A model for flash weakening by asperity melting during - PDF document

Amodelforflashweakeningbyasperitymelting duringhigh-speedearthquakeslip A.W.Rempel 1 andS.L.Weaver 1 Received25February2008;revised22July2008;accepted15August2008;published26November2008. [ 1 ] Recentresultsfromlaboratoryexperimentsonabroadrangeofmineralsystems exhibitdramaticdropsintheeffectivefrictioncoefficient m oncethesliprateexceedsa criticallevel V w ,whichistypically O (0.1)m/s.This‘‘flashweakening’’hasbeen attributedtotheeffectsoflocalizedheatingathighlystressedmicroscopicasperities.We extendpreviousphenomenologicaltreatmentstoassesswhethermeltingatasperity contactscanexplaintheobservedchangesinstrength.Usingphysicalparametersobtained fromtheliteratureonthephasebehaviorandmechanicalpropertiesofquartz,albite, dolomite,gabbro,Westerlygranite,andserpentinite,thepredictionsofoursimplified modelareinreasonableagreementwithavailableexperimentaldata.Wederive approximateanalyticalexpressionsthatsuggestthatstrengthchangesareinsensitivetothe meltviscosityunderconditionsthatlikelyincludethoseduringearthquakeslipalong majorfaultsystems.Instead,theprimarycontrolson m aretheratioofsliprate V to V w and theStefannumber S ,whichisdefinedastheratioofthelatentheatoffusiontothe sensibleheatrequiredtoraisethetemperaturefromambientlevels.Thephasebehavior duringtheshortlifetimesandatthehighconfiningpressuresofasperitycontactsisa significantsourceofuncertaintyintheparameterchoices,asarethepresenceand availabilityofwater.Nevertheless,ourresultsareencouragingforfurthereffortsto incorporatethemicrophysicsoffaultzoneprocessesintoearthquakesimulations. Citation: Rempel,A.W.,andS.L.Weaver(2008),Amodelforflashweakeningbyasperitymeltingduringhigh-speedearthquake slip, J.Geophys.Res. , 113 ,B11308,doi:10.1029/2008JB005649. 1.Introduction [ 2 ]Frictionalbehaviorhasarichhistoryofstudythatcan betracedthroughthe15thcenturyeffortsofLeonardoda VincitolatercontributionsbyAmontons,Euler,andCou- lomb(see,e.g., Scholz [2002]and BowdenandTabor [1950]forhistoricalreviews).Earlyexperimentsdemon- stratedthattheratioofslidingresistance t tonormalstress s n ,thefrictioncoefficient m ,isapproximatelyconstantfora givensolidandindependentoftheapparentcross-sectional area A oftheslidingsurface.Themicroscopicoriginsof frictionalresistanceremainanareaofactiveresearch[e.g., Kilgoreetal. ,1993; Marone ,1998; MairandMarone , 1999; Goldsbyetal. ,2004],butabasicphenomenological modelforthecontrolson m atlowslidingspeedshasbeen established.Asarguedby BowdenandTabor [1950]and demonstratedexperimentallyby LoganandTeufel [1986] and DieterichandKilgore [1994,1996],truecontact betweenslidingsurfacesoccursonlyatmicroscopicasperity junctions,whichhaveanarea A c thatisasmallfractionof A .Sincetheslidingresistanceisdominatedbytheasperity behavior,itfollowsthat t = t c A c / A ,wherethejunction strength t c forbrittlerocksisfoundtypicallytobeapprox- imately10%oftheshearmodulus[ Rice ,2006].The fractionalareaintruecontact A c / A = s n / s c ,wherethe contactindentationstrength s c canberelatedtothemineral hardness.Modelspredictthat s c isapproximatelyconstant bothforjunctionsthatdeformplastically[ Bowdenand Tabor ,1950]andforthosethatdeformelastically [ GreenwoodandWilliamson ,1966].If t c isapproximately constantaswellthisyieldsafrictioncoefficient m
