between5 MPaand6 MPa. Thenumberof
particlesin thePP-basedmodelsdoesnot appearto have muchinfluenceof theeffective properties
generatedby GMC. On the whole, thereis very little differencebetweenthe effective properties
predictedby GMC for thePP-basedmodelsandtheDB-basedmodels.
148
7.2.2.3 RCM Calculations
Theeffectivestiffnessesfrom RCM calculationsonthePP-basedmodelsandthecorresponding
ratioswith valuesfrom FEM calculationsareshown in Figure7.20.
0
2000
4000
6000
8000
10000
1
2
3 41
2
3 4
12
3 4
1 2 3 4
C*11
C*22
C*12
C*66
C* (
RC
M)
(MP
a)
0
0.5
1
1.5
2
2.5
3
1
23
4
1
2
34
1
23
41
2 3 4
C*11
C*22
C*12
C*66
C* (
RC
M)
/ C* (
FE
M)
Figure7.20. Effective stiffnessmatrix componentsfor microstructuresbasedonpressedPBX9501from RCM calculations.
For the 100 particlePP-basedmodel, RCM predictsa valueof 9,000MPa for þ$ÿ��� which is
around3 timesthe FEM prediction. A considerablylower valueof þ ÿ��� is predictedfor the 200
particlemodel - around7,000MPa. This value is about1.7 times the valuepredictedby FEM.
149
For the 500 and1000 particlemodels,RCM predictsvaluesof������ that areabout10,000MPa.
Thoughthesevaluesarehigh, they arecloserto the valuespredictedby FEM calculationsthan
for the smaller RVEs. Exceptingthe 200 particle model, all the modelspredict behavior that
closely approximatessquaresymmetrythoughthe effective behavior is not particularly closeto
beingtransverselyisotropic.
Theeffective shearmodulus������ shows anincreasewith increasingnumberof particles(except
for the 200 particlemodel). The valuespredictedby RCM arelower thanthosepredictedby the
FEM calculations.
It is believed that a two-stepcalculationof the effective properties,similar to that carriedout
in GMC, would serve to reducetheartificial stressbridgingthat leadsto thepredictionof a stiffer
responseby RCM.
7.2.3 Square Particles - PressedPBX 9501
With the increasein particle volume fraction, the numberof contactingparticlesincreases.
If the particlesare circular, the geometryof the particlescannotbe representedaccuratelyby a
rectangulargrid. On theotherhand,if trianglesareusedto discretizetheRVE, thetrianglesclose
to the contactingregionsarepoorly shaped.Finite elementcalculationson suchmeshesproduce
largenumericalerrors.To eliminatethisproblem,we have createddistributionsof squareparticles,
alignedwith therectangulargrid, to representthemicrostructureof PBX 9501.
Theparticlesshown in thethreemicrostructuresin Figure7.21arebasedonthesizedistribution
of pressedPBX 9501. Thesedistributionshave beencreatedby first overlayinga squaregrid on
theRVE andthenplacingparticlessequentiallyin theRVE sothatthey fit to thegrid. Thesmallest
particlesoccupy a single subcellof the grid. Larger particlesare chosenfrom the particle size
distributionsothatthey fit into anintegermultipleof thegrid size.Particlegeometriescantherefore
beexactly representedif theappropriatesquaregrid is chosen.
In the threemodelsshown in Figure7.21, the particlesizedistribution for the pressedpiece
is truncatedso that the largest(360 � m andhigher)andthe smallest(30 � m andlower) particle
sizesin the distribution arenot used. The RVEs arefilled with particlesto volume fractionsof
about86-87%.Theremainingvolumeis assumedto beoccupiedby a “dirty” binder. Thenumber
of particlesin the first model is 700 andthe RVE width is around3,600 � m. The secondmodel
contains2,800particlesand the RVE width is about5,300 � m, while the third model contains
11,600particlesandhasa width of 9,000 � m. TheseRVEs are,therefore,considerablylarger than
thoseusedfor thecircularparticles.SmallerRVEsarenotusedbecauseof thedifficultiesassociated
with fitting particlesinto integermultiplesof subcellwidths.
