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Development of Yield Criteria for Describing the Behavior of Porous Metals with Tension-Compression Asymmetry

Permanent Link: http://ufdc.ufl.edu/UFE0024779/00001

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Title: Development of Yield Criteria for Describing the Behavior of Porous Metals with Tension-Compression Asymmetry
Physical Description: 1 online resource (251 p.)
Language: english
Creator: Stewart, Joel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anisotropy, asymmetry, compression, constitutive, element, finite, gurson, porous, strength, tension, yield
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: A significant difference between the behaviors in tension versus compression is obtained at the polycrystal level if either twinning or non-Schmid effects are included in the description of the plastic deformation at the single crystal level. Examples of materials that exhibit tension-compression asymmetry include hexagonal close packed (HCP) polycrystals and intermetallics (e.g., molybdenum compounds). Despite recent progress in modeling the yield behavior of such materials, the description of damage by void growth remains a challenge. This dissertation is devoted to the development of macroscopic plastic potentials for porous metallic aggregates in which the void-free, or matrix, material displays tension-compression asymmetry. Using a homogenization approach, new analytical plastic potentials for a random distribution of voids are obtained. Both spherical and cylindrical void geometries are considered for void-matrix aggregates containing an isotropic matrix, while spherical voids are considered for the case of an anisotropic matrix material. The matrix plastic behavior in all cases is described by a yield criterion that captures strength differential effects and can account for the anisotropy that may be exhibited in the void-free material. For the case when the matrix material is isotropic, the developed analytical potentials for the void-matrix aggregate are sensitive to the second and third invariants of the stress deviator and display tension-compression asymmetry. Furthermore, if the matrix material has the same yield strength in tension and compression, the developed criteria reduce to the classical Gurson criteria for either spherical or cylindrical voids. It has also been demonstrated that the developed isotropic criterion for porous aggregates containing spherical voids captures the exact solution of a hollow sphere loaded in hydrostatic tension or compression. Finite element cell calculations with the matrix material obeying an isotropic yield criterion and displaying tension-compression asymmetry were performed and the comparison between finite element calculations and theoretical predictions demonstrate the versatility of the proposed formulations. A new anisotropic potential for the porous aggregate was also developed for the case when the matrix material is anisotropic and displays tension-compression asymmetry. If the matrix is isotropic, the proposed analytical anisotropic criterion reduces to the isotropic criterion developed in this dissertation for a void-matrix aggregate containing spherical voids. Comparison between finite element calculations and theoretical predictions show the predictive capabilities of the developed anisotropic formulation. The yield criteria developed in this dissertation are the only criteria available to capture the influence of damage by void growth in HCP metals and other materials that exhibit tension-compression asymmetry.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joel Stewart.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024779:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024779/00001

Material Information

Title: Development of Yield Criteria for Describing the Behavior of Porous Metals with Tension-Compression Asymmetry
Physical Description: 1 online resource (251 p.)
Language: english
Creator: Stewart, Joel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anisotropy, asymmetry, compression, constitutive, element, finite, gurson, porous, strength, tension, yield
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: A significant difference between the behaviors in tension versus compression is obtained at the polycrystal level if either twinning or non-Schmid effects are included in the description of the plastic deformation at the single crystal level. Examples of materials that exhibit tension-compression asymmetry include hexagonal close packed (HCP) polycrystals and intermetallics (e.g., molybdenum compounds). Despite recent progress in modeling the yield behavior of such materials, the description of damage by void growth remains a challenge. This dissertation is devoted to the development of macroscopic plastic potentials for porous metallic aggregates in which the void-free, or matrix, material displays tension-compression asymmetry. Using a homogenization approach, new analytical plastic potentials for a random distribution of voids are obtained. Both spherical and cylindrical void geometries are considered for void-matrix aggregates containing an isotropic matrix, while spherical voids are considered for the case of an anisotropic matrix material. The matrix plastic behavior in all cases is described by a yield criterion that captures strength differential effects and can account for the anisotropy that may be exhibited in the void-free material. For the case when the matrix material is isotropic, the developed analytical potentials for the void-matrix aggregate are sensitive to the second and third invariants of the stress deviator and display tension-compression asymmetry. Furthermore, if the matrix material has the same yield strength in tension and compression, the developed criteria reduce to the classical Gurson criteria for either spherical or cylindrical voids. It has also been demonstrated that the developed isotropic criterion for porous aggregates containing spherical voids captures the exact solution of a hollow sphere loaded in hydrostatic tension or compression. Finite element cell calculations with the matrix material obeying an isotropic yield criterion and displaying tension-compression asymmetry were performed and the comparison between finite element calculations and theoretical predictions demonstrate the versatility of the proposed formulations. A new anisotropic potential for the porous aggregate was also developed for the case when the matrix material is anisotropic and displays tension-compression asymmetry. If the matrix is isotropic, the proposed analytical anisotropic criterion reduces to the isotropic criterion developed in this dissertation for a void-matrix aggregate containing spherical voids. Comparison between finite element calculations and theoretical predictions show the predictive capabilities of the developed anisotropic formulation. The yield criteria developed in this dissertation are the only criteria available to capture the influence of damage by void growth in HCP metals and other materials that exhibit tension-compression asymmetry.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joel Stewart.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024779:00001


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DEVELOPMENTOFYIELDCRITERIAFORDESCRIBINGTHEBEHAVIOROF POROUSMETALSWITHTENSION-COMPRESSIONASYMMETRY By JOELB.STEWART ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c 2009JoelB.Stewart 2

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ACKNOWLEDGMENTS Iwouldliketothankthemembersofmysupervisorycommitteefortheirsuggestions andsupport.IwouldliketoespeciallythankDr.OanaCazacuforhersupport,her patienceandherenthusiasmthroughoutthisstudy.Hercommentsanddirectionhave beeninvaluabletothecompletionofthisresearch. IwouldalsoliketothanktheAirForceResearchLaboratoryforprovidingtheopportunitytopursueagraduateeducation.SpecialrecognitionisextendedtoDr.Lawrence LijewskiandDr.KirkVandenforenthusiasticallysupportingmygraduateresearchgoals fromtheoutset.Iwouldalsoliketoacknowledgethemanyhelpfulconversationsand generousmoralsupportreceivedfromDr.MichaelNixon,Dr.MartinSchmidtandDr. BrianPlunkett.Theirsteadfastsupportandhonestcritiqueswerealwaysappreciated andasourceofencouragement.IwouldliketoextendspecialgratitudetoDr.Michael Nixonforreadingthroughthedissertationandmakinganumberofinsightfulcomments andsuggestions.Dr.StefanSoarewasofgreatassistancewhileheandIwerestudying forthequalifyingexam;hishelpfulsuggestionsandengagingconversationweregreatly appreciated. Finally,Iwouldliketothankmyfriendsandfamilywhosupportedmethroughout boththiscurrentresearcheortandearlieracademicendeavors.Specialappreciation isextendedtomywife,Kelly,forsupportingmethroughoutmyacademicjourney. Theprocesswouldhavebeenmuchmorepainfulwithoutherunfailingsupportand encouragement. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................3 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................11 CHAPTER 1INTRODUCTION..................................13 2HOMOGENIZATIONAPPROACH.........................27 2.1KinematicHomogenizationApproachofHillandMandel..........28 2.2YieldCriterionfortheMatrixMaterial....................30 2.3PlasticMultiplierRateDerivation.......................33 2.3.1Plasticmultiplierratewhen J 3 0..................35 2.3.2Plasticmultiplierratewhen J 3 0..................38 2.3.3Generalplasticmultiplierrateexpression...............41 3PLASTICPOTENTIALFORHCPMETALSWITHSPHERICALVOIDS...46 3.1LimitSolutions.................................48 3.1.1Zeroporosityandequalyieldstrengthslimitingcases........48 3.1.2Exactsolutionforahydrostatically-loadedhollowsphere......49 3.1.2.1Strain-displacementrelations................49 3.1.2.2Straincompatibility.....................50 3.1.2.3Equationsofmotion.....................51 3.1.2.4Elasticconstitutiverelation:Hooke'slaw..........52 3.1.2.5Ultimatepressure.......................52 3.2ChoiceofTrialVelocityField.........................57 3.3CalculationoftheLocalPlasticDissipation.................62 3.4DevelopmentoftheMacroscopicPlasticDissipationExpressions......66 4NUMERICALIMPLEMENTATIONOFTHEMATRIXYIELDCRITERION.78 4.1ReturnMappingProcedure..........................78 4.2FirstDerivatives................................85 4.2.1IsotropicCPB06rstderivatives:generalloading..........87 4.2.2IsotropicCPB06rstderivatives:biaxialloading...........88 4.3SecondDerivatives...............................89 4.3.1Vonmisessecondderivatives......................89 4.3.2IsotropicCPB06secondderivatives:generalloading.........92 4.3.3IsotropicCPB06secondderivatives:biaxialloading.........96 4

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5ASSESSMENTOFTHEPROPOSEDSPHERICALVOIDMODELBYFINITEELEMENTCALCULATIONS........................97 5.1ModelingProcedure..............................97 5.2FiniteElementResults.............................102 5.3ConcludingRemarks..............................104 6PLASTICPOTENTIALSFORHCPMETALSWITHCYLINDRICALVOIDS122 6.1LimitSolutions.................................123 6.1.1Zeroporosityandvonmisesmateriallimitingcases.........123 6.1.2Analysisofahydrostatically-loadedhollowcylinder.........124 6.1.2.1Strain-displacementrelations................125 6.1.2.2Straincompatibility.....................126 6.1.2.3Equationsofmotion.....................127 6.1.2.4Elasticconstitutiverelation:Hooke'slaw..........128 6.1.2.5Relationbetween J 3 and m ................128 6.2ChoiceofTrialVelocityField.........................133 6.3ParametricRepresentationofthePorousAggregateYieldLocusforAxisymmetricLoading...............................137 6.3.1 1:thematrixyieldstrengthintensionisgreaterthanincompression.................................138 6.3.1.1 m > 0and J 3 < 0......................138 6.3.2Discussion................................144 6.4ProposedClosed-FormExpressionforaPlaneStrainYieldCriterion....145 6.4.1Calculationofthelocalplasticdissipation...............145 6.4.2Developmentofthemacroscopicplasticdissipationexpressions...148 7ASSESSMENTOFTHEPROPOSEDCYLINDRICALVOIDMODELBYFINITEELEMENTCALCULATIONS........................162 7.1ModelingProcedure..............................162 7.2FiniteElementResults.............................166 7.3ConcludingRemarks..............................168 8ANISOTROPICPLASTICPOTENTIALFORHCPMETALSCONTAINING SPHERICALVOIDS.................................180 8.1KinematicHomogenizationApproachofHillandMandel..........181 8.2YieldCriterionfortheMatrixMaterial....................183 8.3ChoiceofTrialVelocityField.........................187 8.4CalculationoftheLocalPlasticDissipation.................188 8.5DevelopmentoftheMacroscopicPlasticDissipationExpression......189 8.6AssessmentoftheProposedAnisotropicCriterionthroughComparison withFiniteElementCalculations.......................192 8.7ConcludingRemarks..............................197 5

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9CONCLUSIONS...................................210 APPENDIX APARAMETRICREPRESENTATIONDERIVATIONOFTHEAXISYMMETRICYIELDLOCUS.................................213 A.1GeneralFormofEquations...........................213 A.1.1Plasticmultiplierratebranches....................213 A.1.2Macroscopicplasticdissipationandderivatives............217 A.2 1:TheMatrixYieldStrengthinTensionisGreaterthaninCompression........................................220 A.2.1 J 3 < 0..................................220 A.2.1.1 m > 0............................220 A.2.1.2 m < 0............................221 A.2.2 J 3 > 0..................................223 A.2.2.1 m > 0............................223 A.2.2.2 m < 0............................226 A.3 1:TheMatrixYieldStrengthinTensionisLessthaninCompression.229 A.3.1 J 3 < 0..................................230 A.3.1.1 m > 0............................230 A.3.1.2 m < 0............................233 A.3.2 J 3 > 0..................................236 A.3.2.1 m > 0............................236 A.3.2.2 m < 0............................237 BRELATIONSHIPBETWEENHILL48ANDCPB06COEFFICIENTS.....240 B.1DetermineTheHill48CoecientsGivenTheCPB06Coecients.....245 B.2DetermineTheCPB06CoecientsGivenTheHill48Coecients.....246 REFERENCES.......................................247 BIOGRAPHICALSKETCH................................251 6

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LISTOFTABLES Table page 2-1k-relations.......................................44 2-2z-parameters......................................44 5-1 k =0, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix.......108 5-2 k =0, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix.......108 5-3 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix...108 5-4 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix...109 5-5 k =0 : 3098, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix....109 5-6 k =0 : 3098, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix....109 5-7 k =0, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix.......111 5-8 k =0, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix.......111 5-9 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix...111 5-10 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix...112 5-11 k =0 : 3098, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix....112 5-12 k =0 : 3098, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix....112 5-13 k =0, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix.......114 5-14 k =0, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix.......114 5-15 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix...114 5-16 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix...115 5-17 k =0 : 3098, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix....115 5-18 k =0 : 3098, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix....115 7-1 k =0planestraincylindricalvoidcomputationaltestmatrix...........172 7-2 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098planestraincylindricalvoidcomputationaltestmatrix.......172 7-3 k =0 : 3098planestraincylindricalvoidcomputationaltestmatrix........173 8-1TransverselyisotropicCPB06constants.......................198 7

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LISTOFFIGURES Figure page 1-1Particlecrackingexample...............................24 1-2Particledecohesionexample..............................25 1-3Voidinsinglecrystal..................................26 2-1CPB06yieldcondition -planerepresentation...................43 2-2CPB06plasticmultiplierrate -planerepresentation................45 3-1Ductilecrackinaluminumplate...........................74 3-2Representativevolumeelementforaspherecontainingasphericalvoid......75 3-3Hydrostatically-loadedhollowsphere.........................75 3-4Macroscopicductileyieldsurfaceswithtension-compressionasymmetry......76 3-5Evolutionofmacroscopicductileyieldsurfaceswithporosity............77 5-1Axisymmetricunitcellforthesphericalvoid....................106 5-2 f 0 =0 : 01axisymmetricniteelementmeshfortheunitcell............107 5-3 f 0 =0 : 04axisymmetricniteelementmeshfortheunitcell............110 5-4 f 0 =0 : 14axisymmetricniteelementmeshfortheunitcell............113 5-5 k =0with f =0 : 01:FEversusanalytical.....................116 5-6 k =0with f =0 : 01, f =0 : 04and f =0 : 14:FEversusanalytical........116 5-7 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098with f =0 : 01:FEversusanalytical.................117 5-8 k =0 : 3098with f =0 : 01:FEversusanalytical..................117 5-9 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098with f =0 : 04:FEversusanalytical.................118 5-10 k =0 : 3098with f =0 : 04:FEversusanalytical..................118 5-11 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098with f =0 : 14:FEversusanalytical.................119 5-12 k =0 : 3098with f =0 : 14:FEversusanalytical..................119 5-13 k =0axisymmetricFEdataversusanalyticalyieldcurves............120 5-14 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098axisymmetricFEdataversusanalyticalyieldcurves........120 5-15 k =0 : 3098axisymmetricFEdataversusanalyticalyieldcurves.........121 8

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6-1Voidsandshearbandintitanium..........................157 6-2CylindricalRVE....................................158 6-3 T = C =1 : 21yieldcurveparametricrepresentationforonequadrant.......159 6-4 T = C =1 : 21yieldcurveparametricrepresentation.................159 6-5 T = C =0 : 82yieldcurveparametricrepresentation.................160 6-6 f =0 : 01cylindricalmacroscopicyieldcurves....................160 6-7 f =0 : 04cylindricalmacroscopicyieldcurves....................161 6-8 f =0 : 14cylindricalmacroscopicyieldcurves....................161 7-1Planestraingeometryusedinniteelementcalculations..............169 7-2Validtriaxialityanglesforplanestrain........................170 7-3 f 0 =0 : 01planestrainniteelementmeshfortheunitcell.............170 7-4 f 0 =0 : 04planestrainniteelementmeshfortheunitcell.............171 7-5 f 0 =0 : 14planestrainniteelementmeshfortheunitcell.............171 7-6Plainstrainyieldpointdetermination........................173 7-7 k =0,nottingparameters:FEversusanalytical.................174 7-8 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098,nottingparameters:FEversusanalytical..............175 7-9 k =0 : 3098,nottingparameters:FEversusanalytical...............176 7-10 k =0,withttingparameters:FEversusanalytical................177 7-11 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098,nottingparameters:FEversusanalytical..............178 7-12 k =0 : 3098,nottingparameters:FEversusanalytical...............179 8-1TransverselyisotropicRVE..............................198 8-2Anisotropiceectivestressversuseectivestraincurves..............199 8-3Planestressyieldlociformaterials A B and C with T = C ...........199 8-4Planestressyieldlociformaterials A B and C with T < C ...........200 8-5Planestressyieldlociformaterials A B and C with T > C ...........200 8-6 k =0Material A deviatoricplot...........................201 8-7 k< 0Material A deviatoricplot...........................201 9

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8-8 k> 0Material A deviatoricplot...........................202 8-9 k =0Material B deviatoricplot...........................202 8-10 k< 0Material B deviatoricplot...........................203 8-11 k> 0Material B deviatoricplot...........................203 8-12 k =0Material C deviatoricplot...........................204 8-13 k< 0Material C deviatoricplot...........................204 8-14 k> 0Material C deviatoricplot...........................205 8-15Material A theoreticalyieldcurvesversusFEdata.................206 8-16Material B theoreticalyieldcurvesversusFEdata.................207 8-17Material C theoreticalyieldcurvesversusFEdata.................208 8-18Anisotropicductileyieldsurfaceswithtension-compressionasymmetry......209 10

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy DEVELOPMENTOFYIELDCRITERIAFORDESCRIBINGTHEBEHAVIOROF POROUSMETALSWITHTENSION-COMPRESSIONASYMMETRY By JoelB.Stewart August2009 Chair:OanaCazacu Major:MechanicalEngineering Asignicantdierencebetweenthebehaviorsintensionversuscompressionis obtainedatthepolycrystallevelifeithertwinningornon-Schmideectsareincludedin thedescriptionoftheplasticdeformationatthesinglecrystallevel.Examplesofmaterials thatexhibittension-compressionasymmetryincludehexagonalclosepackedHCP polycrystalsandintermetallicse.g.,molybdenumcompounds.Despiterecentprogressin modelingtheyieldbehaviorofsuchmaterials,thedescriptionofdamagebyvoidgrowth remainsachallenge. Thisdissertationisdevotedtothedevelopmentofmacroscopicplasticpotentialsfor porousmetallicaggregatesinwhichthevoid-free,ormatrix,materialdisplaystensioncompressionasymmetry.Usingahomogenizationapproach,newanalyticalplastic potentialsforarandomdistributionofvoidsareobtained.Bothsphericalandcylindrical voidgeometriesareconsideredforvoid-matrixaggregatescontaininganisotropicmatrix, whilesphericalvoidsareconsideredforthecaseofananisotropicmatrixmaterial.The matrixplasticbehaviorinallcasesisdescribedbyayieldcriterionthatcapturesstrength dierentialeectsandcanaccountfortheanisotropythatmaybeexhibitedinthevoidfreematerial. Forthecasewhenthematrixmaterialisisotropic,thedevelopedanalyticalpotentials forthevoid-matrixaggregatearesensitivetothesecondandthirdinvariantsofthestress deviatoranddisplaytension-compressionasymmetry.Furthermore,ifthematrixmaterial 11

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hasthesameyieldstrengthintensionandcompression,thedevelopedcriteriareduce totheclassicalGursoncriteriaforeithersphericalorcylindricalvoids.Ithasalsobeen demonstratedthatthedevelopedisotropiccriterionforporousaggregatescontaining sphericalvoidscapturestheexactsolutionofahollowsphereloadedinhydrostatictension orcompression.Finiteelementcellcalculationswiththematrixmaterialobeyingan isotropicyieldcriterionanddisplayingtension-compressionasymmetrywereperformed andthecomparisonbetweenniteelementcalculationsandtheoreticalpredictions demonstratetheversatilityoftheproposedformulations. Anewanisotropicpotentialfortheporousaggregatewasalsodevelopedforthecase whenthematrixmaterialisanisotropicanddisplaystension-compressionasymmetry. Ifthematrixisisotropic,theproposedanalyticalanisotropiccriterionreducestothe isotropiccriteriondevelopedinthisdissertationforavoid-matrixaggregatecontaining sphericalvoids.Comparisonbetweenniteelementcalculationsandtheoreticalpredictionsshowthepredictivecapabilitiesofthedevelopedanisotropicformulation.The yieldcriteriadevelopedinthisdissertationaretheonlycriteriaavailabletocapturethe inuenceofdamagebyvoidgrowthinHCPmetalsandothermaterialsthatexhibit tension-compressionasymmetry. 12

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CHAPTER1 INTRODUCTION Theaimofthisdissertationistodevelopanalyticalplasticpotentialsformetalsthat exhibittension-compressionasymmetrye.g.,certainBCCmaterialssuchasmolybdenum andHCPmetalssuchas -titaniumandthatcontaineithercylindricalorsphericalvoids. Closed-formyieldcriteriaforporousmaterialsexhibitingtension-compressionasymmetry i.e.,thathavedierentyieldstrengthsintensionversuscompressiondonotcurrently existintheliterature.Thesuccessfulderivationofanalyticyieldcriteriaforthesetypes ofmaterialsshouldgiveresearchersanadditionalandmoreaccuratetoolfordealingwith damageinthesematerials. Ductilefailureinmetalsoccursduetothenucleation,growthandcoalescenceofvoids e.g.,seeMcClintock,1968;Rousselier,1987.Additionally,somelocalizationphenomena thatcommonlyleadtofailureinmetallicstructurese.g.,adiabaticshearbandformation arethoughttobeinuencedbymicro-voidsinthematrixmaterialsee,forexampleBatra andLear,2005;Tvergaard,1980.Thevoiddistributioninamaterialcanexistbecause ofpre-existingvoidse.g.,manufacturingdefectsorbecauseofnucleationatsecondphaseparticles.Forexample,Figure1-1illustratesvoidnucleationduetocrackingofthe inclusionaswellassomedecohesionattheinclusion-matrixinterface.Figure1-2shows anexampleofvoidnucleationduetodecohesionofthematrixmaterialattheinclusions. Voidscanalsonucleateinsinglecrystalsthatcontainneitherpre-existingvoidsnor inclusionssee,forexample,Cuiti~noandOrtiz,1996;Lubardaetal.,2004.Cuiti~noand Ortiz1996proposedthatvacancycondensationcanactasthenucleatingmechanismin singlecrystalsforlowstrainrates.Lubardaetal.2004foundthatvacancycondensation cannotaccountforvoidnucleationattheextremeconditionsseen,forexample,inlaserdrivenshockexperimentswithstresspulsedurationontheorderof10nsandproposed dislocationemissionasanalternatenucleatingmechanismatthishigherstrainrate 13

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regime.Figure1-3showsanexamplefromLubardaetal.2004ofavoidthathasformed inmonocrystallinecopper. Therehavebeenagreatdealofexperimentalinvestigationstoassessvoidevolution underloadinganditsinuenceontheload-carryingcapacity.Forexample,Benzergaetal. 2004aperformedanumberofexperimentsonmediumcarbonlowalloysteelinorder toinvestigatetheroleofplasticityandporousmicrostructureonthismaterial'sfailure. Thesteelbeinginvestigatedexhibitedanisotropicbehaviorbothduetoanisotropicvoid evolutionandplasticanisotropyaswellasmildtension-compressionasymmetry.Both roundbarspecimenswereusedtocharacterizethematerial'sanisotropyandinvestigate failurepropertiesandnotchedroundbarspecimenstoinvestigatetheinuenceofstress triaxialityonfailure.Inthesetests,theinitialmicrostructurewascharacterizedand interruptedtestswereperformedtotracktheevolutionofporouspropertiesincluding averagevaluesofvoidvolumefraction,voidaspectratioandvoidspacingratio.The experimentsshowvoidnucleationoccurringatverylowstrainlevelsatMnSandoxide inclusionseitherbyparticlecrackingfollowedbydecohesionordebondingaftercracking dependingontheloadingdirection.Atacertainpoint,themicrovoidswereobservedto coalesceduetomicro-neckingoftheligamentsbetweenneighboringvoids.Thespecimens alsoexhibitedanisotropiccrackpropagationi.e.,thepropagationpatterndependedon theloadingdirection.Forallspecimens,thevoidgrowthratewasfoundtobemainly connedtothecenterregionofthespecimenwiththeextensionalvoidgrowthrate dominatingatlowtriaxialitiesandtheradialvoidgrowthratebecomingdominant withincreasingstresstriaxiality.Theauthorsalsocomparedtheirexperimentaldatato theoreticalresultsusingniteelementcalculationsseeBenzergaetal.,2004b,andthe discussionofthemodelusedthereinlaterinthissection. Becauseoftherelationshipbetweenmaterialporosityandductilefailure,theability toaccuratelydescribetheevolutionofvoidsinaductilemetaliscrucialtobeingableto accuratelypredictthefailureofthematerial.Unfortunately,computationalconstraints 14

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makeitprohibitivelyexpensivetomodeleachofthemicro-voidsinmostengineering structures;therefore,themethodofexplicitlytrackingtheevolutionofeachmicro-void isnotpracticalatthistime.Analternativetoexplicitlytrackingtheevolutionofeach voidi.e.,trackingmicroscopic,orlocal,quantitiesistoincorporatetheeectsofthe micro-voidsintothemacroscopic,oraverage,propertiessuchasmacroscopicstress, strain,yielding,etc..Sincetherateofdilatationoftheporoussolidisrelatedtothe voidgrowthrate,plasticpotentialsfortheporoussolidmustbedevelopedinorderto describethevoidgrowth.Themostwidelyusedplasticpotentialforporoussolidswas proposedbyGurson1977.Forthecaseofasphericalvoidgeometry,theunitcellor RepresentativeVolumeElementRVEconsideredwasasphericalshell,whileacylindrical tubeRVEwasusedforthecylindricalvoidanalysis.Toderiveananalyticexpressionfor theplasticpotentialand,thus,fortheyieldcriterion,assumingassociatedplasticity, GursonperformedalimitloadanalysisontheRVE.Theworkinthisdissertationmainly followsthehomogenizationprocedureoutlinedinGurson1977whereGursondeveloped analyticyieldcriteriaforductilematerialscontainingeithersphericalorcylindricalvoids. InGurson'sanalysis,itwasassumedthatthevirginmaterialvoid-freeobeystheclassical vonMisesyieldcriterion.Toobtaintheplasticpotential,minimizationoftheplastic energywasdoneforaspecicvelocityeldcompatiblewithuniformstrainrateboundary conditions.Thus,theobtainedcriterionisanupperboundoftheexactplasticpotential sincetheplasticenergywasminimizedforonlyonevelocityeldratherthanforthe completesetofkinematicallyadmissiblevelocityelds.Gurson'syieldcriterionisgiven forsphericalvoidsas S G = e Y 2 +2 f cosh 3 m 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =0{1 andforcylindricalvoidsas C G = C eqv e Y 2 +2 f cosh p 3 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(f 2 =0{2 15

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where C eqv = 8 > < > : +3 f +24 f 6 2 forplanestrain 1foraxisymmetry : {3 Inthepreviousexpressions, e isthemacroscopicvonMiseseectivestress, Y isthe yieldstrengthofthevirginmaterial, f isthevoidvolumefractionsometimesreferredto astheporosity, m isthemacroscopicmeanstressand isthesumofthein-plane stressese.g., = 11 + 22 ifthe3-directionistheout-of-planedirection.Note thatGurson'scriteriadependsnotonlyonthesecondinvariantofthestressdeviator, butalsoonthepressureorthemeanstressandonthelevelofporosityinthematerial; therefore,theyieldfunctionincorporatestheinuenceofvoidsontheplasticdeformation inamaterial.Themainpremisebehindincorporatingthisvoideectisthatavolumeof virginmaterialwillbehavedierentlyunderanappliedloadthanwillavolumeofmaterial thathasareducedload-bearingareaduetothepresenceofvoidsandtheirsubsequent evolutiongrowthandcoalescence.Gurson'ssphericalvoidcriterionisconsideredmost oftenintheliteraturebutthecylindricalvoidcriterionisapplicabletocertainproblemsas welle.g.,theplanestressanalysisofsheetmetal. Thevoidvolumefraction, f ,theratioofthevoidvolumetothetotalvolume evolvesbothfromthenucleationatsecond-phaseparticlesandthegrowthofexistingvoids see,forexample,ChuandNeedleman,1980;LemaitreandDesmorat,2005suchthat f = f growth + f nucleation {4 wheretherateofchangeduetogrowthisdeterminedfromtheassumptionofplastic incompressibilityas f growth = )]TJ/F21 11.9552 Tf 11.956 0 Td [(f d P kk {5 16

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with d P ij beingtheplasticpartoftherateofdeformationtensor.Thevoidvolumefraction'srateofchangeduetonucleationisgenerallygivenas f nucleation = A Y + B 3 kk {6 forstress-controllednucleation,oras f nucleation = A N {7 forplasticstrain-controllednucleation,where isobtainedfromthefollowingrelation: ij d P ij = )]TJ/F21 11.9552 Tf 11.956 0 Td [(f Y : {8 Inthecaseofplasticstrain-controllednucleation,thefollowingstatisticalexpressionhas beenfrequentlyused: A N = f N s N p 2 exp )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(1 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( N s N 2 # {9 where f N isthevolumefractionofvoid-nucleatingparticles, N isthemeannucleation strainand s N isthestandarddeviation. AmodiedversionofGurson'ssphericalcriterionwasusedinSpitzigetal.1988to comparewithexperimentaltestsoncompactedironspecimens.Themodiedcriterioncan bewrittenas = e 2 +2 f m cosh 3 m m 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 m =0{10 where istheowstressand m isamaterialcoecientrepresentingstrainhardeninga powerlawmatrixmodelwasusedinthepapertocomparewiththeexperimentaldata. AwidelyusedmodicationofGurson'ssphericalyieldcriterionwassuggested inTvergaard1981andTvergaard1982basedoncomparisonswithniteelement calculationsofshearbandinstabilitieswheretheinstabilityisdeterminedbyalossof ellipticityofthegoverningequations.Withniteelementcalculations,theminimization oftheplasticenergyisdoneoveralargersetofkinematicallyadmissiblevelocityelds thaninGurson'sanalysisinwhichasinglekinematicallyadmissiblevelocityeldis 17

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assumed;therefore,adjustmentstoGurson'syieldcriteriacanbeproposedbasedonthese calculations.InTvergaard1982,theniteelementcalculationsinvolvedacylindrical cellcontainingasphericalvoidwhichwascomparedwiththemodiedGursonspherical yieldcriterioni.e.,comparedwiththemodiedformproposedinTvergaard,1981. Specically,thesecalculationsweremeanttorepresentaperiodicarrayofsphericalvoids whichwerearrangedsuchthathexagonalrepresentativevolumeelementsRVEscould bettogethertoformthestructurethecylindricalRVEusedinthecalculationswas anapproximationofthehexagonalRVE.Basedontheseaxisymmetricniteelement calculations,thefollowingmodiedformofGurson'syieldcriterionwassuggested: = e Y 2 +2 fq 1 cosh q 2 3 m 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q 3 f 2 =0{11 wherethe q i arethettingparametersintroducedbyTvergaardallequaltoonein Gurson'soriginalexpression.Tvergaardrecommendedvaluesof q 1 =1 : 5, q 2 =1and q 3 = q 2 1 basedontheniteelementresults.Theintroductionofthesettingparameters q 1 q 2 and q 3 canbethoughtofasanecessaryadjustmentoftheyieldsurfacetoaccount fortheinuenceofneighboringvoids.LeblondandPerrin1990usedaself-consistent analysisofavoidwithinaporousplasticspherethusexplicitlyaccountingfortwo dierentvoidgeometriesandtheirinteractiontorecommendvaluesforTvergaard's ttingparametersas q 1 =4 =e 1 : 47, q 2 =1and q 3 = q 2 1 TvergaardandNeedleman1984furthermodiedGurson'ssphericalyieldcriterion toaccountfortheonsetofvoidcoalescenceleadingtonalmaterialfracture.Theauthors usedthismodiedyieldcriterioninbothnumericalandniteelementcalculations tocomparewithexperimentaldataofacopperrodfracturingunderuniaxialtension exhibitingcup-conefracture.ThisnalmodicationisgenerallyreferredtoastheGTN criterionafterthethreeauthorsintheliteratureandiswhatistypicallyfoundinthe 18

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niteelementcodes.TheGTNcriterionisgivenbelow: = e Y 2 +2 f q 1 cosh q 2 3 m 2 Y )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q 3 f 2 =0{12 where f istheeectivevoidvolumefractionwhichisafunctionoftheactualvoidvolume fraction, f ,andrepresentsthemodicationoftheyieldcriteriontoaccountfornal materialfailure.Theeectivevoidvolumefractionisgivenas f f = 8 > < > : f for f f C f C + f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f C f U )]TJ/F21 11.9552 Tf 11.955 0 Td [(f C f F )]TJ/F21 11.9552 Tf 11.955 0 Td [(f C for f>f C {13 where f C isthecriticalvoidvolumefractionofamaterial, f F isthevoidvolumefraction atnalfailureand f U = f f F istheultimatevalueoftheeectivevoidvolumefraction i.e.,theeectivevoidvolumefractionatwhichthemacroscopicstresscarryingcapacity vanishessuchthat f U =1 =q 1 if q 3 = q 2 1 isusedassuggestedbyTvergaard.Richelsenand Tvergaard1994performedthree-dimensionalunitcellniteelementcalculationswith proportionalloadingi.e.,theappliedmacroscopicstressesontheboundariesoftheunit cellwereconstantmultiplesofeachotherforthedurationofthecalculationtocompare withtheaxisymmetricmodelofEquation1{12notethatGurson'ssphericalmodelis anaxisymmetricmodelsincethegeometryoftheRVEandtheassumedvelocityeldwere bothaxisymmetricinthelimitloadanalysis.Forequalappliedtransversestresses,the GTNmodelusing q 1 =1 : 5wasfoundtoagreewellwiththeunitcellcalculations.Unit cellcalculationswithunequalappliedtransversestressesalsoshowedgoodagreementwith theGTNmodel;specically,theunitcellresultsforlargelyunequaltransversestresses werefoundtoliebetweenthetheoreticalresultsof q 1 =1 : 5and q 1 =1 : 0intheGTN model. Inmanycases,themicro-voidsthatformwithinaloadedmaterialareellipsoidal ratherthansphericalduetotheinuencesofasymmetricalloadingand/orananisotropic 19

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microstructure.Gologanuetal.1993usedahomogenizationprocedureonanaxisymmetric,ellipsoidalRVEcontainingaconfocal,prolateellipsoidalvoidtoarriveatan analyticexpressionforprolate,ellipsoidalvoidsratherthanforsphericalvoids.The generalexpressionisgivenas = e Y 2 +2 fq 1 cosh A : Y )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q 3 f 2 =0{14 where = 1 p 3 + p 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 ln e 1 =e 2 ln f )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 {15 andthevoidanisotropytensor, A isgivenas A = e 2 ~e x ~e x + ~e y ~e y +[1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 e 2 ] ~e z ~e z : {16 Inthepreviousequations, e 1 and e 2 aretheeccentricitiesoftheellipsoidalvoidand RVE,respectively,andshouldnotbeconfusedwiththeCartesianbasisvectors ~e x ~e y and ~e z .Theevolutionoftheeccentricitiesisgovernedbytheevolutionofthevoidshape parameter, S =ln a 1 =b 1 ,where a 1 and b 1 arethemajorandminorsemi-axesofthevoid, respectively.Thefunction, ,isgivenas e i = 1 2 e 2 i )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(e 2 i 2 e 3 i tanh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 e i : {17 GologanushowedthatthemodelreducedtoGurson'ssphericalcriterioninthecaseof asphericalvoid e 1 = e 2 =0andtoGurson'scylindricalcriterioninthecaseofa cylindricalvoid e 1 = e 2 =1.TheformulasgiveninthisparagraphareforGologanu's prolateellipsoidalvoidanalysis;Gologanudevelopedsimilarexpressionsforoblatevoids see,forexample,Gologanuetal.,2001.Garajeuetal.2000extendedtheworkof Gologanubyinvestigatingtheinuenceofvoiddistributionevolutioninadditiontothe inuenceofvoidshapeevolution. Liaoetal.1997developedaGurson-typecriteriafortransverselyisotropicmetal sheetsunderplanestressconditionsusingHill's1948yieldcriterionseeHill,1948,1950 20

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forthematrixmaterial.Theproposedmodelisasfollows: = e Y 2 +2 f cosh s 1+2 R 2+ R 3 m 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =01{18 wheretheanisotropyparameter, R ,isdenedastheratioofthetransverseplastic strainratetothethrough-thicknessplasticstrainrateunderin-planeuniaxialloading conditions."Notethatthepreviousmodelaccountsforporosityintransverselyisotropic materialsandreducestoGurson'scylindricalcriterionforisotropy R =1. BenzergaandBesson2001alsodevelopedaGurson-typecriteria,intendedfor orthotropicporousmetals,byassumingamatrixmaterialthatcouldbecharacterized usingHill's1948criterion.Themodelisgivenas = 3 0 : H : 0 2 2 Y +2 fq 1 cosh q 2 3 m h Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q 3 f 2 =0{19 where h iscalledtheanisotropyfactor h =2forisotropyandisafunctionofthe macroscopicanisotropytensor, H .Equation1{19diersfromEquation1{18inthat itassumessphericalratherthancylindricalvoidsandappliestothemoregeneralcaseof orthotropicmaterialsversustransverselyisotropicmaterials. BessonandGuillemer-Neel2003extendedtheGTNmodelofEquation1{12to includemixedisotropicandkinematichardeningwithinathermodynamicalframework. Statevariablesrelatedtoisotropicandkinematichardeningareemployedinthemodel andareevolvedinathermodynamically-consistentmannerusingadissipationpotential seeLemaitreandChaboche,1990.Theextendedmodelcanbewrittenas = e 2 +2 f q 1 cosh q 2 3 m 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q 3 f 2 =0{20 where isaneectivescalarstressandisafunctionofthestatevariablesrelatedto isotropicandkinematichardening.BessonandGuillemer-Neelobtainedgoodagreement betweenthepreviousmodelandniteelementcalculationsofcyclically-loadedunitcells. 21

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Benzergaetal.2004busedayieldcriterionwhichcombinesmanyoftheproperties ofEquations1{14and1{19inordertocomparewiththeexperimentalresultsgivenin Benzergaetal.2004a.Theyieldcriterion,priortovoidcoalescence,usedinthepaperis asfollows: = 3 0 : H : 0 2 2 +2 fq w cosh A : h )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q w f 2 =0{21 where q w =1+ q 1 )]TJ/F15 11.9552 Tf 12.041 0 Td [(1 = cosh S wasintroducedbyGologanusee,forexample,Gologanu etal.,2001.Inthisyieldcriterion, replaces Y inthedenominatorstoaccountfor hardeningasinEquation1{20.Theauthorsuseadierentcriteriontoaccountforyield aftercoalescencethisapproachcanbeviewedasanalogoustothe f parameterinthe GTNmodel;anaccurateductileyieldcriterionmustconsidervoidcoalescenceseparately frominitialvoidgrowth.Thispost-coalescencecriterionisgivenas = q 3 2 0 : H : 0 + 1 2 j 3 m j )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(3 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 f ;S =0{22 where istheligamentsizeratiorelatedtothevoidspacingand f isafunctionof both andthevoidshapeparameter, S .Theauthorsshowedgoodagreementbetween theexperimentalresultsobtainedinBenzergaetal.2004aandtheyieldcriteriagivenin thisparagraph. ResearchersatLosAlamosNationalLaboratoryLANLhaveextendedGurson's modelsee,forexample,AddesioandJohnson,1993;Bronkhorstetal.,2006;Maudlin etal.,1999tosolvecomputationalproblemsinvolvinghighlydynamicmaterialbehavior i.e.,highstrainratesandhightemperatures.Thismodiedmodelisreferredtoas TEPLAandusesaMie-Gruneisenequationofstatetospecifythepressurealongwitha mechanicalthresholdstrengthMTSmodeltoaccountforrateandtemperatureeects. Theaboveworksandmanyothershavedoneanenormousamountofresearchaimed atextendingGurson'sanalysisbothtomorecomplexmaterialsaswellastomorecomplex voidshapesanddistributions.Itisworthnotingthatallofthemodelsmentionedabove areapplicabletometallicmaterialswithcubicstructure;i.e.,themodelsweredeveloped 22

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forface-centered-cubicmetalse.g.,aluminumandbody-centered-cubicmetalse.g., steel.Theaimofthisresearchistodevelopmodelstodescribetheyieldingandfailure ofhexagonal-closed-packedmetalssuchas -titanium,zirconium,anduraniumi.e., materialsthathavedierentyieldstrengthsintensionversuscompression. Theoutlineofthisdissertationisasfollows.Thekinematichomogenizationapproach ofHill-MandelHill,1967;Mandel,1972thatisusedtodeveloptheanalyticalplastic potentialsofthevoid-matrixaggregateisdescribedinChapter2.Chapter3detailsthe developmentofaclosed-formplasticpotentialforanisotropicporousaggregatecontaining sphericalvoidswhenthematrixdisplaystension-compressionasymmetry.Thereturn mappingalgorithmsusedinimplementingthematrix,orvoid-free,plasticpotential intoaniteelementcodeisoutlinedinChapter4andthenecessaryrstandsecond derivativesaregivenforthechosenmatrixplasticpotentialofCazacuetal.2006,which canaccountfortension-compressionasymmetryinthevoid-freematerial.InChapter 5,comparisonsaremadebetweenthedevelopedisotropicporousplasticpotentialfor sphericalvoidsandniteelementunitcellcalculations.Theniteelementcalculationsare designedsuchthatasphericalvoidisexplicitlymeshedinsideanaxisymmetriccylinder whoseresponseisgovernedbytheCazacuetal.2006plasticpotential.Macroscopic plasticpotentialsforvoid-matrixaggregatescontainingcylindricalvoidswhenthematrix displaystension-compressionasymmetryaredevelopedinChapter6andaproposedplane strainplasticpotentialiscomparedtoniteelementunitcellcalculationsinChapter7. Finally,theisotropicplasticpotentialdevelopedinChapter3foravoid-matrixaggregate containingsphericalvoidsandamatrixexhibitingtension-compressionasymmetryis extendedinChapter8toincludetheeectsofmatrixanisotropyontheyieldingof thevoid-matrixaggregate.Thedevelopedanisotropicmacroscopicplasticpotentialis comparedtotransverselyisotropicunitcellniteelementcalculations. 23

