micromechanics-based prediction of …banerjee/thesis/proposal.pdfmicromechanics-based prediction of...

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



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`�òm_ òp] òp]-_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


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


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�¶DÂ ¦ § ¦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 ¶ ¿ Â (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) ( � ò�� /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 ÁqÂ¨ÃÄ�Å@ÆGÇÈ ÉÉÉ�Ê�Ë ÁqÂ Ä�Å@ÆGÇÈ ÉÉÉÌÊyÍ´Î-Ï (4.13)Á Â Ä ÃÅXÆGÇÈ ÉÉÉ�Ê�Ë Á Â Ä�Å@ÆGÇÈ ÉÉÉÌÊyÍ´Î-Ï (4.14)Á Â Ä�Å ÃÆ�ÇÈ ÉÉÉ�Ê�Ë Á Â 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 ÁqÂ Ä�Å@ÆGÇÈ ÉÉÉÌæ�çÌè@éê`ëíì Ë Á�Â¨ÃÄbÅGÆGÇÈ ÉÉÉ æ�ç�îè@éê�ëðï�ì Í´Î-Ï (4.19)Á Â 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�Â Ä�ÅGÆGÇÈ ÉÉÉ æ�ç ó!éô ëíì Ë Á�Â Ä�Å ÃÆXÇÈ ÉÉÉ æ�ç7îó!éô ëðï�ì Í´Î-Ð (4.21)

Applying thesedisplacementcontinuityequationsonanaveragebasisover theinterfaceswe get,õ-öø÷áù Á Â Ä�Å@ÆGÇÈ ÉÉÉ æ çÌè@éê`ëíì Ë Á Â¨ÃÄbÅ@ÆGÇÈ ÉÉÉ æ ç�îè@éê`ëðï�ì�úüû�ý Â Å�Çþ û�ý Â ÆGÇÿ Í´Î-Ï (4.22)õ ö�÷ ù Á Â Ä�ÅGÆGÇÈ ÉÉÉ æ�ç ñ�éò ëíì Ë Á Â Ä ÃÅGÆGÇÈ ÉÉÉ æ�ç îñ,éò ëðï�ì ú û ý Â Ä�Ç� û�ý Â ÆGÇÿ Í´Î-Ï (4.23)õ ö�÷ ù Á Â Ä�Å@ÆGÇÈ ÉÉÉ æ�çÌó!éô ëíì Ë Á Â Ä�Å ÃÆXÇÈ ÉÉÉ æ�çOîó,éô\ëðï�ì ú û�ý Â Ä�Ç� û ý Â Å�Çþ Í´Î-Ï (4.24)

whereõö�÷ û ý Â �� DÇÈ û�ý Â �� DÇ� � õ ìï�ì õ ìï�ì û ý Â ��aÇÈ û ý Â �� 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 �� Á3ÂÄÃ · � �� Å §Substitutingequation(4.6) into theabove equationwe get¶�· �� º � »Á �� Á Â ®:� �� § (4.61)

Now, if equation(4.57)is multipliedby Æ © ° with ½ � » andsummedover Ç and È we get

»Á �� Á4Â ® � �� ««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)

¶�·ÃÂ ¥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× ÏÞÝ§�§ « (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 ÏâÝ§�© « (4.98)

For theÈ ·

matrices, È ·bÏÑÐ ¥º¹»»»¼ã8Ó ÏäÐ§�§ ã8Ó ÏäÐ§�© ««« ã8Ó ÏäÐ§É®ã8Ó ÏäÐ©�§ ã8Ó ÏäÐ©�© ««« ã8Ó ÏäÐ©]®...

.... . .

...ã8Ó ÏÑÐ®å§ ã8Ó ÏÑÐ®Ê© ««« ã8Ó ÏÑÐ®C®¿!ÀÀÀÁ (4.99)

whereã8Ó ÏÑÐá�æ ¥1ç ÛCÜæfá ÏÞÝ§ ± 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�Â is of thesamedimensionsas �4ÁÃÂ thoughtheelementsin 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 �uÂ ��¨°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� Ì½áãâ�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\]Qî�` ajb WZYS[h\]Qî�i dOe VMWkYS[ ]�î�` acb WkYS[ ]�î�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 \[ ]�î�` dGe VXWZYS[ ]�^n�n acb WZYS[ ]�î�` dOoCp (4.148)

Equations(4.146-4.148)form a systemof qGrts èquationsin 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�[ ]�î�i dRev|�ajb WZYS[ ]�î�i dgf�a�b W �YH[ ]�î�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\]Qî�i dOe}|�ajb WkYS[ ]�î�i dgf�a�b WZYS[ �]Qî�i dgf�a�b WZY \[ ]�^`�` dRev|�ajb WZYS[ ]�^`�` dgf�a�b WZY �[ ]�^`�` d�e� a b WZY \[ ]�î�` d e a b WkY \[ �]�î�` d e a b WZYS[ ]�î�` d e a b WZYS[ �]Qî�` dD� e� a�b WZYS[D\]Qî�` dOe4a�b WkY �[D\]�î�` dRe4acb WZYS[ ]�î�` dGe4ajb WZY �[ ]�î�` 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¿=Â subcellson eachsidewhereÃ 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�Å¨Ø�ÙvÉSÊÚ È (5.2)ÒgÛ Ë Å¨ØÝÜÞÆ«Ê�Å¨Ø�ÙvÉSÊÚ È (5.3)Ògß Ë Å¨ØÝÜÞÆ«Ê�Å¨ØàÜáÉSÊÚ È (5.4)Ò Ì Ë Å¨ØAÙ Æ«Ê�Å¨ØàÜáÉSÊÚ 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 ¦¯ ¨¬¨

