《塑性加工模拟及自动控制》课件：郑江-esson 2-Continuous Media

ContinuousMediaZhengJiangChongqingUniversityApril-13-2016Outline1Objectives2ConservationAndContinuityEquations3ConstitutiveEquations4BoundaryandInitialConditions1ObjectivesIntroducetheequationsofconservationofmass,solute,momentumandenergy.IntroducetheprincipalequationsformaterialsbehaviorDefinetheboundaryconditionsandinitialconditions.介绍质量守恒。。。。。。。。。介绍材料特性（材料力学行为）的主要公式解释边界条件和初始条件Theequationsofelectromagnetismwillnotbecoveredhere;wewilldiscussconservationofmass,ofmomentum,ofenergyandofsolute.Next,wewillcoverthemostimportantconstitutiveequationsformaterialbehaviorthatconnect,forexample:stresstostrainortovelocityindeformationorflowproblems;orenthalpyandheatfluxtotemperatureinheattransfercalculations;or,finally,thefluxofsolutetoconcentrationindiffusion.Thederivationoftheseequationswillbedoneintheperspectiveofsubsequentapplicationsthatmayinvolvetwodifferentscales.方程的推导将运用在两个尺度Atthemacroscopiclevel,theconservationandconstitutiveequationsallowmodeling,orevenoptimization,ofanindustrialprocessorsimulationofthebehaviorofasampleundergoingmechanicaltesting.在宏观尺度，守恒和本构方程允许建模,甚至优化of一个工业过程或行为的模拟of进行机械测试的样本Atthemicroscopicscale,theequationscanbeusedtodescribetheformationandevolutionofthemicrostructure(dendrites,spherulites,lamellae,orfibers,etc.),theinteractionbetweenafiberandthematrixincompositematerials,orthedeformationofacrystallatticearoundadislocation.在微观层面,方程可以用来描述微观结构的形成和演化(树突,球晶、薄片或纤维,等等),纤维和基质之间的相互作用in复合材料or一个位错周围的晶格的变形。HomogenizationRollingAnnealCasting1mEngineBlock1-10mmMacrostructureGrainsMacroporosityPropertiesHigh-cyclefatigueDuctility~100-500mMicrostructureEutecticPhaseMicroporosityIntermetallicPropertiesYieldstrengthTensilestrengthHigh-cyclefatigueLow-cyclefatigueThermalGrowthDuctility~10-100ÅAtomaticStructureCrystalStructureInterfaceStructurePropertiesThermalGrowthYieldstrength~3-100nmNanostructurePrecipitatePropertiesYieldstrengthThermalGrowthTensilestrengthLow-cyclefatigueDuctilityBasedonprocessingflowchartBasedonthemetallurgicallengthscalesFigure1Thedifferentscalesappearinginmaterialsscience.Aturbineblade(a),solidifiedinaceramicmold(investmentcasting),measuresafewdozencentimeters.Itiscomposedofgrainswhichareclearlyvisibleafterchemicaletching(scale:afewmillimeters),whichthemselvesaremadeupofdendriteswhicharespacedatafewdozentohundredsofmicrons(b).In(c)isshownaschematicatatomicscaleofthetransformationfromaliquidtoasolidforametalalloyduringtheformationofsuchabladebyprecisioncasting.1.2ConservationandContinuityEquationsFigure2Inthecontinuouscastingofaluminum(a)，liquidmetalisinjectedfromanozzlethroughadistributionbagwhichfiltersoutinclusionsandoxideskindebris.Themetalcoolsoncontactwithamold,whichitselfiscooledbycirculatingwaterandwaterspraying.Solidmetalisextractedcontinuouslybyajackattachedtoabottomblock.Convectionintheliquid(indicatedbyarrows)cantransportgrowinggrainsofaluminuminthe‘mushy'region(b)Theenvironmentisnothomogeneousasitiscomposedofatleasttwophase,namelysolidandliquid.Thephenomenaoccursatmultiscale.Itcanbedescribedbyfourconservationequations(mass,momentum,energyandsolute)Thetypeofbehavior(elastic,plastic,viscous,etc.)andalsothevaluesofthermo-mechanicalpropertiesofthematerial(specificmass,viscosity,elasticmodulus,strain-hardeningcoefficient,thermalconductivity,etc.)enterintotheconstitutiveequationsforthedifferentphasesofthematerials.NoticeFigure3Afewareasofmaterialssciencewheremodelingplaysanimportantrole:polymerinjection(a),waterdiffusioninconcrete(b)，deformationofatestspecimenundertension(c)1.2.1Definition“Nothingislost,nothingiscreated,everythingistransformed”-------

