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Experimental Characterization and Constitutive Modeling of the High-Pressure Behavior of Dry Sand

Permanent Link: http://ufdc.ufl.edu/UFE0042783/00001

Material Information

Title: Experimental Characterization and Constitutive Modeling of the High-Pressure Behavior of Dry Sand
Physical Description: 1 online resource (174 p.)
Language: english
Creator: MARTIN,BRADLEY ERIC
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: CONFINEMENT -- CONSTITUTIVE -- DYNAMIC -- ELASTIC -- GRANULAR -- HIGH -- MODELING -- PRESSURE -- SAND -- SHPB -- VISCOPLASTIC
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: Key in predicting the stability of a rigid body penetrating sand is gaining fundamental understanding of the physics of deformation and failure at high pressures. However, existing mechanical data and constitutive models are based on low pressures and strain rates, therefore they are applicable mainly to civil engineering applications. The main goal of this dissertation was to extend the knowledge on the mechanical response of sand, to the high pressure (up to 700 MPa) range, which may be encountered in penetration events. To this end, an experimental investigation was conducted on Quikrete #1961 sand and the data used to develop a constitutive model able to describe the observed behavior. For Quikrete #1961 sand, hydrostatic tests were conducted up to 0.5 GPa allowing for accurate determination of the dependence of the bulk modulus on pressure and the correct estimation of the material's compaction properties when subjected to the high pressures in the range encountered in dynamic events. Triaxial compression tests were conducted at a strain-rate of 10^-5 s^-1, and for confining pressures ranging from 0 to 0.3 GPa. During all triaxial compression tests the material exhibited hardening up to failure while both compressibility and dilatancy regimes of the volumetric behavior were observed. Furthermore, the transition from compressibility to dilatancy was found to be highly dependent on the level of confinement. To investigate the influence of loading rate on the material behavior confined Kolsky bar experiments were conducted at a strain rate of 1000 s^-1. The confined Kolsky bar imposes a constant confining pressure on the specimen through implementation of a high pressure confining cell at the specimen/bar interface. Thus, the influence of confining pressure on the dynamic response of the material is investigated for confining pressures ranging from 0 to 0.125 GPa. For all confined Kolsky bar tests no strain rate sensitivities were observed while a clear pressure dependence was observed. Furthermore, in all confined Kolsky bar tests the material exhibited a higly non-linear response with hardening up to the end of loading. For the first time, high quality data that show the influence of confining pressure on the dynamic response were obtained while extending the existing experimental database on sand (the confinements reported in the literature being rather small (less than 6 MPa)). A new elastic/viscoplastic model that captures the compressibility and dilatancy, as well as strain rate effects has been developed for sand. No a priori assumptions regarding the specific mathematical expressions of the yield function or viscoplastic potential were imposed. It was demonstrated that an associated flow rule doesn`t apply. Thus, a new flow rule has been proposed for Quikrete #1961 sand. Comparison between model predictions and data showed the proposed model describes very well the high-pressure behavior of Quikrete #1961 sand.
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 BRADLEY ERIC MARTIN.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042783:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042783/00001

Material Information

Title: Experimental Characterization and Constitutive Modeling of the High-Pressure Behavior of Dry Sand
Physical Description: 1 online resource (174 p.)
Language: english
Creator: MARTIN,BRADLEY ERIC
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: CONFINEMENT -- CONSTITUTIVE -- DYNAMIC -- ELASTIC -- GRANULAR -- HIGH -- MODELING -- PRESSURE -- SAND -- SHPB -- VISCOPLASTIC
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: Key in predicting the stability of a rigid body penetrating sand is gaining fundamental understanding of the physics of deformation and failure at high pressures. However, existing mechanical data and constitutive models are based on low pressures and strain rates, therefore they are applicable mainly to civil engineering applications. The main goal of this dissertation was to extend the knowledge on the mechanical response of sand, to the high pressure (up to 700 MPa) range, which may be encountered in penetration events. To this end, an experimental investigation was conducted on Quikrete #1961 sand and the data used to develop a constitutive model able to describe the observed behavior. For Quikrete #1961 sand, hydrostatic tests were conducted up to 0.5 GPa allowing for accurate determination of the dependence of the bulk modulus on pressure and the correct estimation of the material's compaction properties when subjected to the high pressures in the range encountered in dynamic events. Triaxial compression tests were conducted at a strain-rate of 10^-5 s^-1, and for confining pressures ranging from 0 to 0.3 GPa. During all triaxial compression tests the material exhibited hardening up to failure while both compressibility and dilatancy regimes of the volumetric behavior were observed. Furthermore, the transition from compressibility to dilatancy was found to be highly dependent on the level of confinement. To investigate the influence of loading rate on the material behavior confined Kolsky bar experiments were conducted at a strain rate of 1000 s^-1. The confined Kolsky bar imposes a constant confining pressure on the specimen through implementation of a high pressure confining cell at the specimen/bar interface. Thus, the influence of confining pressure on the dynamic response of the material is investigated for confining pressures ranging from 0 to 0.125 GPa. For all confined Kolsky bar tests no strain rate sensitivities were observed while a clear pressure dependence was observed. Furthermore, in all confined Kolsky bar tests the material exhibited a higly non-linear response with hardening up to the end of loading. For the first time, high quality data that show the influence of confining pressure on the dynamic response were obtained while extending the existing experimental database on sand (the confinements reported in the literature being rather small (less than 6 MPa)). A new elastic/viscoplastic model that captures the compressibility and dilatancy, as well as strain rate effects has been developed for sand. No a priori assumptions regarding the specific mathematical expressions of the yield function or viscoplastic potential were imposed. It was demonstrated that an associated flow rule doesn`t apply. Thus, a new flow rule has been proposed for Quikrete #1961 sand. Comparison between model predictions and data showed the proposed model describes very well the high-pressure behavior of Quikrete #1961 sand.
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 BRADLEY ERIC MARTIN.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Cazacu, Oana.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042783:00001


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EXPERIMENTALCHARACTERIZATIONANDCONSTITUTIVEMODELINGOF THEHIGH-PRESSUREBEHAVIOROFDRYSAND By BRADLEYE.MARTIN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011BradleyE.Martin 2

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Iwouldliketodedicatethisworktomywife,Jessica,whohasgivenmenothingbut supportduringmycareerasagraduatestudent.Sheenduredmanynightsandweekends aloneandwhenoursonwasbornwasrequiredtowatchhimbyherselfandforthisIwill alwaysbegrateful. 3

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ACKNOWLEDGMENTS Thelistofpersonswhohaveinuencedmylifeandthisresearchistoolongto properlydocument,however,Imustthankanumberofpeople.Firstandforemost, Iwouldliketoextendmuchgratitudetomyadvisor,ProfessorOanaCazacuforher support,patience,andgreatfriendshipthroughoutthisstudy.IhaveknownProfessor Cazacuformanyyearsandshehasprovidedmanyinvaluablecommentsanddirection inthecompletionofthisworkandthroughoutmygraduateeducation.Iwouldalsolike tothankProfessorWeinongChenofPurdueUniversityforbeingsuchagoodfriendand supporterconcerningmyprofessionalgrowth.IhavetoextendmanythankstoProfessor Chenforallowingmetocontinouslyusehislabequipment.ImustrecognizeProfessor IoanIonescuforallofhishelpandworkwithimplementingthemodelintothecode FreeFEM.Hewasverypatientwithmewhilelearningtheintricaciesofthenumerical workandIamtrulygratefulforthis.Lastly,IwouldliketothankProfessorsBucklin, Kumar,andBoginskifortherekindwordsandmanyexcellentcommentsconcerningthis research.Itrulyappreciateallofthereeortandsupport. Iwouldliketothankmywifeandbestfriendforbeingsosupportiveandmynumber onefanduringmyacademiccareer.Iamespeciallythankfultoherforsupportingme throughallthetripstoPurdueUniversityandlongeveningsworkingwhileshewatched ourson.Iwouldalsoliketothankmymotherforpushingmesohardtoattendcollege andforreiteratingmanytimestheimportanceofaneducation.Iwouldalsoliketo acknowledgethestrongsupportofmyemployersattheAirForceResearchLaboratory, MunitionsDirectorate,inparticularDr.NormKlausutis.IamverythankfultoDr. Klausutisforhisfriendshipandcontinuoussupportofmyacademicgrowth. Icouldnothaveaccomplishedthisresearchwithoutthecontributionsofmanyofmy professionalcolleagues.FirstmycolleaguesfromtheAirForceResearchLaboratory.Mr. MarkGreenprovidedmemanyinsightsintothebehaviorofgeomaterialsaswellasthe modelingofthesematerials.Hehasbeenagreatinuenceonmycareerandestablished 4

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formetheimportanceofdoinggoodresearchnomatterwhatitrequiresandworryabout everythingelselatter.Dr.BrianPlunkettforhisrelentlesshoursdiscussingcomputational andtheoreticalmodelingandforthemanynon-workrelatedconversations.Dr.Joel Stewartforhismanyphilosophicaldiscussions,butmainlyforhisinvaluableinsightsinto theoreticalmodelingaswellasboostingmycondencewhenneeded.Dr.JoelHousefor hissteadfastsupportandhonestcritiqueswerealwaysappreciated.Iwouldliketoextend specialgratitudetoDr.StephenAkersoftheEngineeringResearchandDevelopment Center,Vicksburg,MSforhismanyhoursofconversationsconcerningmodelingof geomaterialsandinsightfuldiscussions. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................8 LISTOFFIGURES....................................9 NOMENCLATURE....................................16 ABSTRACT........................................19 CHAPTER 1INTRODUCTION..................................21 1.1ProblemBackground..............................21 1.1.1RateIndependentModels........................21 1.1.2RateDependentModels........................30 2QUASI-STATICBEHAVIOROFQUIKRETE R ] 1961SAND..........34 2.1OverviewofQuasi-staticTestingTechniques.................34 2.2MaterialProperties,Quikrete R ] 1961.....................38 2.3ExperimentalInvestigation...........................40 2.3.1TestMatrix...............................42 2.3.2SpecimenPreparation..........................42 2.3.3InstrumentationoftheQuikrete R SandSpecimen..........46 2.3.4TestDevices...............................48 2.4ExperimentalResults..............................48 2.4.1500MPaHydrostaticCompression...................49 2.4.220MPaCTCExperiment.......................49 2.4.350MPaCTCExperiment.......................52 2.4.4100MPaCTCExperiment.......................53 2.4.5150MPaCTCExperiment.......................55 2.4.6200MPaCTCExperiment.......................57 2.4.7300MPaCTCExperiment.......................59 2.4.8FailureSurface&Compressibility/DilatancyBoundary.......62 2.5EvaluationofElasticParameters.......................64 3DYNAMICBEHAVIOROFQUIKRETE R ] 1961SAND.............69 3.1Split-HopkinsonPressureBarorKolskyBarTechnique...........69 3.1.1Background...............................69 3.1.2AnalysisoftheDataUsing1-DStressWaveTheory.........72 3.1.3DynamicTriaxialKolskyBar......................76 3.2DynamicExperimentalProgram........................79 6

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3.3ExperimentalSetup...............................79 3.3.1ConnedKolskyBar..........................79 3.3.2MaterialProperties,Quikrete R ] 1961Sand..............84 3.3.3SpecimenPreparation..........................84 3.4ConnedKolskyBarExperimentsonQuikrete R ] 1961Sand........85 4APPLICATIONOFLADEANDDUNCAN'SMODELTOSAND...96 4.1LadeandDuncanElastic-PlasticModel................96 4.2ParameterIdentication............................99 4.3ApplicationofLadeModeltoQuikrete R Sand............101 4.4Discussion....................................102 5CONSTITUTIVEMODELING...........................106 5.1Elastic-ViscoplasticModelDevelopment...................106 5.2ElasticParameterEvaluation.........................108 5.3YieldFunctionEvaluation...........................109 5.4StrainRateOrientationTensor........................117 5.5StrainRateOrientationTensork T N 1 p ; q .................118 5.6StrainRateOrientationTensork T N 2 p ; q .................133 6SUMMARYANDVALIDATIONOFANELASTIC-VISCOPLASTICMODEL150 6.1Elastic-ViscoplasticModelSummary.....................150 6.2IntegrationAlgorithm.............................160 6.3NumericalSimulationofQuasi-staticCTCTests...............162 7CONCLUSIONS...................................167 REFERENCES.......................................169 BIOGRAPHICALSKETCH................................174 7

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LISTOFTABLES Table page 2-1PropertiesofQuikrete R ] 1961negrainsand....................39 2-2Quasi-staticexperimentmatrixforQuikrete R sand................44 3-1VoidratioandrelativedensityofQuikrete R sand.................84 3-2DynamictestingconditionsforQuikrete R sand..................86 4-1ParametersofLade1975forQuikrete R ] 1961sand.................102 5-1Numericalvaluesofparametersinvolvedintheevolutionlaw5{7forYoung's Modulus, E .......................................109 5-2Peakvaluesof k T N 1 frommonotonicCTCtests...................122 5-3Average k T N 2 valuesfor p;q ............................140 6-1Materialparametervaluesfortheproposedelastic-viscoplasticmodel.......157 8

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LISTOFFIGURES Figure page 1-1Mohr-Coulombyieldsurface..............................22 1-2ModiedDrucker-PragerCapyieldsurface.....................23 1-3VolumeexpansionpredictedbyDruckerandPrager1952modelwithassociated owrule........................................24 1-4Uniaxialstrain-constantvolumeexperimentofQuikrete R ] 1961..........27 1-5RepresentationoftheCapmodelofDiMaggioandSandler1971inthedeviatoric plane..........................................28 2-1Strain-rateregimesfortesting............................35 2-2Hydraulicallydrivenloadframe...........................36 2-3Karmantestingdevice................................37 2-4ParticlesizedistributionforQuikrete R ] 1961....................39 2-5MonotonictriaxialcompressiontestofQuikrete R sandshowinghystereticresponse 3 =150MPa....................................42 2-6HydrostaticcreeptestforQuikrete R sand.....................43 2-7Triaxialcompressioncreeptestduringdeviatoricphaseforconningpressure 3 =50MPa......................................43 2-8Evolutionofvolumetricstrain-ratewithtimemeasuredinacreeptestonQuikrete R sandatapressureof75MPa.............................45 2-9SpecimenofQuikrete R ] 1961sand..........................46 2-10HydrostaticcompressionexperimentonQuikrete R sandupto500MPa.....49 2-11Comparisonbetweenaxialandradialstrainsmeasuredinhydrostaticcompression testes2b14forpressures300to500MPa......................50 2-12Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =20MPaforQuikrete R sand................51 2-13Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =20MPaforQuikrete R sand...........................51 2-14Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =20MPaforQuikrete R sand.....52 9

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2-15Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =50MPaforQuikrete R sand................53 2-16Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =50MPaforQuikrete R sand...........................54 2-17Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =50MPaforQuikrete R sand.....54 2-18Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =100MPaforQuikrete R sand................55 2-19Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =100MPaforQuikrete R sand..........................56 2-20Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =100MPaforQuikrete R sand.....56 2-21Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =150MPaforQuikrete R sand................57 2-22Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =150MPaforQuikrete R sand..........................58 2-23Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =150MPaforQuikrete R sand.....58 2-24Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =200MPaforQuikrete R sand................59 2-25Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =200MPaforQuikrete R sand..........................60 2-26Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =200MPaforQuikrete R sand.....60 2-27Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtestat conningpressure 3 =300MPaforQuikrete R sand................61 2-28Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconningpressure 3 =300MPaforQuikrete R sand..........................62 2-29Comparisonbetweenthevolumetricresponseinthedeviatoricphaseofmonotonic andcyclictestsatconningpressure 3 =300MPaforQuikrete R sand.....63 2-30Compressibility/DilatancyboundaryforQuikrete R ] 1961sand..........64 2-31FailureSurfaceforQuikrete R ] 1961sand......................65 2-32Unloading-reloadingcurvefor 3 =50MPa.....................66 10

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2-33ExperimentalvariationofYoung'sModuluswiththemeanstress.........66 2-34ExperimentalvariationoftheShearmoduluswiththemeanstress........67 2-35ExperimentalvariationoftheBulkmoduluswiththemeanstress.........68 3-1SchematicofHopkinson'sapparatus........................71 3-2Kolskybarset-up...................................73 3-3SchematicrepresentationoftheKolskybar.....................73 3-4PhotographoftheconningcelldevelopedbyMalvernandJenkins1990....78 3-5SchematicoftheconnedKolskybarset-upusedintheexperimentsonQuikrete R sand...........................................81 3-6ConnedKolskybarradialandaxialconningcells................81 3-7ConnedKolskybarapparatusatPurdueUniversity,WestLafayette,IN.....82 3-8ConnedKolskybarTie-rodreactionplate.....................82 3-9SpecimenfabricationandinstallationforConnedKolskytests.........85 3-10ConnedKolskybarhydrostaticcompressionexperimentonQuikrete R sandup to125MPa.......................................87 3-11ComparisonbetweenhydrostaticcompressionexperimentsonQuikrete R sand.87 3-12Osetofincidentandtransmittedsignalsfor 3 =25MPaconnement.....89 3-131-and2-wavestressprolesforapulse-shapedexperimentonQuikrete R sand for 3 =25MPa...................................90 3-14RadialdeformationanddynamicpressuremeasurementsinaconnedKolsky bartestfor 3 =25MPa...............................91 3-15Stresspathsplottedinthedeviatoricplane p;q foralltestsperformed.....91 3-16Comparisonbetweenthequasi-staticanddynamicstress-strainresponseinthe deviatoricphaseoftheCTCtestsatconningpressure 3 =25MPaforQuikrete R sand..........................................93 3-17Comparisonbetweenthequasi-staticanddynamicstress-strainresponseinthe deviatoricphaseoftheCTCtestsatconningpressure 3 =50MPaforQuikrete R sand..........................................93 3-18Comparisonbetweenthequasi-staticanddynamicstress-strainresponseinthe deviatoricphaseoftheCTCtestsatconningpressure 3 =100MPaforQuikrete R sand..........................................94 11

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3-19Dynamicstress-strainresponseinthedeviatoricphaseoftheCTCtestsatconning pressure 3 =125MPaforQuikrete R sandasobtainedinthetwotestsconducted.94 3-20Dynamicstressstrainresponseduringdeviatoricloadingforconningpressures of25,50,100,and125MPaforQuikrete R sand..................95 4-1Variationof 2 withstresslevel, f ..........................100 4-2Stressratio, f ,asafunctionofirreversiblework, W p ................101 4-3 f )]TJ/F21 11.9552 Tf 9.299 0 Td [(f t asafunctionofirreversiblework, W p :Comparisonbetweendatasymbols andtheoreticalvariationEq.4{11........................102 4-4Principalstressdierencevs.volumetricstrain A =0 : 818............103 4-5Principalstressdierencevs.volumetricstrain A =0 : 400............104 5-1Comparisonoftheoreticalandexperimentalvariationsoftheelasticparameters withmeanstress...................................110 5-2Stabilizationboundaryschematic..........................110 5-3Stabilizationboundary H H asafunctionofmeanstress..............112 5-4Irreversiblestressworkatstabilizationasafunctionoftheprincipalstressdierence calculatedfromthedeviatoricphaseofCTCtestsat 3 =20-30MPa.......113 5-5Deviatoricyieldsurfaces, H D p;q ,asafunctionofprincipalstressdierence..114 5-6Compressibility-Dilatancyboundary........................115 5-7Failuresurface....................................116 5-8YieldlociforQuikrete R ] 1961sand.........................116 5-9Variationofthevolumetricirreversiblestrain-rateorientationtensor, k T N 1 ,with pressureasobtainedfromhydrostatictests.....................119 5-10ComparisonbetweentheoreticalEq.5{29andexperimentfortheirreversible volumetricorientationtensor, k T N 1 ,forhydrostaticloading...........120 5-11Experimentalvaluesoftheirreversiblevolumetricorientationtensor, k T N 1 ,calculated forthedeviatoricphaseoftheCTCtestfor 3 =300MPa.............120 5-12Experimentalvaluessmoothedofthestrain-rateorientationtensor, k T N 1 ,calculated forthedeviatoricphasefromalltests........................121 5-13AxialstrainasafunctionofprincipalstressdierenceforallCTCtestswith superimposedpeakvalues..............................122 12

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5-14RadialstrainasafunctionofprincipalstressdierenceforallCTCtestswith superimposedpeakvalues..............................123 5-15Irreversibleaxialstrain-rateasafunctionofprincipalstressdierenceforall CTCtestswithsuperimposedpeakvalues.....................124 5-16Irreversibleradialstrain-rateasafunctionofprincipalstressdierenceforall CTCtestswithsuperimposedpeakvalues.....................124 5-17Comparisonbetweenexperimentandmodelforthevariationof 0 withconning pressure........................................126 5-18Comparisonbetweenexperimentandmodelforthevariationof 1 withconning pressure........................................126 5-19Comparisonbetweenexperimentandmodelforthevariationof 2 withconning pressure........................................127 5-20Comparisonbetweenexperimentandmodelforthevariationof 0 withconning pressure........................................128 5-21Comparisonbetweenexperimentandmodelforthevariationof 1 withconning pressure........................................129 5-22Experimentalvaluesforthecoecients 0 and 2 ofEq.5{39..........130 5-23Experimentalvaluesforthecoecients 1 and 3 ofEq.5{39..........130 5-24Experimentalvaluesforthescaledcoecients 1 0 and 3 2 involvedinEq.5{39.131 5-25Comparisonbetweentheoreticalandexperimentalvariationof 0 withconning pressure........................................131 5-26Comparisonbetweenexperimentalandtheoreticalvariationof 2 ........134 5-27Comparisonbetweenexperimentandtheoreticalvariationof 1 0 .........134 5-28Comparisonbetweenexperimentandtheoreticalvariationof 3 2 .........135 5-29Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =20MPa.........135 5-30Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =50MPa.........136 5-31Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =100MPa........136 13

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5-32Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =150MPa........137 5-33Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =200MPa........137 5-34Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =300MPa........138 5-35Variationoftheshearirreversiblestrainratetensor, k T N 2 ,withloadfor 3 = 300MPa........................................138 5-36Experimentalvaluesoftheshearirreversiblestrain-rateorientationtensor, k T N 2 versusprincipalstressdierencecalculatedfromalltests..............139 5-37Average k T N 2 valuesfor p;q ...........................140 5-38Comparisonbetweentheoreticalandexperimentalvariationof 0 withconning pressure........................................142 5-39Comparisonbetweentheoreticalandexperimentalvariationof 1 withconning pressure........................................142 5-40Comparisonbetweentheoreticalandexperimentalvariationof 2 withconning pressure........................................143 5-41Comparisonbetweentheoreticalandexperimentalvariationof 0 withconning pressure........................................145 5-42Comparisonbetweentheoreticalandexperimentalvariationof 1 withconning pressure........................................145 5-43Comparisonbetweentheoreticalandexperimentalvariationof 2 withconning pressure........................................146 5-44Comparisonbetweentheoreticalandexperimentalvariationof 3 withconning pressure........................................146 5-45Comparisonbetweenexperimentalandtheoreticalvariationoftheshearirreversible strainratetensor, k T N 2 ,withloadfor 3 =50MPa................147 5-46Comparisonbetweenexperimentalandtheoreticalvariationoftheshearirreversible strainratetensor, k T N 2 ,withloadfor 3 =100MPa...............147 5-47Comparisonbetweenexperimentalandtheoreticalvariationoftheshearirreversible strainratetensor, k T N 2 ,withloadfor 3 =150MPa...............148 5-48Comparisonbetweenexperimentalandtheoreticalvariationoftheshearirreversible strainratetensor, k T N 2 ,withloadfor 3 =200MPa...............148 14

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5-49Comparisonbetweenexperimentalandtheoreticalvariationoftheshearirreversible strainratetensor, k T N 2 ,withloadfor 3 =300MPa...............149 6-1Comparisonbetweenexperimentalandtheoreticalresultsforhydrostaticloading ofQuikrete R sand...................................163 6-2Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =20MPa.........................164 6-3Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =50MPa.........................164 6-4Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =100MPa.........................165 6-5Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =150MPa.........................165 6-6Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =200MPa.........................166 6-7Comparisonbetweenexperimentalandtheoreticalresultsfordeviatoricloading ofQuikrete R sandat 3 =300MPa.........................166 15

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NOMENCLATURE A incident/transmissionbararea[m 2 ] A s specimenarea[m 2 ] c 0 steelwavespeed[ms )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ] E Young'sModulus[GPa] E 1 limitvalueof E [GPa] E s initialvalueof E [GPa] G ShearModulus[GPa] H yieldfunction[MPa] I 1 Firstinvariantofstress[MPa] J 2 Secondinvariantofdeviatoricstress[MPa] k Cohesion[MPa] K BulkModulus[GPa] k T Viscositycoecient[-] l s instantaneousspecimenlength[m] N viscoplasticstrain-rateorientation[-] p meannormalstress[MPa] p c characteristicpressure[MPa] q principalstressdierence[MPa] t time[seconds] T r reectedwavestartingtime[seconds] T s transittimethroughspecimen[seconds] 16

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T t transmittedwavestartingtime[seconds] u 1 displacementofincidentbar[m] u 2 displacementoftransmitterbar[m] u i displacementfromincidentwave[m] u r displacementofreectedwave[m] u t displacementoftransmittedwave[m] v 1 particlevelocityofincidentbar[ms )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ] v 2 particlevelocityofincidentbar[ms )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ] W t irreversiblestressworkperunitvolume[MPa] x 1-DLagrangianspatialcoordinate[m] X p;q compressibility/dilatancyboundaryfunction[MPa] angleofinternalfriction[-] KroneckerDelta[-] straintensor[-] 0 straindeviatortensor[-] v volumetricstrain[-] i strainfromincidentwave[-] r strainfromreectedwave[-] t strainfromtransmittedwave[-] s specimenstrain[-] strainrate[s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ] s specimenstrainrate[s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ] 17

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Poisson'sratio[-] stresstensor[MPa] 0 stressdeviatortensor[MPa] s specimenstress[MPa] 1 axialstress[MPa] 2 radialstress[MPa] 3 radialstress[MPa] 18

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy EXPERIMENTALCHARACTERIZATIONANDCONSTITUTIVEMODELINGOF THEHIGH-PRESSUREBEHAVIOROFDRYSAND By BradleyE.Martin May2011 Chair:OanaCazacu Major:MechanicalEngineering Keyinpredictingthestabilityofarigidbodypenetratingsandisgainingfundamental understandingofthephysicsofdeformationandfailureathighpressures.However, existingmechanicaldataandconstitutivemodelsarebasedonlowpressuresandstrain rates,thereforetheyareapplicablemainlytocivilengineeringapplications. Themaingoalofthisdissertationistoextendtheknowledgeonthemechanical responseofsand,tothehighpressureupto700MParange,whichmaybeencountered inpenetrationevents.Tothisend,anexperimentalinvestigationwasconductedon Quikrete R ] 1961sandandthedatausedtodevelopaconstitutivemodelabletodescribe theobservedbehavior. ForQuikrete R ] 1961sand,hydrostatictestswereconductedupto0.5GPaallowing foraccuratedeterminationofthedependenceofthebulkmodulusonpressureandthe correctestimationofthematerial'scompactionpropertieswhensubjectedtothehigh pressuresintherangeencounteredindynamicevents.Triaxialcompressiontestswere conductedatastrain-rateof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ,andforconningpressuresrangingfrom0to 0.3GPa.Duringalltriaxialcompressionteststhematerialexhibitedhardeningupto failurewhilebothcompressibilityanddilatancyregimesofthevolumetricbehaviorwere observed.Furthermore,thetransitionfromcompressibilitytodilatancywasfoundtobe highlydependentonthelevelofconnement. 19

