材料力学性能课件

StressedandStrainedStates

Stress

Stressistheloadappliedtoabodyandrelatedperunitareaofthebody’ssection.Arelativequantity;Thedimensionofstressisdeterminedastheforceactiveperunitareaofthebodysectiontowhichtheforceisapplied.Usuallymeasuredasnewtonspersquaremetre(N/m2)orkgf/mm2;Theunitsofstressexpresstheprincipalmechanicalproperties(ultimatestrength,resistancetoplasticflow,resistancetoindentation,fatiguestrength,creepstrength,etc.)Thecaseofaxialtensionofacylindricalrod)ifS=constant(uniformdistributionofthestressoverthecrosssection)P=SForS=P/FInamoregeneralcase

Thenormalstress(正应力)Theshearstress(剪应力Thus,ifweknowthetensileforcePappliedtotherodandthecross-sectionalareaF.wecandeterminethenormalandshearstressesinanyplanemakinganarbitraryangleαwiththerodaxis.ThedistributionofnormalandshearstressesinvariouslyorientedplanesofatensionedspecimenareillustratedinFig.4.

Engineering/Actual(True)StressF:forceapplied;A0:areabeforedeformation

Theengineeringstressisoftenemployedforelasticstressesorstressesforcomponentsdeformedtosmallplasticstrains.Atlargestrains,thechangeincross-sectionalareasignificantlyalterstheactualstresses.

Thetruestressis:whereAistheinstantaneousarea.

Strain

Strain

istheratioofthechangeindimensiontoitsinitialvalue.Axialtensionofacylindricalrodas;Loadapplied;Roddeformed,thelengthincreasedfroml0

toln;engineeringstrainTheengineeringstrainshouldbeusedonlyifthedeformationstrainsaresmallinmagnitude(e.g.,eeng<0.10)oronlylimitedprecisionisrequired.Calculationsofstrainsfor

multiple-stepformingprocessesorforoperationswithpreciserequirementsonfinaldimensionsrequireameasureoftheinstantaneousortruestrain.Truestrainisbasedoninstantaneoussamplelength.Itcanbeapproximatedbyconsideringthetotalstraintoresultfromaseriesofsmall,incrementalextensions(

l)withthegagelengthateachincrementbeingtheinstantaneoussamplelength.Thus,where

l1=l0+

l,l2=l1+

l,etc.Whenexpressedindifferentialform,itbecomesexact;i.e.Onintegratingitfroml=l0tol=li,Theconstant-volumeconditionofplasticdeformationallowsrelationshipstobedevelopedamongthevariousstressesandstrains,provideddeformationalongthegagelengthisuniform.Truestresscanberearrangedas

Sinceli=l0+li-l0,sothat(A0/Ai)=(li/l0)=[1+(

l/l0)]Clearly,σT>σEforatensiletest,aresultintuitivelydeducedpreviously.Incontrast,ifthematerialwerecompressedsothatthecross-sectionalareaincreasedduringdeformation(withσE<0),wewouldfindσT<σE.WhichshowsthatεT<εE

inatensiontest(i.e.,ln(l+x)<x).Thedifferencebetweenthetrueandengineeringstressesandstrainsincreaseswithplasticdeformation.Atlowstrains,σT≈

σEandεT≈εE

,sothat,forexample,indiscussionofelasticdeformationthereisnoneedtodifferentiatebetweenengineeringandtruestressandstrain.Wealsofindthatforuniformgagelengthdeformation,εTandεEarerelatedthrough

Example:Ifarodhasbeenelongatedtwiceintensionorcompressedtohalfitsinitiallength,itisthenclearthatthedeformationsareequivalentanddifferonlyinsign.

Shear承受的外力是一对大小相等，方向相反，作用线相距很近的平行力剪切面受力非常复杂Shearstressτ=F/A0Anydeformationcanberepresentedasacertaincombinationofshearsandelongations,muchlikeanyintricatestressedstatecanbeexpressedthroughnormalandshearstresses.Elasticity

（1）Thepropertyofreturningtoaninitialformorstatefollowingdeformation.