Lavoisier‘sprincipleFigure4Diagramofthecalculationdomainnanditsboundaryon(a).Theoutgoingnormalvectornandthetangentvectorτarealsoshown.Inthreedimensions(b)，theretwotangentvectors.In(c)，thevolumeelementΔVisshownwiththevelocitiesinthemediaateachfaceforthederivationoftheconservationequations.Ω的计算范围和边界ᵊΩ是图a，向外的法向量和切向量也如图示；在b中的三维图里，有两个切向量；c中，体积元素ΔV和它在介质中每个表面的速度一起显示出来，用于本构方程的推导。Inthefirst,calledtheLagrangian,analogoustotraditionalmechanics,wefollowthematerialelementthroughitsmovements(Forsoliddeformation);拉格朗日，类似于经典力学，我们遵从材料元素的运动Secondly,Eulerian,thereferenceframeisfixedandwewatchwhathappensatapointasafunctionoftime.(Forfluidmechanics)欧拉，修正了参考坐标系，并且我们能观察到随时间发生了什么Twopossibleapproachedtodescribeconservation(ofmass,energy,momentum,etc)AslongasthedomainΔVissmall,wecanmaketheassumptionthat,oneachface,thevaluesconsideredareconstant.

ΔV足够小时，我们能够假设，在每个面（所取的ΔV）上，值是恒定不变的。1.2.2

EquationofconservationofmassThissumincludesonlythetwocontributionsexpressingthatthemassvariationinsidetheelementmustbeduetotransferofmassacrossthefacesbythevelocityfieldv.

Thereisnodiffusion,norproductionofmass.Themassvariationinsidetheelementisgivenby:ThetotalquantityofmaterialleavingthevolumeΔVisgivenby:Theintegral(1.2)canbemanipulatedas:Thereisneitherlossnorcreationofmaterial:Stationarycase:Incompressible:1.2.3ConservationofsoluteAsforthemasssumderivedabove,atemporalvariationofthequantityinsolutioninthevolumeelementaswellasatransporttermcontainingthevelocityfieldwillappearinthesum.Twonewcontributionsneedtobetakenintoaccount.Thefirstisadiffusiveterm扩散项

associatedwithconcentrationgradients.Thesecondisasource

term(orsink)forchemicalreactions.isthenumberofmolesthatappearordisappearlocallyperunittimeandunitvolume.Weobtainthelocalequationfortheconservationofsolute.Formassspecificconcentration.1.2.4ConservationofmomentumFigure5Thesurfaceforces,T,andthegravityforce,ρg,actingonasmallvolumeelementΔV.Takingthesumofthecontributionstotheforcesactingonthesurfaceandonthevolume,weobtainforthecomponentx:Themomentumfluxinthexdirectionenteringorleavingthedomainisgivenby:Weobtainforthexcomponentoftheconservationofmomentum:Allthreedirectionalcomponentsofthemomentumcanbeexpressedasfollows:Forquasi-staticproblem:1.2.5Displacement,strain,strainrateOveraperiodoftime,displacementsofthematerialoccurduetotheappliedstressandvolumeforces.InLagrangiancoordinates,themovementofthematerialisdescribedbythesetoftrajectoriesofallthepointsinthematerial:,wherexoistheinitialpositionofapointattimet=0Thedisplacementu(xo,t)ofapointxoattimetisnaturallydefinedasthedifferencebetweenitspositionattimetanditsinitialposition:Figure7Forgingofasolidtoobtainacomplexform(a)andtwo-dimensionalrepresentationofthedisplacements(b)forsuchaprocessTosimplifytheexpression,itisnecessaryatthisstagetoswitchtothecoordinatenotation(x1,x2,x3)inplaceof(x,y,z).WhereistheKroneckerdeltafunctionThisequationdefinesthegradienttensorofthetransformationF=I+Gradu.Theresultisasymmetricsecondordertensor,a3x3matrixcalledtherightCauchy-Greenstraintensor,orCauchy'sdilatationtensor,Returbingtothe(x,y,z)coordinates,thesymmetricstraintensoriswritten:Figure8Illustrationofthedifferentelementarystraincomponentsforaparallelepiped.Wheredesignatesthevelocityvectorassociatedwiththedisplacementvectoru.ThisvelocityissimplythatofapointinLagrangiancoordinates:EngineeringstrainThe

engineeringnormalstrain

or

engineeringextensionalstrain

or

nominalstrain

e

ofamateriallineelementorfiberaxiallyloadedisexpressedasthechangeinlengthΔL

perunitoftheoriginallength

L

ofthelineelementorfibers.Thenormalstrainispositiveifthematerialfibersarestretchedandnegativeiftheyarecompressed.Thus,wehave

eisthe

engineeringnormalstrain,

Listheoriginallengthofthefiberandislisthefinallengthofthefiber.