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Figure1-1.VoidnucleationatMnSinclusionsinsteel.aandbLongitudinalloading. canddTransverseloading.eDecohesionatthepolesofatinyspherical MnSparticleclosetoagrainboundaryand100lmaheadofthecracktip% Nitalsolutionetched.FieldemissionSEMimagingusingback-scattered electronsinacandeandsecondaryelectronswithanin-lensdetectorin dtorevealthecrack.[Reprintedwithpermission.Benzerga,Besson,and Pineau2004a.Anisotropicductilefracture.PartI:Experiments. Acta Materialia 52 ,4623-4638.] 24

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Figure1-2.Reconstructedimagesofthesameinternalsectionatvedierentrelevant deformationstepsofametalcompositeconsistingofanaluminummatrixwith embeddedsphericalceramicparticlesainitialstate;b =0.065;c = 0.15;d =0.35-0.48;e =0.51-0.81.DetailApre-existingholesinduced bytheextrusionprocesswhichstarttogrowduringtension.DetailB particle/matrixdecohesionduringthetensileloading.DetailCcoalescence. Thetensiledirectionisverticalinthegure.Thevarioustruestrainvalueson dandeconveythepresenceofneckinginthesample.[Reprintedwith permission.Babout,Maire,Buere,andFougeres2001.Characterizationby X-raycomputedtomographyofdecohesion,porositygrowthandcoalescencein modelmetalmatrixcomposites. ActaMaterialia 49 ,2055-2063.] 25

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Figure1-3.TEMmicrographoflaser-shockedmonocrystallinecopper:darkeldimageof anisolatedvoidneartherearsurfaceofthespecimenandassociated work-hardenedlayerwhiterim.[Reprintedwithpermission.Lubarda, Schneider,Kalantar,Remington,andMeyers2004.Voidgrowthby dislocationemission. ActaMaterialia 52 ,1397-1408.] 26

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CHAPTER2 HOMOGENIZATIONAPPROACH Allengineeringmetalsandalloyscontaininclusionsandsecond-phaseparticlesat whichmicro-voidsnucleateeitherbydecohesionoftheparticle-matrixinterfaceorby particlebreaking.Subsequently,voidsgrowduetoplasticdeformationofthesurrounding materialuntilalocalizedinternalneckingoftheintervoidmatrixoccursthatleadsto theformationofsomemacroscopiccrack.Investigationsoftheexpansionofvoidsof cylindricalandsphericalgeometriesinrigididealplasticmatricesbyMcClintock1968 andRiceandTracey1969haveestablishedtheeectsofstressstatestresstriaxiality onthevoidgrowthrate.Gurson1977proposedapproximateyieldcriteriaandowrules forductilematerialscontainingsphericalorcylindricalcavitiesusinganupper-bound approach.Tobetteraccountfortheinteractionbetweenvoids,severalmodicationsof Gurson'scriterionhavebeenproposedbasedonresultsoftwo-dimensionalnite-element studiese.g.,KoplikandNeedleman,1988;Tvergaard,1981;TvergaardandNeedleman, 1984orfromrigorousestimatesoftheexactmacroscopicpotentialsforareviewofthose alternativeapproachespioneeredbyTalbotandWillis1985,seeforexample,Garajeu andSuquet1997;Leblondetal.1994.Recently,analyticalcriteriathataccountforthe combinedeectsofvoidshapeandmatrixanisotropyonthemacroscopicresponseof ductileporoussolidswereproposedseeforexampleBenzergaandBesson,2001;Monchiet etal.,2008. Inallthemodelsmentioneditisassumedthatthematrixhasthesameyieldin tensionandcompression.However,intheabsenceofvoids,somecubicmaterialssuch ashighstrengthsteelsormolybdenumexhibittension-compressionasymmetry.This strength-dierentialS-Deectisaconsequenceofcrystalslipthatdoesnotobeythe well-knownSchmidlawseeforexampleViteketal.,2004.Also,twinningatthesingle crystallevelmayresultinastrongtension-compressionasymmetryattheaggregate levelinsomecubicmetalsseeHosfordandAllen,1973.HexagonalclosepackedHCP 27

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metalsexhibittension-compressionasymmetryasaresultoftwinningactivationatthe singlecrystallevel.HCPmetalscandeformeitherbysliportwinning,withtwinning becomingincreasinglyprominentwithincreasingstrainrate.Ifametaldeformsbyslip aloneareversibleshearmechanism,irreversibleowdependsonlyonthemagnitudeof theresolvedshearstresssuchthatthestrengthsintensionandcompressionareequal; however,iftwinningalsoexistsasadeformationmechanismthenadierencebetweenthe strengthsintensionandcompressionwillexist. Thischapterisorganizedasfollows.Section2.1introducesthehomogenization approachduetoHillandMandelHill,1967;Mandel,1972thatisusedindeveloping themacroscopicplasticpotentialsforthevoid-matrixaggregatesconsideredinthis dissertation.Section2.2presentstheisotropicversionoftheCazacuetal.2006plastic potentialthatisusedtodescribethematrixmaterialinallanalysesexceptforthose discussedinChapter8whichfocusesonanisotropicplasticpotentials.Section2.3 presentsthederivationoftheplasticmultiplierrateassociatedtotheisotropicversionof theCazacuetal.2006plasticpotential.Thederivationofthisplasticmultiplierrateis themainchallengeindeterminingthelocalplasticdissipationinthematrix. 2.1KinematicHomogenizationApproachofHillandMandel ThecurrentworkwillconsiderbothasphericalandacylindricalrepresentativevolumeelementRVEcontainingavoidofsimilargeometryi.e.,sphericalandcylindrical, respectively.TheseRVEswerechosenbothbecausethevoidgeometriesconsideredare typicallyseeninexperimentsandbecausethesymmetryofthevoidandoutersurface greatlysimpliesthehomogenizationanalysis.IfthematrixmaterialintheRVEisassumedtoberigidplastic,thenanyvolumechangeisdueexclusivelytovoidevolutionsuch thattherateofvoidgrowthiseasilyobtainedfromtherateofdilatationseeEquation 1{5. Considerarepresentativevolumeelement V ,composedofahomogeneousrigidplasticmatrixandatraction-freevoid.Thematrixmaterialisdescribedbyaconvexyield 28

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function inthestressspaceandanassociatedowrule d = @' @ ; {1 where istheCauchystresstensor, d = = 2 r v + r v T denotestherateofdeformation tensorwith v beingthevelocityeld,and 0standsfortheplasticmultiplierrate.The yieldsurfaceisdenedas =0.Let C denotetheconvexdomaindelimitedbythe yieldsurfacesuchthat C = f j 0 g : {2 Theplasticdissipationpotentialofthematrixisdenedas w d =sup 2 C : d {3 where:"denotesthetensordoublecontraction.Uniformrateofdeformationboundary conditionsareassumedontheboundaryoftheRVE, @V ,suchthat v = Dx forany x 2 @V {4 with D ,themacroscopicrateofdeformationtensor,beingconstant.Fortheboundary conditionsofEquation2{4,theHill-MandelHill,1967;Mandel,1972lemmaapplies; hence, h : d i V = : D ; {5 where hi denotestheaveragevalueovertherepresentativevolume V ,and = h i V Furthermore,thereexistsamacroscopicplasticdissipationpotential W D suchthat = @W D @ D {6 with W D =inf d 2 K D h w d i V ; {7 29

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where K D isthesetofincompressiblevelocityeldssatisfyingEquation2{4formore detailsseeGologanuetal.,1997;Leblond,2003.Thematrixmaterialbeingconsidered obeystheisotropicversionofthepressure-insensitiveyieldcriterionthatcapturesstrength dierentialeectsofCazacuetal.2006. 2.2YieldCriterionfortheMatrixMaterial Twinningandmartensiticsheararedirectionaldeformationmechanismsand,if theyoccur,yieldingwilldependonthesignofthestressseeHosford,1993.Early polycrystallinesimulationsresultsbyHosfordandAllen1973whoanalyzeddeformation bytwinninginrandomFCCpolycrystals,predictedayieldstressinuniaxialtension22% lowerthanthatinuniaxialcompression.Basedonthesesimulationsandmorerecent resultsconcerningtheeectsofnon-Schmidtypeyieldcriteriaatthesingle-crystallevel onthepolycrystallineresponsee.g.,seeViteketal.,2004,itcanbeconcludedthatyield lociwithastrongasymmetrybetweentensionandcompressionshouldbeexpectedin anyisotropicpressure-insensitivematerialthatdeformseitherbytwinningordirectional slip.Toaccountforstrengthdierentialeectsinpressureinsensitivematerials,Plunkett 2005andCazacuetal.2006proposedthefollowingisotropicformforyielding: F = j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 2 + j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 2 + j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 2 {8 where s i aretheprincipalvaluesoftheCauchystressdeviator.InEquation2{8, k takes intoaccountthetension-compressionasymmetryandisgivenby k = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(h T ; C 1+ h T ; C {9 where h T ; C = v u u u u t 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 {10 intheisotropiccase.Theterms T and C inthepreviousexpressionsaretheyield strengthsintensionandcompression,respectively.Thecurrentworkwillsometimesrefer 30

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toEquation2{8alongwiththedenitionsinEquations2{9and2{10astheCPB06 isotropiccriterion"fornotationalconvenience. Let e 1 ; e 2 ; e 3 beaCartesiancoordinatesystemassociatedwiththeprincipal directionsofthestresstensor.Duetothetension-compressionasymmetry,theprojection oftheyieldsurfaceinEquation2{8inthedeviatoricplanetheplanewithnormal n = 1 p 3 e 1 + 1 p 3 e 2 + 1 p 3 e 3 hasthreefoldsymmetry.Let f i betheprojectionsoftheeigenvectors e i i =1 ::: 3onthedeviatoricplane.Asanexample,Figure2-1illustratesdeviatoric -planerepresentationsoftheyieldcurvesgivenbyEquation2{8for T = C =0 : 82 and T = C =1 : 21,alongwiththevonMisesyieldlocusforcomparison.Noteavery drasticdepartureoftheyieldlocusfromtheVonMisescircle.Thisdepartureisdueto astronginuenceofthethirdinvariantofthestressdeviatoronyielding,whichresults fromtension-compressionasymmetry.Inthe -planerepresentation,theradialcoordinate isrelatedtothesecondinvariantofthestressdeviatorwhiletheangularcoordinateis relatedtothethirdinvariantofthestressdeviator.Therefore,thevonMisescircleis independentofthethirdinvariant,whiletheCPB06criterionfornon-zero k dependson boththeradialandangularcoordinatesi.e.,itisnotacirclesuchthatitisdependent onboththesecondandthirdinvariantsofthestressdeviator. Theyieldfunctiona.k.a.thestresspotential, ,isgivenasfollows: = )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y where isascalareectivestressassociatedtoEquation2{8and Y representsthe material'shardeningcondition.If Y istakentobetheyieldstrengthintension, T ,then thepreviousequationyields s ;k; T = e s ;k )]TJ/F21 11.9552 Tf 11.955 0 Td [( T {11 where e isnowtheeectivestressformulatedsuchthatitreducestotheyieldstrengthin tensionforuniaxialloading.Theexplicitexpressionforthiseectivestressmaybefound 31

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bywritingEquation2{8fortheuniaxialtensioncaseandforcing e equalto T .The resultingexpressionisgivenas e = m p F = m v u u t 3 X i =1 j s i j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks i 2 {12 where m := s 9 2 1 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 : {13 Noticethatfor k =0, m = p 3 = 2and e simplybecomesthevonMiseseectivestress. Also,notethatfortheuniaxialcompressioncase, e = C r 3 k 2 +2 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 = C T C = T suchthattheyieldsurfaceofEquation2{11isindeedsatised. SincetheCazacuetal.2006criterionishomogeneousofdegreeone,Equation2{1 yieldsassumingrigidplasticbehaviorinthematrix w )]TJ/F37 11.9552 Tf 5.48 -9.684 Td [(d P = ij d P ij = ij @f @ ij = T {14 suchthatthedeterminationofthelocalplasticdissipation w d P reducestodetermining anexpressionfortheplasticmultiplierrate, .Inthenextsection,theplasticmultiplier ratecorrespondingtotheisotropicversionoftheCazacuetal.2006criterionwillbe derived. 32

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2.3PlasticMultiplierRateDerivation Akeystepinobtainingtheoverallplasticpotential,W,ofthevoid-matrixaggregate isthecalculationofthelocalplasticdissipation,wseeEquations2{3and2{14, associatedwiththeCazacuetal.2006criterion,theisotropicversionofwhichisshown inEquation2{12.Sincethematrixmaterialisisotropic,theeigenvectors e i i =1 ::: 3 oftheCauchystresstensor arealsoeigenvectorsoftherateofdeformationtensor d SubstitutingintheowruleEquation2{1theexpressioninEquation2{12ofthe plasticpotential, ,itfollowsthattheprincipalvaluesofthestrainratetensor d i are expressedas d P = @ e @ = @ e @s i @s i @ = 2 3 @ e @s )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 @ e @s )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 @ e @s where 6 = 6 = .Makinguseoftherelation @ e @s = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( @ e @s + @ e @s yields d P = @ e @s : {15 ThederivativetermsinEquation2{15mustnowbedetermined.Notingthatthe threestressdeviatorsaredependentfunctionsofeachother,theeectivestressiswritten asfollows: e = m 2 s 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 3 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(s 3 3 2 + 2 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 3 3 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 s 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 3 3 2 + 2 s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 1 3 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 s 3 )]TJ/F21 11.9552 Tf 11.956 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 1 3 2 1 2 33

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wherethefactthat s 1 + s 2 + s 3 =0hasbeenused.Now,takingderivativesoftheabove expressionyields: @ e @s 1 = m p F 2 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.088 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k @ e @s 2 = m p F 2 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.087 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k @ e @s 3 = m p F 2 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.088 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k {16 Notethat @ e @s 1 + @ e @s 2 + @ e @s 3 =0 : Equations2{15and2{16cannowbeusedinordertocomeupwithexpressions fortheprincipalplasticrateofdeformationcomponents, d P ,intermsof s F and k as follows: d P 1 = m p F 2 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.087 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k d P 2 = m p F 2 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.088 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k d P 3 = m p F 2 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.087 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k {17 wherethethreecomponentssumtozerosince d P isadeviatorictensor.Equation2{17 containsthegeneralexpressionsfortheprincipalrateofdeformationcomponentsinterms of s F and k .Inordertosimplifytheexpressionsfurther,someassumptionregarding 34

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thesignoftheprincipalstressdeviators, s ,mustbemade.Therearetwopossiblecases correspondingtothefollowingstressstates: 1.Theloneprincipalstressdeviatorisnegativesuchthat s 1 > 0; s 2 0; s 3 < 0 J 3 0. 2.Theloneprincipalstressdeviatorispositivesuchthat s 1 > 0; s 2 0; s 3 < 0 J 3 0. Theterm J 3 aboveisthethirdinvariantofthedeviatoricstresstensorandcanbe expressedasfollows: J 3 = s 1 s 2 s 3 : Relationshipsbetweenvariousfunctionsof k m andtheratiooftheyieldstrengthsin tensionandcompressioncanbederivedusingEquations2{9,2{10,and2{13.Some usefulrelationsaregiveninTable2-1andwillbeusedthroughouttheremainderofthis document. 2.3.1Plasticmultiplierratewhen J 3 0 Forthecasewhen J 3 0,Equation2{17isusedtoobtain d P 1 = m p F 1 3 s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + s 1 + s 2 + k 2 = m p F 1 3 s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +1 + s 1 + s 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(k 2 +2 k +1 = m p F 1 3 s 1 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 +4 ks 2 {18 and d P 2 = m p F 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + s 1 + s 2 + k 2 = m p F 1 3 s 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 +4 ks 1 : {19 Notethat d P 3 = )]TJ/F26 11.9552 Tf 11.291 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 1 + d P 2 = m p F 1 3 s 3 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(3 k 2 +2 k +3 35

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suchthat d P 3 < 0since s 3 < 0.Also, d P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(d P 2 = m p F s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 suchthat d P 1 >d P 2 since s 1 >s 2 Now,fromEquation2{18, s 1 = 1 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 d P 1 3 p F m )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 ks 2 # andfromEquation2{19, s 2 = 1 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 d P 2 3 p F m )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 ks 1 # : Solvingtheabovetwoexpressionsfor s 1 and s 2 yields, s 1 = p F m )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(c 1 d P 1 + c 2 d P 2 s 2 = p F m )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(c 2 d P 1 + c 1 d P 2 where c 1 = 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 3 k 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 3 +2 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k +3 = 1 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 3 k 2 +2 k +3 = 1 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 C T 2 and c 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 k 3 k 4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k 3 +2 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 k +3 = 1 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 k 3 k 2 +2 k +3 = 1 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 # 36

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usingsomeoftherelationsgiveninTable2-1.Now,usingtheaboveequationsalongwith Equation2{8andtherelation s 3 = )]TJ/F15 11.9552 Tf 11.291 0 Td [( s 1 + s 2 yieldsthefollowing: F = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s 2 1 + s 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + s 2 3 + k 2 = F m 2 2 nh )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(c 1 d P 1 + c 2 d P 2 2 + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(c 2 d P 1 + c 1 d P 2 2 i )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + c 1 + c 2 2 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(d P 1 + d P 2 2 + k 2 o = F m 2 2 n )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 2 2 )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(c 2 1 + c 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 + c 1 + c 2 2 + k 2 + d P 1 d P 2 4 c 1 c 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 +2 c 1 + c 2 2 + k 2 o : Aftersomealgebra,andtakingintoaccountthefollowingrelations, c 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 2 = 1 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 c 1 + c 2 = 3 3 k 2 +2 k +3 onecanobtainthefollowingexpressionfortheplasticmultiplierrate, 1 ,correspondingto thecasewhen J 3 0: 1 2 = 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 2 2 + 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(10 k +3 3 k 2 +2 k +3 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 3 2 : Theaboveequationcanbebrokenoutintoacontributionfromthesecondinvariantofthe plasticrateofdeformationtensor,andacontributionfromthecomponentassociatedwith thelonenegativeprincipalcomponenti.e., d P 3 .Theexpressioniswrittenbelowas 1 2 = A 1 h )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 2 2 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 3 2 i + B 1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 3 2 {20 37

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where A 1 = 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 = 2 3 2 6 4 1 2 )]TJ/F26 11.9552 Tf 11.955 13.271 Td [( T C 2 3 7 5 {21 and B 1 = 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(12 k 3 k 2 +2 k +3 =2 2 6 4 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 3 7 5 : {22 Intheaboveexpressions, m isdenedbyEquation2{13.NotethatEquation2{20 reducestovonMisesif k =0i.e.,if T = C .Also,notethattheplasticmultiplierrate forthiscase, 1 ,issingularfor k =1. 2.3.2Plasticmultiplierratewhen J 3 0 Equation2{17canbeusedagaintodeterminetheexpressionsforthersttwo principalstrainratescorrespondingtothiscaseasfollows: d P 1 = m p F 1 3 2 s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + s 1 + k 2 = m p F 1 3 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 s 1 {23 and d P 2 = m p F 1 3 s 2 + s 1 + k 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 = m p F 1 3 3 s 2 + k 2 + s 1 k : {24 Notethat,similarlytotheprevioussection, d P 1 > 0since s 1 > 0and d P 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(d P 3 = m p F s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(s 3 + k 2 suchthat d P 2 >d P 3 since s 2 >s 3 38

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SolvingEquations2{23and2{24fortheprincipalstressdeviatorssimilartothe procedureusedintheprevioussectionnowyields s 1 = 3 p F m 1 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 d P 1 and s 2 = 1 3 1 1+ k 2 3 p F m d P 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 ks 1 # : PluggingthepreviousequationsintoEquation2{8yieldsthefollowing: F = s 2 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s 2 2 + s 2 3 + k 2 = 2 F m 2 2 3 k 2 +1 d P 1 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 2 + d P 2 + )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 k 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 d P 1 2 1 1+ k 2 + 3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 d P 2 + )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 k 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 d P 1 d P 1 = 2 F m 2 2 1 1+ k 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 1 2 9 k 4 +6 k 3 +10 k 2 +6 k +9 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 2 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 2 2 + d P 1 d P 2 # = F m 2 2 1 1+ k 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 1 2 3 k 2 +10 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 2 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 3 2 : Usingthepreviousrelation,onecannowobtainthefollowingexpressionfortheplastic multiplierrate, 2 ,correspondingtothecasewhen J 3 0: 2 2 = 1 m + k 2 3 k 2 +10 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 2 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 3 2 : Writingthepreviousequationinthesameformasintheprevioussectionyieldsthe following: 2 2 = A 2 h )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(d P 2 2 + )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(d P 3 2 i + B 2 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(d P 1 2 39

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where A 2 = 1 m + k 2 = 2 3 2 6 4 1 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 and B 2 = 1 m + k 2 12 k 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 =2 2 6 4 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 : Notethatthefollowingrelationshipexistsbetweentheparameters A 2 and B 2 abovewith theparameters A 1 and B 1 ofEquation2{20: A 2 = C 2 A 1 B 2 = )]TJ/F21 11.9552 Tf 9.298 0 Td [(C 2 T C 2 B 1 where C 2 = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 1+ k 2 = 2 6 4 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 : {25 Usingtherelationshipbetween A 1 A 2 B 1 and B 2 yieldsthefollowingexpressionforthe plasticmultiplierrate,correspondingtothecasewhen J 3 0,intermsof A 1 and B 1 : 2 2 = C 2 n A 1 h )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 1 2 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 2 2 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 3 2 i )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( T C 2 B 1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(d P 1 2 o : {26 Notethat,aswasthecasewithEquation2{20,Equation2{26reducestovonMisesif k =0i.e.,if T = C .Also,notethattheaboveplasticmultiplierratealsocontainsa singularity,thistimeat k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1. 40

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2.3.3Generalplasticmultiplierrateexpression Itcanbeshownthatbothplasticmultiplierrates2{20and2{26areequivalentfor thecasewhen J 3 =0andreducetothefollowingexpression: 2 =4 k 2 +1 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(d P 1 2 = 2 3 1+ T C 2 # )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(d P 1 2 Thepreviousexpressionfurtherreducesto 2 = = 3 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(d P 1 2 forthevonMisescaseof k =0i.e,ifthematerialhasthesamestrengthsintensionandcompressionaswould beexpectedwhen J 3 =0suchthat d P 2 =0and d P 3 = )]TJ/F21 11.9552 Tf 9.298 0 Td [(d P 1 refertoEquations2{17for k =0. Itisconvenienttowriteasingle,generalexpressionfor thatincludesallpossible stressstates.Equations2{20and2{26canbecombinedtoyieldthefollowinggeneral plasticitymultiplier: 2 = z 1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(d P 1 2 + z 2 tr h )]TJ/F37 11.9552 Tf 5.479 -9.683 Td [(d P 2 i + z 3 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(d P 3 2 {27 where z 1 := 1 2 sgn 2 J 3 +sgn J 3 2 6 4 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 {28a z 2 := 2 6 4 2 = 3 2 )]TJ/F26 11.9552 Tf 11.956 13.27 Td [( T C 2 3 7 5 + 1 2 sgn 2 J 3 +sgn J 3 2 6 4 2 = 3 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 30.822 8.088 Td [(2 = 3 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 3 7 5 {28b z 3 := 2 6 4 2 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 3 7 5 )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 2 sgn 2 J 3 +sgn J 3 2 6 4 2 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 3 7 5 {28c with sgn J 3 =1 sgn J 3 =0 sgn J 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if J 3 > 0 J 3 =0 J 3 < 0 : 41

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Table2-2showsthevaluesfortheparameters z 1 z 2 and z 3 asfunctionsofthematerial yieldstrengthsandsignofthethirdinvariantofthestressdeviator.Notethat,by Equation2{28,Equation2{27reducestotheexpressionfor 1 ,givenbyEquation 2{20,when J 3 =0.Thisequationisvalidsinceboth 1 and 2 arevalidand,indeed, equivalentfor J 3 =0. AnalternatewayofwritingEquation2{27istoexpresseverythingintermsofthe rateofdeformationcomponents.Notingthat s 2 s 3 0for J 3 0and s 1 s 2 0for J 3 0, theplasticmultiplierratecanbeexpressedasfollows: = 8 > > > > > > < > > > > > > : r 2 3 s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d P 1 2 + d P 2 2 + d P 3 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 if d P 1 q d P ij d P ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 r 2 3 s d P 1 2 + d P 2 2 + )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 d P 3 2 = 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 if d P 3 q d P ij d P ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 {29 where d P 1 d P 2 d P 3 aretheorderedprincipalcomponentsoftheplasticrateof deformationtensorand = T = C .NotethattherstbranchinEquation2{29 correspondsto J 3 0andthesecondbranchto J 3 0. Figure2-2showstherepresentationoftheplasticmultiplierrateinthedeviatoric planefor = T = C =0 : 82.Eachofthehalf-lines L )]TJ/F20 7.9701 Tf 0 -7.879 Td [(1 and L )]TJ/F20 7.9701 Tf 0 -7.879 Td [(2 formwith )]TJ/F37 11.9552 Tf 9.298 0 Td [(f 3 anangle )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [(,expressibleintermsof as tan )]TJ/F15 11.9552 Tf 7.085 -4.937 Td [(= p 3 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 2 : {30 Theanglebetween L + 1 and f 3 or L + 2 and f 3 isgivenby tan + = p 3 3 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : {31 Notethatifthematrixmaterialdisplaystension-compressionasymmetry 6 =1then )]TJ/F19 11.9552 Tf 10.406 -4.338 Td [(6 = + .UsingEquations2{30and2{31itcanbeeasilyshownthat )]TJ/F15 11.9552 Tf 9.742 -4.936 Td [(+ + = 3 : {32 42

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Equation2{32showsthattheplasticmultiplierratei.e.theexactdualoftheplastic potentialinEquation2{12hasthree-foldsymmetry,whichisaconsequenceofthe materialisotropy.Notethatwhentheyieldstrengthsintensionandcompressionareequal i.e.,vonMises,theplasticmultiplierratehassix-foldsymmetrysuchthat + = )]TJ/F15 11.9552 Tf 11.682 -4.339 Td [(= = 6.NotealsothatinthiscaseEquation2{29reducesto = r 2 3 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(d P ij d P ij {33 whichistheclassicvonMiseseectiveplasticstrainrate. Figure2-1.-planerepresentationoftheisotropicversionoftheCPB06yieldcriterionfor k =0vonMisescircle, k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C and k =0 : 3098 T > C 43

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Table2-1.Variousrelationshipsbetween k -expressionsandyieldstrengthratios. k -expressionsstrength-ratioexpressions 3 k 2 +2 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 T C 2 3 k 2 +10 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 k +3 3 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(10 k +3 3 k 2 +2 k +3 3 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 k 3 k 2 +2 k +3 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 k 2 +1 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 1 6 T C 2 +1 # 1 m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 2 3 2 6 4 1 2 )]TJ/F26 11.9552 Tf 11.955 13.271 Td [( T C 2 3 7 5 1 m + k 2 2 3 2 6 4 1 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 Table2-2.z-parametersasfunctionsofmaterialyieldstrengthsandsignof J 3 J 3 > 0 J 3 0 z 1 = 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 1 =0 z 2 = 2 = 3 2 T C 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 z 2 = 2 = 3 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 z 3 =0 z 3 = 2 C T 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 )]TJ/F26 11.9552 Tf 11.955 13.27 Td [( T C 2 44

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Figure2-2.-planerepresentationoftheplasticmultiplierrateassociatedwiththe isotropicversionoftheCPB06criterionfor k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C 45

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CHAPTER3 PLASTICPOTENTIALFORHCPMETALSWITHSPHERICALVOIDS Sphericalvoidsarecommonlyobservedinisotropicmetalspriortofailure.Asan example,Figure3-1showsthegrowthandcoalescenceofinitiallysphericalvoidsin analuminumyerplateexperiment.Todescribetheinuenceofvoidsontheplastic owofaporousaggregate,Gurson1977proposedananalyticalmacroscopicyield criterion.Tvergaard1981,1982modiedGurson's1977criterionbyintroducingtting parameters q 1 q 2 and q 3 tobetteragreewithniteelementunitcellcalculations.Leblond andPerrin1990performedaself-consistentanalysisofavoidwithinaporousplastic spheretoprovideaphysicalbasisandarationaleforthenumericalvaluesofthesetting parameters.TherecommendedvaluesforTvergaard'sttingparametersinLeblondand Perrin1990were q 1 =4 =e 1 : 47, q 2 =1and q 3 = q 2 1 .TvergaardandNeedleman1984 replacedtheparameterassociatedwiththevoidvolumefraction, f ,inGurson's1977 criterionwithaneectivevoidvolumefraction, f ,asawaytoaccountfortheonsetof voidcoalescenceleadingtonalmaterialfractureseeChapter1. Ellipsoidalvoidsaregenerallyobservedincoldrolledsheets.Gologanuetal.1993 usedahomogenizationprocedureonanaxisymmetric,ellipsoidalRVEcontaininga confocal,prolateellipsoidalvoidtoarriveatananalyticexpressionfortheyieldcriterion ofaporousaggregatecontainingprolate,ellipsoidalvoids.Gologanualsoconsidered thecaseofoblatevoidsanddevelopedanalyticexpressionsforsuchcriteriasee,for example,Gologanuetal.,2001.Garajeuetal.2000extendedtheworkofGologanuand collaboratorsbyinvestigatingtheinuenceofthisvoiddistributionevolutioninaddition totheinuenceofvoidshapeevolution.Garajeualsocapturedtheeectsofmatrix viscosityonductilefailurebyconsideringamatrixdescribedbyaNorton-typepowerlaw. Asmentionedpreviously,themodelscurrentlyexistingintheliteratureconcern materialsthathavethesameyieldintensionandcompressionintheabsenceofvoids. However,earlyresultsofHosfordandAllen1973haveshownthattension-compression 46

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asymmetryinyieldingatthepolycrystallevelisobservediftwinningisacontributor atthesinglecrystalleveltoplasticdeformation.Furthermore,metalswithhexagonal closepackedstructuredisplayadirectionalorientationpreferredtension-compression asymmetry.Thecentralgoalofthisdissertationistoprovideyieldcriteriaforsuch materials.Thischapterfocusesondevelopingamacroscopicyieldcriterionforavoidmatrixaggregatewherethevoidshavesphericalgeometryandthemetalmatrixis isotropicwithtension-compressionasymmetry. Theoutlineofthischapterisasfollows.Section3.1presentssomelimitingsolutions whichthenewcriterionshouldreducetoforsomeparticularcaseszeroporosity,equal yieldstrengthsandhydrostaticloading,respectively.Secondly,thevelocityeldusedto calculatethelocalplasticdissipationpotentialwillbepresentedinSection3.2.Section 3.3detailsthederivationofthecorrespondinglocalplasticdissipationandSection 3.4developstheexpressionsforthemacroscopicplasticdissipationofthevoid-matrix aggregate. Figure3-2illustratestherepresentativevolumeelementRVEthatwillbeusedin thischapter.Itisaspherecontainingaconcentricalsphericalvoid.Theouterradiusof thesphericalRVEis b whiletheinnerradiusortheradiusofthevoidisdenotedby a UniformrateofdeformationboundaryconditionsareassumedfortheRVE.Thefollowing assumptionsaremadewithrespecttotheloadingandmatrixmaterialresponse: Theaxialdeformationissymmetric;thus,aprincipalsysteminCartesiancoordinates canbefoundsuchthat D 11 = D 22 and D ij =0if i 6 = j withsimilarexpressionsfor Thematrixmaterialisfullyplasticandincompressiblesuchthat d ij = d P ij inEquation 2{1i.e.,rigidplasticbehavior. 47

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Sinceaxisymmetricloadingisassumed,themacroscopicstressstate, ,andassociated deviator, 0 ,aregivenas = 2 6 6 6 6 4 11 00 0 11 0 00 33 3 7 7 7 7 5 {1 and 0 = 2 6 6 6 6 4 1 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 00 0 1 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 0 00 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(2 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 3 7 7 7 7 5 : {2 3.1LimitSolutions Beforeproceedingtothederivationofthemacroscopicyieldcondition,itisprudent todeterminesomelimitingsolutionsthatareavailablefortheproblemofinterest.These limitingsolutionscanbecomparedwiththeeventualyieldcriterionandcanserveas guidesinlatersectionsforanyassumptionsthatneedtobemadeindevelopingthe macroscopicyieldexpressions.Ideally,anyanalyticcriterionthatisdevelopedwillreduce totheselimitingsolutions. 3.1.1Zeroporosityandequalyieldstrengthslimitingcases Therststatewhereananalyticsolutionimmediatelycomestomindiswhenthe materialisvoid-free.Whennovoidexistsinthesphere,thehomogenizedmaterialis exactlythematrixmaterialandtheyieldcriterionshouldreducetotheisotropicCazacu etal.2006yieldcriterionseeEquation2{11.Inotherwords, ~ e = e = T : {3 Secondly,ifthematrixmaterialhasequalstrengthsintensionandcompression,the yieldconditionofEquation2{11reducestothevonMisesyieldconditionsuchthatthe macroscopicyieldconditionfromthehomogenizationperformedinthissectionshould reducetoGurson's1977macroscopicyieldconditiongiveninChapter1andrewritten 48

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belowas S G = e Y 2 +2 f cosh 3 m 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =01{1 where Y istheyieldstrengthofthevonMisesmatrixmaterial. 3.1.2Exactsolutionforahydrostatically-loadedhollowsphere ConsiderthehollowsphereshowninFigure3-3ofinnerradius a andouterradius b thatissubjectedtoahydrostaticpressureonitsoutersurfaceeithertensileorcompressive.Anexactsolutionfortheproblemofanelastic-plastichollowspheresubjectto hydrostaticloadingexistsinthecasewhenthematerialisgovernedbythevonMisesyield criterionsee,forexample,Lubliner,1990.Theexternalpressureatwhichthehollow spherebecomesfullyplastici.e.,theultimate,orlimit,pressureisgivenas m = 2 3 Y ln f {4 where Y istheuniaxialyieldstrengthand f = a 3 =b 3 isthevoidvolumefraction.The choiceof+or )]TJ/F15 11.9552 Tf 13.2 0 Td [(intheequationdependsonwhethertheappliedloadiscompressiveor tensile,respectivelysinceln f < 0. Thechallengehereistodetermineanexactsolutionforahydrostatically-loaded hollowspherewhenthematrixmaterialexhibitsstrengthdierentialeects.Inthe derivationbelow,theproblemofanelastic-plasticmaterialgovernedbytheisotropic CPB06yieldconditionofEquation2{8isconsidered.Therigid-plasticsolutioncanthen bededucedbylettingtheYoungmodulustendtoinnity.Strain-displacementrelations, straincompatibility,Hooke'slaw,theyieldcriterionandtheequilibriumequationwill allbeusedtoobtainanexpressionfortheappliedexternalpressureasafunctionofthe matrixyieldstressandvoidgeometryforafullyplasticsphere. 3.1.2.1Strain-displacementrelations Assumingsmallstrains,themicroscopicstraintensorisdenedas = 1 2 u )]TJ 0.997 -8.17 Td [(r + )778(! r u ; 49

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where,insphericalcoordinates,thegradientofthedisplacementvector u isdenedassee Malvern,1969 [ u )]TJ 0.997 -8.169 Td [(r ]:= 2 6 6 6 6 6 4 @u r @r 1 r @u r @ )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(u r 1 r sin @u r @ )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(u r @u @r 1 r @u @ + u r r 1 r sin @u @ )]TJ/F21 11.9552 Tf 13.15 8.087 Td [(u r cot @u @r 1 r @u @ 1 r sin @u @ + u r cot + u r r 3 7 7 7 7 7 5 with [ u )778(! r ]=[ u )]TJ 0.996 -8.17 Td [(r ] T : Inthegivenproblem,theexternalloadingishydrostaticsuchthat, m = r .Duetothe sphericalsymmetryoftheproblem,onlytheradialdisplacementcomponentisdierent fromzero.Inotherwords, u = u r r ~e r .Theonlynonzerostraincomponentsarethen rr = du dr {5a = = u r : {5b 3.1.2.2Straincompatibility TheSt.Venantcompatibilityequationsarerequiredwhenthestrainsareconsidered knownsince,inthegeneralcase,thestrain-displacementequationsyieldsixequations forthreedisplacementunknowns.Thecompatibilityequationsensurethattheassumed strainsarephysicallypossible.TheSt.Venantcompatibilityequationsareexpressedas seeMalvern,1969 S = )778(! r E )]TJ 0.997 -8.169 Td [(r =0 : Thisexpressionrepresentssixpartialdierentialequations;however,itcanbeshownthat onlythreeofthesixequationsareindependent.Thesixequationsarerathertediousto writeinsphericalcoordinates,soonlytherstequationtheonlyoneneededforthis 50

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hollowsphereproblemisexpandedbelowas S 11 = 1 r 2 sin @ @ @ @ )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin @ @ + r sin + cos )]TJ/F21 11.9552 Tf 19.936 8.087 Td [(@ @ 1 sin @ @ )]TJ/F15 11.9552 Tf 11.955 0 Td [(cos )]TJ/F15 11.9552 Tf 11.956 0 Td [(sin @ @ )]TJ/F21 11.9552 Tf 17.001 8.088 Td [( sin cos )]TJ/F21 11.9552 Tf 11.955 0 Td [( r +sin @ r @ )]TJ/F21 11.9552 Tf 15.951 8.088 Td [(@ @r r + rr )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( sin + r sin @ @r )]TJ/F21 11.9552 Tf 13.815 8.088 Td [(@ r @ + r cos + rr sin : Theotherveequationscansimilarlybedeterminedusingtheprocedureoutlinedin Malvern1969butareunnecessaryforthepresentanalysis.Thus,thecompatibility conditionyieldsthenecessaryrelationship r = d dr r : {6 Inthefollowing,onlyonesubscriptwillbeusedtodenotethestraincomponentssince therearenoshearcomponentspresentsuchthatthesecondsubscriptisextraneous.Note thatforaradialdisplacementeldthecompatibilityequationgivenbyEquation3{6is automaticallysatised. 3.1.2.3Equationsofmotion Neglectingbodyforcesandassumingstaticequilibrium,thebalanceoflinearmomentumyieldsthefollowing: r =0 : Theexpandedformoftheequilibriumequationinsphericalcoordinatesis @ rr @r + 1 r @ r @ + 1 r sin @ r @ + 1 r rr )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [( =0 @ r @r + 1 r @ @ + 1 r sin @ @ + 1 r [3 r + )]TJ/F21 11.9552 Tf 11.955 0 Td [( cot ]=0 @ r @r + 1 r @ @ + 1 r sin @ @ + 1 r r +2 cot =0 : {7 Aswiththedisplacementeld,thestresseldinthespheremustalsobespherically symmetricduetothesphericalgeometry,theisotropicmaterialandtheappliedboundary 51

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conditionsontheboundary r = b ,thestateofstressisradial.Theonlynonzero componentsofthestresstensorarethen rr ,and with = .Usingthese symmetries,therstequationinEquations3{7nowyields d r dr +2 r )]TJ/F21 11.9552 Tf 11.955 0 Td [( r =0{8 whereonlyonesubscriptisnowbeingusedforthestresscomponentsasforthestrain components. 3.1.2.4Elasticconstitutiverelation:Hooke'slaw Intheelasticregime,thestress-strainrelationsare ij = 1 E [+ ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( kk ij ]{9 where E istheYoung'smodulus, isPoisson'sratio,and ij istheKroneckerdelta.Note thatforrigidplasticbehavior,theYoung'smodulus,E,tendstowardinnitysuchthat theelasticstressesproducenostrainsand,hence,nodisplacements.Usingthefactthat = ,Equation3{9canbereducedtothefollowingtworelevantequationsrelating thetwounknownstrainstothetwounknownstresses: r = 1 E [ r )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ] = 1 E [ )]TJ/F21 11.9552 Tf 11.956 0 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 r ] : {10 3.1.2.5Ultimatepressure SubstitutingEquation3{10intoEquation3{6straincompatibility,yields d dr [ )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 11.956 0 Td [( r ]+ 1+ r )]TJ/F21 11.9552 Tf 11.955 0 Td [( r =0 : Now,makinguseoftheequilibriumequation,Equation3{8,theaboveequationsimpliesfurthertoyield d dr r +2 =0 : 52