t / s n = t c / s c thatdoesnotdependon A . Baumberger [1997]and Scholz [2002]giveinformativereviewsofthemicrophysics offrictionthatoutlinemorerecentdiscoveries,including therateandstatedependencethataresoimportanttoslip instabilitiesduringthenucleationofearthquakes[ Dieterich , 1978,1979; Ruina ,1983; RiceandRuina ,1983]. [ 3 ]Theresistancetosliding,orstrength,ofmajorplate- boundingfaultsisthesubjectofvigorousongoingdebate [e.g., Scholz ,2000; TownendandZoback ,2000].Several linesofevidenceindicatethatduringearthquakes,typically t ismuchlowerthanthevalueextrapolatedfromlaboratory measurementsofrockfrictionatlowerslidingspeeds [ Byerlee ,1978].Dynamic-weakeningmechanismsthathave beensuggestedtoexplainthisbehaviorincludethethermal pressurizationofporefluids[ Sibson ,1973],normalstress variations[ Bruneetal. ,1993],acousticfluidization [ Melosh ,1996],elastohydrodynamiclubrication[ Brodsky andKanamori ,2001],theformationofsilicagels[ Goldsby andTullis ,2002],andasperity-scaledecompressional heating[ O’Hara ,2005].Significantly,recentexperiments JOURNALOFGEOPHYSICALRESEARCH,VOL.113,B11308,doi:10.1029/2008JB005649,2008 Click Here for Full A rticl e 1 DepartmentofGeologicalSciences,UniversityofOregon,Eugene, Oregon,USA. Copyright2008bytheAmericanGeophysicalUnion. 0148-0227/08/2008JB005649$09.00 B11308 1of14 [ HiroseandBystricky ,2007; HiroseandShimamoto ,2005; Prakash ,2004; TsutsumiandShimamoto ,1997; Tullisand Goldsby ,2003a,2003b;D.L.GoldsbyandT.E.Tullis, manuscriptinpreparation,2008]revealthattheeffective frictioncoefficientsoffaultrocksthemselvescanfall dramaticallyathighsliprates(i.e.,afewdecimeters persecond). Rice [1999]notedthesimilaritywithflash- weakeningbehavioridentifiedpreviouslyinmetals[e.g., BowdenandThomas ,1954].Theessentialideaisthatthe temperatureriseslocallyatasperitycontactswhentheyare heatedtoorapidlyforconductiontodissipatetheenergy releasedbymechanicalwork.Thestrengthsofasperitiesfall below t c athightemperatures.Aslongasthetruecareof contact A c doesnotchangesignificantly,however,the localizedtemperatureincreaseisnotexpectedtochange s c so m drops.Reasonablefitstoexperimentalobservations havebeenachievedwithelementarymodelsthatquantify theseeffectsbyassumingatemperaturethreshold T w ,upon whichtheasperitystrengthdropsfrom t c toaweakened value t w thatistreatedasaconstant[ Rice ,2006; Beeleret al. ,2008]. [ 4 ]Theassumptionofconstant t w isausefulidealization forassessingthebasicflash-weakeningprocess,butitis morerealistictoexpecttheasperitystrength t a toevolve aftertheweakeningtemperature T w isreached[ Rice ,2006]. Themechanismsthatcontrol t a determinehow m changes withsliprate V andambienttemperature T .Inmetals,flash weakeninghasbeenattributedtoenhancedplasticyielding atelevatedasperitytemperatures[ Molinarietal. ,1999]. Thestiffnessesofmineralsystemstendnottobeas sensitivetoelevatedtemperaturesastheyareformetallic systems,soenhancedplasticyieldingisnotexpectedto dominatetheweakeningbehavior.Extremelocalizedpres- surevariationsmaycauseaglasstransitiontoweaken quartz[ Kingmaetal. ,1993],variousdehydrationreactions toweakenserpentiniteandotherphylosilicates[ Hiroseand Bystricky ,2007],andotherthermallyactivatedreactions