150
700 Particles 2800 Particles
11600 Particles
Figure 7.21. Microstructuresusingsquareparticlesbasedon thepressedpiecesizedistribution of PBX 9501.
The materialpropertiesof the particlesare thoseshown in Table7.1. The propertiesof the
“dirty” binderfor thethreemodelsareshown in Table7.9.
Table7.9. Moduli of the“dirty” binderfor thethreePBX microstructureswith squareparticles.
Model No. of � � Propertiesof “dirty” binderParticles in binder E (MPa) �
1 700 0.868 0.39 2.358 0.47992 2800 0.866 0.40 2.448 0.47953 11600 0.863 0.42 2.588 0.4788
151
Finite element(FEM) andRCM calculationswereperformedon thethreemodelsusingregular
grids of �� ������� �� squareelements. GMC calculationson the threemodelsusedthe two-step
approachon a ��������� grid. The effective propertiescalculatedusingFEM, GMC andRCM are
shown in Table7.10.
Table7.10. Effective stiffnessfor thethreepressedPBX 9501modelmicrostructurescontainingsquareparticles.
FEM CalculationsRVE No. of ������ FEM/ ������ FEM/ ������ FEM/ ������ FEM/
Parts. ��! "��#%$ Expt. &�' "��#($ Expt. &�! "��)*$ Expt. &�! "��)*$ Expt.(MPa) (MPa) (MPa) (MPa)
1 700 1.10 6.8 1.15 7.1 4.04 4.6 3.14 8.42 2800 1.12 6.9 1.13 7.0 4.13 4.7 3.30 8.83 11600 1.22 7.5 1.20 7.4 4.58 5.2 3.50 9.3Mean 1.15 7.1 1.16 7.1 4.25 4.9 3.31 8.8
GMC CalculationsRVE No. of � ���� GMC/ � ���� GMC/ � ���� GMC/ � ���� GMC/
Parts. &�! "� � $ FEM &�' "� � $ FEM &�! "� � $ FEM FEM(MPa) (MPa) (MPa) (MPa)
1 700 1.68 0.015 1.70 0.015 1.38 0.03 6.05 0.0022 2800 1.74 0.016 1.76 0.016 1.34 0.03 6.18 0.0023 11600 1.93 0.016 1.93 0.016 1.17 0.03 6.39 0.002Mean 1.79 0.016 1.80 0.016 1.30 0.03 6.21 0.002
RCM CalculationsRVE No. of ������ GMC/ ������ GMC/ ������ GMC/ ������ GMC/
Parts. &�! "��#%$ FEM &�' "��#($ FEM &�! "��)*$ FEM &�! "��)*$ FEM(MPa) (MPa) (MPa) (MPa)
1 700 1.26 1.1 1.30 1.1 4.28 1.1 2.16 0.72 2800 1.26 1.1 1.29 1.1 4.32 1.0 2.38 0.73 11600 1.27 1.0 1.25 1.0 4.22 0.9 2.29 0.7Mean 1.26 1.1 1.28 1.1 4.27 1.0 2.27 0.7
7.2.3.1 FEM Calculations
Theeffectivepropertiesfor thethreemodels,from FEM calculations,areshown in Figure7.22.
The figure alsoshows ratiosof FEM-basedvaluesto experimentaldataon PBX 9501. It canbe
seenthat,even thoughthemicrostructureis modeledaccurately, thepredictedeffective properties
increaseas the size of the RVE increases.This suggeststhat the actual representative volume
elementmaybeevenlargerthanthosemodeled.
152
0
2000
4000
6000
8000
10000
12000
1 23
1 2 3
1 2 31 2 3
C*11
C*22
C*12
C*66
C* (
FE
M)
(MP
a)
0
1
2
3
4
5
6
7
8
9
10
1 23 1 2 3
1 23
1 23
C*11
C*22
C*12
C*66
C* (
FE
M)
/ C* (
PB
X 9
501)
Figure7.22. Effective stiffnessmatrix componentsfrom FEM calculationsfor microstructurescontainingsquareparticles.