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ThepreviousequationalongwithEquation3{8representasystemoftwoordinary dierentialequationsintwounknownfunctions r and .Inordertosolvethesystem, anoperatormethodwillbeusedseeRoss,1984.Lettheoperator D = d=dr ,suchthat thelastequationalongwiththeequilibriumequationbecome D r +2 D =0 and D + 2 r r )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(2 r =0 : Inordertodetermine r therstequationismultipliedby1 =r ,thesecondequationby D ,andtheresultingequationsareaddedtoobtainanexpressionintermsof r alone. Usingasimilarprocedureyieldsanexpressionintermsof alone.Replacing D with d=dr yieldstheresultingordinarydierentialequationsbelowas r 2 d 2 r dr 2 +3 r d r dr =0 r 2 d 2 dr 2 +3 r d dr =0 : Thepreviousequationsareordinarylineardierentialequationswithvariable coecients.Furthermore,theaboveequationscorrespondtoaspecictypeofordinary dierentialequationknownasCauchy-Eulerequations.Cauchy-Eulerequationsare equationswhosetermsareconstantmultiplesofanexpressionoftheform x n d n y=dx n Themethodofsolutionistomakethesubstitution x = e t inordertoreducetheequation toalineardierentialequationintermsof t withconstantcoecients.Makingthe substitutionintotheabovetwoequationsyields d 2 r dt 2 +3 d r dt =0 d 2 dt 2 +3 d dt =0 ; 53

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withgeneralsolutions r = c 1 + c 2 r 3 = c 3 + c 4 r 3 : Theconstants c 1 { c 4 aredeterminedbysubstitutingtheaboveexpressionsbackintothe equilibriumequation3{8.Doingsoyields c 1 = c 3 and c 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 c 4 .Now,let c 1 = c 3 =: A and c 4 = )]TJ/F15 11.9552 Tf 9.299 0 Td [( = 2 c 2 =: B ,suchthattheexpressionsforthestressesbecome r = A )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(2 B r 3 = A + B r 3 : Theconstants A and B aredeterminedbyimposingtheboundaryconditionsforthe problem.Theboundaryconditionsforthegivenproblemareasfollows: r = 8 > < > : 0at r = a m = r at r = b Theconstants A and B cannowbesolvedforas A = m 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(f B = m a 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f where f := a=b 3 .Finally,theexpressionsforthestressesaregivenas r = m 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1 )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(a 3 r 3 {11a = m 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1+ a 3 2 r 3 : {11b SubstitutingEquations3{11aand3{11bintotheyieldconditiongivesthelevel ofappliedexternalpressure m atwhichthematerialrstyieldsandthelocationat whichtheplasticowrstdevelops.Thedeviatorofthestresstensorwhosenon-zero 54

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componentsaregivenbyEquations3{11aand3{11bisasfollows: s = 2 6 6 6 6 4 s r 00 0 s 0 00 s 3 7 7 7 7 5 suchthat s = 1 3 2 6 6 6 6 4 2 r )]TJ/F21 11.9552 Tf 11.955 0 Td [( 00 0 )]TJ/F15 11.9552 Tf 11.291 0 Td [( r )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 00 )]TJ/F15 11.9552 Tf 11.291 0 Td [( r )]TJ/F21 11.9552 Tf 11.955 0 Td [( 3 7 7 7 7 5 = m 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 6 6 6 6 4 )]TJ/F22 7.9701 Tf 10.494 4.707 Td [(a 3 r 3 00 0 a 3 2 r 3 0 00 a 3 2 r 3 3 7 7 7 7 5 {12 where m = 1 3 r +2 = m 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f : NotethatEquation3{12dictatesthatthesignofthemicroscopicthirdinvariantof thedeviatoricstresstensor, J 3 = s r s 2 ,isoppositeinsigntotheexternalpressurei.e., sgn J 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn m .Thisisaveryimportantpointsincethesignof J 3 willbeusedto determinewhichbranchoftheplasticmultiplierrateexpressionwhichisrelatedtothe localplasticdissipationgiveninEquation2{27mustbeusedincomputingtheoverall plasticdissipation. ForastateofstressdescribedbyEquationeq:stressallelast,theisotropicCPB06yield criterionofEquation2{12resultsinthefollowingexpressions: r )]TJ/F21 11.9552 Tf 11.955 0 Td [( = T if J 3 > 0 r )]TJ/F21 11.9552 Tf 11.955 0 Td [( = )]TJ/F21 11.9552 Tf 9.299 0 Td [( C if J 3 < 0 : 55

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NotethattheaboveexpressionsarethesameaswhatonewouldobtainwithavonMises materialunderhydrostaticloadingiftheyieldstrengthsintensionandcompressionare equali.e.,if T = C .Also,notethat J 3 < 0impliesthat r )]TJ/F21 11.9552 Tf 12.886 0 Td [( < 0suchthat yieldingisalwaysgovernedbytheabsolutevalueofthedierencebetweenthetwostress components.ThisquantityisdeterminedusingEquation3{11as j r )]TJ/F21 11.9552 Tf 11.955 0 Td [( j = j m j 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(f 3 a 3 2 r 3 whichhasobviouslyamaximumontheinnerboundary r = a .Inconclusion,plasticow rstdevelopsattheinnerboundary r = a thevoidboundary. Iftheassumptionofrigidplasticbehaviorisusedalongwiththeassumptionthatthe sphereisentirelyplastic,Equation3{8canbeintegrateddirectlyasfollows: Z m 0 d r =2 c e Z b a dr r whichyields m = 2 3 c e ln f 3{13 where c e isaconstantthee"istodenotethatitisdependentontheeectivestressand isgivenas c e = 8 > < > : )]TJ/F21 11.9552 Tf 9.298 0 Td [( C if m > 0 T if m < 0 : Intheaboveexpressions,therelationsgn J 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn m hasbeenusedtoshift thedependencefromamicroscopicquantityi.e., J 3 toamacroscopicquantitythe macroscopicmeanstress, m .NotethatEquation3{13reducestoEquation3{4when theyieldstrengthsintensionandcompressionareequali.e., T = C = Y Equations3{3,1{1and3{13providelimitingsolutionsthatthemacroscopic yieldcriteriondevelopedinthenextfewsectionscanbecomparedagainst. 56

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3.2ChoiceofTrialVelocityField AsmentionedinSection2.1,themacroscopicplasticdissipationpotential W D is denedas W D =inf d 2 K D h w d i V 2{7 where K D isthesetofincompressiblevelocityeldssatisfyingtheuniformrateof deformationboundaryconditionsontheboundaryoftheRVE, @V ,suchthat v = Dx forany x 2 @V: 2{4 Obtainingqualityvelocityeldswhichsatisfytheuniformstrainrateboundaryconditions ischallenging.Gurson1977proposedanincompressiblevelocityeldcompatiblewitha constantmacroscopicrateofdeformationtensor D .Forthesakeofclarity,theexpression ofthisvelocityeldwillbededucedinthefollowing. Themacroscopicrateofdeformationeld, D ij ,isrelatedtothelocalrateofdeformationeld, d ij asfollows: D ij = 1 V Z V d ij dV: Thelocal,ormicroscopic,rateofdeformationtensorisbydenitionseeMalvern,1969 d ij = 1 2 @v i @x j + @v j @x i where ~x isthepositionofamaterialpointinCartesiancoordinates. Thevelocityeld, ~v ,intheRVEisassumedtosatisfythecompatibilityconditions displacementboundaryconditionsaswellastheincompressibilitycondition.These conditionsare ~v j ~x = b~e r = D ~x = D ij x j div ~v =0 : {14 57

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ThemainassumptionofGurson1977isthatasolutionofEquation3{14canbe obtainedbyusinganadditivedecompositionofthevelocityeld.Therefore, ~v = ~v V + ~v S where ~v V isassociatedwithvolumechangesand ~v S isassociatedwithshapechanges. Thedeviatoricportionofthemacroscopicrateofdeformationis D 0 ij = D ij )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 D kk = D ij )]TJ/F21 11.9552 Tf 11.956 0 Td [(D m : TheboundaryconditioninEquation3{14canbeexpressedintermsofthevolumetric andshape-changingpartsofthevelocityeldas ~v S i j ~x = b~e r = D 0 ij x j ~v V i j ~x = b~e r = D m x i : {15 Theportionofthevelocityeldresponsibleforthevolumechange, ~v V ,shouldbea purelyradialeldbecauseofsymmetry.Inotherwords, ~v V = v V r ~e r : Iftheincompressibilityconstraintisimposedonthevolumetricportionofthevelocity eld,thefollowingexpressionresults: @ @r )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r 2 v V r =0 : Asolutiontotheequationaboveis v V r = c 1 r 2 where c 1 isaconstant.EnforcingtheboundaryconditionofEquation3{15,yieldsan expressionfor c 1 asfollows: c 1 = D m b 3 58

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Usingtheresultabovefor c 1 yieldstheexpressionforthevolumetricpartoftheradial velocitycomponentas v V r = D m b 3 r 2 : Theshape-changingpartofthevelocityeldisfoundbyassumingthatitisofthe followingform: ~v S = B ~x where B isaconstant,symmetricanddeviatorici.e., tr B =0tensor.Equation3{15 showsthatthetensor B isidenticallyequaltothedeviatoricpartofthemacroscopicrate ofdeformationtensor, D 0 .Notethatduetoaxisymmetry, D 0 ij =0if i 6 = j inCartesian coordinates. Insummary,avelocityeldcompatiblewiththeuniformrateofdeformationboundaryconditionsis ~v = ~v V + ~v S where ~v V = D m b 3 r 2 ~e r and ~v S = D 0 ~x {16 Now,thedeviatoricportionofthelocalrateofdeformationtensorissimply d S = D 0 .The volumetricpartoftherateofdeformationeldcanbefoundbyusingthedenitionofthe rateofdeformationinsphericalcoordinatesseeMalvern,1969alongwiththeassumption ofaxisymmetricdeformationasfollows: d V rr = @v V r @r = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D m b r 3 and d V = d V = v V r r = D m b r 3 59

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Notethat ~v V isexpressedinsphericalcoordinates,while ~v S isinCartesiancoordinatessuchthatachangeofreferenceframeisnecessarytowrite ~v inasinglecoordinate system.Inordertotransformtheexpressionfortheshape-changingpartoftherateof deformationtothesphericalcoordinatesystem,thefollowingtransformationequation mustbeused: d S r;; = A T d S ; 2 ; 3 A where A isnowthespherical-to-Cartesiantransformationmatrix.Since x 1 = r sin cos x 2 = r sin sin x 3 = r cos and ~e r = @~x @r =sin cos ~ i 1 +sin sin ~ i 2 +cos ~ i 3 ~e = @~x @ =cos cos ~ i 1 +cos sin ~ i 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin ~ i 3 ~e = @~x @ = )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin ~ i 1 +cos ~ i 2 thetransformationmatrixisthen A =[ a r c ] = h ~ i r ~e c i = 2 6 6 6 6 4 sin cos cos cos )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin sin sin cos sin cos cos )]TJ/F15 11.9552 Tf 11.291 0 Td [(sin 0 3 7 7 7 7 5 : 60

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Thefollowingexpressionsforthenonzerocomponentsoftheshape-changingpartofthe microscopicrateofdeformationtensorinsphericalcoordinatesarenowobtained: d S rr = D 0 rr = D 0 11 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(sin 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2cos 2 = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 D 0 11 +3cos2 d S = D 0 = D 0 11 )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(cos 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2sin 2 = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 D 0 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3cos2 d S = D 0 = D 0 11 d S r = D 0 r =3 D 0 11 sin cos = 3 2 D 0 11 sin2 : Therefore,thenonzerocomponentsinsphericalcoordinatesofthemicroscopicrateof deformationtensorbecome d rr = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D m b r 3 + D 0 rr d = D m b r 3 + D 0 d = D m b r 3 + D 0 d r = D 0 r Theprincipalvaluesfortheratesofdeformationareneededintheexpressionforthe plasticmultiplierEquation2{27.Theunorderedprincipalvaluesare, ~ d 2 ; 3 = d rr + d 2 s d rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(d 2 2 + d r 2 ~ d 1 = d = D m b r 3 + D 0 : Theaboveexpressionsresultin ~ d 2 ; 3 = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(D m 2 b r 3 )]TJ/F21 11.9552 Tf 13.151 9.321 Td [(D 0 2 3 2 s D 2 m b r 6 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 D m b r 3 D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(D 0 2 ~ d 1 = D m b r 3 + D 0 : {17 61

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3.3CalculationoftheLocalPlasticDissipation AsinthecaseofGurson'sanalysis,forarbitraryloadingthemacroscopicplastic dissipation W D cannotbedeterminedanalyticallywithoutmakingsomesimplifying assumptions.Duetothetension-compressionasymmetryofthematrix,freshdiculties areencounteredwhenestimatingthelocalplasticdissipation, w d .Thisisbecausethe plasticmultiplierrate hasmultiplebranchesseeEquation2{29anddependsonall theprincipalvaluesofthelocalrateofdeformationtensor, d .However,forthespecial casesofpurelyhydrostaticorpurelydeviatoricloading,themacroscopicplasticdissipation canbesolvedforexplicitly.Thecorrespondingmacroscopicstressesatyieldingaregiven inSection3.4. Theexpressionsforthemicroscopicratesofdeformationgivenby3{17arein principalcomponentsasneededforEquation2{27;however,identicationofthelone positiveornegativeprincipalcomponentisalsoneeded.Inordertodeterminewhichis theonlypositiveornegativeprincipalcomponenthereafterreferredtoasthelone" component,theproduct ~ d 2 ~ d 3 willbeinvestigatedsuchthatiftheproductispositivethen d lone = ~ d 1 ,whereasiftheproductisnegativetheneither ~ d 2 or ~ d 3 isthelonecomponent. Accordingly, ~ d 2 ~ d 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D 2 m u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(9sin 2 D 0 11 D m u )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 D 0 11 2 {18 where u = b 3 =r 3 .Takingthediscriminantwithrespectto D m ofthisquadraticpolynomial yieldsthefollowing: D m = b 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 ac = )]TJ/F15 11.9552 Tf 11.955 0 Td [(9sin 2 2 D 0 11 2 u 2 +4 u 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 D 0 11 2 =9 u 2 D 0 11 2 sin 2 sin 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(8 : 62

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Now,therearethreecasescorrespondingtothediscriminantbeingpositive,negativeor zero.Thesethreecasescanberepresentedas sgn ~ d 2 ~ d 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 u 2 =1if D m > 0 sgn ~ d 2 ~ d 3 =sgn )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 u 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1if D m < 0 sgn ~ d 2 ~ d 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1if D m =0 wherethelastrelationresultsfromeither D m =0or D 0 11 =0i.e.,azerodiscriminant correspondstoeitherthepurelyhydrostaticorpurelydeviatoriccase.Notethatinthe rstcase d lone = ~ d 1 whereasinthenaltwocasesthelonecomponentiseither ~ d 2 or ~ d 3 Notealsothatthesignofthediscriminantissimplyafunctionoftheangularcoordinate, 2 ; ,suchthat 2 1 ; 2 if D m > 0 2 ; 1 [ 2 ; if D m < 0 2 ; if D m =0 where 1 =arcsin p 8 = 3and 2 = )]TJ/F15 11.9552 Tf 11.955 0 Td [(arcsin p 8 = 3. Notethattheseresultsimplythatoneexpressionforthemicroscopicplasticmultiplierrateisvalidovertheentiredomainof forthepurelyhydrostaticandpurely deviatoricsolutions,whereasthereexisttwoexpressionsforthemicroscopicplasticmultiplierrateforthemoregeneralcaseofnon-zero D m and D 0 11 ,dependingonthevalueof .Thisbecomesanissueinthenextsectionwhenthemacroscopicplasticdissipationis derivedwhichinvolvesintegratingthemicroscopicplasticmultiplierrateover since therearetwoseparateexpressionswithnoapparentmethodofcombiningthem.However, sincethepurelydeviatoricandpurelyhydrostaticcasesprovidetheinterceptsoftheyield criterioninthe m e -plane,themicroscopicplasticmultiplierratecorrespondingto thesespecialcaseswillbedeveloped.Thereasoninghereisthat1itiseasiertond analyticalsolutionsforthesespecialcases,and2theyieldcriterion,attheveryleast, shouldsatisfythesespecialcases. 63

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Keepingthepreviousargumentinmind, d lone correspondstoeither ~ d 2 or ~ d 3 since theirproductisnegativewhenthecoupledterminEquation3{18iszero.Recognizing that d lone willipinsignwhenever ~ d 1 ipsinsign,theloneprincipaldeviatorcanbe writtenas d lone = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(D m 2 )]TJ/F21 11.9552 Tf 13.15 9.322 Td [(D 0 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(sgn D m + D 0 3 2 s D m 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 D m D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(D 0 2 {19 where = r 3 =b 3 Equation2{27requiressquaredvaluesoftheprincipalratesofdeformation.First notethatforthecaseofaxiallysymmetricdeformationwhere D 0 33 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 D 0 11 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D 0 22 in Cartesiancoordinates,theterm D e canbedenedasameasureofthesecondinvariantof thedeviatoricpartofthemacroscopicrateofdeformationtensorasfollows: D e = r 2 3 D 0 ij D 0 ij =2 j D 0 11 j =2 D 0 : {20 Notethat D e asdenedcorrespondstotheequivalentrateofdeformationforavonMises material.Inthissection, D e issimplyusedfornotationalconvenienceandasameasure forthesecondinvariantofthedeviatoricrateofdeformationtensor. UsingEquation3{20torelate D 2 e intermsof D 0 ,Equation3{17yields ~ d 2 2 ; 3 = 3 2 D m + D 0 s D m 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 D m D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + D e 2 2 + 1 4 D m + D 0 2 + 9 4 D m 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 D m D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + D e 2 2 # ~ d 2 1 = D m + D 0 2 suchthatthesecondinvariantofthemicroscopicrateofdeformationtensorbecomes d ij d ij = 3 2 D 2 e +6 D m 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 D m D 0 rr 64

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whichcanbeshownasbeingequivalenttotheexpressiongiveninGurson1977for d ij d ij .Also,thesquareofthelonepositiveornegativemicroscopicrateofdeformation principalcomponentcanbeexpressedasfollows: d lone 2 = 3 2 D m + D 0 sgn D m + D 0 s D m 2 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 D m D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(D 0 2 + 1 4 D m + D 0 2 + 9 4 D m 2 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 D m D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(D 0 2 # : {21 Now,theplasticmultiplierratefromEquation2{27canbeexpressedas = q z 2 d ij d ij + z 1 + z 3 d lone 2 {22 where z 6 =4 z 1 +6 z 2 +4 z 3 and z 1 { z 3 weredenedinTable2-2ofChapter2.Notethat thesecondterminthepreviousexpressionisduetothetension-compressionasymmetryof Equation2{8anddoesnotappearintheGursonanalysisusingavonMisesmaterial. ThecomplexityofEquations3{19or3{21makesthepracticeofobtainingan analyticsolutionfromthevolumeintegralinthenextsectionanexceedinglycomplicated task.However,notingthatinthespecialcasesofpurelyhydrostaticandpurelydeviatoric loadingseetherationalepresentedearlierinthissectionforfocusingontheseparticular cases, d lone canbeexpressedsimplyas d lone = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D m + D 0 {23 suchthat d lone 2 =4 D m 2 +4 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(D 0 2 {24 sincethecrossedtermiszeroforbothpurelyhydrostaticandpurelydeviatoricloading. Now, v u u t z 6 D m 2 + D e 2 2 # )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 z 2 D m D 0 rr : {25 65

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Notethat z 6 canbeexpressedintermsofthetensileandcompressiveyieldstrengthsas follows: z 6 = 8 > < > : 4if J 3 > 0 4 C T 2 if J 3 0 : {26 3.4DevelopmentoftheMacroscopicPlasticDissipationExpressions Let W + D denotethemacroscopicplasticdissipationcorrespondingtotheparticular velocityeld v givenbyEquation3{16.Inotherwords, W + D = 1 V Z V w d dV where V = = 3 b 3 isthevolumeofthesphericalRVEconsidered.Notethat W + isnot thetrueplasticdissipationsinceaspecicvelocityeldisbeingassumed.Themicroscopic plasticdissipationisdeterminedasfollows: w = ij d ij = ij @' @ ij = T where T istheyieldstrengthinuniaxialtensionassumedconstantand istheplastic multiplierrateassociatedtotheCazacuetal.2006yieldcriterionseeEquation2{27 or,forthespeciccasepresentedinthissection,Equation3{25. Notethattheexpressionfor inEquation3{25containsanon-invarianttermi.e., the D 0 rr termunderasquarerootwhichdoesnotlendtheexpressiontobeingeasily integrated.TheapproachofLeblond2003willbefollowed,involvingtheapplication oftheCauchy-Schwartzinequality,toobtainanupper-boundestimateoftheyieldlocus. 66

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Thisprocedurewillresultinanewupperbound, W ++ ,for W + asfollows: W + = 1 V Z V w d dV = 1 V Z b a Z 2 0 Z 0 wr 2 sin dddr = 4 V Z b a r 2 1 4 r 2 Z 2 0 Z 0 wr 2 sin dd dr = 4 V Z b a r 2 1 S Z S wdS dr = 4 V Z b a h w i S r r 2 dr wherethesurfaceareaofthesphereatalocation r is S =4 r 2 withadierentialelement of dS = r 2 sin dd .Now,usingtheCauchy-Schwartzinequalityof Z V fg dV Z V fdV 1 = 2 Z V gdV 1 = 2 anupperboundfortheaverageof w onasurfaceofradius r canbedeterminedas h w i S r = 1 S Z S wdS 1 S Z S 1 dS 1 = 2 Z S w 2 dS 1 = 2 = s 1 S Z S w 2 dS = h w 2 i S r 1 = 2 : Since, h w 2 i S r = 1 S Z S w 2 dS = 2 T S Z S 2 dS 67

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theupperboundmacroscopicplasticdissipationbecomes W ++ = 4 V Z b a r 2 h w + 2 i S r 1 = 2 dr = p 4 T V Z b a r 2 Z 2 0 Z 0 2 sin dd 1 = 2 dr = p 4 T 4 3 b 3 Z 1 f r 2 Z 2 0 Z 0 2 sin dd 1 = 2 b 3 3 r 2 d = T p 4 Z 1 f Z 2 0 Z 0 2 sin dd 1 = 2 d where f = a 3 =b 3 istheporosityintheRVEand = r 3 =b 3 wasdenedintheprevious subsection. Notethattheintegralof D 0 rr = D 0 11 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(sin 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2cos 2 overthesurfaceofthesphere isequaltozero.Also,noticethatinGurson'sanalysis,theupperboundassumption resultingin 2 intheintegrandinsteadof eectivelyremovesthedependenceofthe macroscopicthirdinvariantofthestressdeviator, J 3 ,onporousyielding.Inthepresent analysis,a J 3 -dependenceonyieldingstillexistsduetothemacroscopiceectivestress beingdenedintermsoftheisotropicCPB06criterionofEquation2{8anda J 3 dependencestillexistsdueto z 6 althoughthedependenceon J 3 willneedtobeshiftedto adependenceonmacroscopicquantitiesinthelatersections. Usingthepreviousassumption,theintegralexpressionfortheupperboundmacroscopicplasticdissipationbecomes W ++ = T p 4 Z 1 f Z 2 0 Z 0 2 sin dd 1 = 2 d = T Z 1 f p z 6 D m 2 + D e 2 2 # 1 = 2 d = p z 6 j D m j T Z 1 =f 1 r u 2 + 1 4 D e D m 2 u 2 du wherethesubstitution u = )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 hasbeenused. 68

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Theaboveintegralisoftheform I = Z p u 2 + A 2 u 2 du whichhasthesolutionsee,forexample,Zwillinger,2003 I = )]TJ 10.494 17.963 Td [(p u 2 + A 2 u +ln u + p u 2 + A 2 = )]TJ 10.494 17.964 Td [(p u 2 + A 2 u +sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 u A +constant Theexpressionfortheupperboundmacroscopicplasticdissipationisnowobtainedby replacing A intheexpressionabovewith D e = j D m j andisgivenas W ++ = p z 6 2 T 2 j D m j sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 j D m j D e u )]TJ/F26 11.9552 Tf 11.955 19.229 Td [(r D 2 e u 2 +4 D 2 m # 1 =f 1 : {27 Theaboveequationfortheupper-boundmacroscopicplasticdissipationdiersfromthat obtainedusingavonMisesmatrixmaterialby p z 6 = 2whichisequaltoonefor k =0. TheisotropicCPB06eectivestressisgivenbyEquations2{12,and2{13.For axisymmetricloading,theformoftheeectivestressdependsonwhether 11 or 33 is thelargerstresscomponent;alternatively,itcouldbestatedthattheformoftheeective stressdependsonthesignofthemacroscopicthirdinvariantofthestressdeviator, J 3 Thefollowingexpressiongivestheeectivestressdependingonthesignof J 3 : ~ e = z e j 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 j = z e r 3 2 0 ij 0 ij where z e = T C + 1 2 1+sgn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J 3 1 )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( T C : {28 Expressionsfor ~ e and m intermsoftheupper-boundmacroscopicplasticdissipationofEquation3{27nextneedtobedeveloped.Thisisdonebyusingthefollowing 69

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relation: ij = @W @D ij @W ++ @D ij = @W ++ @D m @D m @D ij + @W ++ @D e @D e @D ij = 1 3 ij @W ++ @D m + 2 3 D 0 ij D e @W ++ @D e : Notethat ij = 0 ij + m ij ;comparingthiswiththelastlineoftheequationaboveyields droppingthe m = 1 3 @W ++ @D m 0 ij = 2 3 D 0 ij D e @W ++ @D e : Alsonotethat 0 ij 0 ij = 2 3 @W ++ @D e 2 ~ e = z e @W ++ @D e : Thetwoderivativetermsthatappearinthepreviousequationsfor ~ e and m are obtainedfromEquation3{27as @W ++ @D e = p z 6 2 T 2 6 6 4 2 j D m j r 2 D m D e u 2 +1 )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(2 j D m j u D 2 e )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 2 D e =u 2 q D 2 e u 2 +4 D 2 m 3 7 7 5 1 =f 1 = p z 6 2 T )]TJ/F21 11.9552 Tf 9.298 0 Td [( 2 u 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 u p 2 u 2 +1 # 1 =f 1 = p z 6 2 T )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p 2 u 2 +1 u # 1 =f 1 70

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and @W ++ @D m = p z 6 T 2 6 6 4 sgn D m sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 j D m j D e u + 2 D m u=D e r 2 D m D e u 2 +1 )]TJ/F15 11.9552 Tf 33.199 8.088 Td [(2 D m q D 2 e u 2 +4 D 2 m 3 7 7 5 1 =f 1 =sgn D m p z 6 T sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 u 1 =f 1 wherethesubstitution =2 j D m j =D e hasbeenused.Thestressinvariantscannowbe expressedasfollows: ~ e = z e @W ++ @D e = z e p z 6 2 T )]TJ/F26 11.9552 Tf 10.494 18.53 Td [(p 2 u 2 +1 u 1 =f 1 = z e p z 6 2 T )]TJ/F26 11.9552 Tf 9.298 10.949 Td [(p 2 1+ f 2 + p 2 +1 = z e p z 6 2 T p 2 +1 )]TJ/F26 11.9552 Tf 11.955 10.95 Td [(p 2 + f 2 and m = 1 3 @W ++ @D m = p z 6 3 sgn D m T sinh )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 u 1 =f 1 = p z 6 3 sgn D m T sinh )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 f )]TJ/F15 11.9552 Tf 11.955 0 Td [(sinh )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 : Eliminating fromtheequationsfor ~ e and m givenattheendoftheprevious subsectioncanbedonebyrearrangingtheexpressionforthemeanstressandtakingthe hyperboliccosineofbothsidesoftheequationasfollows: sgn D m 3 m p z 6 T =sinh )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 f )]TJ/F15 11.9552 Tf 11.955 0 Td [(sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 cosh 3 m p z 6 T =cosh sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 f )]TJ/F15 11.9552 Tf 11.955 0 Td [(sinh )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = 1 f p 2 + f 2 p 2 +1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 71

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wheretherelationscosh x + y =cosh x cosh y +sinh x sinh y andcoshsinh )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 x = p 1+ x 2 havebeenused.Now,squaringtheexpressionfortheeectivestressyields ~ e T 2 = z e 2 z 6 4 p 2 +1 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p 2 + f 2 2 = z e 2 z 6 4 1+ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 p 2 +1 p 2 + f 2 +2 2 : Thelasttwoequationscannowbecombinedtoobtain ~ e T 2 4 z e 2 z 6 =1+ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f cosh 3 m p z 6 T {29 Notethattheparameters z e and z 6 arefunctionsofsgn )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(J 3 andsgn J 3 ,respectively, aswellasbothbeingfunctionsoftheratiooftheyieldstrengths.Anymacroscopicyield criterionshouldonlybeafunctionofmacroscopicquantitiessincethereasonforusinga macroscopicyieldcriterionbecomesunclearifonehasaccesstothemicroscopicquantities. Therefore,thedependenceof z 6 onthemicroscopicquantity J 3 needstobeshiftedtoa dependenceonmacroscopicquantities.Thefollowingassumptionswillbeusedtoshiftthe dependenceofsgn J 3 tothemacroscopicquantities m and J 3 : 1.Thesignofthemicroscopicquantity J 3 ontherighthandsideofEquation3{29is equaltothenegativesignofthemacroscopicmeanstress, m ,suchthatsgn J 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn m 2.Thesignofthemicroscopicquantity J 3 onthelefthandsideofEquation3{29is equaltothesignofthemacroscopicthirdinvariantofthestressdeviator, J 3 ,such thatsgn J 3 =sgn )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(J 3 NotethattherstassumptionensuresthatthehydrostaticsolutiongivenbyEquation 3{13isrecoveredwhilethesecondassumptionensuresthatthecriterionmatchesthe limitingsolutionofEquation3{3. Equation3{29describesthemacroscopicyieldingandcannowbewritteninthe formofayieldfunctionasfollows: = ~ e T 2 +2 f cosh z s 3 m 2 T )]TJ/F26 11.9552 Tf 11.956 9.684 Td [()]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ f 2 =0{30 72

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wheretheparameter z s isdenedas z s =1+ 1 2 sgn 2 m )]TJ/F15 11.9552 Tf 11.956 0 Td [(sgn m T C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 {31 andtheisotropicCPB06eectivestressisgivenbyEquations2{12and2{13andis rewrittenbelowforthegeneralloadingcaseas ~ e = s 9 2 j S 1 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(kS 1 2 + j S 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(kS 2 2 + j S 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(kS 3 2 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 where S i arethemacroscopicprincipalstressdeviators. Equation3{30reducestoGurson'sresultasshowninEquation1{1for T = C since z s =1forthatcase.Theyieldfunctioncanbebrokenoutintothefollowing expressionswhichexplicitlyshowtheinuenceof T and C onyielding: = 8 > > > > > > > < > > > > > > > : ~ e T 2 +2 f cosh 3 m 2 T )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =0if m > 0 ~ e T 2 +2 f cosh 3 m 2 C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =0if m < 0 and ~ e = )]TJ/F21 11.9552 Tf 11.955 0 Td [(f T if m =0 : NotethatEquation3{30reducestothetwoexactsolutionsderivedearlierandgivenin Equations3{3and3{13forthecaseswhen f =0and ~ e =0,respectively.Figure 3-4showsacomparisonbetweentheyieldcurvesrepresentedbyEquation3{30fortwo dierentmaterials k = 0 : 3098assumingaporosityof f =0 : 01.Theexacthydrostatic solutionsforbothtensilemeanstressandcompressivemeanstressarecapturedbythe yieldcurvesforthetwodierentmaterialsandarelabeledinFigures3-4Aand3-4B. Noticeinbothguresthatthenewyieldcurvesarenotsymmetricaboutthevertical axis.Thisistobeexpectedwhenconsideringtheexacthydrostaticsolutionshownin theguresandrepresentedbyEquation3{13sinceforasymmetricyieldstrengths,a 73

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materialthatbehavesaccordingtotheisotropicCPB06criterionofEquation2{8has hydrostaticsolutionswithdierentmagnitudes.Figure3-5showstheyieldcurvesfor amaterialwith T < C k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098butforthreedierentvoidvolumefractions f =0 : 01, f =0 : 04and f =0 : 14.Notethatasthevoidvolumefractionincreases,the materialsoftensduetothedecreaseinload-bearingarea.InChapter5,niteelementcell calculationswillbeconductedtovalidatethedevelopedcriterionfortheporousaggregate. A B Figure3-1.Ductilecrackinaluminumplateduetoyerplateimpact.ADuctilecrackby voidcoalescence.BTipofductilecrackshowninAathighermagnication. Source:Antoun,Seaman,andCurran1998,pg.35.Dynamicfailureof materials:Volume1-experimentsandanalyses.Tech.Rep. DSWA-TR-96-77-V1,DefenseSpecialWeaponsAgency. 74

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Figure3-2.Representativevolumeelementforaspherecontainingasphericalvoid. Figure3-3.Hydrostatically-loadedhollowsphere. 75

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A B Figure3-4.MacroscopicyieldsurfacesdenedbyEquation3{30foramatrixmaterial containingasphericalvoidofporosity f =0 : 01.A k =0 : 3098 T > C .B k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098 T < C 76

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A B Figure3-5.MacroscopicyieldsurfacesdenedbyEquation3{30foramatrixmaterial with k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C andcontainingsphericalvoids:normalizedvon Misesmacroscopiceectivestressversusnormalizedmacroscopicmeanstress forvariousporosities.A J 3 > 0.B J 3 < 0. 77

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CHAPTER4 NUMERICALIMPLEMENTATIONOFTHEMATRIXYIELDCRITERION TheanalyticalexpressionforthestresspotentialderivedinSection3.4wasobtained usingthetrialvelocityeldduetoGurson1977andgivenbyEquation3{16.However, thetruemacroscopicplasticdissipationisdenedinEquation2{7andrewrittenbelow forconvenience: W D =inf d 2 K D h w d i V 2{7 where K D isthesetofincompressiblevelocityeldssatisfyingtheuniformrateof deformationboundaryconditionsontheboundaryoftheRVE, @V .Thus,theproposed criterionisanupper-boundoftheexactstresspotentialsinceoneparticularvelocityeld wasconsideredinsteadofthesetofkinematicallyadmissibleelds. Inordertoassessthevalidityoftheyieldcriteriaproposedinthisdissertation, calculationsof W D forarbitraryincompressiblevelocityeldsneedtobeperformed. Thesecalculationscannotbedoneanalytically;hence,niteelementcalculationsofunit cellswillbeperformedinstead.Inthesecalculations,thematrixmaterialismodeled usingtheisotropicCPB06criterionandthevoidboundaryisexplicitlymeshedintothe geometry.Theaverageyieldbehavioroftheunitcellundervariousloadingconditionscan thenbecomparedtothemacroscopicyieldfunctionsdevelopedinthisdissertation. Akeystepinperformingtheseniteelementcalculationsistheimplementationofthe specicyieldcriterionforthematrix.Thischapterdetailsthestressupdatealgorithms andthederivativesoftheisotropicCPB06criterionnecessarytoimplementtheyield criterionintoaniteelementcode.Thecomputationaltestingprocedureusedintheunit cellniteelementcalculationsisgivenforaxisymmetriccalculationsinChapter5andfor planestraincalculationsinChapter7. 4.1ReturnMappingProcedure Beforeniteelementcalculationscanbeperformed,theyieldcriterionofEquation 2{8mustbeimplementedintoaniteelementframeworke.g.,intothecommercial 78

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niteelementcodeAbaqus,2008.Inordertoimplementacriterionintoaniteelement framework,astressupdatealgorithmmustbeusedtocorrectlyupdatethestressesateach timeincrementbasedontheamountofplasticowthathasoccurredintheincrement. Returnmappingschemesarewidely-usedfornumericallyimplementingthesestress updates.Returnmappingschemesrstpredictthetrialstressesbasedonthecurrenttotal strainincrementanelasticpredictorstepandthenreturnthestressesbacktotheyield surfaceusinganiterativeprocedureaplasticcorrectivestep. Thenextfewparagraphsoutlinethereturnmappingprocedureusedtoimplement theisotropicCPB06criterionofEquation2{8intoaniteelementframeworkinto ABAQUSinthiscase,althoughthereturnmappingschemeissimplyanumericalalgorithmforintegratingtheconstitutiveequationsandassuchisindependentofthenite elementcodebeingused.Theschemeispresentedbelowasasmallstrainalgorithm; however,sincecommercialniteelementcodesgenerallyupdatethestresstensortoaccountforlargedeformationsandrotationsbeforecallingthestressupdatesubroutines, theschemepresentedbelowisvalidalsoforlargedisplacementswiththeCauchystress replacedbysomeobjectivestressmeasureandthesmallstrainincrementsreplacedbythe rateofdeformationincrements.SeeBelytschkoetal.2000orSimoandHughes1998 formoredetails. Theyieldconditioncanbestatedbelowas = e )]TJ/F21 11.9552 Tf 11.956 0 Td [( T =0{1 where e istheeectivestressofthematrixmaterialand T isthematrixmaterial'syield strengthintension.Forassociatedowi.e.,theplasticowpotentialcoincideswiththe yieldfunction, andconstantyieldstrength, d P = @' @ = @ e @ 79

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where d P istheplasticpartofthestrainratetensor, theplasticmultiplierrateand istheCauchystresstensor.Theloading-unloadingconditionsalsoknownasthe Kuhn-Tuckerconditionsaregivenasfollows: 0 ;' 0 ; =0 ; wheretherstexpressionstatesthattheplasticmultiplierrateisalwayspositiveorzero, thesecondexpressionstatesthatthestressisalwaysonorwithintheyieldsurface,and thelastexpressionstatesthatthestressesremainontheyieldsurfaceduringplastic loadingi.e.,when > 0.Thislastconditioncanalsobestatedas =0{2 since =0duringplasticloading.Equation4{2isknownastheconsistencycondition. Fortherate-independentmacroscopicyieldfunctiongivenbyEquation3{30,all oftheunitcellcalculationswillbeperformedasquasi-static.Inquasi-staticanalyses, therateofdeformationisnotphysicallymeaningfulsincethereisnotime-dependence althoughonecouldstilldeneapseudo-ratebyusingthestepsize.Becauseofthe rate-independenceofthisproblem,theplasticmultiplier, ,willbeemployedfortherest ofthissectionasopposedtotheplasticmultiplierrate, .Theplasticmultiplierisdened forassociatedowas P = @' @ = @ e @ where P istheplasticpartofthestraintensor.Eventhoughsmallstrainsareimplied byusing insteadof E ,thisisnotanassumptionneededforthereturnmappingscheme; iftheniteelementcodecomputesnitestrainsandobjectivestressesi.e.,astress measurethatisframe-invariantsuchthatlargerotationsandstretchingsareaccounted 80

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forcorrectlyandpassesthesetothestressupdatesubroutine,thenthereturnmapping schemepresentedbelowwillalsobevalidforlargedeformations. Thereturnmappingschemeiterativelyupdatesthestressessuchthattheyield conditionofEquation4{1issatised.Therefore,therststepinderivingthereturn mappingschemeistoexpandEquation4{1usingaTaylorseriesexpansionandsetthe resultequaltozero.Doingso,whiledroppingsecond-orderandhigherterms,yields j +1 = j + d' d j j = j + j =0 {3 wherethesubscriptnotationhasbeenusedtodenoteaderivativewithrespecttothe subscriptand j =0correspondstothetrialvaluesobtainedfromtheelasticpredictor step.Thepreviousequationcontainstheunknownplasticcorrectortothestresstensor, j .Thefollowingsetofequationscanbeusedinordertodeterminethisunknown: n +1 = P n +1 + E n +1 P n +1 = P n + n +1 r {4 wherethe n indicatetheincrementnumberoftheconvergedquantitiesandshouldnotbe confusedwiththeiterativecounter j .Also,theplasticowdirection, r ,isdenedfrom therelation P = r andisdistinguishedfrom sincethetwoareonlyequalforassociatedow.Theconvergedstresstensorcannowbeexpressedasfollows: n +1 = C E n +1 = C )]TJ/F41 11.9552 Tf 5.479 -9.684 Td [( n +1 )]TJ/F41 11.9552 Tf 11.955 0 Td [( P n +1 = C )]TJ/F41 11.9552 Tf 5.479 -9.683 Td [( n + n +1 )]TJ/F41 11.9552 Tf 11.955 0 Td [( P n )]TJ/F15 11.9552 Tf 11.955 0 Td [( P n +1 = n + C n +1 )-222(C P n +1 81

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suchthat n +1 = trial n +1 )-222(C P n +1 = trial n +1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( n +1 C r = trial n +1 + n +1 where C istheelasticitytensor. Notethatnothinghasyetbeensaidregardingwhethertheplasticowdirection, r istakenasthepreviouslyconvergeddirection, r n ,orasthecurrentincrement'sdirection, r n +1 .Itturnsoutthateithervaluecanbeused,althoughthesimplicityandaccuracyof theroutinewilldependonthischoice.Ifthepreviouslyconvergedvaluefortheplastic owdirectionisusedtoupdatethestressesthenthereturnmappingschemepresented belowisreferredtoasasemi-implicitbackwardEulerschemesincethealgorithmis implicitintheplasticmultiplier n +1 butexplicitintheplasticowdirection r n Alternatively,ifthecurrentincrement'svalue, r n +1 ,isusedfortheplasticowdirection intheplasticcorrectorphase,thenthereturnmappingalgorithmpresentedhereis referredtoasafullyimplicitbackwardEulerschemesinceitisimplicitinboththe plasticmultiplierandtheplasticowdirection.Forassociatedplasticity,ageometric interpretationcanbemaderegardingthedierencebetweenthetwoschemes.The semi-implicitroutineprojectsthepredictedstressesontotheyieldsurfacealongthe normaltothepreviousyieldsurfacewhilethefullyimplicitroutineprojectsthepredicted stressesontotheyieldsurfacealongthenormaltothecurrentyieldsurfaceclosestpoint projection.Obviously,asthesteporincrementsizedecreases,thetwoschemesconverge tothesameresultsincethetwonormalswillbecomeapproximatelythesame.Thesemiimplicitschemeiscommonlyusedinexplicitniteelementmethodssincethetimestep requiredisverysmallcomparedtoimplicitmethodsduetotheCourant-Friedrichs-Lewy conditionforexample,theCFLconditiongenerallylimitsthetimestepsuchthatawave cannotpassthroughthesmallestelementinthegridinlessthanonetimeincrement. 82