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ToinvestigatetheinuenceofloadingrateonthematerialbehaviorconnedKolsky barexperimentswereconductedatastrainrateof1000s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .TheconnedKolskybar imposesaconstantconningpressureonthespecimenthroughimplementationofahigh pressureconningcellmountedoverthetestingsectionwherethespecimenislocated. Thus,theinuenceofconningpressureonthedynamicresponseofthematerialwas investigatedforconningpressuresrangingfrom0to0.125GPa.ForallconnedKolsky bartestsnostrainratesensitivitieswereobservedwhileaclearpressuredependencewas observed.Furthermore,inallconnedKolskybarteststhematerialexhibitedahigly non-linearresponsewithhardeninguptotheendofloading.Forthersttime,high qualitydatathatshowtheinuenceofconningpressureonthedynamicresponsewere obtainedwhileextendingtheexistingexperimentaldatabaseonsandtheconnements reportedintheliteraturebeingrathersmalllessthan6MPa. Anewelastic-viscoplasticmodelthatcapturesthecompressibilityanddilatancy, aswellasstrainrateeectshasbeendevelopedforsand.No apriori assumptions regardingthespecicmathematicalexpressionsoftheyieldfunctionorviscoplastic potentialwereimposed.Itwasdemonstratedthatanassociatedowruledoesn`t apply.Thus,anewowrulehasbeenproposedforQuikrete R ] 1961sand.Comparison betweenmodelpredictionsanddatashowedtheproposedmodeldescribesverywellthe high-pressurebehaviorofQuikrete R ] 1961sand. 20

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CHAPTER1 INTRODUCTION Aconstitutivemodelconsistsofasetofequationsthatdescribesthetotalstress-strain responseofagivenmaterial.Aconstitutivemodelisconstructedsuchastorepresentthe behaviorforacertainrangeofstress,strain,andstrain-rateconditions.Inthischapter, areviewofsoilcontinuumconstitutivemodelswillbeprovided.Thediscussionwill encompassawidespectrumofmodelsincluding:perfectly-plastic,elastic-plastic,and elastic-viscoplasticmodels. 1.1ProblemBackground 1.1.1RateIndependentModels Themodelingofsoilsbeganin1773withtheMohr-Coulombfailurecriterion.The rststress-strainmodeldevelopedintheframeworkofplasticitytheorywasproposedin the1950'sbyD.C.Drucker.Thesemodelsweredevelopedtopredict/estimatethestability ofslopes,bearingcapacitiesoffoundations,andpressuresonretainingwalls.Thus,the eectsofthehydrostaticstressonthesoilresponsehadtobeaccountedfor.Accordingto theMohr-Coulombtheory,failureoccurswhen, = c )]TJ/F21 11.9552 Tf 11.955 0 Td [( n tan {1 where istheshearstressonthefailureplane, n thenormalstressonthatplane,while and c arematerialparameterscalledangleofinternalfrictionandcohesion,respectively. Intermsofprincipalstressesthecriterionwrites: f = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 3 + 1 + 3 sin )]TJ/F15 11.9552 Tf 11.956 0 Td [(2 c cos {2 where 1 and 3 arethemaximumandminimumprincipalstresses,respectively.Inthe plane I 1 ;J 2 ,where I 1 =tr {3 21

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Figure1-1.Mohr-Coulombyieldsurface. and J 2 = 1 2 tr 0 2 : {4 Equation1{2representsastraightlineofslopegivenby andinterceptwiththe deviatoricaxis I 1 =0givenby c "Figure1-1.Thus,thedeterminationofthese parametersfromconventionaltriaxialcompressiontestsCTCisstraightforwardi.e. failuredatacorrespondingtoCTCtestsatvariousconningpressuresareapproximated withtheexpression1{1. TheDrucker-Pragermodelimpliesperfectly-plasticresponse,i.e.theyieldsurfaceis xedinstressspaceandcoincideswiththefailuresurfaceDruckerandPrager1952. Theexpressionofthissurfaceisgivenby: f = p J 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(I 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k {5 where and k arematerialconstantswhiletheinvariants I 1 and J 2 aregivenbyEqs. 1{3and1{4,respectively.NotethatthismodelisanextensionoftheclassicalVon Misescriterionsuchastoincludetheeectsoftherstinvariantofstress.Equation1{5 mayalsobewrittenintermsofprincipalstressesbysubstitutingthedenitionsfor J 2 and 22

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I 1 intoEq.1{5. f = r 1 3 [ 2 1 + 2 2 + 2 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 3 ] )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 + 2 + 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [(k {6 Forconnedtriaxialcompression 1 ; 2 = 3 ,failureoccursif 1 p 3 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 + )]TJ/F15 11.9552 Tf 15.475 8.088 Td [(1 p 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 3 = k: {7 Thematerialparameters and k areexpressedintermsoftheangleofinternalfriction andcohesivestrength c involvedinEq.1{1fortheMohr-Coulombcriterion. = 2sin )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin p 3 k = 6 c cos )]TJ/F15 11.9552 Tf 11.955 0 Td [(sin p 3 Hence,theseparametersmaybedeterminedfromfailuredataobtainedinCTCtestssee thediscussionontheevaluationofthematerialparametersinEq.1{1.Theprojection oftheDrucker-PragerfailuresurfaceABAQUS2008intheplane p;J 2 ,where p denotesthemeanstress p = I 1 = 3,isshowninFigure1-2.NotethatMohr-Coulomb Figure1-2.ModiedDrucker-PragerCapyieldsurface.[RecreatedfromABAQUSVersion 6.8Referencemanuals,2008.Providence,RI.] 23

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andDrucker-Pragerfailuresurfacesareopenalongthehydrostaticaxis.However, experimentallyitwasobservedthatforlargevaluesofpressurewhenmostofthepores havebeenclosedfullycompactedstate,thesoilresponseshouldoccurwithoutfurther volumetricdeformatione.g.,dataonYumaclayeysandreportedbyRubin1990.In otherwords,theasymptoticresponseshouldbedescribedbyavonMisesexpression. Despitetheselimitations,theMohr-CoulombandDrucker-Pragerarestillthemostwidely usedmodelsfordescribingfailureofsoils.WhentheDrucker-PragerandMohr-Coulomb modelsareusedtodescribethedirectionofplasticow,thenonlyvolumeexpansionor dilatancyofthematerialcanbepredictedFigure1-3.Also,formostsoilsthepredicted levelsofdilatancyarefarinexcessofwhatisobservedexperimentally.Asalready Figure1-3.VolumeexpansionpredictedbyDruckerandPrager1952modelwith associatedowrule. mentioned,theDrucker-Pragermodelisperfectly-plastic,i.e.theyieldlocusisxed anddoesnotevolvewithaccumulateddeformation.Laterin1957,Druckerdeveloped amodeltoincludeaxedfailuresurfaceandmovablehardeningyieldsurfaceDrucker etal.1957,referredtoascapFigure1-2.ThexedfailuresurfaceisaDrucker-Prager surfaceandgivenbyEq.1{5withthecapallowedtoexpandorcontractasafunctionof 24

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thevolumetricplasticstrain.Thecapfunctionisexpressedas, F c = s p )]TJ/F21 11.9552 Tf 11.955 0 Td [(p a 2 + R p J 2 1+ )]TJ/F21 11.9552 Tf 11.956 0 Td [( cos 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(R k + p a tan ; {8 where R isamaterialparameterinuencingtheshapeofthecap, denesasmall numberthatfacilitatesdeningthetransitionsurface F t inFigure1-2betweenthe failuresurfaceandthecap,and p a denesthecenteroftheellipseandisafunctionof volumeplasticstrain.Theexpressionsforthetransitioningsurface, F t ,andparameter, p a aredenedas F t = s p )]TJ/F21 11.9552 Tf 11.955 0 Td [(p a 2 + p J 2 )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( 1 )]TJ/F21 11.9552 Tf 20.862 8.088 Td [( cos k + p a tan 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( k + p a tan {9 and p a = p b )]TJ/F21 11.9552 Tf 11.955 0 Td [(Rk + R tan : {10 InEq.1{10, p b givesthenalvalueofhydrostaticstresspriortoshearloadinginCTC testsandisafunctionofvolumeplasticstraini.e. p b p v .Thehardeningiscaptured duringshearloadingimplicitlythroughtheparameter R ,whichdescribestheratio betweenthemajorandminoraxesoftheellipse.Themajoraxismaybedenedbythe variable a whiletheminoraxisisgivenby b .Theanalyticalformsofthemajorandminor axesaredeterminedfromFigure1-2andexpressedas, a = p b )]TJ/F21 11.9552 Tf 11.955 0 Td [(p a b = k + p b tan : Therefore,Eq.1{10directlyfollowsfromtheequationoftheratio, R = a=b .This ratiocanbedescribedexplicitythroughexperimentsorimplicitlythroughaniterative methodintheabsenceofexperimentsforfurtherdetailsconcerningcapmodels,see DiMaggioandSandler1971,ChenandBaladi1985,andHuangandChen1990. Theiterativemethodismorecosteectivethantheexperimentalapproach.Theiterative methodcanbeperformedutilizingstressloadingdatafromuniaxialstraintestsinthe I 1 ;J 2 plane.Thestressloadingunderuniaxialstrainconditionsallowsboth I 1 and 25

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J 2 tochangesimultaneouslyallowingstressstatestooccuronthehardeningcap.The ratioinuencesthestressloading,thereforeadjusting R willallowreplicationofthe data.Furthermore,changing R willincurrminorchangestotheuniaxialstrainresponse i.e. 1 ; 1 plane.Toallowthebestevaluationof R itisrecommendedthatallother parametersbeevaluatedrst. Alessintrusivemethodistoconducttheappropriateexperimentstoexplicitlydene R .Thisisaccomplishedbyconductingeitherhydrostaticcompression-constantvolumeor uniaxialstrain-constantvolumetests.Inhydrostaticcompression-constantvolumetests thematerialisloadedtothedesiredlevelofhydrostaticstressfollowedbyaconstant volumei.e. v =constantboundaryconditionimposedduringshearloading.The constantvolumeboundaryconditionwillallowthematerialtofollowthecapsurfaceuntil failureisreached.Asimilarprocedureisfollowedfortheuniaxialstrain-constantvolume caseandispresentedinFigure1-4. Byintroducinghardeningthedeformationhistoryeectsonthebehaviorare accountedfor.Thevolumechangehistoryenablesthematerialtocompactduring hydrostaticloadingwhiledilatingduringshearloading.Whenthematerialyieldsunder hydrostaticloading 1 = 2 = 3 thecapispushedoutalongthehydrostaticaxis decreasingthevolumetricstraini.e.compaction.Thus,undershearloadingthematerial willdilateuntiltheshearfailuresurfaceisreached.Furthermore,theaccumulationof plasticvolumestrainenablesthecaptoincreaseordecreaseallowingforbettercontrolof dilatancy.Despitetheserevolutionarystridesinsoilplasticity,Drucker'scapmodelutilizes anassociatedowrule.Enforcingassociativityoverestimatesthevolumetricresponse ofthematerialandprohibitsthemodelfromcapturingrealisticallythetransitionfrom compressibilitytodilatancy. FollowingtheworkofDrucker,DiMaggioandSandler1971proposedacapmodel withafailuresurfacedescribedbyaDrucker-Pragerfailurecriteriongivenas, f 1 = p J 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [([ A )]TJ/F21 11.9552 Tf 11.955 0 Td [(C exp BJ 1 ]{11 26

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Figure1-4.Uniaxialstrain-constantvolumeexperimentofQuikrete R ] 1961sand;Failure surfaceline-symbol;experimentsolidline. whiletheplasticvolumestraindependentyieldsurfaceand/orcapisexpressedby, f 2 = L P )]TJ/F21 11.9552 Tf 11.955 0 Td [(J 1 L P )]TJ/F21 11.9552 Tf 11.955 0 Td [(X P 2 + p J 2 A )]TJ/F21 11.9552 Tf 11.955 0 Td [(C exp[ BL P ] 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1{12 where X P = L P )]TJ/F21 11.9552 Tf 11.955 0 Td [(R [ A )]TJ/F21 11.9552 Tf 11.955 0 Td [(C exp[ BL P ] : {13 TheparametersinEqs.1{11and1{12aredepictedinFigure1-5.InEq.1{11, A is anasymptoticvalue,describingthatthematerialinafullycompactedstateapproachesa VonMisestypebehaviorwithnoadditionalhardening; C and B arematerialconstants. TheseparametersareevaluatedbyplottingthestressloadingdataforCTCtestsinthe I 1 ;J 2 planeandttingEq.1{11.InEqs.1{12and1{13, L P and X P arethe sameas p a and p b intheDruckerCapmodel,respectively.Theratio, R ,representsthe 27

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majortominoraxesoftheellipse,asintheDruckerCapmodel,andmaybeevaluatedin thesamemanner. Figure1-5.RepresentationoftheCapmodelofDiMaggioandSandler1971inthe deviatoricplane.[RecreatedfromDiMaggio,F.L.andSandler,I.S.,1971. Materialmodelforgranularsoils.J.Eng.Mech.Division97,935-950.] TheproposedmodelofDiMaggioandSandler1971generalizedthecapmodel ofDruckerbyproposingmoregeneralformsforboththefailuresurfaceandthe strain-hardeningcap,thusallowingthemodeltotawiderangeofmaterials.However, thismodelisformulatedusingclassicalplasticitysatisfyingDrucker'sPostulatefor uniqueness,continuitiy,andstability.Here,stabilityinfersthatthematerialisnotallowed tostrainsoften,thusonlyhardeningispermitted.ThegeneralizedcapmodelDiMaggio andSandler1971hassincebeenextendedtoincluderatedependencies,non-associated ow,anisotropicyielding,kinematichardening,andviscoplasticbehaviorNelson1978; NelsonandBaladi1977;SandlerandBaron1979. ThepreviouslyreviewedCapmodelsprimarilyuseplasticvolumestrainastheir hardeningparameterandadheretothenormalitycondition.ToimproveupontheCap 28

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models,Ladeandhiscolleaguesdevelopedelastic-plasticmodelsthatincorporatedplastic work-hardeningandnon-associatedowrulesLade1977;LadeandDuncan1975; LadeandKim1988a,b.Furthermore,severalofthesemodelspredictsofteningbehavior Lade1977;LadeandKim1988a,b.LadeandDuncan1975proposedamodelthat predictssoilbehaviorfordrainedandundrainedconditionsatdenseandloosestates.This modelallowsgoodpredictionofsoilbehavioratlowconningpressures.However,the mechanicalbehavioratlowconningpressuresaresignicantlydierentfromthatofhigh conningpressures.Thepresentationofthismodelanditsapplicationtothematerial underinvestigationisgiveninChapter4. Cohesionlessgranularmaterialsimmediatelyyielduponloadingwhetherisotropicor deviatoric,thusplasticworkmustbeaccountedforundereachtypeofloading.Dueto thelowconningpressuresinvestigatedinLadeandDuncan1975thematerialacquires verylittleplasticworkinthehydrostaticphaseofthetriaxialexperiments.Thus,the LadeandDuncan1975modelneglectsplasticworkduringisotropicloading.However, highconningpressuresacquirelargeamountsofplasticworkduringhydrostaticloading andwouldbeneglectedinthemodelofLadeandDuncan1975.Furthermore,high stressesimposedduringhydrostaticloadingrequirelargevaluesofplasticworktoacquire additionalplasticdeformationduringdeviatoricloading.ThisisshowninFigure4-3of Chapter4.ThehyperbolicmodelproposedbyLadeandDuncan1975Eq.4{11in Chapter4signicantlyunderestimatestheplasticworkattheonsetofdeviatoricloading, thereforepredictingunrealisticallysmallvaluesofplasticstrain.Thisisdiscussedindetail inChapter4. Later,Lade1977proposedadouble-hardeningelastic-plasticmodeltoaccountfor plasticworkduringisotropicanddeviatoricloadingaswellaswork-softeningbehavior. Themodelproposedseparateyieldlociandplasticpotentialsforthehydrostaticand deviatoricphasesofthetriaxialexperiments.Thehydrostaticportionoftheexperiment isdescribedbyaconicalfailuresurfaceandassociatedowrulewhilethedeviatoric 29

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phaseincorporatesafailuresurfacewithsofteningbehaviorandamovablecapwitha non-associatedowrule.Thismodelwhileaccountingfortheplasticworkduringisotropic loadingstillunderestimatestheplasticdeformation.ThisisbecauseLade1977reduces toLadeandDuncan1975whennosofteningbehaviorisevidentthefailuresurfacefor ourmaterialinFigure2-31ofChapter2. 1.1.2RateDependentModels Theprevioussectiondiscussedmodelsthatdescribewelltherate-independent behaviorofgranularmaterialsatlowpressures.However,granularmaterialscanbe strain-ratedependent.Tomodeltheinuenceofstrain-rateonplasticdeformationoften timeselastic-viscoplasticmodelsareproposed.Thefoundationofthemathematicaltheory ofviscoplasticitywasestablishedbyBingham1922.Binghamassumedthatinorderto describerateeectsonplasticowonehastodescribe1theonsetofplasticowand2 howtheplasticowdependsonthedynamicalstateofthesystemmobilityoruidity. ThemodelproposedbyBinghamisrigid/viscoplastic,i.e.elasticdeformationisneglected whiletheonsetofplasticowisdependentontheshearstress.TheBinghamconstitutive relationisexpressedas, D = 8 > < > : 1 2 1 )]TJ/F21 11.9552 Tf 20.488 8.088 Td [(k p J 2 0 if J 2 >k 2 0if J 2 k 2 {14 where D istherateofdeformationtensor, thematerialviscosity,and J 2 thesecond invariantofthestressdeviator.Thevalue, k ,representstheconstantyieldstressinshear. Accordingtothemodelplasticow D 6 =0manifestsonlyiftheshearstressexceedsthe constantyieldstress. In1932,HohenemserandPrager1932generalizedtheBinghammodeltodevelop therstelastic-viscoplasticmodel.Thismodelassumesadditivedecompositionofthetotal strain-ratetoincludeanelasticcomponent, E ,andviscoplasticcomponent, vp ,such 30

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that, = E + vp where E = = 2 G: {15 Theviscoplasticstrain-rateisgivenas, vp = 1 2 1 )]TJ/F21 11.9552 Tf 20.488 8.088 Td [(k p J 2 0 {16 where k istheyieldstressinshearand 0 isthestressdeviator.Notethattheviscoplastic strain-rate, vp ,isgovernedbytheBinghamtypeconstitutivelawEq.1{14where viscoplasticdeformationoccurswhenthestressstateexceedstheyieldlimit, k .The limitationofthismodelisitsinabilitytodescribehardeningbehaviorduetoutilizinga VonMisesyieldconditionwithaxedyieldlimit k =constant. Inthe1960'sthedevelopmentoftherst3-Delastic-viscoplasticmodelswith hardeningwereproposedbyPerzyna1966.Thesemodelsincorporateadditive decompositionofthetotalstrain-rateasgiveninEq.1{15.AsintheoriginalBingham modelitisassumedthatthereisplasticowonlyifthestressstateissuchthatthe quasi-staticyieldlimit,whichisafunctionofstressonly,isexceeded.Thus,therateof irreversibledeformationis vp = h F i @f @ {17 where isaviscosityparameter, hi representstheMacaulaybracketdenotingthe positivepartofafunctioni.e. h A i = = 2 A + j A j andisamonotonicfunction.Thus: h F i = 8 > < > : 0if F 0 ; F if F> 0 : {18 Intheaboveequations,thefunction F isdenedas: F ; vp = f ; vp )]TJ/F15 11.9552 Tf 11.955 0 Td [(1{19 where f dependsonthestressstate, ,andtheviscoplasticstrain, vp .Theparameter, ,describesthehardeningbehaviorandisdependentonthestrainhistory.Thus,the 31

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function F ,inEq.1{19,representstheboundarybetweenelasticandviscoplasticstress states. Thisformulationhasbeenutilizedextensivelyoverthelast50yearstodescribe rate-sensitivitiesformanydierentsoilsunderdrainedandundrainedconditionswith bothassociatedandnon-associatedowrulesBaladiandRohani1982;Boukpetietal. 2004;DesaiandZhang1987;diPriscoetal.2000;YinandGraham1999.Many dierentelastic-viscoplasticmodelsexist.However,thesemodelsareapplicabletothe rangeofstressesandstrain-ratesassociatedwithcivilengineeringapplications. Theelastic-viscoplasticformulationofPerzyna1966utilizesanassociatedow rule,thusprohibitinganaccuratedescriptionofthevolumetricdeformationofsoilsdue tolargeratesofdilation.Aspreviouslydiscussed,anassociatedowruleassumptionis inadequateforthedescriptionofsomesoilbehavior.Furthermore,ifannon-associated owruleisassumedthemathematicalformoftheplasticpotentialistypicallygiven similartothatoftheyieldfunction.Toeliminatethesediculties,Cristescuproposeda modelthatdetermines,foragivenmaterial,thespecicexpressionsoftheconstitutive functionsbasedonexperimentaldataCristescu1989,1991,1994.Thegeneralformof thiselastic-viscoplasticconstitutivemodelis: = 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.767 8.087 Td [(1 2 G p + k T 1 )]TJ/F21 11.9552 Tf 14.331 8.087 Td [(W t H @F @ {20 where W IrreversibleStressWork K BulkModulus F ViscoplasticPotential G ShearModulus H YieldFunction StrainTensor k T ViscosityCoecient StressTensor p MeanNormalStress KroneckerDelta h A i 1 2 A + j A j : 32

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Thisformulationmakesno apriori assumptionsconcerningtheexpressionsof theyieldfunctionorplasticpotential.Theyieldfunctionandplasticpotentialmay beconstructedusingdatafromthequasi-staticCTCteststhatwillbedescribedin Chapter2.Furthermore,theyieldfunctionisdependentontherstinvariantofstress, I 1 ,secondinvariantofdeviatoricstress, J 2 ,andinternalhardeningvariable, ,afunction ofplasticwork.Furthermore,theformulationcanincorporatedamage,failure,and creepbehaviors.Theformulationcouplestheshearandvolumetriceectsanddoesnot requireanadditionalequationofstatetodescribethepressure-volumebehavior.Granular geomaterialsshowbothcompressibleanddilatantbehaviorsrequiringadescription ofthetransitionfromastateofcompressiontooneofdilatancy.Thistransitionis integratedintotheformulationthroughtheplasticpotential,thusestablishingasmooth transitionfromacompressiblebehaviortoadilatantone.Thisallowsforamoreaccurate modelingofthecompressiblity/dilatancyboundaryincontrasttootherrate-sensitive models.Lastly,duetothedicultiesassociatedwithdeterminingtheexpressionofthe plasticpotentialCazacuetal.1997proposedanalternativeapproachusingtheoremsof representationofisotropictensorfunctions. ThisformulationhasbeenutilizedforawiderangeofmaterialsCazacuetal.1997; Cristescu1991;CristescuandHunsche1998;Schmidtetal.2009,however,hasnot beenexpandedtothehighconningpressuresanddynamiceventsforgranulargeological materials. 33

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CHAPTER2 QUASI-STATICBEHAVIOROFQUIKRETE R ] 1961SAND 2.1OverviewofQuasi-staticTestingTechniques Tounderstandthemechanicalbehaviorofthematerialofinterestitisrequired toconductexperimentsthatreplicatethestressstatesandstrain-ratesobservedfor theapplicationofinterest.Indynamiceventssuchasprojectilepenetration,pressures canapproachtheGigapascalrangenearthenoseoftheprojectilewhileapproaching quasi-staticconditionsnearthefreesurfacesCurranetal.1993.Furthermore,the strain-ratescanbeveryhighatimpactbutessentiallyquasi-staticmanyprojectile diametersaway.Currently,nooneexperimentaltechniquecanreplicateallofthesestress conditionsrequiringvariousexperimentaltechniquestocovertheentirepressurerange. Figure2-1showsthecommonexperimentaltechniquesutilizedandthecorresponding rangesofstrain-ratescoveredLindholm1971. Thetimeeectsonthequasi-staticresponsecanbeinvestigatedusingahydraulically drivenloadframedeviceFigure2-2thatistypicallyservo-controlled.Theservo-controlled capabilityallowstheusertocontroltheratesofload,conningpressure,andaxial displacementtoacquiretheappropriatestressconditions.Thedynamicresponseis investigatedbytheuseofaKolskyorsplit-Hopkinsonpressurebarandyerplateimpact experiments. Tofullycharacterizeamaterial,thequasi-staticbehaviorunderathree-dimensional stateofstressmustbeinvestigated.Toapplyathree-dimensionalstressstateaconning orpressurecellisrequired.Twotypesofpressurecellsmaybeutilized:1CubicalorTrue triaxialcelland2Karmancell.Thetruetriaxialcellisofacubicalshapeandusessix individualplatestoapplyloadstothespecimens.Theadvantageofthistechniqueisthat allthreeprincipalstressescanbecontrolledindependently.Thisexperimentaltechnique isusuallymoreexpensiveandcomplex.Furthermore,thespecimenfabricationismore complicatedandexpensivesinceallsixsurfaceshavetobemachinedtowellcontrolled, 34

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Figure2-1.Strain-rateregimesfortesting.[ReprintedfromLindholm,U.S.,1971.High strainratetesting.In:Bunshah,R.F.Ed.,TechniquesinMetalsResearch. Interscience,NewYork] smalltolerances.ThemorecommonlyusedtechniqueisthatoftheKarmancell,named afterTheodorevonKarman,whoin1911conductedtriaxialcompressiontestsonmarble VonKarman1911.InaKarmanexperiment,thespecimeniscylindricalandisheld betweentwosteelplatensofaloadframeandplacedinaconningcell. AKarmanexperimentconsistsoftwophases,hydrostaticanddeviatoric.Inthe hydrostaticphasethespecimenisloadedhydrostatically 1 = 2 = 3 ,withtheall aroundpressureincreasedslowlytoapredeterminedlevel.Thehydrostaticpressureis appliedviaaconninguid,generallyamixtureofhydraulicoilandkerosene.When thepredeterminedconninglevelisreachedthehydrostaticphaseiscompleteandthe deviatoricphasestarts.Inthedeviatoricphasetheconningorradialpressureisheld constantwhiletheaxialloadisincreasedataconstantloadingrate.Theaxialloadis increased,viathesteelpistonfromtheloadingframe,untilfailureisreached.Torecord 35

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Figure2-2.EngineeringResearchandDevelopmentCenter8.9MNhydraulicallydriven loadframeusedintheexperimentalinvestigation. thedeformationprocesstheaxialandradialstrains,conningpressure,andaxialload aremeasured.Today,theuseofservo-controlledloadersareprevalentandenableoneto controlwelltheloadingrate,conningpressure,andaxialdisplacement.Furthermore, theservo-controlledloadersenabletheexperimentstobeconductedundereitherstress orstraincontrolandrunmonotonicallyorwithcreepand/orrelaxationcycles.Figure 2-3givesasectionedviewofatypicalKarmandeviceChinnandZimmerman1965. 36

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Byconductingaseriesofexperimentsatvariouslevelsofconningpressure,onecan Figure2-3.Karmantestingdevice.[ReprintedfromChinn,J.,Zimmerman,R.M.,1965. Behaviorofplainconcreteundervarioushightriaxialcompressionloading conditions.TechnicalReportWL-TR-64-163,AirForceWeaponsLaboratory] studytheinuenceofthemaximumandminimumprincipalstressesonthemechanical behavior.Basedonthedataobtained,continuummodelsmaybedevelopedandutilized forpredictingmaterialbehaviorundervariousloadingconditionsthatcannotbereplicated throughexperiments.Discussiononhowtointerpretandevaluatetheconstitutive parametersarewelldocumentedinliteratureCazacuetal.1997;Cristescu1989; CristescuandHunsche1998,etc..Thehydrostaticpressureinsuchexperimentsis limitedduetothedesignoftheKarmancellandpossiblyonthematerialofinterest. However,Karmancellshavebeendesignedtoaccommodatehydrostaticpressuresonthe orderofagigapascalGPaFrewetal.1993. ConningpressuresexceedingaGigapascalhavebeenachievedthroughtheuseof auniaxialstraindevice.Itconsistsofplacingthematerialofinterestinarigidsteel 37