StretchstrainTheextensionratioisapproximatelyrelatedtotheengineeringstrainbyThisequationimpliesthatthenormalstrainiszero,sothatthereisnodeformationwhenthestretchisequaltounity.TheextensionratioisapproximatelyrelatedtotheengineeringstrainbyThe

truestrain

ε,(althoughnothingisparticularly"true"aboutitcomparedtoothervaliddefinitionsofstrain).Consideringanincrementalstrain.thetruestrainisobtainedbyintegratingthisincrementalstrain:where

e

istheengineeringstrain.Thetruestrainprovidesthecorrectmeasureofthefinalstrainwhendeformationtakesplaceinaseriesofincrements,takingintoaccounttheinfluenceofthestrainpath.TruestrainTheGreenstrainisdefinedas:GreenstrainAlmansistrainTheAlmansistrainisdefinedas:NormalstrainConsideratwo-dimensionalinfinitesimalrectangularmaterialelementwithdimensions

,whichafterdeformation,takestheformofarhombus.FromthegeometryoftheadjacentfigurewehaveForverysmalldisplacementgradientsthesquaresofthederivativesarenegligibleandwehaveThenormalstraininthex

directionoftherectangularelementisdefinedbySimilarly,thenormalstrainintheydirection,andzdirectionbecomes

ShearstrainTheengineeringshearstrain()isdefinedasthechangeinanglebetweenlinesand,thereforeFromthegeometryofthefigure,wehaveForsmalldisplacementgradientswehaveForsmallrotations,i.e.αandβare,wehavetanα≈

α,tanβ

≈

β,thereforeByinterchangingxandyanduxanduy,itcanbeshownthat,similarly,forthey-zplaneandz-xplane,wehaveGeneralthree-dimensionalbodywithan8-nodethree-dimensionalelement1.2.6VirtualpowerThebodyislocatedinthefixed(stationary)coordinatesystemX,Y,ZConsideringthebodysurfacearea,thebodyissupportedontheareaSuwithprescribeddisplacementsUSuandissubjectedtosurfaceforcesfsf(forcesperunitsurfacearea)onthesurfaceareaSf.ThebodyissubjectedtoexternallyappliedbodyforcesfB(forcesperunitvolume)andconcentratedloadsRc(whereidenotesthepointofloadapplication).WeintroducetheforcesRcasseparatequantities,althougheachsuchforcecouldalsobeconsideredsurfacetractionsfsfoveraverysmallarea.Ingeneral,theexternallyappliedforceshavethreecomponentscorrespondingtotheX,Y,Zcoordinateaxes:U=USuonthesurfacearea.ThestrainscorrespondingtoUareCisthestress-strainmaterialmatrixandthevector

denotesgiveninitialstresses1.2.7ConservationofEnergyIngeneral,thefirstlawofthermodynamicstellsusthatthevariationoftotalenergyofadomainunderconsiderationisduetothemechanicalpoweroftheexternalforces,Pmech,andthecaloricpowerapplied,PcalThetotalenergyiscomposedofthekineticenergy,Ebandtheinternalenergy,Ej.Puttingallthetermstogether:Thesumofthefirstthreetermsontherighthandsideoftheequationisthedeformationpower.1.2.8UnifiedformoftheconservationequationsTheconservationequationsderivedabovearesimilarinthattheyallcontainatemporalvariationterm,anadvectivetransportterm,andsomehaveadiffusiontermandasourceterm.Theycanbereducedtoasinglegeneralequation:1.3ConstitutiveEquationsIn