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Asemi-implicitschemecanalsobeusedinimplicitniteelementmethodsalthoughcare mustbetakentoensurethatthestepsizeusedissmallenoughtoachieveacceptable results.Obviously,asemi-implicitschemewillrequiresmallerstepsizesforagiven problemthanwillafullyimplicitschemesincetheplasticowdirectionisexplicitinthe semi-implicitscheme. Anexpressionfortheplasticcorrectortothestresstensor, n +1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [( n +1 r still needstobedeterminedfortheiterativeplasticcorrectorphase.Inordertodoso,the secondexpressioninEquation4{4willberewrittenasfollows: a n +1 := )]TJ/F41 11.9552 Tf 9.299 0 Td [( P n +1 + n + n +1 r = C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 n +1 + n +1 r = 0 : {5 Inordertoobtainanexpressionfortheincrementintheplasticmultiplierduringthe plasticcorrectioni.e., j ,aTaylorseriesexpansionofthevectorequation a willbe performedasfollowsagaindroppingsecond-ordertermsandhigher: a j +1 = a j + d a d j j + d a d j j Thepreviousexpressionrepresentsavectorequationandresultsintwoseparateexpressionsdependingonwhethertheplasticloadingdirectionisexplicitorimplicit.Thetwo expressionsareasfollows: a j +1 FI = a j + r j j + C )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 j + j r j j =0 a j +1 SI = a j + r n j + C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 j =0 wherethesubscriptsFI"andSI"refertothefullyimplicitandsemi-implicitsolution, respectively.Thepreviousequationscanbesolvedfortheplasticcorrectortothestress 83

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tensor, j .Doingsoyields, j FI = )]TJ/F26 11.9552 Tf 11.291 9.684 Td [( C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + j r j )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 )]TJ/F37 11.9552 Tf 5.479 -9.684 Td [(a j + r j j =: )]TJ/F15 11.9552 Tf 11.528 3.022 Td [( C j )]TJ/F37 11.9552 Tf 5.48 -9.684 Td [(a j + r j j j SI = C r n j {6 Finally,thepreviousresultscanbesubstitutedbackintotheTaylorseriesexpansionofthe yieldcondition,Equation4{3,andsolvedfortheplasticcorrectoriteration'sincrement intheplasticmultiplier, j .Doingsoyields j FI = j )]TJ/F21 11.9552 Tf 11.955 0 Td [(' j C j a j j C j r j j SI = j j C r n {7 where C = C )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + j r j )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and a j isgivenbyEquation4{5.Oncetheiteration'sincrementintheplasticmultiplier, j ,isdeterminedfromEquation4{7,theiteration'sincrementintheplastic correctortothestresstensor, j canbedeterminedfromEquation4{6.Theiterative loopisstoppedoncetheupdatedstressessatisfytheyieldconditionofEquation4{1 andonceEquation4{5issatisedwithinagiventolerancenotethatEquation4{5 isalwayssatisedateveryiterationforthesemi-implicitscheme.NotefromEquation 4{7thatthefullyimplicitroutinerequiresthesecondderivativeoftheyieldfunction withrespecttothestresstensorintheformof r embeddedin a j ,andassuming associativeplasticitywhilethesemi-implicitroutineonlyrequirestherstderivative.The second-ordertensor r = assumingassociativeplasticityhereandthefourth-order tensor r willbederivedinthefollowingsectionsthesetwotensorswillbecomeavector andamatrix,respectively,duringtheniteelementimplementationsinceinthisphasethe symmetricstressandstraintensorswillbevectorizedtoreducecomputationalcost. 84

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ImplicitniteelementcodessuchastheimplicitversionofABAQUSrequirean algorithmictangentmodulustobecomputedeveryincrement.Thealgorithmictangent modulusthatwillbeusedforboththesemi-implicitandfullyimplicitschemesisas follows: C alg = C)]TJ/F15 11.9552 Tf 25.03 11.11 Td [( C r C C r wheretheplasticowdirection, r ,istakentobe r n forthesemi-implicitschemeand r n +1 forthefullyimplicitschemewhile C = C forthesemi-implicitscheme.Additionally,for thesemi-implicitscheme,thevalueof istakentobethepreviousincrement'svaluesuch that = r n andthealgorithmictangentmodulusissymmetric.NotethatBelytschko etal.2000givesanalternateunsymmetricalgorithmictangentmodulusforthesemiimplicitscheme. 4.2FirstDerivatives Boththesemi-implicitandfullyimplicitreturnmappingschemesusetherst derivative = r inordertoperformtheplasticcorrectiontotheelasticpredictorsee Equation4{7.Thissectiondetailsthederivationoftherstderivativeoftheyield criterion.TakingtherstderivativeofEquation4{1gives = d' d = d e d wheretheyieldstrength, T ,hasbeentakenasaconstant.Applyingthechainruletothe lastterminthepreviousexpressionyields @ e @ ij = @ e @s @s @ ij = @ e @s @s @J 2 @J 2 @ ij + @s @J 3 @J 3 @ ij {8 with =1 ; 2and3andwith J 2 and J 3 beingthesecondandthirdinvariantsofthe deviatoricstresstensor,respectively.Theexpressionsfor J 2 and J 3 intermsofthe 85

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principaldeviatoricstressesarewrittenbelowas J 2 = 1 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s 2 1 + s 2 2 + s 2 3 J 3 = 1 3 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s 3 1 + s 3 2 + s 3 3 : TherstderivativetermontherighthandsideofEquation4{8wasderivedinSection 2.3andisrepeatedbelowforconvenienceas @ e @s 1 = m p F 2 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.088 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k @ e @s 2 = m p F 2 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.087 Td [(1 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k @ e @s 3 = m p F 2 3 j s 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 3 sgn s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 1 sgn s 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.088 Td [(1 3 j s 2 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks 2 sgn s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k : 2{16 Expressionsfor @s =@J 2 and @s =@J 3 cannowbefoundbydierentiatingthe characteristicequation 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 3 =0 wheretheprincipalstressdeviators s 1 s 2 and s 3 aretherootsofthepreviousthirdorderalgebraicexpression.Dierentiatingthecharacteristicequationwithrespectto yieldsthefollowing: 3 2 d )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 d )]TJ/F21 11.9552 Tf 11.955 0 Td [(dJ 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(dJ 3 =0 @ @J 2 = 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 @ @J 3 = 1 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 86

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where canbeanyoftheprincipalstressdeviators, s ,with =1 ; 2 ; 3.Thus, @s @J 2 = s 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 @s @J 3 = 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 {9 Notethatthederivativesinthepreviousexpressionhavesingularitiesat s = p J 2 = 3. Thisconditionholdswhenanytwoofthe s 'sareequalbiaxialloadingorwhenthey areallequali.e., s =0asinpurehydrostaticloading.Onlytherstcaseisofinterest heresinceplasticitydoesnottypicallyoccurinnon-porousmetalsduringpurehydrostatic loading.Forthetwosingularcasescorrespondingtobiaxialloading s 1 = s 2 = p J 2 = 3 and s 2 = s 3 = )]TJ/F26 11.9552 Tf 9.298 10.222 Td [(p J 2 = 3, @ e =@ ij mustbecalculateddirectly.Theexpressionsforthe @ e =@ ij correspondingtothetworelevantloadingstatesgeneralandbiaxialwillnowbe presented. 4.2.1IsotropicCPB06rstderivatives:generalloading Forthegeneralloadingsituationwhen s 1 6 = s 2 6 = s 3 ,Equation4{8canbeused directlyalongwithEquation4{9toobtainthefollowing: @ e @ ij = @ e @s s 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 @J 2 @ ij + 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 @J 3 @ ij with @J 2 @ ij = 0 ij @J 3 @ ij = 0 ip 0 pj )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 J 2 ij {10 where 0 ij isthedeviatoricstresstensornottobeconfusedwiththeprincipaldeviatoric stresscomponents, s .Therefore,thepreviousexpressioncanbewrittenasfollows: @s @ ij = 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 0 ij s + 0 ip 0 pj )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(2 3 J 2 ij {11 suchthat @ e @ ij = @ e @s 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 0 ij s + 0 ip 0 pj )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 J 2 ij {12 87

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wherethederivativetermsontherighthandsideinthepreviousexpressionaregivenby Equation2{16. 4.2.2IsotropicCPB06rstderivatives:biaxialloading Inthebiaxialloadingcase,let~ s denotetheloneprincipalstressdeviatorsuchthat ~ s = 2 r J 2 3 withtheothertwoprincipalstressdeviatorsbeingidenticalandequaltominusonehalfof ~ s .Inthisloadingstate,theloadingdirection, @ e =@ ij ,canbefounddirectlyasfollows: @ e @ ij = @ e @F @F @ ij = @ e @F @F @ ~ s @ ~ s @J 2 @J 2 @ ij {13 where F wasdenedinEquation2{8andcanbewrittenfortheparticularcaseof biaxialloadingas F = ~ s 2 2 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2sgn~ s k +3 : Also,notethat e = m p F suchthat @ e @F = m 2 p F : ThepreviousequationscanbeusedtoevaluateEquation4{13andyieldthefollowing result: @ e @ ij = m 3 p F 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2sgn~ s k +3 0 ij : {14 Since k =0yields m = p 3 = 2and F =2 J 2 ,Equation4{14reducesto @ e @ ij = p 3 2 p J 2 0 ij 88

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for k =0,whichistheknownresultforavonMisesmaterialseeEquation4{15.Note thatthepreviousresultforavonMisesmaterialholdsforallloadingconditions,notjust thespecialcaseofbiaxialloading. Therstderivativesderivedinthissectionyieldallthenecessaryingredientsfor implementingasemi-implicitreturnmappingalgorithmintoaniteelementcodesee Section4.1.Thenextsectionwilldetailthesecondderivativeswhichwillbeneeded alongwiththerstderivativesderivedinthissectiontoimplementafullyimplicitreturn mappingalgorithmintoaniteelementcode. 4.3SecondDerivatives Thefullyimplicitschemerequiresthesecondderivativeoftheyieldconditionin Equation4{1withrespecttothestresstensorinadditiontotherstderivativederived intheprevioussection.AswasthecasefortherstderivativesinSection4.2,thesecond derivativeswillalsoneedtobehandleddierentlyforthespecialcaseofbiaxialloading. ThissectionrstderivesthesecondderivativesforavonMisesmaterial,thenderivesthe secondderivativesforthegeneralloadingcasewhen s 1 6 = s 2 6 = s 3 ,andlastlyderivesthe derivativesforthespecialcaseofbiaxialloading. 4.3.1Vonmisessecondderivatives BeforederivingtheisotropicCPB06secondderivatives,itisinstructivetoderivethe secondderivativesforavonMisesmaterialsinceisotropicCPB06reducestovonMisesfor k =0andsince,atleastnumerically,thesevonMisessecondderivativescanvalidatethe longerandmorecomplexisotropicCPB06secondderivatives.Therefore,thevonMises relationscanbewrittenas, VM e = r 3 2 0 ij 0 ij = p 3 J 2 89

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suchthat @ VM e @ ij = @ VM e @J 2 @J 2 @ ij = p 3 0 ij 2 p J 2 {15 and @ @ mn @ VM e @ ij = @ @ mn p 3 0 ij 2 p J 2 = p 3 2 p J 2 im jn )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 ij mn )]TJ/F21 11.9552 Tf 13.151 9.167 Td [( 0 ij 0 mn 2 J 2 = 3 2 VM e ^ I {16 where ^ I = I dev )]TJ/F15 11.9552 Tf 12.765 0.166 Td [(^ n ^ n with ^ n = r k r k = r 2 3 r = 0 p 2 J 2 and I dev = I)]TJ/F15 11.9552 Tf 23.199 8.088 Td [(1 3 II where I isthefourth-orderunittensorgivenby I ijmn = im jn : Equation4{16isgivenintensorialformasafourth-ordertensor.Commercialnite elementcodestypicallyhavethesymmetricstressesandstrainsinvectorformfor computationaleciency.Equation4{16canbetransformedusingVoightnotationtoa 90

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formthatcanbereadilyimplementedintoaniteelementcodeasfollows: @ 2 VM e @ ij @ mn = p 3 2 p J 2 2 6 4 V 11 V 12 V T 12 V 22 3 7 5 : {17 Notethatintherstquadrantwhere V 11 resides, i = j and m = n with i = j = m = n on thediagonal.Therefore, V 11 = 2 6 6 6 6 4 2 3 )]TJ/F20 7.9701 Tf 13.15 5.698 Td [( 0 x 2 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.151 6.861 Td [( 0 x 0 y 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.15 5.698 Td [( 0 x 0 z 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.151 6.861 Td [( 0 x 0 y 2 J 2 2 3 )]TJ/F15 11.9552 Tf 13.151 6.695 Td [( 0 y 2 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.15 6.861 Td [( 0 y 0 z 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.15 5.699 Td [( 0 x 0 z 2 J 2 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(1 3 )]TJ/F22 7.9701 Tf 13.151 6.861 Td [( 0 y 0 z 2 J 2 2 3 )]TJ/F20 7.9701 Tf 13.151 5.699 Td [( 0 z 2 2 J 2 3 7 7 7 7 5 : {18 Similarly,inthesecondandthirdquadrantswhere V 12 resides, i = j and m 6 = n suchthat V 12 = 2 6 6 6 6 4 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( 0 x xy 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.699 Td [( 0 x xz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( 0 x yz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 6.861 Td [( 0 y xy 2 J 2 )]TJ/F22 7.9701 Tf 10.494 6.861 Td [( 0 y xz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 6.861 Td [( 0 y yz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.864 Td [( 0 z xy 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.698 Td [( 0 z xz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.864 Td [( 0 z yz 2 J 2 3 7 7 7 7 5 : {19 Lastofall, i 6 = j and m 6 = n inthefourthquadrantwhere V 22 resides.Goingthroughthe necessarymanipulationyields V 22 = 2 6 6 6 6 4 1 )]TJ/F20 7.9701 Tf 13.151 5.865 Td [( xy 2 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( xy xz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( xy yz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( xz xy 2 J 2 1 )]TJ/F20 7.9701 Tf 13.151 5.699 Td [( xz 2 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.865 Td [( xz yz 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.864 Td [( yz xy 2 J 2 )]TJ/F22 7.9701 Tf 10.494 5.864 Td [( yz xz 2 J 2 1 )]TJ/F20 7.9701 Tf 13.151 5.864 Td [( yz 2 2 J 2 3 7 7 7 7 5 : {20 ThepreviousexpressionsdetailthesecondderivativescorrespondingtoavonMises material.ThesesecondderivativesarenotofmuchpracticalusesincethevonMisesyield surfaceisacircleindeviatoricstressspaceand,thus,thestressescanbeupdatedby usingtheradialreturnalgorithmwhichdoesnotexplicitlyusethesecondderivatives. However,thesecondderivativesofavonMisesmaterialcanbeusedtocheckforerrors inthesecondderivativesderivedinthenextfewparagraphsforanisotropicCPB06 materialsincethetwoshouldbeequivalentwhentheyieldintensionisequaltotheyield incompressioni.e.,when k =0. 91

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4.3.2IsotropicCPB06secondderivatives:generalloading Thenextfewparagraphswillfocusonderivingtheexpressionsforthesecond derivativesofanisotropicCPB06materialgivenbyEquation4{1forthegeneralloading situationwhen s 1 6 = s 2 6 = s 3 .Therefore,thefollowingderivativesareneeded: = @ @ mn @ e @ ij = @ @ mn @ e @s @s @ ij = @ 2 s @ mn @ ij @ e @s + @ 2 e @s @s @s @ mn @s @ ij = T 1 + T 2 wherethetensors T 1 and T 2 aredenedas T 1 = @ 2 s @ mn @ ij @ e @s T 2 = @ 2 e @s @s @s @ mn @s @ ij : {21 Notethattherstderivativetermin T 1 isgivenbyEquation2{16andtherstderivativesin T 2 aregiveninSection4.2;therefore,theonlyunknownsinEquation4{21are thesecondderivatives. Thesecondderivativetermappearingintheexpressionfor T 1 inEquation4{21is foundasfollows: @ 2 s @ mn @ ij = @ @ mn @s @J 2 @J 2 @ ij + @s @J 3 @J 3 @ ij = @s @J 2 @ 2 J 2 @ ij @ mn + @s @J 3 @ 2 J 3 @ ij @ mn + @J 2 @ ij @ 2 s @ mn @J 2 + @J 3 @ ij @ 2 s @ mn @J 3 = T A + T B + T C + T D 92

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wherethetensors T A T B T C and T D aredenedas T A = @s @J 2 @ 2 J 2 @ ij @ mn T B = @s @J 3 @ 2 J 3 @ ij @ mn T C = @J 2 @ ij @ 2 s @ mn @J 2 T D = @J 3 @ ij @ 2 s @ mn @J 3 : {22 Therstderivativetermsintensors T A { T D werealreadyderivedinSection4.2.Only thesecondderivativesarecurrentlyunknown.Thesecondderivativetermintensor T A is foundbytakingthederivativeoftherstexpressioninEquation4{10.Doingsoyields @ 2 J 2 @ ij @ mn = @ @ mn ij )]TJ/F21 11.9552 Tf 13.151 8.088 Td [( kk 3 ij = im jn )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 mn ij : Thepreviousexpressionisintensorialformandmust,asbefore,betransformedtomatrix formusingVoight'snotationasfollows: @ 2 J 2 @ ij @ mn = 2 6 4 S 1 0 0 I 3 7 5 {23 where S 1 = 1 3 2 6 6 6 6 4 2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(12 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(12 3 7 7 7 7 5 {24 and I istheidentitymatrix. Similarly,thesecondderivativetermoftensor T B canbefoundbyusingthesecond expressioninEquation4{10whichyields, @ 2 J 3 @ ij @ mn = @ @ mn 0 ip 0 pj )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 J 2 ij = im 0 nj + jn 0 im )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 3 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( mn 0 ij + ij 0 mn : 93

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UsingVoightnotation,thepreviousexpressioncanbepresentedas, @ 2 J 3 @ ij @ mn = 2 6 4 S 11 S 12 S T 12 S 22 3 7 5 {25 where S 11 = 2 3 2 6 6 6 6 4 0 x )]TJ/F26 11.9552 Tf 11.291 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( 0 x + 0 y )]TJ/F15 11.9552 Tf 11.291 0 Td [( 0 x + 0 z )]TJ/F26 11.9552 Tf 11.291 9.684 Td [()]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( 0 x + 0 y 0 y )]TJ/F26 11.9552 Tf 11.291 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( 0 y + 0 z )]TJ/F15 11.9552 Tf 11.291 0 Td [( 0 x + 0 z )]TJ/F26 11.9552 Tf 11.291 9.683 Td [()]TJ/F21 11.9552 Tf 5.479 -9.683 Td [( 0 y + 0 z 0 z 3 7 7 7 7 5 ; {26 S 12 = 1 3 2 6 6 6 6 4 xy xz )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 yz xy )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 xz yz )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 xy xz yz 3 7 7 7 7 5 {27 and S 22 = 2 6 6 6 6 4 0 x + 0 y yz xz yz 0 x + 0 z xy xz xy 0 y + 0 z 3 7 7 7 7 5 : {28 Thesecondderivativetermsoftensors T C and T D forthegeneralloadingstateof s 1 6 = s 2 6 = s 3 canbedeterminedusingthechainrule.Doingsoforthe J 2 -termyields @ 2 s @ mn @J 2 = @ @ mn s 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 = 1 3 s 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(J 2 @s @ mn )]TJ/F21 11.9552 Tf 35.903 8.088 Td [(s s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 2 6 s @s @ mn )]TJ/F21 11.9552 Tf 17.439 8.088 Td [(@J 2 @ mn : {29 Similarly,forthederivativewithrespectto J 3 @ 2 s @ mn @J 3 = @ @ mn 1 3 s 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 s 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(J 2 2 6 s @s @ mn )]TJ/F21 11.9552 Tf 17.44 8.087 Td [(@J 2 @ mn : {30 Theequationsderiveduptothispointalongwiththerstderivativesderivedin Section4.2determinethetensor T 1 inEquation4{21.Thetensor T 2 inEquation4{21 mustnowbedetermined.TherstderivativetermwasderivedinSection4.2;therefore, 94

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theonlyunknownisthesecondderivativetermanditisfoundusingEquation2{16 whichisrewrittenasfollows: @ e @s = m p F 2 3 j s j)]TJ/F21 11.9552 Tf 17.933 0 Td [(ks sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 j s j)]TJ/F21 11.9552 Tf 17.932 0 Td [(ks sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k )]TJ/F15 11.9552 Tf 14.346 8.087 Td [(1 3 j s j)]TJ/F21 11.9552 Tf 17.932 0 Td [(ks sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k with 6 = 6 = inthepreviousequationand ;; =1 ; 2 ; or3.Usingtheprevious equationyieldsanexpressionforthesecondderivativeterminthetensor T 2 ofEquation 4{21as H := @ 2 e @s @s {31 suchthat H = m p F n 2 3 sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + 1 3 sgn s )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 + 1 3 sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 )]TJ/F15 11.9552 Tf 16.665 8.087 Td [(1 m 2 @ e @s 2 o {32 for = ,and H = m p F h )]TJ/F15 11.9552 Tf 11.593 8.088 Td [(2 9 sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 )]TJ/F15 11.9552 Tf 11.593 8.088 Td [(2 9 sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 + 1 9 sgn s )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 )]TJ/F15 11.9552 Tf 16.152 8.088 Td [(1 m 2 @ e @s @ e @s i {33 for 6 = Thepreviousequationsfor H completethederivationofthesecondderivativesof theyieldfunctiongivenbyEquation4{1forthecaseofgeneralloadingwhen s 1 6 = s 2 6 = s 3 .Thederivativespresentedinthissectioncanbeeasilyimplementedintoasubroutine usingthefullyimplicitschemepresentedinSection4.1forthepurposeofrunningnite elementcalculations. 95

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4.3.3IsotropicCPB06secondderivatives:biaxialloading Inthebiaxialloadingcase,thesecondderivativescanbecalculateddirectlyusing Equation4{14asfollows: @ @ mn @ e @ ij = @ @ mn m 3 p F 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(sgn~ s k +3 0 ij = m 3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(sgn~ s k +3 1 p F @ 0 ij @ mn )]TJ/F21 11.9552 Tf 25.868 9.167 Td [( 0 ij 2 F 3 = 2 @F @ mn # whichleadstotheresult @ @ mn @ e @ ij = m 3 p F 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(sgn~ s k +3 im jn )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 3 ij mn )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 3 3 k 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(sgn~ s k +3 0 ij 0 mn F {34 forbiaxialloadingconditions.Similartowhatwasseenwiththebiaxialrstderivatives, Equation4{34reducestothevonMisesresultofEquation4{16when k =0. Equation4{34alongwiththesecondderivativesderivedpreviouslyforthegeneral loadingcaseandtherstderivativesofSection4.2yieldallthatisnecessarytoimplement afullyimplicitreturnmappingalgorithmintoacommercialniteelementcodesee Section4.1. 96

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CHAPTER5 ASSESSMENTOFTHEPROPOSEDSPHERICALVOIDMODELBYFINITE ELEMENTCALCULATIONS Theanalyticexpressionforthemacroscopicyieldfunctionofaporousaggregate containingsphericalvoidsseeEquation3{30wasobtainedbyassumingaspecicRVE geometry,specicloadingconditions,andanumberofapproximations.Theseassumptions werenecessarytoobtainananalyticsolution;however,itisnowdesirabletodetermine howwelltheseanalyticyieldcurvescomparetomoregeneralloadingconditionsand RVEs.Onewaythatthisvalidationcanbeconductedisbycomparingtheanalyticyield curvestoniteelementunitcellcalculations.Intheniteelementunitcellcalculations, thevoidboundaryisexplicitlymeshedandthematrixmaterialismodeledasanelasticplasticmaterialwiththeplasticresponsegovernedbytheisotropicCPB06criterionof Equation2{8. Sinceinniteelementcalculations,theminimizationofthemacroscopicplastic dissipation W D isperformedoveralargesetofkinematicallyadmissiblevelocityelds, theaccuracyoftheyieldsurface,developedforasingleassumedvelocityeld,canbe assessed.Theaxisymmetricunitcellcalculationsdetailedinthissectionareunder constantstraintriaxialityandsimilartothoseperformedbyRistinmaa1997toassess thepredictivecapabilitiesofacyclicmodelforaporousaggregate. 5.1ModelingProcedure TheaimofthissectionistocomparetheanalyticalyieldlocusgivenbyEquation 3{30withnumericalyieldpointsobtainedbyperformingaxisymmetricniteelementcell calculations.Thecontinuumisconsideredtoconsistofaperiodicassemblageofhexagonal cylindricalunitcellswhichareapproximatedbyrightcircularcylindersasshowninFigure 5-1A.Duetosymmetry,onlyonehalfoftheunitcellismeshedseeFigure5-1B.Every cellofinitiallength L 0 andradius R 0 containsasphericalvoidofradius a andissubject tohomogeneousradialandaxialdisplacements.Theboundaryconditionsforthisunitcell 97

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areasfollows: u 1 = ur 2 = ur 3 =0for x 1 =0 u 3 = ur 1 = ur 2 =0for x 3 =0 u 1 = U 1 for x 1 = A u 3 = U 3 for x 3 = B where u i arethedisplacementsinthe x i directionsand ur i aretherotationsaboutthe x i axes.Thevoidisconsideredtobetractionfree.Forthisaxisymmetricunitcell,theinitial porosity f 0 isdenedas f 0 = V void V total = 2 a 3 0 3 A 2 0 B 0 : Aconstantstraintriaxialityisimposedonthesurfaceoftheunitcell.Theprocedurethat isusedwasproposedbyRistinmaa1997andisdetailedinthefollowing.Themainidea isthataconstantstraintriaxialityisequivalenttomaintainingaconstantstresstriaxiality duringtheelasticportionofthecalculation. Themacroscopicprincipalstrainsareasfollows: E 1 =ln A A 0 =ln A 0 + U 1 A 0 E 3 =ln B B 0 =ln B 0 + U 3 B 0 where A 0 and B 0 aretheinitialsidelengthsoftheunitcellasshowninFigure5-1B while U 1 and U 3 aretheprescribeddisplacementsonthosesides.Thesedisplacements willbeprescribedsuchthataconstantstraintriaxialityismaintainedthroughoutthe calculations.Notethatforaxisymmetricloading, E 1 = E 2 .Thestraintriaxiality, T E ,is 98

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denedas T E = E kk 3 E e = 2 E 1 + E 3 2 j E 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(E 3 j where E kk isthetraceofthemacroscopicstraintensorand E e isthemacroscopiceective straindenedas E e = r 2 3 E 0 ij E 0 ij {1 with E 0 ij beingthemacroscopicdeviatoricstraintensor.Theequationaboveshowsthat thestrains E 1 and E 3 mustbelinearlyrelatedinorderforthestraintriaxiality T E tobe constant.Indeed, T E = 2 c E +1 2 j c E )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 j sgn E 3 {2 where c E = E 1 =E 3 .Alternatively, c E = 2 T 2 E +1 3 T E 2 T 2 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 {3 Notethatifthestraintriaxiality T E isconsideredconstantforagivencalculation,then c E maybecalculatedusingtheaboveequation;however,thesignofthestraincomponents e.g., E 3 arenowxedsuchthatthesignofthestraintriaxialitygivenbyEquation5{2 usingthecalculatedvaluefor c E mustagreewiththesignofthestraintriaxialitythat waschosen. Theprescribedboundarydisplacements, U 1 and U 3 cannowberelatedusingthe aboveequations.Doingsoyields U 1 = A 0 B 0 + U 3 B 0 c E )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Themacroscopicstress, ,isalsoaxisymmetricsuchthat 1 = 2 = F 1 2 AB 3 = F 3 A 2 99

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where F 1 isthetotalradialforceat X 1 = A and F 3 isthetotalaxialforceat X 3 = B Intheelasticregime,themacroscopicstrainsarerelatedtothemacroscopicstresses viaHooke'slawsuchthat EE 1 = )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 3 EE 3 = 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 1 where E isYoung'smodulusand isPoisson'sratio.Theaboveexpressioncanbesolved for 1 and 3 usingthefactthat E 1 = c E E 3 toyieldthefollowing: 1 = c E + 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 2 EE 3 3 = 2 c E )]TJ/F21 11.9552 Tf 11.955 0 Td [( +1 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 EE 3 suchthat 1 = c 3 with c = c E + 2 c E )]TJ/F21 11.9552 Tf 11.955 0 Td [( +1 : Now,noticethat sgn J 3 =sgn 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 1 =sgn )]TJ/F21 11.9552 Tf 11.955 0 Td [(c E sgn E 3 =sgn J E 3 suchthat sgn J 3 =sgn J E 3 =sgn T E sgn )]TJ/F21 11.9552 Tf 11.955 0 Td [(c E sgn c E +1 whichresultsinthethirdinvariantofthemacroscopicstressbeingcontrolledbythe signinEquation5{3atleastuntilmacroscopicyield.Inotherwords, J 3 inthe elasticportionofthecalculationispositiveornegativeifthenegativeorpositivesign, respectively,ischoseninEquation5{3.Thisisanimportantpointthatwillbeusedto 100

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controlthesignofthethirdinvariantintheniteelementcalculationspresentedinthe nextsection. Themacroscopicstresstriaxialityisdenedanalogoustothemacroscopicstrain triaxialityas T = kk 3 e = 2 1 + 3 3 j 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 1 j = 2 c +1 3 j 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(c j sgn 3 where e isthevonMiseseectivestressdenedas e = r 3 2 0 ij 0 ij {4 TheisotropicCPB06eectivestress, ~ e ,isrelatedtothevonMiseseectivestress,for theaxisymmetriccase,by ~ e = z s e e where z s e isgivenbyEquation3{28.Notethatthe macroscopicstresstriaxialityasgivenbytheaboveequationisconstantforagivenstress ratio, c ,sincethesignsofthemacroscopicstressesarealsoconstantforadisplacement controlledcalculationintheelasticregion.Alsonotethatthestressequationsinthis sectionarevalidonlyfortheelasticregionsinceHooke'slawwasusedtorelatethe macroscopicstressestothemacroscopicstrains.Thestresstriaxialitywilltypicallynotbe constantoncemacroscopicyieldingoccurs,andthismacroscopicyieldbehavioriswhat willbecomparedtotheanalyticyieldcurvegivenbyEquation3{30. Arelationshipwhichisusefulforobtainingthedesiredspacingofdatapointsisthe slopeofagivencalculationintheelasticregioninthe0 : 5 m e plane: slope = 6 j 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(c j 2 c +1 sgn 3 suchthat =tan )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 slope 101

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where istheanglethattheelasticcurvemakeswithrespecttothehorizontali.e., 0 : 5 m axis.Thisisused,alongwiththeotherequationspresentedinthissection,to obtainthecomputationaltestmatrixpresentedinthenextsection. 5.2FiniteElementResults Finiteelementresultsforthreedierentmaterialsunderaxisymmetricloading willbepresentedinthissection.AllcalculationswereperformedusingaCPB06fully implicitusermaterialsubroutinethatwasimplementedintotheABAQUSniteelement codeAbaqus,2008.Thethreematerialshavethesameyieldstrengthintension; however,onematerialhasanequalyieldstrengthincompression k =0,onehasa yieldincompressionthatislessthantheyieldintensioncorrespondingtoabccmaterial k =0 : 3098andonehasayieldincompressionthatisgreaterthantheyieldintension correspondingtoafccmaterial k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098.Thesespecicvaluesfor k weretakenfrom polycrystalcalculationsdonebyCazacuetal.2006forrandomly-orientedpolycrystals eitherfccorbccinwhichtwinningwastheonlydeformationmodeatthesinglecrystal level.APoisson'sratioof =0 : 32wasusedforeachmaterialalongwithatensileyield strengthtoYoung'smodulusratioof T =E =0 : 00124.Allstressandstrainquantities presentedinthissectionaremacroscopicoraveragequantitiesunlessotherwisenoted. Thecomputationaltestmatrixusedtoobtainthedatapresentedinthissection isshowninTables5-1|5-18.Figures5-2,5-3and5-4showtheniteelementmeshes usedforthecalculationsandcorrespondtoinitialporositiesof f 0 =0 : 01, f 0 =0 : 04 and f 0 =0 : 14,respectively.Theunitcellshavedimensionsof A 0 = B 0 =6inches theunitsareoflimitedpracticalimportancebutareprovidedtoaddmeaningtothe prescribeddisplacements U 1 and U 3 giveninTables5-1|5-18.Themeshesarecomprised ofaxisymmetricquadrilateralelementswithreducedintegrationanduseABAQUS' enhancedhourglasscontrol.Ameshrenementstudywasperformedonasampleof thecalculationstoensuremeshconvergenceandtheselectedmesheswerefoundto providesatisfactoryresultsnotethatsinceallquantitiesbeinginvestigatedareaverage 102

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quantities,evenfairlycoarsemeshescanprovidereasonablycloseagreementintermsof theaverageresponse;themeshesshownwerealsofoundtoprovideadequatelocaldetail withreasonablecomputationalcost. ThemacroscopicyieldsurfacedenedinEquation3{30canbemodiedtoinclude theparametersintroducedinTvergaard1981toaccountforvoidinteractionasfollows: = e T 2 +2 fq 1 cosh z s q 2 3 m 2 T )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ q 3 f 2 =0{5 where z s wasdenedbyEquation3{31as z s =1+ 1 2 sgn 2 m +sgn m T C )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : 3{31 Theparameters q i aretakentobethevaluessuggestedinLeblondandPerrin1990; thus, q 1 =4 =e 1 : 47, q 2 =1and q 3 = q 2 1 Figures5-5and5-6showtheniteelementresultsofavonMisesmaterial k =0 versustheGTNcriterionofEquation1{11forvariousvoidvolumefractionsusing q 1 =4 =e q 2 =1and q 3 = q 2 1 .Sincethemacroscopicstresstriaxialityintheniteelement calculationsremainsconstantonlyintheelasticregion,themacroscopicyieldpointswere identiedwhenthestresstriaxialitywouldchangefromitsinitialconstantvalue.Note thegoodagreementbetweenthetheoreticalyieldcurvesandtheniteelementresults. Thepositivemeanstresshalfoftheguresiswhathasbeentypicallyreportedinthe literature,althoughnoindicationregardingthesignofthethirdinvariantistypically given.NoticethatevenforthevonMisescalculations k =0showninFigures5-5 and5-6wherethematrixmaterial'syieldstrengthsareequal,thersthalfofthecurve alonedoesnotcontainthecompleteinformation.Forexample,noticethattheGTN yieldcurveseemstomatchreasonablywellwiththeniteelementdatacorrespondingto sgn J 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn m .Thisconclusionbasedontheniteelementdatacannotbedrawn byinvestigatingsolelytheregionwithpositivemeanstress. 103

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Figures5-7to5-12comparetheniteelementresultsformaterialsexhibitingtensioncompressionasymmetryusingCPB06valuesof k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098and k =0 : 3098,respectively whentheinitialporosityis f 0 =0 : 01, f 0 =0 : 04and f 0 =0 : 14withtheproposedmodel ofEquation5{5 q 1 =4 =e q 2 =1and q 3 = q 2 1 .Theresultsseemtoindicatedistinct trends;namely,thattheyieldsurfacedependsstronglyonthesignofthethirdinvariant ofthestressdeviator, J 3 .Also,notethattheguresillustratematerialsofteningasthe voidvolumefractionincreases;thisbehavioristobeexpectedsincetheload-bearingarea decreaseswithincreasingvoidvolumefraction. Figures5-13,5-14and5-15showcomparisonsofEquation5{5withtheunitcell axisymmetricniteelementcalculationswhen k =0, k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098and k =0 : 3098, respectively,forinitialporositiesof f 0 =0 : 01, f 0 =0 : 04and f 0 =0 : 14;thegures showtheyieldpointsalongwiththecompletecurvesobtainedfromtheniteelement calculations.NoticethattheselastguresusethenormalizedCPB06eectivestress astheverticalaxissuchthatthethirdinvariantisimplicitandtheyieldcurvesfora particularporositywhichdependonthesignof J 3 fornon-zero k collapsetoasingle curvethischoiceofverticalaxiswasmadesimplytoprovidealloftheinformationfora givenmaterialononegurewithminimalconfusion. Noticethatanimplicitassumptioninalloftheseguresisthattheinitialporosity, f 0 ,isapproximatelyequaltothenalporosity, f ,atyieldforallcalculations.This assumptionshouldnotbetoobadsinceonlythebehavioruptoinitialyieldisofinterest hereand,thus,theloadingonalloftheunitcellsisfairlysmall. 5.3ConcludingRemarks Yieldingofmaterialsinwhichthematrixdisplaystension-compressionasymmetry andcontainssphericalvoidshasbeenstudied.Ananalyticalyieldcriterionwhichisan upper-boundestimatewasdevelopedinChapter3byextendingGurson's1977analysis ofthehollowspheretothecasewhenthematrixplasticbehaviorisdescribedbyCazacu etal.'s2006isotropicyieldcriterion.Duetothetension-compressionasymmetryofthe 104

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matrixresponse,freshdicultieswereencounteredwhenestimatingthelocalplasticdissipation, w d .ThisisbecausetheplasticmultiplierrateassociatedtotheCazacuetal. 2006yieldcriterionhasmultiplebranchesseeEquation2{29anddependsoneach oftheprincipalvaluesofthelocalrateofdeformationtensor, d .Certainapproximations wereintroducedinordertoobtaintheanalytical,closed-formexpressioninEquation 3{30oftheoverallplasticpotential.Yet,theapproximateyieldcriterioninEquation 3{30hasthepropertythatitreproducestheexactsolutionforallkinematicallyadmissiblevelocityeldsinthecaseofpurelyhydrostaticloadingandtheexactsolution fortheassumedGurson1977velocityeldinthecaseofdeviatoricloading. ComparisonbetweenthetheoreticalpredictionsusingthecriterionofEquation3{30 andresultsofniteelementcellcalculationsshowanoverallgoodagreement.Thederived criterionofEquation3{30issensitivetothethirdinvariantofthestressdeviatorand exhibitstension-compressionasymmetryi.e.,itisnolongersymmetricwithrespectto theverticalaxis, m =0.Althoughtheexpressionoftheproposedcriterionforthe void-matrixaggregateissimilartothatofGurson's1977criterion,therearedistinct dierences: First,thevonMisesequivalentstressisreplacedby ~ e = m q P 3 i =1 j 0 i j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k 0 i 2 whichdependsonallprincipalvaluesofthestressdeviator 0 andontheratio betweentheuniaxialyieldintensionandtheuniaxialyieldincompressionofthe matrixthroughtheparameter m seeEquation2{13forthedenitionofthe constant m Secondly,Equation3{30involvesanewcoecient z s denedinEquation3{31 suchthatfortensilehydrostaticloading,yieldingofthevoid-matrixaggregateoccurs when m = )]TJ/F15 11.9552 Tf 9.298 0 Td [( = 3 C ln f ,whileforcompressivehydrostaticloadingyieldingoccurs when m = = 3 T ln f bothcorrespondtotheexactsolutionsobtainedinSection 3.1.2.Thus,forarbitraryloadingstheeectiveyieldlocusisnolongersymmetric withrespecttotheverticalaxis, m =0,asitisinthecaseofGurson's1977yield locus. Ifthereisnodierenceinresponsebetweentheyieldintensionandcompression,the coecient z s =1and ~ e = e sinceCazacuetal.'s2006isotropiccriterionreduces 105

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tothevonMisescriterion;hence,theproposedcriterionofEquation3{30reducesto theclassicalanalyticcriterionofGurson1977.Intheabsenceofvoids,theproposed criterionreducestoCazacuetal.'s2006yieldcriterion.Theaccuracyoftheanalytical criterionwasassessedthroughcomparisonwithnite-elementcellcalculations.Toimprove theagreement,theproposedanalyticyieldcriterionofEquation3{30wasmodied toincludeadditionalparameters, q i ,aswasdonebyTvergaard1981andTvergaard andNeedleman1984fortheGurson's1977yieldcriterionseeEquation5{5.In thismanner,for k =0VonMisesmatrix,thecriterionreducestotheGTNmodel commonlyfoundinthecommercialniteelementcodes.Theagreementbetweenthe theoreticalpredictionsusingthiscriterioninEquation5{5andresultsofniteelement cellcalculationsarequitegood. A B Figure5-1.Axisymmetricunitcellforthesphericalvoid.AUnitcellgeometry.BSlice neededforniteelementcalculations. 106

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Figure5-2. f 0 =0 : 01axisymmetricniteelementmeshfortheunitcell. 107

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Table5-1. k =0, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S10.2500-0.003000.0150273.0 S20.75000.002140.0150247.5 S31.10000.004290.0150236.6 S41.50000.006000.0150228.6 S52.75000.009010.0150216.6 S6-0.2500-0.011240.01126-73.0 S7-0.7500-0.017770.00356-47.5 S8-1.1000-0.01799-0.00113-36.6 S9-1.5000-0.01895-0.00474-28.6 S10-2.7500-0.01565-0.00843-16.6 Table5-2. k =0, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S110.25000.01130-0.0112073.0 S120.75000.01780-0.0036047.5 S131.10000.018000.0011036.6 S141.50000.019000.0047028.6 S152.75000.015700.0084016.6 S16-0.25000.00300-0.01500-73.0 S17-0.7500-0.00210-0.01500-47.5 S18-1.1000-0.00430-0.01500-36.6 S19-1.5000-0.00600-0.01500-28.6 S20-2.7500-0.00900-0.01500-16.6 Table5-3. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S210.2500-0.003000.0150273.0 S220.75000.002140.0150247.5 S231.10000.004290.0150236.6 S241.50000.006000.0150228.6 S252.75000.009010.0150216.6 S26-0.2500-0.011240.01126-73.0 S27-0.7500-0.017770.00356-47.5 S28-1.1000-0.01799-0.00113-36.6 S29-1.5000-0.01895-0.00474-28.6 S30-2.7500-0.01565-0.00843-16.6 108

