工學類地盤工學特論

地盤工學特論

第10回

Stress,Strain,ElasticityandPlasticity廣岡明彥ahirooka@civil.kyutech.ac.jphttp://geo.civil.kyutech.ac.jp2004/6/231地盤工學特論前回の講義のまとめ-1粘土の排水せん斷強度エネルギー補正2004/6/232地盤工學特論前回の講義のまとめ-2排水せん斷強度圧密非排水三軸試験で求めたf’

と圧密排水三軸試験で求めたfd

との比較2004/6/233地盤工學特論前回の講義のまとめ-3粘土の殘留強度一面せん斷あるいはリングせん斷試験における応力～変位関係の特徴2004/6/234地盤工學特論前回の講義のまとめ-4Stress,Strain,ElasticityandPlasticitymodel:physicalrepresentationinthesenseofascalemodel?aconceptualideaoranumberofmathematicalequationsForSoilsbasedontheoriesofelasticity,plasticity,andfrictionmathematicalmodels:CriticalStateModelSchofieldandWroth,1968AtkinsonandBransby,19782004/6/235地盤工學特論前回の講義のまとめ-5PlainStrainAxialSymmetrySpecialcase:Solidcylindersofsoilintriaxialtestss’2=s’3de2=de3Soil:anaggregateofmineralgrainswithafluidfillingtheporespacesdry ;porefluidisairalonesaturated ;porefluidiswateralone2004/6/236地盤工學特論Theprincipleofeffectivestress(statedbyTerzaghi1936)?Allmeasurableeffectsofachangeofstress,suchascompression,distortion,andachangeofshearingresistance,aredueexclusivelytochangesofeffectivestress.Mathematicalmodelsforsoilbehaviormustbewrittenintermsofeffectivestress,nottotalstress.前回の講義のまとめ-6effectivestresss’=s

–ucontrolssoilbehavior2004/6/237地盤工學特論1.3 AnalysisofDeformationandStrainModeofdeformation.(a)Stressesonaplaneelement.(b)Continuousstraining.(c)Discontinuousslipping.ds’v,ds’h

:stressincrementdv,dh

:smallchangeof dimensionsnodiscontinuitiesintheelement.thebehaviorineachandeverypartoftheelementisthesame.alldeformationsareduetoverylargestrainwithintheslipzone. →SlipPlane(inthelimit,thezone:zerothickness)(b)and(c)oftenoccursimultaneously，but(b)or(c)usuallypredominates.2004/6/238地盤工學特論Incrementofplanedisplacementtheengineer’sshearstrain2004/6/239地盤工學特論Incrementofstrainincrementsofstraininaplaneelement(positive)principalstraincounterclockwiseshearstrains:positiveMohr’scircleofplanestraincorrespondingtotheincrementsofstrainaboveP:pole2004/6/2310地盤工學特論Planesofzerostrainincrement.theincrementofminorprincipalstrainisnegativeandtensiley:anangleofdilationeg:maximumshearstrainev:volumetricstrain(ife2=0)P:pole=45°－y/22004/6/2311地盤工學特論,inthelimit,,inthelimit,Forthespecialcaseofaxialsymmetry(de2=de3):specificvolume2004/6/2312地盤工學特論DirectionsofzerostrainIncrementthesamevaluefortheplaneofzerostrainincrementFromthegeometry,thetangenttothepointAmakesanangleytothedeaxis.b=45°＋y/2P:poleA2004/6/2313地盤工學特論Analysisofdiscontinuousslippingalldeformationsandstrainsarecontainedwithinslipline?discontinuousslipping2004/6/2314地盤工學特論1.4 AnalysisofStateofStressStateofstressintwodimensionsMohr’scircleofstresscorrespondingtothestateofstressaboveP:pole2004/6/2315地盤工學特論stressparameterst’s’mobilizedangleofshearingresistance

:r’Forthespecialcaseofaxialsymmetry(s’2=s’3)2004/6/2316地盤工學特論P:poleStressonplanesforwhichtangentsfromtheorigintouchtheMohr’scircletheplanesfordiscontinuousslipping2004/6/2317地盤工學特論Mohr’scirclesoftotalandeffectivestress2004/6/2318地盤工學特論1.5 RelationshipsbetweenstressandstrainprincipalplainsofstressandprincipalplainsofstrainandstrainincrementcoincideCoaxialitycondition:;notageneraltheoreticalrequirementofamaterialbehavior;agoodapproximationformaterialbehaviorUnitWork（共軸）incrementofenergytransferredfromtheexternalforcesdE:apartofenergystoredandrecoveredonunloadingdU:apartofenergydissipatedinthematerialdW:→dissipative:ex.plasticmaterial→conservative:ex.elasticmaterial2004/6/2319地盤工學特論←continuousstrain←discontinuousslipping:strainincrementvectors:effectivestressincrementvectors:compliancematrixcontainingamaterial←determinantof[C’]isnon-zero2004/6/2320地盤工學特論foraxialsymmetryforplanestrainIncrementalModel↓alternativeformulationwhichrelatesincrementsofstraintostates,ratherthanincrements,ofstress2004/6/2321地盤工學特論1.6 ElasticandPlasticDeformationsstatesandincrementsofstress?statesandincrementsofstrainMathematicalmodel