153
Thevaluesof +�,-�- and +�,.�. arecloseto eachother, meaningthat squaresymmetryis obtained
from theparticledistributions.However, thevaluesof + ,-�- , at11,000MPato 12,000MPa,areabout
7 timeshigherthat theexperimentalvaluesfor PBX 9501. The valuesof +�,-�. arearound5 times
that of PBX 9501andthe valuesof + ,/�/ arearound9 timesthat of PBX 9501. Thesevaluesare
higherthanthoseobtainedfrom thecircularparticledistributions.
The higherstiffnessobtainedfrom the FEM calculationsis partly becausea 0�1�24350�1�2 grid
that is alignedwith theparticleshasbeenusedto discretizethegeometry. As a result,theparticle
boundariesarenot“seen”by thesemethodsandthecomputedresultsareessentiallyfor acontinuous
particlephasecontainingpocketsof binder. This canbeobserved from thegeometry, for the700
particlemodel,shown in Figure7.23.
Figure 7.23. Microstructurefor the700particlemodelof PBX 9501usingsquare,alignedparticles.
Theproblemcanberesolvedif theparticlephaseis notmodeledasacontinuousmaterial.This
canbeachieved by modelingparticleinterfaceswith a zerovolumeinterfaceelementthathasthe
stiffnessof thebinder. Theinterfaceelementcanbeusedto connectadjacentelementsthatarenot
perfectlybondedsincethey arepartof differentparticles.
7.2.3.2 GMC Calculations
Effective propertiespredictedby GMC, for the threemodels,areshown in Figure7.24. The
valuesof +�,-�- and +�,.�. arearound180MPa, +�,-�. is around140MPaand +�,/�/ is around6 MPa.
154
0
20
40
60
80
100
120
140
160
180
200
1 23
1 23
1 23
1 2 3
C*11
C*22
C*12
C*66
C* (
GM
C)
(MP
a)
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
1 2 3 1 2 3
1 2
3
1 2 3
C*11
C*22
C*12
C*66
C* (
GM
C)
/ C* (
FE
M)
Figure7.24. Effective stiffnessmatrix componentsfrom GMC calculationsfor microstructurescontainingsquareparticles.
155
If wecomparetheeffectivepropertiespredictedby GMC with thosefrom FEM calculations,we
seethatGMC predictsvaluesthatarearound1.5%of theFEM valuesfor 6�78�8 and 6�79�9 . The 6�78�9valuesarerelatively higher- ataround3%of theFEM values.Thevaluesof 6 7:�: areabout0.2%of
theFEM predictions.
Thereasonfor theselow valuesis thattherearenocontinuousstressbridgesin theRVE thatare
alignedwith theaxesof theRVE. Hence,GMC doesnot take theinteractionsbetweensubcellsinto
accountaccuratelyenough,leadingto low stiffnesspredictions.
7.2.3.3 RCM Calculations
Interestingly, theeffective propertiespredictedfor thethreemodelsby RCM arequitecloseto
thosepredictedby FEM. Thiscanbeseenfrom Figure7.25.
The predictedvaluesof 6 78�8 and 6 79�9 arearound13,000MPa. Thesearealmost11 timesthe
experimentalvaluesfor PBX 9501but only around10%larger thanthevaluespredictedby FEM.
Thevaluesof 6�78�9 arearound4,500MPa - around1.05timestheFEM values.On theotherhand,
thepredictedshearmodulus6 7:�: is around2,000MPa - 5 timestheexperimentalvaluesandlower
thantheFEM predictions(about70%of theFEM values).
Theincorporationof aninterfaceelementin RCMwouldserveto reducetheeffectivestiffnessof
theRVEs containingsquareparticlesby allowing greaterdisplacement.A joint element,described
by Beer [134], canbe usedto model the effect of a thin layer of binderbetweenparticleswhen
square,alignedparticlesareusedto modela PBX in two dimensions.However, the complexity
involved in modelinginterfaceelementscould leadto lossof computationalefficiency andhence
will notbeexploredfurtherin this research.