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Table5-4. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S310.20700.01004-0.0112473.0 S320.62000.01868-0.0063247.5 S330.90900.01746-0.0011336.6 S341.24000.019390.0026728.6 S352.27200.013810.0063316.6 S36-0.20700.00364-0.01498-73.0 S37-0.6200-0.00111-0.01498-47.5 S38-0.9090-0.00321-0.01498-36.6 S39-1.2400-0.00495-0.01498-28.6 S40-2.2720-0.00812-0.01498-16.6 Table5-5. k =0 : 3098, J 3 > 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S410.2500-0.003000.0150273.0 S420.75000.002140.0150247.5 S431.10000.004290.0150236.6 S441.50000.006000.0150228.6 S452.75000.009010.0150216.6 S46-0.2500-0.011240.01126-73.0 S47-0.7500-0.017770.00356-47.5 S48-1.1000-0.01799-0.00113-36.6 S49-1.5000-0.01895-0.00474-28.6 S50-2.7500-0.01565-0.00843-16.6 Table5-6. k =0 : 3098, J 3 < 0and f 0 =0 : 01sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S510.30300.01297-0.0112473.0 S520.90700.01706-0.0011347.5 S531.33100.014790.0026736.6 S541.81500.018000.0063328.6 S553.32800.013890.0084416.6 S56-0.30300.00227-0.01498-73.0 S57-0.9070-0.00320-0.01498-47.5 S58-1.3310-0.00535-0.01498-36.6 S59-1.8150-0.00700-0.01498-28.6 S60-3.3280-0.00979-0.01498-16.6 109

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Figure5-3. f 0 =0 : 04axisymmetricniteelementmeshfortheunitcell. 110

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Table5-7. k =0, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S610.2500-0.003650.0183073.0 S620.75000.002610.0183047.5 S631.10000.005220.0183036.6 S641.50000.007310.0183028.6 S652.75000.009870.0164616.6 S66-0.2500-0.013300.01333-73.0 S67-0.7500-0.018780.00376-47.5 S68-1.1000-0.01886-0.00118-36.6 S69-1.5000-0.01855-0.00464-28.6 S70-2.7500-0.01621-0.00873-16.6 Table5-8. k =0, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S710.25000.01333-0.0133073.0 S720.75000.01884-0.0037647.5 S731.10000.018920.0011836.6 S741.50000.018600.0046528.6 S752.75000.016250.0087416.6 S76-0.25000.00365-0.01824-73.0 S77-0.7500-0.00261-0.01824-47.5 S78-1.1000-0.00522-0.01824-36.6 S79-1.5000-0.00730-0.01824-28.6 S80-2.7500-0.00986-0.01642-16.6 Table5-9. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S810.2500-0.003650.0183073.0 S820.75000.002610.0183047.5 S831.10000.005220.0183036.6 S841.50000.007310.0183028.6 S852.75000.009870.0164616.6 S86-0.2500-0.013300.01333-73.0 S87-0.7500-0.018780.00376-47.5 S88-1.1000-0.01886-0.00118-36.6 S89-1.5000-0.01855-0.00464-28.6 S90-2.7500-0.01621-0.00873-16.6 111

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Table5-10. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S910.20700.01321-0.0147873.0 S920.62000.01880-0.0063747.5 S930.90900.01830-0.0011836.6 S941.24000.017920.0024728.6 S952.27200.017160.0078716.6 S96-0.20700.00444-0.01824-73.0 S97-0.6200-0.00135-0.01824-47.5 S98-0.9090-0.00391-0.01824-36.6 S99-1.2400-0.00603-0.01824-28.6 S100-2.2720-0.00890-0.01642-16.6 Table5-11. k =0 : 3098, J 3 > 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1010.2500-0.003650.0183073.0 S1020.75000.002610.0183047.5 S1031.10000.005220.0183036.6 S1041.50000.007310.0183028.6 S1052.75000.009870.0164616.6 S106-0.2500-0.013300.01333-73.0 S107-0.7500-0.018780.00376-47.5 S108-1.1000-0.01886-0.00118-36.6 S109-1.5000-0.01855-0.00464-28.6 S110-2.7500-0.01621-0.00873-16.6 Table5-12. k =0 : 3098, J 3 < 0and f 0 =0 : 04sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1110.30300.01382-0.0119773.0 S1120.90700.01987-0.0013147.5 S1131.33100.018750.0033936.6 S1141.81500.018120.0063728.6 S1153.32800.015990.0097216.6 S116-0.30300.00276-0.01824-73.0 S117-0.9070-0.00390-0.01824-47.5 S118-1.3310-0.00651-0.01824-36.6 S119-1.8150-0.00768-0.01642-28.6 S120-3.3280-0.01073-0.01642-16.6 112

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Figure5-4. f 0 =0 : 14axisymmetricniteelementmeshfortheunitcell. 113

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Table5-13. k =0, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1210.2500-0.005030.0252373.0 S1220.75000.003600.0252347.5 S1231.10000.007200.0252336.6 S1241.50000.009070.0227028.6 S1252.75000.013610.0227016.6 S126-0.2500-0.018330.01838-73.0 S127-0.7500-0.025860.00519-47.5 S128-1.1000-0.02597-0.00163-36.6 S129-1.5000-0.02554-0.00640-28.6 S130-2.7500-0.02232-0.01203-16.6 Table5-14. k =0, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1310.25000.01838-0.0183373.0 S1320.75000.02597-0.0051847.5 S1331.10000.026080.0016336.6 S1341.50000.025650.0064028.6 S1352.75000.022400.0120516.6 S136-0.25000.00504-0.02512-73.0 S137-0.7500-0.00360-0.02512-47.5 S138-1.1000-0.00719-0.02512-36.6 S139-1.5000-0.00906-0.02262-28.6 S140-2.7500-0.01358-0.02262-16.6 Table5-15. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1410.2500-0.004530.0227073.0 S1420.75000.003240.0227047.5 S1431.10000.006480.0227036.6 S1441.50000.009070.0227028.6 S1452.75000.012250.0204316.6 S146-0.2500-0.016500.01654-73.0 S147-0.7500-0.023280.00467-47.5 S148-1.1000-0.02338-0.00146-36.6 S149-1.5000-0.02299-0.00576-28.6 S150-2.7500-0.02009-0.01083-16.6 114

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Table5-16. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1510.20700.01639-0.0183373.0 S1520.62000.02333-0.0079047.5 S1530.90900.02271-0.0014636.6 S1541.24000.024710.0034028.6 S1552.27200.021290.0097616.6 S156-0.20700.00550-0.02262-73.0 S157-0.6200-0.00168-0.02262-47.5 S158-0.9090-0.00485-0.02262-36.6 S159-1.2400-0.00748-0.02262-28.6 S160-2.2720-0.01103-0.02036-16.6 Table5-17. k =0 : 3098, J 3 > 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1610.2500-0.004530.0227073.0 S1620.75000.003240.0227047.5 S1631.10000.006480.0227036.6 S1641.50000.009070.0227028.6 S1652.75000.012250.0204316.6 S166-0.2500-0.016500.01654-73.0 S167-0.7500-0.023280.00467-47.5 S168-1.1000-0.02338-0.00146-36.6 S169-1.5000-0.02299-0.00576-28.6 S170-2.7500-0.02009-0.01083-16.6 Table5-18. k =0 : 3098, J 3 < 0and f 0 =0 : 14sphericalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] S1710.30300.01715-0.0148573.0 S1720.90700.02466-0.0016347.5 S1731.33100.023270.0042036.6 S1741.81500.022480.0079128.6 S1753.32800.019830.0120516.6 S176-0.30300.00343-0.02262-73.0 S177-0.9070-0.00483-0.02262-47.5 S178-1.3310-0.00807-0.02262-36.6 S179-1.8150-0.01058-0.02262-28.6 S180-3.3280-0.01331-0.02036-16.6 115

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Figure5-5.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e for k =0i.e.,vonMisesandaninitialporosityof f 0 =0 : 01. Figure5-6.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e for k =0i.e.,vonMiseswithinitialporositiesof f 0 =0 : 01, f =0 : 04and f =0 : 14. 116

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Figure5-7.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foranFCCmaterialwith k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C andaninitialporosityof f 0 =0 : 01. Figure5-8.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foraBCCmaterialwith k =0 : 3098 T > C andaninitialporosityof f 0 =0 : 01. 117

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Figure5-9.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foranFCCmaterialwith k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C andaninitialporosityof f 0 =0 : 04. Figure5-10.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foraBCCmaterialwith k =0 : 3098 T > C andaninitialporosityof f 0 =0 : 04. 118

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Figure5-11.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foranFCCmaterialwith k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C andaninitialporosityof f 0 =0 : 14. Figure5-12.Axisymmetricniteelementresultsversusanalyticyieldcurveusing q 1 =4 =e foraBCCmaterialwith k =0 : 3098 T > C andaninitialporosityof f 0 =0 : 14. 119

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Figure5-13.Axisymmetricniteelementyieldpointsversusanalyticalyieldcurvefor k =0i.e.,vonMisesandvariousporositiestheverticalaxisisthevon Miseseectivestress. Figure5-14.AxisymmetricniteelementresultsversusanalyticalyieldcurveforanFCC materialwith k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 T < C andvariousporositiestheverticalaxis istheisotropicCPB06eectivestress,containinganimplicitdependenceon thethirdinvariant, J 3 120

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Figure5-15.AxisymmetricniteelementresultsversusanalyticalyieldcurveforaBCC materialwith k =0 : 3098 T > C andvariousporositiestheverticalaxisis theisotropicCPB06eectivestress,containinganimplicitdependenceonthe thirdinvariant, J 3 121

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CHAPTER6 PLASTICPOTENTIALSFORHCPMETALSWITHCYLINDRICALVOIDS Thischapterfocusesondevelopingamacroscopicyieldcriterionforavoid-matrix aggregatewherethevoidshavecylindricalgeometryandthemetalmatrixisisotropic withtension-compressionasymmetry.Nominallycylindricalvoidsareoftenfoundin experimentalspecimensdue,forexample,tothecrackingofcylindricalinclusionsasin Figure1-1,theelongationofvoidsinneckingregionsortothelinkingupofelliptical voidsinsomepreferreddirectionsee,forexample,Figure6-1wheresphericalvoidsinside ashearbandhavecoalescedtoformalong,cylindricalvoid.Gurson1977developeda macroscopicyieldcriterionforavoid-matrixaggregatecontainingcylindricalvoidsand withthematrixmaterialobeyingavonMisesyieldcriterionseeChapter1.Leblond etal.1994extendedbothGurson's1977cylindricalandsphericalcriteriatothecase whenthematrixmaterialobeysaNortonowrule.Liaoetal.1997extendedGurson's 1977cylindricalcriterioninthecaseoftransverselyisotropicmetalsheetsunderplane stressloadingconditionswhenthematrixmaterialobeysHill's1948yieldcriterionsee Hill,1950. Theoutlineofthechapterisasfollows.Section6.1discussessomelimitsolutions whichthedevelopedcriterionshouldreducetoundertherespectivescenarios.Section 6.2presentsthechoiceofvelocityeld.Section6.3detailsthederivationofparametricrepresentationsoftheyieldlocusforthevoid-matrixaggregatewhenthevoidsare cylindrical,thematrixexhibitstension-compressionasymmetryandtheloadingisaxisymmetric.Section6.4.1developsthelocalplasticdissipationforeitherplanestressorplane strainloadingandSection6.4.2developsthecorrespondingapproximateanalyticaland macroscopicyieldcriterion. Inthischapter,therepresentativevolumeelementRVEisaxisymmetricand containsathrough-thickness,circular,cylindricalvoidasshowninFigure6-2.The followingassumptionsaremadeconcerningtheloadingandmatrixmaterialresponse: 122

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Theaxialdeformationisuniform;i.e., d 33 = D 33 =constant. Theaxialdeformationissymmetric;thus,aprincipalsysteminCartesiancoordinates canbefoundsuchthat D 11 = D 22 and D ij =0if i 6 = j withsimilarexpressionsfor Thematrixmaterialisfullyplasticandincompressiblesuchthat d ij = d P ij inEquation 2{1i.e.,rigidplasticbehavior. Intheaboveassumptions, d representsthemicroscopicrateofdeformationtensorand D representsthemacroscopicrateofdeformationtensor.TheouterradiusoftheRVEis b whiletheinnerradiusortheradiusofthevoidisdenotedby a .Themacroscopicstress state, ,andassociateddeviator, 0 ,correspondingtotheseassumptionsare = 2 6 6 6 6 4 11 00 0 11 0 00 33 3 7 7 7 7 5 {1 and 0 = 2 6 6 6 6 4 1 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 00 0 1 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 0 00 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(2 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 3 7 7 7 7 5 : {2 6.1LimitSolutions Beforeproceedingtothederivationofthemacroscopicyieldcriterionfortheporous aggregate,variouslimitingsolutionswillbeinvestigatedinordertoprovidedirection forcertainassumptionsmadeinthelatersectionsinordertoarriveataclosed-form expressionforthecriterion.Themacroscopicanalyticcriteriontobedevelopedinthe latersectionsshouldagreewiththelimitingsolutionslaidoutinthefollowing. 6.1.1Zeroporosityandvonmisesmateriallimitingcases Therststatewhereananalyticsolutionisimmediatelycleariswhentheporosity tendstowardzero.Inthiscase,thehomogenizedmaterialandthematrixmaterialbecome 123

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thesameandtheyieldcriterionreducestoEquation2{11.Inotherwords, e = e = T : {3 AnotherlimitingcaseofinterestisthemacroscopicyieldcriterionobtainedbyGurson 1977inwhichthevoid-matrixaggregateiscomprisedofcylindricalvoidsandavon Misesmatrixmaterial.Gurson'smacroscopicyieldcriterionforcylindricalvoidswasgiven inChapter1seeEquation1{2andisrewrittenbelowforthereader'sconvenience: C G = C eqv e Y 2 +2 f cosh p 3 2 Y )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(f 2 =01{2 where Y istheyieldstrengthofthevonMisesmaterialand C eqv = 8 > < > : +3 f +24 f 6 2 forplanestrain 1foraxisymmetry : 1{3 SincetheisotropicCPB06yieldcriterionreducestothevonMisesyieldcriterionfor k =0,themacroscopicyieldcriterionfortheporousaggregatecontainingcylindricalvoids andwiththematrixdescribedbytheisotropicCPB06shouldalso,for k =0,reduceto Gurson'smacroscopicyieldcriterion. 6.1.2Analysisofahydrostatically-loadedhollowcylinder Theproblemofacylindercontainingacylindricalvoidandunderaxisymmetric loadinghasbeeninvestigatedovertheyearsinsomedetailwhenthematrixmaterialis governedbyavonMisescriterionsee,forexample,Kachanov,2004;Lubliner,1990. Ratherthanobtainthefullsolutiontotheelastic-plasticproblemofanisotropicCPB06 thick-walledcylinder,thenextfewparagraphswillbedevotedtotheassessmentof thesignsoftheprincipaldeviatorstressesthatarethesolutiontotheproblem.This informationiscrucialindeterminingthelocalplasticdissipation|specically,theordering oftheprincipalvaluesofthemicroscopicrateofdeformationtensorseeEquation2{27. 124

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Considerathick-walledtubeofinnerradius a andouterradius b thatissubjected toahydrostaticpressureonitsoutersurface.Asalreadymentioned,themoregeneral problemofanelastic-plasticmaterialobeyingtheisotropicCPB06yieldconditionof Equation2{8willbeassumedeventhoughinthederivationofthemacroscopicyield criterionforporousaggregates,thelimitingsolutionwillcorrespondtotherigid-plastic case.Thisisdonebecausetheelastic-plasticsphereisamoregeneralproblemwiththe rigidplasticstateresultingastheYoung'smodulustendstowardinnity.Thederivation carriedoutbelowwillfollowmuchofthesameprocedureusedinLubliner1990withthe variousgoverningequationsobtainedfromMalvern1969.Strain-displacementrelations, straincompatibility,Hooke'slaw,andtheequilibriumequationwillallbeusedtoobtaina relationshipbetweenthethirdinvariantofthelocaldeviatoricstresstensorandtheglobal appliedpressureormeanstress. 6.1.2.1Strain-displacementrelations Assumingsmallstrains,themicroscopicstraintensorisdenedas = 1 2 u )]TJ 0.997 -8.17 Td [(r + )778(! r u ; where,incylindricalcoordinates,thegradientofavector u isdenedas [ u )]TJ 0.997 -8.17 Td [(r ]:= 2 6 6 6 6 4 @u r @r 1 r @u r @ )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(u r @u r @z @u @r 1 r @u @ + u r r @u @z @u z @r 1 r @u z @ @u z @z 3 7 7 7 7 5 with [ u )778(! r ]=[ u )]TJ 0.996 -8.169 Td [(r ] T : Inthegivenproblem,theonlyexternalloadinginthe r planeisthein-planemean stress, = r ,whichactsnormaltothecylinder'soutersurface.Therefore,the displacementeldmustbeaxiallysymmetricandtheonlynonzerodisplacementinthe r planeistheradialdisplacementwhichvarieswith r aloneduetotheaxialsymmetryof 125

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theproblemi.e., u r = u r r .Also,notethatifthecylinderissucientlylong,theaxial stressesandstrainsawayfromtheendsmayberegardedasindependentof z .Withthe conditionsofaxialsymmetryandalong"cylinder,theresultisthatalltheshearstrains andstressesvanishandthestrain-displacementrelationaboveyieldstheonlynonzero componentsofthestraintensorasfollows: r = du r dr {4a = u r r {4b z = du z dz {4c 6.1.2.2Straincompatibility TheSt.Venantcompatibilityequationsarerequiredwhenthestrainsareconsideredknownsincethenthestrain-displacementequationsyieldsixequationsforthree displacementunknownsinthegeneralcase.Thecompatibilityequationsensurethatthe assumedstrainsarephysicallypossible.Ifthedisplacementsareincludedasunknowns intheproblem,asinNavier'sdisplacementequationsofmotion,thenthecompatibility equationsarenotneeded.TheSt.Venantcompatibilityequationsareexpressedbelowas S = )778(! r E )]TJ 0.997 -8.17 Td [(r =0 : Theaboveexpressionrepresentssixpartialdierentialequations.However,itcanbe shownthatonlythreeofthesesixequationsareindependent.Thesixequationsarerather tedioustowriteincylindricalcoordinates,soonlytheequationneededforthepresent problemisexpandedbelowas S zz = 2 r @ 2 r @@r )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(@ 2 @r 2 )]TJ/F15 11.9552 Tf 15.391 8.088 Td [(1 r 2 @ 2 rr @ 2 + 1 r @ rr @r + 2 r 2 @ r @ )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 r @ @r =0 : 126

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TheothervecompatibilityequationsincylindricalcoordinatescanbefoundinMalvern 1969.Thecompatibilityconditionaboveyieldstherelationship r = d dr r {5 whereonlyonesubscriptisnowbeingusedforthestraincomponentssincethereareno shearspresentandthesecondsubscriptisextraneous.Equation6{5isobviouslysatised byEquations6{4. 6.1.2.3Equationsofmotion NeglectingbodyforcesandassumingstaticequilibriuminCauchy'sequationsof motionalsoknownasCauchy'srstlawofmotionorsimplyasthebalanceoflinear momentum,yieldsthefollowingequations: r =0 : Theaboveresultiscommonlyreferredtoastheequilibriumequationsandrepresentsin R 3 threerst-order,linearpartialdierentialequationswithsolution ij = ij x 1 ;x 2 ;x 3 Theexpandedformoftheequilibriumequationincylindricalcoordinatesislistedbelowas @ rr @r + 1 r @ r @ + @ zr @z + 1 r rr )]TJ/F21 11.9552 Tf 11.955 0 Td [( =0 @ r @r + 1 r @ @ + @ z @z + 2 r r =0 @ rz @r + 1 r @ z @ + @ zz @z + 1 r rz =0 : {6 Aswiththedisplacementeld,themicroscopicstresseldmustalsobeaxiallysymmetric since,inthe r plane,theproblemcontainsonlyboundarystressesactingnormaltothe outersurfaceofanisotropiccylinderandsincethecylinderisconsideredlongenoughsuch thatthestrainsandstressesintheinteriorawayfromtheedgesareindependentof z Theonlynonzerocomponentsofthestresstensorarethen r ,and z .Usingthese symmetries,therstequationinEquations6{6istheonlynon-trivialequationand 127

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yields: d r dr + r )]TJ/F21 11.9552 Tf 11.956 0 Td [( r =0{7 whereonlyonesubscriptisnowbeingusedforthestresscomponentsaswiththestrain components. 6.1.2.4Elasticconstitutiverelation:Hooke'slaw TheelasticresponseofthematerialisassumedtobegovernedbyHooke'slaw. Therefore,thestress-strainrelationbecomes, ij = 1 E [+ ij )]TJ/F21 11.9552 Tf 11.955 0 Td [( kk ij ]{8 where E istheYoung'smodulus, isPoisson'sratio,and ij istheKroneckerdelta.Note thatforrigidplasticbehavior,theYoung'smodulus,E,tendstowardinnitysuchthatthe elasticstressesproducenostrainsand,hence,nodisplacements.Equation6{8canbe reducedtothefollowingrelevantequationsrelatingtheunknownstrainstotheunknown stresses: r = 1 E [ r )]TJ/F21 11.9552 Tf 11.956 0 Td [( + z ] = 1 E [ )]TJ/F21 11.9552 Tf 11.955 0 Td [( r + z ] z = 1 E [ z )]TJ/F21 11.9552 Tf 11.955 0 Td [( r + ] : Theaboveexpressionscanbesimpliedbysolvingthelastequationfor z andsubstitutingtheresultantexpressionintotheothertwo.Doingsoyields, r = 1+ E [ )]TJ/F21 11.9552 Tf 11.955 0 Td [( r )-222()]TJ/F21 11.9552 Tf 21.254 0 Td [( ] )]TJ/F21 11.9552 Tf 11.955 0 Td [( z = 1+ E [ )]TJ/F21 11.9552 Tf 11.955 0 Td [( )-222()]TJ/F21 11.9552 Tf 21.254 0 Td [( r ] )]TJ/F21 11.9552 Tf 11.955 0 Td [( z : {9 6.1.2.5Relationbetween J 3 and m Inordertoderivetheequationsfortheelasticsolution,thecylinderwillberegarded aspurelyelastic.Thein-planeproblemcontainstwounknownstressesandtwounknown strainswhichwillbesolvedforusingtwoequationsfromHooke'slaw,oneequationfrom 128

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compatibilityandoneequationfromequilibriumi.e.,fourequationsforfourunknowns. Theaxialstrainandaxialstresswillbedeterminedbyassumingeitherplanestress z =0orplanestrain z =0loadingconditions. SubstitutingtheequationsfromHooke'slaw,Equation6{9,intotheequationfrom straincompatibility,Equation6{5,yields d dr [ )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [( r ] )]TJ/F21 11.9552 Tf 13.151 8.087 Td [( r )]TJ/F21 11.9552 Tf 11.955 0 Td [( r =0 : Now,makinguseoftheequilibriumequation,Equation6{7,theaboveequationsimpliesfurthertoyield d dr r + =0 : TheaboveequationalongwithEquation6{7representasystemoftwoordinary dierentialequationsintwounknownfunctions r and .Inordertosolvethesystem, anoperatormethodwillbeusedseeRoss,1984.Lettheoperator D = d=dr ,suchthat thelastequationalongwiththeequilibriumequationbecome, D r + D =0 and D + 1 r r )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 r =0 : Inordertodetermine r therstequationismultipliedby1 =r ,thesecondequationby D ,andtheresultingequationsareaddedtoobtainanexpressionintermsof r alone. Usingasimilarprocedureyieldsanexpressionintermsof alone.Replacing D with d=dr yieldstheresultingordinarydierentialequationsbelowas, r 2 d 2 r dr 2 +2 r d r dr =0 r 2 d 2 dr 2 +2 r d dr =0 : Theaboveequationsareordinarylineardierentialequationswithvariablecoecients.Furthermore,theaboveequationscorrespondtoaspecictypeofordinary 129

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dierentialequationknownasCauchy-Eulerequations.Cauchy-Eulerequationsareequationswhosetermsareconstantmultiplesofanexpressionoftheform x n d n y=dx n .The methodofsolutionforaCauchy-Eulerequationistomakethesubstitution x = e t in ordertoreducetheequationtoalineardierentialequationintermsof t withconstant coecients.Makingthesubstitutionintotheabovetwoequationsyields, d 2 r dt 2 +2 d r dt =0 d 2 dt 2 +2 d dt =0 ; withgeneralsolutions r = c 1 + c 2 r 2 = c 3 + c 4 r 2 : Theconstants c 1 to c 4 aredeterminedbysubstitutingtheaboveexpressionsbackintothe equilibriumequation6{7.Doingsoyields c 1 = c 3 and c 2 = )]TJ/F21 11.9552 Tf 9.298 0 Td [(c 4 .Now,let c 1 = c 3 =: A and c 4 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(c 2 =: B ,suchthattheexpressionsforthestressesbecome, r = A )]TJ/F21 11.9552 Tf 13.569 8.087 Td [(B r 2 = A + B r 2 : Theconstants A and B aredeterminedbyimposingtheboundaryconditionsforthe problem.Theboundaryconditionsforthegivenproblem,againassumingthatthe cylinderistotallyelastic,areasfollows: r = 8 > < > : 0at r = a r = = 2at r = b Theconstants A and B cannowbesolvedforandaregivenas, A = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f B = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f a 2 2 130

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where f := a=b 2 .Finally,theexpressionsforthestressesaregivenasfollows: r = = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1 )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(a 2 r 2 {10a = = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1+ a 2 r 2 : {10b Now,letplanestressloadingconditionsbeassumedsuchthat z =0.Ifthisisthe case,thenthemicroscopicmeanstress, m ,isgivenas PST m = 2 3 = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f andthemicroscopicstressdeviatorfortheplanestresscase, s PST ,becomes s PST = 2 6 6 6 6 4 s r 00 0 s 0 00 s z 3 7 7 7 7 5 = = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 6 6 6 6 4 1 3 )]TJ/F22 7.9701 Tf 13.15 4.707 Td [(a 2 r 2 00 0 1 3 + a 2 r 2 0 00 )]TJ/F20 7.9701 Tf 10.494 4.707 Td [(2 3 3 7 7 7 7 5 : {11 NotethatEquation6{11dictatesthefollowing: sgn s r = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn for r< p 3 a =: c PST sgn s =sgn for r 2 a;b sgn s z = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn for r 2 a;b whichimpliesthat sgn J 3 = 8 > < > : sgn if rc PST where c PST = p 3 a: 131

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Ifplanestrainloadingconditionsareassumedinsteadsuchthat z = r + ,then themicroscopicmeanstress, m ,willbecome PSN m = 2 3 + = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f andthemicroscopicstressdeviatorfortheplanestraincase, s PSN ,becomes s PSN = 2 6 6 6 6 4 s r 00 0 s 0 00 s z 3 7 7 7 7 5 = = 2 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 6 6 6 6 4 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 3 )]TJ/F22 7.9701 Tf 13.151 4.707 Td [(a 2 r 2 00 0 1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 3 + a 2 r 2 0 00 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 )]TJ/F20 7.9701 Tf 6.675 -4.977 Td [(1 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 3 3 7 7 7 7 5 : {12 Notethattheterm1 )]TJ/F15 11.9552 Tf 12.438 0 Td [(2 isalwayspositiveorzerosince 0 : 5.Therefore,Equation 6{12impliesthefollowing: sgn s r = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn for r< q 3 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 a =: c PSN sgn s =sgn for r 2 a;b sgn s z = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn for r 2 a;b whichimpliesthat sgn J 3 = 8 > < > : sgn if rc PSN where c PSN = r 3 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 a: Notethatinboththeplanestressandplanestraincase,thefollowingisobserved: sgn s r = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn for r
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whichimpliesthat sgn J 3 = 8 > < > : sgn if rc {14 where c = 8 > < > : c PST = p 3 a forplanestressconditions c PSN = q 3 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 a forplanestrainconditions : {15 Equations6{3,1{2and6{14providelimitingsolutionsthatcanbeusedtoguidein theorderingoftheprincipalvaluesofthelocalrateofdeformation.Asalreadymentioned, thecrucialchallengeinobtainingaclosed-formexpressionforthemacroscopiccriterionof theporousaggregateistheestimateoftheexpressionoftheplasticdissipationinamatrix exhibitingstrengthdierentialeects. 6.2ChoiceofTrialVelocityField TherststepindeterminingthemacroscopicyieldfunctionistodetermineaparticularformforthemicroscopicvelocityeldintheRVEanddeterminethecorresponding microscopicrateofdeformation.Themacroscopicrateofdeformationeld, D ij ,isdened bythefollowingequationseeGurson,1977: D ij = 1 V Z S 1 2 v i n j + v j n i dS where V isthevolumeofaunitcubeRVE, S istheoutersurfaceoftheRVE, ~n is theunitoutwardnormaltothesurface S ,and ~v isthemicroscopicvelocityeld.The aboveequationcanberewrittenusingGauss'stheoremandthedenitionfortherateof deformationas D ij = 1 V Z V d ij dV: Intheaboveexpression, d ij isthemicroscopicrateofdeformationtensor,whichisby denitionseeMalvern,1969 d ij = 1 2 @v i @x j + @v j @x i where ~x isthepositionofamaterialpointinCartesiancoordinates. 133

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Thevelocityeld, ~v ,intheRVEisrequiredtobekinematicallyadmissible.In otherwords,thevelocityeldshouldsatisfythecompatibilityconditionsdisplacement boundaryconditionsaswellastheincompressibilitycondition.Theseconditionsare writtenasfollows: ~v j ~x = b~e r = D ~x = D ij x j div ~v =0 : {16 Therstrequirementisastatementofthedisplacementboundarycondition.Thesecond requirementstatesthatthematrixmaterialisincompressible,whichisgenerallythe assumptionmadeforafullyplasticstate. Toobtaintheupperboundestimateoftheoverallplasticpotential,theclassicallocal velocityeld v proposedbyGurson1977;RiceandTracey1969willbeused.This velocityeldcontainsapartthatcontrolsthevolumechange, v V ,andapartthatcontrols thechangeinshape, v S : v = v V + v S : Thein-planedeviatoricportionofthemacroscopicrateofdeformationisnowdenedas follows: D 0 = D )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 D : Intheaboveequationandthroughouttheremainderofthissection,Greeksubscripts rangefrom1to2.Notethatinthespecialcaseofaxisymmetricdeformationthatwas assumed, D 0 =0since D 11 = D 22 .Theterm = 2 D inthepreviousexpression representsthemeanin-planerateofdeformation.Thisnomenclatureisconsistentwith thatusedinGurson1977andLiaoetal.1997.TheboundaryconditioninEquation 6{16cannowbeexpressedintermsofthevolumetricandshape-changingpartsofthe 134

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velocityeldasfollows: v S j ~x = b~e r = D 0 x v V j ~x = b~e r = 1 2 D x : {17 Theincompressibilityconstraintcannowbeenforcedonthevolumetricportionof thevelocityeldnotingthatthevelocityeldinthez-directionispurelyvolumetricto obtain 1 r @ @r )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(rv V r + 1 r @v V @ + @v z @z =0 @v V r @r + v V r r + D 33 =0 or @v V r @r = )]TJ/F26 11.9552 Tf 11.291 16.856 Td [( v V r r + D 33 : Asolutiontotheequationaboveis v V r = c 1 r )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(D 33 2 r where c 1 isaconstant.EnforcingtheboundaryconditionofEquation6{17,yieldsan expressionfor c 1 asfollows: c 1 = b 2 2 D 11 + D 22 + D 33 : Usingthepreviousresultfor c 1 ,thevolumetricpartoftheradialvelocitycomponentcan beexpressedas v V r = 1 2 D 11 + D 22 + D 33 b 2 r )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 33 r 2 = D b 2 r )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 33 r 2 wherethenewquantity D isdenedasfollows: D = 1 2 D 11 + D 22 + D 33 : 135

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Theradialandangularportionsofthevolumetricpartoftherateofdeformationeldcan nowbewrittenasseeMalvern,1969 d V rr = @v V r @r = )]TJ/F15 11.9552 Tf 12.044 3.022 Td [( D b r 2 )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(D 33 2 and d V = v V r r = D b r 2 )]TJ/F21 11.9552 Tf 13.15 8.087 Td [(D 33 2 : Theshape-changingpartofthevelocityeldisfoundbyassumingthatitisofthe followingform: ~v S = B ~x where B isaconstant,symmetricanddeviatorici.e., tr B =0tensor.UsingEquation6{17,thetensor B isshowntobeidenticallyequaltothedeviatoricpartofthe macroscopicrateofdeformationtensor, D 0 ,whichiszeroduetotheassumptionofaxiallysymmetricanduniformdeformationpurein-planevolumechange.Therefore,the velocityeldcanbeexpressedas v V = D b 2 r )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 33 r 2 e r and v S = D 0 x {18 where,onceagain, D = = 2 D 11 + D 22 + D 33 and D 0 = D )]TJ/F15 11.9552 Tf 12.457 0 Td [( = 2 D 11 + D 22 .The principalvaluesunorderedofthelocalstrainrateeld d correspondingtothevelocity eld v givenbyEquation6{18areasfollows: d r = )]TJ/F15 11.9552 Tf 12.045 3.022 Td [( D b r 2 )]TJ/F21 11.9552 Tf 13.15 8.087 Td [(D 33 2 d = D b r 2 )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(D 33 2 d z = d 33 = D 33 : {19 136

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Sinceprincipalvaluesofasecond-ordertensorareinvariantseeMalvern,1969,theabove componentsyieldtheprincipalmicroscopicrateofdeformationcomponentswithrespect toanyarbitrarychoiceofinitialcoordinatesystem.Theunorderedprincipalcomponents ofthemicroscopicrateofdeformationtensor, ~ d ,canthenbeexpressedasfollows: ~ d 1 = d r ~ d 2 = d ~ d 3 = d z = D 33 : {20 6.3ParametricRepresentationofthePorousAggregateYieldLocusfor AxisymmetricLoading Thissectiondetailsthedevelopmentofanexactfortheassumedvelocityeld givenbyEquation6{18,parametricrepresentationfortheyieldlocusofavoid-matrix aggregatewhenthematrixisgovernedbytheCazacuetal.2006criterion.Herethe voidgeometryiscylindricalandtheloadingisaxisymmetric.Thefollowingconstantsare denedforuseintheanalysis: = T C and p = 3 4 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [( 2 +1 m =3 )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 q =3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 g = s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 4 2 : Also, s 3 = 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 2 2 and s 3 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 3 : 137

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Thenecessaryderivativesofthemacroscopicplasticdissipation, W + ,areobtainedusing thechainruleonthetwovariables D and B = D 33 suchthat 11 = @W + @D 11 = @W + @ D @ D @D 11 = 1 2 @W + @ D = 11 + 22 =2 11 = @W + @ D and 33 = @W + @D 33 = @W + @ D @ D @D 33 + @W + @B @B @D 33 = 1 2 @W + @ D + @W + @B = 11 + @W + @B e = j 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 j = @W + @B : Lastly,lettheparameter u bedenedsuchthat u = D j B j : {21 Thisratioandthepreviously-denedconstantswillbeusedinthesubsequentanalysis. 6.3.1 1 :thematrixyieldstrengthintensionisgreaterthanincompression Aderivationoftheparametricrepresentationsisgiveninthefollowingforamaterial where = T = C 1with m > 0and J 3 < 0.Thederivationsfortheothercasesand for 1aregiveninAppendixA.Theparametricrepresentationsarethenplottedfor thismaterial 1formultiplevaluesofvoidvolumefraction. 6.3.1.1 m > 0 and J 3 < 0 If m > 0and J 3 < 0then D> 0and B = D 33 < 0suchthat u = )]TJ/F15 11.9552 Tf 13.24 11.11 Td [( D B byEquation6{21.TherelevantbranchesoftheplasticmultiplierrateseeEquation 2{29mustnowbedetermined.Theexpressionfortheplasticmultiplierrate is 138

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rewrittenbelowforthereader'sconvenienceas = 8 > > > > < > > > > : r 2 3 s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d 2 1 + d 2 2 + d 2 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 if d 1 p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 r 2 3 s d 2 1 + d 2 2 + )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 d 2 3 = 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 if d 3 p d ij d ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 : {22 Let x denotethequantity x = Db 2 =r 2 suchthat d r = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( x + B 2 d = x )]TJ/F21 11.9552 Tf 13.15 8.088 Td [(B 2 d z = B: Evidently, d d r d z forthiscase D> 0and B = D 33 < 0.Theresultisthatthe relevantbranchfromEquation6{22eithercorrespondsto d = d 1 beingtheprincipal valuethatisaloneinsignwithnotation = or d z = d 3 beingtheprincipalvaluethat isaloneinsignwithnotation = z Let z betherelevantbranch.Therangewherethisbranchisvalidisdeterminedby solvingtherelevantinequalityfromEquation6{22foranyroots, x i .Thus,for d z p d ij d ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 ; thevalidrootnotingthat x 0, > 1, D> 0and D 33 < 0isobtainedas x 3 = s 3 B: Theplasticmultiplierrateforthisrangeis z = 2 p 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p x 2 + g 2 B 2 Similarly,for = ,theinequality d p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 139

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yieldsthefollowingvalidroot: x 3 = s 3 B: Theplasticmultiplierrateforthisbranchcanbewrittenas = s 2 3 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 p pB 2 + mBx + qx 2 : Insummary,itcanbeseenthatthevalidrangefor = z is0 u s 3 andthevalid rangefor = is s 3 u 1 Ingeneral,themacroscopicplasticdissipationcanbewrittenas W + D = 1 V Z V w d dV = T V Z V dV: Inordertodetermineanexpressionforthemacroscopicplasticdissipation, W + D ,there arethreerangesthatmustbelookedatseparatelythatcorrespondtotherelevantlimits ofintegration. Firstrange:0 u s 3 f Inthisrange,themacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z D=f D p x 2 + g 2 B 2 dx x 2 = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h h 4 x i D=f D where h 4 x = Z p x 2 + g 2 B 2 x 2 dx = )]TJ/F26 11.9552 Tf 10.494 18.53 Td [(p x 2 + g 2 B 2 x +ln x + p x 2 + g 2 B 2 : 140

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Thederivativescannowbewrittenas @W + @ D = W + D + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 @h 4 @x @x @ D D=f D @W + @B = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 @h 4 @x @x @B + @h 4 @B D=f D with @h 4 @x = p x 2 + g 2 B 2 x 2 and @h 4 @B = )]TJ/F21 11.9552 Tf 47.966 8.088 Td [(Bg 2 x 2 + x p x 2 + g 2 B 2 suchthat,atyielding, = @W + @ D = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln u + p u 2 + g 2 f 2 u + p u 2 + g 2 1 f and e = @W + @B = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 p u 2 + g 2 f 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p u 2 + g 2 : Secondrange: s 3 f u s 3 Inthisnextrange,theexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 qh 1 x + mB 2 h 2 x + h 3 x D=f x 3 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h h 4 x i x 3 D where x 3 = s 3 B .Inthepreviousequation,theintegralsolutionof Z p X x 2 dx = qh 1 x + mB 2 h 2 x + h 3 x 141

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hasbeenusedwith X = pB 2 + mBx + qx 2 and h 1 x = Z dx p X = 1 p q ln 2 p qX +2 qx + mB h 2 x = Z dx x p X = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 p pB 2 ln 2 p pB 2 X + mBx +2 pB 2 x h 3 x = )]TJ 9.299 10.122 Td [(p X x : Therelevantderivativescannowbewrittenas @W + @ D = W + D + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @ D D=f x 3 + mB 2 @h 2 @x @x @ D D=f x 3 + @h 3 @x @x @ D D=f x 3 + s 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 @h 4 @x @x @ D x 3 D 9 = ; @W + @B = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @B + @h 1 @B D=f x 3 + mB 2 @h 2 @x @x @B + @h 2 @B D=f x 3 + @h 3 @x @x @B + @h 3 @B D=f x 3 + s 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 @h 4 @x @x @B + @h 4 @B x 3 D with @h 1 @x = 1 p X @h 2 @x = 1 x p X @h 3 @x = )]TJ/F15 11.9552 Tf 11.291 0 Td [( mBx= 2+ qx 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(X x 2 p X and @h 1 @B = 1 p q p q mx +2 pB + m p X 2 p qX + qx + mB p X @ Bh 2 @B = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn B p p pB pB 2 +3 mBx +2 qx 2 + mx +4 pB p pB 2 p X 2 pB 2 X + mBx +2 pB 2 p X @h 3 @B = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(pB + mx= 2 x p X : 142

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Thisyieldsthefollowingstressinvariantsatyielding: = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln )]TJ/F21 11.9552 Tf 10.494 8.136 Td [(s 3 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p s 2 3 + g 2 u + p u 2 + g 2 + p q ln 2 p q p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(mf 2 p q p p + ms 3 + qs 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qs 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 pf 2 p p p p + ms 3 + qs 2 3 + ms 3 +2 p s 3 u !# and e = r 2 3 T p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p 2 u 2 + g 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : Thirdrange: s 3 u 1 Inthislastrange,themacroscopicplasticdissipationisasfollows: W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f D p qx 2 + mBx + pB 2 dx x 2 = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 qh 1 x + mB 2 h 2 x + h 3 x D=f D withderivatives @W + @ D = W + D + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @ D D=f D + mB 2 @h 2 @x @x @ D D=f D + @h 3 @x @x @ D D=f D @W + @B = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @B + @h 1 @B D=f D + mB 2 @h 2 @x @x @B + @h 2 @B D=f D + @h 3 @x @x @B + @h 3 @B D=f D suchthat,atyielding, = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(mf 2 p q p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 f + m 2 p p ln 2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(mu +2 pf 2 p p p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 p !# 143