形变后恢复到原来形状或状态的性质Stress-straindiagram(tensiletestofacylindricalspecimen)

Threedistinctlydifferentportions:a.

Adirectproportionalitybetweentheloadandelonglationandthereversibilityofdeformation;b.

Theloadincreasesfurther,butnotsosharplyasbefore,thedeformationisirreversiblebutdistributedevenlyoverthelength;c.

Theloaddecreases,a“neck”formsandfinallyruptureoccurs.Hooke’sLaw（1）

RobertHooke(1678)Acylindricalmetalroadwithoriginallengthl0

andcross-sectionareaF0:

Hooke’sLaw

（2）a.

ifl0andF0are

constant：b.

ifPandF0are

constant：c.

ifPandl0are

constant：Then(1-1)Young’smodulus

(1)TheproportionalityfactorEwhichrelatesthestressandstrainiscalledthemodulusofelasticityoryoung’smodulusinuniaxialtension,itisameasureofstiffnessofagivenmaterial.

ThomasYoung(1773～1829)(1-2)Young’smodulus(2)

Echaracterizestheintensityofloadgrowthwithincreasingelongation(orofgrowthofthestresswithanincreaseofstrain),

Itdeterminestheslopeoftheinitialsegmentofthestress-straindiagramYoung’smodulusisindependentofthesignofdeformationandhasthesamevalueintensionandcompression.Forceofinteractionbetweentwoatoms

Energyofinteractionbetweentwoatoms

Ifaforceappliedtoabodycausesanelasticdeformation,thedistancebetweenatomschangestoanewequilibriumpositioninwhichthenewforcesofatomicinteractioncounterbalancetheappliedexternalforce.Thedisplacementofatomsfromtheinitialequilibriumpositionischaracterizedbythedifferencebetweentheoriginalandnewinteratomicdistanceu=a-a0whereaisthenewdistancebetweenatoms.

Iftheforcefappliedisatensileforce,a>a0

anduispositive.Incompression,a<a0andu<0.Theequilibriumconditioncanbewritten,asfollows:

whereΦ(u)isthebondenergyondisplacementu.Byanalysingthesystemoftwoatoms,itisalsopossibletoderiveHooke'slawwhichestablishestherelationshipbetweentheexternalforceappliedandtheresultingdisplacement.

ForHooke'slawtobevalid,thefollowingthreeconditionsmustbesatisfied:

(1)thefunctionΦ(u)mustbecontinuous;(2)thefunctionΦ(u)musthaveaminimumdΦ/du=0atu=0;and(3)thedisplacementumustbemuchlessthana0.ThefirstconditionmakesitpossibletoexpandtheinteractionenergyfunctionintoaTaylorseries:

Inthisequation,Φ0istheinteractionenergyatu=0and,allthederivativesareobtainedforthepointu=0.

SincedΦ/duisequaltozeroatu=0,and,thetermswiththethirdandhigherpowersofucanbeneglected(asuissmall),weobtain:

Thesecondderivative(d2Φ/du2)oisthecurvatureofthefunctionΦ(u)inpointu=0,and,therefore,itdoesnotdependonuandisaconstant.Thus,weobtainf=constu,

i.e.theforceisproportionaltodisplacement(Hooke'slaw).

Itshouldberecalledthattheregionofadirectproportionalitybetweentheforceanddisplacementislimitedtoslightdeformations.Withanappreciablemagnitudeofdisplacementu,thetermsofhigherpowersofucannotbeneglectedand,therefore,thef(u)curvedeviatesfromthestraightline.Thisphenomenonisneverencounteredinpractice,sinceanirreversibleplasticdeformationbeginsinmetalevenatlowerstresses.Thelawofdirectproportionalityisthendisturbedbutfordifferentreasons.

Perfectthread-shapedmetalcrystalsofadiameterofaround2um(called'whiskers'),inwhichplasticflowisimpeded,can,however,bedeformedelasticallybyafewpercentand,athighelasticdeformations,adeviationfromHooke'slawcanbeobservedexperimentally

Inshearstress

Theshearstressisrelatedwithacorrespondingsheardeformationbysimilarexpression:

whereGistheshearmodulus(orthemodulusofelasticityinshear)

(1-3)Inhydrostaticcompression(ortension)

Hooke’slawexpressesadrectproportionalitybetweenthehydrostaticpressurePandthevolumechangex:whereKisthemodulusofbulkdeformation.