156
0
2000
4000
6000
8000
10000
12000
14000
1 2 3 1 2 3
1 2 3
1 2 3
C*11
C*22
C*12
C*66
C* (
RC
M)
(MP
a)
0
0.2
0.4
0.6
0.8
1
1.2
1 23
1 23 1 2
3
1 23
C*11
C*22
C*12
C*66
C* (
RC
M)
/ C* (
FE
M)
Figure 7.25. Effective stiffnessmatrix componentsfrom RCM calculationsfor microstructurescontainingsquareparticles.
CHAPTER 8
PROPOSEDRESEARCH
Thegoalof this researchis to explorecomputationallyinexpensive techniquesof determining
thermoelasticpropertiesof PBXs. To date,the GMC techniquehasbeenstudiedin somedetail.
Additionally, a new RCM techniquehasbeendevelopedfor two-dimensionalanalysis.Both these
methodsshow promisefor determiningeffective properties,but arenotsufficiently accuratein their
presentform. The proposedresearchprogramwill attemptto remedythe presentshortcomings
of RCM. Themethodwill thenbe usedto predictthebehavior of PBX-like materialsat different
temperaturesandstrainrates.
8.1 Curr ent Statusof ResearchIn this research,datahave beencollectedon thebehavior of PBX 9501andit’s componentsat
varioustemperaturesandstrainrates.Dataon mockpropellants,containingglassbeadsandsugar,
have alsobeencollected.
Variousmethodsof determiningthe thermoelasticpropertiesof two-componentcomposites
have beenstudied. Rigorouslydeterminedboundson the effective propertieson PBX 9501have
beenfound to be too far apartto be of practicaluse,except for the boundson the coefficientsof
thermalexpansion.Analyticalmodels,whenappliedto PBX 9501,predictvaluesof effectiveelastic
propertiesthatareeithertoolow or toohightobeof use.Numericaltechniquesfor thedetermination
of effective elasticpropertiesof materialssuchasPBX 9501,tendto betime consuming.This has
led to theinvestigationof computationallyinexpensive numericaltechniques.
Thegeneralizedmethodof cells (GMC) techniquehasbeenexploredbecauseof it’s computa-
tional efficiency andaccuracy underspecialcircumstances.The discovery of situationsin which
GMC is notaccurateenoughhasled to thedevelopmentof therecursive cellsmethod(RCM).
The effective propertiescalculatedfrom GMC and RCM have beenvalidatedwith detailed
finite element(FEM) calculations.Someexact relationsfor theeffective propertiesof composites
have beenusedto validatethe approachtaken for the finite elementcalculations. Additionally,
someaccuratenumericalcalculationsby other researchershave beenusedto validatethe FEM
calculations.TheGMC andRCM calculationshavebeenfoundto bequiteaccuratefor low modulus
158
contrasts,thoughGMC hasbeenfound to underestimatethe effect of stressbridging in certain
situations.For highmoduluscontrasts(greaterthan1000),bothGMC andRCM predictinaccurate
effective properties.
Microstructurescontainingcircularparticleshave beengenerated,bothmanuallyandautomat-
ically, to modelparticlesizedistributionsin PBX 9501. Themanuallygeneratedmicrostructures,
which do not have any contactbetweenparticles,have effective elasticmoduli that are around
10%thatof PBX 9501(basedon FEM calculations).This resultsuggeststhat thereis somestress
bridging in PBX 9501. It hasalsobeenfoundFEM calculationsthatusea rectangulargrid to ap-
proximatethemicrostructurepredicthighereffective propertiesthanmicrostructuresapproximated
with triangles.However, rectangulargridsaretheonly wayto discretizetheautomaticallygenerated
microstructuresbecauseof thecloseproximity of theparticles.
GMC calculationson the simulatedmicrostructurespredicteffective propertiesthat arelower
than thosepredictedby FEM calculationsbecausestressbridging in not modeledproperly. On
theotherhand,RCM calculationspredicteffectivepropertiesthatarehigherthanthosepredictedby
FEM. In theremainderof thisresearch,weproposeto improvetheRCM techniquesothatpredicted
effective propertiesarecloserto theFEM-basedpredictions.