maybeimportantforcarbonatemineralssuchascalcite [e.g., Hanetal. ,2007].Thesespecialcasesareclearly deservingoffurtherconsideration,butflash-weakening observationsinothermineralsystemsmotivatetheinvesti- gationofamoregeneralmechanism.Iflocaltemperature increasesaresufficientformeltingtooccur,then t a is expectedtobecontrolledbytheresistancetoviscousshear ofthinmoltenlayers.Thedynamicsofflashmeltingarethe focusofthispaper. [ 5 ]Wepresentanelementarytheoryofflashmeltingnext andderiverelationshipsfor m ( V )whenheatflowcanbe treatedasonedimensionalandmeltextrusionisnegligible. Severalrecentexperimentalandtheoreticaleffortsprobethe effectsofpervasivemeltingalongthefaultsurface[e.g., Di Toroetal. ,2006a,2006b; HiroseandShimamoto ,2005; Nielsenetal. ,2008; Sironoetal. ,2006].Incontrasttothose efforts,thoughlocalizedheatingisassumedtocausemelt- ingatasperities,themodelsherearelimitedtoconditionsin whichtheaveragefaulttemperatureremainsbelowthat requiredforapervasivemeltlayertocovertheentirefault surface.Ourgoalistoassesswhetheramodelthatissimple enoughtobeeasilyimplementedasacomponentofmore complicateddynamicrupturecalculationsisconsistentwith observedfrictionalbehaviorinthelaboratory.Insection3 wediscussourchoicesforthevaluesofrelevantparameters andsomeoftheuncertaintiesthatarise.Wefocusin particularonslidingsystemsforwhichrecentexperimental high-speedfrictiondataisavailable(D.L.GoldsbyandT.E. Tullis,manuscriptinpreparation,2008),namely,novaculite (quartz),Tancofeldspar(albite),dolomite,gabbro,Westerly granite,andserpentinite(chrysotile).Foragiventime- dependentslidingrate V ( t )wecalculatetheevolutionof backgroundtemperature T ( t )andgeneratepredictionsfor m ( t ).Theseresultsarediscussedinthecontextofrecent experimentalobservations[ HiroseandBystricky ,2007; D.L.GoldsbyandT.E.Tullis,manuscriptinpreparation, 2008],geologicobservations,andpotentialcomplicating processes. 2.ElementaryModelforFlashMelting [ 6 ]Asperitycontactsareassumedtoweakenwhenthey areraisedfromthebackgroundtemperature T toacritical weakeningtemperature T w intheweakeningtime q w .A largepopulationofasperitycontactsispresentatalltimes andtheirindividualstrengthsevolveindependentlyasthey areloaded,heatedandquicklyslideoutofcontact.The time-averagedstrengthofacontactwithdiameter D a and lifetime q = D a / V � q w isgivenby
t c  t c q w 
q q w t a d t q
 1  where t c isthelow-speedcontactstrengthand t a isthe evolvingcontactstrengthaftertheweakeningtemperatureis reached.Assumingcontactlifetimesremainsufficiently shortthat q (d V /d t )  V ,thetime-averagedstrengthofa representativecontactcanbeequatedwiththespatially averagedstrengthofthecontactpopulationsotheoverall frictionalresistanceisproportionalto
t c .Asinglevalueof D a isadoptedhereforsimplicity;extensionstotreatmore realisticdistributionsofasperitysizescanbemade followingthetreatmentof Beeleretal. [2008]. [ 7 ]Itisinstructivetoreviewfirsttheidealizedcase [ Beeleretal. ,2008; Rice ,2006]inwhichthecontact strengthdropsabruptlytoaweakenedstate t a = t w
m w s c at T w andremainsconstantthereaftersothattheaverage contactstrengthis
t c = t c q w / q + m w s c (1  q w / q ).Assuming A c remainslinearlydependenton s n ,whichisheldconstant, then s c doesnotdependonthelocaltemperatureandthe effectivefrictioncoefficientis m 