(1-4)Hooke’slaw(3)Formulae(1-2),(1-3)and(1-4)expresswhatiscalledHook’slaw.DeterminestherelationshipbetweenstressandstrainactinginthesamedirectionWhendeformationappearinadirectiondifferentfromthatofthestressaction,itdoesnotwork.Elementaryform

nomenclature(1)Poisson’sratioIsotropicAnisotropicModuliCoefficientPolymorphoustransformationPhasetransformation术语（1）泊松比各向同性的各向异性的modulus的复数系数多形态转变相变nomenclature（2）RecrystallizationSubstantiallyPreferableorientationTextureRadiographicHeterophaseAnomaly,（anomalies,anomalous）PeculiarMagneticeffectElinvar术语（2）重结晶充分地择优取向织构辐射照相的异质相（名）不规则，异常的人或物罕见的、特殊的；特权磁效应恒弹性镍铬钢Poisson’sratioArodsubjectedtouniaxialtensionnotonlyincreasesinlength(achangeinthesizealongtheaxisX)butalsodiminishesindiameter(compressionalongthetwootheraxes).Thus,auniaxialstressedstateresultsinatridimensionaldeformation.TheratioofthesizeschangeinthelateraldirectiontotheirchangeinthelongitudinaldirectioniscalledPoisson'sratio:

visPoisson'sratioandisamaterialelasticproperty;thenegativesigninEq.indicatesthatthesampledimensionsnormaltotheprimaryextensiondecrease(increase)astheaxiallengthofthesampleincreases(decreases).Formetals,thevalueofvisoftenontheorderof1/3.Thechangeinvolumeassociatedwiththesmallstrainsoflinearelasticdeformationcanbeobtainedbydifferentiatingtheexpressionforthevolume(V=l1l2l3)andkeepingtermsonlytofirstorder.Theresultis

Foruniaxialdeformation,ΔV/V=

(l-2

).Giventhat

=1/3,anelasticuniaxialstrainof0.5%wouldproduceavolumechangeofca.0.2%.Sincelinearelasticstrainsaretypicallysmallerthanthis,thevolumechangeduringthistypeofdeformationisusuallyquitesmall.Theelasticvolumechangedecreasesas

increases.Foranincompressiblematerial,suchasaplasticallydeformingmetalforwhichthevolumechangeiszero,theratiooflateraltouniaxialstrainis–1/2.Suchavaluedoesnotimplythat

,anelasticproperty,hasavalueof0.5forametalduringplasticdeformation.long-chainpolymerstypicallyhavevaluesofvgreaterthanmetals.Hence,andasnotedintheprevioussection,thesematerialsdiffersubstantiallyfromotherlinearelasticmaterials.FourelasticconstantsofanisotropicbodyEffectofvariousfactorsonelasticmoduliTemperatureWorkhardeningAlloyingAnomalousTemperatureeffectSinceelasticmoduliareassociatedwithinteratomicforcesandthelatterdependonthedistancesbetweenatomsinthecrystallattice,elasticconstantsdependontemperature.Thetemperaturedependenceofelasticmoduliisveryweak;Asmaybeseen,themagnitudeofmodulusdecreaseswithincreasingtemperature,withtheE(T)relationshipbeingalmostlinear.Ontheaverage,theelasticmodulusdecreasesby2-4percentbyevery100°C.Thetemperaturecoefficientoftheelasticmodulusofametaldependsonthemeltingpointofthatmetal.Forthatreasonitissometimesconvenienttoconsiderthedependenceofthemodulusonhomologoustemperature.Inthispresentation,thetemperaturerelationshipofthemodulusisnearlylinear.