8.2 RemainingResearchThe approachusedin the recursive methodof cells, thoughnot expectedto generateexact
effectivepropertiesfor agivenmicrostructure,hasthepotentialof generatingacloseupperboundon
theeffective properties.This methodis a definiteimprovementover GMC, asfar aspredictingthe
responseof high volumefraction composites,with stressbridging, is concerned.The remaining
researchwill, therefore,involve the exploration of approachesto improve the RCM technique.
Effectivepropertiescalculatedfor PBX 9501from RCM will becomparedwith FEM-basedresults
andexperimentaldata. Theshearcoupledmethodof cells, thoughan improvementover GMC, is
computationallyexpensiveandhencewill notbeexploredfurther. In addition,only microstructures
containingcircularparticleswill beexploredfurther.
8.2.1 Impr ovementsto RCM
For moduluscontrastsof less than 1000 betweenthe particlesand the binder, the effective
propertiespredictedby RCM at eachrecursionarequite accurate.However, for PBX 9501, the
contrastin theYoung’s modulusis around20,000andRCM canbequite inaccuratein predicting
effective propertiesin this situation,ashasbeenshown in Chapter7. Preliminaryinvestigationson
the reasonsfor the increasedinaccuracy of RCM for PBX 9501microstructureshave shown that
159
errorsaccumulateat eachlevel of recursion,leadingto substantialerrorsin the predictionof the
overall propertiesof a representative volumeelement(RVE).
As partof the remainingresearch,theaccumulationof errorsat eachstepof recursionwill be
explored. This investigationwill beperformedfor oneof themanuallygeneratedmicrostructures
andoneof theautomaticallygeneratedmicrostructuresthatrepresentPBX 9501.Thestepsinvolved
in this processare:
1. Comparisonof blockscontaining ;=<>; , ?@<�? , A=<�A , BDC=<EBDC , and F�;=<>F�; subcellswith
detailedfinite elementcalculationsto determinethemagnitudeof error in theestimationof
effective propertiesat eachlevel of recursion.At present,we useblocksof ;=<�; subcells
for the first level of recursion. Four suchblocksarehomogenizedat eachhigher level of
recursion.Theerrorsaccumulateat eachlevel andhencetheneedfor thecomparisonsof the
propertiescalculatedateachlevel with finite elementcalculations.
2. Calculationof thetotal strainenergy at eachlevel of recursionandcomparisonwith thetotal
strainenergy at thenext level to determineif theenergy is conserved.
Thesecalculationswill provide a guidelineregardingtheoptimalnumberof subcellsto beusedto
homogenizeablock andtherebytheoptimalnumberof recursive steps.
In orderto improvethecalculationof effectivepropertiesby RCM ateachlevel of recursion,the
nine-nodeddisplacementbasedelement,discussedin Chapter5 will beutilized to modelsubcells.
For subcellscontainingthe bindermaterial,the mixed nine-nodeddisplacement-pressureelement
will beused.Themixedelementcanbeusedto modelmaterialsthathave Poisson’s ratiosgreater
than0.49andlessthan0.5. Theaccumulationof errorat eachrecursive stepwill alsobeexplored
for thenine-nodedelementsin amannersimilar to thatfor thefour-nodedelements.
TheRCM techniquewill bemodifiedto allow for morethan ;G<>; subcellsin a block during
recursion.Themodifiedmethodwill beusedto determinethegainin accuracy from theutilization
of blockscontaining?G<4? , AH<�A , and BDCI<JBDC subcells.As thethenumberof subcellsin a block
increases,the techniquebecomesmorecomputationallyexpensive. Hence,it is possiblethat the
error in total strainenergy betweenrecursionstepscanbe usedto determinethe optimal number
of subcellsthatshouldcomposea block at any stageof therecursion.This possibilitywill alsobe
exploredasapossibleimprovementto RCM.