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¡£¢¡£¤¥ ¢ ¦¡ §® ©¡ ª« ¨¦¬ ¢ ¦« ¢¬¢® ¨¡ ª¦¯ ¨¦¬¨

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¡£¢ ¦¡ ¤® © ¦¡ 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 °¨ ®± ²ÃÂ �Ä ³ ¥}³ °¨ ® ± ²$² � ¥}³¨ ¸�¶ ° ® ± ²Ã· � ¥ ¢ ³ § ¡ ³ ° ®±d²Ã¹ � ³ (¸ ¢ °i® ±h²Å» � µ ¢ §!µª¨ °6® ±h²Ã¼ � ¥}³´(¨h¥�¶ °6®±d²Ã½ � ¥}³>§Æ¥ª¥ °i® ±h² ¾ � ¸ ¢ §Æ¥ª¥}¸ °6® ±h² À � ¥ ¢ ³Ç(¨ ¢ ³ °i®±d² Á � ¥}³´ °6® ±�·$Â � ¥}¨ ¢ §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

ö ö ö ö ö öö ö ö ö ö öö ö ö ö ö öö ö ö ö ö öö ö ö ö ö öö ö ö ö ö ö÷ ÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷÷ ÷ ÷ ÷ ÷ ÷

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

ü ü ü ü ü üü ü ü ü ü üü ü ü ü ü üü ü ü ü ü üü ü ü ü ü üü ü ü ü ü üý ý ý ý ý ýý ý ý ý ý ýý ý ý ý ý ýý ý ý ý ý ýý ý ý ý ý ýý ý ý ý ý ý

þ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þþ þ þ þ þ þÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿÿ ÿ ÿ ÿ ÿ ÿ

� � � � � ��

� � � � � ��

� � � � � ��

� � � ��

� � � ��

� � � ��

� � � ��

� � � � � ��

� � � � � ��

� � � � � � ��

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 Â (circles)and Á¾Ã(diamonds)show thatthesearetheclosestto theexactresults.Theratioof Á Ã to Á¼Â increasesasthe

moduluscontrastthetwo componentsincreases.For relatively low moduluscontrast,theeffective

shearmoduluspredictedby the phaseinterchangeidentity is approximatelyequalto the meanofÁ Â and Á¾Ã predictedby FEM. The effective Á Â and Á¾Ã predictedby RCM arehigher than those

predictedby FEM while thosepredictedby GMC arelower.

The ratio of Á Â 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 Ã 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.TheeffectiveÍ¼Î and Í¾Ï of a checkerboardwith a shearmoduluscontrastof about25,000(correspondingto the

highestmoduluscontrastshown in Figure6.4) have beencalculatedusingvariouslevels of mesh

refinementandplottedin Figure6.5. Theseplotsshow that theeffective Í¼Î convergesto a steady

valuewhenabout Ð<ÆÈÑÒÊÓÐ<ÆÈÑ elementsareusedwhile Í Ï convergesto asteadyvaluewhen ÆÈÇÈÉÒÊ¤ÆÈÇÈÉelementsareusedto 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�»�»�ºcÂ¤ºk½±¸,º�»�»�Ã

where,¿ À º ´'®"Äoriginal

Â¤º ´'®"Ätranslated

Ãor,¿ À º ´ÆÅ%ÇÉÈ

translatedÂ¤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.

� � � � � � � ��

� � � � � � � ��

� � � � ��

� � � � ��

� � � � � � � ��

¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡¡ ¡ ¡ ¡ ¡

¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢¢ ¢ ¢ ¢ £ £ £ ££ £ £ ££ £ £ ££ £ £ ££ £ £ ££ £ £ ££ £ £ ££ £ £ ££ £ £ £

¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤¤ ¤ ¤ ¤

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥¥ ¥ ¥ ¥ ¥ ¥ ¥ ¥

¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦¦ ¦ ¦ ¦ ¦ ¦ ¦ ¦

§ § § § § § § §§ § § § § § § §§ § § § § § § §§ § § § § § § §§ § § § § § § §§ § § § § § § §§ § § § § § § §§ § § § § § § §

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©© © © © ©

ª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ªª ª ª ª

« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «« « « « «

¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬¬ ¬ ¬ ¬

® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®® ® ® ® ® ® ® ®

¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯¯ ¯ ¯ ¯ ¯

° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °° ° ° °

± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±± ± ± ±

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�ÁKÂ 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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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


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|ÁÂ°�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 ¸�´�´ÃÁÄ¸�´�´ 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.



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.


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.


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


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


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


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.


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.


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'Ã and Ä�Ã thesetwo equationscanbesolved iteratively to determine

®�®and

¯�® ±. Thevaluesof

¶ ®�®and

¶ ® ±canbedeterminedusingtheprocedureoutlinedat thebeginningof

this section.It remainsto bediscussedhow ÂÅÃ and Ä�Ã areto bedetermined.

A.4 Determination of Æ~Ç and ÈPÇTwo methodscanbeusedto determinethevaluesof Â'Ã and Ä�Ã 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, ÂÅÃ · É ® Â ® ³ É ± Â ± » (A.21)Ä�Ã · 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

¯�® Ã , ¯ ± Ã and

¯ Ã�Ã obtainedfrom GMC sincetheseare

alsoquiteaccuratefor theoutof planedirection.Thus,we have,Â'Ã · Ì¯ÎÍPÏÑÐÃ�Ã »(A.23)Ä Ã · ½ ¯ ÍPÏÑÐ® Ã Â Ã ¼ (A.24)

This is theprocedurewe have useto determinetheeffective compliancematricesdiscussedin

Chapter6.

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