t c / s c = m 0 q w / q + m w (1  q w / q ),where m 0
t c / s c istheconventionalfrictioncoef- ficientthatpertainstoslidingratesthatareslowenoughthat q q w .Theweakeningvelocity V w
D a / q w V = V w isdefined asthethresholdslidingspeedatwhichflashheatingfirst beginstoaffectthefrictionalresistance.Analysisofthe temperatureevolutionpriortoweakening[ Rice ,2006] suggeststhat V w  ( r C ) 2 ( T w  T ) 2 pa th /( t c 2 D a ),where r C and a th arethevolumetricheatcapacityandthermaldiffu- sivity.Theeffectivefrictioncoefficientcanbewrittenas m  m w  m 0  m w  V w V
 2  whichtendstotheweakenedvalue m w when V V w .The valueof m w inequation(2)canberegardedasafitting parameterthatshoulddependonthepropertiesofthe B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 2of14 B11308 slidingsystemwhenasperitytemperaturesareelevated.A valueof m w  0.2istypical(e.g.,D.L.GoldsbyandT.E. Tullis,manuscriptinpreparation,2008). [ 8 ]Inprinciple,itshouldbepossibletocalculatethe weakenedshearresistancefromthemechanicalproperties andphasebehavioroftheasperitiesandtheirmelts.The resistancetoshearingmeltatstrainrate  g is t a = h  g where h isthemeltviscosity.Inmostoftheworkthatfollowswe treatthemeltasisothermalso h isconstantandauniform t a isachievedacrossthemeltthickness h m with  g = V / h m .A scalinganalysissuggeststhattemperaturevariationsinthe meltaresmallwhen h m  4  a th q  .Whenthisconditionis notmet,significantgradientsinmelttemperaturecanarise so h isnolongerspatiallyuniformand  g isexpectedtovary laterallyaswelltosatisfyforceequilibriumconstraints. Keepingthisinmind,westilltake t a = h V / h m ,buttreat h as anaveragemeltviscositywhenrheologicallysignificant temperaturevariationsoccuracrossthelayerthickness h m (seeAppendixA). [ 9 ]Theonsetofmeltinginpolycrystallinematerials beginsatnodesandthree-grainjunctionswhen T T m , thenproceedstograinboundariesandfreesurfacesbefore achievingbulkcoexistenceat T = T m [e.g., Smith ,1948; Dash ,2002; Dashetal. ,2006].Onamicroscopicscale,the meltingtransitionoccursatlowertemperaturesinlocations ofelevateddisorder,suchasthatwhichaccompanies collisionaldamage.Meltonsetatgrainboundariescanbe abrupt,inthesensethatmeltfilmsofmicron-scalethickness appearimmediatelyonceathresholdtemperatureiscrossed [ BenatovandWettlaufer ,2004].Asperitycontactscanbe thoughtofassimilartoimperfectgrainboundariesthatare heatedrapidlyandsubjecttoenormousstressgradients.For thespecificmineralsystemsconsideredherewehaveno firmexperimentalevidencetoguideus,andsowedonot attempttoresolvethedetailsofmeltonsetitself.Insteadwe makethecrudeassumptionthatfinitemoltenlayersare generatedimmediatelyoncethetemperaturereaches T w  T m ,andwechoosetheinitiallayerthickness h m ( q w )= h 0 to ensurethatthevariationinasperitystrengthiscontinuous, with t a ( q w )= t c .Thissetsthelevelfromwhichflash weakeningcausestheeffectivefrictioncoefficienttodrop whenthecontactlifetimeexceeds q w .Usingequation(1), wefindthatfor q � q w m  m 0 q w q  h 0 q  q q w d t h m 
 3  where h 0