8.2.2 Further FEM Calculations
The manuallygeneratedmicrostructureshown in Figure 7.1 containapproximately90% by
volumeof particles. However, the effective propertiesfor thesemicrostructuresareconsiderably
160
lowerthanthatof PBX 9501.Thisdifferenceis becausethereis nostressbridgingbetweenparticles
in thesemicrostructures.Themanuallygeneratedmicrostructureswill bemodifiedsothat thereis
somecontactbetweenparticles. The contactswill be chosenso that stressbridging occurs. The
effect of stressbridgingon theeffective propertiesof themodifiedform of the microstructuresin
Figure7.1 canthenbe explored. This analysiswill not be performedon the randomlygenerated
microstructuresdiscussedin Chapter7 asthenumberof particlesin eachRVE is quitelarge.
In addition,the effective propertiesof a compositecontaininga squarearrayof disks(in two
dimensions)and a cubic array of spheres(in three dimensions)will be calculatedusing finite
elementanalysis. The materialpropertiesof the componentsof PBX 9501at room temperature
and low strain rate will be usedfor thesecalculations. Thesecalculationswill provide an es-
timate of the differencebetweentwo-dimensionaland three-dimensionaleffective propertiesfor
particulatematerials. The effect of particlevolumefractionsfrom 75% to 90% on the two- and
three-dimensionaleffective propertieswill alsobe modeledfor a squarearrayof disks/spheresby
placingcircular/sphericalparticles,with increasingradii, at thecornersof theRVEs.
8.2.3 Calculations for PBX 9501
Finally, theexperimentalelasticpropertiesof thecomponentsof PBX 9501atvarioustempera-
turesandstrainrates,asdiscussedin Chapter2, will beusedto computetheeffectiveelasticproper-
tiesof PBX 9501.Therandomlygeneratedmicrostructureswill beusedfor thesecalculations.The
calculationswill beperformedusingbothRCM anddetailedfinite elementanalyses.Thecalculated
effective propertieswill be comparedwith experimentaldataon PBX 9501. Similar comparisons
will be madefor the Estane-glassandEstane-sugarmock propellantsdiscussedin Chapter2. It
shouldbenotedthat theeffective coefficient of thermalexpansionis predictedquiteaccuratelyby
theHashin-Rosenboundsandwill notbecomputednumerically.
APPENDIX A
PLANE STRAIN STIFFNESSAND COMPLIANCE
MATRICES
Thecomponentsof thetwo-dimensionalstiffnessmatrixcanbecomputedfromtwo-dimensional
planestrainfinite elementanalyses.However, thecomponentsof thetwo-dimensionalcompliance
matrix cannotbe directly determinedfrom two-dimensionalplanestrain finite elementanalyses.
Thereasonsfor thesearediscussedin this appendix.Theapproachtaken to approximatethetwo-
dimensionalcompliancematrix is alsodiscussed.
A.1 Two-DimensionalStiffnessMatrixThestress-strainrelationfor ananisotropiclinearelasticmaterialis givenbyKLLLLLLM
NPO�ONRQ�QNRS�ST Q�ST(O�ST O�QUWVVVVVVXEY
KLLLLLLMZ O�O Z O�Q Z O�S Z O\[ Z O�] Z O�^Z O�Q Z Q�Q Z Q�S Z Q_[ Z Q�] Z Q�^Z O�S Z Q�S Z S�S Z S_[ Z S�] Z S�^Z O\[ Z Q_[ Z S_[ Z [�[ Z [`] Z [`^Z O�] Z Q�] Z S�] Z [`] Z ]�] Z ]�^Z O�^ Z Q�^ Z S�^ Z [`^ Z ]�^ Z ^�^
UWVVVVVVXKLLLLLLMaO�Oa`Q�Qa`S�SbcQ�Sb O�SbRO�QUWVVVVVVX4d (A.1)
For theplanestrainassumption,we have,aeS�S Y b Q�S Y b O�S Ygfhd (A.2)
Therefore,thestress-strainrelationcanbereducedtoKM NPO�ONRQ�QT O�Q UX Y KM Z O�O Z O�Q Z O�^Z O�Q Z Q�Q Z Q�^Z O�^ Z Q�^ Z ^�^ UX KM aO�Oa`Q�QbRO�Q UX d (A.3)
The six termsin the apparenttwo-dimensionalstiffnessmatrix reduceto four is the material is
orthotropic,i.e., KM NPO�ONRQ�QT(O�Q UX Y KM Z O�O Z O�Q fZ O�Q Z Q�Q ff f Z ^�^ UX KM aO�Oa`Q�Qb O�Q UX d (A.4)
ThethreeconstantsZ O�O , Z O�Q and
Z Q�Q canbedeterminedby theapplicationof normaldisplacements
in the’1’ and’2’ directionsrespectively. TheconstantZ ^�^ canbedeterminedusingsheardisplace-
mentboundaryconditionsin afinite elementanalysis.Hence,it canbeseenthatthestiffnessmatrix
162
canbecalculateddirectly from two-dimensionalplanestrainbasedfinite elementanalyses.This is
not truefor thecompliancematrix.