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collarandplacingitbetweenanupperandlowersteelassembly.Thetwoassembliesare fastenedtogetherwhiletheloadisappliedthroughaloadframequasi-staticconditions orexplosivessubmillisecondloading.Thematerialisprohibitedfromexpandinginthe lateraldirection,thusuniaxialstrainconditionsmaybeassumed.Insuchexperiments, onlyaxialloadingandstrainaremeasured.Descriptionsofthisexperimentaltechnique aregivenintheliteratureBridgman1952;Cristescu1991;FarrandWoods1988. Thedisadvantagesofthistestingtechniqueincludeprecisetolerancesofthespecimens thatneedtobeachievedwhichisespeciallydiculttoobtainwithgeomaterialsandthe abilitytoquantifyfrictionbetweentherigidsteelcollarandspecimen. Inthisresearch,adeviceoftheKarmantypewasutilizedattheU.S.Army EngineeringResearchandDevelopmentCenterERDC,Vicksburg,MStocharacterize thequasi-staticbehavior.TheKarmandeviceatERDCoeredalargerangeof conningpressuresanddirectmeasurementofboththeaxialandradialstrains,hence acharacterizationofthevolumetricdeformationbehavior.Note,theKarmanexperiment willbereferredtoasaconventionaltriaxialcompressionCTCtestthroughoutthetext. Insubsequentsectionswewilldiscussthematerialofinterest,experimentalplan,analysis ofresults,andmodelingoftheresponseintheelasticrange. 2.2MaterialProperties,Quikrete R ] 1961 ThesandselectedforthisresearchisreferredtoasQuikrete R ] 1961,purchasedfrom QuikreteCompany,Atlanta,GA.Thesandissilicabased,kilndried,andpoorlygraded. Poorlygraded"indicatesthematerialhasasmallrangeofparticlediametersasdened bytheUniedSoilClassicationSystemUSCSASTM-D24872006;Craig1987.The physicalproperties,whicharesummarizedinTable2-1,weredeterminedusingstandard laboratoryproceduresrecommendedbytheAmericanSocietyofTestingandMaterials ASTM.Furthermore,theparticlesizedistributionisgiveninFigure2-4. 38

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Table2-1.PropertiesofQuikrete R ] 1961negrainsand. PropertyValue USCSClassication 1 SP SpecicGravity2.69 D 50 2 [mm]0.323 Max.DryDensity 3 [kg/m 3 ]1760 Min.DryDensity 3 [kg/m 3 ]1480 Max.VoidRatio0.789 Max.Porosity[%]44.2 Min.VoidRatio0.505 Min.Porosity[%]33.6 Figure2-4.ParticlesizedistributionforQuikrete R ] 1961. Allspecimenshadaninitialnominaldensityof1.65g/cm 3 withanassociatedvoid ratio,percentporosity,andrelativedensityof0.62,38.0,and0.61respectively.The 1 UniedSoilClassicationSystem 2 Denotestheaverageparticlesize 3 ASTMD4254 39

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voidratioisdenedastheratioofthevolumeofvoidstothevolumeofsolidswhilethe relativedensityofasoilinitsdenseststateis1andinitslooseststate0.Thespecimens forallquasi-staticCTCtestswerenominally50mmindiameterandapproximately110 mminlength.Furthermore,allspecimensweretestedunderdryundrainedconditions. 2.3ExperimentalInvestigation Themainobjectiveoftheexperimentalinvestigationwastodeterminethequasi-static behaviorofQuikrete R ] 1961sandunderavarietyofloadingconditions.Theprimary interestwastostudytheinuenceoftheconningpressureonthematerialstrengthand deformationproperties.Typicallyunconnedtestsareperformed,however,duetoour materialbeingcohesionlesssomeamountofconningpressureisalwaysrequired.The materialstudiedinthisworkwasinvestigatedinhydrostaticcompression,CTC,and uniaxialstraintests.UndertheconditionsofCTCtests,thestresstensorisgivenbyEq. 2{1,where 3 istheradialloadappliedtothespecimenusingthehydraulicconning uid.Theradialloadwillbereferredasconningpressureor 3 throughoutthetext. Thus,theCauchystress e appliedtothespecimenis: = 0 B B B B @ 1 00 0 3 0 00 3 1 C C C C A {1 Thehydrostaticpressureormeanstressisdenedas, p = tr 3 {2 wheretr"denotesthetraceofthetensorsumofitsdiagonalcomponents.Ameasureof the2 nd invariantofthestressdeviator, 0 ,is: q = r 3 2 tr 0 2 : {3 ForCTCtests, q = 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 3 ,i.e.representstheprincipalstressdierence. 40

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Allexperimentswereconductedunderundrainedconditionsprohibitinggasfrom escapingthemembraneenclosingthespecimens,withstrain-ratesontheorderof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 to 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Theaxialandradialstrains,aswellastheappliedaxialandradialloadswere measured.Thus,theinformationregardingthevolumetricdeformationofthematerial andthefailuresurfaceasafunctionof p and q maybeobtained.Theformulationof theconstitutivemodelmaybedevelopedentirelybasedontheCTCtestsdescribed. Furthermore,uniaxialstrainanduniaxialstrain/constantvolumetestswereconductedas avalidationtoolfortheconstitutivemodelandwillnotbediscussedherein. Inpractice,unload-reloadcyclesareincludedintheteststoallowforevaluationof theelasticpropertiesdeterminedbytheslopesoftheseunload-reloadcycles.Typicallyin unload-reloadcyclessignicanthysteresisexistsmakingthedeterminationoftheslopes oftheunload-reloadcurvessubjective.Figure2-5showsthehystereticbehaviorofthe materialunderinvestigationinaCTCtest 3 =150MPawithunload-reloadcycles. NotethatCristescu1991hasshownthattheintroductionofacreepcyclei.e.stress heldconstantmayeliminateviscouseectsandensureelasticunloading.Duringcreep thespecimendeformsunderaconstantstateofstresswhiletheevolutionofstrainwith timeismonitoreduntilthestrain-rateofthematerialapproacheszero.Atthispoint, thematerialhasreachedastabilizedstate.Fromthisstateunloadingisentirelyelastic whilefurtherloadingmaycreateadditionalplasticdeformation.Inordertoevaluate theviscosityofthematerialandminimizeitseectontheresponseduringunloading itwasdeterminedthatacreepcycleneededtobeintroducedpriortoeachunloading. Thus,ineachexperimentthreeseparatecreepcycleswereperformed.Inthisway,the elasticmodulicanbeevaluatedatthreedierentstresslevelsduringanyexperiment. Asanexample,Figures2-6and2-7showresultsfromahydrostaticanddeviatoric experimentwithtypicalcreepunload-reloadcycles.Thematerialwasallowedtocreep forapproximately25minutesi.e.whenthestrain-rateisontheorderof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 Figure2-8presentsthevolumestrain-rateasafunctionoftimemeasuredduringcreepat 41

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1 = 3 =75MPa.Notethatunderconstantload,theevolutionofdeformationstopped after25minutes_ v = 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 Figure2-5.MonotonictriaxialcompressiontestofQuikrete R sandshowinghysteretic response 3 =150MPa. 2.3.1TestMatrix AseriesoftestswereconductedatERDCinVicksburg,MStofullycharacterizethe quasi-staticbehaviorofQuikrete R ] 1961sand.Theexperimentalmatrixissummarized inTable2-2.Inthistable,HCdenotesahydrostaticcompressiontest,TXCdenotes atriaxialcompressiontest,UXdenotesauniaxialstraintest,andUX/CVreferstoa uniaxialstrain/constantvolumetest.Creep"indicatesthatcreepcycleswereintroduced intotheexperiment. 2.3.2SpecimenPreparation Thetestspecimenswereremoldedusingdry,looseQuikrete R ] 1961sandand fabricatedusingseverallifts.Aliftcontainingtheappropriateamountofsandispoured intothespecimenxtureandtampereduntilaheightofapproximatelyoneinchis acquired.Thisisrepeateduntiltheappropriatemassandheightareobtained.Priorto 42

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Figure2-6.HydrostaticcreeptestforQuikrete R sandtestes2b14inTable2-2. Figure2-7.Triaxialcompressioncreeptestduringdeviatoricphaseforconningpressure 3 =50MPa. 43

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Table2-2.ExperimentalMatrixforcharacterizationofthequasi-staticbehaviorofQuikrete R sand. ConningMoistureAsTestedDry TestPressureHeightDiameterMassContentDensityDensityPorosity NumberTestDateTestType[MPa][mm][mm][g][%][kg/m 3 ][kg/m 3 ][%] es2b011/23/2009HC500108.15350.876360.160.061638.091637.1038.69 es2b021/23/2009TXC10110.36350.673365.230.061640.971639.9838.58 es2b031/26/2009TXC20108.83951.105368.730.051651.621650.8038.17 es2b041/26/2009TXC50109.09350.902365.740.051647.491646.6738.33 es2b051/27/2009TXC100108.20450.978364.800.071651.811650.6538.18 es2b061/27/2009TXC150109.22050.698364.670.061653.941652.9538.09 es2b071/28/2009TXC200109.34750.724365.510.071654.171653.0138.09 es2b081/28/2009TXC300109.09350.904368.250.051658.631657.8037.91 es2b091/29/2009UX{108.71251.257368.890.061644.451643.4638.45 es2b101/29/2009UX/CV75109.60150.698368.480.051665.411664.5837.66 es2b111/30/2009UX/CV150108.20451.156365.050.061641.471640.4938.56 es2b121/30/2009UX-Creep{109.47450.825368.300.061658.211657.2137.93 es2b132/2/2009HC-Creep500108.71250.546362.090.051659.881659.0537.86 es2b142/3/2009HC-Creep500109.47450.724367.070.061659.301658.3037.89 es2b152/4/2009TXC-Creep50108.96650.749364.610.051654.201653.3838.08 es2b162/4/2009TXC-Creep100109.85550.495364.010.041654.641653.9838.05 es2b172/5/2009TXC-Creep150109.22050.876365.260.051645.061644.2338.42 es2b182/5/2009TXC-Creep200109.98250.800365.770.041640.851640.1938.57 es2b192/6/2009TXC-Creep300108.96650.775368.640.061670.821669.8137.46 es2b202/6/2009TXC-Creep20109.09350.673365.060.051659.301658.4737.89 es2b212/9/2009TXC150107.95051.029365.330.061654.801653.8138.06 es2b222/10/2009UX-Creep{109.98250.063360.670.071665.931664.7737.65 es2b232/11/2009UX-Creep{108.96650.927367.350.071655.021653.8638.06 44

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Figure2-8.Evolutionofvolumetricstrain-ratewithtimemeasuredinacreepteston Quikrete R sandatapressureof75MPa. testing,thepreparedspecimenmassanddimensionsweremeasured.Thisinformationis utilizedtocalculatethedensityofeachspecimenTable2-2.Allpreparedtestspecimens hadanominalheightof110mmandadiameterof50mm.Posttest,allspecimenswere driedinanoventoremoveallmoisture.Thisallowscalculationoftheastested"density aswellasthedry"density.Figure3-9givesanillustrationofthespecimenassembly. Allpreparedspecimenswereplacedbetweenahardenedsteeltopandbasecaps Figure3-9.Two0.6mmthicksyntheticlatexmembraneswereplacedaroundthe specimenwiththeexterioroftheoutsidemembranecoatedwithaliquidsyntheticrubber. Theliquidsyntheticrubberprohibitsdeteriorationcausedbytheconninguid.Finally, thespecimenwithitstopandbasecapassemblyisplacedontheinstrumentationstand ofthetestapparatusandinstrumentationmounted.Theinstrumentationofthespecimen willbediscussedinthefollowingsection. 45

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Figure2-9.SpecimenofQuikrete R ] 1961sand. 2.3.3InstrumentationoftheQuikrete R SandSpecimen Theaxialdeformationwasmeasuredinallthetestsbytwolinearvariabledierential transformersLVDTsmountedverticallyonthespecimenandpositioneddiametrically apart.TheLVDTisaelectromagneticdevicethatproducesanelectricalvoltage proportionaltothedisplacementofamovablemagneticcore.TheLVDTswereoriented tomeasurethedisplacementbetweenthetopandbasecaps,thusprovidingameasureof theaxialdeformation, 1 ,ofthespecimen.TheLVDT'sarecalibratedtomeasureatotal axialstrainofapproximately11%.Duetotheporosityofourmaterial,thislimitstheuse 46

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oftheLVDTsinceitcannotbesubsequentlyzeroedattheendofthehydrostaticphase. Thus,theLVDTwillbeexclusivelyusedduringthehydrostaticphaseandpartiallyused inthedeviatoricphaseoftheCTCtests.Alinearpotentiometerwasmountedexternalto thepressurevesseltomeasurethedisplacementofthepistonthroughwhichaxialloads areapplied.ThisprovidestheaxialdeformationmeasurementwhentheLVDTexceedsits calibratedrange 1 =11%.Note,thatthelinearpotentiometerhasloweraccuracythan theLVDTs. Tworadialdeformationmeasurementsystems,referredtoaslateraldeformeter", wereutilizedinthisexperimentalprogram.Theoutputofeachlateraldeformeterwas calibratedtotheradialdisplacementoftwofootingsmountedatthespecimenmidpoint, diametricallyopposed,andbetweenthetwolatexmembranesFigure3-9.Thefooting facesweremachinedtomatchthecurvatureofthetestspecimenwhileathreadedpost extendsfromtheoutsideofeachfootingandprotrudesthroughtheoutermembrane.Steel capsarescrewedontothethreadedpoststosealthemembranetothefootingandthe lateraldeformeterringmountedtothesteelcapswithsetscrews. Thersttypeoflateraldeformeter,denotedUX/BX,consistedofanLVDTmounted onahingedringmeasuringtheexpansionorcontractionofthering.TheUX/BXwas usedforsmallerrangesofradialdeformationwhenthegreatestmeasurementaccuracy wasrequired.Thus,theUX/BXwasusedforalloftheuniaxialstrainUXanduniaxial strain/constantvolumeUX/CVtests.Thisdesignissimilartotheradialdeformeter designofBishopandHenkel1962.Whenthespecimenexpandsorcontracts,thehinged deformeterringopensand/orclosescreatingachangeintheelectricaloutputofthe horizontallymountedLVDT. Thesecondtypeoflateraldeformeter,spring-arm,wasusedforallhydrostatic compressionandCTCtests.Itusestwostraingauged,spring-steelarmsmountedona double-hingedringFigure3-9andisusedwhenthegreatestradialdeformationrange isrequired.Note,thistypeofdeformeterislessaccuratethantheUX/BXdeformeter. 47

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Whenthespecimenexpandsorcontractstherigiddeformeterringexesatitshinge allowingachangeintheelectricaloutputofthestraingaugedspring-arm.Furthermore, theoutputofthespring-armsiscalibratedtothespecimen'sdeformation. 2.3.4TestDevices Alltestswereconductedina600MPacapacityKarmancellwhiletheaxialload wasappliedbyan8.9MNMTSloader.The8.9MNloaderFigure2-2wasregulatedby aservo-controlleddataacquisitionsystemthatmonitoredtheapplicationofaxialload, conningpressure,andaxialdisplacement.Thisservo-controlledsystemallowedtheuser toprogramratesofload,pressure,andaxialdisplacementinordertoachievethedesired stressorstrainpath.Conningpressurewasmeasuredexternaltothepressurevesselby apressuretransducermountedtotheconninguidline.Aloadcellmountedinthebase ofthespecimenpedestalwasutilizedtomeasuretheappliedaxialloads.Outputsfrom alltheinstrumentationsensorswereelectronicallyampliedandlteredtoensurethat conditionedsignalswererecordedbythedataacquisitionsystem.Thedataacquisition systemwasprogrammedtosamplethedatachannelsevery1to5sec. 2.4ExperimentalResults Inthissectiontheresultsofthequasi-static_ =10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 hydrostaticcompression andCTCtestsonQuikrete R ] 1961sandarepresentedanddiscussed.Foralltest conditions,bothexperimentswithandwithoutcreepcyclesminimumofthreeloading -creep-unloading-reloadingcycleswereconducted.IntheCTCtests,onecreepcycle wasimposedattheendofthehydrostaticphaseandthreeimposedduringthedeviatoric phase.Creepcycledurationwasapproximately25minutestoallowthematerialto achievestabilization.Thecompressibility/dilatancyboundaryofthematerialwas determinedfromthevolumetricdataacquiredinthedeviatoricphaseoftheCTCtests. TheCTCtestsrunundertheseconditionswereperformedatconningpressuresof20, 50,100,150,200,and300MPa.Furthermore,stressesaretruestressandstrainsare engineeringstrainwithcompressivestressesandstrainspositive. 48

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2.4.1500MPaHydrostaticCompression Theresultsfromahydrostaticcompressiontestupto500MPatestes2b14inTable 2-2arepresentedinFigures2-10and2-11.Threeload-creep-unload-reloadcycleswere conducted.Thepressurewasheldconstantforapproximately33minutesforthersttwo creepcycleswhilethethirdcreepcycledurationwasapproximately10minutes.Note,the materialbehaviorisisotropicupto300MPa,however,followingthispressuretheradial strainissmallerthantheaxialstrainFigure2-11.Thiscouldbeanindicationthatthe specimenisbecomingmoreconsolidatedradiallythanaxially.However,thebehaviormay beassumedisotropicsincetheaxialandradialstrainmeasurementsareoverallinclose agreement.Furthermore,theslopesoftheaxialandradialunloadcurvesappeartobethe same.Notethatthecreepcyclessignicantlyreducehystereticeects,i.e.theunloading curvesbeingquasi-linearFigures2-10and2-11. Figure2-10.HydrostaticcompressionexperimentonQuikrete R sandupto500MPa. 2.4.220MPaCTCExperiment ResultsoftheCTCtestat20MPaconningpressuretesttestes2b20inTable 2-2arepresentedinFigures2-12through2-14.Figure2-12presentstheresultsofthe 49

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Figure2-11.Comparisonbetweenaxialandradialstrainsmeasuredinhydrostatic compressiontestes2b14forpressures300to500MPa. monotonichydrostaticcompressionphaseoftheexperiment.Thecorrelationbetween theaxialandradialstrainsduringthehydrostaticphaseisnotgoodFigure2-12.The dierenceisassociatedwiththeradialstrainsduetoutilizingastraingagetomeasure theradialdeformationwhiletheaxialdeformationismeasuredusingtheLVDTs.Thus, theerrorinmeasuringtheradialstrainmaybeassociatedwiththecalibrationofthe instrumentation.Figure2-13presentstheaxial,radial,andvolumetricstrainsforthe deviatoricportionoftheexperiment.Thequalityofthedataappearstobeexcellentwith minimalhystereticeects.ThevolumetricresponseobtainedinamonotonicCTCtest atthesameconningpressureof20MPabluelineinFigure2-14aresuperposedon theresultsobtainedinthecyclictest.Itbecomesapparentthattheinstabilitiesobserved inthecyclictestatabout33MPaprincipalstressdierencecorrespondstothepointof transitionfromcompressibilitytodilatancy.Furthermore,thematerialshowsnon-linear behaviorandcontinuouslyhardenswithincreasingload. 50

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Figure2-12.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =20MPaforQuikrete R sand. Figure2-13.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =20MPaforQuikrete R sand. 51

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Figure2-14.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =20MPaforQuikrete R sand. 2.4.350MPaCTCExperiment Resultsofthe50MPaexperimenttestes2b15inTable2-2arepresentedinFigures 2-15through2-17.Figure2-15presentstheresultsofthemonotonichydrostatic compressionphaseoftheexperiment.Notetheexcellentagreementbetweentheaxial andradialstrains,indicativeofahighlyisotropicresponse.Figure2-16presentstheaxial, radial,andvolumetricstrainsforthedeviatoricportionoftheexperiment.Thequalityof thedataappearstobeexcellentwithminimalifanyhysteresisintheunload-reloadcycles. ResultsfromthevolumedeformationFigure2-17wereutilizedtodeterminethepoint oftransitionfromcompressibilitytodilatancy.Itisreachedpriortofailureataprincipal stressdierenceof87MPa.Notethatthestresscouldnotbemaintainedconstantat80 MPaprincipalstressdierence.ThisisduetothefactthatthematerialreachestheC/D boundaryseemonotonicresponse,bluelineinFigure2-17.Thus,unloadinghadtobe 52

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performed.Notethatuponreloadingthematerialstress-volumestrainresponseisnearly identicaltothatobtainedinthemonotonictest. Figure2-15.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =50MPaforQuikrete R sand. 2.4.4100MPaCTCExperiment Resultsofthe100MPaexperimenttestes2b16inTable2-2arepresentedin Figures2-18through2-20.Figure2-18presentstheresultsofthemonotonichydrostatic compressionphaseoftheexperiment.Initially,theredoesnotappeartobegood agreementbetweentheaxialandradialstrains,however,thedierenceisfairlysmall andontheorderofanhalfpercentstrain.Figure2-19presentstheaxial,radial,and volumetricstrainsforthedeviatoricportionoftheexperiment.Thequalityofthedata appearstobeexcellentwithminimalifanyhysteresisintherstunload-reloadcyclewhile thesecondunload-reloadcycleexhibitsincreasingamountsofhysteresis.Asmentioned inSection2.4.3,thisislargelyduetotheinstrumentation.Thethirdcreepcyclewas attempted,however,itwasendedquicklyduetooperatorerror.Thecreepcycleshaveto beconductedinteractivelybypushingabuttontellingthedataacquisitionsystemwhen 53

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Figure2-16.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =50MPaforQuikrete R sand. Figure2-17.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =50MPaforQuikrete R sand. 54

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tobeginandend.ResultsfromthevolumedeformationaregiveninFigure2-20.The transitionfromcompressibilitytodilatancyisreachedpriortofailureataprincipalstress dierenceof171MPa.Ataprincipalstressdierenceof80MPathemeasurementofthe axialdeformationtransitionedfromtheLVDTtothelinearpotentiometer.Again,small oscillationsandincreasedhysteresisarepresent. Figure2-18.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =100MPaforQuikrete R sand. 2.4.5150MPaCTCExperiment Resultsofthe150MPaexperimenttestes2b17inTable2-2arepresentedin Figures2-21through2-23.Figure2-21presentstheresultsofthemonotonichydrostatic compressionphaseoftheexperiment.Thereisexcellentagreementbetweentheaxial andradialstrainsindicativeofahighlyisotropicmaterial.Figure2-22presentstheaxial, radial,andvolumetricstrainsforthedeviatoricportionoftheexperiment.Thequality ofthedataappearstobeexcellentwithminimalhysteresis.Resultsfromthevolume deformationaregiveninFigure2-23.Thetransitionfromcompressibilitytodilatancyis reachedpriortofailureataprincipalstressdierenceof240MPa.Ataprincipalstress 55

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Figure2-19.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =100MPaforQuikrete R sand. Figure2-20.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =100MPaforQuikrete R sand. 56

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dierenceof90MPathemeasurementoftheaxialdeformationtransitionedfromthe LVDTtothelinearpotentiometer.Again,thepresenceofsmalloscillationsareapparent. Concerningthesecondcycle,itisveryinterestingtonotethatthemonotonicandcyclic responseareveryclose.Thisindicatesthatathighpressurestheinuenceoftimeonthe responseisnotveryimportant. Figure2-21.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =150MPaforQuikrete R sand. 2.4.6200MPaCTCExperiment Resultsofthe200MPaexperimenttestes2b18inTable2-2arepresentedin Figures2-24through2-26.Figure2-24presentstheresultsofthemonotonichydrostatic compressionphaseoftheexperiment.Initially,theredoesnotappeartobegood agreementbetweentheaxialandradialstrains,however,thedierenceisfairlysmall andontheorderofanhalfpercentstrain.Figure2-25presentstheaxial,radial,and volumetricstrainsforthedeviatoricportionoftheexperiment.Thequalityofthe dataappearstobeexcellentwithminimalifanyhysteresisintheaxialdeformation unload-reloadcycleswhiletheradialdeformationunload-reloadcyclesexhibitssmall 57

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Figure2-22.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =150MPaforQuikrete R sand. Figure2-23.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =150MPaforQuikrete R sand. 58

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amountsofhysteresis.ResultsfromthevolumedeformationaregiveninFigure2-26.The transitionfromcompressibilitytodilatancyisreachedpriortofailureataprincipalstress dierenceof336MPa.Ataprincipalstressdierenceof100MPathemeasurementof theaxialdeformationtransitionedfromtheLVDTtothelinearpotentiometer.Again,the presenceofsmalloscillationsareapparent. Figure2-24.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =200MPaforQuikrete R sand. 2.4.7300MPaCTCExperiment Resultsofthe300MPaexperimenttestes2b19inTable2-2arepresentedin Figures2-27through2-29.Figure2-27presentstheresultsofthemonotonichydrostatic compressionphaseoftheexperiment.Theredoesnotappeartobegoodagreement betweentheaxialandradialstrains,however,thedierenceisfairlysmalloftheorderof anhalfpercentstrain.Figure2-28presentstheaxial,radial,andvolumetricstrainsfor thedeviatoricportionoftheexperiment.Thequalityofthedataisexcellentwithminimal ifanyhysteresisintheaxialdeformationunload-reloadcycleswhiletheradialdeformation unload-reloadcyclesexhibitssmallamountsofhysteresis.Resultsfromthevolume 59

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Figure2-25.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =200MPaforQuikrete R sand. Figure2-26.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =200MPaforQuikrete R sand. 60

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deformationaregiveninFigure2-29.Thetransitionfromcompressibilitytodilatancyis reachedpriortofailureataprincipalstressdierenceof504MPa.Ataprincipalstress dierenceof100MPathemeasurementoftheaxialdeformationtransitionedfromthe LVDTtothelinearpotentiometer.Again,thepresenceofsmalloscillationsareapparent. ThevolumeresponseFigure2-29showsanabruptchangefollowingthesecondcreep cycle.Thiscouldbeduetosignicantvoidcollapseatornearthelocationofthelateral deformeter. Figure2-27.Stress-strainresponseinthehydrostaticcompressionphaseoftheCTCtest atconningpressure 3 =300MPaforQuikrete R sand. TheresultsofthemonotonicandcyclicCTCtestsconductedforconningpressures intherange20-300MPashowedthatthematerialexhibitsanisotropicstrongly non-linearbehavior.Thematerialdisplaysbothcompressiveanddilatantbehavior,which isstronglydependentontheloadinghistory.Byperformingcreeppriortounloading, hystereticeectsweresignicantlyreducedallowingamoreaccurateestimateofthe elasticparameters.Thereappearstobesomeissuesregardingtheaccuracyofthe radialstrainmeasurements,however,thisislikelyduetothesensitivityofthelateral 61

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Figure2-28.Stress-strainresponseinthedeviatoricphaseoftheCTCtestatconning pressure 3 =300MPaforQuikrete R sand. deformeterstraingage.Thus,ahigherdegreeofcondencewasplacedontheaxial strainmeasurements.Basedonthisexperimentalinvestigation,itmaybeconcludedthat inordertodescribethemainfeaturesoftheobservedbehavioranelastic/viscoplastic modelingapproachshouldbeadopted. 2.4.8FailureSurface&Compressibility/DilatancyBoundary Thissectionpresentsthefailuresurfaceandcompressibility/dilatancyboundary evaluatedfromtheCTCexperimentaldatapresentedinSections2.4.2thru2.4.7. Thecompressibility/dilatancyboundarydescribesthetransitionofthevolumetric behaviorfromcompressibilitytodilatancyorexpansion.Thecompressibility/dilatancy boundaryalsoaidsindeterminingwhetheranassociatedornon-associatedowruleis appropriate.Thelatteraspectwillbediscussedinthedevelopmentoftheconstitutive modelChapter5.Sincethesecondcreepcyclehappenedtooccureitherclosetoor atthecompressibility/dilatancyC/Dboundarytheevaluationofthislocationwas subjective.Thus,theC/DboundaryforeachCTCexperimentwasdeterminedfrom 62

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Figure2-29.Comparisonbetweenthevolumetricresponseinthedeviatoricphaseof monotonictestbluelineandcyclictestredlineatconningpressure 3 =300MPaforQuikrete R sand. experimentsperformedwithnocreepcycles.Thisisreasonablesincebothtypesof CTCtestsessentiallyoverlap.Asmentionedinprevioussections,thetransitionfrom compressiontodilatancyisdeterminedbyplotting,inthe v q plane,thedatameasured inthedeviatoricphaseoftheCTCtestFigure2-29for 3 =300MPa.Theprincipal stressdierencewherethistransitionoccursisthenplottedasafunctionofmeanstress, p ,asshowninFigure2-30. Thefailuresurfaceisthelimitorultimatesurfacethatdescribesallpossiblestress statesforthegivenmaterial.Thefailuresurfaceisevaluatedusingdatafromallthe CTCexperiments.ThethirdcreepcycleforallCTCexperimentsoccurredduringfailure, therefore,theCTCexperimentswithnocreepcycleswereutilizedforconstructingthe failuresurface.Failureisdeterminedbythemaximumprincipalstressdierenceorthe principalstressdierenceat15%axialstrain,whicheveroccursrst.Thus,thefailure surfaceisconstructedbyestablishingthefailurepointforeachconningconditionand 63

