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Earth and Planetary Science Letters 506 (2019) 282–291Contents lists available atScienceDirectEarthandPlanetaryScienceLetterswww.elsevier.com/locate/epslDislocationdynamicsmodellingofthepower-lawbreakdowninolivinesinglecrystals:TowardaunifiedcreeplawfortheuppermantleKarine Gourieta,Patrick Cordiera,,Fanny Garelb,Catherine Thoravalb,Sylvie Demouchyb,Andréa Tommasib,Philippe CarrezaaUniv.Lille,CNRS,INRA,ENSCL,UMR8207UMETUnitéMatériauxetTransformations,F-59000Lille,FrancebGéosciencesMontpellierUniversitéMontpellier&CNRS,F-34095Montpellier,FrancearticleinfoabstractArticlehistory:Received12January2018Receivedinrevisedform28October2018Accepted31October2018Availableonline16November2018Editor:J.BrodholtKeywords:olivinerheologycreepdislocationglidedislocationclimbEarthmantlenumericalmodellingInthepresentwork,weuseanumericalmodellingapproachbasedon2.5-dimensionaldislocationdynamicssimulationstoinvestigatethetransitionbetweenthepowerandtheexponentiallawsinolivinefortemperaturesrangingbetween800 Kand1700 Kandstressesbetween100and500 MPa.Wemodelthedeformationofanolivinecrystalbytheinterplayofglideandclimbofdislocations.Plasticstrainisproducedbyglide,theamountofglidingdislocationsbeingcontrolledbyclimbactingasarecoverymechanism.Withinthisframework,andwithouttheneedofintroducinganyothermechanism,ourmodelreproducesapowerlawbreakdownabove200 MPa.Consequently,weconcludethattheuseoftworheologicallawstodescribethecreepofolivinecanbemotivatedbyconvenience,butisnotimposedbytheoreticalneeds.Alternativelyaunifiedcreeplawcanbeproposedtodescribetherheologyofolivineinawiderangeoftemperaturerelevantfortheuppermantle.Thisflowlawmayhaveanexponentialformanddescribetheentirerangeofexperimentaldata,fromroomtemperatureto1800 Katbothlowandhighstresses,usingasingleadjustingparameter,theso-calledmechanicalresistance ̃σ.Wealsoproposeanalternativemathematicalexpressionbasedonasigmoidfunction,whichismoresuitableforimplementationingeodynamicalmodels.©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).1.IntroductionUppermantlerocksarepolymineralicaggregates,mostlycom-posedofolivine,pyroxenes,andaminorAl-richcomponent(i.e.,plagioclase,spinel,orgarnet).Studyofmantle-derivedrocks,suchasperidotiticxenolithstransportedinlavasortectonically-exhumedperidotitemassifs,indicatesthattheolivineabundanceexceedsbyfartheabundanceofpyroxenes(olivine>60 vol.%).Therefore,therheologicalpropertiesofperidotiticuppermantleislikelytobecontrolledbyolivinerheology.Olivinewasthesub-jectofnumerousplasticdeformationexperimentsduringthepastcoupleofdecades.Experimentswereperformedonsinglecrys-tals(e.g.Raleigh,1968;Baietal.,1991;Demouchyetal.,2013;Tielkeetal.,2016),monomineralicaggregates(e.g.Meietal.,2010;Demouchyetal.,2014;Thiemeetal.,2018),andpolymineralicaggregates(e.g.Bystrickyetal.,2006)atawiderangeofther-modynamicsconditions:hightemperatureandambientpressure(e.g.Baietal.,1991);hightemperatureandhighpressure(e.g.*Correspondingauthor.E-mailaddress:patrick.cordier@univ-lille1.fr(P. Cordier).Couvyetal.,2004;Raterronetal.,2004;Jungetal.,2009);variousoxygen(e.g.Ryersonetal.,1989; Keefneretal.,2011)andwaterfugacities(e.g.MeiandKohlstedt,2000a;Demouchyetal.,2012;Jungetal.,2009;Tielkeetal.,2017);andfordifferentchemicalcompositionsofolivine(e.g.Zhaoetal.,2009;Fauletal.,2016).Inaddition,variousdeformationgeometriesweretested:axialcompression(e.g.Tielkeetal.,2016;ChopraandPaterson,1981),simpleshear(e.g.JungandKarato,2001;Bystrickyetal.,2000),andindentation(e.g.EvansandGoetze,1979;Baietal.,1991;Kumamotoetal.,2017).Athightemperatures(T>0.7Tm,Tmbeingthemeltingtem-perature)andlowstresses(<100 MPa),apowerlawforthestrainrateasafunctionofstressdescribeswelltheolivinerheology(e.g.,Baietal.,1991;DarotandGueguen,1981),whichisconsis-tentwithalowermostlithosphereandconvectivemantleexhibit-ingaductile,non-linearviscousrheology.Atlowertemperatures(T<0.6Tm)andhigherstresses(>100 MPa)relevantfortheshal-lowmantle,astrongerdependenceofthestrainrateonstressisobserved,whichisusuallydescribedbyanexponentiallaw(EvansandGoetze,1979; Phakeyetal.,1972;Goetze,1978;Demouchyetal.,2009,2013).Thechange fromapowerlawtoanexpo-https://doi.org/10.1016/j.epsl.2018.10.0490012-821X/©2018TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