formaterialbehavior:

basedon

thetheoriesofelasticityandplasticityExaminethebehaviorofanidealsoil-likematerialsubjectedtoprincipaleffectivestresssa’,sb’andsc’?theprincipleofeffectivestress“Strainsdependsoneffective,nottotal,stresses.”2004/6/2322地盤工學特論Stress-straincurvesforanidealsoil-likematerialforuniaxialcompressionStrainhardeningOY:linearlyelastic, strainsarefullyrecoveredforstatesbeyondY:

irrecoverable

plasticstrainsoccurthestressisreducedfromG:

linearlyelasticintherangeBGY:yieldpointG:yieldpoint↓Yieldstressincreaseswithplasticstrain?StrainhardeningG→F:sufferingelasticandplasticstrainsF:failure

point2004/6/2323地盤工學特論Stress-straincurvesforanidealsoil-likematerialforuniaxialcompressionStrainsofteningOY:linearlyelastic, strainsarefullyrecoveredforstatesbeyondY:

irrecoverable

plasticstrainsoccurthestressisreducedfromG:

linearlyelasticintherangeBGY:yieldpointG:yieldpoint↓Yieldstressdecreaseswithplasticstrain?StrainsofteningG→F:sufferingelasticandplasticstrains2004/6/2324地盤工學特論1.7 Yielding,HardeningandPlasticFlowvariouscombinationsofsa’andsc’atyieldandatfailurefor

astrainhardeningmaterial

YaYc

:yieldcurveFaFc

:failureenvelopeYaYc

～FaFcgeometricallysimilarornotYaYc

→GaGcStrainHardeningchangeofyeildcurve?

Hardeninglawplasticstrain2004/6/2325地盤工學特論yieldsurfaceplasticstrainyieldsurfaceThestateofthematerialcanbeonorwithin

theyieldsurfacebutcannotlieoutsideit….forastrainhardeningmaterialforastrainsofteningmaterial?????2004/6/2326地盤工學特論OgGaGc

:elasticwallPathsforloadingandunloadingHardeningmaterialloadingpathA→BA→CA→D：traversetheyieldsurfaceandplasticstrainsoccurunloadingpathA→E：traverseanelasticwallandplasticstrainsarezeropurelyelasticstrainsoccurunloadingpathA→E’

Impossible!!!!2004/6/2327地盤工學特論PathsforloadingandunloadingSofteningmaterial2004/6/2328地盤工學特論FlowruleofplasticitytheoryPlasticpotentialNormalityconditiontheratioofincrementsofplasticstrain?thestateofstresscausingtheplasticdeformationplasticitytheorythevectorofplasticstrainincrement?thevectorofstressflowruleplasticpotential=yieldcurve?associatedflowrule?normalitycondition2004/6/2329地盤工學特論1.8 IdealElasticBehaviorthebehaviorofisotropicelasticmaterial←Hooke’slawE’：Young’smodulusn’：Poisson’sratioreversiblerecoverable→conservative2004/6/2330地盤工學特論ontheprincipalstress(strain)planeforthespecialcaseofaxialsymmetry(s’2=s’3,de2=de3)K’：bulkmodulusG’：shearmodulus2004/6/2331地盤工學特論forplanestrain(de2=0)K’ps

：bulkmodulusappropriateforplanestrainG’ps

：shearmodulusappropriateforplanestrain2004/6/2332地盤工學特論forthespecialcaseofaxialsymmetry des:dependentonlyondq’

incrementofshearstress→incrementofshearstrain dev:dependentonlyondp’ incrementofnormalstress→incrementofvolumetricstrainforplanestrain deg:dependentonlyondt’ dev:dependentonlyonds’2004/6/2333地盤工學特論theratioisdependentontheratiotherelationshipbetweentworatios ←thevalueofPoisson’sratio independentofYoung’smodulus?plasticitytheory（1.7）2004/6/2334地盤工學特論1.9 Elasto-PlasticBehaviorelasticityyieldinghardeningplasticflowgeneralstress-strainequationstotalstrains =elasticcompon