Empiricalcorrelationindicatesthattheappropriatescalingconstantisabout100(whenSIunitsareused;i.e.,kTm

inJand

inm3).Thus,K=Boltzmannconstant,Tm=absolutemeltingtemperature,=volumeperatomThemodulusdecreasesconcurrentwiththeincreasedatomicseparation.Thisdecreaseisessentiallylinearwithtemperature,andanapproximateequationdescribingthemodulus-temperaturerelationshipiswhereEisthemodulusattemperatureTandE0themodulusat0K.Theproportionalityconstantaformostcrystallinesolidsisontheorderof0.5.Thus,forsucha"typical"material,themodulusdecreasesbyabout50%asthetemperatureincreasesfrom0Ktothematerial'smeltingpoint.Alloying(1)Alloying(2)inAlTheeffectofalloyingonelasticconstants,liketheeffectoftemperature,canbeassociatedwithvariationsintheinteratomicdistancesandinteratomicforcesinthecrystallattice.Ashasbeendemonstratedinradiographicstudies,thelatticeparameterofasolventvariesalmostlinearlywiththeconcentrationofanalloyingelement.Thedependenceoftheelasticmodulusofanalloyontheconcentrationofanalloyingelementisalsoclosetolinear.Asmaybeseenfromthefigure,alloyingcanincreasetheelasticmodulusinsomecasesanddecreaseitinothers,dependingontherelationshipbetween

thebondforcesofatomsofthesoluteandsolvent,ontheonehand,and

theforcesofatomicinteractioninthesolventlattice,ontheother.Iftheformeraregreaterthanthelatter,alloyingwillincreasetheelasticmoduli.Apartfromthevariationsoftheinteratomicforcesinthelatticeofthebasecomponent,alloyingcanalsocausecertainstructuralchangeswhichcaninfluenceappreciablythemagnitudeoftheelasticconstants.Forinstance,ifalloyingaboveadefinitelimitresultsintheformationofasecondphase,theelasticmodulusmaychangeadditionallycomparedwithitsvalueinasingle-phasesolidsolution.Ifthesecondphasehasahighermodulusthanthatofthebasemetal,itspresencewillincreasethemodulusoftheheterophasealloy.WorkhardeningWorkhardeninghasnoessentialeffectonelasticmoduli.Aslightdecreaseofelasticmoduli(usuallybelow1percent)onworkhardeningisusuallyassociatedwithdistortionsofthecrystallatticeofametaloralloy.Workhardeningcanresultintheformationofpreferableorientations,ortextures,whichmakethematerialanisotropicandcanchangesubstantiallytheelasticmoduli.Recrystallizationduringheatingofadeformedmetalalsoformstexturesandchangesappreciablytheelasticmoduli.Variationsinelasticmoduliandduetotheformationanddestructionofpreferableorientationsmayreachafewtenspercent.Intexturedpolycrystallinematerials,themagnitudeofanelasticmodulusdependsonthedirectionofmeasurement.AnomalousElinvarMagneticeffectscompensatethenormaldropofmoduliwithtemperature.Therangeofclimaticvariationsoftemperature.Review

Stress(relative/engineeringoractual/true)Strain(relative/engineeringoractual/true)Hooke’slawYoung’smodulus（Stiffness)ShearmodulusBulkmodulusShearstrainBulkStrainelasticmoduli

nomenclature(1）AnelasticityHysteresisMicroscopicMacroscopicCoordinatesThermodynamicLinearityQuasi-术语（1）n.滞弹性n.滞后现象微观的宏观的坐标热力学的线性准、伪，类似nomenclature（2）InstantaneouslyReciprocityMicroplasticallyMacroplasticallyHysteresisloopElasticaftereffectsStressrelaxationInternalfrictionDissipate术语（2）即时地，瞬时地互惠微观塑性（地）宏观塑性（地）滞后环弹性后效应力松弛内摩擦、内耗消耗IdealelasticbodiesAuniquerelationshipbetweenstressandstrainintheelasticregionAssumption:theloadisincreasedinfinitelyslowsothatthestateofthesystemhasthetimetofollowloadvariations.Or:achangeinthestateofasystemoccursinstantaneouslywithachangeintheload.Theprocessofloadingandunloadingcanberegardedenergeticallyreversible.AnelasticityInrealbodies,thedirectrelationshipbetweenstressanstrainisdisturbedandahysteresisloopappearsontheStress-StraindiagramStress-straindiagramincyclicloadingandunloading