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Figure2-30.Compressibility/DilatancyboundaryforQuikrete R ] 1961sand. plottingthemasafunctionofmeanstress, p .TheplottedfailuresurfaceisgiveninFigure 2-31andshowsaverydistinctandremarkablyconsistenttrendinthedata. Thevaluesdescribingthetransitionfromcompressibilitytodilatancyandfailurewere plottedasafunctionofmeannormalstress.Testresultsshowaremarkablylineartrend forthecompressibility/dilatancyboundaryandfailuresurface. 2.5EvaluationofElasticParameters Theelasticmoduliareevaluatedfromtheslopesoftheunload-reloadcycles,following creep,intheCTCexperiments.TheYoung'smodulus, E ,isdeterminedfromtheslopes ofunload-reloadcyclesinthe 1 q plane.Thehistoryoftheprincipalstressdierence isrecorded,thustheentireunloading-reloadingcurveisknown.Theinitialandmiddle portionsoftheunloading-reloadingcurveisutilizedinevaluatingtheYoung'smodulus. Theseportionsofthecurvearegenerallyquasi-linearasshowninFigure2-32for 3 =50 MPa.Inevaluatingtheslopes,aleastsquarestwasmadetothedata.Valuesof E at varyingconningpressuresarepresentedasafunctionofmeanstressinFigure2-33. Uponinspectiontheredoesnotappeartobeareasonabletrend,however,atrendcan 64

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Figure2-31.FailureSurfaceforQuikrete R ] 1961sand. beextractedfromthepresenteddatawithseveralreasonableassumptionsconcerningthe bulkmodulusandPoissonratio.TheseassumptionswillbediscussedinChapter5. Theshearmodulus, G ,isevaluatedinthesamemannerasYoung'smodulus,however, thetheslopesoftheunloading-reloadingcurvesinthe 1 )]TJ/F21 11.9552 Tf 10.221 0 Td [(" 3 ;q planeareutilized.Values of G atvaryingconningpressuresarepresentedasafunctionofmeanstressinFigure 2-34.Again,nodiscernibletrendisseen,however,thebehaviorwillbeknownoncethe trendfor E isdetermined. Thebulkmodulus, K ,wasevaluatedfromtheunloadingcurvesofthehydrostatic compressiontestsFigure2-10.Thebulkmodulusmayalsobedeterminedinthe v ;p planeinthehydrostaticphaseofCTCexperiments.However,unload-reloadcycleswere notperformedfollowingcreepinthehydrostaticphaseoftheCTCtests.Therefore,the hydrostaticcompressiondatawasutilizedtoevaluate K .Furthermore,thehydrostatic compressiontestsFigure2-10utilizetheLVDTsduringtheentireexperimentdeeming thedatamoreaccurate.Thisincreasedaccuracyallowsthebulkmodulustoserveasa guideindeterminingthecorrectYoung'sandshearmodulus.Note,thebulkmodulus 65

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Figure2-32.Unloading-reloadingcurvefor 3 =50MPa. Figure2-33.ExperimentalvariationofYoung'sModuluswiththemeanstress. 66

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Figure2-34.ExperimentalvariationoftheShearmoduluswiththemeanstress. wasnotevaluatedfortheinitialunloadingattheendofthehydrostaticcompressiontest. Valuesof K arepresentedasafunctionofmeanstressinFigure2-35.Thedatashows agradualsaturationofthebulkmodulusasthepressureincreases.Thisiscommonfor granularmaterialsduetotheircompressivenature.Therefore,athighlevelsofpressure thematerialwillapproachafullycompactedstatewithnoadditionalvolumechangeand allelasticmoduliremainingconstant. Figures2-33through2-35showthattheelasticparametershaveastrongdependence onpressure.WhileacleartrendwasnotapparentconcerningtheYoung'sandshear modulusatrendmaybefoundusingthevaluesofthebulkmodulusasaguide.Thiswill bediscussedinChapter5. 67

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Figure2-35.ExperimentalvariationoftheBulkmoduluswiththemeanstress. 68

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CHAPTER3 DYNAMICBEHAVIOROFQUIKRETE R ] 1961SAND InChapter2,thequasi-staticcharacterizationforQuikrete R ] 1961sandunderhigh pressureswasdiscussed.Inthischapter,thehighstrainratebehaviorofthismaterialis investigated.Thehighstrainratecharacterizationofmaterialsi.e.100s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 andhigher isgenerallyobtainedthroughtheutilizationoftheSplit-HopkinsonpressurebarSHPB orKolskybar.Althoughotherhighrateexperimentaltechniquesexiste.g.Charpytests, drop-weighttowertests,andultrasonicevaluation;onlybyusingtheKolskybarmethod canthefullstress-straincurveforagivenstrainratebeobtained.Usingsuchdata,strain rateeectsondeformationandstrengthcanbeincorporatedintoconstitutivemodels. InSection3.1ageneralpresentationoftheKolskybartechniqueanditsapplicability togeomaterialsisgiven.TheexperimentaltestsconductedonQuikrete R sandunder connedconditionsandtheexperimentalsetuparepresentedinSections3.2and3.3, respectively.Theresultsofthetestsconductedalongwithconcludingremarksaregivenin Section3.4. 3.1Split-HopkinsonPressureBarorKolskyBarTechnique 3.1.1Background UsingKolskybartechniques,typicallystrain-ratesof10 2 -10 4 s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 canbeachieved Nemat-Nasser2000.TheKolskybarwasoriginallydevelopedtostudythedynamic responseofmetallicmaterialsandwithinthelastdecadeshasbeenadaptedtoallow thestudyofgeologicandcementitiousmaterialsi.e.,concrete,soils,ceramics,etc..In thissectionabriefhistoryoftheKolskybarmethodandthemaincontributorstothe developmentofthisexperimentaltechniquearepresented. ThehistoryoftheKolskybarbeginswiththepioneeringworkofHopkinson1914. Apressureloadingisimposedbydetonationofguncottonplacedinacylinder,denoted AinFigure3-1,andmountedtothebarend,denotedBinFigure3-1.Thisimposed pressureloadingcanbegeneratedbytheimpactofariebulletontheendofthebar,B. 69

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Thus,thetechniquewasoriginallyknownastheHopkinsonbarFigure3-1.Theimposed pressureloadingand/ordetonationcreatesacompressivewavethatpropagatesalongthe rod,denotedBinFigure3-1,andimpactsapiece,referredasthetimepiece",denoted CinFigure3-1,withthesamematerialanddiameterastherodgenerally,steeland magneticallymountedtotheendoftherod.Thecompressivewavereectsattheend ofthetimepieceasatensionwaveandpropagatesbacktowardstherodseparatingthe twobarsatthemagneticinterfacebecausetheinterfacecannotsustainanytension.The momentumfromthecompressivewaveistrappedinthetimepieceandonceseparated iesintoaballisticmomentumtrap,denotedDinFigure3-1,allowingmeasurement ofthetimepiecemomentum.Fromthemomentummeasurementtheaveragepressure imposedbytheimpactand/ordetonationisknownwhenthedurationofthepulsei.e. wavelengthisdetermined.Thistechniqueallowsthedeterminationofthemaximum pressureandpulsedurationbyvaryingthetimepiecelength,however,acomplete pressure-timehistoryofthewaveasitpropagatesthroughtherodcouldnotberecorded. Toovercomethisdeciency,Daviesin1948,modiedtheHopkinsonbarby incorporatingaparallelplateandcylindricalcondenserstomeasurethedynamicradial andaxialdisplacementtimehistoriesinthebarusingacathode-rayoscilloscopeDavies 1948.Utilizingone-dimensionalwavepropagationinabarwithafreeend,Davies showedthattheparticlevelocity, ,oftheendofthebarandtheradialdisplacement, wererelatedtothecompressivepressureinthebar, p ,as = 2 p c 0 = a p E where isthebardensity, c 0 thebarwavespeed, isPoisson'sratio, E theYoung's modulus,and a thebarradius.Thus,theincorporationoftheparallelplateand cylindricalcondensersallowedthepressure-timehistoryofthecompressivewavetobe determined. 70

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Figure3-1.SchmaticoftheHopkinson'sapparatus.[ReprintedfromHopkinson,B.,1914. Amethodofmeasuringthepressureproducedinthedetonationofhigh explosivesorbytheimpactofbullets.Philos.Trans.R.Soc.London,Ser.A 213,437-456.];Aguncottonplacedincylinderandmountedtotheendofan elasticsteelbar;Belasticsteelbar;Ctimepiecemountedtotheendofthe elasticsteelbaroppositetotheguncotton;Dmomentumtrap. LaterKolskymodiedtheDavie'sbartoallowforstress-strainmeasurementsby extendingthelengthofthetimepiecerenamedtheextensionbar"andplacedathin discofmaterialbetweentheextensionbari.e.,transmitterbarandelasticsteelbari.e. incidentbarKolsky1949.Thedynamiccompressiveloadingisappliedbyringa detonatorplacedagainstahardenedsteelanvil.Thecompressivewavepropagatesdown thepressurebarwhereitpassesthroughacylindricalcondensermicrophonemeasuring theamplitudeofthecompressivewaveasafunctionoftime,beingrecordedusinga cathoderayoscilloscope.Thecompressivewaveatthespecimen-pressurebarinterface reectsasatensilewaveintothepressurebar;acompressivewavepropagatesthrough thespecimenandthroughtheextensionbar.Theextensionbarismountedwithaparallel platecondenserallowingtheaxialdisplacementsattheextensionbarfreeendtobe measuredandcollectedbythecathoderayoscilloscope.Thisapparatusbecameknown 71

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asthesplit-HopkinsonPressureBar"butisalsoreferredtoasaKolskybarwithinthe literature.Usingthemeasureddisplacement-timehistories,Kolskywasabletoobtainthe mechanicalresponseofthespecimenplacedbetweenthepressureandextensionbarsusing one-dimensionalelasticwavetheory. TheoriginalKolskybarhasbeenmodiedsuchastoimposeloadinginuniaxial tension,torsion,simultaneoustorsioncompression/tensionandsimultaneouscompression/ torsionGray2000.TheclassicalKolskybarhasadditionallybeenadaptedby implementingpulseshapingtechniquessuchastocontrolloading.Furthermore,pulse shapingallowsdynamicstressequilibriumandconstantstrain-ratedeformationin specimensofvariousmaterialsandgeometriesthatwouldotherwisenotdeformunder thedesiredconditionsusingtheclassicalKolskybar.Morerecently,theKolskybarhas beenmodiedtoallowaxisymmetrictriaxialstressloading.Thistechniqueincorporates hydrostaticallaroundequalpressureanddeviatoricshearloadingphases,respectively. Thus,theinuenceofconningpressurecanbeassessedfordynamicloadingconditions. 3.1.2AnalysisoftheDataUsing1-DStressWaveTheory Asalreadymentioned,theKolskyapparatusconsistsofastrikerbar,anincidentbar, andatransmitterbar.Thespecimenisplacedbetweentheincidentandtransmitterbars Figure3-2.Thestrikerbarimpactstheincidentbarproducinganelasticcompressive wavethattravelstowardsthespecimen.Whentheimpedanceofthespecimenisless thantheimpedanceofthebars,partofthecompressivewaveistransmittedthrough thespecimenandthroughthetransmitterbarwhileanotherpartisreectedbackinto theincidentbarasatensilewave.Iftheelasticwavesinthebarsarenon-dispersive,the signalsmeasuredwithstraingagesmountedonthebarsurfacesawayfromthespecimen canbeassumedtobethesameasthoseattheinterfacesbetweenthebarsandspecimen. Figure3-3showsanexpandedviewofaspecimenplacedbetweentheincidentand transmitterbars.Here,thesubscripts1and2denotetheincidentandtransmitterbar ends,respectively.Let i r ,and t denotetheincident,reected,andtransmittedstrains, 72

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Figure3-2.Kolskybarset-up. respectively,while u 1 and u 2 denotethedisplacementsoftherespectivebarends.It shouldbenoted,thattheincidentandtransmitterbarsareofthesamematerialand diameter.Thus,bardensity, ,Young'sModulus, E ,cross-sectionalarea, A ,andwave speed, c 0 arethesame.Furthermore,stressesandstrainsareconsideredtobepositivein compression. Figure3-3.SchematicrepresentationoftheKolskybar. Thedataanalysisisdoneassuminganelasticresponse,inthebars,and1-Dwave propagation.Thus,therodsareconsideredtobehomogeneous,isotropicandlinear elastic.Furthermore,anytransversedeformationisneglectedandparallelcross-sections remainplane.Thisimpliesthatthebarpropertiesarethesamethroughoutitsentire lengthwithnodispersion.Thespecimenendsoftheincidentandtransmissionbarsare conned,solateraleectswillbepresent,however,forsimplicitytheseeectswillbe ignored.Thus,accordingto1-Dwavepropagationtheorythedisplacementsatisesthe equation @ 2 u @x 2 = 1 c 2 0 @ 2 u @t 2 : {1 73

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Intheaboveequation,timeisdenotedby t x isthe1-DLagrangianspatialcoordinate, and c 0 istheelasticwavespeedinanyofthebarswhichisdenedas: c 0 = s E : TheclassicalD'AlembertsolutionofEq.3{1isexpressedas, u 1 x;t = f x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t + g x + c 0 t u 2 x;t = h x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t {2 where u 1 isthedisplacementintheincidentbarwhile u 2 isthedisplacementinthe transmitterbarand f g ,and h arearbitraryfunctionstobespeciedbasedoninitial conditions.Thefunctions f and h representwavestravelinginthepositive x directioni.e. totherightinourframeofreference,while g representsawavetravelingintheopposite direction.Thus, f and g maydenotetheincidentandreectedwaves,respectively,while h denotesthetransmittedwave.Thus,Eq.3{2mayberewrittenas: u 1 x;t = f x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t + g x + c 0 t = u i + u r u 2 x;t = h x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t = u t : {3 Thestrainsintheincidentandtransmissionbarsarecalculatedusingthesmallstrain assumption, = @u=@x ,bydierentiatingEq.3{3withrespectto x .Thus, @u 1 @x = f 0 x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t + g 0 x + c 0 t = i + r @u 2 @x = h 0 x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t = t : {4 ThebarparticlevelocitiesmaybeobtainedbytakingthetimederivativeofEq.3{4,i.e. @v 1 @t = )]TJ/F21 11.9552 Tf 9.298 0 Td [(c 0 f 0 x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t + c 0 g 0 x + c 0 t = c 0 r )]TJ/F21 11.9552 Tf 11.955 0 Td [(" i @v 2 @x = )]TJ/F21 11.9552 Tf 9.298 0 Td [(c 0 h 0 x )]TJ/F21 11.9552 Tf 11.955 0 Td [(c 0 t = )]TJ/F21 11.9552 Tf 9.299 0 Td [(c 0 t : {5 74

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Forhomogeneousdeformationthestrainrateinthespecimenisgivenby, = v 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(v 2 l s {6 where l s istheinstantaneouslengthofthespecimen.Equation3{6mayberewrittenby substitutingEq.3{5as, = c 0 l s )]TJ/F21 11.9552 Tf 9.298 0 Td [(" i + r + t : {7 Next,usingHooke'slaw,thestressesoneithersideofthespecimenareobtained: 1 = A A s E i + r 2 = A A s E" t ; {8 where A and A s arethecross-sectionalareasofthebarandspecimen,respectively.The generalassumptionisthatthespecimenisunderastateofuniformstress,i.e.theloading onbothendsofthespecimenareequal.Itfollowsthat: t = i + r {9 withthespecimenaveragestrainrate,strain,andstressundercompressionloading expressedas, s = 2 c 0 l s r ; {10 s = 2 c 0 l s Z t 0 r ; {11 s = E A A s t ; {12 wherethesubscript s referencesthespecimen, A representsthebarcross-sectionalarea, A s thespecimencross-sectionalarea, E thebarYoung'smodulus,and l s thespecimen length.However,ifthespecimenandbarcross-sectionalareasareequalthentheratioof 75

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theseareasinEq.3{12becomeunity.Thus,Eqs.3{10thru3{12yieldtheaverage mechanicalresponseofthespecimenandmayonlybeutilizedwhenthespecimenisina stateofstressequilibriumasdiscussedbyGray2000;GrayandBlumenthal2000. 3.1.3DynamicTriaxialKolskyBar ThepreviouslydiscussedtechniqueissometimestermedaClassical"Kolskybar experiment.AswasdiscussedinChapter2,thebehaviorofmostgeologicmaterialsis highlysensitivetohydrostaticpressure,however,usingtheclassicalKolskybar,such materialscanbecharacterizedonlyunderuniaxialstressconditions.Dataobtainedon cementitiousmaterialsusingtheclassicalKolskybartechniquewerereportedinRoss 1989;Rossetal.1996,1995.Incontrasttoconcrete,cohesionlesssoilsaretypically characterizedundernearlyuniaxialstrainconditions.Underuniaxialstrainconditions theconningpressure, 3 ,constantlyincreasesandthereisalinearrelationbetweenthe principalstressdierenceandmeanstressLuetal.2009.However,developmentof constitutivemodelsforgeo-materialsrequiretriaxialoraxisymmetricstressdata,suchas CTCdata.Furthermore,inapplications,themechanicalloadsappliedtogeo-materialsare notcommonlyuniaxial.Forexample,duringprojectilepenetrationofgeomaterials,the stressenvironmentsnearthewarheadareverycomplexandmulti-axialinnatureCazacu etal.2008.Therefore,itisdesirabletoacquiredynamicdataunderconnedconditions similartothatdiscussedforquasi-staticCTCtestsChapter2. SuchCTCexperimentsathighloadingrateswerereportedintheliterature. CasagrandeandShannon1949andSeedandLundgren1954performedsomeof thepioneeringworkinthisareabetweenthelate1940'sandearly1950's.Inthecourse oftheseinvestigationstheydevelopedvariousloadingmethodsforobtaininghighloading ratesontheorderof2s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 whileimposingconningpressuresupto0.6MPa.Later WhitmanandHealy1962conductedrapidCTCexperimentsondryOttawasand withconningpressuresupto0.10MPa.Highloadingrateswereimposedusinga hydraulicallydrivenloadingframethatcouldacquirestrainratesof35s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Inmorerecent 76

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investigationstriaxialcellsandservo-hydraulicloadingframeswereutilizedtostudy thehigh-frequencycyclicCTCresponseofsandHaerietal.2008;Wangetal.2007; WichtmannandTriantafyllidis2004.Despitetheseeorts,theconningpressures andstrainratesinvestigatedaremoderate;thustheconditionsreplicatedinthetestsare typicalforcivilapplicationsratherthanimpactevents. Toinvestigatethedynamicmechanicalresponseofacohesionlessmateriallikesand, thespecimenneedstobeconned.Therefore,inpreviousstudiestheKolskybarwas modiedinanattempttoapplystressstatessimilartothatimposedinaquasi-static CTCexperimentforsoilsBragovetal.1996,2008;Feliceetal.1987a,b;Luetal. 2009;Martinetal.2009;Songetal.2009.Intheabovereferences,ahardened steelsleevewasusedtoapplymechanicalconnementtothespecimen.Thistechnique presentsseveralchallengesthatlimititseectivenessandaccuracy:1thestressdata undermechanicalconnementusingthesleeveidoesnotcorrespondtoastraightline inthedeviatoricplane p;q ,thusmakingitdiculttoinferanymaterialproperties, thelateralconnementappliedbythesteelsleevedoesnotremainconstantduring theexperimentbutcontinuouslychanges,andfrictioneectsbetweenthesleeveand specimenareimportantandcannotbetakenintoaccountproperlywhenreducingthe data.Becauseoftheselimitationstheaccuracyisquestionableandthedatacanprovide onlyqualitativeinformationonthematerialbehavior. ToimproveuponthesedecienciesMalvernandJenkins1990developeda connementcellabletoimposelow-conningpressures < 20MPaon76mmdiameter concretecylindersatstrainratesbetween10and100s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 .Theconnementcellencloses thespecimenandbarendsandisthenlledwithwaterandsubsequentlypressurizedin ordertoapplylateralconnementFigure3-4.Thus,thisconnementcellcanbeeasily mountedtoanyKolskybar.Thistechniquehasbeenusedinsubsequentinvestigations tocharacterizethepressuredependentbehaviorofmortarandconcreteunderdynamic loadingatvaryingstrainratesandconningpressuresSchmidtandCazacu2006; 77

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Schmidtetal.2009.However,themannerinwhichtheconningcellisimplemented intotheKolskybarintroducesseverallimitations.Whenloadingisappliedareaction momentisgeneratedattheendsoftheincidentandtransmissionbarsoppositeofthe specimenthatcausenon-normalimpactontheincidentbarfromthestrikerbar, possiblebucklingoftheincidentandtransmissionbars,andincreasedfrictionatthe pointsonthebarswheremountsarelocated. Figure3-4.PhotographoftheconningcelldevelopedbyMalvernandJenkins1990. TheselimitationswereeliminatedbytheexperimentaltechniqueofFrewetal. 2010wheretheKolskybarismodiedtoincludeahigh-pressurehydraulicconning cellcapableofimposingconningpressures, 3 ,upto400MPatothetestspecimen. Thus,anallaroundequalpressure 1 = 2 = 3 canbeappliedtothespecimen duringthehydrostaticphasewhileimposingshearloading 1 ; 2 = 3 inthedeviatoric phase.Furthermore,theconningcellissucientlylargetoensurethattheconning pressureremainsnearlyconstantduringthedeviatoricphase.Thespecimensdeformation historyismeasuredduringbothphasesoftheexperimentallowingthematerialsshearand volumetricbehaviortobecharacterizedKabirandChen2009.Lastly,thedynamic shearfailuresurfaceorultimatesurfacemaybedeterminedutilizingdatafromthe deviatoricphaseoftheexperiment. 78

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Aspartofthisdissertation,theexperimentaltechniquesofFrewetal.2010and KabirandChen2009wereusedtocharacterizethedynamicresponseofdrysand underconnement.Itisworthnotingthatusingthistechniqueitispossibletogenerate highqualitymechanicaldataunderthesamestressconditionsasinquasi-staticCTC experimentsandthususethedatatodevelopconstitutivemodels. 3.2DynamicExperimentalProgram TocharacterizethedynamicbehaviorofQuikrete R ] 1961sandaseriesofconned Kolskybarexperimentswereconductedatastrainrateof1000s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 undervaryinglevels ofconnement.Inalltests,theinitialdrydensityofthespecimenwas1.65g/cm 3 Experimentswereconductedforconningpressures, 3 ,of25,50,100MPaandrepeated 4times,whileonlytwotestswereconductedfor125MPaconnement.Todetermine theeectsofmeanpressureonthevolumetricbehaviorasinglehydrostaticcompression experimentupto125MPawasconducted.AllthesetestswereconductedatPurdue UniversityinthelaboratoryofProfessorWeinongWayne"Chen. 3.3ExperimentalSetup 3.3.1ConnedKolskyBar Theprimarygoalwastostudytheinuenceoftheconningpressureonthe material'sdynamicstrengthanddeformationpropertiesundertriaxialloadingconditions. AlldynamicCTCexperimentswereinvestigatedinanundrainedstateprohibitingair andwaterfromescapingthespecimen.AsalreadymentionedtheconnedKolskybar techniqueofFrewetal.2010wasusedforimposingconningpressurewhileradial measurementsweredoneusingthetechniqueofKabirandChen2009.Theuniaxial stressisgivenbyEqs.2{1thru2{3. PhotographsoftheconnedKolskybarintegratedwithhydrostaticcompression cellsareshowninFigures3-5thru3-7.TheconnedKolskybarhastwohighpressure conningcellsmountedatthespecimenlocationandthefreeendofthetransmissionbar, respectively.Hydraulicpressureisappliedtothechambersthroughuidportsattachedto 79

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ahydraulicpressuresupplysystem.Eachconningcellhasendcapsonthefrontandrear ofthechambersandtranslatealong19.0mm,centerlessgroundR c 53C-350maraging steel,incidentandtransmissionbars,respectively. Tomountthetestspecimenthehydraulicuidfromtheradialconningcellispurged andthefrontendcapremoved.Oncetheendcapisremovedtheradialconningcell translatesalongfoursupporttie-rodstoexposethespecimenendsoftheincidentand transmissionbars.Thepreparedspecimenspecimenpreparationtoensureagiveninitial densitywillbedescribedinSection3.3.3;aphotographisshowninFigure3-9Disthen putbetweentheincidentandtransmissionbars.Acapacitor,consistingofaradialspring andcoppertubingFigure3ofKabirandChen2009,ismountedaroundthespecimen measuringtheradialdeformationwhileamanganinpressuregageismountedinsidethe radialconningcellmeasuringthehydraulicpressureKabirandChen2009.Once theinstrumentationismountedtothespecimen,theradialconningcellisreassembled, hydraulicuidaddedtotheradialconningcelltoremoveanyair,andthenboth conningcellsattachedtothehydraulicpressuresupplysystem.Lastly,aLinearVariable DierentialTransformerLVDTismountedtotheincidentandtransmissionbarsto measureaxialdeformationduringthehydrostaticphaseofeachexperiment. Theaxialconningcellappliesaforceatthefreeendofthetransmissionbarequalto theconningpressuremultipliedbythebarscross-sectionalareaFigure3-5.Theaxial forcewillacttopushthetransmissionbar,specimen,andincidentbarintothebarrelof theairgun.Toneutralizethismovementareactionforceisneededandissuppliedby asymmetrictie-rodassemblythatholdsthetie-rodreactionplateandaxialconning celltogetherFigure3-7.Thetie-rodstravelfromtheaxialconningcelldowntothe tie-rodreactionplateontheoppositesideoftheradialconningcellFigure3-8.The incidentbar,onthefrontsideofthetie-rodreactionplateandoppositetothestriker bar,ismountedtothereactionplatediskusingcyanoacrylate.Thereactionplatedisk isdiametricallyalignedinarecessedhole,inthetie-rodreactionplate,andrestson 80

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amachinedshoulderinthetie-rodreactionplateFigure3-8.Thus,whenconning pressureisapplied,themachinedshoulderwillaccepttheforceanddistributeitthrough thetie-rodreactionplateandintothetie-rods.Furthermore,sincethetie-rodsare mountedsymmetricallyaroundtheincidentandtransmissionbarsnobendingmoments areappliedtothesystem. Figure3-5.SchematicoftheconnedKolskybarset-upusedintheexperimentson Quikrete R sand. Figure3-6.Radialandaxialconningcellschematics. 81

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Figure3-7.ConnedKolskybarapparatusatPurdueUniversity,WestLafayette,IN.;A Tie-rodreactionplate;BRadialconningcell;CIntensier;DTie-rods; EAxialconningcell;FBarsupports. Figure3-8.Tie-rodreactionplate 82

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Asalreadymentioned,connedKolskybarexperimentsconsistoftwodistinctphases. Intherstphase,ahydrostaticorallaroundequalpressure 1 = 2 = 3 isimposedon thespecimen.Duringhydrostaticloadingtheaxialandradialdeformationsaremeasured usingthecapacitorandLVDT,respectively,whilethehydrostaticpressureismeasured bythemanganinpressuregage.Whenthepredeterminedconninglevelisreachedthe hydrostaticphaseiscompleteandthedeviatoricphasebegins.Inthedeviatoricphase i.e.dynamicloadingtheconningpressure, 3 ,isheldconstantwhiletheaxialload isincreasedcreatingshearloading.Thisaxialloadingisappliedbylaunchingastriker bar,viaanairgun,intopulseshapersontheimpactendoftheincidentbarcreatinga compressivewavethattravelsdowntheincidentbarandloadsthespecimen.Sincethe impedanceofthesandspecimenislessthanthatofthebars,partofthecompressive wavepropagatesthroughthespecimentothetransmissionbarandpartisreected backintotheincidentbarasatensilewave.Toimposethedesiredtestingconditions onthespecimenandreducehigh-frequencycomponentsintheincidentwave,thepulse shapingtechniquewasimplemented,thusenablingtheshapedincidentwavetopropagate alongthebarwithoutdispersionwhiledeformingthespecimenuniformly.Thenearly non-dispersivewavesmeasuredatstraingagelocationsonthebarsurfacesawayfromthe specimenareutilizedtodeterminethespecimenresponse.Straingagesmountedonthe incidentbarmeasuretheincident, i ,andreected, r ,strainsandgagesmountedonthe transmissionbarmeasurethetransmittedstrain, t .Thedataanalysismethodisbasedon one-dimensionalwavetheoryandwasdiscussedinSection3.1.2. Duringdynamicloading,fromtheincidentbar,thetotalvolumeofthespecimen willchangecausingapossiblevariationinthehydraulicuidpressurewithintheradial conningcell.Iftheconningpressurechangessignicantly,thespecimenresponsefrom thisexperimentaldevicecannotbecompareddirectlytoquasi-staticdataobtainedunder aconstantconningpressure, 3 ,duringdeviatoricloading.Toensuretheconning pressureremainsconstanttheradialconningcellvolumewasmaximizedinaneortto 83