A.2 Two-DimensionalComplianceMatrixThestrain-stressrelationfor ananisotropiclinearelasticmaterialcanbewrittenasijjjjjjk
l"m�mlen�nleo�op n�op m op m nqWrrrrrrsEt
ijjjjjjku m�m u m n u m o u m\v u m�w u m�xu m n u n�n u n�o u n v u n w u n xu m o u n�o u o�o u o v u o w u o xu m\v u n v u o v u v�v u v`w u v`xu m�w u n w u o w u v`w u w�w u w�xu m�x u n x u o x u v`x u w�x u x�x
qWrrrrrrsijjjjjjkyzm�my{n�ny{o�o|"n�o| m o| m nqWrrrrrrs~} (A.5)
Therelationshipbetweenthestiffnessmatrixandthecompliancematrix isijjjjjjku m�m u m n u m o u m\v u m�w u m�xu m n u n�n u n�o u n v u n w u n xu m o u n�o u o�o u o v u o w u o xu m\v u n v u o v u v�v u v`w u v`xu m�w u n w u o w u v`w u w�w u w�xu m�x u n x u o x u v`x u w�x u x�x
qWrrrrrrs�tijjjjjjk� m�m � m n � m o � m\v � m�w � m�x� m n � n�n � n�o � n v � n w � n x� m o � n�o � o�o � o v � o w � o x� m\v � n v � o v � v�v � v`w � v`x� m�w � n w � o w � v`w � w�w � w�x� m�x � n x � o x � v`x � w�x � x�x
qWrrrrrrs� m
(A.6)
or, �t�� � m } (A.7)
It is obvious from theabove equationthat theapparenttwo-dimensionalcompliancematrix is not
equalto theinverseof theapparenttwo-dimensionalstiffnessmatrix, i.e.,ik u m�m u m n u m�xu m n u n�n u n xu m�x u n x u x�x qs��t ik � m�m � m n � m�x� m n � n�n � n x� m�x � m�x � x�x qs � m } (A.8)
Hence,we cannotdeterminethe two-dimensionalcompliancematrix if we only know the two-
dimensionalstiffnessmatrix.
Let usagainexaminetheeffect of theplane-strainassumptionon thestress-strainrelation.We
thenhave ijjk l m�ml n�n�p m nqWrrs t
ijjk u m�m u m n u m o u m�xu m n u n�n u n�o u n xu m o u n�o u o�o u o xu m�x u n x u o x u x�xqWrrs ijjk y m�my n�ny{o�o| m n
qWrrs } (A.9)
For orthotropicmaterials,this relationsimplifiestoijjk l"m�mlen�n�p m nqWrrs t
ijjk u m�m u m n u m o �u m n u n�n u n�o �u m o u n�o u o�o �� � � u x�xqWrrs ijjk yzm�my{n�ny{o�o| m n
qWrrs } (A.10)
This equationshows thatwe needto know thestressy{o�o to determinethetermsof thecompliance
matrix and hencethree-dimensionalanalysesare necessary. If we assumeplanestress,we can
163
determinethe termsof thematrix � directly. However, theapparenttwo-dimensionalcompliance
matrix for planestressis not equalto that for planestrainandhencewe cannotapply this method
for our purposes.This is why theplanestraincompliancematrix cannotbedeterminedusingtwo-
dimensionalfinite elementanalysesonly.