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and e = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 : Figure6-3illustratesthethreebranchesderivedhereforthecaseof 1, J 3 < 0 and m > 0.Thehorizontalaxisinthegureisthein-planemeanstress0 : 5 = 0 : 5 11 + 22 normalizedbythetensileyieldstrength.Thecurvesshownarefora materialwithayield-strengthratioof =1 : 21.Curvescorrespondingtothreedierent voidvolumefractionsaregiveninFigure6-3andshowadecreasingyieldlocuswith increasingporosity.Thistrendistobeexpectedsinceavoid-matrixaggregatehasless load-bearingareaasthevoidvolumefractionincreases. 6.3.2Discussion Section6.3.1detailedthedevelopmentofparametricrepresentationsforamaterial whoseyieldstrengthintensionwasgreaterthantheyieldstrengthincompressionandfor theparticularcaseof J 3 < 0and m > 0.Forthistypeofmaterial 1thereare threeadditionalcasesthatneedtobeconsideredcorrespondingtopermutationsonthe signof J 3 and m .Thederivationofandexpressionsfortheseparametricrepresentations aregiveninAppendixA. Figure6-4plotsthecompleteparametricrepresentationoftheyieldcurveforthe particularstrengthratioof =1 : 21and f =0 : 01.Ascanbeseen,thecurvesare symmetricabouttheverticalaxisanddependonthesignofthethirdinvariantofthe macroscopicstressdeviator J 3 Parametricrepresentationswerealsoderivedforamaterialwhosetensileyield strengthislessthanthecompressiveyieldstrengthi.e., < 1suchthat k< 0.The relevantequationsarepresentedinAppendixA.Figure6-5plotsthecompleteparametric representationoftheyieldcurvecorrespondingtoaparticularstrengthratioof =0 : 82 andvoidvolumefraction f =0 : 01.Again,thecurvesaresymmetricaboutthevertical axisbutareippedwithrespecttoFigure6-4.Inotherwords,the J 3 > 0curveisthe 144

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onethatreachesthehighestpointontheverticalaxisinFigure6-4whereasthe J 3 < 0 curveistheonethatreachesthehighestpointontheverticalaxisinFigure6-5. Lastly,theCPB06isotropiccriterionreducestothevonMisesyieldcriterionwhenno tension-compressionasymmetryexistsinthematrixi.e.,for =1suchthat k =0.This resultsintheexactparametricrepresentationspresentedherereducingtothecylindrical criteriondevelopedbyGurson1977usingthesamevelocityeldgivenbyEquation 6{18. 6.4ProposedClosed-FormExpressionforaPlaneStrainYieldCriterion Intheprevioussection,aparametricrepresentationoftheyieldsurfacewasdeveloped.Itistobenotedthattheintegralrepresentingthemacroscopicplasticdissipation forthegivenvelocityeldwascalculatedexactly.Themaindrawbackwiththeparametricrepresentationsderivedintheprevioussectionisthatmultiplebranchesappearand u = D= j D 33 j cannotbeeasilyeliminatedfromanyofthem. Foreaseinapplicationsandniteelementimplementation,aclosedformofthe surfaceratherthanaparametricformismoredesirable.Withthegoalofderivingsucha closed-formexpression,certaintermswillbeneglectedintheexpressionofthelocalplastic dissipationinthissection. Thefollowingwillfocusonderivinganalyticalexpressionsfortheyieldlocuswhenthe void-matrixaggregatecontainscylindricalvoids,thematrixexhibitstension-compression asymmetryandtheloadingconditioncorrespondstoplanestraini.e., u = 1 usingthe terminologyoftheprevioussection.Itisdemonstratedthatthisapproximationresultsin amuchsimplerformfor e and .Thevalidityofthesimpliedanalyticalformofthe criterionforcylindricalvoidswillbefurtherassessedbycomparisonswithniteelement calculations. 6.4.1Calculationofthelocalplasticdissipation Equation2{27containsthesquaredvaluesoftherateofdeformationcomponents;therefore,theexpressionsforthesesquaredvaluesaredevelopedbelow,usingthe 145

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substitution = r 2 =b 2 andEquations6{19and6{20: ~ d 2 1 = D 2 + D 33 2 2 +2 D D 33 2 ~ d 2 2 = D 2 + D 33 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 D D 33 2 ~ d 2 3 = D 2 33 suchthatthesecondinvariantoftherateofdeformationtensorcanbeexpressedas d ij d ij =2 D 2 +6 D 33 2 2 : NowtheplasticmultiplierratefromEquation2{27canbewrittenas 2 = z 2 d ij d ij + z 1 + z 3 d 2 lone where z 1 { z 3 weredenedinTable2-2ofChapter2.Thepreviousexpressionfor requires thattheprincipalrateofdeformationcomponentcorrespondingtothelonepositiveor lonenegativeprincipalstressdeviatorbeidentiedi.e., d lone .NoticefromEquation 6{13thattheloneprincipalrateofdeformationcomponentsare d z and d for r>c and rc or r
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moreuncertain.Inordertosimplifythesubsequentcalculations,thecoupledtermwill beneglected;thevalidityofthisapproximationcanbeassessed,forexample,usingnite elementcalculations.Neglectingthelastterminthepreviousexpressionyields, s z 1 +2 z 2 + z 3 D 2 + z 1 +6 z 2 + z 3 D 33 2 2 = z 1 p x 2 + g {23 where z 1 := p z 1 +2 z 2 + z 3 g := z 1 +6 z 2 + z 3 z 1 +2 z 2 + z 3 D 33 2 2 = z 2 D 33 2 2 and x := D : Likewise, z 2 = z 2 d ij d ij + z 1 + z 3 d 2 z = z 2 2 D 2 +6 D 33 2 2 # + z 1 + z 3 D 2 33 : Therefore, z = s 2 z 2 D 2 + z 1 +6 z 2 +4 z 3 D 33 2 2 = z z 1 p x 2 + g z {24 147

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where z z 1 := p 2 z 2 g z := 4 z 1 +6 z 2 +4 z 3 2 z 2 D 33 2 2 = z z 2 D 33 2 2 and x = D asbefore.Notethatthesedenitionsof x and g ;z reducetotheonesusedinGurson 1977when k =0sincethen z 1 = z 3 =0and z 2 =2 = 3.Equations6{23and6{24 providetractableformsfortheplasticmultiplierrate, ,whichcanbeusedinderivingthe macroscopicplasticdissipationintheRVE. 6.4.2Developmentofthemacroscopicplasticdissipationexpressions Theupper-bound,macroscopicplasticdissipation, W + ,isdenedintermsofthe microscopicplasticdissipation, w ,as W + = 1 V Z V w d dV: Themicroscopicplasticdissipationisdeterminedasfollows: w = ij d ij = ij @f @ ij = T where T istheyieldstrengthinuniaxialtensionassumedconstantand isgivenby Equation2{27or,specicallyforthecasepresentedinthissubsection,byEquations 6{23and6{24. 148

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Therefore,theupperboundmacroscopicplasticdissipationbecomes W + = T V Z V dV = T b 2 L Z L 0 Z 2 0 Z b a r rdrddz = T Z ^ f f d + Z 1 ^ f z d # = T D Z D=f D= ^ f x dx x 2 + Z D= ^ f D z x dx x 2 # = T D z 1 Z D=f D= ^ f p x 2 + g x 2 dx + z z 1 Z D= ^ f D p x 2 + g z x 2 dx where f = a 2 =b 2 isthevoidvolumefractionintheRVEwith = r 2 =b 2 and x = D= being denedintheprevioussection.Theparameter ^ f isthelocationwherethethirdinvariant ofthestressdeviatoripssigninthehydrostaticanalysisofSection6.1.2andisdenedas ^ f := 8 > < > : c 2 b 2 if c
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Theequationabovewillnowbeusedinordertoderiveexpressionsforthemacroscopic eectivestress, e ,andthein-planemeanstress, = 2 TheCPB06eectivestressisgivenbyEquations2{12and2{13.FortheparticularproblemillustratedbyEquation6{2,theformoftheeectivestressdependson whether 11 or 33 isthelargerstresscomponent;alternatively,itcouldbestatedthatthe formoftheeectivestressdependsonthesignofthemacroscopicthirdinvariantofthe stressdeviator, J 3 .Thefollowingexpressiongivestheeectivestressdependingonthe signof J 3 : ~ e = 8 > > > > < > > > > : )]TJ/F15 11.9552 Tf 11.291 0 Td [( 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 T C if J 3 < 0 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 if J 3 > 0 0if J 3 =0 : Theseexpressionscanbecombinedintoasingleexpressionusingthesignfunctionas follows: ~ e = )]TJ/F15 11.9552 Tf 10.494 8.087 Td [( 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 2 2 T C )]TJ/F26 11.9552 Tf 11.956 9.683 Td [( sgn 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(J 3 +sgn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J 3 T C +1 = z c e 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 {26 where z c e = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( T C + 1 2 sgn 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J 3 +sgn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J 3 T C +1 : {27 Anexpressionforthein-planemeanstress, = 2 ,isnowneeded.Thisisdone simplybynotingthat 11 = 22 byEquation6{1;therefore, = 2 = 11 .Next, expressionsfor 11 and 33 intermsoftheupper-boundmacroscopicplasticdissipationof Equation6{25needtobedeveloped.Thisisdonebyusingthefollowingrelation: ij = @W @D ij @W + @D ij : {28 150

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UsingthechainruleinEquation6{28yieldsthefollowing: @W + @D 11 = @W + @ D @ D @D 11 + @W + @x @x @ D @ D @D 11 @W + @D 33 = @W + @ D @ D @D 33 + @W + @x @x @ D @ D @D 33 + @W + @g @g @D 33 + @W + @g z @g z @D 33 wheretheindividualderivativesare @ D @D 11 = @ D @D 33 = 1 2 @x @ D = 1 @g ;z @D 33 = z ;z 2 D 33 2 @W + @ D = W + D with @W + @x = T D 8 < : z 1 g x 2 p x 2 + g + p x 2 + g + x x p x 2 + g + x 2 + g # D=f D= ^ f + z z 1 g z x 2 p x 2 + g z + p x 2 + g z + x x p x 2 + g z + x 2 + g z # D= ^ f D 9 = ; = T D 8 < : z 1 p x 2 + g x 2 # D=f D= ^ f + z z 1 p x 2 + g z x 2 # D= ^ f D 9 = ; and @W + @g ;z = T 2 Dz ;z 1 )]TJ/F15 11.9552 Tf 38.231 8.087 Td [(1 x p x 2 + g ;z + 1 x p x 2 + g ;z + x 2 + g ;z # = T 2 Dz ;z 1 1 g ;z )]TJ/F26 11.9552 Tf 13.15 18.53 Td [(p x 2 + g ;z g ;z x # : 151

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Theexpressionsfor 11 and 33 cannowbewrittenasfollows: 11 @W + @D 11 = T 2 z 1 h ln x + p x 2 + g i D=f D= ^ f + z z 1 h ln x + p x 2 + g z i D= ^ f D 33 @W + @D 33 = @W + @D 11 + T 2 D 8 < : z 1 z 2 D 33 2 )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p x 2 + g g x # D=f D= ^ f + z z 1 z z 2 D 33 2 )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p x 2 + g z g z x # D= ^ f D 9 = ; : Thereforedroppingthe ,thein-planemeanstressandtheeectivestressaredeterminedasfollows: 2 = T 2 8 < : z 1 ln 2 4 0 @ D + p D 2 + g f 2 D + q D 2 + g ^ f 2 1 A ^ f f 3 5 + z z 1 ln 2 4 0 @ D + q D 2 + g z ^ f 2 D + p D 2 + g z 1 A 1 ^ f 3 5 9 = ; {29 and ~ e = z c e T 2 z 1 p z 2 p g p D 2 + g f 2 )]TJ/F26 11.9552 Tf 11.955 15.969 Td [(q D 2 + g ^ f 2 + z z 1 p z z 2 p g z q D 2 + g z ^ f 2 )]TJ/F26 11.9552 Tf 11.956 11.541 Td [(p D 2 + g z : {30 Themacroscopicyieldfunctiongivesarelationshipbetweenthemacroscopiceective stressfromCazacuetal.2006, ~ e ,andthein-planemeanstress, = 2.First,Equations 6{29and6{30canbewrittenasfollows: = + z ~ e = ~ e + ~ z e {31 wherethetermsdueto rc aredenedas ~ z e and z .BreakingupEquations6{29 and6{30inthiswayallowstheinvarianttermsineachoftheregions rc tobegroupedtogether.Forexample,considertheregionwhere r
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substitutions: Y 1 := q g ^ f 2 + D 2 + D Y 2 := p g f 2 + D 2 + D T II := ~ e z c e T 2 p g z 1 p z 2 T I := z 1 T =ln ^ fY 2 fY 1 suchthatT II = Y 1 )]TJ/F21 11.9552 Tf 13.151 8.088 Td [(f ^ f Y 1 exp )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(T I Y 1 = T II 1 )]TJ/F22 7.9701 Tf 13.151 5.256 Td [(f ^ f exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T I Y 2 = f ^ f T II exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T I 1 )]TJ/F22 7.9701 Tf 13.151 5.256 Td [(f ^ f exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T I : Theexpressionsfor Y 1 and Y 2 cannowbecombinedasfollows: )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(Y 1 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(g ^ f 2 =2 DY 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(Y 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(g f 2 =2 DY 2 whichyields )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(Y 1 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(g ^ f 2 Y 1 = )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(Y 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(g f 2 Y 2 : Thepreviousrelationscanbecombinedtoobtain )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T II 2 g = f 2 + ^ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f ^ f 1 2 exp )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T I +exp )]TJ/F19 11.9552 Tf 5.479 -9.684 Td [()]TJ/F21 11.9552 Tf 9.299 0 Td [(T I = f 2 + ^ f 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 f ^ f cosh )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(T I suchthat ~ e z c e T 2 2 z 1 p z 2 2 = f 2 + ^ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f ^ f cosh z 1 T {32 153

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and,usingasimilarprocedureintheregionwhere r>c ~ z e z c e T 2 2 z z 1 p z z 2 2 =1+ ^ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ^ f cosh z z z 1 T : {33 WhilethetworelationsinEquations6{32and6{33haveaverysimilarformto thecylindricalyieldcriterioninGurson1977,thecriterionmustbeexpressedinterms ofthemacroscopicinvariants ~ e and .Havingtheyieldcriterionbrokenupbyregion i.e., rc isnotpracticallyusefulsince,ingeneral,onlythemacroscopic quantitiesontheboundaryoftheRVEareknowne.g.,thevaluefor calculated ontheboundaryoftheRVEistypicallyknownratherthanthevaluesofeither or z .Therefore,Equations6{32and6{33willbetreatedasparticularlimitingcases andthesetwoexpressionswillbecombinedtohaveageneralcriterion.Thevalidityof thisgeneralcriterioncanthenbeassessedusingniteelementunitcellcalculations,for example. Therstcasetobeconsiderediswhen ^ f f ;inotherwords, c a .Basedonthe elasticanalysisofSection6.1.2,thisisonlyanapproximateconditionwhichshouldhold forverysmallvoidvolumefraction.Underthisscenario,themacroscopicyieldcriterionis givenbyEquation6{33andcanbewrittenasfollows: p 3 J 2 T 2 =1+ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f cosh T z z 1 for J 3 > 0 p 3 J 2 T 2 2 =1+ f 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f cosh T z z 1 for J 3 < 0 {34 where = T = C TherearetwoissueswithEquations6{34.First, J 3 isamicroscopicquantityand mustbeexpressedintermsofaveragequantitiestobeusefulinniteelementcalculations whereonlyaveragequantitiesontheboundaryandnotthemicroscopiceldsinthe RVEareknown.Secondly,neitherofEquations6{34containthesolutionfor J 3 =0. Thisisbecauseundertheassumptionofaxisymmetricloading, J 3 =0isequivalentto 154

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=0whichisthetrivialcaseofnoloading.Eventhoughthiscriterionisderivedunder axisymmetricloadingassumptions,theanalyticalformmakesitanattractivecriterion touseformoregeneralloadingcasessuchastheplanestrainnon-axisymmetricloading analyzedinChapter7.Theprevioustwoissueswillbeaddressedasfollows.First,note thatfor f =0, J 3 = J 3 ;thisensuresthatEquation6{34reducestotheCazacuetal. 2006criterionofEquation2{12for f =0.Secondly,for f 6 =0thetwoexpressions giveninEquations6{34willbecombinedinordertohaveasingleequationforgeneral usesincethesetwoequationsshouldagreewhen =0.Thecombinedformis obtainedbytakingtheaverageoftheconstantontheleft-hand-sideofeachexpressionin Equations6{34toyield z = p 3 J 2 T 2 c z )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 +2 f cosh T z z 1 =0{35 where c z = 8 > > > > > > > < > > > > > > > : 2 +1 1 2 for f 6 =0 sgn 2 J 3 +sgn J 3 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 1 2 + 2 for f =0 : {36 Thesecondcasetobeconsideredoccurswhen ^ f =1suchthat c = b .Thisscenariois governedbyEquation6{32andyields p 3 J 2 T 2 4 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 +1 =1+ f 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 f cosh T z 1 for J 3 > 0 p 3 J 2 T 2 4 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 4 2 +1 =1+ f 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 f cosh T z 1 for J 3 < 0 : {37 Followingasimilarrationaleasbeforeexceptthat f =0isnolongeraconcernsince ^ f =1,ageneralexpressionfortheyieldconditioncanbeobtainedas = p 3 J 2 T 2 c )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 +2 f cosh T z 1 =0{38 155

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where c = )]TJ/F21 11.9552 Tf 9.298 0 Td [( 4 +4 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 +1 : {39 Now,forthegeneralcasewhen a c b ,Equations6{35and6{38canbe combinedasfollows: = w + w z z {40 where w and w z areweightingconstantsgivenas w = ^ f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f w z = 1 )]TJ/F15 11.9552 Tf 14.503 3.155 Td [(^ f 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f {41 suchthat= if ^ f =1and z if ^ f f .Thegeneralyieldconditioncannowbe writteninamorecompleteexpressionas: = p 3 J 2 T 2 c w + c z w z )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ f 2 +2 f w cosh z 1 T + w z cosh z z 1 T =0 {42 where z z 1 and z 1 weredenedintheprevioussection, c z isdenedinEquation6{36, c is denedinEquation6{39while w z and w aregiveninEquation6{41. Notethatboth c and c z reducetounitywhen T = C suchthatEquation6{42 reducestothecylindricalyieldcriterionobtainedinGurson1977whenthereisno tension-compressionasymmetryinthematrix. ItshouldbenotedthattheexpressionobtainedinEquation6{42wasderivedunder theassumptionofaxisymmetricloading.Inthenextsectiontheanalyticalresultwillbe comparedtoniteelementcalculationsofaunitcellunderplanestrainloadingconditions. TheaimofthiscomparisonistoexplorethevalidityoftheproposedcriterionofEquation 6{42formoregeneralnon-axisymmetricloadingconditions. Figures6-6{6-8showacomparisonbetweentheyieldcurvesrepresentedbyEquation 6{42forthreedierentmaterialscorrespondingto k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098, k =0and k =0 : 3098 156

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usingtheplanestrainversionofthetransitionboundary c inSection6.1.2and =0 : 32. TheverticalaxisintheguresisthevonMiseseectivestress e = p 3 J 2 .The sametensileyieldstrengthwasusedforallmaterialsi.e.,onlythecompressiveyield strengthswerevaried.Asexpected,thecriterionpredictsthatincreasingporosityresults insofteningofthematerial.NoticethattheGurson1977yieldcurvescorresponding to k =0aresymmetricabouttheverticalaxiswhilethenewyieldcurvesarenot.This dierenceistobeexpectedsincetheCazacuetal.2006yieldcriterionaccountsfor asymmetricyieldstrengthswhilethevonMisesand,byextension,Gursonyieldcriterion doesnot.Also,notethattheyieldcurvesofFigure6-6andFigure6-8arequalitatively dissimilarfor k 6 =0.Thisisbecausefor =0 : 32,thetransitionboundary c equalsthe outerboundary b whentheporosityreachesalevelof f =0 : 12seeEquation6{15in Section6.1.2suchthattheyieldcriterioninFigure6-6hasnon-zeroweightsonboth and z since ab A B Figure6-1.Voidsinsideashearbandcoalescingintoacrackintitaniumduetotension. AVoids.BCrack.[ReprintedwithpermissionofJohnWiley&Sons,Inc. Meyers1994. Dynamicbehaviorofmaterials .Page457,Figure15.9.New York:JohnWiley&Sons,Inc.Copyright1994byJohnWiley&Sons,Inc.] 157

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Figure6-2.Axisymmetriccylindricalrepresentativevolumeelementcontaininga through-thicknesscylindricalvoid. 158

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Figure6-3.Parametricrepresentationofthevoid-matrixaggregatefor T = C =1 : 21 k =0 : 3098, J 3 < 0and m > 0.Curvespresentedareforvoidvolume fractionsof f =0 : 01, f =0 : 04and f =0 : 14. Figure6-4.Parametricrepresentationofthevoid-matrixaggregatefor T = C =1 : 21 k =0 : 3098and f =0 : 01. 159

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Figure6-5.Parametricrepresentationofthevoid-matrixaggregatefor T = C =0 : 82 k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098and f =0 : 01. Figure6-6.MacroscopicyieldsurfacesfromEquation6{42correspondingtoavoid volumefractionof f =0 : 01foracylindricalRVEcontainingalongcylindrical void:materialsshowncorrespondtoaratiobetweentheyieldstrengthin tensionandcompressionof T = C =0 : 82, T = C =1vonMisesand T = C =1 : 21i.e., k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098,0and0 : 3098,respectively. 160

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Figure6-7.MacroscopicyieldsurfacesfromEquation6{42correspondingtoavoid volumefractionof f =0 : 04foracylindricalRVEcontainingalongcylindrical void:materialsshowncorrespondtoaratiobetweentheyieldstrengthin tensionandcompressionof T = C =0 : 82, T = C =1vonMisesand T = C =1 : 21i.e., k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098,0and0 : 3098,respectively. Figure6-8.MacroscopicyieldsurfacesfromEquation6{42correspondingtoavoid volumefractionof f =0 : 14foracylindricalRVEcontainingalongcylindrical void:materialsshowncorrespondtoaratiobetweentheyieldstrengthin tensionandcompressionof T = C =0 : 82, T = C =1vonMisesand T = C =1 : 21i.e., k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098,0and0 : 3098,respectively. 161

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CHAPTER7 ASSESSMENTOFTHEPROPOSEDCYLINDRICALVOIDMODELBYFINITE ELEMENTCALCULATIONS Theanalyticalexpressionofthemacroscopicyieldfunctionforaporousaggregate withrandomly-oriented,cylindricalvoidsseeEquation6{42inChapter6wasobtained byassumingaspecicgeometryfortherepresentativevolumeelementRVE,specic loadingconditionsandanumberofapproximations.Theseassumptionswerenecessaryto obtainaclosed-formexpressionfortheyieldsurface.Onewaytovalidatethedeveloped criterionistocomparethetheoreticalyieldsurfacetoniteelementunitcellcalculations. Intheniteelementunitcellcalculations,thevoidboundaryisexplicitlymeshedand thematrixmaterialismodeledasanelastic-plasticmaterialwiththeplasticresponse governedbytheisotropicCPB06criterionofEquation2{8.Thissamematrixyieldcriterionwasusedinthehomogenizationproceduretoderivethemacroscopicyieldcriterion inChapter6.Thepurposeoftheunitcellcalculationsistoperformaminimizationof theplasticdissipationforalargesetofvelocityeldscompatiblewiththeuniformstrain rateboundaryconditionsandthustestwhetherthedevelopedyieldsurfaceadequately capturestheyieldbehaviorforvelocityeldsotherthantheoneassumedinSection6.2. 7.1ModelingProcedure Theunitcellcalculationsdetailedinthischapterareanalogoustothosedone byRichelsenandTvergaard1994foracylindricalvoidunderplanestrainloading conditionsandRistinmaa1997forasphericalvoidunderaxiallysymmetricloading conditions.Thissectiondetailsthecomputationaltestingprocedureusedinsettingup andrunningtheniteelementunitcellcalculationsdescribedintheintroductionof Chapter4.Theproceduresusedfortheplanestraincylindricalvoidcalculationsandthe axisymmetricsphericalvoidcalculationsareverysimilar.Thebasicideainbothcasesis tokeepthestraintriaxialityconstantthroughoutaparticularcalculationandtodenethe yieldpointforthatparticularcalculationastheincrementinwhichtheglobalmaximum occursontheeectivestress-straincurve.Therstissuethenisobviouslytheneedto 162

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ensurethatthestraintriaxialityisconstantintheelasticregime.Itwillbedescribedin thefollowinghowthisrequirementofaconstantstraintriaxialityisensuredforplane strainloading. ThegeometryoftheunitcellforthecylindricalvoidgeometryisshowninFigure7-1. Notethatforthisunitcell,theinitialporosity f 0 isdenedas f 0 = V void V total = a 2 0 4 A 0 B 0 Aconstantstresstriaxialityforthisproblemismaintainedinasimilarmannertothat proposedbyRistinmaa1997foraxisymmetricloadingconditionsandasphericalvoid.In theelasticregime,maintainingaconstantstresstriaxialityisequivalenttomaintaininga constantstraintriaxialityinthecalculationthiscanbeseenintheequationspresented below. First,letthemacroscopicstrainsbedenedasfollows: E 1 =ln A A 0 =ln A 0 + U 1 A 0 E 2 =ln B B 0 =ln B 0 + U 2 B 0 where A 0 and B 0 aretheinitialsidelengthsoftheunitcellasshowninFigure7-1while U 1 and U 2 aretheprescribeddisplacementstothosesides.Notethatforplanestrain loading, E 3 =0.Thestraintriaxiality, T E ,isdenedas T E = E kk 3 E e = E 1 + E 2 2 q E 1 + E 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 E 1 E 2 where E kk isthetraceofthemacroscopicstraintensorand E e isthemacroscopiceective straindenedas E e = q = 3 E 0 ij E 0 ij with E 0 ij beingthemacroscopicdeviatoricstrain tensor.Theequationaboveshowsafteralittlealgebrathatthestrains E 1 and E 2 must belinearlyrelatedinorderforthestraintriaxialitytobeconstant.Thisrelationisstated 163

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asfollows: E 1 = c E E 2 with c E = 2 T 2 E +1 2 p 3 T E p 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 2 E 4 T 2 E )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Notethattheaboveequationimpliestheconstraint T E 1for c E tobeareal.The givetwovaluesfor c E whicharetheinverseofeachother;thus,thetwo c E valuesgive thesameresultsinceplanestraincalculationsaresymmetricwithrespecttothe x 1 and x 2 directions.Analternativetothepreviousexpressionwith T E nowintermsof c E isas follows: T E = 1+ c E 2 p 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(c E + c 2 E sgn E 2 : Theprescribedboundarydisplacements, U 1 and U 2 cannowberelatedusingthe aboveequations.Doingsoyields U 1 = A 0 B 0 + U 2 B 0 c E )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : Intheelasticregime,themacroscopicstrainsarerelatedtothemacroscopicstressesvia Hooke'slawsuchthat EE 1 = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 1 )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( 2 + 2 EE 2 = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 2 )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( 2 + 1 where E isYoung'smodulusand isPoisson'sratio.Theaboveexpressioncanbesolved for 1 and 2 usingthefactthat E 1 = c E E 2 andthusobtain 1 = c 2 with c = c E )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 + 2 + c E 2 + +1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 : 164

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Themacroscopicstresstriaxialityisdenedanalogoustothemacroscopicstraintriaxiality as T = kk 3 e = + c + g 1 k;c ; sgn 2 = g 2 k;c ; where g i k;c ; aresimplyconstantfunctionsof k c and andare,therefore,constant intheelasticregionofagivencalculationifthestraintriaxialityand,thus, c E and c areconstant.Thestresstriaxialitywilltypicallynotbeconstantoncemacroscopicyielding occurs;hence,thedeviationfromaconstanttriaxialityintheniteelementunitcell calculationscorrespondtotheonsetofyielding. ArelationshipwhichisusefulforobtainingadesiredspacingoftheFEdatapointsin the0 : 5 e planeistheslope,for k =0,ofagivencalculationintheelasticregionin therespectiveplane: slope = p 6+ c 2 + 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2+ c 2 + c sgn 2 : {1 Equation7{1canbeconvertedtoanangle =atan slope ,andisplottedversusthe strainratio E 1 =E 2 inFigure7-2.Figure7-2illustratesthattheseplanestraincalculations areconstrainedregardingwhatnumericalpointscanbeobtainedi.e.,theslopescannot gobelowtheminimumobtainedwhen c E =1.Notethatwhiletheaboveequationwas derivedforavonMisesmaterial k =0,thecalculationswhere k =0willbeusedas baselinecalculationssuchthatthespacingofthedatapointsissetforthebaselinecase andallthe c E valuesusedinthecomputationaltestmatrixforthisbaselinecasewillthen beusedfortheothervaluesof k .Also,thepreviousequationdoesprovideaquickand easyestimatefortheslopewhentherearenon-zerovaluesof k 165

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Lastly,themacroscopicstressesobtainedfromtheniteelementcalculationsare denedasfollows: 1 = F 1 B 2 = F 1 A 3 = avg 3 where F 1 and F 2 arethetotalforcesonthecellboundariesand 3 aretheindividual out-of-planestressesintheniteelements.Allofthesequantitiesarereadilyavailable fromanystandardniteelementcode. 7.2FiniteElementResults Finiteelementresultsforthreedierentmaterialsunderplanestrainloadingwillbe presentedinthissection.ForthispurposeausermaterialroutineUMATwasdeveloped seealsoChapter4toimplementtheCazacuetal.2006yieldcriterion.Allcalculations wereperformedinABAQUSAbaqus,2008usingafullyimplicitreturnmappingscheme. Ameshrenementstudywasperformedtoensuremeshconvergence.Figures7-3to 7-5showtheniteelementmeshesusedwithinitialvoidvolumefractionsof f =0 : 01, f =0 : 04and f =0 : 14,respectively. Allthreematerialsconsideredhavethesameyieldstrengthintension;however, onematerialhasanequalyieldstrengthincompression k =0,onehasayieldin compressionthatislessthantheyieldintensioncorrespondingtoarandomlyoriented BCCpolycrystal k =0 : 3098andonehasayieldincompressionthatisgreaterthanthe yieldintensioncorrespondingtoarandomlyorientedFCCpolycrystal k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098. Thesespecicvaluesfor k werereportedinCazacuetal.2006.APoisson'sratioof =0 : 32wasusedforeachmaterialalongwithatensileyieldstrengthtoYoung'smodulus ratioof T =E =0 : 00124i.e.,theelasticpropertieswerearbitrarilychosentocorrespond toasteel.Tables7-1to7-3tabulatethestraintriaxialitiesandresultingdisplacements prescribedintheunitcellcalculationsfor k =0, k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098and k =0 : 3098,respectively. 166

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Figures7-7to7-9showacomparisonbetweentheniteelementdataandtheanalyticalyieldsurfacesobtainedusingEquation6{42forthreedierentinitialporosities f =0 : 14, f =0 : 04and f =0 : 01andthethreedierentmaterials k =0, k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098 and k =0 : 3098.AscanbeseeninFigures7-7to7-9,theagreementbetweenthenite elementdataandEquation6{42islessthanideal.Inordertoprovidebetteragreement, theproposedanalyticalyieldcriterionofEquation6{42canbemodiedinthefollowing way: = C eqv p 3 J 2 T 2 c w + c z w z )]TJ/F26 11.9552 Tf 11.955 9.683 Td [()]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1+ q 2 f 2 +2 qf w cosh z 1 T + w z cosh z z 1 T =0 {2 with w = q f ^ f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f w z = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(q f ^ f 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f : {3 and ^ f = 3 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 f {4 forplanestrainloadingseeEquation6{15inSection6.1.2. ThreemodicationswereintroducedinEquation7{2.First,theterm C eqv is includedasintroducedbyGurson1977forcylindricalvoids.Theformof C eqv beingused inthissectionis C eqv = )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+3 f )]TJ/F15 11.9552 Tf 11.955 0 Td [(81 f 3 +24 f 6 2 {5 whichdiersfromtheplanestraintermintroducedinGurson1977bytheadditionofa f 3 term.Thisadditionaltermwasintroducedbasedonthe k =0i.e.,vonMisesnite elementcalculationstoobtainbetteragreementbetweentheniteelementresultsand analyticalyieldsurfacesforincreasingporosity.Secondly,theparameter q wasintroduced byTvergaard1981andisalsousedinEquation7{2with q =4 =e thisisthesame 167

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valueasthatusedinChapter5for q 1 .Lastly,thettingparameter q f =0 : 55was introducedasamultiplieron ^ f .Therationalefortheinclusionof q f isthat ^ f wasderived foracylindricalRVEratherthanthecubicRVEbeingemployedhere.Theparameter q f allowsforbetteragreementastheporosity, f ,increases. Figures7-10to7-12nowshowacomparisonbetweentheniteelementdataandthe analyticalyieldcurvesobtainedusingEquation7{2forthreedierentinitialporosities f =0 : 14, f =0 : 04and f =0 : 01andthreedierentmaterials k =0, k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098 and k =0 : 3098.Ascanbeseen,theagreementwiththeniteelementdatausingthe modiedcriterionofEquation7{2isfairlygood. 7.3ConcludingRemarks Ananalyticalyieldcriterionwhichisanupper-boundestimatehasbeendeveloped byextendingGurson's1977analysisofthehollowcylindertothecasewhenthematrix plasticbehaviorisdescribedbyCazacuetal.'s2006isotropicyieldcriterion.Due tothetension-compressionasymmetryofthematrixresponse,freshdicultieswere encounteredwhenestimatingthelocalplasticdissipation, w d .Thisisbecausethe plasticmultiplierrateassociatedtotheCazacuetal.2006yieldcriterionhasmultiple branchesseeEquation2{29anddependsoneachoftheprincipalvaluesofthelocal rateofdeformationtensor, d .Ifthereisnodierenceinresponsebetweentheyieldin tensionandcompressiontheproposedcriterioninEquation6{42reducestotheclassical analyticalcylindricalcriterionproposedinGurson1977sincetheisotropiccriterionof Cazacuetal.2006reducestovonMisesfor k =0.Intheabsenceofvoids,theproposed criterionreducestoCazacuetal.'s2006yieldcriterion.Theaccuracyoftheanalytical criterionwasassessedthroughcomparisonwithnite-elementcellcalculations.Toimprove theagreement,theproposedanalyticalyieldcriterionofEquation6{42wasmodied toincludeadditionalparameters, q q f and C eqv seeEquation7{2basedpartlyon suggestionsmadeinGurson1977;RichelsenandTvergaard1994;Tvergaard1981. 168

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TheagreementbetweenthetheoreticalpredictionsusingthecriteriongiveninEquation 7{2andresultsofniteelementcellcalculationsisfairlygood. Figure7-1.Planestraingeometryusedinniteelementcalculations. 169

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Figure7-2.Validtriaxialityanglesavailablefor k =0usingtheprocedureoutlinedin Section7.1seeEquation7{1;anglesarewithrespecttothe0 : 5 )]TJ/F15 11.9552 Tf 11.955 0 Td [( e planeandmeasuredfromthe0 : 5 axiswith+ referringtoslopesinthe positive domainand )]TJ/F21 11.9552 Tf 9.299 0 Td [( referringtoslopesinthenegative domain. Figure7-3. f 0 =0 : 01planestrainniteelementmeshfortheunitcell. 170

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Figure7-4. f 0 =0 : 04planestrainniteelementmeshfortheunitcell. Figure7-5. f 0 =0 : 14planestrainniteelementmeshfortheunitcell. 171

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Table7-1. k =0planestraincylindricalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] C1-1.0000-0.0135-0.0135-19.7989 C2-0.7100-0.0192-0.0052-26.8869 C3-0.5300-0.0198-0.0008-34.1861 C4-0.3200-0.01800.0047-48.3665 C5-0.1900-0.01780.0089-62.1759 C6-0.0900-0.01500.0109-75.9638 C70.0900-0.01090.015075.9638 C80.1900-0.00750.015062.1759 C90.3200-0.00390.015048.3665 C100.53000.00060.015034.1861 C110.71000.00410.015026.8869 C121.00000.01350.013519.7989 Table7-2. k = )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 3098planestraincylindricalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] C13-1.0000-0.0135-0.0135-19.7989 C14-0.5900-0.0192-0.0023-26.8805 C15-0.4400-0.01800.0015-34.1198 C16-0.2800-0.01770.0058-47.4640 C17-0.1700-0.01640.0089-61.5341 C18-0.0800-0.01440.0109-76.0444 C190.0800-0.01130.015076.6962 C200.1900-0.00750.015061.3407 C210.3200-0.00390.015048.0132 C220.53000.00060.015034.1758 C230.70000.00390.015026.7508 C240.90000.00850.015019.7240 172

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Table7-3. k =0 : 3098planestraincylindricalvoidcomputationaltestmatrix. Test# T E U 1 [in] U 3 [in]Angle[deg] C25-1.0000-0.0135-0.0135-19.7989 C26-0.8100-0.0175-0.0072-26.9583 C27-0.6400-0.0189-0.0034-33.9966 C28-0.3900-0.01810.0028-48.0278 C29-0.2200-0.01820.0080-62.5359 C30-0.1000-0.01550.0109-76.3462 C310.1000-0.01050.015075.5189 C320.2000-0.00720.015062.0157 C330.3300-0.00370.015047.9415 C340.53000.00060.015034.2013 C350.67000.00330.015028.7276 C361.00000.01350.013523.5401 Figure7-6.Planestraineectivestresscriterionusedtopicktheyieldpointdatashown fromTest#C1. 173

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Figure7-7.Yieldsurfaceofthevoid-matrixaggregatefor T = C =1vonMiseswith f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation6{42. 174

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Figure7-8.Yieldsurfaceofthevoid-matrixaggregatefor T = C =0 : 82 k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098 with f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation6{42. 175

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Figure7-9.Yieldsurfaceofthevoid-matrixaggregatefor T = C =1 : 21 k =0 : 3098with f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation6{42. 176

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Figure7-10.Yieldsurfaceofthevoid-matrixaggregatefor T = C =1vonMiseswith f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation7{2. 177

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Figure7-11.Yieldsurfaceofthevoid-matrixaggregatefor T = C =0 : 82 k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3098 with f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation7{2. 178

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Figure7-12.Yieldsurfaceofthevoid-matrixaggregatefor T = C =1 : 21 k =0 : 3098 with f =0 : 01, f =0 : 04and f =0 : 14.Comparisonbetweentheniteelement resultsandtheproposedcriterionofEquation7{2. 179

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CHAPTER8 ANISOTROPICPLASTICPOTENTIALFORHCPMETALSCONTAINING SPHERICALVOIDS Thischapterfocusesonextendingthemacroscopicyieldcriteriondevelopedin Chapter3foravoid-matrixaggregatewherethevoidshavesphericalgeometryandthe metalmatrixisisotropicwithtension-compressionasymmetrytothecasewherethemetal matrixstillexhibitstension-compressionasymmetrybutisnowanisotropic.Webegin withabriefreviewofthemaincontributionsinmodelingyieldingofporousaggregates containingananisotropicmatrixmaterial. Aspreviouslymentioned,Gurson1977developedwidelyusedmacroscopicyield criteriaforvoid-matrixaggregatescontainingeithersphericalorcylindricalvoidsand withthematrixobeyingavonMisesisotropicyieldcondition.Liaoetal.1997extended Gurson's1977cylindricalcriteriontoaccountfortransverseisotropyusingHill's1948 yieldcriterionseeHill,1948,1950forthematrixmaterial;theproposedcriterion is2Dplanestressconditionsandwasappliedtomodelthebehaviorofthinmetal sheetssteelsandaluminum.BenzergaandBesson2001extendedGurson's1977 sphericalcriterionfororthotropicmetalsbyassumingamatrixmaterialthatcouldbe characterizedusingHill's1948criteria.UsingaHill-Mandelhomogenizationprocedureon anaxisymmetric,ellipsoidalRVEcontainingaconfocal,prolateellipsoidalvoid,Gologanu etal.1993arrivedatananalyticexpressionforprolate,ellipsoidalvoids.Benzergaetal. 2004busedayieldcriterionwhichcombinedpropertiesfrombothGologanuetal.'s 1993andBenzergaandBesson's2001criteriatoaccountforbothvoidshapeand orthotropy.ThiscriterioncompareswellwiththeexperimentalresultsgiveninBenzerga etal.2004a. Allthepreviousworkscitedinvolvecubicmetals.Modelingtheyieldingbehaviorof hexagonalclosepackedHCPmetalsposestremendouschallengesduetotheirunusual deformationcharacteristics|namely,strongasymmetrybetweentensileandcompressive behaviorandhighlypronouncedanisotropy.Rigorousmathematicalmethodstointroduce 180