K. Gouriet et al. / Earth and Planetary Science Letters 506 (2019) 282–291283nentiallawwasdescribedas“thepowerlawbreakdown”(Goetze,1978).Thisbehaviourisnotsingulartoolivineandhaslongbeenobservedanddiscussedinplasticity.In1954,Dorn alreadyques-tionedthepossibilitythatthepowerlawandtheexponentiallawmightnotreflectdistinctmechanisms:“Itisindeedrationaltosus-pectthatthehighandlowstresscreeplaws[...]areinnowayduetoanyabruptchangeinthemechanismofcreepandthereforemerelyreflecttwolimitingconditionsofasinglelawforcreep”.However,fiftyyearslater,thequestionwasstillpresentedasanopenissuebyNabarro:“Thereislittleevidencewhetherthephys-icalprocessesoccurringinthepower-lawregionandtheregionofpower-lawbreakdownarethesameordifferent”(Nabarro,2004).Severalattemptshavebeenmadetoestablishtheoreticalcon-ceptstodescribethecreepbehaviour.Inmostcases,thesteadystatecreepbehaviourisdescribedasacompetitionbetweenworkhardeningandrecovery.Bailey (1926) andOrowan (1946)describetheseeffectsthroughhandrcoefficients(thestrainhardeningandrecoveryrates),whichrespectivelyincreaseordecreasetheflowstressbyafiniteamountσ.Steadystateisreachedwhenthoseeffectscompensate,leadingtothestrain-rategivenbytheBailey–Orowanequation( ̇εs=rh).Othermodels(e.g.Barrett,1972)describethesystembasedontheevolutionofdislocationdensity(dρdt= ̇ρ++ ̇ρ).Twodislocationdensitytemporalevolutions: ̇ρ+and ̇ρ,describestrain-hardeningandrecovery,respectively.Atsteadystate,dynamicequilibriumisreachedandthedislocationdensityisconstant.Thefollowingstepwastodescribehowele-mentarymechanisms(glide,climb,subgrainboundarymigration,cross-slip,etc.)couldconstrainhardeningandrecoveryrates(e.g.GottsteinandArgon,1987).Dependingontheunderlyingrecoverymechanismsandonphenomenologicalrelationsusedtodescribework-hardening,differentregimescharacterisedbydifferentstresssensitivitieswereproposed.Thedrawbackofthesemodelsistorelyon“ad-hocassumptionsonthestressdependenceofthevari-ousingredientsofcreepmodels”(PoirierandDuba,1997).Indeed,GottsteinandArgon (1987)discussthefactthatnounequivocalrelationshipcanbedrawnbetweencreepparameters(activationenthalpyandstresssensitivity)andunderlyingelementarymecha-nisms.However,somerobusttrendsemerge.Athigh-temperaturewhenonlyrecoverybyclimbistakenintoaccount,apower-lawdependencewithastressexponentn=3isobserved.Atlowtem-perature,thestresssensitivityisfoundtoincrease.Uncertaintyonthedeterminationofthestresssensitivityis,nevertheless,critical,sincelaboratorycreepdataneedtobeextrapolatedoverseveralordersofmagnitudeofstrainrateformodellingmantleflow(e.g.,HirthandKohlstedt,2015).Arobust,quantitativeandpredictivetheoryisthusneededtoallowextrapolationfromlaboratorytonaturalconditions.Toinvestigatetherespectiveroleandrelevanceofthepowerandtheexponentiallaws,wemodelleddislocationcreepinolivinesinglecrystalsfortemperatureandstressrangesrelevantforboththelithosphericandtheasthenosphericmantle.Ourstudytakesadvantageofthecapabilitiesofferedbytherecentlyde-veloped2.5-Dimensional-DislocationDynamics(2.5D-DD)simula-tions.Thesemodelsallowtoaccountforcomplexcollectivebe-havioursofdislocationswithoutad-hocassumptions,basedonthesimulationofwell-understoodelementarymechanisms(dis-locationglideandclimb,aswellasmultiplication,interactions,andannihilation).Thepresentworkfollowsrecent2.5D-DDmod-elsofsteady-statecreepofolivinefor[100]dislocationsathightemperatures(T1400 K)(Boiolietal.,2015a)andfor[001]dislocationsatlowertemperatures(800 KT1200 K),whichprevailinthelithosphericmantle(Boiolietal.,2015b).Inthesestudies,thenumericalapproachintroducedbyKeralavarmaetal.