h V / t c . [ 10 ]Toevaluatetheintegralterminequation(3),we approximatethevariationsin h m byconsideringamodelfor isothermalmelting.Theheatflowfromtheshearzoneafter meltonsetisapproximatedasthatduetoaplanarheat sourceonthesymmetryplane,withalltheremainingheat excessconvertedtomeltingofthefilmwalls.InAppendix Awecomparethepredictedfrictionalbehaviorwithresults fromasimilaritysolutioninwhichtheviscousshearofa meltfilmofuniform,buttime-varyingviscosityactsasa heatsourceandwetrackthechangesinmelttemperatureas thefilmthicknessevolves.Thegoodagreementbetweenthe twomodelsprovidesincreasedconfidencethattheapprox- imatetreatmentofthemeltfilmevolutiondoesnotsignif- icantlyinfluencetheessentialmodelpredictions. [ 11 ]Toapproximatetheeffectsofheatlossdueto conduction,withsolidconductivity k
r C a th ,wetake thedifferencebetweentherateofheatproductionatthe asperitycontact t a V andtheheatconductedawayasbeing equaltotherateoflatentheatreleaseasthemeltfilm thickenssothat t a V  2 k  T  y     h  m  2  r L  h m  q
 4  where r L isthelatentheatoffusionperunitvolume.Prior totheonsetofmelting,for q  q w thecontactstrengthis t a = t c andtheconductivefluxat y = h m /2is[ Carslaw andJaeger ,1948] 2 k  T  y  V t c erfc h m 4  a th q   Treatingtherateatwhichsubsequentconductioncontinues tocarryheatawayinasimilarfashionaftermelting begins,therateoffilmgrowthisapproximatedas r L d h m d q  V 2 h h m erf h m 4  a th q    5  Equation(5)wasintegratedsubjectto h m ( q w )= h 0 andthe resultingeffectivefrictioncoefficientwascalculatedfrom equation(3)toobtaintheresultsplottedinFigure1. [ 12 ]Theeffectivefrictioncoefficientfallsoncetheslid- ingrateissufficienttocausemeltingatasperities.Increases in V causethemeltfilmtothickenand m evolvesasshown inFigure1.Itisusefultocalculateanalyticalapproxima- tionsfor m tobetterunderstandthesystembehavior.When Figure1. Thepredictedfrictionalbehaviorforflash meltingatasperitycontactsinasimplifiedmodelwhere themelttemperatureistreatedasconstant(solidcurve).The dottedanddashedcurvesshowanalyticalapproximations fromequations(6)and(7)thatarevalidforfilmthicknesses thataremuchsmallerandlargerthanthecharacteristic distanceforthermaldiffusion.Parametersforalbiteandan ambienttemperatureof T 0 =210
Cwereused(seeTables1 and3fordetails). B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 3of14 B11308 thefilmthicknessissmallenoughthat h m  4  a th q  wecan makeuseofthesingle-termexpansionerf u  (2/  p  ) u to integrateequation(5)andevaluateequation(3)tofindthat m  m s  m 0 V w V 1  2 S  V V w   1  1  S  ln1   V V w   1 S           
 6  wheretheStefannumber S
L /[ C ( T w  T )].Significantly, theeffectivefrictioncoefficientisindependentofmelt viscositybecausethefilmthicknessincreasesinproportion totheviscosityinthisregime.Belowweshowthatthisis theregimeinwhichmanymineralslidingsystemsare expectedtooperate.Whenthefilmthicknessissufficiently largethat h m 4  a th q  ,theconductivefluxisnegligible andequation(5)predictsthatthefilmthickensinproportion tothesquarerootoftime.Inthislimit,equation(3)gives theapproximatefrictioncoefficientas m  m l  m 0 V w V 1  V V h  2  1  2 V 2 h V 2 V V w  1    1     
 7  where V h