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eliminateanyissuesconcerninguctuationsintheconningpressure.Itshouldbenoted, inalltestsconducteditwasfoundthatthemaximumvariationinconningpressureison theorderof3MPa. 3.3.2MaterialProperties,Quikrete R ] 1961Sand Specimenswerepreparedtotheinitialnominaldrydensityof1.65g/cm 3 .The associatedvoidratioandrelativedensityaretabulatedinTable3-1.Thespecimensforall dynamictestswerenominally19.0mmindiameterandapproximately9.3mminlength. Table3-1.VoidratioandrelativedensityofQuikrete R sandforconnedKolskybar experiments. PropertyValue Density[g/cm 3 ]1.65 Sandmass[g]4.38 VoidRatio0.63 RelativeDensity0.56 3.3.3SpecimenPreparation Thespecimenswereremoldedusingdry,looseQuikrete R sand.Theremolded specimenisconnedusingathinpieceofheatshrinktubingmoldedtoasteelrodof thesamediameterastheincidentandtransmissionbarsFigure3-9A.Theheatshrink tubingisthenremovedandasinglesteelplateninsertedintothetubingFigure3-9B andmountedtoaxturewiththesamediameterastheincidentandtransmission bars.Theappropriatemassofsandisweighedandpouredintothesteelplaten/tubing conguration.Asecondsteelplatenisthenplacedontopofthesandandlightlytapped. Onedropofcyanoacrylateisplacedontheincidentbarandspreadoverapproximately 70%ofthebardiameter.Thisensuresthesteelplatenisheldinplacewhenthereected tensilewavearrivesattheincidentbar/steelplateninterface.Thefabricatedspecimen Figure3-9DisthenmountedbetweentheincidentandtransmissionbarsFigure3-9E withlightpressureappliedtothetransmissionbartoensuretheappropriatespecimen thickness.Finally,theheatshrinktubingisshrunkaroundthefreeendsoftheincident 84

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andtransmissionbarstoprohibithydraulicuidfromenteringthespecimenFigure 3-9F. AHeatshrinktube BSteelPlaten CWeighingsand DFabricatedspecimen EInstalledspecimen FHeatshrinkingtubeto bars Figure3-9.Specimenfabricationandinstallation:AHeatshrinktube;BSteelplaten; CWeighedsand;DFabricatedspecimen;ESpecimeninstalledbetween theincidentandtransmissionbars;FHeatshrinktubingshrunktothe incidentandtransmissionbars. 3.4ConnedKolskyBarExperimentsonQuikrete R ] 1961Sand Asalreadymentioned,allexperimentswereconductedunderundrainedconditions andasinglestrain-rateof 1000s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Thepulseshapersutilizedfortheexperiments aregiveninTable3-2.Allexperimentswereconductedusing19.0-mmdiameterstriker, incident,andtransmissionbarsmadeofhighstrengthmaragingC-350maragingsteelwith density, =8100kg/m 3 ,Young'smodulus, E =207GPa,andbarwavespeed, C 0 =5080 m/s.Thestriker,incidentandtransmissionbarswere304mm,2312mm,and1144mmin length,respectively.Toensurestressequilibriumanduniformdeformationofthespecimen 85

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atwolayerpulseshaperconsistingofM-2annealedsteelandC11000annealedcopper wereutilized.Thespecimensusedinthedynamictestsareshorterthanthoseusedinthe quasi-statictests.Thereasonisthelowwavespeedofsand,hencetheneedtoconsider shortspecimensinordertoensureuniformdeformation. Table3-2.TestingconditionsforQuikrete R sandfor =1.65g/cm 3 and_ =1000s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ConningStrikerPulseShapers PressureVelocityDiameter 1 Thickness 1 Diameter 2 Thickness 2 MPam/smmmmmmmm 2512.210.301.303.200.50 5015.511.101.304.000.60 10023.011.101.404.000.60 12528.99.501.304.000.80 Ahydrostaticexperimentupto125MPaandstrainrateof 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 wasconducted tocharacterizethevolumetricresponseunderuniformpressureconditions.Theresultsof thehydrostaticexperimentareshowninFigure3-10. Thehydrostaticcompressionexperimentindicatesthematerialrespondedsomewhat isotropically,however,theradialdeformationshowsaverylineartrendasopposed totheobservedcurvatureintheaxialdeformationresponsetypicalofhydrostatic compressiontestsChapter2,Figure2-10.Thedierencebetweenaxialandradialstrain measurementsiswithin1%.Thisispossiblyduetothespecimens'smallaspectratio. Itshouldbenotedthatthisaspectratiowasfoundtosatisfythegoverningassumptions oftheKolskybarandwillbediscussedfurtherinsubsequentparagraphs.Theconned Kolskybarhydrostaticresponseiscomparedtothehydrostaticcompressionresponse obtainedunderquasi-staticconditions.TheconnedKolskybarresponseisstier,yetit canbeconcludedthatthedierenceissmallunderhydrostaticcompression. 1 M-2annealedsteel 2 C11000annealedcopper 86

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Figure3-10.ConnedKolskybarhydrostaticcompressionexperimentonQuikrete R sand upto125MPa. Figure3-11.ComparisonofhydrostaticcompressionexperimentsonQuikrete R sand; ConnedKolskybardashedline;quasi-staticsolidline. 87

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Whenreducingdata,itisassumedthespecimenacquiresstressequilibriumearly duringloading.However,ifthespecimen'swavespeedisamagnitudelowerthanthatof thebarmaterial,asisthecaseforsand,additionaltimeisrequiredforstressequilibrium whichcannotbeobtainedwithaconventionalKolskybar.Furthermore,bynotusing pulseshapingthespecimenwilldeformnon-uniformlyintroducingmulti-axialstresses. Therefore,thepulseshapingtechniqueFrewetal.2002,2005;Nemat-Nasseretal. 1991wasutilizedtofacilitatestressequilibriumanduniformdeformation.Further explanationconcerningthespecimenaspectratioandpulse-shapingrequirementscanbe foundinRefs.Frewetal.2010;Martinetal.2009;Songetal.2009. InconnedKolskybarexperimentstheallaroundpressureimposedinthehydrostatic phaseoftheexperimentcreatesanosetwithintheincidentandtransmissionbarsignals Figure3-12.Thisosetiscausedbytheinitialhydrostaticpressureappliedtothe incidentbarFrewetal.2010.Duringthedeviatoricphaseoftheexperimentthestriker barimpactstheincidentbarunloadingthetie-rodsandsubsequentlytheincidentbar signalFrewetal.2010.Thisunloadingoftheincidentbarsignalprohibitsonefrom verifyingstressequilibrium,for 3 > 25MPa,usingthemeasuredwavesignals.However, aspartofmyMaster'sresearchi.e.,publicationMartinetal.2009issuedbasedonthe workitwasdemonstratedthatequilibriumisestablishedinsandspecimens.Despite thesediculties,stressequilibriumismoreeasilyobtainedwithincreasinglevelsof 3 sincethespecimenbecomesmoreconsolidatedandsubsequentlythespecimenwavespeed becomeslarger. One-dimensionalwavetheory,discussedinSection3.1.2,isutilizedtoanalyzeall data.Thestartingpointofthereectedpulseiscalculatedusingthedistancefromthe incidentbarstraingagetothespecimenandthelongitudinalwavespeedofthebar material,i.e. T r =2 l=C 0 .Here, l isthedistancefromthestraingagecentertothe bar/specimeninterfaceand C 0 isthewavespeedofthebarmaterial.Thisisthetime requiredfortheincidentpulsetotravelfromthestraingagetothespecimen,bereected 88

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Figure3-12.Osetofincidentsolidlineandtransmitteddashedlinesignalsfor 3 =25 MPaconnement. atthebar/specimeninterface,andtravelbacktothestraingage.Thestartingpoint forthetransmittedsignaliscomputedas T = T s + T t + 1 2 T r where T s isthetransit timethroughthespecimen, T t thetimetotravelfromthespecimentothetransmission barstraingageand T r thetransittimeforthereectedpulse.Oncethestartingtimes arecalculated,stressequilibriumischeckedusingthecommon-wave",2-wave" analysis.The-wave"representsthestressatthetransmissionbar/specimeninterface measuredfromthetransmittedsignalwhilethe-wave"isthetime-resolvedsummation oftheincidentandreectedsignalsattheincidentbar/specimeninterface.Figure3-13 illustratesthe-wave",-wave"methodforverifyingstressequilibriumintheconned Kolskybarexperimentfor 3 =25MPa.Aspreviouslymentioned,allexperimentswere conductedusingatwo-layerpulseshaperwiththeirrespectivedimensionsgiveninTable 3-2. Figure3-13showsthatstressequilibriumhasbeenobtainedsincethe-wave"and -wave"signalsarecomparable.Alsonotethat,stressequilibriumisobtainedveryearly 89

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Figure3-13.1-and2-wavestressprolesforapulse-shapedexperimentonQuikrete R sandfor 3 =25MPa;-wave"solidline;-wave"dashedline. withintheexperiment.Thisispossibleonlybyimposinghydrostaticpressureandthe utilizationofthepulse-shapingtechnique.Thus,pulse-shapinginuencestheincident signalsloadingratewhiletheappliedhydrostaticpressureincreasesthespecimenwave speedcompactingthespecimen. Alldatacanbeplottedastruestress-engineeringstrain.Typicalsignalsforthe capacitorandmanganingageobtainedinthedynamictestunder 3 =25MPaisshown inFigure3-14.Note,thatthespecimenisinastateofexpansionduringdeviatoric loading.Also,thechangeinconningpressureduringthetestshowninFigure3-14 issmallandisontheorderof 1MPa.Toverifythattheappliedstressstateinthe dynamictestissimilartothatofaquasi-staticCTCtestweplottedthestressdatainthe deviatoricplane p;q Figure3-15.Ifthestresspathcanbeapproximatedbyastraight lineofslopeequalto3duringdeviatoricloadingthenCTCstressstatesareachieved. Notethat,thestressdatashowninFigure3-15closelyapproximatesCTCdeviatoric loadingconditions. 90

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Figure3-14.RadialdeformationanddynamicpressuremeasurementsinaconnedKolsky bartestat 3 =25MPaforQuikrete R sand;Diameterchangesolidline; conningpressurechange, 3 dashedline. Figure3-15.Stresspathsplottedinthedeviatoricplane p;q foralltestsperformed; 3 = 25MPasolidline; 3 =50MPadashedline; 3 =100MPadottedline; 3 =125MPadashdotline. 91

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ThemechanicalresponseforQuikrete R sandundervaryinglevelsofconnement, 3 ,andastrainrateof1000s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 isshowninFigures3-16thru3-19incomparisonwith theresponseunderquasi-staticconditionsobtainedunderthesameconnement.The dynamicresponsefor 3 =25MPaFigure3-16showsthatunderdynamicconditions theyieldstressandfailurestressarelargerthanforquasi-staticconditions.Thismaybe duetothefactthatinrealitythequasi-statictestcorrespondsto 3 =20MPawhereas thedynamicresponsecorrespondsto 3 =30MPaconnement.Arepeatofbothtests needstobeperformedtoelucidatetheobserveddierenceinresponse.For50and100 MPaconnements,theresponseofthematerialindynamicconditionsissimilartothat forquasi-staticconditionsFigures3-17and3-18.Indeed,thedatashowninFigure 3-17,for 3 =50MPa,indicatethatthreeoftheobtainedmechanicalresponsesarevery similartothatobtainedforquasi-staticconditions,whereasthe4 th repeatofthetest showsaweakerresponsethanthatforthequasi-staticcase.Notealsothatthespecimen aspectratiois 0.5inthedynamictestswhiletheaspectratiois2forthequasi-static tests.Figure3-18presentsthemechanicalresponsefor 3 =100MPa.Again,minimal discrepanciesareobservedbetweenthedynamicandquasi-staticresponses.Lastly, Figure3-19showstheobservedbehaviorfor 3 =125MPa.Itshouldbenoted,thatno quasi-statictestwasperformedforthisconningpressure.Thelargediscrepancybetween thetworepeatsofthistest,denotedastest A and B ,respectivelyFigure3-19,iscaused byadierenceinconningpressureobtainedattheendofthehydrostaticphase.While intest A ,thepressureachievedattheendofthehydrostaticphasewasof130MPa,only apressureof110MPawasachievedfortest B .Finally,toassesstheinuenceofconning pressureonthedynamicresponsetheaveragedynamicresponseateachconningpressure ispresentedinFigure3-20.Thus,thedatashowninFigure3-20indicateaclearpressure eectoftheconningpressureonthedynamicresponse,i.e.thehighertheconnement thehigheristheyieldlimitandtheultimatestrength. 92

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Figure3-16.Comparisonbetweenthequasi-staticanddynamicstress-strainresponsein thedeviatoricphaseoftheCTCtestsatconningpressure 3 =25MPafor Quikrete R sand;quasi-staticresponsesolidline;dynamicresponse dash-dotline. Figure3-17.Comparisonbetweenthequasi-staticanddynamicstress-strainresponsein thedeviatoricphaseoftheCTCtestsatconningpressure 3 =50MPafor Quikrete R sand;quasi-staticresponsesolidline;dynamicresponse dash-dotline. 93

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Figure3-18.Comparisonbetweenthequasi-staticanddynamicstress-strainresponsein thedeviatoricphaseoftheCTCtestsatconningpressure 3 =100MPafor Quikrete R sand;quasi-staticresponsesolidline;dynamicresponse dash-dotline. Figure3-19.Dynamicstress-strainresponseinthedeviatoricphaseoftheCTCtestsat conningpressure 3 =125MPaforQuikrete R sandasobtainedinthetwo testsconducted. 94

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Figure3-20.Dynamicstressstrainresponseduringdeviatoricloadingforconning pressuresof25,50,100,and125MPaforQuikrete R sand;25MPasolid line;50MPadashline;100MPadottedline;125MPadash-dotline. Basedonthetestresults,itcanalsobeconcludedthatthematerialdoesnotexhibit strainratesensitivity.Notethatthesedynamictestsareofhighquality,giventhe dicultiesrelatedtotestingsuchaheterogeneousmaterialundercombinedloading involvinghighstrainratesandconnement.Furthermore,dynamicallyitwaspossible toachievethesamestressstatesasinquasi-staticCTCtests.Therefore,itwaspossible togeneratedynamicdatathatwerenotpreviouslyavailableforgeologicmaterialsthat canbefurtherusedfordevelopmentofcomputationalmodelsthatwouldpredictwith increasedaccuracytheresponseofsuchmaterialstocomplexloading. 95

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CHAPTER4 APPLICATIONOFLADEANDDUNCAN'SMODELTOSAND 4.1LadeandDuncanElastic-PlasticModel TheobjectiveofthischapteristoutilizethemechanicaldatapresentedinChapter 2toexploreoneofthemostperformantmodelswithintheliteratureusedtodescribethe behaviorofgranularmaterials.Therefore,itwasdecidedtoexploretheelastic-plastic modelofLadeandDuncan.Thismodeliswidelyusedtodescribethelow-pressure behaviorofmanygranularmaterials,however,itneedstobeinvestigatedtodetermine ifthemodelcanbeextendedtodescribethehigh-pressurebehaviorofQuikrete R ] 1961 sand. Wewillbeginbybrieypresentingthismodel.Tosimplifythesubsequentdiscussion, thismodelwillbereferredtoasLade1975".Lade1975isanelastoplasticstress-strain modeldevelopedbasedontheexperimentalresultsofcubicaltriaxialcompressiontestson acohesionlessdrysoil.Incubicaltriaxialcompressiontestsallthreeprincipalstressesare appliedindependentlytothespecimens.IncontrasttoCTCexperiments,whereonlytwo principalstressesarecontrolledindependently,basedoncubicaltriaxialcompressiontests onecandeterminetheinuenceofthethirdinvariantofstressonthemechanicalbehavior. Forfurtherinformationregardingcubicaltriaxialcompressiontests,thereaderisreferred toLadeandDuncan1973. Theconstitutivehypothesesofthismodelare: H1Thematerialisassumedtobehomogeneousandisotropic H2Thedisplacementsandmaterialrotationsbeingsmall,theelasticandplastic componentsoftherateofdeformationtensorareadditive. = E + I {1 TheelasticcomponentisdeterminedfromaHooketypelaw E = 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.767 8.088 Td [(1 2 G p I {2 96

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wherethedotstandsforthederivativewithrespecttotimei.e.increment, K and G are thebulkandshearmodulus,respectively, I denotesa2 nd orderidentitytensor,and p is themeanstressi.e. p = = 3 I 1 .Theelasticparametersareconsideredtovarywiththe stateofstressaccordingwiththefollowingevolutionlaw: E ur = K ur P a 3 p a n {3 where K ur and n arematerialconstantsthatcanbedeterminedfromtriaxialcompression testsperformedatvariouslevelsoftheconningpressure, 3 .InEq.4{3, p a isthe atmosphericpressureexpressedinthesameunitsas E ur and 3 .Furthermore,Poisson's ratio, ,isassumedtobezero. H3Theplasticcomponentoftherateofdeformationisobtainedfromtheowrule p = @g @ ; {4 whentheyieldconditionissatised.InEq.4{4 istheplasticmultiplierand g the plasticpotential. H4Isotropichardeningrepresentedbytheaccumulatedworkperunitvolume W T isconsidered.ForanytimeT, W T = Z T 0 3 I v t dt + 0 : 0 I dt {5 where_ I v istheirreversiblevolumetricstrain-rateand 0 I theirreversibledeviatoric strain-rate. H5Thespecicmathematicalexpressionsfortheyieldfunction,plasticpotential, andfailuresurfaceareconsideredtobesimilarinform.Theeectofthethirdinvariantof stressisexplicitlymodeled. Theyieldsurfaceisallowedtoexpandasthematerialhardenswhilethefailure surfacerepresentstheultimatestateofthematerial.Thus,atyielding: I 3 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 I 3 =0{6 97

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where I 1 and I 3 aretherstandthirdstressinvariants,respectively,givenbythefollowing expressions: I 1 = 1 + 2 + 3 I 3 = 1 2 3 : InEq.4{6 1 isamaterialparameterdependentontheaccumulatedplasticdeformation. Atfailure 1 reachesaconstant,criticalvalue.Letdenote f = I 3 1 I 3 : {7 NotethataccordingtoEq.4{7forhydrostaticcompression 1 = 2 = 3 f isequalto 27. Theplasticpotentialisexpressedinasimilarmannerasthefailureandyieldsurface, respectively.Thus: g = I 3 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 I 3 {8 where 2 isaconstantforanygivenstresslevel, f Eq.4{7.Usingtheowrulegiven byEq.4{4,itfollowsfromEq.4{8that: 2 = 3 I 2 1 + p 3 1 + p 3 ; {9 where p istheratiobetweentheaxialplasticstrainincrement, p 1 ,andthelateralplastic strainincrement, p 3 ,forCTCtests.Usingtheexperimentaldata,thefollowinglawof variationof 2 with f wasproposed: 2 = Af +27 )]TJ/F21 11.9552 Tf 11.955 0 Td [(A ; {10 where A istheslopeofastraightlinewhichcanbeevaluatedbyplottingtheexperimental datainthe f; 2 plane.Sinceforhydrostaticconditions f =27,accordingtoEq.4{10 underhydrostaticconditions 2 =27. 98

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Theexpressionofthework-hardeninglawisdeterminedbasedontheexperimental variationoftheirreversibleworkwiththestressratio, f .Tothisend, f ,iscalibrated fromCTCtestsandthenplottedasafunctionoftheirreversiblework.Thefollowing hyperbolicvariationisproposed: f )]TJ/F21 11.9552 Tf 11.956 0 Td [(f t = W p a + d e W p ; {11 where f t isathresholdstresslevel,with a and d e materialparameterstobeidentied. Notethataccordingtothislaw,forstresslevelsbelowthethresholdstresslevel, f t ,no plasticdeformationoccurs.Thereciprocaloftheparameter a describestheinitialslope ofthehyperbola.Itsdependenceupontheconningpressure, 3 ,wasmodeledbythe followingrelation, a = p a M 3 p a l {12 where p a istheatmosphericpressurewiththesameunitsas a and 3 while M and l arematerialconstants.Thereciprocalofparameter d e describestheasymptoticvaluefor f )]TJ/F21 11.9552 Tf 10.84 0 Td [(f t .Theasymptoticvalueisrelatedtothevalueof f )]TJ/F21 11.9552 Tf 10.841 0 Td [(f t atfailurethroughafailure ratioexpressedas, r f = 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(f t f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f t ult {13 where f )]TJ/F21 11.9552 Tf 12.752 0 Td [(f t ult =1 =d e .Therefore, r f mustbelessthanunity.Lastly,theincrement ofirreversibleworkmaybedeterminedbysolvingEq.4{11for W p andtakingthe derivativewithrespectto f .Theexpressionfortheincrementofirreversibleworkisgiven as, dW p = a df 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(r f f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f t 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(f t 2 {14 where df istheincrementinstresslevel, f ,betweentosuccessivestressstates. 4.2ParameterIdentication Inthissectiontheparametersofthemodelwillbeidentiedusingexperimentaldata fromCTCtestsonQuikrete R ] 1961sand.Ourapplicationofinterestrequireshighlevels 99

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ofconningpressuretobeapplied,thereforeallparameterswillbeidentiedusingdata forthehighestconningpressurei.e. 3 =300MPa.Thus,theparameter a involvedin Eq.4{11isconsideredconstant.Parameter A inEq.4{10isdeterminedbycalculating 2 usingEq.4{9andthenplotting 2 asafunctionofthestresslevel, f ,asshown inFigure4-1.Aspreviouslymentioned 2 shouldhaveavalueof27forhydrostatic Figure4-1.Variationof 2 withstresslevel, f ;Experimentaldatasymbolsin comparisonwithlinearmodelredlineandLademodelblueline. conditions.Forthematerialofinterestthevalueof 2 doesindeedhaveaninitialvalueof 27forastateofhydrostaticstress,however,variationinthedatamakesdeterminationof theslope, A ,subjective.TheredlineinFigure4-1showsthesloperepresentingthetrend oftheexperimentalvalues.ThisslopevalueisusedinEq.4{10todeterminetheblue lineinFigure4-1.Thebluelineensuresthat 2 =27forhydrostaticconditions.However, duetothevariationintheexperimentaldata,parameter A willbevariedwhenreplicating thestress-strainresponsetoensurethebestrepresentationofthedata. Theremainingparameterstobeidentiedareinvolvedintheexpressionofthe work-hardeninglaw.Todeterminethethresholdstress, f t f isplottedasafunctionof 100

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thecalculatedirreversibleworkgivenbyEq.4{5.ThisisillustratedinFigure4-2.As previouslymentioned,accordingtothemodelif f 27, thus,thethresholdstresswillbeequaltothestresslevelforhydrostaticconditions. Tocompletethedeterminationofthework-hardeninglawparameters a and d e needto beidentied.Theseparametersareobtainedbyplotting f )]TJ/F21 11.9552 Tf 12.77 0 Td [(f t asafunctionofthe irreversiblework, W p ,andmodelingitutilizingEq.4{11.Thecomparisonbetween experimentandtheoryisgiveninFigure4-3.TheparametersobtainedforQuikrete R ] 1961sandaresummarizedinTable4-1. Figure4-2.Stressratio, f ,asafunctionofirreversiblework, W p 4.3ApplicationofLadeModeltoQuikrete R Sand ToevaluateLade1975,wedevelopedastressintegrationroutineusingMATLAB. Sincethecompressiontestsareperformedunderstresscontroltheinputtotheprogramis thestresshistoryandtheoutputarethestrainsaccordingtothemodel.Onlytheloading partsofthedataareused,thus,thefailurecriteriondoesnotneedtobeevaluatedat 101

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Figure4-3. f )]TJ/F21 11.9552 Tf 11.955 0 Td [(f t asafunctionofirreversiblework, W p :Comparisonbetween datasymbolsandtheoreticalvariationEq.4{11. Table4-1.ParametersofLade1975forQuikrete R ] 1961sand. ParameterValueUnits 1 45.452{ f t 27.000{ f )]TJ/F21 11.9552 Tf 11.956 0 Td [(f t ult 28.570{ A 0.818{ a 1.364MPa d e 0.035{ eachtimestepsince df> 0foreachtimestep.Thisiswarrantedsincegranularmaterials yieldwithanyamountofloadingduetotheirinherentporosity.Thesimulationswererun usingtheparametervaluesgiveninTable4-1.Severalvaluesfortheparameter A were considered.Figures4-4and4-5showthecomparisonbetweendataandmodel. 4.4Discussion ParametersfortheelastoplasticmodelLadeandDuncan1975wereidentiedfor Quikrete R ] 1961negrainsandfollowedbymodelimplementationintoMATLAB.The modelwasutilizedtoreplicateexperimentaldatafromthedeviatoricphaseofanCTC 102

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Figure4-4.Principalstressdierenceasafunctionofthevolumetricstrainfor 3 =300 MPausingmodelparametersinTable4-1;Experimentsolidline;Lade 1975symbols. testwith 3 =300MPa.Thissectionofthedocumentwilldiscusstheresultsobtainedin comparisontoexperimentaldata. Figure4-4clearlyillustratesthatthemodeldoesnotrepresentwellthedata.The modelunderestimatesthetotalvolumetricstrainbecauseitpredictshighratesofdilation. Thus,thetransitionfromcompressibilitytodilatancyisnotcaptured.Themainreason forthisinaccuracyinrepresentingthematerialbehavioris 2 .Itisclearlyseenthat alinearvariationfor 2 Eq.4{10doesnotrepresenttheexperimentaldatawell. Furthermore,thescatteroftheexperimentaldatainFigure4-1doesnotallowforan objectiveevaluationoftheparameter A .Todeterminetheinuenceoftheparameter A onthesimulationresults,weperformedsimulationswhereallotherparameterswerekept constantandonlyparameter A wasvaried.Itwasobservedthatparameter A controlsthe 103

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Figure4-5.Principalstressdierenceasafunctionofthevolumetricstrainfor 3 =300 MPausingmodelparametersinTable4-1 A =0 : 400;Experimentsolidline; Lade1975symbols. ratesofdilationand/orcompressibility. 0 : 00
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pressuresissignicantlydierent.ThisdierenceprohibitstheextrapolationofLade1975 tohighconningpressures.Whenapplyinghighconningpressuresthematerialbecomes veryconsolidatedwithsignicantparticlefracturing.Theconsolidationandparticle fracturingrequireslargerirreversibleworkincrementstoaccumulateadditionalirreversible strainsincontrasttothebehaviorunderlowerconningpressures.Thisiseasilyseenby comparingFigure6ofLadeandDuncan1975withFigure4-2.Figure6ofLadeand Duncan1975showssmallamountsofirreversibleworkattheonsetofloading,whereas, inFigure4-2largeirreversibleworkisevident.Again,referringtoFigure4-3,Lade1975 underestimatestheirreversibleworkattheonsetofloading,thussmallerirreversiblestrain valuesarepredictedintheinitialstageofloading.Thisisintuitivesincetheirreversible work, W p ,inuences inEq.4{4givenas, = dW p 3 g : {15 ThisexplanationisalsosupportedbyFigures4-4and4-5.Ithasbeenshownthat thepresentedelastoplasticmodeldoesnotdescribeaccuratelythebehaviorofQuikrete R ] 1961.SeveralimprovementsofLade1975havebeenimplementedanddiscussedinLade 1977.Thechangesincludeaseparateyieldfunctionandplasticpotentialforhydrostatic loadingandawork-hardeninglawthatcapturesapost-peakwork-softeningbehavior. However,theLademodelisunabletocaptureourmaterialresponse.Indeed,since Quikrete R ] 1961hasastraightfailuresurfaceathighconningpressures,Lade1977 reducestoLadeandDuncan1975forplasticdeformationunderdeviatoricloading. 105