AnelasticityAnirreversibledissipationofenergyduringtheprocessesofloadingandunloading;Theenergydissipatedinonecycleisdeterminedastheareaofthehysteresisloopintheσ-εcoordinatesandisthemeasureofinternalfrictioninthematerial.在弹性极限内应变落后于应力的现象称为滞弹性。Threedifferentmeaningsofanelasticdeformation:Anelasticdeformationispossiblewithoutparticipationofdislocations;(belowmicroscopicelasticlimit)Anelasticdeformationcanbeduetoenergeticallyirreversiblemovementofdislocation;(betweenmicroscopicelasticlimitandmacroscopicelasticlimit)Atstillhigherstresses,movementofdislocationsceasedtobemechanicallyreversible.Elasticaftereffectsandstressrelaxationσ(t)=Mε(t)whereMisthestaticmodulusofelasticity.Relaxationatconstantstress(a)andconstantstrain(b)Elasticaftereffectsandstressrelaxation(2)Thegradualriseofstraininloadingandgradualdisappearanceuponunloadingarecalledrespectivelythedirectandthereverseelasticaftereffect.ThegradualvariationofthestresstothevaluecorrespondingtoHooke’slawiscalledstressrelaxationElasticandplasticstraininstressrelaxationnomenclature（1）BauschingereffectInhomogeneuosDampingPrecipitationDissolutionAmplitudeResonanceAcoustic术语（1）包申格效应不均匀的阻尼、衰减沉淀、析出分解、溶解振幅共振声学的nomenclature（2）Pseudo-PseudoelasticityThermoelasticMartensiteTubularAnnealingDeviateSuccessive术语（2）伪、假、虚伪弹性热弹性的马氏体管状的退火偏离继承的、连续的InternalfrictionInternalfrictionistheabilityofmaterialstodissipatethemechanicalenergyobtainedonloadapplication;Theareaofthehysteresisloopintheσ-εcoordinatesisthemeasureofinternalfrictioninthematerial.TypesofhysteresisWhyinternalfriction?应力感生有序产生内耗；位错内耗；热流产生内耗；磁致伸缩内耗；非共格晶界内耗应力感生有序产生内耗SuccessivestagesofdeflectionofalockeddislocationlineatincreasingstressStress-dislocationstrainrelationshipforthemodel

TheBauschingereffect金属材料经过预先加载产生少量塑性变形（残余应变小于4%），而后再同向加载，规定残余伸长应力增加；反向加载，规定残余伸长应力减少的现象叫做包申格效应；包申格应变：在给定应力条件下，拉伸卸载后第二次拉伸与拉伸卸载后第二次压缩两曲线之间的应变差。Bauschingereffectintwistedtubularsteelspecimen

AnisotropyofslipbarrierscausingBauschingereffectSignificanceofanelasticphenomenaInstrument-making,elasticelement,bellsormusicalinstrumentsHighdampingcapacity:diminishnoise,avoidfailuresduetoresonanceInhomogeneity,localmicroplasticdeformation,internaltransformation,superplasticalloysetcforHigh-dampingapplication.NomenclaturePlasticdeformationStrainhardeningSlipResolvedshearstressCriticalresolvedshearstressAusteniticIntersectLacquer塑性变形应变硬化滑移分切应力临界分切应力奥氏体的相交、交叉、横断漆、涂漆于…使表面光滑术语Nomenclature术语PeculiarityPolycrystallinePeriodicallyIsothermalCrystallographyHexagonal

Syngony

特性多晶的周期性地等温的结晶学、晶体学六角形的，六边形的晶系Themechanicalbehaviourofmetalsandalloys

Themechanicalbehaviourofmetalsandalloysisdescribedbythefollowinglawsoftheirresistancetoelasticandplasticdeformationandfracture.Theisothermalmechanicalbehaviourofametalisdeterminedbyfourfactors:Stress,time,shape,andstructure.PeculiaritiesofMechanicalbehaviour

(i)howhighcanbeperiodicallyorconstantlyappliedloadssothatanobjectcouldrestoreitsshapeandsizeupontheirremoval;(ii)howhighistheresistanceofanobjecttoplasticflowatashort-termorlong-termloadapplied,whatistherateofvariationoftheshapeanddimensionsoftheobject,andwhatcharacteristicsandparticularconditionsofloadapplicationdeterminethecourseofplasticflowatadesiredrate;(iii)howlargeistheforcetocausefractureoftheobjecttopieces.