 pr L a th  h S 2   isacharacteristicvelocitythat increaseswithdecreasedmeltviscosity.Atlarge V ,an expansionoftherightsideofequation(7)revealsthat m l increasesbacktowarditsinitialvalueof m 0 .Wenotethat significanttemperaturevariationscanoccuracrossthefilm when h m 4  a th q  sotheassumptionsmadeherethatthe meltviscosityisuniformandallexcessheatgoesto thickeningthemeltfilmarenotstrictlyjustified.These effectstendtocounteracteachothertosomeextentand favorablecomparisonswithasimilaritysolutiondiscussed inAppendixAthataccountsfortemperature-induced variationsinmeltviscositysuggestthatthesimplified treatmentleadingtoequation(7)approximatestheexpected frictionalbehaviorreasonablywell.Asdiscussedfurther below,fortheslidingsystemsconsideredherethebehavior typicallyisexpectedtofallwithintheregimewhere equation(6)applies. [ 13 ]Thepredictionsofequations(6)and(7)arecom- paredwiththefrictioncoefficientderivedfromthenumer- icalsolutiontoequation(5)forarepresentativesetof parametervaluesinFigure1.Exceptforasmallrangeof slidingratesnear V  2m/s,where h m iscomparabletothe conductivelengthscale4  a th q  ,theeffectivefrictioncoef- ficientisapproximatedwellby m  max( m l , m s ).Equation(6) ismorerepresentativeofthepredictedbehaviorforsmaller viscositiesandslidingrates;equation(7)givesabetter approximationatlarge h and V .Figure2showstherangesof parametervaluesforwhich m s = m l ,with S =1(solid), S =0.1 (dashed),and S =0.01(dotted).Thismarkstheboundary betweenparameterregimesinwhichequation(6)isa betterapproximation(inthelowerright)andthosein whichequation(7)isexpectedtobemoreaccurate(in theupperleft). 3.PhysicalParameters [ 14 ]Thethermodynamicandmechanicalpropertiesofthe slidingsystemmustbeevaluatedinordertopredictthe frictionalbehavior.Table1summarizesthenominalparam- etervaluesthatweusedtomodelarangeofslidingsystems. Asperitycontactsaresubjecttolargeandrapidchangesto boththetemperatureandtheeffectivenormalstress.Listed valuesforthedensities r ,thermalconductivities k ,heat capacities C ,andcontactindentationstrengths s c areall givenatstandardtemperatureandpressureconditions. Variationsintheseparameterswithchangesinthepressure andtemperaturestateareassumedtobesmallincompar- isontotheuncertaintiesinsomeoftheotherparameter choices.Similarly,thelatentheats L listedinTable1arefor phasechangesatatmosphericpressureunlessotherwise notedbelow.Therangesof m 0 and D a areestimatesby D.L.Goldsby(personalcommunication,2007)forhis experimentalconditions.The m 0 valuesfordolomiteand serpentiniteweretakenfromtheliterature[ WeeksandTullis , 1985; Mooreetal. ,2004].Asdiscussedinsection1,when asperitycontactsareloadedtheeffectivenormalstressis expectedtoapproach s c .Weuseanempiricalfitreportedby Beeleretal. [2008]tothedataof Brozetal. [2006]to estimate s c fromMohshardness H as s c  0.123 H 2.3 GPa. Themeltingtemperatureatthiselevatedpressureislisted foreachcaseas T m s .Foreachoftheseslidingsystemsthere areextraconsiderationsthatcomplicatethechoicesfor Figure2. Regimediagramshowingtherangesofnormal- izedslidingrate V / V w asafunctionof V h / V w forwhich m l is biggerorsmallerthan m s ,with S =1(solid), S =0.1 (dashed),and S =0.01(dotted).Highslidingrateshave m l � m s ,andequation(7)betterapproximatestheeffective frictionalbehavior.Lowslidingrateshave m s � m l ,and equation(6)givesabetterapproximation.When S =0.1or 0.01,therearesmallrangesofslidingvelocitieswith V / V w closetounity(i.e.,belowthehorizontaldashedanddotted lines)where m l exceeds m s slightly,butbothapproximations give m  m 0 V w / V inthisparameterrange.Labeledpoints correspondtoexpectedparametersforvarioussliding systems,asdiscussedfurtherinsection3andsummarized inTable3. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 4of14 B11308 someoftheparametervalues.Weoutlinetheseconsider- ationsandexplainourassumptionsnext. [ 15 ]Arapidsolid-statetransformationfrom a -to b - quartzoccursatmodesttemperatureswellbeforemelting. Underequilibriumconditionsatlow(e.g.,atmospheric) pressuresandelevatedtemperatures, b -quartztransforms furthertocristobalitesothemeltingtemperatureofthis phaseislistedas T ma .Underequilibriumconditionsatthe highpressuresexpectedofasperitycontacts b -quartzis convertedtocoesitepriortomelting.Thetabulatedvalueof T m s isforthishigh-pressurepolymorph(forcomparison, thetriplepointforcoexistenceofcoesite, b -quartzandmelt isatapproximately2450
Cand4.4GPa[ Zhangetal. , 1993]).Weapproximate L fromthemetastablecongruent meltingofquartzandignorethesmallenthalpychanges associatedwithsolid-statephasetransitions[ Spera ,2000]. Giventheshortlifetimesofasperitycontacts(typicallytens ofmicroseconds),however,thereisreasontosuspectthat insufficienttimemaybeavailableforthe b -quartztocoesite transitiontotakeplace.Theoreticalcalculations[ Zhanget al. ,1993]suggesttheexistenceofaglasstransitionwitha negativeClapeyronslopewhenthequartzmeltingcurveis extrapolatedtohigherpressures.Experimentalevidencefor suchatransitionislimitedtoanintriguingsetofSEMand photomicrographimagesby Friedmanetal. [1974]that showglassyfilamentsinquartzgougethatformedduring low-speedslidingexperimentsonsandstone.Withoutfur- therempiricaldatatoguideus,hereweusethecoesite meltingtemperaturetocharacterizeflash-meltingbehavior atquartzasperitycontacts. [ 16 ]Underequilibriumconditionsalbiteistransformedto jadeiteandquartzabove3GPa.Significantchangesin crystalstructurearerequired,however.Comparisonsbe- tweendifferentsources[e.g., Holland ,1980; Morse ,1980; WilliamsandKennedy ,1970]revealsomeuncertaintyinthe detailsofthephasediagramnearthesemineralphase transitionsathighpressures.Here,weassumethat q w is sufficientlyshortthatalbitedoesnotinfactchangephase. Instead, T m s isextrapolatedfromthelower-pressuremelting curveforalbite. [ 17 ]Gabbroandgranitebothcontainmanydifferent minerals,andasaconsequencethepropertiesofasperity contactsareexpectedtobewidelyvaried.Forasamplethat contains N differentmineralphases,eachwithvolume fraction y i ,ifasperitycontactsinvolveonlyasinglemineral phaseoneachsurfacethenarandomsamplingwouldbe expectedtoincludeproportion2 y i y j ofcontactsbetween the i thand j thmineralsfor i  j and y i 2 otherwise.The softermineralisexpectedtocontrolthelevelofthe confiningstressateachasperitycontact[e.g., Scholz , 2002].Wedonottreatthephasebehaviorateachpotential contactseparately,butinsteadevaluatetheaverageconfin- ingstressoverapopulationofasperitiesas
s c  y 2 1 s 1   N i  2 y i s i y i  2  i  1 j  1 y j 