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anisotropyinanygivenisotropicyieldcriterionexistintheliteraturee.g.,thelinear transformationapproachandthegeneralizedinvariantsapproach.Amajordiculty encounteredinformulatinganalyticalexpressionsfortheyieldbehaviorofHCPmetals isrelatedtothedescriptionofthetensionversuscompressionasymmetrywhichisdue tothecombinedeectsofcrystalstructureandtwinning.Inthefollowingsections,the isotropicyieldcriteriondevelopedinChapter3whichalreadyaccountsforthepresenceof voidsandtension-compressionasymmetryinthematrixwillbeextendedtoaccountfor anisotropyinthematrix. Thestructureofthischapterisasfollows.InSection8.1,thehomogenization approachduetoHillandMandelHill,1967;Mandel,1972thatisusedindevelopingthe macroscopicplasticpotentialforthevoid-matrixaggregateisbrieyrecalled.Next,the anisotropicversionoftheCazacuetal.2006yieldcriterionthatdescribesthematrix materialbehaviorinthevoid-matrixaggregateispresentedinSection8.2.Thirdly,the choiceofvelocityeldusedtocalculatethelocalplasticdissipationpotentialisintroduced inSection8.3.ThecorrespondinglocalplasticdissipationisthenderivedinSection8.4 andtheexpressionsforthemacroscopicplasticdissipationofthevoid-matrixaggregate aredevelopedinSection8.5.Section8.6showscomparisonsbetweenthemacroscopic yieldcriteriondevelopedinthischapterforavoid-matrixaggregatewithaxisymmetric transverselyisotropicniteelementunitcellcalculations. 8.1KinematicHomogenizationApproachofHillandMandel Considerarepresentativevolumeelement V ,composedofahomogeneousrigidplasticmatrixandatraction-freevoid.Thematrixmaterialisdescribedbyaconvexyield function inthestressspaceandanassociatedowrule d = @' @ ; {1 where istheCauchystresstensor, d = = 2 r v + r v T denotestherateofdeformation tensorwith v beingthevelocityeld,and 0standsfortheplasticmultiplierrate.The 181

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yieldsurfaceisdenedas =0.Let C denotetheconvexdomaindelimitedbythe yieldsurfacesuchthat C = f j 0 g : {2 Theplasticdissipationpotentialofthematrixisdenedas w d =sup 2 C : d {3 where:"denotesthetensordoublecontraction.Uniformrateofdeformationboundary conditionsareassumedontheboundaryoftheRVE, @V ,suchthat v = Dx forany x 2 @V {4 with D ,themacroscopicrateofdeformationtensor,beingconstant.Fortheboundary conditionsofEquation2{4,theHill-MandelHill,1967;Mandel,1972lemmaapplies; hence, h : d i V = : D ; {5 where hi denotestheaveragevalueovertherepresentativevolume V ,and = h i V Furthermore,thereexistsamacroscopicplasticdissipationpotential W D suchthat = @W D @ D {6 with W D =inf d 2 K D h w d i V ; {7 where K D isthesetofincompressiblevelocityeldssatisfyingEquation8{4formore detailsseeGologanuetal.,1997;Leblond,2003.Thematrixmaterialbeingconsidered obeystheanisotropicversionofthepressure-insensitiveyieldcriterionthatcaptures strengthdierentialeectsofCazacuetal.2006. 182

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8.2YieldCriterionfortheMatrixMaterial TheCazacuetal.2006criterionusing a =2canbewrittenas F = j ^ I j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k ^ I 2 + j ^ II j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k ^ II 2 + j ^ III j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k ^ III 2 {8 where^ I ,^ II and^ III aretheprincipalcomponentsof^ .Theanisotropicversionof theCazacuetal.2006yieldcriterioninvolvesalineartransformation, L ,ofthestress deviator, 0 .Inthisworkanadditionalconstraintthatthetransformedstress ^ = L 0 {9 bedeviatoricwillbeimposed.AppendixBshowsthattheCPB06criterioncanreduceto theHill48criterioniftheconstraintthatthetransformedstressbedeviatoricisimposed andif k =0suchthatthereisnotension-compressionasymmetrysince,inthiscase, bothcriterionhavethesamenumbersofdegreesoffreedom.Notethatthisconditionof adeviatorictransformedstressismorerestrictivethantheoriginalanisotropiccriterion presentedinCazacuetal.2006.Withrespecttothecoordinatesystemassociatedwith orthotropy,thelineartransformation, L ,canbewritteninVoightnotationasfollows: L = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 L 11 L 12 L 13 000 L 12 L 22 L 23 000 L 13 L 23 L 33 000 000 L 44 00 0000 L 55 0 00000 L 66 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : 183

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Notethat,inorderfor^ tobetraceless,thefollowingrestrictionsonthecomponentsof L musthold: L 11 + L 12 + L 13 =1 L 12 + L 22 + L 23 =1 L 13 + L 23 + L 33 =1 : Theconstantontherighthandsideofthepreviousequationshasbeenchosentobeunity suchthatthelineartransformationreducestotheidentitytensorforthecaseofisotropy. TheCPB06anisotropicyieldcriterioncanbewrittenas = e )]TJ/F21 11.9552 Tf 11.955 0 Td [( T 1 =0{10 where e =^ m p F =^ m v u u t 3 X i =1 j ^ i j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k ^ i 2 {11 and T 1 istheuniaxialyieldstrengthalonganaxisoforthotropyofthematerial,saythe rollingdirection,whichisdenotedthe1-direction.Theconstant,^ m ,istheanisotropic versionoftheeectivestressconstantandisgivenby ^ m = s 1 j 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k 1 2 + j 2 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k 2 2 + j 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k 3 2 {12 where 1 = 2 3 L 11 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 L 12 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 13 2 = 2 3 L 12 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 L 22 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 23 3 = 2 3 L 13 )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 L 23 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 33 : {13 RecallfromSection8.1thatdeterminingthemacroscopicplasticdissipation W D requiresthedeterminationofthelocalplasticdissipationexpression, w d .Sincethe 184

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anisotropiccriterionisahomogeneousfunctionofrstorderinstresses,thelocalplastic dissipationreducesto w = T 1 where istheanisotropicplasticmultiplierrateorthe dualoftheCazacuetal.,2006,anisotropicyieldcriterionand T 1 isthetensileyield strengthintherollingdirection.ThedualoftheCazacuetal.2006anisotropicyield criterioncanbedeterminedasfollows.Notethat ^ mn = L mnkl 0 kl = L mnkl J klij ij {14 where,inVoightnotation, J = 1 3 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 2 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1000 )]TJ/F15 11.9552 Tf 9.299 0 Td [(12 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1000 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 )]TJ/F15 11.9552 Tf 9.299 0 Td [(12000 000300 000030 000003 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Theplasticrateofdeformationtensor d P cannowbewrittenasfollows: d P ij = @ @ ij = @ @ ^ mn @ ^ mn @ ij = @ @ ^ mn L mnrs J rsij = @ @ ^ mn L mnij {15 where L J = L if L isadeviatorictensor.Let b denotethefollowingrstordertransformedrateofdeformationtensor: b rs = d P ij L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ijrs : {16 Therefore, b rs = @ @ ^ rs : {17 185

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Thus,theexpressionfor intheanisotropiccasecannowbefoundinthesamemanner asintheisotropiccaseseeSection2.3suchthat d intheisotropicexpressionfor see Equation2{29isreplacedwith b toobtaintheanisotropicexpressiongivenas 2 = b ij b ij + b 2 lone {18 where b lone istheprincipalcomponentof b associatedwiththelonepositiveornegative principaltransformedstressdeviatorsi.e.,^ I ,^ II or^ III .Theparameters and are functionsofthetension-compressionasymmetryparameter k andcanbeexpressedas follows: = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : 1 ^ m )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 if J ^ 3 0 b III p b 2 I + b 2 II + b 2 III )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 +2 k +3 p 6 k 2 +3 k 2 +1 1 ^ m + k 2 if J ^ 3 0 b I p b 2 I + b 2 II + b 2 III 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 p 6 k 2 +3 k 2 +1 {19 and = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : )]TJ/F15 11.9552 Tf 9.299 0 Td [(12 k 3 k 2 +2 k +3 if J ^ 3 0 b III p b 2 I + b 2 II + b 2 III )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 +2 k +3 p 6 k 2 +3 k 2 +1 12 k 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 if J ^ 3 0 b I p b 2 I + b 2 II + b 2 III 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 p 6 k 2 +3 k 2 +1 {20 where b I b II and b III aretheorderedprincipalcomponentsof b ij andassociatedwiththe orderedtransformedstressdeviators,^ I ^ II ^ III 186

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Alternatively, canbeexpressedinthesamemanneraswasdoneinSection2.3: = 8 > > > > < > > > > : 1 ^ m + k s 3 k 2 +10 k +3 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 b 2 I + b 2 II + b 2 III if b I p b ij b ij 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 p 6 k 2 +3 k 2 +1 1 ^ m )]TJ/F21 11.9552 Tf 11.955 0 Td [(k s b 2 I + b 2 II + 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(10 k +3 3 k 2 +2 k +3 b 2 III if b III p b ij b ij )]TJ/F15 11.9552 Tf 11.291 0 Td [( k 2 +2 k +3 p 6 k 2 +3 k 2 +1 {21 where b I b II and b III aretheorderedprincipalcomponentsof b ij 8.3ChoiceofTrialVelocityField Thevelocityeld, ~v ,intheRVEistakentobethesameincompressiblevelocity eldproposedbyGurson1977i.e.,thesamevelocityeldassumedinChapter3.The velocityeldiswrittenasfollows: ~v = ~v V + ~v S where ~v V = D m b 3 r 2 ~e r and ~v S = D 0 ~x {22 suchthat d rr = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 D m b r 3 + D 0 rr d = D m b r 3 + D 0 d = D m b r 3 + D 0 d r = D 0 r {23 withprincipalvalues ~ d 2 ; 3 = )]TJ/F21 11.9552 Tf 10.494 8.087 Td [(D m 2 b r 3 )]TJ/F21 11.9552 Tf 13.151 9.321 Td [(D 0 2 3 2 s D 2 m b r 6 )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(2 3 D m b r 3 D 0 rr )]TJ/F21 11.9552 Tf 11.955 0 Td [(D 0 + )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(D 0 2 ~ d 1 = D m b r 3 + D 0 : {24 187

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8.4CalculationoftheLocalPlasticDissipation Thecalculationofthelocalplasticdissipation w = T 1 evenintheisotropiccaseis non-trivialseeSection3.3duetothemultiplebranchesintheexpressionfortheplastic multiplierrate, seeEquation2{29or8{21.RecallfromSection3.3thataformfor anapproximatemacroscopicplasticdissipation W ++ D whichcapturedthehydrostatic anddeviatoricsolutionswasobtainedandthatitcouldbewrittenas W ++ D = T p 4 Z 1 f h h 2 i S r i 1 = 2 d = + 2 3 T p 4 Z 1 f h d ij d ij i S r 1 = 2 d {25 where h x i S r = 1 4 Z 2 0 Z 0 x sin dd: {26 Thisexpressionfor W ++ D resultsfromthefactthat h h 2 i S r i 1 = 2 = + 2 3 h d ij d ij i S r 1 = 2 sincethesurfaceintegralof D 0 rr iszeronotethattheparameter z 6 denedinSection3.3 isrelatedtotheparameters and as z 6 =6 +4 .Inordertoobtainananisotropic extensionoftheyieldcriteriondevelopedinChapter3,theresultfromSection8.2that theanisotropicplasticmultiplierrate isthesameformastheisotropicversionbut withtherateofdeformationtensor d replacedwiththetransformedrateofdeformation tensor b = L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 d willbeused.Similarly,inthissection, d willbereplacedwith b inthe expressionforthesurfaceintegraloftheplasticmultiplierratetoobtain h 2 i S r = + 2 3 h b ij b ij i S r : {27 Thisexpressionwillbeusedinthenextsectiontodetermineanapproximatemacroscopic plasticdissipation W ++ D thatwillincorporatetheeectsoftension-compression asymmetrythrough k andanisotropythrough L whilereducingtotheisotropic expressionobtainedinChapter3for L = I 188

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8.5DevelopmentoftheMacroscopicPlasticDissipationExpression Thissectionfocusesondevelopinganexpressionforthemacroscopicplasticdissipationfortheanisotropiccase.Separatingtherateofdeformationtensorintoavolumetric partandadeviatoricpart,thevolumetricportioncanbewrittenasfollowsinthesphericalreferenceframeseeEquation8{23: d V = D m u 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : d V rr d V d V 0 0 0 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; = D m u 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 1 1 0 0 0 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; where u = b 3 =r 3 with r beingtheradialcoordinateand b beingtheouterradiusofthe sphericalRVE.Now,notethatthescalar b ij b ij canberewrittenasfollows: b ij b ij = d P kl L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 klij d P pq L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 pqij = d P kl ^ L klpq d P pq {28 where ^ L klpq isadiagonalmatrixwhenwritteninVoightnotationsuchthat ^ L = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 ^ l 1 00000 0 ^ l 2 0000 00 ^ l 3 000 000 ^ l 4 00 0000 ^ l 5 0 00000 ^ l 6 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : {29 189

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Thus,themacroscopicplasticdissipationexpressioncanbewrittenas W ++ = T 1 V Z b a 4 r 2 h h 2 i S r i 1 = 2 dr = T 1 V Z b a 4 r 2 + 2 3 h b : b i S r 1 = 2 dr = T 1 V Z b a 4 r 2 + 2 3 h d V : ^ L : d V + d S : ^ L : d S +2 d V : ^ L : d S i S r 1 = 2 dr: {30 Notingthat h d V : ^ L : d S i S r = h d V i S r : ^ L : d S andthat h d V i S r = h [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 ~e r ~e r + ~e ~e + ~e ~e ] i S r =0 thefollowingexpressionisobtainedfortheestimatedmacroscopicplasticdissipation: W ++ = T 1 4 3 b 3 Z b a 4 r 2 + 2 3 h d V : ^ L : d V + d S : ^ L : d S i S r 1 = 2 dr = s + 2 3 3 T 1 b 3 Z b a r 2 s D 2 m b 6 r 6 h ^ l i ;; i S r + D 2 e + 2 3 dr = T 1 D m h Z 1 =f 1 p u 2 + A 2 du u 2 {31 where u = b 3 =r 3 D 2 e = + 2 3 h d S : ^ L : d S i S r ; {32 A = D e D m 1 h {33 and h = s + 2 3 h ^ l i ;; i S r : {34 Thefunction isdependenton and alongwiththeanisotropicparameters, ^ l i ,and isdened,fortheparticularvelocityeldbeingused,as =[ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 ~e r ~e r + ~e ~e + ~e ~e ]: ^ L :[ )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 ~e r ~e r + ~e ~e + ~e ~e ] : {35 190

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Theanisotropyfactor, h ,whichisappliedtothehydrostaticpartofthemacroscopicyield criterioncannowbedeterminedinasimilarmannerasinBenzergaandBesson2001. Thesecond-ordertensor ^ L mustbetransformedfromthematerialaxeswhereitisa diagonalmatrixinVoightnotationtothesphericalcoordinatesystemwhereitisafull matrix.Thenewnotationwillbe ^ L MN M;N = I;:::;VI = ^ L MN ^ l i ;; .Now, h canbe determinedasfollows: h 2 = + 2 3 h i S r = + 2 3 h ^ L I;I )]TJ/F15 11.9552 Tf 7.545 -6.529 Td [( d V rr 2 + ^ L II;II )]TJ/F15 11.9552 Tf 7.546 -6.529 Td [( d V 2 + ^ L III;III )]TJ/F15 11.9552 Tf 7.546 -6.529 Td [( d V 2 +2 ^ L I;II d V rr d V +2 ^ L I;III d V rr d V +2 ^ L II;III d V d V i S r = + 2 3 h 4 ^ L I;I + ^ L II;II + ^ L III;III )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 ^ L I;II )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 ^ L I;III +2 ^ L II;III i S r {36 suchthat h 2 = + 2 3 4 5 ^ l 1 + ^ l 2 + ^ l 3 + 6 5 ^ l 4 + ^ l 5 + ^ l 6 {37 whereMatlabwasusedtoperformthecoordinatetransformationof ^ L neededtoobtain Equation8{37. When k =0, =2 = 3with =0and ^ L = ^ H suchthatEquation8{37reducesto h 2 = 8 15 ^ l 1 + ^ l 2 + ^ l 3 + 4 5 ^ l 4 + ^ l 5 + ^ l 6 {38 whichmatchestheresultobtainedinBenzergaandBesson2001.Also,forisotropyi.e., ^ L = ^ I ,Equation8{37yields h 2 =6 +4 = z 6 {39 whichmatchestheresultobtainedinChapter3seeEquation3{29. Theresultingmacroscopicyieldcriterionincorporatinganisotropyisasfollows: = ~ e T 1 2 +2 f cosh 3 m h T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f 2 =0 : {40 191

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8.6AssessmentoftheProposedAnisotropicCriterionthroughComparison withFiniteElementCalculations Transverselyisotropicbehaviorisoftenseen,forexample,inrolledsheetswherethe in-planepropertiesareisotropicwiththethrough-thicknessyieldbehaviorbeingeither weakerorstronger.Inthissectiontheproposedcriteriondevelopedintheprevioussection willbeassessedbycomparingthetheoreticalpredictionswithniteelementunitcell calculations.Thisin-planesymmetryallowsfortheassessmentofmaterialbehaviorby conductingaxisymmetriccellcalculations. Table8-1tabulatestheanisotropicparametersusedinallmaterialsbeingconsidered inthissection.Thespecicplasticityparameterscorrespondtoanisotropicmaterial materialA,amaterialwhosethrough-thicknessyieldstrengthisgreaterthanitsin-plane yieldstrengthmaterialBandamaterialwhosethrough-thicknessyieldstrengthisless thanitsin-planeyieldstrengthmaterialC.ThelasttwomaterialsmaterialsBandC correspondfor k =0totwoofthetransverselyisotropicmaterialsconsideredinBenzerga andBesson2001.Figure8-1showstheaxisymmetricsectionusedintheniteelement calculationsforthetransverselyisotropicmaterialsconsideredinthissection.Notethat undertheseaxisymmetricloadingconditions,thethirdinvariantofthemacroscopicstress deviator, J 3 ,isnegativewhen 3 < 1 andpositiveotherwise.Figure8-2illustratesthe eectivestress-straincurvesforthematerialsbeingconsideredinthissectionwhen k =0 astraintriaxialityof1.5wasusedinthecalculationsillustratedbyFigure8-2.Note thatmaterialBtendstoyieldlastduetoitslargerthrough-thicknessyieldstrengthwhile materialCyieldsrstduetoitslowerthrough-thicknessyieldstrengththein-planeyield strengthswereequalamongthethreecalculations. Figures8-3to8-5showtheplanestressyieldlocifortheninedierentmaterials lookedatinthisstudydierentsetsof L -coecientscombinedwith3dierent k values.Notethatallmaterialshaveroughlythesamein-planeyieldstrengthsforxed valuesof k andthesamein-planetensileyieldstrengththroughout.Allmaterialshave 192

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eitherequalyieldstrengthsintensionandcompression k =0,smalleryieldstrengths intensionversuscompressionbothintheplaneandinthethrough-thicknessdirection orlargeryieldstrengthsintensionversuscompressionagain,bothintheplaneaswellas throughthethickness. Figures8-6to8-14showthe -planerepresentationsoftheductilecriteriongivenby Equation8{40forthe9materialsconsideredinthisstudy.Notethattheyieldsurface shrinksinthedeviatoricplaneasthepressureincreasestowardthetensilehydrostatic limitpressure, P max .Figures8-6to8-14implythattheanisotropyofthematrix,together withanystrengthdierential,inuencestheshapeoftheyieldlocusinthedeviatoric planewhilethepresenceofvoidsintheaggregateleadstoadecreaseinthesizeofthe yieldlocuswithincreasingpressurepositiveornegative.Theplotsonlyshowcurves forasinglevoidvolumefraction f =0 : 01;however,theyieldlocussimplydecreases insizewithincreasingvoidvolumefractionforaxedpressure.Insummary,the shapeoftheyieldlocusbutnotthesizeisindependentofthevoidvolumefractionand independentoftheappliedpressure;theshapeisgovernedbythematrixanisotropyand tension-compressionasymmetry. Figures8-15to8-17showtheniteelementresultswherethe9dierentmaterials areusedasthematrixmaterialinaunitcellcontainingasphericalvoid%voidvolume fraction.Theanalyticalyieldcurvesareobtainedusingthefollowingequation: = ~ e T 1 2 +2 qf cosh kk h T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( qf 2 {41 where q =4 =e aswasusedinChapter5.Also,thersttermcontainstheanisotropic CPB06eectivestresswhichisgivenas ~ e =^ m p F: Forthespecialcaseoftransverseisotropywiththe3-directionbeingthethroughthicknessdirection,thediagonalcomponentsofthematrix ^ L givenbyEquation8{29 193

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are ^ l 1 = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 12 2 = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 + L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 ^ l 2 = ^ l 1 ^ l 3 =3 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 2 + )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 33 2 +2 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 12 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 12 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 33 =13 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(8 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 13 +2 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 2 +1 {42 and ^ l 4 = ^ l 5 = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 44 2 ^ l 6 = ^ l 1 {43 where L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 = L 11 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 L 11 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 2 13 4 L 13 +2 L 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 L 11 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 L 2 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 = L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L 11 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 2 13 4 L 13 +2 L 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(6 L 11 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(3 L 2 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 44 = 1 L 44 : Again,fortransverseisotropy,thehydrostaticparameter, h ,isgivenas h = r 2 15 +2 7 ^ l 1 +2 ^ l 3 +6 ^ l 4 = r 2 15 +2 h 24 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 2 +33 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 2 +24 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 11 L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(24 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(30 L )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 13 +6 )]TJ/F21 11.9552 Tf 5.479 -9.683 Td [(L )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 44 2 +9 i : {44 The and parametersaregivenbyEquations8{19and8{20andeachcontaintwo branchesdependingonthesignof J 3 .However,inthissection,thebranchingdependence willbetransferredtothesignofthemacroscopicmeanstresssgn J 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn m since h appearsinthehydrostatictermoftheyieldconditionthisissimilartowhatwas donewhenderivingtheisotropicyieldcriterionandensuresthattheanisotropiccriterion presentedherereducestotheisotropiccriterionpresentedinChapter3. 194

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Figures8-15to8-17showthenormalizedmacroscopicvonMiseseectivestress e = 1 T versusthenormalizedmacroscopicmeanstress m = 1 T usingCPB06 L -coecients correspondingtomaterials A B and C ofTable8-1with k variedasshowninthe gures.NotethatthedatapresentedinFigure8-15forMaterial A isthesamedatapresentedinChapter3sinceMaterial A isisotropicthedataisreplottedinthissectionto comparewiththetransverselyisotropicbehaviorofMaterials B and C .Thehydrostatic solutionsinthenegativeplanetheleftsideareseentobeindependentof k inFigure 8-15.Thisisbecausetheisotropichydrostaticsolutioninthenegativeplaneissimply = 3ln f asderivedinSection3.1.2ofChapter3. ThetransverselyisotropicmaterialsillustratedinFigures8-16and8-17showvery weakdependenceon k forthehydrostaticsolutioninthenegativeplanesimilartothe isotropiccasewherethereisno k -dependenceonthenegativehydrostaticsolution. Thehydrostaticsolutionsforthesetransverselyisotropiccasescanbedeterminedfrom Equation8{41as m T 1 = h 3 ln qf suchthat m T 1 = 8 > > > > > > > > < > > > > > > > > : )]TJ/F26 11.9552 Tf 9.299 25.151 Td [(v u u t 2 15^ m 2 7 ^ l 1 +2 ^ l 3 +6 ^ l 4 3 k 2 +2 k +3 ln qf for m > 0 v u u t 2 15^ m 2 7 ^ l 1 +2 ^ l 3 +6 ^ l 4 3 k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 k +3 ln qf for m < 0 {45 wherethetwoexpressionsdierbythe 2 k inthedenominator.Figures8-18Aand8-18B showtheproposedcriterionwiththehydrostaticsolutionsofEquation8{45explicitly labeledformaterial B withthetensileyieldstrengthsgreaterthanthecompressiveyield strengthsandviceversa,respectively.Noticethatthecurvesarenotsymmetricwith respecttotheverticalaxis.ThisresultsfromEquation8{45wherethetwosolutionsare onlyequivalentifthereisnotension-compressionasymmetryinthematrix. 195

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ThemacroscopicvonMiseseectivestress e canberelatedtothemacroscopic CPB06eectivestress ~ e asfollows.Ifthe3-directionistakentobetheout-of-plane,or through-thickness,directionthenthestressdeviatorsfortheaxisymmetricloadingcase canbewrittenas 0 11 = 0 22 = 1 3 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 0 33 = )]TJ/F15 11.9552 Tf 10.494 8.088 Td [(2 3 11 )]TJ/F15 11.9552 Tf 11.956 0 Td [( 33 suchthatthevonMiseseectivestressbecomes e = q 3 J 2 = j 11 )]TJ/F15 11.9552 Tf 11.956 0 Td [( 33 j : {46 TheCPB06eectivestresscanbedeterminedas ~ e =^ m p F =^ m q 2 j ^ 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k ^ 1 2 + j ^ 3 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k ^ 3 2 : {47 Notethatforthecaseofaxisymmetricloadingunderconsideration ^ 1 =^ 2 = 1 3 L 11 + L 12 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L 13 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 ^ 3 = 2 3 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 33 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 suchthatforallmaterialsunderconsiderationinthissectioni.e.,MaterialsA,BandC giveninTable8-1 sgn^ 1 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(sgn 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 sgn^ 3 =sgn 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 and sgn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J 3 =sgn )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(J ^ 3 : ThevonMiseseectivestressgiveninEquation8{46cannowberelatedtotheCPB06 eectivestressgiveninEquation8{47asfollows: e = j 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 33 j = g ~ e 196

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where g = 8 > > > > < > > > > : 3 ^ m q 2 L 11 + L 12 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L 13 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k 2 +4 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 33 2 + k 2 for 11 > 33 3 ^ m q 2 L 11 + L 12 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 L 13 2 + k 2 +4 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 33 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(k 2 for 11 < 33 : 8.7ConcludingRemarks Ananalyticalyieldcriterionhasbeendevelopedbyextendingthesphericalvoid analysisofChapter3toaccountformatrixanisotropy.Theproposedcriterionreduces tothatgiveninBenzergaandBesson2001ifthereisnodierenceinresponsebetween theyieldintensionandcompression.Intheabsenceofvoids,theproposedcriterion reducestotheanisotropiccriterionofCazacuetal.2006ifthereistension-compression asymmetryinthematrixandtoHill's1948yieldcriterionwhennostrengthdierential exists.Thevalidityoftheproposedanisotropiccriterionwasassessedthroughcomparisons withtransverselyisotropicniteelementcalculations.Thetheoreticalpredictionsprovided bytheanisotropicyieldcriterionforthevoid-matrixaggregatecomparewellwiththe predictionsobtainedbytheniteelementunitcellcalculations. 197

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Table8-1.TransverselyisotropicCPB06constantsusedintheaxisymmetricniteelement calculations:MaterialAhasthesamein-planeandthrough-thicknessyield strengthsisotropic,materialBhasathrough-thicknessyieldstrengthgreater thanthein-planeyieldstrength T 1 < T 3 andmaterialChasa through-thicknessyieldstrengthlowerthanthein-planeyieldstrength T 1 > T 3 CPB06ParametersMaterialAMaterialBMaterialC L 11 = L 22 1.0001.0540.963 L 33 1.0000.8501.064 L 13 = L 23 0.0000.075-0.032 L 12 0.000-0.1290.069 L 44 = L 55 a 1.0000.7751.817 L 66 a 1.0001.1830.894 a ForaxisymmetriccalculationsABAQUSdoesnotuseeither L 55 or L 66 Figure8-1.AxisymmetricsectionoftheRVEforthetransverselyisotropicmaterialused intheniteelementcalculations.Axis X 3 isthethrough-thicknessdirection forthetransverselyisotropicmaterialwiththe X 1 X 2 planebeingtheplaneof symmetry. 198

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Figure8-2.Typicaleectivestressversuseectivestraincurvesillustratingtheinuence ofthedierentthrough-thicknessyieldstrengthsthein-planeyieldstrengths areheldconstantformaterialsA,BandCofTable8-1curvesshownarefor k =0suchthatthereisnotension-compressionasymmetry.MaterialAhas thesamein-planeandthrough-thicknessyieldstrengthsisotropic,materialB hasathrough-thicknessyieldstrengthgreaterthanthein-planeyieldstrength T 1 < T 3 andmaterialChasathrough-thicknessyieldstrengthlowerthan thein-planeyieldstrength T 1 > T 3 Figure8-3.Planestressyieldlociforvoid-freematerials A isotropic, B and C transverselyisotropicaccordingtotheCPB06yieldcriterion. x isan in-planedirectionwith z beingthethrough-thicknessdirection.Itisassumed herethatallthesematerialsdonothavestrengthdierentialeects k =0. 199

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Figure8-4.Planestressyieldlociforvoid-freematerials A isotropic, B and C transverselyisotropicaccordingtotheCPB06yieldcriterion. x isan in-planedirectionwith z beingthethrough-thicknessdirection.Itisassumed herethatallthesematerialsdisplaytension-compressionasymmetrywiththe yieldstrengthsintensionlessthantheyieldstrengthsincompression. Figure8-5.Planestressyieldlociforvoid-freematerials A isotropic, B and C transverselyisotropicaccordingtotheCPB06yieldcriterion. x isan in-planedirectionwith z beingthethrough-thicknessdirection.Itisassumed herethatallthesematerialsdisplaytension-compressionasymmetrywiththe yieldstrengthsintensiongreaterthantheyieldstrengthsincompression. 200

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Figure8-6.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foranisotropicmaterialmaterial A inTable8-1withthe yieldstrengthsintensionandcompressionequal. P max isthetensile hydrostaticyieldpressure. Figure8-7.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foranisotropicmaterialmaterial A inTable8-1withthe yieldstrengthsintensionlessthantheyieldstrengthsincompression. P max is thetensilehydrostaticyieldpressureand f isthevoidvolumefraction. 201

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Figure8-8.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foranisotropicmaterialmaterial A inTable8-1withthe yieldstrengthsintensiongreaterthantheyieldstrengthsincompression. P max isthetensilehydrostaticyieldpressureand f isthevoidvolumefraction. Figure8-9.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrength greaterthanthein-planeyieldstrengthmaterial B inTable8-1andwiththe yieldstrengthsintensionandcompressionequal. P max isthetensile hydrostaticyieldpressureand f isthevoidvolumefraction. 202

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Figure8-10.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrength greaterthanthein-planeyieldstrengthmaterial B inTable8-1andwith theyieldstrengthsintensionlessthantheyieldstrengthsincompression. P max isthetensilehydrostaticyieldpressureand f isthevoidvolume fraction. Figure8-11.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrength greaterthanthein-planeyieldstrengthmaterial B inTable8-1andwith theyieldstrengthsintensiongreaterthantheyieldstrengthsincompression. P max isthetensilehydrostaticyieldpressureand f isthevoidvolume fraction. 203

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Figure8-12.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrengthless thanthein-planeyieldstrengthmaterial C inTable8-1andwiththeyield strengthsintensionandcompressionequal. P max isthetensilehydrostatic yieldpressureand f isthevoidvolumefraction. Figure8-13.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrengthless thanthein-planeyieldstrengthmaterial C inTable8-1andwiththeyield strengthsintensionlessthantheyieldstrengthsincompression. P max isthe tensilehydrostaticyieldpressureand f isthevoidvolumefraction. 204

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Figure8-14.Representationinthedeviatoricplaneoftheductileyieldcriteriongivenby Equation8{40foramaterialwiththethrough-thicknessyieldstrengthless thanthein-planeyieldstrengthmaterial C inTable8-1andwiththeyield strengthsintensiongreaterthantheyieldstrengthsincompression. P max is thetensilehydrostaticyieldpressureand f isthevoidvolumefraction. 205

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A B Figure8-15.Anisotropic,axisymmetricniteelementyieldpointsversusanalyticcurves wheninitialvoidvolumefractionis f 0 =0 : 01andtheyieldstrengthinthe in-planeandthrough-thicknessdirectionsareequali.e.,MaterialAinTable 8-1.A J 3 > 0 3 > 1 .B J 3 < 0 3 < 1 206

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A B Figure8-16.Anisotropic,axisymmetricniteelementyieldpointsversusanalyticcurves wheninitialvoidvolumefractionis f 0 =0 : 01andtheyieldstrengthinthe through-thicknessdirectionisgreaterthantheyieldstrengthinthein-plane directioni.e.,MaterialBinTable8-1.A J 3 > 0 3 > 1 .B J 3 < 0 3 < 1 207

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A B Figure8-17.Anisotropic,axisymmetricniteelementyieldpointsversusanalyticcurves wheninitialvoidvolumefractionis f 0 =0 : 01andtheyieldstrengthinthe through-thicknessdirectionislessthantheyieldstrengthinthein-plane directioni.e.,MaterialCinTable8-1.A J 3 > 0 3 > 1 .B J 3 < 0 3 < 1 208

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A B Figure8-18.AnisotropicyieldsurfacesdenedbyEquation8{41with q =1foramatrix materialcontainingasphericalvoidofporosity f =0 : 01andwiththe through-thicknessyieldstrengthgreaterthanthein-planeyieldstrength.The hydrostaticparameter h isdenedbyEquation8{44fortransverseisotropy. AYieldstrengthsintensiongreaterthantheyieldstrengthsincompression. BYieldstrengthsintensionlessthantheyieldstrengthsincompression. 209

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CHAPTER9 CONCLUSIONS Thisdissertationwasdevotedtothedevelopmentofmacroscopicplasticpotentialsfor void-matrixaggregateswiththematrixdisplayingplasticanisotropyanddierentyield strengthsintensionversuscompression.Amicromechanically-basedapproachwasadopted formodelingthemechanicalresponseoftheporousductilemetals.UsingaHill-Mandel homogenizationapproachwithhomogeneousrateofdeformationboundaryconditionssee Chapter2,anupperboundoftheyieldsurfacewasderivedforahollowsphere,anda hollowcylinder,madeofaperfectlyplasticmatrixobeyingtheCazacuetal.2006yield criterion. Avoid-matrixaggregateconsistingofsphericalvoidsinanisotropicmatrixexhibiting tension-compressionasymmetrywasrstconsideredseeChapter3.Ananalytical criterionthatdependsonthesecondandthirdinvariantsofthestressdeviatorandthat accountsforthetension-compressionasymmetryoftheplasticowofthematrixwas developed.Thedevelopedcriterioncapturesthederivedexacthydrostaticsolution.If thematrixhasthesameyieldintensionandcompression,thenewcriterionreducesto Gurson's1977criterionforaductilemetalwiththematrixobeyingthevonMisesyield criterionandarandomdistributionofsphericalvoids. Avoid-matrixaggregateconsistingofcylindricalvoidsinanisotropicmatrixwasalso consideredseeChapter6.Foraxisymmetricloading,aparametricrepresentationofthe yieldsurfacewasobtained.Anexpressionoftheyieldsurfaceapplicabletoplanestrain conditionswasalsodeveloped,whichreducestoGurson's1977cylindricalcriterionwhen theyieldintensionisequaltotheyieldincompression. Lastly,thecombinedeectsofmatrixplasticanisotropyandtension-compression asymmetryontheplasticowoftheporousaggregatewasinvestigatedinChapter 8.Themicromechanicalanalysiswasdoneforarbitraryloadingconditionsbutthe principalaxesofloadingwereassumedtoalignwiththesymmetryaxesofthematrix. 210

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TheproposedanisotropiccriterionreducestothatofBenzergaandBesson2001if thereisnotension-compressionasymmetryinthevoid-freematerial.Thisdeveloped modelprovidesinformationabouthowtheplasticanisotropyinuencesthestrengthof theporousmaterial.Namely,theanisotropyofthematrix,togetherwithanystrength dierential,inuencestheshapeoftheyieldlocusinthedeviatoricplanewhilethe presenceofvoidsintheaggregateleadstoadecreaseinthesizeoftheyieldlocuswith increasingpressurepositiveornegative.Theevolutionofthevoidsresultintheyield locusdecreasinginsizewithincreasingvoidvolumefractionstheshapeoftheyield locus,bycontrast,isindependentofthevoidvolumefractionandgovernedbythematrix anisotropyandtension-compressionasymmetry. Theanalyticalexpressionsdevelopedforthemacroscopicyieldfunctionswereobtainedbyassumingaspecicgeometryfortherepresentativevolumeelement,specic loadingconditionsandothersimplifyingassumptions.Theseassumptionsandsimplicationswerenecessarytoobtainanalyticalsolutionsbutneededtobevalidatedagainstmore generalscenarios.Thevalidityoftheproposedyieldcriteriawasassessedbycomparing theanalyticalyieldcurvestoniteelementunitcellcalculations.Intheniteelement unitcellcalculations,thevoidboundarywasexplicitlymeshedandthematrixmaterial wasmodeledasanelastic-plasticmaterialwiththeplasticresponsegovernedbythe void-freeyieldcriterion. ThetheoreticalyieldcriteriondevelopedinChapter3foravoid-matrixaggregate containingsphericalvoidswascomparedtoaxisymmetricunitcellniteelementcalculationsinChapter5.Theagreementbetweentheniteelementresultsandtheproposed criterionwasfoundtobequitegood.PlanestrainunitcellcalculationswerealsoperformedandcomparedinChapter7withtheplanestraincriteriondevelopedinChapter 6.Again,theagreementwasfoundtobesatisfactory.Theproposedanalyticalyield curvedevelopedinChapter8compareswellwithtransverselyisotropicniteelement calculationsi.e.,thevoid-freematerialwasconsideredtobetransverselyisotropic. 211

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Alloftheyieldcriteriaproposedinthisdissertationaresignicantinthattheyare theonlymodelsavailablethataccountforbothvoidgrowthandtension-compression asymmetryinthevoid-freematerialcommonlyseeninhexagonalclosepackedmetals. Alloftheproposedcriteriadependonthethirdinvariantofthestressdeviatorandon boththetensileandcompressiveyieldstrengths.Theyallreducetowell-acceptedmodels whennostrengthdierentialeectispresentinthevoid-freematerial.Thematerial modelsthatresultedfromthiseortshouldprovidetheengineeringcommunitywitha moreaccuratetoolforanalyzingthefailureofporoushexagonalclosepackedmetals. Futureresearchcouldextendthemodelsdevelopedhereininasimilarmannerto howGurson's1977criteriahavebeenextendedsee,forexample,theliteraturereview doneinChapter1.Specically,theyieldcriteriadevelopedinChapters3,6and8could beextendedtocapturethebehaviorofmaterialsduringlargedeformationsandhigh strainratesbyincorporatingtemperaturesensitivity,strainhardeningandstrain-rate dependence.Also,whilethevoid-growthstageofductilefailurehasbeentheprimary focushere,bothanucleationandacoalescencecriterionmaybeconsideredinconjunction withtheyieldcriteriadevelopedinthisdissertationinordertoprovidecompletemodels forductiledamage.Eitherexistingcriteriaforvoidnucleationandcoalescencecouldbe employedsee,forexample,thecriteriapresentedinChapter1oradditionalcriteria,if required,couldbedeveloped.Still,thecriteriadevelopedinthisdissertationllagapin theliterature;thesecriteriaaretheonlyonesthatcapturetheparticularitiesofdamageby voidgrowthintextured,hexagonalclosepackedmaterials. 212

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APPENDIXA PARAMETRICREPRESENTATIONDERIVATIONOFTHEAXISYMMETRICYIELD LOCUS Thisappendixdetailsthedevelopmentofanexactfortheassumedvelocityeld givenbyEquation6{18,parametricrepresentationfortheyieldlocusofavoid-matrix aggregatewhenthematrixisgovernedbytheCazacuetal.2006criterion,thevoid geometryiscylindricalandtheloadingisaxisymmetric. A.1GeneralFormofEquations Thegeneralformsoftherelevantequationsusedthroughouttheparametricrepresentationderivationfortheyieldsurfaceoftheporousaggregatecylindricalvoidsand axisymmetricloadingaregiveninthefollowing.First,thedierentbranchesofthe plasticmultiplierrate, ,willbegivenfollowedbythegeneralformoftheupper-bound macroscopicplasticdissipation, W + ,itisanupperboundsinceonlyonevelocityeldis consideredandassociatedderivatives. A.1.1Plasticmultiplierratebranches Inordertodeterminethemacroscopicplasticdissipationandassociatedderivatives,theexpressionoftheplasticmultiplierratemustrstbedetermined.Theplastic multiplierassociatedwiththeCazacuetal.2006yieldcriterionhasmultiplebranches dependingonthesignandorderingoftheprincipalvaluesoftherateofdeformationtensorseeEquation2{29.Iftheparameter x isdenedas x := Db 2 =r 2 where D := D kk = 2, thenthenon-zeromicroscopicrateofdeformationcomponentscanbeexpressed,usingthe assumedvelocityeldofEquation6{18,asfollows: d r = )]TJ/F26 11.9552 Tf 11.291 16.857 Td [( x + B 2 d = x )]TJ/F21 11.9552 Tf 13.15 8.087 Td [(B 2 d z = B 213

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where B := D 33 .Now,theplasticmultiplierrateassociatedwiththeCazacuetal.2006 isotropicyieldcriterioncanbewrittenasfollowswhere d P i = d i forrigidplasticow: = 8 > > > > < > > > > : r 2 3 s 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 d 2 1 + d 2 2 + d 2 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 if d 1 p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 r 2 3 s d 2 1 + d 2 2 + )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 d 2 3 = 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 if d 3 p d ij d ij )]TJ/F21 11.9552 Tf 9.298 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 A{1 where d 1 d 2 d 3 aretheorderedprincipalcomponentsofthemicroscopicrate ofdeformationtensorand = T = C .NotethattherstbranchinEquationA{1 correspondsto J 3 0andthesecondbranchto J 3 0. Now,thefollowingconstantsaredenedforuseinthelatersections: p = 3 4 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [( 2 +1 m =3 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 q =3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 g = s 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 4 2 with m = )]TJ/F21 11.9552 Tf 9.299 0 Td [(m q 2 =3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 g 2 = r 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 4 and s 1 = 1 2 2 +1 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 s 2 = 1 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 s 3 = 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 2 2 214