Moredeepanalysisofthemechanicalbehaviourofmetalsandalloysinthelasttwoorthreedecadesisassociatedwiththedevelopmentofthetheoryofdislocationsandthedescriptionofthephenomenaobservedontheatomiclevelandalsowithimprovementsinthemethodsofcontinuummechanics.Thisassociationbetweenvariouslevelsofdescriptionofthemechanicalbehaviourofmaterialsseemstobefruitful.Themechanicalbehaviouratthemacroscopiclevelisstudiedinothercourses;weshalldealwiththemechanismofplasticflowatthedislocationlevel.CarbonsteelintheelasticregionLinearelasticityandsubsequentplasticityUnstablecreepinannealedcopperPlasticdeformationThedeformationwhichisindependentoftimeandisretaineduponstressreleaseiscalledplasticdeformation.Effectofdeformationrateonstress-straincurveSlipofmetalcrystals

a–Zn,b–Cd,c-Sn,d-BiVariationofsliporientationindeformedtungstensinglecrystalatadifferentdirectionofexternalshearstress

Slipoflow-carbonsteel

(polycrystalline)Slip

Slipisthedisplacementofaportionofacrystalrelativetoanotherportionwiththecrystalstructureofbothportionsremainingunchanged.a-undeformed,b-elasticallydeformed,c-elasticallyandplasticallydeformed,d-plasticallydeformationinwhichsliphastakenplace;AB–slipplaneSlipplanesinthreetypicallatticeofmetalcrystals(Slipplainsusuallyhavetheclosestpackingofatoms)Threepossibleslipdirectionsinα-Fe;theshortest<111>directionispreferable

MicrostructureofausteniticCr-Ni-Mosteeldeformed25%(a.)and50%(b.)Strainbandsoflow-carbonsteelStretchedgrainsinlow-carbonsteelCrystallographyofslipinsinglecrystalsFractureofzincsinglecrystalIncubicsyngonycrystalsthissituationisimpossible,i.e.theultimatestrengthintensioncannotbeattainedearlierthanplasticflowbegins.

Forinstance,inf.c.c.crystalswherethefoursystemsof{111}planesintersectoneanother,itisimpossibletoorientatethecrystalrelativetothetensileorcompressiveaxissothattheshearstressbezeroinalltheseplanes.Atleastoneoftheplanesystemsturnsouttobeorientatedforfavourableslip.Withf.c.c.metals(aluminium,copper,lead,gold,silver)subjectedtotensionorcompression,fractureisalwaysprecededbyaplasticdeformation.B.c.c.crystalshavenoplaneswithsuchadensepackingofatomsasthebasalplanesinc.p.h.crystalsoroctahedralplanesinf.c.c.crystals.Forinstance,the{110}planesinb.c.c.crystals,thoughtheyarecharacterizedbytheclosestpackingofatoms,differinthisparameteronlyslightlyfromotherfamiliesofplanesinthatlattice.Themostessentialstructuralfeatureofb.c.c.crystals,whichcaninfluencethecourseofslip,istheexistenceofafamilyofclose-packeddirections,<111>cubediagonals.Thesedirectionsplayevenagreaterpartinslipthantheclose-packeddirectionsinhexagonalorface-centeredcubiccrystals.

