Simulation of grain boundary diffusion creep analy

ISSN1360-1725UMISTSimulationofgrainboundarydiffusioncreep:analysisofsomenewnumericaltechniquesJudithM.Ford,NevilleJ.FordandJohnWheelerNumericalAnalysisReportNo.435ManchesterCentreforComputationalMathematicsNumericalAnalysisReportsDEPARTMENTSOFMATHEMATICSReportsavailablefrom:DepartmentofMathematicsUniversityofManchesterManchesterM139PLEnglandAndovertheWorld-WideWebfromURLs://ftp.ma.man.ac.uk/pub/narepSimulationofgrainboundarydiffusioncreep:analysisofsomenewnumericaltechniques¤JudithM.Fordy,NevilleJ.FordzandJohnWheelerxOctober21,2003AbstractWeconsiderthesimulationofdeformationofpolycrystallinematerialsbygrainboundarydiffusioncreep.Foragivennetworkofgrainboundariesin-tersectingatnodes,withappropriateboundaryconditions,wecancalculatetherateatwhichmaterialwillbedissolvedordepositedalongeachgrainboundaryandhencepredicttherateatwhicheachgrainwillmovetoaccom-modatethisdissolution/deposition.Wediscusstwonumericalmethodsforsimulatingthenetworkchangesoverafinitetimeinterval,basedonusingthemovementofadjacentgrainboundariesoverasmalltimeintervaltoestimatethevelocitiesofthenodes.(Thesecondofthesemethodshasenabledustospeedupsolutionby100timesintypicalexperimentscomparedwithana¨ıveforward-Eulerapproach.)Weshowthattheaccuracywithwhichthenodevelocitiescanbeestimatedisdependentonlyontheprecisionofthemachinewithwhichtheyarecomputedanddeducethat,forallpracticalpurposes,thelackofprecisenodevelocityvaluesdoesnotdetractfromthequalityofoursolution.Finally,weconsidertheunderlyingstabilityoftheproblemundervariousdifferentboundaryconditionsandconcludethatourmethodshavethepotentialforprovidingusefulinsightintotheeffectofgrainsizeandshapeondeformationinpolycrystallinematerials.Keywords:diffusioncreep,microstructure,grainboundaries,stability,predictor-corrector,Euler’smethod,Runge-Kuttamethods1IntroductionCrystallinematerialshaveamicrostructureof“grains”,separatedfromonean-otherby“grainboundaries”.Theprecisenatureofthesediffersfromonematerialtoanotherandalsovariesaccordingtotheenvironmentalconditions.Inthisar-ticleweusea2-dimensionalmodelinwhicheachgrainisconsideredtobearigid¤ThisworkwassupportedbyNERCResearchGrantNER/B/S/2000/[e-mail]s,[e-mail]s,[e-mail]eanSciences,UniversityofLiverpool.12Diffusioncreepsimulationbodyseparatedfromtheadjoininggrainsbyinfinitesimallynarrowstraightgrainboundaries.Coblecreep(see[3])occurswhenexternalstressescausemattertobetransportedbygrain-boundarydiffusionfromboundariesundercompressiontoboundariesun-dertension.Asaresult,individualgrainstendtoelongateproducingmacroscopicstrains.Pressuresolutioncreep,wherematerialdiffusesalongfluid-filledgrain-boundariesisverysimilarandcanbemodelledusingthesameequations(see[5,13]).BothCoblecreepandpressuresolutioncreepareimportantmechanismsinrockdeformation(see[4,5,11,16])andCoblecreepisalsoafactordeterminingthestrengthofmetals,ceramicsandothermaterials(see[2]).Simulationofgrain-boundarydiffusioncreepcanprovideusefulinsightsintohowthemicrostructureofamaterialaffectsitsbehaviourunderstress.Hazzledine&Schneibel[9]derivedasystemofequationsrelatingthestressdistributionandinitialgrainvelocitiestothenetworkconfigurationandboundaryconditions.Morerecently,Kim&Hiraga[10]usedaverymuchsimplifiedmodel,inwhichthestressisassumedconstantalongeachgrainboundary,tosimulatedeformationovertime.Fordetal.[6]ex-tendedtheHazzledine&Schneibelmodeltoincludeagreatervarietyofboundaryconditionsandintroducedatime-steppingalgorithmtoenabledeformationtobesimulatedovertimemorerealisticallythanusingtheKim&Hiragamodel.Herewepresenttwonumericalmethodsforimplementingthetime-steppingprocedureandanalysethemmathematically.Thepaperisstructuredasfollows.Wedescribethemathematicalmodelinx2andshowhowitcanbeusedtodefineadifferentialequationdescribingthenodemovementsovertimeinx3.Inx4wepresentnumericalschemesforperformingthesimulationandanalysetheirconvergenceproperties.Theperformanceofthesemethodsisillustratedbyexampleresultspresentedinxx5and6.Inx7wediscussthestabilityoftheunderlyingproblemundervarioustypesofboundaryconditions.Weconcludewithadiscussionofsomepossiblefuturedevelopments.2DefiningtheproblemThefluxinatoms/m2/salongagrainboundaryisgiven(see,forexample,[14])byJ(x)=¡DkT@¾n(x)@x;(2.1)whereDisthegrain-boundaryself-diffusivity,kisBoltzmann’sconstant,Tistheabsolutetemperature,and¾n(x)isnormalstressalongtheboundary,xbeingthedistancealongtheboundaryfromagivenpoint.(Compressivestressesaretakentobepositiveandtensilestressesnegative.)Considerapolycrystal(suchasthatillustratedinfigure2.1)madeupofapre-definednumberofgrainswithstraightgrainboundariesmeetingattriple-points(3-nodes)anddouble-points(2-nodes).Theconfigurationofthenetworkatagivenc°Copyrightreserved3timecanbedefinedbythepositionsofthenodesandtopologicalinformationaboutwhichnodesarejoinedtowhichothers.Figure2.1:10-grainpolycrystalWedefinethefollowingconstantsrelatingtothenetworkofgrainboundaries:ngnumberofgrainsngbnum