多尺度位错动力学框架(MDDP)

1Tri-LabShortCourseonDislocationsinMaterialsPleasanton,CAJune8-10,1998Lectureon:3DDislocationDynamics:NumericalTreatmentTri-LabShortCourseonDislocationsinMaterialsPleasanton,CAJune8-10,1998Lectureon:3DDislocationDynamics:NumericalTreatmentH.M.Zbib,M.Rhee&J.P.HirthSchoolofMechanicalandMaterialsEngineeringWashingtonStateUniversity2ContentsContents–Discretizationofdislocationcurves–IdentificationofSlipgeometry(bcc)–Longrangeinteraction–EquationofMotion:Glide,climb,cross-slip,multiplication–Short-rangeinteractions:Annihilation-ProductionandFrank-Readsources,Junctionformation:Co-planarandnon-coplanar,Jogs,andDipoles•BasicStructureofDislocationsDynamics•Numericalissues:•Long-rangeInteractions:Superdislocations•TimestepandSegmentlength•Parallelprocessing:Familydecomposition•CriticalIssues•Movie(Typicalsimulations)3FLOWCHARTOFDislocationDynamicsFLOWCHARTOFDislocationDynamics4I.BasicGeometry(bcc)I.BasicGeometry(bcc)SimulationCell(5-20)[100][010][001]Slipplane(101)bdislocationb=Burgersvector=linesensevector[]111μm5DiscretizationDiscretization•StressFieldofa3DStraightdislocationsegmentisknownexplicitly(Hirth&Lothe,1982).•Discretizeeachcurveintoasetofmixedsegments.bbxyzIdentificationofbasicgeometryIdentificationofbasicgeometryi(x,yz)jkForeachnodeidentify:•Coordinates,Burgersvector•slipplaneindex•neighboringnodes(k&j)•Nodetype(free,fixed,junction,jog,boundary,etc.b6SlipSystems&Cross-slipplanesSlipSystems&Cross-slipplanes7InitialConfiguration*MoveNodes*MaintaincontinuityNodalVelocity=averagevelocity“V”ofadjacentsegments“V”isintheglideplaneandnormaltothedislocationsegmentxyzVggglideFTMV),(θ=bθGlideMobilityNetGlideForce/unitlengthII.EquationofMotionII.EquationofMotion8MacroscopicStrainMacroscopicStrain)(iii1pnbbn2D⊗+⊗=∑=iNigiiVvl)(iii1pnbbn2W⊗-⊗=∑=iNigiiVvlCellvolumeStrainratetensor:SpinTensor:Segmentlength90=+++aiFFMvvm/*&Effectivemass=dvdWv1Forscrewdislocation:()3120--+-=γγvWm*()21221//Cv-=γ⎟⎟⎠⎞⎜⎜⎝⎛=0204rRnbWlπμ(Hirth,ZbibandLothe,1998)Foredgedislocation:[]53131420622501484016-----+-+++--=γγγγγγγlllvCWm*()21221//llCv-=γInthelimitofsmallvelocity,theyreducetostandardforms,e.g.Gilman(1997),Beltz(1968),Weertman(1961)vEquationofMotion:InertiaEquationofMotion:Inertia10InertiaeffectisverysmallforV0.5CRise-timetoforVtoreachsteadystateMDcalculationsbyShastry,1998EffectofInertiaEffectofInertiaτ11III.DrivingForce:Peach-KoehlerForceIII.DrivingForce:Peach-KoehlerForce•Averagestressiscalculatedatthecenterofeachsegmentandincludes5contributions:•a)self-forceduetoadjacentsegments•b)forceduetootherremotesegments,•c)forceduetotheappliedstress,•d)forceduetothePeierlsstress(),,,,,11111-+-≠+≠≠=++×⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎝⎛⋅+=∑iiiiiNijijijjiaDjiFFbFξσσ12ForcefromaremotesegmentForcefromaremotesegmentxyzABCDbbABABCDABbFζσ×=).