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CHAPTER5 CONSTITUTIVEMODELING 5.1Elastic-ViscoplasticModelDevelopment Thefocusoftheworkpresentedhereistoconstructacontinuummodeltorepresent thebehaviorofQuikrete R ] 1961sandunderstressconditionscommoninhighlydynamic eventsi.e.shockloading,projectileimpactetc..Toreplicatethesedynamicevents theconstitutivemodelmustincorporateadescriptionoftheelasticresponse,strain-rate eects,hardening,nonassociatedplasticow,andmodelingofthecompressibility/dilatancy boundaryrepresentingthetransitionfromcompressibilitytodilatancy.Furthermore,the modelmustbeentirelydevelopedfromexperimentaldataconductedwithintherange ofpressuresthatoccurintheseevents.Furthermore,nomodelsexistinopenliterature thatcovertherangeofpressuresinvestigatedinthisresearchChapter1.Basedonthe experimentalresultspresentedinChapter2,anelastic-viscoplasticmodelingframeworkis developed.Themainconstitutiveassumptionsarepresentedbelow. Thematerialisassumedtobehomogeneousandisotropic.Thereferenceconguration isthecurrentcongurationwithdisplacementsandrotationsassumedtobesmall.Thus, additivedecompositionofthestrainrateisutilizedwiththetotalstrain-ratecomposedof anelasticpart, E ,andanirreversiblei.e.plasticpart, I expressedas = E + I {1 TheelasticorreversiblecomponentofthestrainrateisgovernedbyHooke'slaw, E = 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.767 8.088 Td [(1 2 G p I {2 wherethedotstandsforthederivativewithrespecttotime, K and G arethebulkand shearmoduli,respectively, I isa2 nd orderidentitytensor,and p themeannormalstress i.e. p = = 3 I 1 .Theirreversiblestrain-rateisgivenbyanoverstresstypelawofthe 106

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form: I = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N {3 where t istheactualtime, k t isaviscositycoecient, W t theirreversiblestressworkper unitvolume,and N theorientationoftheviscoplasticstrain-rate.Theonlyrestriction onthetensorvaluedfunction N istobeisotropic.Aspreviouslymentioned,the bracketinEq.5{3istheMacaulayBracket,denedas h x i = x + j x j = 2.Intheabove equation, H representsthestabilizationboundary,i.e.thelocusofpointsinstress spacewheretransientcreependsaftersomenitetimeinterval.Thus, H = W t {4 andonlywhen H >W t viscoplasticdeformationoccurs.Hardeningofthistypeis appropriateforsandsincebothvolumetricandshearinelasticeectsarethusmodeled. Indeed,inaCTCtesttheirreversiblestressworkperunitvolumemaybedecomposedas: W T = Z T 0 t : I t dt = W H + W D {5 where W H representsirreversiblestressworkrelatedtothevolumechangeofthematerial duringthehydrostaticphaseofCTCtests,and W D representstheirreversiblestresswork associatedtoshapechangeduringthedeviatoricphaseofCTCtests.Themathematical formfortheirreversiblestressworkduringthehydrostaticanddeviatoricphasesaregiven below: W H = Z t H 0 p t I v t dt 0
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First,evaluationoftheelasticparametersi.e. K and G inEq.5{2mustbe performed.Anaccuratemeasureoftheelasticparameterswillensurethattheirreversible strain-rateiscalculatedcorrectly.Oncetheirreversiblestrain-rateisdetermined,theyield function, H ,orstabilizationboundaryisobtained.Finally,theyieldfunctionandthe experimentalirreversiblestrain-ratequantitiesi.e._ I v ,_ I 1 ,etc.areutilizedtodetermine theorientationoftheviscoplasticstrain-rate, N 5.2ElasticParameterEvaluation Thedeterminationoftheelasticparametersbecomesnon-trivialwhenthematerial exhibitsviscouseects.Inthiscase,theunloadingportionsoftheload-unload-reload cyclesarenotpurelyelastic.Toseparateviscouseectsfromunloading,eachCTCtest onsandhadthreecreepcyclesasdiscussedinChapter2.Thecreepcyclespermitviscous eectsinthematerialtostabilize.Unloadingfromthestabilizedstate,isthusquasi-linear. TheYoung'smodulus, E ,isthenevaluatedfromtheseslopesofthequasi-linearportions oftheunloadingsintheuniaxialstrainvs.principalstressdierencecurves.Thevariation ofthevaluesof E asafunctionofmeanstresswasgiveninFigure2-33ofChapter2.Itis intuitivethatanasymptoticlimitof E exists,as p !1 ,becauseforhighpressuresvoid closureorfullcompactionofthespecimenmaybeachieved.Therefore,anexponentiallaw ofvariationwaschosen,i.e. E p = E 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(E s exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(bp p c {7 where p isthemeanstress, p c isacharacteristicpressuresetto1MPa,and E 1 the asymptoticlimitof E theseparametervaluesaregiveninTable5-1.Thequantities E s and b arematerialconstantstobedetermined.Usingtherelations G = E 2+ ;K = E 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 ; {8 theshearandbulkmodulicanbecalculated.ThescatterwithintheYoung'sandshear modulusvaluesmakesthedeterminationoftheparametersinEq.5{7subjective.Thus, 108

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basedontheexperimentalvaluesof E and G correspondingtoeachcycle, and K are calculatedusingEq.5{8.ThevaluesofthePoisson'sratiovaryfrom0.19-0.25.The K valuesarecomparedtothatobtainedfromthehydrostaticcompressiontests.As discussedinSection2.5ofChapter2,theexperimentalbulkmodulusmeasurementsare moreaccuratethanthatoftheotherexperimentalelasticmoduli.AYoung'sandshear modulusmeasurementsaredeemedgoodifthecalculatedPoisson'svalueiswithinthe documentedrangeandcalculatedbulkmodulusapproximatestheexperimentalbulk modulus.Satisfyingthesecriteriawillsignicantlyreducetheobservedscatterandexpose alogicaltrendwithinthedataFigure5-1.Toeliminatethescatter wasconsidered constant, =0 : 220withtheassociatedparametervaluesgiveninTable5-1.Acomparison betweencalculatedandexperimentalelasticmoduliareshowninFigure5-1. Table5-1.Numericalvaluesofparametersinvolvedintheevolutionlaw5{7forYoung's Modulus, E ParameterValue E 1 [GPa]23.00000 E s [GPa]22.00000 b 0.00619 5.3YieldFunctionEvaluation Thespecicexpressionforthestabilizationboundary, H Eq.5{4,is determinedbasedonthedatafromaseriesofcontrolledcreepexperiments.Increep experiments,thestressstateisheldconstantforaniteamountoftimewhilestrains aremeasured.Whenthestrain-rateapproacheszeroitisassumedthatcreependsand thestabilizationboundaryisreachedwithnoadditionalirreversiblestrains.Figure 5-2showsschematicallyina ;" planehowthestabilizationboundaryisdetermined experimentally.Thematerialbeingisotropic, H dependson throughitsinvariants. Weneglectthedependenceonthethirdinvariant.Thus, H isconsideredtodepend onlyon I 1 =tr {9 109

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Figure5-1.Comparisonoftheoreticalandexperimentalvariationsoftheelastic parameterswithmeanstress;Young'sModulusdiamondsymbols;Shear Modulussquaresymbols;BulkModulustrianglesymbols;modelsolid blacklines. and q = r 3 2 tr 0 2 {10 where I 1 istherstinvariantofstresswhile q isproportionaltothe2 nd invariantofthe Figure5-2.Stabilizationboundaryschematic. 110

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stressdeviator.InaCTCexperiment, q reducestotheprincipalstressdierence,i.e. q = 1 )]TJ/F21 11.9552 Tf 12.111 0 Td [( 3 ,and I 1 =3 p = 1 +2 3 with p themeanstress.Letusassumethattheyield functionmaybeexpressedasthesumoftwoterms: H p;q = H H p + H D p;q {11 with H D p; 0=0 : {12 Since H = W t ,itfollowsEq.5{6that H H p = W H and H D p;q = W D ,with W H and W D representingthehydrostaticanddeviatoriccontributionsoftheirreversible stressworkatcreepstabilization.Thehydrostaticpart, H H p ,isdeterminedusingthe dataobtainedinthehydrostaticphaseofCTCexperiments.Ineachtest,themeasured volumetricstrainsareusedtocalculatetheirreversiblestressworkusingEq.5{6a.The valuesoftheirreversiblestressworkcorrespondingtothestabilizationboundary,fromall tests,arethenplottedasafunctionofthecorrespondingpressure, p .Theexpressionthat bestapproximatesthedatais H H p = a h 2 6 6 4 1+ b h exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(c h p p c )]TJ/F21 11.9552 Tf 11.955 0 Td [(c h exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(b h p p c c h )]TJ/F21 11.9552 Tf 11.955 0 Td [(b h 3 7 7 5 {13 where a h =30 : 0MPa, b h =3 : 452 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 c h =4 : 830 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 ,and p c acharacteristic pressure p c =1MPa.Figure5-3comparestheexperimentalvaluesof W H determined fromthehydrostaticphaseofCTCtestsatconningpressuresrangingfrom20-300 MPawithexpression5{13for H H p .Notethatirreversibleworkshowsahighrateof increasefromtheverybeginningofloading.Thisindicatesthatthematerialisprimarily respondingplasticallyevenforverylowvaluesoftheappliedpressure.Thisisexpected foraporousmaterial.Ifthepressureisallowedtocontinuouslyincreaseallporeswill closeandthematerialbecomesfullycompacted.Whenfullycompactednoadditional volumechangemayoccurunderincreasingpressure.Therefore,irreversibleworkwill 111

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Figure5-3.Stabilizationboundary H H asafunctionofmeanstresssolidline; Experimentalvaluesofirreversiblestressworkatcreepstabilizationunder hydrostaticloadingsymbols. eventuallyreachaconstantvaluee.g.,Figure5-3.Thefullycompactedvalue, a h ,should bedeterminedexperimentally,however,thehydrostaticpressuresrequiredtoreachfull compactionarebeyondthepressurelimitoftheequipmentavailable.Inpenetrationtests, incertainregionsaroundtheprojectile,thematerialreachesafullycompactedstate. Hence,thevalueof a h canbeestimatedfromsuchtests.Itwasassumedthat a h =30 : 0 MPa. InthedeviatoricphaseofaCTCtest,theconningpressure, 3 ,remainsconstant whiletheaxialstress, 1 ,isincreasedcontinuouslyuntilfailure.Thestateofstressis = 0 B B B B @ 1 00 0 3 0 00 3 1 C C C C A ;t t H : {14 TheirreversiblestressworkisdeterminedusingEq.5{6b.Thevaluesoftheirreversible stressworkcorrespondingtostabilizationarethenplottedasafunctionofthesecond 112

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invariant, q Eq.5{10.Notethattheworkisdependentonboth q andconning pressure, 3 Figure5-4.ThetrendsinFigure5-4showlargevaluesofirreversiblestress Figure5-4.Irreversiblestressworkatstabilizationasafunctionoftheprincipalstress dierencecalculatedfromthedeviatoricphaseofCTCtestsat 3 =20-30 MPa. workareacquiredattheonsetofloadingduringthedeviatoricphaseoftheexperiments. Thisresponseistobeexpectedsincethematerialexhibitsverylittleelasticbehavior.To capturethebehaviorgiveninFigure5-4asecondorderpolynomialfunctionwasselected. Thus,foranyconningpressure, 3 ,thedataisapproximatedwiththeexpression, H D p;q = d 0 + d 1 q p c 2 {15 where d 0 and d 1 arecoecientsdependentontheconningpressure, 3 ,while p c =1MPa. Thefollowingexpressionsapproximateswellthevariationofthesecoecientswith conningpressure, 3 : d 0 = d 0 ;a 3 p c 1+ d 0 ;b 3 p c + d 0 ;c 3 p c 2 {16 113

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d 1 = d 1 ;a 1+ 1 d 1 ;b 3 p c d 1 ;c {17 where d 0 ;a =1 : 875 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 MPa, d 0 ;b = )]TJ/F15 11.9552 Tf 9.298 0 Td [(7 : 688 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 d 0 ;c =3 : 692 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 d 1 ;a = 9 : 075 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 MPa, d 1 ;b =14 : 45, d 1 ;c =1 : 444. Comparisonbetweentheexperimentalresultsforallconningpressuresandthe theoreticalexpressionforthedeviatoricyieldfunction, H D p;q ,arepresentedinFigure 5-5.Figure5-5showsgoodcorrelationbetweenexperimentaldataandtheory. Figure5-5.Deviatoricyieldsurfaces, H D p;q ,asafunctionofprincipalstressdierence; Experimentsymbols;TheoreticalsolidlinesgivenbyEq.5{15. Theexperimentalvaluesrepresentingthecompressibility/dilatancyboundarydenein the p;q planetheboundarybetweenthedomainsofcompressibleanddilatantbehavior. Forourmaterial,thisboundarycanbeexpressedas X p;q =0,where X p;q =1 : 075 p )]TJ/F21 11.9552 Tf 11.955 0 Td [(q: {18 Stress-statesforwhich X> 0willproducecompressibility,whilestress-statesfor X< 0 producedilatancy.Equation5{18approximatesthisboundaryfortherangeofpressures 114

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consideredinallourtests.Thecomparisonbetweenexperimentsandtheoryisshownin Figure5-6.ThematerialfailureisapproximatedbyaDrucker-Pragertypesurface,but Figure5-6.Compressible/DilatancyBoundary;Experimentssymbols;Theoreticalgiven byEq.5{18solidlines. withnocohesion.Theexpressionforthefailuresurfaceis, f = q )]TJ/F21 11.9552 Tf 11.955 0 Td [(p )]TJ/F21 11.9552 Tf 11.955 0 Td [(k {19 where =1 : 301istheangleofinternalfrictionand k =0nocohesion.Thecomparison betweenexperimentsandtheoryisshowninFigure5-7. Theyieldlociforseveralvalues H =constanti.e.constantvaluesofirreversible stressworkareshowninFigure5-8.Theuppermostsolidlinedescribesthefailure orultimatesurfaceforthematerialwiththeexperimentalcompressibility/dilatancy boundaryshownbythedash-dotline.Ifanassociativeowruleisassumedhighratesof dilationwillbepredictedcorrespondingtothestatesonthedottedlineinFigure5-8. Thus,toaccuratelydescribethecompressibility/dilatancyboundaryanon-associatedow rulemustbeconsidered. 115

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Figure5-7.Failuresurface;Experimentssymbols;TheoreticalgivenbyEq.5{19solid lines. Figure5-8.YieldlociforQuikrete R ] 1961sand;Yieldsurfaces H =constantsolidlines; Compressible/DilatancyBoundarydash-dotline; @H=@ =0dottedline. 116

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5.4StrainRateOrientationTensor Tothispointwehavedeterminedthepressuredependentelasticparametersandyield function.Tocompletetheformulationwenowneedtoconstructtheexpressionfor N orthestrainrateorientationtensor.Foranisotropicmaterial, N mustsatisfythe invariancerequirement N QQ T = QN Q T {20 foranyorthogonaltransformation Q .Fromclassicalresultsregardingtherepresentation ofisotropictensorfunctionse.g.Wang1970,itfollowsthat N canberepresented as N = N 1 I + N 2 + N 3 2 {21 where N 1 N 2 ,and N 3 arescalar-valuedfunctionsofallstressinvariants.Thirdinvariant dependencecannotbeinferredwithoutdataforwhichallthreeprincipalstressesare distinct,andsuchdatawerenotavailable.Thus,weconsider N tobeoftheform N = N 1 p;q I + N 2 p;q 0 q {22 Hence, I = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N 1 p;q I + N 2 p;q 0 q {23 Toestablishtheirreversiblestrain-rateorientationtensorwemustdeterminethe appropriateexpressionsfor N 1 and N 2 inEq.5{23.ForthecaseofCTCtests,the loadingaxesareprincipaldirectionsand_ I 2 =_ I 3 .Thus,Eq.5{23yields: I 1 = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N 1 p;q + 2 3 N 2 p;q {24 I 3 = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.087 Td [(W t H N 1 p;q )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(1 3 N 2 p;q : {25 Todeterminetheexpressionsfor N 1 and N 2 wehavetoperformseveralalgebraic manipulations.First,wesubtractEqs.5{24and5{25toisolate N 2 p;q .Wethen substitutetheexpressionfor N 2 p;q intoEq.5{24toisolate N 1 p;q .Performingthese 117

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steps,yieldsthefollowingexpressions, k T N 1 p;q = I v 3 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H {26 k T N 2 p;q = I 1 )]TJ/F15 11.9552 Tf 14.044 0 Td [(_ I 3 1 )]TJ/F21 11.9552 Tf 14.331 8.087 Td [(W t H : {27 Equation5{26indicatesthat N 1 p;q controlstheirreversiblevolumetricresponse,while N 2 p;q Eq.5{27controlstheirreversibleshearresponse. 5.5StrainRateOrientationTensor-k T N 1 p ; q InCTCtestsirreversiblevolumetricdeformationsoccurduringboththehydrostatic anddeviatoricphases,thusitcanbeassumedthatEq.5{26maybedecomposedinto hydrostaticanddeviatoriccomponentsgivenby k T N 1 p;q = p + p;q ; {28 suchthat p; 0=0.Therefore, p isdeterminedusingdatafromthehydrostaticphase ofCTCtests,while p;q utilizesdatafromthedeviatoricphaseofCTCtests.The variationof k T N 1 withtheappliedhydrostaticpressureforconningpressuresof20,50, 100,150,200,and300MPaisshowninFigure5-9.Itshouldbenotedthatthematerial compactstoapeak k T N 1 valueandsubsequentlydecreaseswithcontinuedloadinguntil I v =0andthematerialisfullycompacted.Thisisalsoshownbytheplateauinthe stabilizationfunction H H Figure5-3.Thisobservedvariationisapproximatedby: p = 0 + 1 4 2 6 4 1+exp 0 B @ p p c + 3 ln 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(' 2 3 1 C A 3 7 5 )]TJ/F23 5.9776 Tf 5.756 0 Td [(' 4 )]TJ/F18 5.9776 Tf 5.757 0 Td [(1 4 exp 0 B @ p p c + 3 ln 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(' 2 3 1 C A 4 +1 4 +1 4 ; {29 118

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Figure5-9.Variationofthevolumetricirreversiblestrain-rateorientationtensor, k T N 1 withpressureasobtainedfromhydrostatictests. where 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(7 : 56 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 1 =4 : 051 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 2 =68 : 245, 3 =11 : 588, 4 =76 : 303, and p c =1MPa.Acomparisonofthemodelwiththeexperimentaldataisgivenin Figure5-10.Todetermine p;q ,weutilizedEq.5{26inconjunctionwiththedata fromthedeviatoricphaseoftheCTCtests.TheresultscalculatedfromCTCtestsshow scatterofthedataforallvaluesof 3 withthedata.Asanexample,thedatafor 3 = 300MPaconnementarepresentedinFigure5-11.Thescatterinthedataoccursdue totransitioningfromtheLVDTtothelinearpotentiometerwhenmeasuringtheaxial deformation.AsdiscussedinChapter2,thelinearpotentiometerhaslessaccuracyandis utilizedoncethemeasurementrangeoftheLVDTisexceeded.Thisscattercanintroduce complexitieswhenmodelingthedata,therefore,wedecidedtosmooththedatared symbolsinFigure5-11.Thedatasmoothingminimizedthescatterwhileretainingthe qualityofthedataallowingittobeutilizedwhenmodeling p;q .Thesmootheddata fromtestsconductedatconningpressuresof20,50,100,150,200,and300MPaare presentedinFigure5-12. 119

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Figure5-10.ComparisonbetweentheoreticalEq.5{29andexperimentforthe irreversiblevolumetricorientationtensor, k T N 1 ,forhydrostaticloading; Experimentaldatasymbols;Eq.5{29solidline. Figure5-11.Experimentalvaluesoftheirreversiblevolumetricorientationtensor, k T N 1 calculatedforthedeviatoricphaseoftheCTCtestfor 3 =300MPa; Experimentdatasymbols;Smootheddatasolidline. 120

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Figure5-12.Experimentalvaluessmoothedofthestrain-rateorientationtensor, k T N 1 calculatedforthedeviatoricphasefromalltests. ThecreepcyclesconductedduringtheCTCtestswereremovedfromthedata, priortosmoothing,hencethegapsinthedatashowninFigure5-12.Asmentionedin Chapter2,themonotonictestscorrelatewellwiththecyclictestswithcreep,thususing theCTCtestswithcreepshouldbesucientformodeling p;q .ThedatainFigure 5-12initiallyrampsuptoapeakvalue,thentransitionstoalinearvariationwitha negativeslope,eventuallybecomingnegativeandtransitioningtoanasymptoticbehavior. Thevalueoftheprincipalstressdierence, q ,where k T N 1 becomeszeroindicatesthe compressibility/dilatancyboundaryrefertothedataanalysispresentedinChapter2. ThetrendsinFigure5-12makeitdicultifnotimpossibletomodelthebehaviorwitha singlemathematicalfunction,therefore, p;q isdecomposedintothefollowingbranches p;q = 8 > > < > > : p;q for q p )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 q p;q for q>p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q: {30 121

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Asanexample,Figure5-34showstherespectivebranchesforaCTCtestat 3 =300 MPa.Furthermore,thetransitionfromcompressibilitytodilatancyismodeledthrough p;q p;q models:theramptoapeak,andthelinearlydecreasingportion ofthedatapresentedinFigure5-12.Tounderstandtherstfeatureinthematerial behavior,i.e.thepeak,wemustdeterminewhereandwhythesepeaksoccur.Thisis accomplishedbydeterminingthepeak k T N 1 valueandthecorrespondingvalueof q Table 5-2.Additionally,thelocationsofthesepeaksarerepresentedbythestarsymbolsin Table5-2.Peakvaluesof k T N 1 frommonotonicCTCtests. 3 [MPa] q [MPa] k T N 1 2015.1920.02427 5027.1520.02751 10049.9700.02411 15093.1880.02256 20095.0780.02173 300133.0170.02575 Figure5-13.AxialstrainasafunctionofprincipalstressdierenceforallCTCtests; Experimentdatasolidlines;Peakvaluesfor k T N 1 symbols. 122

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Figure5-14.RadialstrainasafunctionofprincipalstressdierenceforallCTCtests; Experimentdatasolidlines;Peakvaluesfor k T N 1 symbols. thestress-straincurvesshowninFigures5-13and5-14.Itcanbeseenthatthesepeaksdo notexplicitlyrelatetotrendsinthevariationofthetotalstrainswiththeappliedstress. Togainabetterunderstandingtheirreversibleaxialstrain-rate,_ I 1 ,andirreversibleradial strain-rate,_ I 3 ,asfunctionsofprincipalstressdierencewerealsoplottedinFigures5-15 and5-16. Notethatthelocationswherethepeaksoccurrepresentthestresslevelwhere theradialstrain-rateincreasesmorerapidlythantheaxialstrain-rate.Moreprecisely, theaxialstrain-rateremainslargelyconstantFigure5-15whiletheradialstrain-rate continuouslyincreases. Aspreviouslymentioned,therawdatashowsalotofscatterandthusitwas smoothed.Figure5-11showstypicalsmoothingofthedataforaconningpressureof 123

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Figure5-15.Irreversibleaxialstrain-rateasafunctionofprincipalstressdierenceforall CTCtests;Experimentdatasolidlines;Peakvaluesfor k T N 1 symbols. Figure5-16.Irreversibleradialstrain-rateasafunctionofprincipalstressdierenceforall CTCtests;Experimentdatasolidlines;Peakvaluesfor k T N 1 symbols. 124

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300MPaalongwiththeproposedapproximation: p;q = 4 0 q p c )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 +1 1 2 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 2 q p c )]TJ/F22 7.9701 Tf 6.587 0 Td [( 2 2 1 + q p c )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 2 1 2 : {31 InEq.5{31,coecients 0 1 2 dependontheconningpressure 3 while p c =1 MPa.InEq.5{31, 0 controlsthepeak k T N 1 p;q valuewhile 1 istheprincipalstress dierencevaluewherethepeakoccurs.Aspreviouslydiscussed,thismaximumorpeak valuedoesnotexplicitlyrelatetoanycriticalfeaturewithintheexperimentaldata. However,accuracywasretainedwhenmodelingthisparameterwhileminimizingerror associatedwithmodelingthetransitionfromcompressibilitytodilatancy.Toreplicatethe trendswemodeledthevariationof 0 1 ,and 2 withconningpressure, 3 ,forallCTC testsconducted.Theexpressionsthatbestrepresentthevariationoftheseparametersare 0 = 0 ;a 3 p c + 0 ;b 3 p c 2 1+ 0 ;c 3 p c + 0 ;d 3 p c 2 ; {32 1 = 1 ;a 3 p c + 1 ;b 3 p c 2 1+ 1 ;c 3 p c + 1 ;d 3 p c 2 ; {33 2 = 2 ;a + 2 ;b 1+ 2 ;b 2 ;c 3 p c ; {34 where 0 ;a =6 : 096 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 0 ;b =3 : 101 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 0 ;c = )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 : 787 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 0 ;d =1 : 771 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 1 ;a =8 : 062 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 1 ;b =1 : 005 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 1 ;c =2 : 586 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 ;d = )]TJ/F15 11.9552 Tf 9.299 0 Td [(7 : 498 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 2 ;a =1 : 900, 2 ;b =2 : 111, 2 ;c =4 : 422 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ,and p c =1MPa.Equation5{32 waschosenduetoitsabilitytomodeltheentirematerialbehaviorforallvaluesof 3 Notethatforlowconningpressures, 0 isamonotonicallyincreasingfunctionwitha maximumoccurringat35MPa.Beyondthisconninglevel,thefunctionismonotonically 125

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decreasing.Comparisonbetweenexperimentandmodelfortheparameters 0 1 ,and 2 areshowninFigures5-17thru5-19whilecomparisonofexperimentandmodelfor p;q isgiveninFigures5-29thru5-34. Figure5-17.Comparisonbetweenexperimentandmodelforthevariationof 0 with conningpressure;Datasymbols;Eq.5{32solidline. Figure5-18.Comparisonbetweenexperimentandmodelforthevariationof 1 with conningpressure;Datasymbols;Eq.5{33solidline. 126

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Figure5-19.Comparisonbetweenexperimentandmodelforthevariationof 2 with conningpressure;Datasymbols;Eq.5{34solidline. Tocompletemodelingof k T N 1 p;q weneedtoapproximatethedatafor q> 3 suchastoensurecontinuitywiththebranch p;q ;whichcorrespondsto q 3 .Thisis accomplishedbyutilizingthesecondhalfofthesmoothedexperimentaldataforallCTC testsi.e.,Figure5-11.Theresponseofthesandmaterialforlowconningpressures 3 20MPaisdrasticallydierentthanthatfor 3 > 20MPa.Thus: p;q = 8 > < > : p;q for 3 < 20MPa p;q for 3 20MPa : {35 Wecanapproximatethebehaviorfor 3 < 20MPabyutilizingthetransitionfrom compressibilitytodilatancy.Theexpressionthatbestapproximatesthistransitionwhile ensuringcontinuitywithbranch p;q isgivenby p;q = 0 q p c + 1 ; {36 wherecoecients 0 and 1 dependontheconningpressure, 3 ,while p c =1MPa.The expressionsthatbestrepresentthevariationofthesecoecientswithconningpressure, 127

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3 ,are 0 = 0 ;a + 0 ;b 3 p c + 0 ;c 3 p c 2 + 0 ;d 3 p c 2 : 5 ; {37 1 = 1 ;a + 1 ;b 3 p c + 1 ;c 3 p c 2 + 1 ;d 3 p c 2 : 5 ; {38 where 0 ;a = )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 : 524 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 0 ;b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 554 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 0 ;c =8 : 913 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 0 ;d = )]TJ/F15 11.9552 Tf 9.298 0 Td [(8 : 066 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 1 ;a =2 : 158 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 1 ;b =5 : 538 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 1 ;c =3 : 371 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 1 ;d = )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 576 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 ,and p c =1MPa.ComparisonbetweenexperimentandmodelisgiveninFigures5-20and5-21. Figure5-20.Comparisonbetweenexperimentandmodelforthevariationof 0 with conningpressure;Datasymbols;Eq.5{37solidline. For q> 3 and 3 > 20MPa,asinmodeling k T N 1 p;q ,wealsousethetransition fromcompressibilitytodilatancy.Theexpressionthatbestdescribes p;q is p;q = 0 + 1 q p c 1+ 2 q p c + 3 q p c 2 {39 wherecoecients 0 1 2 ,and 3 dependonconningpressure, 3 ,with p c =1MPa. 128