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with s 1 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 1 s 2 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 2 s 3 = )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 3 : If z istherelevantbranch,thentherearetwocasesdependingonthesignof J 3 correspondingto d z = d 1 ,themaximumprincipalvalue,and d z = d 3 ,theminimum principalvalue.Thevalidrangeisthendeterminedbysolvingtherelevantinequalityfrom EquationA{1fortheroots, x notethatoneormorerootsmaybeinvalidforspecic loadingscenariosdependingonthevalueof andtheknownsignof x .Thus,for d z p d ij d ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 theplasticmultiplierratebecomes z = 2 p 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p x 2 + g 2 B 2 withroots x 3 = s 3 B x 3 = s 3 B: Likewise,for d z p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 theplasticmultiplierrateiswrittenas z = 2 p 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q x 2 + g 2 2 B 2 withroots x 2 = s 2 B x 2 = s 2 B: 215

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Iftheloadingconditionissuchthat istherelevantbranch,thentheanalysis proceedsinasimilarmannerasthatusedpreviously.Inotherwords,thereareagaintwo casesdependingonwhether d = d 1 or d = d 3 withacorrespondingrangeofvalidity. When d p d ij d ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 theplasticmultiplierrateis = s 2 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p pB 2 + mBx + q 2 x 2 withroots x 1 = s 1 B x 2 = s 2 B: Similarly,if d p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 then = s 2 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p pB 2 + mBx + qx 2 withroots x 3 = s 3 B x 1 = s 1 B: If r istherelevantbranchthen J 3 0with d r = d 3 isequivalentto d r p d ij d ij )]TJ/F21 11.9552 Tf 9.299 0 Td [( 2 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 suchthat r = s 2 3 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p pB 2 + mBx + q 2 x 2 216

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withroots x 1 = s 1 B x 2 = s 2 B: Similarly,whenthesignof J 3 ispositiveand d r = d 1 ,thisisequivalentto d r p d ij d ij 1 p 2 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 +1 suchthat r = s 2 3 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p pB 2 + mBx + qx 2 withroots x 3 = s 3 B x 1 = s 1 B: A.1.2Macroscopicplasticdissipationandderivatives Thenecessaryderivativesofthemacroscopicplasticdissipation, W + ,areobtained usingthechainruleonthetwovariables D and B = D 33 suchthat 11 = @W + @D 11 = @W + @ D @ D @D 11 = 1 2 @W + @ D = 11 + 22 =2 11 = @W + @ D and 33 = @W + @D 33 = @W + @ D @ D @D 33 + @W + @B @B @D 33 = 1 2 @W + @ D + @W + @B = 11 + @W + @B e = j 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 11 j = @W + @B : Thegeneralformoftheintegralsassociatedwiththemacroscopicplasticdissipation remainsthesameregardlessofwhichcaseisunderconsideration.Lettheparameter u be denedsuchthat u = D j B j : 217

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Fordemonstrationpurposes,theintegralsolutionsusingtheintegrationlimitsforone speciccasewillbegiveninthefollowingequationstheexampleshowncorrespondsto = T = C > 1, D> 0and B< 0where s 3 f u s 3 .Themacroscopicplastic dissipationiswrittenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 qh 1 x + mB 2 h 2 x + h 3 x D=f x 3 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h h 4 x i x 3 D where x 3 = s 3 B andwheretheintegralsolutionof Z p X x 2 dx = qh 1 x + mB 2 h 2 x + h 3 x hasbeenusedwith X = pB 2 + mBx + qx 2 and h 1 x = Z dx p X = 1 p q ln 2 p qX +2 qx + mB h 2 x = Z dx x p X = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 p pB 2 ln 2 p pB 2 X + mBx +2 pB 2 x h 3 x = )]TJ 9.299 10.122 Td [(p X x : Also, h 4 x = Z p x 2 + g 2 B 2 x 2 dx = )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p x 2 + g 2 B 2 x +ln x + p x 2 + g 2 B 2 : 218

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Therelevantderivativescannowbewrittenas @W + @ D = W + D + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @ D D=f x 3 + mB 2 @h 2 @x @x @ D D=f x 3 + @h 3 @x @x @ D D=f x 3 + s 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 @h 4 @x @x @ D x 3 D 9 = ; @W + @B = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q @h 1 @x @x @B + @h 1 @B D=f x 3 + mB 2 @h 2 @x @x @B + @h 2 @B D=f x 3 + @h 3 @x @x @B + @h 3 @B D=f x 3 + s 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 @h 4 @x @x @B + @h 4 @B x 3 D with @h 1 @x = 1 p X @h 2 @x = 1 x p X @h 3 @x = )]TJ/F15 11.9552 Tf 11.291 0 Td [( mBx= 2+ qx 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(X x 2 p X @h 4 @x = p x 2 + g 2 B 2 x 2 and @h 1 @B = 1 p q p q mx +2 pB + m p X 2 p qX + qx + mB p X @ Bh 2 @B = )]TJ/F15 11.9552 Tf 9.298 0 Td [(sgn B p p pB pB 2 +3 mBx +2 qx 2 + mx +4 pB p pB 2 p X 2 pB 2 X + mBx +2 pB 2 p X @h 3 @B = )]TJ/F21 11.9552 Tf 10.494 8.088 Td [(pB + mx= 2 x p X @h 4 @B = )]TJ/F21 11.9552 Tf 47.967 8.087 Td [(Bg 2 x 2 + x p x 2 + g 2 B 2 : Theresultantsolutionsusingthepreviousderivativesaregivenforthevariousbranchesof D and B inthefollowingsections. 219

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A.2 1 :TheMatrixYieldStrengthinTensionisGreaterthanin Compression. WhendealingwiththeyieldcriterionofCazacuetal.2006,therearemultipleregionsandbrancheswithinregionstoconsiderwhenderivingtheparametricrepresentation oftheporousyieldsurface.Thissectionwillfocusonthosematerialsforwhichthematrix hasayieldintensiongreaterthanincompressioni.e., > 1suchthat T > C A.2.1 J 3 < 0 If J 3 < 0thentherearetwocasestoconsider:1 m > 0and2 m < 0.Also,when J 3 < 0,notethat B = D 33 < 0suchthat u = )]TJ/F15 11.9552 Tf 12.045 3.022 Td [( D=B A.2.1.1 m > 0 When m > 0,notethat D> 0.Therearethreedierentrangesof u toconsider inthiscasewhichdictatethelimitsofintegrationusedtoevaluate W + D and,thus, determinethenalformofthestressinvariants. FirstBranch:0 u s 3 f Inthisrstbranchthemacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z D=f D p x 2 + g 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 ln u + p u 2 + g 2 f 2 u + p u 2 + g 2 1 f and @W + @B = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h p u 2 + g 2 f 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p u 2 + g 2 i : SecondBranch: s 3 f u s 3 Inthisnextbranchtheexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 220

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where x 3 = s 3 B .Thisyieldsthefollowingderivatives: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln )]TJ/F21 11.9552 Tf 10.494 8.136 Td [(s 3 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p s 2 3 + g 2 u + p u 2 + g 2 + p q ln 2 p q p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(mf 2 p q p p + ms 3 + qs 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qs 3 )]TJ/F21 11.9552 Tf 11.956 0 Td [(m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 pf 2 p p p p + ms 3 + qs 2 3 + ms 3 +2 p s 3 u !# and @W + @B = r 2 3 T p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p 2 u 2 + g 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 3 u 1 Lastly,when s 3 u 1 themacroscopicplasticdissipationisasfollows: W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f D p qx 2 + mBx + pB 2 dx x 2 suchthat @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(mf 2 p q p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 +2 qu )]TJ/F21 11.9552 Tf 11.955 0 Td [(m 1 f + m 2 p p ln 2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 pf 2 p p p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 p !# and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 h p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 i : A.2.1.2 m < 0 When m < 0thisresultsin D< 0.Inthiscase,thereareagainthreerangesof u for whichdistinctexpressionsofthemacroscopicplasticdissipationexist. FirstBranch:0 u s 3 f Inthisrstbranchthemacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z D=f D p x 2 + g 2 B 2 dx x 2 221

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suchthat @W + @ D = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 ln u + p u 2 + g 2 f 2 u + p u 2 + g 2 1 f and @W + @B = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h p u 2 + g 2 f 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p u 2 + g 2 i : SecondBranch: s 3 f u s 3 Forthesecondbranch s 3 f u s 3 suchthatthemacroscopicplasticdissipationis givenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 where x 3 = s 3 B .Thisyieldsthefollowingderivatives: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln )]TJ/F15 11.9552 Tf 10.975 8.136 Td [( s 3 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p s 2 3 + g 2 u + p u 2 + g 2 + p q ln 2 p q p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 +2 qu )]TJ/F15 11.9552 Tf 14.148 0 Td [( mf 2 p q p p + m s 3 + q s 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q s 3 )]TJ/F15 11.9552 Tf 14.149 0 Td [( m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.149 0 Td [( muf + qu 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 pf 2 p p p p + m s 3 + q s 2 3 + m s 3 +2 p s 3 u !# and @W + @B = r 2 3 T p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p 2 u 2 + g 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 3 u Thelastbranchhasthefollowingmacroscopicplasticdissipation: W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f D p qx 2 + mBx + pB 2 dx x 2 222

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suchthat @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 +2 qu )]TJ/F15 11.9552 Tf 14.149 0 Td [( mf 2 p q p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + qu 2 +2 qu )]TJ/F15 11.9552 Tf 14.148 0 Td [( m 1 f + m 2 p p ln 2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 pf 2 p p p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + qu 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 p !# and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 h p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + qu 2 i : A.2.2 J 3 > 0 When J 3 > 0then B = D 33 > 0suchthat u = D=B .Asbefore,therearetwocases thatmustbetreatedindividually:thecaseof m > 0andthecaseof m < 0. A.2.2.1 m > 0 Forthecaseof m > 0, D> 0.Therearevedierentrangesof u thatmust beaccountedforandthatresultinseparateexpressionsforthemacroscopicplastic dissipation. FirstBranch:0 u s 2 f Themacroscopicplasticdissipationforthisrangeisgivenas W + = 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f D q x 2 + g 2 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ln u + p u 2 + g 2 2 f 2 u + p u 2 + g 2 2 1 f and @W + @B = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 14.565 Td [(q u 2 + g 2 2 f 2 : SecondBranch: s 2 f u s 1 f 223

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Inthesecondbranch,theexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 where x 2 = s 2 B .Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 + p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + ms 2 + q 2 s 2 2 +2 q 2 s 2 + m 1 f )]TJ/F21 11.9552 Tf 18.877 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + ms 2 + q 2 s 2 2 + ms 2 +2 p s 2 u !# and @W + @B = r 2 3 T p 2 u 2 + g 2 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.531 Td [(p pf 2 + muf + q 2 u 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 1 f u s 2 Thethirdbranchcontainstherange s 1 f u s 2 suchthattheexpressionforthe macroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f x 1 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 1 x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 : 224

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Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + ms 1 + qs 2 1 +2 qs 1 + m 1 f )]TJ/F21 11.9552 Tf 18.877 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + ms 1 + qs 2 1 + ms 1 +2 p s 1 u + p 2ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 )]TJ 11.955 10.473 Td [(p 2 p s 2 2 + g 2 2 s 2 + p p + ms 1 + qs 2 1 s 1 # + r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p p + ms 1 + q 2 s 2 1 +2 q 2 s 1 + m 2 p q 2 p p + ms 2 + q 2 s 2 2 +2 q 2 s 2 + m )]TJ/F21 11.9552 Tf 18.877 8.088 Td [(m 2 p p ln 2 p p p p + ms 1 + q 2 s 2 1 + ms 1 +2 p 2 p p p p + ms 2 + q 2 s 2 2 + ms 2 +2 p s 2 s 1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p p + ms 1 + q 2 s 2 1 s 1 + p p + ms 2 + q 2 s 2 2 s 2 # and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 2 u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + qu 2 : FourthBranch: s 2 u s 1 Inthefourthbranchthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 1 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 1 D p q 2 x 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + ms 1 + qs 2 1 +2 qs 1 + m 1 f )]TJ/F21 11.9552 Tf 18.877 8.087 Td [(m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + ms 1 + qs 2 1 + ms 1 +2 p s 1 u !# + r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p p + ms 1 + q 2 s 2 1 +2 q 2 s 1 + m 2 p q 2 p p + mu + q 2 u 2 +2 q 2 u + m )]TJ/F21 11.9552 Tf 18.877 8.088 Td [(m 2 p p ln 2 p p p p + ms 1 + q 2 s 2 1 + ms 1 +2 p 2 p p p p + mu + q 2 u 2 + mu +2 p u s 1 !# 225

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and @W + @B = r 2 3 T p p + ms 1 + qs 2 1 )]TJ/F26 11.9552 Tf 11.956 10.365 Td [(p pf 2 + muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + p p + mu + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.394 Td [(p p + ms 1 + q 2 s 2 1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : FifthBranch: s 1 u 1 Thefthandnalbranchcontainstherange s 1 u 1 suchthattheexpressionfor themacroscopicplasticdissipationbecomes W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f D p qx 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + mu + qu 2 +2 qu + m 1 f )]TJ/F21 11.9552 Tf 18.876 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + mu + qu 2 + mu +2 p !# and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 h p p + mu + qu 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + qu 2 i : A.2.2.2 m < 0 If m < 0then D< 0resultinginverelevantbranchesdependingontherangeof u beingconsidered. FirstBranch:0 u s 2 f Inthisrstbranchthemacroscopicplasticdissipationisgiven as W + = 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f D q x 2 + g 2 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ln u + p u 2 + g 2 2 f 2 u + p u 2 + g 2 2 1 f 226

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and @W + @B = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 14.565 Td [(q u 2 + g 2 2 f 2 : SecondBranch: s 2 f u s 1 f Intherange s 2 f u s 1 f ,theexpressionforthemacroscopicplasticdissipation becomes W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 where x 2 = s 2 B .Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 + p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + m s 2 + q 2 s 2 2 +2 q 2 s 2 + m 1 f )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + m s 2 + q 2 s 2 2 + m s 2 +2 p s 2 u !# and @W + @B = r 2 3 T p 2 u 2 + g 2 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p pf 2 + muf + q 2 u 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 1 f u s 2 Thethirdbranchresultsinamacroscopicplasticdissipationgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f x 1 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 1 x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 : 227

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Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + m s 1 + q s 2 1 +2 q s 1 + m 1 f )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + m s 1 + q s 2 1 + m s 1 +2 p s 1 u + p 2ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 )]TJ 11.955 10.473 Td [(p 2 p s 2 2 + g 2 2 s 2 + p p + m s 1 + q s 2 1 s 1 # + r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p p + m s 1 + q 2 s 2 1 +2 q 2 s 1 + m 2 p q 2 p p + m s 2 + q 2 s 2 2 +2 q 2 s 2 + m )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p p + m s 1 + q 2 s 2 1 + m s 1 +2 p 2 p p p p + m s 2 + q 2 s 2 2 + m s 2 +2 p s 2 s 1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p p + m s 1 + q 2 s 2 1 s 1 + p p + m s 2 + q 2 s 2 2 s 2 # and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q 2 u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + qu 2 : FourthBranch: s 2 u s 1 Thefourthbranchyieldsthefollowingexpressionforthemacroscopicplasticdissipation: W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 1 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 1 D p q 2 x 2 + mBx + pB 2 dx x 2 : 228

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This,inturn,yieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + m s 1 + q s 2 1 +2 q s 1 + m 1 f )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + m s 1 + q s 2 1 + m s 1 +2 p s 1 u !# + r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p p + m s 1 + q 2 s 2 1 +2 q 2 s 1 + m 2 p q 2 p p + mu + q 2 u 2 +2 q 2 u + m )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p p + m s 1 + q 2 s 2 1 + m s 1 +2 p 2 p p p p + mu + q 2 u 2 + mu +2 p u s 1 !# and @W + @B = r 2 3 T p p + m s 1 + q s 2 1 )]TJ/F26 11.9552 Tf 11.956 10.366 Td [(p pf 2 + muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + p p + mu + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p p + m s 1 + q 2 s 2 1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : FifthBranch: s 1 u Thenalbranchcontainstherange s 1 u suchthatthemacroscopicplastic dissipationbecomes W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f D p qx 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln 2 p q p pf 2 + muf + qu 2 +2 qu + mf 2 p q p p + mu + qu 2 +2 qu + m 1 f )]TJ/F15 11.9552 Tf 21.069 8.087 Td [( m 2 p p ln 2 p p p pf 2 + muf + qu 2 + mu +2 pf 2 p p p p + mu + qu 2 + mu +2 p !# and @W + @B = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 h p p + mu + qu 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + qu 2 i : A.3 1 :TheMatrixYieldStrengthinTensionisLessthanin Compression. Thissectionfocusesonthosematerialswhere < 1suchthat T < C 229

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A.3.1 J 3 < 0 If J 3 < 0then B< 0suchthat u = )]TJ/F15 11.9552 Tf 12.044 3.022 Td [( D=B .Thetwocasesthatmustbetakeninto considerationhereare m > 0and m < 0. A.3.1.1 m > 0 When m > 0then D> 0andtherearevedomainsof u thatmustbetreated individually. FirstBranch:0 u s 3 f Intherstbranchthemacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z D=f D p x 2 + g 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 ln u + p u 2 + g 2 f 2 u + p u 2 + g 2 1 f and @W + @B = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h p u 2 + g 2 f 2 )]TJ/F26 11.9552 Tf 11.955 10.95 Td [(p u 2 + g 2 i : SecondBranch: s 3 f u s 1 f Thesecondbranchcontainstherange s 3 f u s 1 f suchthattheexpressionforthe macroscopicplasticdissipationinthisregionisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 where x 3 = s 3 B .Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 3 + p s 2 3 + g 2 u + p u 2 + g 2 + p q ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qu + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p p + ms 3 + qs 2 3 +2 qs 3 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 pf 2 p p p p + ms 3 + qs 2 3 + ms 3 +2 p s 3 u !# 230

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and @W + @B = r 2 3 T )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p 2 u 2 + g 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 11.9552 Tf 13.151 18.531 Td [(p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 # : ThirdBranch: s 1 f u s 3 When s 1 f u s 3 ,themacroscopicplasticdissipationbecomes W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 1 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z x 1 x 3 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 Z x 3 D p 2 p x 2 + g 2 B 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p + ms 1 + q 2 s 2 1 +2 q 2 s 1 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(mu +2 pf 2 p p p p + ms 1 + q 2 s 2 1 + ms 1 +2 p s 1 u + p 2 2 ln )]TJ/F21 11.9552 Tf 9.299 0 Td [(s 3 + p s 2 3 + g 2 u + p u 2 + g 2 + p 2 2 p s 2 3 + g 2 s 3 )]TJ/F26 11.9552 Tf 13.15 18.531 Td [(p p + ms 1 + q 2 s 2 1 s 1 # + r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 p q ln )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + ms 1 + qs 2 1 +2 qs 1 + m )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + ms 3 + qs 2 3 +2 qs 3 + m + m 2 p p ln 2 p p p p + ms 1 + qs 2 1 + ms 1 +2 p 2 p p p p + ms 3 + qs 2 3 + ms 3 +2 p s 3 s 1 + p p + ms 1 + qs 2 1 s 1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p p + ms 3 + qs 2 3 s 3 # and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h )]TJ/F26 11.9552 Tf 9.298 10.806 Td [(p 2 2 u 2 + g 2 + p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 i : FourthBranch: s 3 u s 1 231

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Inthisbranchtheexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 1 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 1 D p qx 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p + ms 1 + q 2 s 2 1 +2 q 2 s 1 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(mu +2 pf 2 p p p p + ms 1 + q 2 s 2 1 + ms 1 +2 p s 1 u !# + r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + ms 1 + qs 2 1 +2 qs 1 + m )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qu + m + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p p + ms 1 + qs 2 1 + ms 1 +2 p 2 p p p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 p u s 1 !# and @W + @B = r 2 3 T p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p p + ms 1 + q 2 s 2 1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 + p p + ms 1 + qs 2 1 )]TJ/F26 11.9552 Tf 11.955 10.366 Td [(p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 # : FifthBranch: s 1 u 1 Inthislastbranch,themacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f D p q 2 x 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + m 1 f + m 2 p p ln 2 p p p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 pf 2 p p p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + q 2 u 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu +2 p !# 232

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and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h p pf 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(muf + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p p )]TJ/F21 11.9552 Tf 11.955 0 Td [(mu + q 2 u 2 i : A.3.1.2 m < 0 Thereareanothervebranchesthatmustbetakenintoaccountwhen m < 0such that D< 0. FirstBranch:0 u s 3 f First,attentionwillbegiventotherangeof0 u s 3 f .Inthisbranch,the macroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 Z D=f D p x 2 + g 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 ln u + p u 2 + g 2 f 2 u + p u 2 + g 2 1 f and @W + @B = 2 p 3 T p 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 h p u 2 + g 2 f 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p u 2 + g 2 i : SecondBranch: s 3 f u s 1 f Inthesecondbranchtheexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z D=f x 3 p qx 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z x 3 D p x 2 + g 2 B 2 dx x 2 where x 3 = s 3 B .Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 ln )]TJ/F15 11.9552 Tf 9.78 0 Td [( s 3 + p s 2 3 + g 2 u + p u 2 + g 2 + p q ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p pf 2 )]TJ/F15 11.9552 Tf 14.149 0 Td [( muf + qu 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qu + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p p + m s 3 + q s 2 3 +2 q s 3 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.69 Td [(2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.149 0 Td [( muf + qu 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 pf 2 p p p p + m s 3 + q s 2 3 + m s 3 +2 p s 3 u !# 233

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and @W + @B = r 2 3 T )]TJ/F26 11.9552 Tf 10.494 18.531 Td [(p 2 u 2 + g 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 11.9552 Tf 13.151 18.531 Td [(p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 # : ThirdBranch: s 1 f u s 3 When s 1 f u s 3 themacroscopicplasticdissipationisasfollows: W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 1 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z x 1 x 3 p qx 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 Z x 3 D p 2 p x 2 + g 2 B 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p + m s 1 + q 2 s 2 1 +2 q 2 s 1 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 14.149 0 Td [( mu +2 pf 2 p p p p + m s 1 + q 2 s 2 1 + m s 1 +2 p s 1 u + p 2 2 ln )]TJ/F15 11.9552 Tf 9.779 0 Td [( s 3 + p s 2 3 + g 2 u + p u 2 + g 2 + p 2 2 p s 2 3 + g 2 s 3 )]TJ/F26 11.9552 Tf 13.15 18.531 Td [(p p + m s 1 + q 2 s 2 1 s 1 # + r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 p q ln )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + m s 1 + q s 2 1 +2 q s 1 + m )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + m s 3 + q s 2 3 +2 q s 3 + m + m 2 p p ln 2 p p p p + m s 1 + q s 2 1 + m s 1 +2 p 2 p p p p + m s 3 + q s 2 3 + m s 3 +2 p s 3 s 1 + p p + m s 1 + q s 2 1 s 1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p p + m s 3 + q s 2 3 s 3 # and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h )]TJ/F26 11.9552 Tf 9.298 10.806 Td [(p 2 2 u 2 + g 2 + p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 i : FourthBranch: s 3 u s 1 234

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Thefourthbranchyieldsanexpressionforthemacroscopicplasticdissipationas W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 1 p q 2 x 2 + mBx + pB 2 dx x 2 + r 2 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 1 D p qx 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p + m s 1 + q 2 s 2 1 +2 q 2 s 1 + m 1 f + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 14.149 0 Td [( mu +2 pf 2 p p p p + m s 1 + q 2 s 2 1 + m s 1 +2 p s 1 u !# + r 2 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p q ln )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 p q p p + m s 1 + q s 2 1 +2 q s 1 + m )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + qu 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 qu + m + m 2 p p ln )]TJ/F15 11.9552 Tf 10.494 8.691 Td [(2 p p p p + m s 1 + q s 2 1 + m s 1 +2 p 2 p p p p )]TJ/F15 11.9552 Tf 14.149 0 Td [( mu + qu 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 p u s 1 !# and @W + @B = r 2 3 T p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.395 Td [(p p + m s 1 + q 2 s 2 1 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 + p p + m s 1 + q s 2 1 )]TJ/F26 11.9552 Tf 11.955 10.366 Td [(p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + qu 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 # : FifthBranch: s 1 u Inthefthandnalbranchtheexpressionforthemacroscopicplasticdissipation becomes W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f D p q 2 x 2 + mBx + pB 2 dx x 2 : Thisyieldsthefollowingderivativeexpressions: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + mf )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 p q 2 p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + q 2 u 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 q 2 u + m 1 f + m 2 p p ln 2 p p p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 pf 2 p p p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + q 2 u 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu +2 p !# 235

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and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h p pf 2 )]TJ/F15 11.9552 Tf 14.148 0 Td [( muf + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p p )]TJ/F15 11.9552 Tf 14.148 0 Td [( mu + q 2 u 2 i : A.3.2 J 3 > 0 When J 3 > 0thenthetwocasestobeconsideredseparatelyare m > 0and < 0. Likewisefor J 3 > 0, B> 0suchthat u = D=B A.3.2.1 m > 0 If m > 0then D> 0andtherearethreebranchescontainingdistinctexpressionsfor themacroscopicplasticdissipation W + D FirstBranch:0 u s 2 f Intherstbranchthemacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f D q x 2 + g 2 2 B 2 dx x 2 suchthat @W + @ D = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ln u + p u 2 + g 2 2 f 2 u + p u 2 + g 2 2 1 f and @W + @B = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 14.565 Td [(q u 2 + g 2 2 f 2 : SecondBranch: s 2 f u s 2 Inthesecondbranchtheexpressionforthemacroscopicplasticdissipationisgivenas W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 236

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where x 2 = s 2 B .Thisyieldsthefollowingderivatives: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 + p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + ms 2 + q 2 s 2 2 +2 q 2 s 2 + m 1 f )]TJ/F21 11.9552 Tf 18.877 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + ms 2 + q 2 s 2 2 + ms 2 +2 p s 2 u !# and @W + @B = r 2 3 T p 2 u 2 + g 2 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p pf 2 + muf + q 2 u 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 2 u 1 Finally,when s 2 u 1 themacroscopicplasticdissipationisasfollows: W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f D p q 2 x 2 + mBx + pB 2 dx x 2 suchthat @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + mu + q 2 u 2 +2 q 2 u + m 1 f )]TJ/F21 11.9552 Tf 18.876 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + mu + q 2 u 2 + mu +2 p !# and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h p p + mu + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + q 2 u 2 i : A.3.2.2 m < 0 If m < 0then D< 0and,again,therearethreedierentbranchesthatmustbe described. FirstBranch:0 u s 2 f When0 u s 2 f themacroscopicplasticdissipationisgivenas W + = 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 Z D=f D q x 2 + g 2 2 B 2 dx x 2 237

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suchthat @W + @ D = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ln u + p u 2 + g 2 2 f 2 u + p u 2 + g 2 2 1 f and @W + @B = 2 p 3 T p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 q u 2 + g 2 2 )]TJ/F26 11.9552 Tf 11.955 14.564 Td [(q u 2 + g 2 2 f 2 : SecondBranch: s 2 f u s 2 Inthesecondbranchthemacroscopicplasticdissipationbecomes W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f x 2 p q 2 x 2 + mBx + pB 2 dx x 2 + 2 p 3 T D p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z x 2 D q x 2 + g 2 2 B 2 dx x 2 where x 2 = s 2 B .Thisyieldsthefollowingderivatives: @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [( 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ln s 2 + p s 2 2 + g 2 2 u + p u 2 + g 2 2 + p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + m s 2 + q 2 s 2 2 +2 q 2 s 2 + m 1 f )]TJ/F15 11.9552 Tf 21.07 8.088 Td [( m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + m s 2 + q 2 s 2 2 + m s 2 +2 p s 2 u !# and @W + @B = r 2 3 T p 2 u 2 + g 2 2 p 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 )]TJ/F26 11.9552 Tf 13.151 18.53 Td [(p pf 2 + muf + q 2 u 2 p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 # : ThirdBranch: s 2 u Thethirdbranchcontainstherange s 2 u andyieldsanexpressionforthe macroscopicplasticdissipationasfollows: W + = r 2 3 T D p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 Z D=f D p q 2 x 2 + mBx + pB 2 dx x 2 238

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suchthat @W + @ D = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 p q 2 ln 2 p q 2 p pf 2 + muf + q 2 u 2 +2 q 2 u + mf 2 p q 2 p p + mu + q 2 u 2 +2 q 2 u + m 1 f )]TJ/F21 11.9552 Tf 18.876 8.088 Td [(m 2 p p ln 2 p p p pf 2 + muf + q 2 u 2 + mu +2 pf 2 p p p p + mu + q 2 u 2 + mu +2 p !# and @W + @B = r 2 3 T p 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 h p p + mu + q 2 u 2 )]TJ/F26 11.9552 Tf 11.955 10.949 Td [(p pf 2 + muf + q 2 u 2 i : 239

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APPENDIXB RELATIONSHIPBETWEENHILL48ANDCPB06COEFFICIENTS ThefollowingdemonstratesthattheCPB06seeCazacuetal.,2006criterion canreducetoHill's1948anisotropiccriterionseeHill,1948,1950forthecasewhere theyieldstrengthsintensionareequaltotheyieldstrengthsincompressioni.e., k = 0.Specically,thecaseoftransverseisotropywiththe1-2planebeingtheplaneof symmetryandthe3-directionbeingtheout-of-planedirectionwillbeconsideredin thefollowinganalysissinceanalyticalexpressionsbecomemoretractable;however,the procedureisthesameforthemoregeneralcaseoforthotropy.Inordertoshowthis reductiontoHill48,let^ = L 0 withtheconstraintthatthetransformedstress,^ ,be deviatoricnotethatthisconditionismorerestrictivethantheoriginalCPB06criterion presentedinCazacuetal.,2006.Thelineartransformation, L ,canbewritteninVoight notationasfollows: L = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 L 11 L 12 L 13 000 L 12 L 22 L 23 000 L 13 L 23 L 33 000 000 L 44 00 0000 L 55 0 00000 L 66 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 which,fortransverseisotropy,becomes L = 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 L 11 L 12 L 13 000 L 12 L 11 L 13 000 L 13 L 13 L 33 000 000 L 44 00 0000 L 44 0 00000 L 11 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 12 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : 240

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Notethat,inorderfortheprevioustransformationtoyieldadeviatorictensorwhen appliedtothestressdeviator,thefollowingequationsmusthold: L 11 + L 12 + L 13 =1 L 12 + L 22 + L 23 =1 L 13 + L 23 + L 33 =1 wheretheconstantontherighthandsideofthepreviousequationshasbeenchosentobe 1suchthatthelineartransformationreducestotheidentitytensorinthecaseofisotropy. Ingeneral,theCPB06criterionusing a =2canbewrittenas F = j ^ I j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k ^ I 2 + j ^ II j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k ^ II 2 + j ^ III j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k ^ III 2 suchthat,for k =0, F =^ 2 I +^ 2 II +^ 2 III =2 J ^ 2 where^ I ,^ II and^ III aretheprincipalcomponentsof^ and J ^ 2 isthesecondinvariant ofthetransformedstresstensor,^ notethattheprincipalcomponentsaredeviators giventheimposedconstraintthattheresultofapplyingthelineartransformation, L ,to adeviatorictensorisitselfadeviatorictensor.Now,theCPB06yieldcriterioncanbe writtenas =^ m 2 F )]TJ/F26 11.9552 Tf 11.955 9.684 Td [()]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( T 1 2 =0B{1 suchthat,for k =0, ^ m 2 )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(2 J ^ 2 = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [( T 1 2 B{2 where T 1 istheuniaxialyieldstrengthinthe1-directiontension-compressionsymmetry isassumedherebutthenotationof Y = T 1 willberetainedthroughoutthefollowing paragraphstomaintainnotationalconsistencywiththecasewhen k 6 =0.Theconstant, 241

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^ m ,istheanisotropicversionoftheeectivestressconstantandisgivenby ^ m = s 1 j 1 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k 1 2 + j 2 j)]TJ/F21 11.9552 Tf 17.932 0 Td [(k 2 2 + j 3 j)]TJ/F21 11.9552 Tf 17.933 0 Td [(k 3 2 where 1 = 2 3 L 11 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 12 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 13 2 = 2 3 L 12 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 22 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 L 23 3 = 2 3 L 13 )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 3 L 23 )]TJ/F15 11.9552 Tf 13.15 8.087 Td [(1 3 L 33 suchthat,assuming k =0, ^ m = s 9 L 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 + L 12 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 2 + L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 whichcanbefurtherreducedtothefollowingfortransverseisotropyusingtheconstraint thatthe L -tensortransformsadeviatorictensortoanotherdeviatorictensor: ^ m = s 3 6 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 11 + L 13 +6 L 2 11 +2 : B{3 Hill's1948criterioncanbewrittenas F 0 22 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 33 2 + G 0 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 11 2 + H 0 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 22 2 +2 L 2 23 +2 M 2 13 +2 N 2 12 = 2 Y B{4 wheretheparameters, F G H L M and N aredenedintermsoftheuniaxialyield strengthsinthe1,2and3directionsas Y T 1 2 =1= G + H = 1 R 11 2 Y T 2 2 = H + F = 1 R 22 2 Y T 3 2 = F + G = 1 R 33 2 242

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and Y 13 2 = M = 3 2 R 13 2 Y 23 2 = L = 3 2 R 23 2 Y 12 2 = N = 3 2 R 12 2 where Y = T 1 isusedandwherethe ij aretheyieldstressesinshearwithrespecttothe respectiveprincipalaxesofthematerial.Theparameters, R ij ,areintroducedheresince ABAQUSusesthisterminologyinitsinternalmateriallibraryforHill48.Forreferences seeHill1948,Hill1950andAbaqus2008.NotethatthescalarHill48Parameter, L isanunfortunatecoincidentalnotationandshouldnotbeconfusedwiththefourth-order lineartransformationtensor, L ,oritsVoight-notationcounterpart, L .Also,theHill48 parameter, F ,shouldnotbeconfusedwiththenotationusedtodescribetheCPB06yield functioninEquationB{2intheremainingparagraphs,` F 'willsolelyrefertotheHill48 parameter. EquatingEquationB{2withEquationB{4yields 2^ m 2 J ^ 2 = F 0 22 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 33 2 + G 0 33 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 11 2 + H 0 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 0 22 2 +2 L 2 23 +2 M 2 13 +2 N 2 12 = H +2 G )]TJ/F21 11.9552 Tf 11.956 0 Td [(F 0 11 2 + H +2 F )]TJ/F21 11.9552 Tf 11.955 0 Td [(G 0 22 2 + F +2 G )]TJ/F21 11.9552 Tf 11.955 0 Td [(H 0 33 2 +2 L 23 2 +2 M 13 2 +2 N 12 2 = H + G 0 11 2 + H + G 0 22 2 + G )]TJ/F21 11.9552 Tf 11.955 0 Td [(H 0 33 2 +2 M 23 2 +2 M 13 2 +2 H + G 12 2 B{5 wherethelastofthepreviousequationsusesthefactthat F = G L = M and N = G +2 H forthecaseoftransverseisotropyunderconsideration.Themaintasknowistoevaluate theleft-hand-sideofEquationB{5andequatethetermsof 0 ij withthoseobtained intheright-hand-sideofthelastexpressioninEquationB{5.Withthatinmind,the 243

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transformedstresscanbeexpressedas ^ = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : L 11 0 11 + L 12 0 22 + L 13 0 33 L 12 0 11 + L 11 0 22 + L 13 0 33 L 13 0 11 + L 13 0 22 + L 33 0 33 L 44 23 L 44 13 L 11 )]TJ/F21 11.9552 Tf 11.956 0 Td [(L 12 12 9 > > > > > > > > > > > > > > = > > > > > > > > > > > > > > ; B{6 wheretransverseisotropyhasbeenassumed.Now,thesecondinvariant, J ^ 2 ,ofthe transformedstressdeviatorcanbeexpressedasfollows: 2 J ^ 2 =^ :^ =^ ij ^ ij = )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(L 2 11 + L 2 12 + L 2 13 +2 L 12 L 13 +2 L 11 L 13 +2 L 13 L 33 0 11 2 + )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(L 2 11 + L 2 12 + L 2 13 +2 L 12 L 13 +2 L 11 L 13 +2 L 13 L 33 0 22 2 + )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(4 L 2 13 + L 2 33 +4 L 11 L 12 0 33 2 +2 L 2 44 23 2 +2 L 2 44 13 2 +2 L 11 )]TJ/F21 11.9552 Tf 11.956 0 Td [(L 12 2 12 2 : Equatingthe 0 ij termsbetweenthepreviousequationandEquationB{5yields 2 H + G =^ m 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(L 2 11 + L 2 12 + L 2 13 +2 L 12 L 13 +2 L 11 L 13 +2 L 13 L 33 4 G )]TJ/F21 11.9552 Tf 11.955 0 Td [(H =^ m 2 )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(4 L 2 13 + L 2 33 +4 L 11 L 12 M =^ m 2 L 2 44 B{7 which,alongwith L 11 + L 12 + L 13 =1 2 L 13 + L 33 =1 B{8 fromthedeviatoricconstrainton L yields5equationsforthe5unknowns L 11 L 33 L 12 L 13 and L 44 assumingthattheHill48coecients, H G and M areknown.Theprevious 244

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5equationscanbeusedtodeterminetheHill48parametersgiventheCPB06coecients ortodeterminetheCPB06coecientsgiventheHill48coecients. B.1DetermineTheHill48CoecientsGivenTheCPB06Coecients GivenasetofCPB06coecients,theHill48parametersarereadilydeterminedfrom EquationsB{7andB{8.Neglectingthealgebra,thefollowingrelationsareultimately obtained: G = 2 L 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 L 13 )]TJ/F21 11.9552 Tf 11.956 0 Td [(L 11 +2 L 2 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 L 2 11 +1 6 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 11 + L 13 +6 L 2 11 +2 H = )]TJ/F15 11.9552 Tf 9.298 0 Td [(4 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 13 )]TJ/F21 11.9552 Tf 11.955 0 Td [(L 11 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 L 2 13 +4 L 2 11 +1 6 L 13 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 L 11 + L 13 +6 L 2 11 +2 M = 3 L 2 44 6 L 13 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 L 11 + L 13 +6 L 2 11 +2 : B{9 Notethatfortransverseisotropy,onlytwoanisotropiccoecientsareindependent;either H or G canbecalculatedgiventheotherusing G + H =1. Fortransverseisotropy,thematerialanisotropyisoftenreportedintermsofthe plasticstrainratios R and R h .TheHill48coecientsarereadilyobtainedfromthese plasticstrainratiosasfollows: R = R L = d 22 d 33 = H G and Y T 1 2 =1= G + H frombefore,suchthat, G = F = 1 R +1 and H = R R +1 : Also, R h = R TS = 2 M )]TJ/F15 11.9552 Tf 11.956 0 Td [( H + G 2 H + G : 245

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B.2DetermineTheCPB06CoecientsGivenTheHill48Coecients IfasetofHill48coecientsaregiven,bothEquationsB{7andB{8mustbe usedtodeterminetheCPB06coecients.Neglectingthetediousalgebraonceagain,the followingrelationsareobtained: L 11 = 1 2 + 1 4 r 8 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 G G +1 )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 G +1 )]TJ/F26 11.9552 Tf 11.955 10.221 Td [(p 3 G G +1 3 G +1 L 13 = G +1 )]TJ/F26 11.9552 Tf 11.955 10.222 Td [(p 3 G G +1 3 G +1 L 44 = r M G +1 : B{10 ThelastofthepreviousequationsyieldsseeEquationB{7 ^ m 2 = G +1B{11 fortransverseisotropy. 246

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Ross,S.L.,1984.DierentialEquations,3rdEdition.JohnWiley&Sons,Inc,NewJersey. Rousselier,G.,1987.Ductilefracturemodelsandtheirpotentialinlocalapproachof fracture.NuclearEngineeringandDesign105,97{111. Simo,J.C.,Hughes,T.J.R.,1998.ComputationalInelasticity.Springer,NewYork. Spitzig,W.A.,Smelser,R.E.,Richmond,O.,1988.Theevolutionofdamageandfracture inironcompactswithvariousinitialporosities.ActaMetallurgica36,1201{1211. Talbot,D.R.S.,Willis,J.R.,1985.Variationalprinciplesforinhomogeneousnon-linear media.IMAJournalofAppliedMathematics35,39{54. Tvergaard,V.,1980.Onlocalizationinductilematerialscontainingsphericalvoids. InternationalJournalofFracture18,237{252. Tvergaard,V.,1981.Inuenceofvoidsonshearbandinstabilitiesunderplanestrain conditions.InternationalJournalofFracture17,389{407. Tvergaard,V.,1982.Onlocalizationinductilematerialscontainingsphericalvoids. InternationalJournalofFracture18,237{252. Tvergaard,V.,Needleman,A.,1984.Analysisofthecup-conefractureinaroundtensile bar.ActaMetallurgica32,157{169. Vitek,V.,Mrovec,M.,Bassani,J.L.,2004.Inuenceofnon-glidestressesonplasticow: fromatomistictocontinuummodeling.MaterialsScienceandEngineeringA365,31{37. Zwillinger,D.Ed.,2003.StandardMathematicalTablesandFormulae,31stEdition. Chapman&Hall/CRCPressLLC,NewYork. 250

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BIOGRAPHICALSKETCH InMayof2002,JoelBenjaminStewartgraduatedfromtheUniversityofFloridawith aBachelorofScienceinaerospaceengineering.HebeganworkinginOctoberof2002for theAirForceResearchLaboratoryatEglinAirForceBaseinFlorida.InDecemberof 2005whileworkingatEglin,hecompletedaMasterofEngineeringdegreeinaerospace engineeringattheResearchandEngineeringEducationFacilityinShalimar,Florida.Joel StewartwasadmittedinAprilof2008,whileworkingasafull-timegraduatestudenton anAirForcefellowship,asaUniversityofFloridadoctoralcandidateintheDepartmentof MechanicalandAerospaceEngineering. 251