physics

and

engineering,a

constitutiveequation

or

constitutiverelation

isarelationbetweentwophysicalquantities.1.3.1Constitutiveequationsformass1.3.2ConstitutiveequationsforsoluteFigure9DiagramofdiffusionaccordingtoFick'sfirstlawrelatingthechemicalfluxtotheconcentrationgradientThefluxcangenerallybeexpressedintermsofthegradientoftheconcentrationbyFick’sfirstlaw:whereDjisthediffusioncoefficient.1.3.3ConstitutiveequationsforEnergyConsideringthepressureconstant,whichisagoodapproximationforcondensedmatter,thespecificenthalpyofaphase,H,isgivenby:whereKisthethermalconductivityofthematerial.Thediffusiveheatflux,jT,isgivenbyanequationsimilartoFick'sfirstlaw(Fourier’slaw):whereEistheelectricalfield(Voltage),jEtheelectricalcurrentdensity,andρEelectricalresistivity.Inthecaseofchemicalreactions,theheatsourcetermbecomesasumoverallthechemicalspecies:Thesourcetermintroducedbyaphasetransformationwillbetreatedindetailinchapter5.Strictlyspeaking,thelatentheatperunitmass,Lα/β,associatedwithaphasetransformationα→β，isnotavolumetricsourcetermasitisproducedatthemovinginterfaceα/β.1.3.4ConstitutiveequationsforMaterials:quasi-staticcaseInthesimplestcase,thatoflinearelasticity,wetrytorelatethestresstensorσtothestrainε(x,y,z).Beingsymmetric,(σij=σjiandεij=εji),thesetwotensorsonlyhavesixindependentcomponents.Inthecoordinates(x,y,z)，Hooke’slawrelatingthesixcomponentstakestheform:Thematrix[Del]istheelasticitymatrix.whereEistheelasticmodulusandvpisPoisson’scoefficient.Assumingthatthematerialisstretchedorcompressedalongtheaxialdirection：Acubewithsidesoflength

L

ofanisotropiclinearlyelasticmaterialsubjecttotensionalongthexaxis,withaPoisson‘sratioofν.Thegreencubeisunstrained,theredisexpandedinthe

x

directionbyduetotension,andcontractedinthe

y

and

z

directionsby∆Lduetotension,andcontractedinthe

y

and

z

directionsby

∆L’.Therelativechangeofvolume

ΔV/V

ofacubeduetothestretchofthematerialcannowbecalculated.UsingV=L3andV+ΔV=(L+ΔL)(L-ΔL’)2(x+Δx)1-2ν

=x1-2ν+(1-2ν)·x-2ν

·Δx-

2ν(1-2ν)·x-2ν-1

·Δx2………

Theproblemencounteredinmaterialsscienceisthatmostofthetimethematerialdoesnothavelinearelasticbehavior;rather,itcanundergoplasticstrain,εpl.Inadditiontothiscomponent,othersofathermalnature，εth，orassociatedwithphasetransformations，εtr(volumechanges),cancontributetothelocaltotalstrain,ε,givenby(1.36).Ingeneral,onehas:1.4BoundaryandInitialConditions1.4.1GeneralitiesInanonstationaryproblem,thefirsttermoftheequationrequiresthespecificationofthevaluesofthefield(x,t=0)ateverypointinthedomainΩattimet=0.Thisistheinitialcondition.Inthesameequation,therearetwointegralsontheboundaryofthedomain:thefirstinvolvesthenormalvelocitycomponent,vn=v·n,andcorrespondstothetransportofthequantityacrossthesurface,whereasthesecondisrelatedtothefluxenteringorleavingthedomainatthesurfacebydiffusion,j=j·n.n.Itcanbeseenthatitwillbenecessarytospecifythesetermsatthesurfacetodeterminewhathappensintheinterior.Theseareboundaryconditions.Theboundaryconditionsfordiffusiveproblemsaredividedintotwotypes:I.Anaturalconditionexpressesdirectlythevalueofjontheboundary.Threecasesofthistypearegenerallyconsidered:I.1ThehomogeneousNeumannconditionisanullflux:I.2TheNeumannconditioncorrespondstoanonzerofluxgivenontheboundary:I.3TheCauchyormixedconditionconsistsofalinearrelationbetweenthefluxandthevalueofthevariableitself:II.AnessentialconditionoraDirichletconditionconsistsofspecifyingthevariabledirectlyratherthanthediffusiveflux,Thebaroverthevariableontherightindicatesthatitisanimposedvalue.Thisvaluecanbeafunctionofpositionandtime.1.4.2SolutetransferAnessentialconditionmeansspecifyingthevalueofconcentrationofsoluteionapartofthedomainboundary.ThenaturalhomogeneousNeumanncondition,AnaturalnonhomogeneousNeumannconditioncorrespondstoafixed,nonzero,solutefluxatthesurface.Amixed,Cauchy,boundaryconditioncanbeexpressedintheform:whereαisachemicaltransfercoefficientbetweenthesur