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Figure5-21.Comparisonbetweenexperimentandmodelforthevariationof 1 with conningpressure;Datasymbols;Eq.5{38solidline. Thetrendsofthesecoecientsarenon-trivialandsomewhatinconsistentFigures 5-22and5-23,thus,toapproximatethesecoecientsamodiedmodelingapproachwas utilized.Thisrequiresthecoecients 1 and 3 tobescaledbycoecients 0 and 2 respectively.Asanexample,experimentalvaluesforthescaledcoecientsareshown inFigure5-24.Itisclearthatscalingthecoecientscreatesamorereasonabletrend. Furthermore,theinconsistencyintrendsfor 0 and 2 Figure5-22makesitdicultto extrapolatetohigherpressures. Thetrendfor 0 Figure5-22issomewhatinconsistent,however,itsvaluesincrease toamaximumthatoccursat50MPawhilebeyondthisconninglevelthevaluesof 0 consistentlydecrease.Therefore,itwasdecidedtoconsideramonotonicallyincreasing functionupto 3 =50MPafollowedbyamonotonicallydecreasingone.Thefunction thatbestrepresentsthevariationof 0 withconningpressure, 3 ,isexpressedas 0 = 0 ;a + 0 ;b 3 p c + 0 ;c 3 p c 2 1+ 0 ;d 3 p c + 0 ;e 3 p c 2 + 0 ;f 3 p c 3 ; {40 129

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Figure5-22.Experimentalvaluesforthecoecients 0 and 2 ofEq.5{39. Figure5-23.Experimentalvaluesforthecoecients 1 and 3 ofEq.5{39. 130

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Figure5-24.Experimentalvaluesforthescaledcoecients 1 0 and 3 2 involvedinEq. 5{39. where 0 ;a =2 : 217 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 0 ;b = )]TJ/F15 11.9552 Tf 9.298 0 Td [(6 : 037 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 0 ;c =2 : 715 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 0 ;d = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 465 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 0 ;e =5 : 656 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 0 ;f =1 : 556 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 ,and p c =1MPa.Thecomparisonbetween experimentandtheproposedvariationisshowninFigure5-25. Figure5-25.Comparisonbetweentheoreticalandexperimentalvariationof 0 with conningpressure;Datasymbols;Eq.5{40solidline. 131

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Thevariationof 2 1 0 ,and 3 2 withconningpressure, 3 ,weremodeledwiththe followingexpressions 2 = 8 > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > : 2 ;a + 2 ;b 3 p c 1 : 5 for 3 < 50MPa 2 6 6 4 2 ;c + 2 ;d 3 p c 1+ 2 ;e 3 p c 3 7 7 5 2 for50 3 297MPa 2 ;f + 2 ;g 3 p c for297 < 3 400MPa 2 ;h + 2 ;i 3 p c 0 : 5 1+ 2 ;j 3 p c 0 : 5 + 2 ;k 3 p c for 3 > 400MPa {41 where 2 ;a = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 254 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 2 ;b =1 : 481 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 2 ;c =4 : 183 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 2 ;d =1 : 892 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 2 ;e =3 : 922 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 2 ;f =6 : 600 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 2 ;g = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 00 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 2 ;h =5 : 528 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 2 ;i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 844 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 2 ;j = )]TJ/F15 11.9552 Tf 9.299 0 Td [(4 : 505 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 2 ;k = )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 : 160 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ,and p c =1MPa.The expressionsthatapproximateswellthevariationofthescaledcoecientswithconning pressure, 3 ,aregivenby 1 0 = 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 1 ;a + 1 ;b 3 p c 0 : 5 1+ 1 ;c 3 p c 0 : 5 for 3 < 100MPa 1 ;d + 1 ;e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F21 11.9552 Tf 9.298 0 Td [( 1 ;f 3 p c + + 1 ;g 2 6 6 4 1 )]TJ/F15 11.9552 Tf 55.036 8.087 Td [(1 1+ 1 ;g 1 ;h 3 p c 3 7 7 5 for 3 100MPa {42 132

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3 2 = 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : 3 ;a + 3 ;b 3 p c 2 1+ 3 ;c 3 p c 2 + 3 ;d 3 p c 4 for 3 < 100MPa 0 + 4 X k =1 k 3 p c k 1+ 5 X k =1 k 3 p c k for 3 100MPa {43 where 1 ;a =5 : 104 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 ;b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 611 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(3 1 ;c = )]TJ/F15 11.9552 Tf 9.298 0 Td [(5 : 337 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 1 ;d = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 334 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 ;e =1 : 079 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 1 ;f = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 039 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 1 ;g =1 : 352 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 1 ;h =5 : 952, 3 ;a = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 979 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 3 ;b = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 734 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 3 ;c =1 : 011 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 3 ;d = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 257 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(9 0 =5 : 082 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 879 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 458 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 3 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 539 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(9 4 =3 : 915 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(12 1 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 198 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 2 =9 : 435 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 3 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(3 : 652 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 4 =8 : 545 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(9 5 = )]TJ/F15 11.9552 Tf 9.298 0 Td [(8 : 900 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(12 ,and p c =1MPa.Thecomparisonbetween modelandexperimentfor 2 1 0 ,and 3 2 areshowninFigures5-26and5-28whileFigures 5-29thru5-34showcomparisonbetweenmodeli.e., p;q and p;q anddatafor k T N 1 p;q calculatedusingtheexpressions5{31thru5{43. TheresultsgiveninFigures5-29thru5-34indicatetheproposedmodelfor k T N 1 p;q modelswelltheobservedexperimentalbehavior.Thedierencebetweenexperimentand modelfor 3 =50and100MPaisaresultofthetheoreticalfailuresurfaceslightlyover predictingfailurefor 3 < 100MPaFigure5-7.Despitethesesmalldierences,the modelresponsepredictswellthetransitionfromcompressibilitytodilatancywiththe modeledresponsebecomingmoreaccurateas 3 increases. 5.6StrainRateOrientationTensor-k T N 2 p ; q Toconstructtheshearirreversiblestrainratetensor, k T N 2 p;q ,weutilizedEq. 5{27inconjunctionwithdatafromthedeviatoricphaseofallCTCtests.Asan example,thevariationof_ 1 )]TJ/F15 11.9552 Tf 16.421 0 Td [(_ 3 withtheappliedloadprincipalstressdierence correspondingtoaconningpressure, 3 =300MPa,isshowninFigure5-35.Note, 133

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Figure5-26.Comparisonbetweenexperimentalandtheoreticalvariationof 2 ;Data blacksymbols;Eq.5{41solidline. Figure5-27.Comparisonbetweenexperimentandtheoreticalvariationof 1 0 ;Datablack symbols;Eq.5{42solidline. 134

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Figure5-28.Comparisonbetweenexperimentandtheoreticalvariationof 3 2 ;Datablack symbols;Eq.5{43solidline. Figure5-29.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =20MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q 135

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Figure5-30.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =50MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q Figure5-31.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =100MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q 136

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Figure5-32.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =150MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q Figure5-33.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =200MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q 137

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Figure5-34.Comparisonbetweenexperimentalandtheoreticalvariationofthevolumetric irreversiblestrainratetensor, k T N 1 ,withloadfor 3 =300MPa;Experiment blacksymbols;Eq.5{30withthetwobranches p;q and p;q Figure5-35.Variationoftheshearirreversiblestrainratetensor, k T N 2 ,withloadfor 3 = 300MPa;Datasymbols;smootheddatasolidline. 138

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thatthereisscatterwithinthedata.However,thisscatterisobservedwhenanalyzingall otherCTCdata.Thisscatterintroducescomplexitieswhenmodelingthedata,therefore, itwasdecidedtosmooththedataFigure5-35.Thedatasmoothingminimizesthe scatterwhileretainingtheoveralltrend.Thesmootheddataforallconningpressuresare presentedinFigure5-36. Figure5-36.Experimentalvaluesoftheshearirreversiblestrain-rateorientationtensor, k T N 2 ,versusprincipalstressdierencecalculatedfromalltests. Thegapsbetweendatapoints,thatcanbeobservedinFigure5-36,areduetothe factthatonlytheloadingportionsofeachCTCtestwereanalyzedi.e.thecreepand unloadingdatapointswereremoved.ThedatainFigure5-36showanexponential variationreachingapeakvalue,followedbyalinearvariationwithasmallpositiveslope andanasymptoticbehaviorasfailureisapproached.ThevariationshowninFigure5-36 isverycomplexmakingitdicultifnotimpossibletomodelitwithasinglemathematical function.Thus, k T N 2 p;q isdecomposedintoseveralbranches. For 3 < 10MPa : k T N 2 p;q = p;q {44 139

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For 3 10MPa : k T N 2 p;q = 8 > > < > > : p;q for q p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p;q for q> p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : {45 p;q modelstheexponentialvariationofthedatainFigure5-36toapeakvalue.To ensurecontinuitybetweenthisbranchandthatmodelingthelinearvariationof k T N 2 ,it wasdecidedtosetthebranchingpointat q = 3 )]TJ/F15 11.9552 Tf 12.083 0 Td [(1.Theaverage k T N 2 valueswithinthe linearportionofthedataandthecorrespondingvalueofprincipalstressdierence, q are giveninTable5-3andrepresentedbystarsymbolsinFigure5-37. Table5-3.Average k T N 2 valuesfor p;q 3 [MPa]Average q [MPa]Average k T N 2 5069.000.1316 100114.000.1113 150169.000.1072 200238.000.1089 300371.000.1018 Figure5-37.Average k T N 2 valuesfor p;q ;Experimentdatasymbols;Average k T N 2 valuesstarsymbols 140

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ThemodelthatbestapproximatesthesmootheddataFigure5-36isexpressedby p;q = 0 )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [( 2 0 + 1 2 q p c )]TJ/F21 11.9552 Tf 11.956 0 Td [( 1 q p c 1 1 )]TJ/F23 5.9776 Tf 5.756 0 Td [( 2 {46 where 0 1 ,and 2 arecoecientsthatdependonconningpressure, 3 ,while p c =1 MPa.InEq.5{46, 0 controlsthepeakvaluewith 1 and 2 bothinuencingthegrowth rate.Themodelsthatbestrepresentthevariationofthecoecientswithconning pressureareexpressedas 0 = 0 ;a + 0 ;b exp )]TJ/F15 11.9552 Tf 15.5 8.088 Td [(1 0 ;c 3 p c ; {47 1 = 1 ;a + 1 ;b 3 p c + 1 ;c 3 p c 2 1+ 1 ;d 3 p c + 1 ;e 3 p c 2 ; {48 2 = 2 ;a 1+ 1 2 ;b 3 p c 2 ;c ; {49 where 0 ;a =9 : 598 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 0 ;b =4 : 454 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 0 ;c =135 : 95, 1 ;a =1 : 251 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 ;b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 584 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 1 ;c =1 : 038 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(6 1 ;d = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 667 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 1 ;e =9 : 674 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 2 ;a =9 : 987 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 ;b =37 : 784,and 2 ;c = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 987.Thecomparisonbetweenexperiment andmodelfor 0 1 ,and 2 areshowninFigures5-38thru5-40.Comparisonbetween experimentandmodeloftherstbranch p;q isshowninFigures5-45thru5-49. Tocompletemodelingof k T N 2 p;q weneedtoapproximatethedatafor q> 3 )]TJ/F15 11.9552 Tf 12.192 0 Td [(1, suchastoensurecontinuitywiththebranch p;q ;whichcorrespondsto q 3 )]TJ/F15 11.9552 Tf 11.962 0 Td [(1.This isaccomplishedbyutilizingthesecondhalfofthesmoothedexperimentaldataforCTC testsfor 3 =50,10,150,200,and300MPai.e.,Figure5-35.Thetrendfor 3 =20 MPavariesfromthatof 3 > 20MPa,therefore,experimentaldatafor 3 =20MPawas excludedwhenmodeling p;q .Theexpressionof p;q thatbestapproximatesthedata 141

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Figure5-38.Comparisonbetweentheoreticalandexperimentalvariationof 0 with conningpressure;Experimentaldatasymbols;Eq.5{47solidline. Figure5-39.Comparisonbetweentheoreticalandexperimentalvariationof 1 with conningpressure;Experimentaldatasymbols;Eq.5{48solidline. 142

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Figure5-40.Comparisonbetweentheoreticalandexperimentalvariationof 2 with conningpressure;Experimentaldatasymbols;Eq.5{49solidline. isgivenas p;q = 0 + 1 1+ 1 3 q p c )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 2 {50 where 0 1 2 ,and 3 arecoecientsthatdependonconningpressure, 3 with p c =1 MPa.InEq.5{50, 0 representstheintercept, 1 thepeak k T N 2 valuewhilethe correspondingprincipalstressdierence, q wherethepeakoccurs,andgrowthrateare representedby 2 and 3 ,respectively.Note,thepeaksrepresentedby 1 occurfollowing failureEq.5{19,thereforetheexperimentalpeakand/orasymptotic k T N 2 values occurpriorto 1 .However,wesystematicallyvariedthecoecientstoapproximatethe experimentalasymptotic k T N 2 values.Toreplicatethetrendswemodeledthevariation of 0 1 2 ,and 3 withconningpressure, 3 ,forCTCtestsconductedfor 3 =50,100, 150,200,and300MPa.Themodelschosenthatbestapproximatethesecoecientsare expressedas 0 = 0 ;a + 0 ;b exp )]TJ/F15 11.9552 Tf 15.5 8.088 Td [(1 0 ;c 3 p c ; {51 143

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1 = 8 > > > > > > > < > > > > > > > : 1 ;a exp )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 1 1 ;c 3 p c )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 ;b 2 for 3 97MPa 1 ;d 1+ 1 1 ;e 3 p c 1 ;f )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 ;g for 3 > 97MPa ; {52 2 = 2 ;a 1 )]TJ/F15 11.9552 Tf 54.809 8.088 Td [(1 1+ 2 ;a 2 ;b 3 =p c ; {53 3 = 3 ;a 3 p c + 3 ;b 3 p c 2 + 3 ;c 3 p c 3 1+ 3 ;d 3 p c + 3 ;e 3 p c 2 + 3 ;f 3 p c 3 : {54 where 0 ;a =9 : 966 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 0 ;b =1 : 244 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 0 ;c =36 : 756, 1 ;a =63 : 025, 1 ;b =52 : 448, 1 ;c =13 : 882, 1 ;d =206 : 39, 1 ;e =412 : 17, 1 ;f = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 : 649, 1 ;g =7 : 191 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 ;a =92226, 2 ;b =2 : 872 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(10 3 ;a =6 : 769 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 3 ;b = )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 631 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 3 ;c =2 : 860 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 3 ;d = )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 351 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 3 ;e = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 814 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 ,and 3 ;f =6 : 857 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(7 .Thecomparison betweenexperimentandmodelfor 0 1 2 ,and 3 areshowninFigures5-41thru5-44. Comparisonbetweenexperimentandmodeli.e. p;q isshowninFigures5-45thru 5-49. TheresultsgiveninFigures5-45thru5-49indicatethattheproposedmodelfor k T N 2 p;q modelswelltheobservedexperimentalbehavior.Thedierencebetween experimentandmodelfor 3 =50and100MPaisaresultofthetheoreticalfailure surfaceslightlyoverpredictingfailurefor 3 < 100MPaFigure5-7.Therefore,the modelof k T N 2 p;q mustsatisfythetheoreticalfailuresurfacewiththemodeledresponse onlyrequireduptotheoreticalfailure.Despitethesesmalldierences,themodelresponse predictswelltheexperimentalbehaviorwiththemodeledresponsebecomingmore accurateas 3 increases. 144

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Figure5-41.Comparisonbetweentheoreticalandexperimentalvariationof 0 with conningpressure;Experimentaldatasymbols;Eq.5{51solidline. Figure5-42.Comparisonbetweentheoreticalandexperimentalvariationof 1 with conningpressure;Experimentaldatasymbols;Eq.5{52solidline. 145

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Figure5-43.Comparisonbetweentheoreticalandexperimentalvariationof 2 with conningpressure;Experimentaldatasymbols;Eq.5{53solidline. Figure5-44.Comparisonbetweentheoreticalandexperimentalvariationof 3 with conningpressure;Experimentaldatasymbols;Eq.5{54solidline. 146

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Figure5-45.Comparisonbetweenexperimentalandtheoreticalvariationoftheshear irreversiblestrainratetensor, k T N 2 ,withloadfor 3 =50MPa;Experiment blacksymbols;Eq.5{45withthetwobranches p;q and p;q Figure5-46.Comparisonbetweenexperimentalandtheoreticalvariationoftheshear irreversiblestrainratetensor, k T N 2 ,withloadfor 3 =100MPa;Experiment blacksymbols;Eq.5{45withthetwobranches p;q and p;q 147

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Figure5-47.Comparisonbetweenexperimentalandtheoreticalvariationoftheshear irreversiblestrainratetensor, k T N 2 ,withloadfor 3 =150MPa;Experiment blacksymbols;Eq.5{45withthetwobranches p;q and p;q Figure5-48.Comparisonbetweenexperimentalandtheoreticalvariationoftheshear irreversiblestrainratetensor, k T N 2 ,withloadfor 3 =200MPa;Experiment blacksymbols;Eq.5{45withthetwobranches p;q and p;q 148

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Figure5-49.Comparisonbetweenexperimentalandtheoreticalvariationoftheshear irreversiblestrainratetensor, k T N 2 ,withloadfor 3 =300MPa;Experiment blacksymbols;Eq.5{45withthetwobranches p;q and p;q 149

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CHAPTER6 SUMMARYANDVALIDATIONOFANELASTIC-VISCOPLASTICMODEL Intheprevioussectionanewelastic-viscoplasticconstitutivemodelforthehighpressureresponseofQuikrete R sandwasdeveloped.Theframeworkusedtodevelopthe modelwastheelastic-viscoplasticframeworkproposedinCristescu1989,1991,1994 whereno apriori assumptionsaremadeconcerningthemathematicalformoftheyield functionorplasticpotential.Therefore,theyieldfunctionandplasticpotentialcanbe constructeddirectlyfromtheexperimentaldataforquasi-staticCTCtestswithcreep. Thedetailsconcerningtheconstructionofthemodelwaspreviouslypresented,thusonly abriefsummarywillbegivenhere.Note,allmaterialparameters,values,andunitswill beprovideddirectlyfollowingthemodelsummary.Thesummaryisfollowedbydiscussion ofthenumericalalgorithmutilizedtosimulatetheexperimentaldatausingtheproposed elastic-viscoplasticmodelanditsimplementationinMatLab R .Finally,comparison betweenthesimulatedresponseanddataarepresented. 6.1Elastic-ViscoplasticModelSummary Theconstitutivehypothesesofthismodelare: H1Thematerialisassumedtobehomogeneousandisotropic H2Thedisplacementsandmaterialrotationsbeingsmall,theelasticandplastic componentsoftherateofdeformationtensorareadditive. = E + I {1 TheelasticcomponentisdeterminedfromaHooketypelaw E = 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.767 8.088 Td [(1 2 G p I {2 wheretheover-dotstandsforthederivativewithrespecttotimei.e.increment, K and G thebulkandshearmoduli,respectively, I isthe2 nd orderidentitytensor,and p the meanstressi.e. p = I 1 = 3= tr = 3.InEq.6{2 K and G aredeterminedfromthe 150

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slopesofthequasi-linearportionsoftheunloading-reloadingcyclesperformedattheend ofeachcreepcycleintheCTCtestsfor 3 =20,50,100,150,200,and300MPa. Young'sModulus E : E p = E 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(E s exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(bp p c {3 where p isthemeannormalpressure.ParametersinvolvedinEq.6{3are: p c =1MPa, E 1 theasymptoticlimitof E E s ,and b .TheirvaluesaregiveninTable6-1. Poissoncoecient, =0 : 22 : while G and K are: G = E 2+ ;K = E 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 {4 H3Theplasticcomponentoftherateofdeformationisobtainedfromtheowrule I = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N ; {5 whentheyieldconditionissatised.Here t istheactualtime, W t theirreversible stressworkperunitvolume,andthetensor N givestheorientationoftheviscoplastic strain-rate.Theonlyrestrictiononthetensorvaluedfunction N istobeisotropic. ThebracketinEq.6{5istheMacaulayBracket,denoting h x i = x + j x j = 2,while k T is aviscositycoecient. H representsthestabilizationboundary,i.e.thelocusofpoints instressspacewheretransientcreependsaftersomenitetimeinterval. H4Isotropichardeningrepresentedbytheaccumulatedirreversiblestressworkper unitvolume W isconsidered. H5Theyieldfunctiondependson throughinvariants p I 1 = 3and q Eq.5{10 suchthat: H p;q = H H p + H D p;q {6 H H p ,iscalibratedbyapproximatingtheirreversiblestressworkEq.5{6awith pressure, p ,atcreepstabilizationcorrespondingtothehydrostaticphaseofCTCtestsat 151

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3 =20,50,100,150,200,and300MPa. H H p = a h 2 6 6 4 1+ b h exp )]TJ/F21 11.9552 Tf 9.299 0 Td [(c h p p c )]TJ/F21 11.9552 Tf 11.955 0 Td [(c h exp )]TJ/F21 11.9552 Tf 9.298 0 Td [(b h p p c c h )]TJ/F21 11.9552 Tf 11.955 0 Td [(b h 3 7 7 5 : {7 Itinvolvesparameters a h b h c h ,and p c Table6-1. H D p;q ,isconstructedbyapproximatingtheirreversiblestressworkvs.principal stressdierencei.e.eectivestresscorrespondingtocreepstabilizationinthedeviatoric phaseoftheCTCtestsatallvaluesof 3 =20,50,100,150,200,and300MPa. H D p;q = d 0 + d 1 q p c 2 {8 where d 0 3 = d 0 ;a 3 p c 1+ d 0 ;b 3 p c + d 0 ;c 3 p c 2 {9 d 1 3 = d 1 ;a 1+ 1 d 1 ;b 3 p c d 1 ;c : {10 Parameters d 0 ;a d 0 ;b d 0 ;c d 1 ;a d 1 ;b d 1 ;c ,and p c aregiveninTable6-1. Failurestressisdeterminedbythemaximumprincipalstressdierenceorthe principalstressdierenceat15%axialstrain,whicheveroccursrst,inCTCtests. f = q )]TJ/F21 11.9552 Tf 11.955 0 Td [(p )]TJ/F21 11.9552 Tf 11.956 0 Td [(k {11 where istheangleofinternalfrictionand k thecohesivestrength.Parameters and k aregiveninTable6-1. Irreversiblevolumetricresponse, k T N 1 p;q : k T N 1 p;q = p + p;q {12 152

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Thehydrostaticcomponent, p ,iscalibratedbyapproximatingthevariation ofthemeannormalstress, p ,withtheirreversiblevolumestrain-rate,_ I v ,fromthe hydrostaticphaseofallCTCexperiments. p = 0 + 1 4 1+exp p p c + 3 ln 4 )]TJ/F21 11.9552 Tf 11.956 0 Td [(' 2 3 !# )]TJ/F23 5.9776 Tf 5.756 0 Td [(' 4 )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 4 exp p p c + 3 ln 4 )]TJ/F21 11.9552 Tf 11.955 0 Td [(' 2 3 4 +1 4 +1 4 {13 Thedeviatoriccomponent, p;q ,determinestheirreversiblestrain-rateorientations duringdeviatoricloadingandadditionallycontrolsthematerialstransitionfroma stateofcompressiontooneofdilatancy.Thedeviatoriccomponentiscalibratedby approximatingthevariationwiththeprincipalstressdierence, q ,withtheirreversible volumestrain-rate,_ I v ,correspondingtothedeviatoricphaseofallCTCexperiments. p;q = 8 > > < > > : p;q for q p )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 q p;q for q>p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q: {14 where p;q p;q = 4 0 q p c )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 2 +1 1 2 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 2 q p c )]TJ/F22 7.9701 Tf 6.587 0 Td [( 2 2 1 + q p c )]TJ/F22 7.9701 Tf 6.586 0 Td [( 2 2 1 2 ; {15 wherethepressuredependentparametersareexpressedas 0 3 = 0 ;a 3 p c + 0 ;b 3 p c 2 1+ 0 ;c 3 p c + 0 ;d 3 p c 2 ; {16 1 3 = 1 ;a 3 p c + 1 ;b 3 p c 2 1+ 1 ;c 3 p c + 1 ;d 3 p c 2 ; {17 153

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2 3 = 2 ;a + 2 ;b 1+ 2 ;b 2 ;c 3 p c : {18 while p;q hasseveralbranches: For 3 < 20MPa: p;q = 0 q p c + 1 {19 where 0 3 = 0 ;a + 0 ;b 3 p c + 0 ;c 3 p c 2 + 0 ;d 3 p c 2 : 5 {20 1 3 = 1 ;a + 1 ;b 3 p c + 1 ;c 3 p c 2 + 1 ;d 3 p c 2 : 5 {21 For 3 20MPa: p;q = 0 + 1 q p c 1+ 2 q p c + 3 q p c 2 {22 wherethepressuredependentparametersaregivenby 0 3 = 0 ;a + 0 ;b 3 p c + 0 ;c 3 p c 2 1+ 0 ;d 3 p c + 0 ;e 3 p c 2 + 0 ;f 3 p c 3 {23 154

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1 3 = 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 2 6 6 6 4 1 ;a + 1 ;b 3 p c 0 : 5 1+ 1 ;c 3 p c 0 : 5 3 7 7 7 5 0 for 3 < 100MPa 1 ;d + 1 ;e 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F21 11.9552 Tf 9.298 0 Td [( 1 ;f 3 p c 0 + + 0 B B @ 1 ;g 2 6 6 4 1 )]TJ/F15 11.9552 Tf 55.037 8.088 Td [(1 1+ 1 ;g 1 ;h 3 p c 3 7 7 5 1 C C A 0 for 3 100MPa {24 2 3 = 8 > > > > > > > > > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > > > > > > > > > : 2 ;a + 2 ;b 3 p c 1 : 5 for 3 < 50MPa 2 6 6 4 2 ;c + 2 ;d 3 p c 1+ 2 ;e 3 p c 3 7 7 5 2 for50 3 297MPa 2 ;f + 2 ;g 3 p c for297 < 3 400MPa 2 ;h + 2 ;i 3 p c 0 : 5 1+ 2 ;j 3 p c 0 : 5 + 2 ;k 3 p c for 3 > 400MPa {25 3 3 = 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : 0 B B B @ 3 ;a + 3 ;b 3 p c 2 1+ 3 ;c 3 p c 2 + 3 ;d 3 p c 4 1 C C C A 2 for 3 < 100MPa 0 B B B B @ 0 + 4 X k =1 k 3 p c k 1+ 5 X k =1 k 3 p c k 1 C C C C A 2 for 3 100MPa {26 155

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Shearirreversibleresponse, k T N 2 p;q : determinestheirreversiblestrain-rate orientationsduringdeviatoricloadingonlyandmaybecalibratedbyapproximatingthe variationoftheprincipalstressdierence, q ,withtheirreversibleshearstrain-rate,_ I 1 )]TJ/F15 11.9552 Tf 13.992 0 Td [(_ I 3 fromthedeviatoricphaseofallCTCexperiments. For 3 < 10MPa: k T N 2 p;q = p;q {27 For 3 10MPa: k T N 2 p;q = 8 > > < > > : p;q for q p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 p;q for q> p )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 3 q )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : {28 where p;q = 0 )]TJ/F26 11.9552 Tf 11.955 16.857 Td [( 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [( 2 0 + 1 2 q p c )]TJ/F21 11.9552 Tf 11.956 0 Td [( 1 q p c 1 1 )]TJ/F23 5.9776 Tf 5.756 0 Td [( 2 {29 with 0 3 = 0 ;a + 0 ;b exp )]TJ/F15 11.9552 Tf 15.5 8.088 Td [(1 0 ;c 3 p c ; {30 1 3 = 1 ;a + 1 ;b 3 p c + 1 ;c 3 p c 2 1+ 1 ;d 3 p c + 1 ;e 3 p c 2 ; {31 2 3 = 2 ;a 1+ 1 2 ;b 3 p c 2 ;c : {32 while p;q p;q = 0 + 1 1+ 1 3 q p c )]TJ/F21 11.9552 Tf 11.955 0 Td [( 2 2 {33 wherethepressuredependentmaterialparametersareexpressedas 0 3 = 0 ;a + 0 ;b exp )]TJ/F15 11.9552 Tf 15.5 8.088 Td [(1 0 ;c 3 p c ; {34 156