(StressfieldofsegmentCDEvaluatedattheCenterofAB.(VariationoverABisverysmall)ζAB13zxyABbx,y,zp)()(ABijijPijσσσ-=()()()()zzzRyxRRxbRxbRyRbRRybRxRbRxbxRxRybxRxRxbRxRxbRyRybyRyRxbyRyRybxRxRxbxRxRybzyxyzzyxxzyxxyyxzzyxyyyxxx-′=+=+=+--+⎟⎟⎠⎞⎜⎜⎝⎛-=+-+⎟⎟⎠⎞⎜⎜⎝⎛+-+-=⎟⎟⎠⎞⎜⎜⎝⎛+--⎟⎟⎠⎞⎜⎜⎝⎛--=⎟⎟⎠⎞⎜⎜⎝⎛--+⎟⎟⎠⎞⎜⎜⎝⎛+-=⎟⎟⎠⎞⎜⎜⎝⎛+++⎟⎟⎠⎞⎜⎜⎝⎛--=⎟⎟⎠⎞⎜⎜⎝⎛+--⎟⎟⎠⎞⎜⎜⎝⎛++-=λρρλνλλνσσλνλνλσσρρλρρλσσλρνλρνσσρρλρρλσσρρλρρλσσ,,22222233203230222222222203232022222222220222222222201121212221212121OtherformsaregiveninHirthandLothe(1982,p.134)ThisformismostconvenienttouseRemotestressFieldRemotestressField•IntrinsiccoordinatesystemººRequiresmatrixtransformation•MorenumericallyefficientformhasbeendevelopedbyDevincre(1995)14Self-ForceSelf-Force1ld2ld∫×=22221dlbFselfξσ.21CABDldForceatsub-segmentld=ForcefromsegmentCA+ForcefromsegmentBD+ForcefromsegmentAB(seeHirthandLothe,1982,p.131)15Self-ForceperunitlengthSelf-ForceperunitlengthCABD∞λ∞gF()()()()∞-∞---⎥⎦⎤⎢⎣⎡---+-+=DCABBDBACAgFFbLbfLbfFααλλνπμνθλπμθπλμcossin,,2111444AθαABbzxBθExplicitexpression,moreefficient16()ννθθν-+-⎥⎦⎤⎢⎣⎡+-+=1111ABzCAxAAAByCAyABxCAxABzCAzCAbbbbbbbbfsincosAverageforceperunitlength:()()[]∞-∞---⎟⎟⎠⎞⎜⎜⎝⎛+=DCBDBACAgFFLnbfbfLFρθθπμl,,4Cut-offparameter;numericalparameterwhichcanbeadjustedtoaccountforcoreenergy•Similarexpressionsareobtainedforthenormalforce.•TheseexpressionsreducetothosegiveninHirthandLothe(1982,p.138)fore.g.CAABbb=17ABbForpureedgepurescrewdislocations,thisreducestoe.g.bgFgF(forceperunitlength)⎟⎟⎠⎞⎜⎜⎝⎛=ρπμLnLbFgl42()⎟⎟⎠⎞⎜⎜⎝⎛-=ρπμνLnLbFgl411218BowoutofedgeandmixeddislocationsExamples:Prismaticloop19Example:StressfieldDemir,ZbibandHirth(1992)ExactApproximate20IV.ShortRangeInteractionsIV.ShortRangeInteractionsBCC:4Burgersvectors,withoutregardtoslipplanesºº8possibledistinctreactions,-4repulsive,3attractive,and-oneannihilation.Whenslipplanesareconsidered:Forthe{110}and{112}planesºº420attractivereactionsallofwhicharesessile(BairdandGalye,1965)dAshort-rangeinteractionoccurswhenthedistance“d”betweentwodislocationsbecomescomparabletothesizeofthecore*annihilation,*formationofdipoles,*jogs,and*junctions.Detailedinvestigationofeachpossibleinteractioncanbecomeverycumbersome211.Criticaldistancecriterion(EssmanandMughrabi,1979)Implicitlytakesintoaccounttheeffectofthelocalfieldsarisingfromallsurroundingdisloc