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 ThomsonLearning isatrademarkusedhereinunderlicense ThetensiontestisthestandardtestfordetermineE theelasticorYoung smodulus TestthatloadacylindricalspecimenintorsionareusedtomeasuretheshearmodulusG KnowingEandG Poisson sratiomaybeobtainedfromtherelationshipwederivedintheprevioussection 杨氏 弹性 模量E MetalsAlloys GraphiteCeramicsSemicond Polymers Composites fibers E GPa BasedondatainTableB2 Callister6e Compositedatabasedonreinforcedepoxywith60vol ofalignedcarbon CFRE aramid AFRE orglass GFRE fibers YOUNG SMODULI COMPARISON Plastictensilestrainatfailure Anotherductilitymeasure Note ARand ELareoftencomparable Reason crystalslipdoesnotchangematerialvolume AR ELpossibleifinternalvoidsforminneck 4 Ductileandbrittlematerials 韧性与脆性材料 Energytobreakaunitvolumeofmaterial Approximatebytheareaunderthestress straincurve TOUGHNESS 韧性 Low carbon mild steelisdifferentfrommostothermetalsinthatthereisasuddensmalldropofloadattheyieldpointfollowedbyanextensionatconstantstress Thelowerloadisusuallyreferredtoastheyieldpointformildsteel Theactualpointofyieldisoftendifficulttoidentify Anumberoftechniquesareusedtolocate y 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 AppliedMechanics RigidBodyMechanicsStaticsDynamics Kinematics Kinetics DeformableBodyMechanicsElasticityPlasticityViscoelasticity FluidMechanicsLiquidsGases 1 2Researchobjectsandcontents Manufacturingprocessesthatmakeuseofcoldworkingaswellashotworking Commonmetalworkingmethods 轧制 挤压 锻造 冲压 拉拔等 2003Brooks Cole adivisionofThomsonLearning Inc ThomsonLearning isatrademarkusedhereinunderlicense 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 Considerarectangularblockofunitthicknessandsupposeshearstresses 1 actonBCandAD Theforces 1 ADand 1 BCformacoupleofmagnitude 1 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 Nowsupposewecuttheprismaticbaratanangle asshownbelow Howdothenormalandshearcomponentsofstressactingonaplaneatagivenpointchangeaswechangetheorientationoftheplaneatthepoint 2 GeneralStressSystemsin2 Dimensions 双向应力状态 ThestressesontheelementABCDinacomponentsubjectedtocombined2Dloading assumingnothroughthicknessstresses i e planestress areschematicallyshownintheFigure Thereferencesystemofcoordinateaxesareasshownalso Whatisthestressstateonachosenplaneofinterest ConsiderrotatingtheelementABCDbyanangle tothex axissothatitnowhasaxesofx andy orientatedatangle tothexandyaxes Todeterminethenewstresses x y and x y ontheelementintermsoftheoriginalstressesconsiderthefreebodydiagramofaprismaticelementADEandthestressesactingonitareasshownintheFigure Thenormalstress x andshearstress x y actontheplaneAEandmaintaintheequilibriumoftheprismaticelement Thestresses x and x y areobtainedbyresolutionofforcesintherespectivedirections 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l m n 0是一组解 但l2 m2 n2 1 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 l2 12 m2 22 n2 32 1l2 2m2 3n2 2 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 Eachoftheoctahedralp