弹性力学基础-中英ppt课件

.,Elastic-plasticMechanicsofMaterialsMing-anCHEN（陈明安）SchoolofMaterialsScienceandEngineeringCentralSouthUniversity,.,.,Typicaltensilespecimen,Typicaltensiletestmachine,gauge,length,(portionofsamplewith,reducedcrosssection),Chapt.1Introduction,1.1Elasticityandplasticity（弹性与塑性）1.STRESS-STRAINTESTING（应力应变曲线拉伸试验）,gauge,length,=,.,.,2.ELASTICDEFORMATIONandELASTICITY（弹性变形与弹性）,Elasticmeansreversible!,Itisreversibleandtimeindependent.Thedeformationvanishesinstantaneouslyassoonasforcesareremoved.,.,3.PLASTICDEFORMATION(METALS)andPLASTICITY（塑性变形与塑性）,Plasticmeanspermanent!,Plasticdeformation-itisirreversibleorpermanent.,.,.,2003Brooks/Cole,adivisionofThomsonLearning,Inc.ThomsonLearningisatrademarkusedhereinunderlicense.,.,ThetensiontestisthestandardtestfordetermineE,theelasticorYoungsmodulus.TestthatloadacylindricalspecimenintorsionareusedtomeasuretheshearmodulusG.KnowingEandG,Poissonsratiomaybeobtainedfromtherelationshipwederivedintheprevioussection.,.,杨氏（弹性）模量E,.,MetalsAlloys,GraphiteCeramicsSemicond,Polymers,Composites/fibers,E(GPa),BasedondatainTableB2,Callister6e.Compositedatabasedonreinforcedepoxywith60vol%ofalignedcarbon(CFRE),aramid(AFRE),orglass(GFRE)fibers.,YOUNGSMODULI:COMPARISON,.,Plastictensilestrainatfailure:,Anotherductilitymeasure:,Note:%ARand%ELareoftencomparable.-Reason:crystalslipdoesnotchangematerialvolume.-%AR%ELpossibleifinternalvoidsforminneck.,4.Ductileandbrittlematerials（韧性与脆性材料）,.,EnergytobreakaunitvolumeofmaterialApproximatebytheareaunderthestress-straincurve.,TOUGHNESS（韧性）,.,Low-carbon(mild)steelisdifferentfrommostothermetalsinthatthereisasuddensmalldropofloadattheyieldpointfollowedbyanextensionatconstantstress.Thelowerloadisusuallyreferredtoastheyieldpointformildsteel.,.,.,.,.,.,.,.,.,.,.,.,Theactualpointofyieldisoftendifficulttoidentify.Anumberoftechniquesareusedtolocatey.Thetangentmethod(orkneemethod)locatestheyieldstrengthattheintersectionoftheelasticslopeandtheinitialportionoftheplasticregion(notreliably).Thepreferredmethodisthepercentageoffsetmethodwhereyieldstrengthisobtainedbydrawingalineparalleltotheinitialelasticregiondataat0.2%strain(0.002)offset.Wherethislineintersectsthestress-straincurvethenbecomesknownasthe0.2%yieldstrength.,5.Determinationofyieldstrength屈服强度,.,Formostmetals,loadingbeyondtheyieldpointcausesapermanentdeformation.WhenamaterialisloadedtopointBandthenunloaded,itreturnstoazerostressstatealongalineparalleltotheinitialelasticregionbutdirectlyfromB.ThestrainremaininginthematerialatpointDisknownastheplasticdeformation.,OnreloadingfromDthereisadeparturefromlinearityatC,slightlybelowB,andthestress-straincurvebecomesthesameastheoriginalstress-straincurve(atE).Notethatthepointofdeparturefromlinearityonthereloadcurve(C)isslightlyhigherthanforthefirstloadingcurve.,6.Unloadingandreloading（卸载与再加载）,.,7.Idealizationsofstress-straincurves应力应变曲线简化,.,Idealrigid-plastic,Rigid-plastic,.,Mechanics:branchofphysicsconcernedwithmotionandbodydeformationcreatedbymechanicaldisturbanceorforces.AppliedMechanics:scienceofapplyingtheprinciplesofmechanicstodesignandanalysisofmechanicalsystem.AppliedMechanicsRigidBodyMechanicsStaticsDynamicsKinematics,KineticsDeformableBodyMechanicsElasticityPlasticityViscoelasticityFluidMechanicsLiquidsGases,1.2Researchobjectsandcontents,.,.,.,.,Manufacturingprocessesthatmakeuseofcoldworkingaswellashotworking.Commonmetalworkingmethods,轧制、挤压、锻造、冲压、拉拔等,.,.,.,.,.,.,.,2003Brooks/Cole,adivisionofThomsonLearning,Inc.ThomsonLearningisatrademarkusedhereinunderlicense.,Anisotropicbehaviorinarolledaluminum-lithiumsheetmaterialusedinaerospaceapplications.Thesketchrelatesthepositionoftensilebarstothemechanicalpropertiesthatareobtained,.,.,.,RectangularCoordinates,.,ThesystemofparticlesintheFigureissaidtobeinequilibriumifeveryoneofitsconstitutiveparticlesisinequilibrium.Consequently,thefirstconditionforequilibrium:thevectorsumofalltheforcesiszero.,wherernextendsfrompoint0toanarbitrarypointonthelineofactionofforceFn.,Ifthesurfaceandbodyforcesareinbalance,thebodyisinstaticequilibrium.,Thesecondconditionforequilibrium:thetotalmomentofalltheexternalforcesaboutanarbitrarypoint0mustbezero.,AnIsolatedSystemofParticlesShowingExternalandInternalForces,Foranobjecttobeatrest(iestaticeq),netforceandnetmomentmustbezero.,.,Sinceforcesandmomentsarevectors,withinherentdirectionality,itisfrequentlyusefulfordecomposeintoindividualcomponents:,.,.,.,ShearStress（剪应力）Shearstressescanalsobegeneratedbyappliedshearloads.ConsidertwoequalandoppositeshearforcesVactingonarectangularblockasshown.,.,应力点的概念:不同点处应力不同。应力面的概念：同一点处不同截面上的应力不同。应力必须指明是哪点、哪个截面上的应力。,.,.,.,.,.,.,Apositivecomponentofstressactsonapositivefaceinapositivecoordinatedirectionoronanegativefaceinanegativecoordinatedirection.,.,ComplementaryShearStress（剪应力互等）Considerarectangularblockofunitthicknessandsupposeshearstresses1,actonBCandAD.Theforces1*ADand1*BCformacoupleofmagnitude1*AD*ABandtheblockisnotinequilibrium.TheremustbeanequalandoppositecoupleformedbyshearstressesonABandCD.,Thusanappliedshearstressisautomaticallyaccompaniedbyashearstressofequalintensityatrightangles(andcausinganoppositeturningmoment)totheoriginalshearstress.Thesearecalledcomplementaryshearstresses.,.,.,Thestateofstressatapointcannormallybedeterminedbycomputingthestressesactingoncertainconvenientlyorientedplanespassingthroughthepointofinterest.Stressesactingonanyotherplanescanthenbedeterminedbymeansofsimple,standardizedanalyticalorgraphicalmethods.Ifsowecanusethestresses,actingontheseconvenientlyorientedplanespassingthroughthepoint,forrepresentingthestressstateofthegivenpoint,andthatthestressstateatthispointisknown.,Theselectionofdifferentcuttingplanesthroughagivenpointwould,ingeneral,resultinstressesdifferinginbothdirectionandmagnitude.Acompletedescriptionofthemagnitudesanddirectionsofstressesonallpossibleplanesthroughthegivenpointconstitutesthestateofstressatthegivenpoint.,Problem:Thestresscomponents,onwhichofandhowmuchdifferentplanes,canbeusedforrepresentingthestressstateofthegivenpoint?,2.2.2Stateofstressatapoint（点的应力状态）,.,一点可以用无穷个微元表示，找出之间应力的关系，称为应力状态分析。,应力状态的概念：过一点不同截面上应力的的集合，称为这一点的应力状态。,.,Stateofuniaxialstress:单向应力状态,Thestressnormaltothecross-sectionalsurface:,.,Stressesonobliqueplanes:（斜面上的应力）,Stresses:,Forces:,Nowsupposewecuttheprismaticbaratanangleasshownbelow.,Howdothenormalandshearcomponentsofstressactingonaplaneatagivenpointchangeaswechangetheorientationoftheplaneatthepoint.,.,2.GeneralStressSystemsin2-Dimensions（双向应力状态）,ThestressesontheelementABCDinacomponentsubjectedtocombined2Dloading(assumingnothroughthicknessstresses,i.e.planestress)areschematicallyshownintheFigure.Thereferencesystemofcoordinateaxesareasshownalso.,Whatisthestressstateonachosenplaneofinterest?,.,ConsiderrotatingtheelementABCDbyanangletothex-axissothatitnowhasaxesofxandyorientatedatangletothexandyaxes.Todeterminethenewstressesx,yandxyontheelementintermsoftheoriginalstressesconsiderthefreebodydiagramofaprismaticelementADEandthestressesactingonitareasshownintheFigure.ThenormalstressxandshearstressxyactontheplaneAEandmaintaintheequilibriumoftheprismaticelement.,.,Thestressesxandxyareobtainedbyresolutionofforcesintherespectivedirections.,.,TransformationofStresses（应力变换）,x-y,xp-yp,.,.,Theseresultsclearlyillustratehowthevaluesforthenormalandshearstresscomponentsofaforcedistributedoveraplaneinsideofanobjectdependsuponhowyoulookatthepointinsidetheobjectinthesensethatthevaluesoftheshearandnormalstressesatapointwithinacontinuumdependupontheorientationoftheplaneyouhavechosentoview.,n=Sxl+Sym+Szn=xl2+ym2+zn2+2(xylm+yzmn+zxnl),l=cos(n,x),m=cos(n,y)andn=cos(n,z),n2=S2-n2,.,Theabovementionedshowesthatifweknowthe9stressescomponentsonthethreemutuallyperpendicularplanesasfacesofacubeofinfinitesimalsize(element)whichsurroundthegivenpointwecandeterminethestersscomponentsactingonanyplanethroughthepoint.Sothese9stressescomponentscanbeusedtorepresentthestressstateofapoint.,.,一点应力状态可表示为：,.,2.3Stresstensorandprincipalstresses（应力张量与主应力）2.3.1Stresstensor（应力张量）,Tensoristhegeneralisedtermforavector.Itsfullmathematicaldefinitionis:Amathematicalentityspecifiablebyasetofcomponentswithrespecttoasystemofco-ordinatesandsuchthatthetransformationthathastobeappliedtothecomponentstoobtaincomponentswithrespecttoanewsystemofco-ordinatesisrelatedinacertainwaytothetransformationthathastobeappliedtothesystemofcoordinates.Thecomponentsofavectorchangewhentheco-ordinatesystemisrotated.However,thevectorstillhasthesamemagnitudeanddirectionasitdidbeforetheco-ordinatesystemwasrotated.Secondranktensors(e.g.stress,inertia)seetheircomponentschangewhenaco-ordinatesystemisrotatedandunlikevectorsthemagnitudeandorientationofthetensormayalsochange.,.,Theoremofconjugateshearingstresses,Thereforeonly6independentstresscomponents,Stresstensorissymmetric.,.,.,Couldyouwriteoutthestresstensorscorrespondingtothefollowingfigures？,.,.,Couldyoushowthestresstensorinacorrespondingelement？,.,.,三向应力状态下的应力变换,.,.,.,.,2.3.2Principalstresses（主应力）,.,Theactualvaluesofthe6stresscomponentsinthestressmatrixforagivenbodysubjectedtoloadingwilldependontheorientationofthecubeinthebodyitself.Ifwerotatethecube,itshouldbepossibletofindthedirectionsinwhichthenormalstresscomponentstakeonmaximumandminimumvalues.Itisfoundthatinthesedirectionstheshearcomponentsonallfacesofthecubebecomezero.Theprincipalstressesaredefinedasthosenormalcomponentsofstressthatactonplanesthathaveshearstresscomponentswithzeromagnitude.,.,一点处一般有三个主平面，互相垂直。,.,.,假设该斜微分面即为待求的主平面，面上0，正应力全应力S。全应力S在3个坐标轴上的投影为：,以l、m、n为未知数的齐次线性方程组，其解就是应力主轴的方向。显然lmn0是一组解，但l2m2n21，故应求其非0解。,.,.,Stressinvariants(应力张量不变量),第一、第二、第三应力不变量,.,.,1.可以证明，在应力空间，主应力平面是存在的；2.三个主平面是相互正交的；3.三个主应力均为实根，不可能为虚根；4.应力特征方程的解是唯一的；5.对于给定的应力状态，应力不变量也具有唯一性；6.应力第一不变量I1反映变形体体积变形的剧烈程度，与塑性变形无关；I3也与塑性变形无关；I2与塑性变形有关;7.应力不变量不随坐标而改变，是点的确定性的判据。,.,.,用主应力表示的各种应力状态的图示：,.,.,.,.,.,.,.,.,.,.,2.3.3Principalshearstresses（主剪应力）,剪应力取极值的平面上的剪应力-主剪应力；主剪应力所在的平面-主剪应力平面；主剪应力平面的法线方向-主剪应力方向。,n=Sxl+Sym+Szn=xl2+ym2+zn2+2(xylm+yzmn+zxnl),n2=S2-n2,S,l2+m2+n2=1,.,现考虑主应力空间下主剪应力、主剪应力平面的求解：,2=l212+m222+n232-1l2+2m2+3n22,将n2=1-l2-m2代入上式，取2对l和m的偏导数并令其为零，可解出对应的l、m、n和极值剪应力。,.,n=0，l=1/(2)，m=1/,m=0，l=1/(2)，n=1/,l=0，m=1/(2)，n=1/,三组（6个）主剪应力平面分别与一个主应力平面垂直，与另两个主应力平面呈45。,.,最大剪应力(maximunshearstress),.,.,.,.,2.3.4Decompositionofstresstensor（应力张量分解）,Deviatoricstresscomponents（偏应力分量）,.,(i,j=x,y,zor1,2,3),Thesphericalorhydrostaticstresstensor(球应力张量),Thedeviatoricstresstensor（偏应力张量）,.,.,.,1、分解的依据：静水压力实验证实，静水压力不会引起变形体形状的改变，只会引起体积改变，即对塑性条件无影响。2、为引起形状改变的偏应力张量(deviatoricstresstensor)，为引起体积改变的球张量(sphericalstresstensor)（静水压力）。3、与应力张量类似，偏应力张量也存在相应的不变量：,.,.,OneDimensionalStateofStresses,.,ShearingStateofStresses,.,.,2.4TheMohrcircleofstress（应力莫尔圆）,.,.,.,.,.,.,.,.,.,.,.,应力极值,哪个几何图形可代表该点的应力状态？,.,画出球应力状态的Mohr圆？,.,.,TheFigureillustratestheorientationofoneoftheeightoctahedralplaneswhichareassociatedwithagivenstressstate.Eachoftheo