Inb.c.c.crystals,however,the<111>directionofpreferableslipcanbefoundinseveralfamiliesofplanes:ina-iron,forinstance,itisfoundin{110},{112}and{123}.Inthatcase,slipoccurssimultaneouslyinanumberoffamiliesofplanes,intheexamplediscussed,intwooreventhreefamilies;inthegeneralcase,itisimpossibletopredictreliablywhichoftheslipplanesinb.c.c.metalswillbeoperative.Ontheotherhand,thesemetalshavealargernumberofintersectingsystemsofprobableslipplanesthanc.p.h.metalsandforthatreasontheyaremoreplasticthanthelatter.Ascomparedwithf.c.c.metals,theslipplanesinb.c.c.metalsdifferlessappreciablyfromtheotherplanesoftheb.c.c.latticeandhavealowerdensityofatomspackingthantheslipplanesinthef.c.c.lattice.Forthatreason,ahighershearstressisrequiredtoinitiateslipinb.c.c.crystalsbuttheyofferalowerresistancetothedevelopmentofplasticdeformationbeforefracture.SlipsystemsinmetalliccrystalstructuresIngeneral,theductilityofb.c.c.metals,suchasa-iron,tungsten,molybdenum,or|

-brasshasintermediatevaluesbetweenthoseoff.c.c.andc.p.h.metals.

nomenclature术语Schmid-BoaslawResolvedshearstressCriticalresolvedshearstressTwinningOctahedralIndeterminacyIncoherentboundaries分切应力临界分切应力孪生八面体的不确定不连贯界面PartialcoherentboundariesIncoherentboundaryResolvedshearstressSchmid-BoaslawOrientationfactor(Schmidfactor)EXAMPLEPROBLEM1.Hexagonalclose-packedzincslipsbybasalplaneslip.Azincsinglecrystalisorientedsothatthenormaltoitsslipplanemakesanangleof60°withthetensileaxis.Ifthethreeslipdirectionshaveanglesof38°,45°,and84°withrespecttothisaxis,andthecriticalresolvedshearstressforZnis2.3MN/m2,determinethetensilestressatwhichplasticdeformationcommences.EXAMPLEPROBLEM2.Asinglecrystalhavingasimplecubicstructure(slipplanes{100},slipdirections<100>)isorientedsuchthatthetensileaxisisparalleltothe[010]crystalaxis.MakealistoftheslipsystemsinthiscrystalandcalculatetheSchmidfactorforthisloadinggeometry.

T.A.

[010][001]Considerthisproblemforasituationwherethetensileaxisisparalleltothe[011]crystalaxis.EffectoforientationfactoronslipstressEffectoftemperatureoninMgsinglecrystal1、BurkeandHibbard2、SchmidandSiebelEffectoftemperatureonandinironsinglecrystals1、upperyieldlimit2、loweryieldlimit3、criticalresolvedshearstressEffectoftemperatureoninCuandCualloys1、pureCu2、Cu-Ag(0.1%)3、Cu-Ge(0.33%)4、Cu-Ag(0.2%)Therelationshipbetweenandcompositionoff.c.c.singlecrystalsEffectofconcentrationofalloyingelementsoninMgalloys1--Mg2--Mg-In3--Mg-Cd4--Mg-Ti5--Mg-Al6--Mg-ZnEffectofalloyingelementsondependingonthedifferenceinatomicdiametersTherelationshipbetweenandcompositionoff.c.c.singlecrystalsCrystallographicdiagramoftwinningTwinningtakesplacewheretheshearstressattainsthecriticalvalueand,likeslip,obeyscertaincrystallographicrelationships.Themirrorimageplaneiscalledthetwinningplaneandthedirectionofdisplaceiscalledthetwinningdirection.Thetwinningdirectionispolar;Twinningshearcanoccurinonedirectiononly;Atomicplanesaredisplacedintwinningthroughthesameverysmalldistance(smallerthantheinteratomicdistance),sothatnoindividualvisiblestraintracesformonthesurfaceofatwinband.Theroleofthetwinningprocessusuallyincreaseswithdecreasingtemperatureofdeformationand/orincreasingrateofdeformation.Sincethestressneededforpropagationofatwinismuchhigherthantheslipstress,itisclearthattwinningispossibleunderparticularconditionswhentheresolvedshearstressturnsouttobehigh.Twinninginb.c.c.andf.c.c.crystalsisusuallyobservedatlowtemperaturesandhighdeformationratesandinc.p.h.crystal