A.3 Approximation of ComplianceMatrixThe two-dimensionalcompliancematrix canbe determinedapproximatelyfor materialswith
squaresymmetryby assumingthat ����� , ����� and ����� areknown. Let,� ���!� � ���'���J� �� � (A.11)� ���'� �� � (A.12)
where, � � is the Poisson’s ratio in the out-of-planedirectionand� � is the Young’s ratio in that
direction.Then,for amaterialwith squaresymmetry,���� � ���� ����� ����W��� �
����������� ��� � ��� � � �� � �� ��� � ��� �J� �� � �� � �� � � � �� � �� � �� � � �����
�W������������� � ���� ���� ���� ���
�W���4� (A.13)
Invertingtherelation,we have,���� � ���� ���� ���� ����W��� � ���� ��� ��� ��� � ��� ��� ��� � ��� ��� ��� �� � � ���
�W��� ���� � ���� ����� ����W���~� (A.14)
where, ��� � � � � ���¡� � ��� �*� ���� ��¢ � �� �£��� � � �D� ����¥¤ ¢ � � �£���'¦ ���§� � � � � ��� ¤ � ��� � � ���� ��¢ � �� � ���¥� � � � ����¥¤ ¢ � � � ��� �
Notethatit is notnecessaryto know ��� , ��� and
��� to determine� ��� and � ��� .Wecanwrite theabove relationsbetween
��� ¦ ��� and � ��� ¦ � ��� in theform� �*� ���� �©¨ � � ��� ¤ ¢ � ���ª ����� �©¨ � �*� ���� ��¢ � �� ����� �«� �� ��� ª � � ¦ (A.15)� � � ���� �©¨ � � ��� ¤ ¢ � ���ª � ���¬�©¨ � � � ���� ��¢ � �� � ��� ¤ � �� ��� ª � � � (A.16)
164
In simplifiedform, �®°¯²±®�®£³µ´ ®°¯�®�® ³E¶ ®¸·º¹h»(A.17)
± ¯²±® ± ³µ´ ± ¯ ® ± ³E¶ ± ·º¹h¼(A.18)
Wecansolve thesequadraticequationsto getexpressionsfor
¯ ®�®and
¯ ® ±as¯ ®�® · ½ ´¾³�¿ ´ ± ½�À ¶Á »
(A.19)¯ ® ± · ½ ´ ½ ¿ ´ ± ½�À ¶Á ¼(A.20)
Knowing
¶ ®�®,
¶ ® ±, Â'à and Ä�à thesetwo equationscanbesolved iteratively to determine
®�®and
¯�® ±. Thevaluesof
¶ ®�®and
¶ ® ±canbedeterminedusingtheprocedureoutlinedat thebeginningof
this section.It remainsto bediscussedhow ÂÅà and Ä�à areto bedetermined.
A.4 Determination of Æ~Ç and ÈPÇTwo methodscanbeusedto determinethevaluesof Â'à and Ä�à for our calculations.Thefirst
methodis to assumethattheruleof mixturesis accurateenoughto determinetheeffectiveproperties
in the’3’ direction.Thus,if thevolumefractionof thefirst componentis É ® andthatof thesecond
componentis É ± , we have, ÂÅà · É ®  ® ³ É ±  ± » (A.21)Ä�à · É ® Ä ® ³ É ± Ä ± ¼ (A.22)
whereÂ!Ê and Ä(Ê aretheYoung’s modulusandthePoisson’s ratioof the Ë th component.
The otheroption is to usethevaluesof
¯�® à , ¯ ± à and
¯ Ã�à obtainedfrom GMC sincetheseare
alsoquiteaccuratefor theoutof planedirection.Thus,we have,Â'à · ̯ÎÍPÏÑÐÃ�à »(A.23)Ä Ã · ½ ¯ ÍPÏÑЮ à  à ¼ (A.24)
This is theprocedurewe have useto determinetheeffective compliancematricesdiscussedin
Chapter6.
REFERENCES
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