(2012)onaluminiumhasbeenextendedtodescribethermallyac-tivatedglideofdislocations.Boiolietal. ( 2015a)haveshownthattheinterplaybetweenglideandclimbdislocationmotionleadstoasteady-statecreepbehaviour,whichcanbedescribedbyapowerlaw,inagreementwithhigh-temperatureexperimentaldata.Boiolietal. ( 2015b)haveshownthatanexponentiallawdescribeswithabetteraccuracythecreepbehaviouratlowertemperatureandhigherstresseswhere[001]glidedominates.However,beingcon-ductedwithdifferentslipsystemsanddisconnectedtemperatureintervals,thesemodelscouldnotshedlightonthetransitionbe-tweenthosetworegimes.Inthisstudy,weproposeacreepmodel,inwhichthesteadystatedeformationregimeofanolivinecrystalwellorientedforsingleslipismodelledinalargetemperaturerange,from800to1700 K.Thebasicingredientsofthemodelareverysimples.Dislo-cationcanmovebyglideandclimb.Work-hardeningresultsfromdislocation–dislocationsinteractions(long-rangeandshort-range).Recoveryresultsfromdipoleannihilationcontrolledbyclimb.Al-thoughparameterizedonolivine,ourmodelismoregeneralinessencesinceitfundamentallydescribesdislocationcreepcon-trolledbythermallyactivatedglideandclimb.2.Dislocationdynamicsmodelling2.1.DescriptionofthemodelDislocationDynamics(DD)isawell-establishedsimulationtechniquetodescribethecollectivebehaviourofdislocationsandtomodelviscoplasticflowincrystallinesolids(DevincreandKu-bin,1997;Durincketal.,2007).Inthismethod,timeandspacearediscretized.Theforcesactingondislocationsarecalculatedusinglinearelasticitytheory.Thisprovidesadescriptionoftheelastic,long-rangestrainfieldinducedbydislocationsinthecrystalasafunctionoftheirinteractionswitheachotherandoftheinter-actionofthedislocationswithanexternal(applied)stressfield(seethelinearelasticitymodelofdislocationsasdescribedforinstanceinHirthandLothe,1982).Theseinteractionsarekeyin-gredientstodescribethecollectivebehaviourofdislocations.Ateachcalculationstep,dislocationsaremovedinthedirectionoftheresultingappliedforceaccordingtomobilitylawswhichtakeintoaccounttherelevantatomisticprocessescontrollingdisloca-tionmotion,suchasglideandclimbprocesses(Devincreetal.,2011).In2.5D-DD(Gomez-Garciaetal.,2006),dislocationsaremodelledasparallelstraightsegmentsperpendiculartoarefer-enceplane,whichcontainstheglideandtheclimbdirections,andtheirdynamicsisfollowedonlyinthisreferenceplane.Additionalrulesareincludedtoreproduceimportant3Ddislo-cationmechanisms:dislocationmultiplication,dipoleannihilation,orjunctionformation.Amultiplicationruleisusedtoreproducethegeneralobservationthatdislocationloopsexpandunderanexternalloading.Inthesimulation,multiplicationisrepresentedbytheinsertionofdislocationsatrandompositionsinthesimu-latedvolume,providedthatthelocaleffectivestresshasthesamesignasthe externalstress.Themultiplicationrateofthedislo-cationdensityasafunctionofstrain(dρdε=m)isheldconstant(Gomez-Garciaetal.,2006).Thisreflectstheobservationthatmostdislocationsemerginginareferenceplane,orslice,ofa3Dvolumeoriginatefromsourcesinthesurroundingvolume.Inthepresentwork,aconstantmultiplicationratem=2×1015m2allowsre-producingtheevolutionofthedislocationdensitypreviouslymod-elledin3D-DDsimulations(Durincketal.,2007).Dislocationsinvolvedinadipoleareallowedtomutuallyan-nihilatewhenthedistancebetweenthemissmallerthanacrit-icaldistancera=10b(bistheBurgersvectormodulus).Theseannihilationeventsresultinreductionofthedislocationdensity.Contrarytoface-centredcubicmetals,junctionformationisarareeventinolivine(Durincketal.,2007).Byconsequencethepossibil-ityfordislocationstoreactandformajunctionisnotintroducedinthepresentsimulations.Moredetailsonthe2.5D-DDapproach