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1 3 = 8 > > > > > > > < > > > > > > > : 1 ;a exp )]TJ/F15 11.9552 Tf 10.494 8.087 Td [(1 2 1 1 ;c 3 p c )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 ;b 2 for 3 97MPa 1 ;d 1+ 1 1 ;e 3 p c 1 ;f )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 ;g for 3 > 97MPa ; {35 2 3 = 2 ;a 1 )]TJ/F15 11.9552 Tf 54.81 8.088 Td [(1 1+ 2 ;a 2 ;b 3 =p c ; {36 3 3 = 3 ;a 3 p c + 3 ;b 3 p c 2 + 3 ;c 3 p c 3 1+ 3 ;d 3 p c + 3 ;e 3 p c 2 + 3 ;f 3 p c 3 : {37 Thisconcludesthesummaryoftheproposedelastic-viscoplasticmodelwithallmaterial parametersdocumentedinthefollowingtable. Table6-1.Materialparametervaluesfortheproposedelastic-viscoplasticmodel. ParameterValueUnits E 1 23.00GPa E s 22.00GPa b 0.00619{ 0.22{ p c 1.00MPa k 0.00MPa 1.301{ a h 30.0MPa b h 3.45 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { c h 4.830 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { d 0 ;a 1.875 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 MPa d 0 ;b -7.688 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { d 0 ;c 3.692 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 { d 1 ;a 9.075 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 MPa d 1 ;b 14.45{ d 1 ;c 1.444{ 157

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Table6-1.Continued ParameterValueUnits 0 -7.560 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 1 4.051 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 2 68.245{ 3 11.588{ 4 76.303{ 0 ;a 6.096 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 0 ;b 3.101 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 0 ;c -3.787 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;d 1.771 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 1 ;a 8.062 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 1 ;b 1.005 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;c 2.586 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;d -7.498 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 { 2 ;a 1.900{ 2 ;b 2.111{ 2 ;c 4.422 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;a -3.524 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 0 ;b -1.554 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 0 ;c 8.913 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 0 ;d -8.066 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 { 1 ;a 2.158 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 1 ;b 5.538 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 1 ;c 3.371 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 1 ;d -5.576 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 { 0 ;a 2.217 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;b -6.037 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 0 ;c 2.715 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 { 0 ;d -3.465 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;e 5.656 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 0 ;f 1.556 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 { 1 ;a 5.104 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;b -2.611 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 1 ;c -5.337 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 1 ;d -1.334 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;e 1.079 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;f -1.039 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;g 1.352 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 1 ;h 5.952{ 2 ;a -2.254 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 2 ;b 1.481 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 2 ;c 4.183 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 2 ;d 1.892 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 158

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Table6-1.Continued ParameterValueUnits 2 ;e 3.922 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 2 ;f 6.600 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 2 ;g -2.000 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 2 ;h 5.528 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 2 ;i -2.844 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 2 ;j -4.505 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 2 ;k -3.160 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 3 ;a -2.979 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 3 ;b -2.734 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 3 ;c 1.011 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 3 ;d -3.257 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(9 { 0 5.082 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 -3.879 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 2 -1.458 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 3 -3.539 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(9 { 4 3.915 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(12 { 1 -1.198 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 2 9.435 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 3 -3.652 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 4 8.545 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(9 { 5 -8.900 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(12 { 0 ;a 9.598 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;b 4.454 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;c 135.95{ 1 ;a 1.251 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;b -1.584 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 1 ;c 1.038 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 1 ;d -1.667 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 1 ;e 9.674 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 { 2 ;a 9.987 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 2 ;b 37.784{ 2 ;c -1.987{ 0 ;a 9.966 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 0 ;b 1.244 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 0 ;c 36.756{ 1 ;a 63.025{ 1 ;b 52.448{ 1 ;c 13.882{ 1 ;d 206.39{ 1 ;e 412.17{ 1 ;f -3.649{ 1 ;g 7.191 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 { 159

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Table6-1.Continued ParameterValueUnits 2 ;a 92226{ 2 ;b 2.872 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(10 { 3 ;a 6.769 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 { 3 ;b -2.631 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 { 3 ;c 2.860 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(6 { 3 ;d -1.351 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 { 3 ;e -2.814 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(5 { 3 ;f 6.857 10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(7 { 6.2IntegrationAlgorithm Theelastic-viscoplasticmodelwasimplementedintothecommercialsoftwareMatLab toallowustosolvethegoverningdierentialequationsnumercially.Thedierential equationswerediscretizedusingarstorderEulerexplicitnumericalschemehaving asinputthestressdatafromthedeviatoricphaseofquasi-staticCTCandhydrostatic compressionHCexperimentsundermonotonicloading.Thedetailsconcerningthe implementationofthemodelinMatLabwillbesummarizedherefollowedbyacomparison betweentheexperimentsandmodelattheendofthesection.Tovalidatetheproposed modelwemustreplicatethequasi-staticCTCtestnumericallyusingthesameboundary conditionsappliedduringtheexperiments.Allquasi-statictests,bothCTCandHC,are stresscontrolled.Thus,weinputstressesandcalculatestrainsbysolvingthefollowing systemofequations: 1 = 1 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.768 8.088 Td [(1 2 G p + k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N 1 p;q + 2 3 N 2 p;q ; {38 3 = 3 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.768 8.088 Td [(1 2 G p + k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H N 1 p;q )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 3 N 2 p;q : {39 where_ 1 istheaxialstrain-rateand_ 3 theradialstrain-ratewith p = I 1 = 3and q givenby Eq.5{10.ForagivenCTCtest 3 = constant inthedeviatoricphaseand q isincreased linearlywithtime. 160

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Tosolvethesetofequationsfortheproposedmodelnumericallywemustestablisha setofinitialconditions.Aspreviously,mentionedtoobtainthecorrectvalueofirreversible stresswork,forCTCtests,wemustinitializethestressdatatoincludethehydrostatic eects.Thisensuresthattheinitialvalueofirreversiblestressworkisequaltothevalue ofirreversiblestressworkattheendofthehydrostaticphase.Theinitialconditionsfor thehydrostaticcompressiontestsare W j t =0 = W 0 =0 j t =0 = 0 =0 j t =0 = 0 =0 ; {40 whileforthedeviatoricphaseofCTCtests W j t =0 = W 0 = W t = t H j t =0 = 0 = 3 j t =0 = 0 =0 : {41 ToevaluatetheMacaulaybracketwemustestablishthenecessaryconditionstodetermine whetherthematerialisdeformingelasticorplastic.Theseare: if H n +1 W n W n +1 = W n Elastic if H n +1 >W n Evaluate W: Plastic {42 ThesystemissolvedusinganexplicitEulerscheme.Thus, W n +1 = W n + t W {43 n +1 = n + t {44 where t isthetimeincrement, W theincrementofirreversiblestresswork,and_ the incrementofstrain.Theincrementofirreversiblestressworkcanbedeterminedby rearrangingEq.5{5andsubstitutingEq.6{5.Thus W = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.087 Td [(W t H : N {45 161

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wheretheinnerproduct, : N ,hasyettobedetermined.Todeterminetheinner product,wesubstituteEq.5{22andobtain : N = N 1 p;q : I + N 2 p;q q : 0 ; {46 wherefurthersimplicationforCTCtests : N =3 pN 1 p;q + 2 3 qN 2 p;q : {47 Theincrementofirreversiblestressworkisthusgivenby W = k T 1 )]TJ/F21 11.9552 Tf 14.331 8.088 Td [(W t H 3 pN 1 p;q + 2 3 qN 2 p;q : {48 Note,forHCexperimentsthelatterterminEq.6{47willbezero.Thegeneralized discretizedformsofthetotalstrainandirreversiblestressworkaregivenbelow n +1 = n + n +1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( n 2 G + 1 3 K )]TJ/F15 11.9552 Tf 17.767 8.087 Td [(1 2 G tr n +1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( n 3 + + t k T 2 1 )]TJ/F21 11.9552 Tf 20.217 8.088 Td [(W n +1 H n +1 N n +1 + 1 )]TJ/F21 11.9552 Tf 20.217 8.088 Td [(W n H n N n ; {49 W n +1 = W n + t k T 1 )]TJ/F21 11.9552 Tf 21.488 8.088 Td [(W n H n n : N n : {50 6.3NumericalSimulationofQuasi-staticCTCTests Numericalsimulationswereconductedfortwodierentcases,monotonichydrostatic compressionupto0.5GPameanstress,andmonotonicdeviatoricCTCcasefor20 3 300MPa.ForeachcasethenumericalschemepresentedinSection6.2was usedandcomparisonsweremadewithexperimentaldata.Notethattheproposedmodel captureswellthehydrostaticcompressionresponseshowninFigure6-1,however,the modelpredictsamorecompliantresponse.Despitebeingmorecompliantthedierence involumestrainisontheorderof2%foragivenvalueofthemeanstresswhichiswell withinexperimentalerror.Possiblyotherapproximationsof H H p and p during 162

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hydrostaticloadingEqs.6{7and6{13couldprovidebetteragreementinthelow pressurerangei.e. p 20MPa,however,theemphasisinmodelingwasplacedonthe highpressurebehaviorsinceitismorerepresentativeforfuturepenetrationapplications. TheresponseaccordingtothemodelduringthedeviatoricphaseofCTCtestsis givenFigures6-2thru6-7.Itisclearthatthetheoreticalresponsepredictsverywellthe highpressurebehaviorwhiletheagreementissatisfactoryforvaluesof 3 50MPa. Thisisexpectedduetothemodelemphasizingthehighpressureresponse,aspreviously mentioned.Despitethisshortcoming,themodelpredictswellthetransitionfromastate ofcompressibilitytooneofdilatancyforallvaluesof 3 Figure6-1.Comparisonbetweenexperimentalandtheoreticalresultsforhydrostatic loadingofQuikrete R sand.Experimentaldatasolidline;Theoreticalresults dashedline. 163

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Figure6-2.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =20MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. Figure6-3.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =50MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. 164

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Figure6-4.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =100MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. Figure6-5.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =150MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. 165

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Figure6-6.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =200MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. Figure6-7.Comparisonbetweenexperimentalandtheoreticalresultsforthedeviatoric phaseofCTCtestsat 3 =300MPaforQuikrete R sand.Experimentaldata solidline;Theoreticalresultsdashedline. 166

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CHAPTER7 CONCLUSIONS Tocharacterizethequasi-staticmechanicalresponseofQuikrete R ] 1961sandat highpressurestypicallyencounteredinpenetrationevents,acomprehensiveexperimental investigationhasbeenconducted.Thequasi-staticexperimentswereconductedwith strainratesontheorderof10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 to10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(5 s )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 .Themechanicalresponsewasinvestigated underavarietyofloadingconditionstoestablishtheinuenceofconningpressureon strength,elasticproperties,andcompressibility/dilatancybehavior.Thehydrostatic experimentshaverevealedthatthepressure-volumeresponseishighlycompressible.The maximumvolumetricstrainat300MPahydrostaticpressurewasapproximately30%. Bothmonotonicandcyclictriaxialcompressiontestswereperformedforconning pressuresrangingfrom20-300MPa.Toreducehystereticeectsassociatedtotheviscosity ofthematerial,priortounloadingshorttermcreepstageswereperformed.Thisallowed amoreaccurateestimateoftheelasticmoduli.Itwasobservedthattheelasticmoduli dependonthestressstate.Evolutionlawsthatrepresentthisdependencehavebeen established. Alltheconningtestsshowthatthematerialdisplaysastrongnon-linearbehavior. Underdeviatoricconditions,thematerialfurthercompactsandthendilateswiththe transitionfromcompressibilitytodilatancybeingafunctionoftheconningpressure. Furthermore,creeptestsshowthattherearetimeeectsontheirreversibleresponse. ToinvestigatetheinuenceofloadingrateonthematerialbehaviorconnedKolsky bartestswereconducted.Alldynamictestswereconductedatastrainrateof1000s )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 Forthersttime,highqualitydatathatshowtheinuenceofconningpressureonthe dynamicresponsewereobtained.Thehighconnementpressurewasappliedusinga conningcellmountedatthebar/specimeninterface.Conningpressuresof25,50,100, and125MPawereimposed.Thisdataextendstheexistingexperimentaldatabaseonsand theconnementsreportedintheliteraturebeingrathersmalllessthan6MPa.While 167

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itwasshownthatminimalstrainrateeectsexist,theobservedresponsewasfoundtobe highlynon-linearandpressuredependent. Tomodeltheobservedresponse,themostperformantmodelsexistingintheliterature havebeenapplied.Itwasshownthatthebasichypothesisonwhichsuchmodelsrelydo notapplytotherangeofpressuresinvestigated.Therefore,amodelingeorthasbeen undertaken.Sincetestresultshaveshownthatthematerialdisplaysirreversibletime dependentshort-termcreepeectsbehavior,anelastic-viscoplasticmodelingframework hasbeenadopted.No apriori assumptionsregardingthespecicexpressionsoftheyield functionandviscoplasticpotentialweremade. Theyieldfunctionexpressioninvolvesahydrostaticanddeviatoriccomponent, respectively.Theeectofaccumulatedirreversibledeformationonyieldinghasbeen modeledusingtheirreversibleplasticworkperunitvolumeasinternalstatevariable. Itwasdemonstratedthatanassociatedowruledoesnotapply.Comparisonof theexperimentalcompressibility/dilatancyboundarywiththeboundarydetermined assumingassociatedowshowsclearlytheneedofannon-associatedowrule.Irreversible behaviorwasdescribedthrougha2 nd orderorientationtensor.Comparisonbetweenmodel predictionsanddatashowedtheproposedmodeldescribesverywellthehigh-pressure behaviorofQuikrete R ] 1961sand. 168

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REFERENCES ABAQUS,2008.Referencemanuals.DassaultSystemesSimuliaCorporation,Providence, RI,version6.8. ASTM-D2487,2006.StandardPracticeforClassicationofSoilsforEngineering PurposesUniedSoilClassicationSystem.ASTMInternational,WestConshohocken, PA,version:2006e1,DOI:10.1520/D2487-06E01,www.astm.org. Baladi,G.Y.,Rohani,B.,1982.Anelastic-viscoplasticconstitutivemodelforearth materials.Tech.Rep.SL-82-10,WaterwaysExperimentationStation. Bingham,E.C.,1922.FluidityandPlasticity.McGrawHill,NewYork. Bishop,A.W.,Henkel,D.J.,1962.Themeasurmentofsoilpropertiesinthetriaxialtest. EdwardArnold,Ltd.,London. Boukpeti,N.,Mroz,Z.,Drescher,A.,2004.Modelingrateeectsinundrainedloadingof sands.CanadianGeotechnicalJournal41,342{350. Bragov,A.M.,Grushevsky,G.M.,Lomunov,A.K.,1996.Useofthekolskymethodfor connedtestsofsoftsoils.Exp.Mech.36,237{242. Bragov,A.M.,Lomunov,A.K.,Sergeichev,I.V.,Tsembelis,K.,Proud,W.G.,2008. Determinationofphysicomechanicalpropertiesofsoftsoilsfrommediumtohighstrain rates.InternationalJournalofImpactEngineering35,967{976. Bridgman,P.W.,1952.Studiesinlargeplasticowandfracture.McGraw-HillBook Company,NewYork. Casagrande,A.,Shannon,W.L.,1949.Strengthofsoilsunderdynamicloads.Trans. ASCE114,755{772. Cazacu,O.,Ionescu,I.R.,Perrot,T.,2008.Numericalmodelingofprojectilepenetration intocompressiblerigidviscoplasticmedia.Int.J.Numer.MethodsEng.74, 1240{1261. Cazacu,O.,Jin,J.,Cristescu,N.D.,1997.Anewconstitutivemodelforaluminapowder compaction.In:PowderParticle.Vol.15.KONA,pp.103{112. Chen,W.F.,Baladi,G.Y.,1985.SoilPlasticity:TheoryandImplementation. DevelopmentsinGeotechnicalEngineeringVol.38.Elsevier,Amsterdam,The Netherlands. Chinn,J.,Zimmerman,R.M.,1965.Behaviorofplainconcreteundervarioushigh triaxialcompressionloadingconditions.Tech.Rep.WL-TR-64-163,AirForceWeapons Laboratory. Craig,R.F.,1987.SoilMechanics,4thEdition.ChapmanandHall,LondonSE18HN. 169

PAGE 170

Cristescu,N.D.,1989.RockRheology.KluwerAcademicPublishers,Dordrecht,The Netherlands. Cristescu,N.D.,1991.Nonassociatedelastic/viscoplasticconstitutiveequationsforsand. InternationalJournalofPlasticity7-2,41{64. Cristescu,N.D.,1994.Aproceduretodeterminenonassociatedconstitutiveequationsfor geomaterials.InternationalJournalofPlasticity10,103{131. Cristescu,N.D.,Hunsche,U.,1998.Timeeectsinrockmechanics.JohnWileyandSons, WestSussex,England. Curran,D.R.,Seaman,L.,Cooper,T.,Shockey,D.A.,1993.Micromechanicalmodelfor comminutionandgranularowofbrittlematerialunderhighstrainrateapplication topenetrationofceramictargets.InternationalJournalofImpactEngineering13, 53{83. Davies,R.,1948.Acriticalstudyofthehopkinsonpressurebar.Philos.Trans.R.Soc. London,Ser.A240,375{457. Desai,C.S.,Zhang,D.,1987.Viscoplasticmodelforgeologicmaterialswithgeneralized owrule.InternationalJournalforNumericalandAnalyticalMethodsinGeomechanics 11,603{620. diPrisco,C.,Imposimato,S.,Vardoulakis,I.,2000.Mechanicalmodellingofdrainedcreep triaxialtestsonloosesand.Geotechnique50,73{82. DiMaggio,F.L.,Sandler,I.S.,1971.Materialmodelforgranularsoils.Journal EngineeringMechanicsDivision97,935{950. Drucker,D.C.,Gibson,R.E.,Henkel,D.J.,1957.Soilmechanicsandworkhardening theoriesofplasticity.Trans.ASCE122,338{346. Drucker,D.C.,Prager,W.,1952.Soilmechanicsandplasticanalysisorlimitdesign. Quart.Appl.Math.10,157{165. Farr,J.V.,Woods,R.D.,1988.Adeviceforevaluatingone-dimensionalcompressive loadingrateeects.GeotechnicalTestingJournal,GTJODJ11,269{275. Felice,C.,Brown,J.,Ganey,E.,Olsen,J.,1987a.Aninvestigationintothehigh strain-ratebehaviorofcompactedsandusingthesplit-hopkinsonpressurebartechnique. In:Proc.2ndSymp.ontheInteractionofNon-NuclearMunitionsWithStructures. PanamaCityBeach,FL,pp.391{396. Felice,C.,Ganey,E.,Brown,J.,Olsen,J.,1987b.Dynamichighstressexperimentson soil.GeotechnicalTestingJournal,GTJODJ10,192{202. Frew,D.J.,Akers,S.A.,,Chen,W.,Green,M.L.,2010.Developmentofadynamic triaxialkolskybar.MeasurementScienceandTechnology21. 170

PAGE 171

Frew,D.J.,Cargile,J.D.,Ehrgott,J.Q.,1993.Wesgeodynamicsandprojectile penetrationresearchfacilities.Proceedingsfromthe1993ASMEWinterAnnual Meeting. Frew,D.J.,Forrestal,M.J.,Chen,W.,2002.Pulseshapingtechniquesfortestingbrittle materialswithasplithopkinsonpressurebar.ExperimentalMechanics42,93{106. Frew,D.J.,Forrestal,M.J.,Chen,W.,2005.Pulseshapingtechniquesfortesting elastic-plasticmaterialswithasplithopkinsonpressurebar.ExperimentalMechanics 45,186{195. Gray,G.T.,2000.Classicalsplit-hopkinsonpressurebartesting.In:ASMHandbook. Vol.8ofMechanicalTestingandEvaluation.AmericanSocietyforMetals,Materials Park,OH,pp.462{476. Gray,G.T.,Blumenthal,W.R.,2000.Split-hopkinsonpressurebartestingofsoft materials.In:ASMHandbook.Vol.8ofMechanicalTestingandEvaluation.Materials Park,OH,pp.488{496. Haeri,S.M.,Shakeri,M.R.,Shahcheraghi,S.A.,2008.Evaluationofdynamicproperties ofacalcitecementedgravelysand.In:GeotechnicalSpecialPublication. Hohenemser,K.W.,Prager,W.,1932.uberdieansatzedermechanikisotroperkontinua. Z.Angew.Math.Mech.12,215{226. Hopkinson,B.,1914.Amethodofmeasuringthepressureproducedinthedetonationof highexplosivesorbytheimpactofbullets.Philos.Trans.R.Soc.London,Ser.A213, 437{456. Huang,T.K.,Chen,W.F.,1990.Simpleprocedurefordeterminingcap-plasticymodel parameters.JournalofGeotechnicalEngineering116,492{513. Kabir,M.E.,Chen,W.W.,2009.Measurementofspecimendimensionsanddynamic pressureindynamictriaxialexperiments.Rev.Sci.Instrum.80. Kolsky,H.,1949.Aninvestigationofthemechanicalpropertiesofmaterialsatveryhigh ratesofloading.Proc.R.Soc.London,Ser.BB62,676{700. Lade,P.V.,1977.Elasto-plasticstress-straintheoryforcohesionlesssoilwithcurveyield surfaces.InternationalJournalofSolidsandStructures13,1019{1035. Lade,P.V.,Duncan,J.M.,1973.Cubicaltriaxialtestsoncohesionlesssoil.ASCEJournal SoilMechanicsFoundationDivision99SM10,793{781. Lade,P.V.,Duncan,J.M.,1975.Elastoplasticstress-straintheoryforcohesionlesssoil. ASCEJournalGeotechnicalEngineeringDivision101,1037{1053. 171

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Lade,P.V.,Kim,M.K.,1988a.Singlehardeningconstitutivemodelforfrictional materialsii.yieldcritirionandplasticworkcontours.ComputersandGeotechnics 6,13{29. Lade,P.V.,Kim,M.K.,1988b.Singlehardeningconstitutivemodelforfrictional materialsiii.comparisonswithexperimentaldata.ComputersandGeotechnics6, 31{47. Lindholm,U.S.,1971.Highstrainratetesting.In:Bunshah,R.F.Ed.,Techniquesin MetalsResearch.Vol.5.Interscience,NewYork. Lu,H.,Luo,H.,Komaduri,R.,2009.Dynamiccompressiveresponseofsandunder connements.In:SocietyforExperimentalMechanics-SEMAnnualConferenceand ExpositiononExperimentalandAppliedMechanics2009.Vol.2.pp.1046{1052. Malvern,L.E.,Jenkins,D.A.,1990.Dynamictestingoflaterallyconnedconcrete.Tech. Rep.ESL-TR-89-47,TyndallAirForceBaseTechnicalReport. Martin,B.E.,Chen,W.,Song,B.,Akers,S.A.,2009.Moistureeectsonthehigh strain-ratebehaviorofsand.MechanicsofMaterials41,786{798. Nelson,I.,1978.Constitutivemodelsforuseinnumericalcomputations.Proc.Int.Symp. onDynamicMethodsinSoilandRockMechanics,45{97. Nelson,I.,Baladi,G.Y.,1977.Outrunninggroundshockcomputedwithdierentmodels. ASCEJEngMechDiv103,377{393. Nemat-Nasser,S.,2000.Introductiontohighstrainratetesting.In:ASMHandbook. Vol.8.MaterialsPark,OH,pp.427{428. Nemat-Nasser,S.,Isaacs,J.B.,Starrett,J.E.,1991.Hopkinsontechniquesfordynamic recoveryexperiments.Proc.R.Soc.London,Ser.AA435,371{391. Perzyna,P.,1966.Fundamentalproblemsinviscoplasticity.AdvancesinApplied Mechanics9,243{377. Ross,C.A.,1989.Split-hopkinsonpressurebartests.Tech.Rep.ESL-TR-88-92,Tyndall AirForceBaseTechnicalReport. Ross,C.A.,Jerome,D.M.,Tedesco,J.W.,Hughes,M.L.,1996.Moistureandstrainrate eectsonconcretestrength.ACIMater.J.93,293{300. Ross,C.A.,Tedesco,J.W.,Kuennen,S.T.,1995.Eectsofstrainrateonconcrete strength.ACIMater.J.92,37{47. Rubin,M.B.,1990.Elastic-viscoplasticmodelforlargedeformationofsoils.Journalof EngineeringMechanics116,1995{2016. Sandler,I.S.,Baron,M.L.,1979.Recentdevelopmentsintheconstitutivemodelingof geologicalmaterials.SAEPreprints1,363{376. 172

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Schmidt,M.J.,Cazacu,O.,2006.Behaviourofcementitiousmaterialsforhigh-strainrate conditions.JournaldePhysique.IV:JP134,1119{1124. Schmidt,M.J.,Cazacu,O.,Green,M.L.,2009.Experimentalandtheoretical investigationofthehigh-pressurebehaviorofconcrete.InternationalJournalNumerical AnalysisMethodsGeomechanics33,1{23. Seed,H.B.,Lundgren,R.,1954.Investigationoftheeectoftransientloadingon thestrengthanddeformationcharacteristicsofsaturatedsands.In:Proceedingsof AmericanSocietyforTestingMaterials.Vol.54.pp.1288{1306. Song,B.,Chen,W.,Luk,V.,2009.Impactcompresssiveresponseofdrysand.Mechanics ofMaterials41,777{785. VonKarman,T.,1911.Festigkeitsversucheunterallseitigemdruck.Z.Ver.Dt.Ing. 55,1749{1757. Wang,C.C.,1970.Anewrepresentationtheoremforisotropicfunctions,partiandii. ArchiveforRationalMechanicsandAnalysis365,1467{1486. Wang,M.,Yang,Q.,Luan,M.T.,2007.Experimentalstudyondynamicelasticmodulus ofunsaturatedsiltyclay.ElectronicJournalofGeotechnicalEngineering12E. Whitman,R.V.,Healy,K.A.,1962.Shearstrengthofsandsduringrapidloadings. JournaloftheSoilMechanicsandFoundationsDivision88SM2,99{132. Wichtmann,T.,Triantafyllidis,T.,2004.Inuenceofacyclicanddynamicloadinghistory ondynamicpropertiesofdrysand,partii:Cyclicaxialpreloading.SoilDynamicsand EarthquakeEngineering24,789{803. Yin,J.H.,Graham,J.,1999.Elasticviscoplasticmodellingofthetime-dependent stress-strainbehaviourofsoils.CanadianGeotechnicalJournal36,736{745. 173

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BIOGRAPHICALSKETCH InMayof2001,BradleyEricMartingraduatedwithaBachelorofSciencein MechanicalEngineeringfromtheUniversityofSouthAlabama.Upongraduation,he beganworkingatIngallsShipbuildinginPascagoula,Mississippi,asaLifeCycleEngineer. InAugust2001,washiredbyGeneralDynamics-OTSinNiceville,FloridaasaDesign Engineer.WhileatGeneralDynamics-OTSheprimarilyconductedpenetrationand sledtracktestingofmunitions.InJune2002,hewashiredbytheAirForceResearch LaboratoryatEglinAFB,FloridaattheDamageMechanismsBranchwherehestill presentlyworks.In2007,whileworkingatEglin,heobtainedhisMasterofEngineering degreeinMechanicalEngineeringfromtheUniversityofFloridaResearchandEngineering EducationFacilityinShalimar,Florida.HecompletedhisPh.D.workin2011atthe UniversityofFloridaResearchandEngineeringEducationFacilityinShalimar,Florida. 174