where s i isthecontactindentationstrengthofthe i thhardest mineral.InTable1,thecontactstrengthsofgabbroand Westerlygraniteareapproximatedinthisway.Other tabulatedpropertiesofthesetwoslidingsystemsare representativeofthepropertiesofbulkgabbroandgranite samples.Forthecaseofgabbro, T ma and T m s are approximatedusingtheexperimentaleutectictemperatures forabasalt[ WangandTakahashi ,1999]withasimilar compositiontothegabbrousedinfrictionalexperimentsby HiroseandShimamoto [2005]andD.L.GoldsbyandT.E. Tullis(manuscriptinpreparation,2008).Forthecaseof granite, T m s isextrapolatedto7.2GPausingthedrysolidus fromexperimentsthatwereperformedtoamaximum pressureof3.5GPa[ SternandWyllie ,1973]. [ 18 ]High-pressurephasetransformationsconvertdolo- mitetoanumberofdifferentminerals.Therelativepro- portionsofCO 2 andH 2 Ostronglyinfluencethemineral assemblagethatispresentpriortomelting.Weassumedry conditionshere,andfor T m s weusethetemperatureofthe liquidusminimumformixturesofCaCO 3 andMgCO 3 , whichweinterpolateto2.2GPausingpublishedvaluesof 1075
Cat1GPaand1290
Cat2.7GPa[ Byrnesand Wyllie ,1981].At s c  2.2GPa,theeutecticcomposition hasacalcium:magnesiumratioofapproximately3:2.The solidustemperatureforthe1:1calcium:magnesiumratioof puredolomiteisapproximately20–30
Chigher[ Byrnes Table1. NominalParameterValuesUsedtoModeltheEffectiveFrictionalBehaviorofVariousSlidingSystems a PropertyQuartzAlbiteGabbroGraniteDolomiteSerpentiniteSource b r (g/cm 3 )2.652.622.592.352.862.551 k (W/(m
C))4.31.352.242.55.52.592 C (kJ/(kg
C))0.7320.7721.481.380.8460.9783 s c (GPa)10.97.66.17.22.27.44 m 0 0.63–0.710.82–0.880.76–0.880.73–0.820.560.555 t c = m 0 s c (GPa)7.36.55.05.61.24.1 D a ( m m)10–2010–2010–2010–2020–5020–506 L (kJ/kg)14824639622025010107 T ma (
C)1710110012009001000 c 1560 T m s (
C)2800171014001800116020508 h ma (Pas)1.8  10 5 3.0  10 7 1.2  10 3 1.3  10 10 0.0200.082 h m s (Pas)121.1  10 3 29980.0122.0  10  4
Seetextforfurtherdetails.  Sourcesare1, Deeretal. [1966]and Spera [2000];2, ClauserandHuenges [1995], Ennisetal. [1979],and VosteenandSchellschmidt [2003];3, Robie andHemingway [1995]and Anderson [2005];4,empiricalfitby Beeleretal. [2008]tothedataof Brozetal. [2006]: s c  0.123 H 2.3 GPa,where H is mineralhardness;5,D.L.Goldsby(personalcommunication,2007), WeeksandTullis [1985],and Mooreetal. [2004];6,D.L.Goldsby(personal communication,2007);7, RobieandHemingway [1978], Spera [2000],and Langeetal. [1994];8, Zhangetal. [1993], Morse [1980], SternandWyllie [1973], WangandTakahashi [1999],and Presnalletal. [1998].  Meltingtemperatureat0.5GPa[ IrvingandWyllie ,1975]. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 5of14 B11308 andWyllie ,1981].Forthepurposeofcomparison,thevalue of T ma listedinTable1correspondstothesolidustemper- atureat0.5GPa[ IrvingandWyllie ,1975],whichwasthe lowest-pressureexperimentalmeltingtemperaturethatwe wereabletofindreportedforCaCO 3 andMgCO 3 mixtures. [ 19 ]Asitisheated,serpentinite(chrysotile,orcrtwith MW277g/mol)undergoesacomplicatedseriesofbreak- downreactionsthatinvolveintermediatephasesofantigor- iteandtalcbeforeproducingacombinationofforsterite (fo),enstatite(en),andwaterimmediatelypriortomelt onset[ BucherandFrey ,2002].Becausecontactlifetimes areshort,itisnotclearwhetherthereactionseriesactually proceedstocompletionbeforemeltingbegins.Here,we assumethatitdoesaccordingto crt  fo  en  2H 2 O
withanetenthalpychangeofapproximately64.5kJ/mol. Thoughsignificantwaterproductionoccurs,weassumethat itescapestheasperitycontactspriortomeltonset. Accordingly,weusetheanhydrousenstatite-forsterite eutecticmeltingtemperaturefor T m s andassumeperitectic meltingat1atmfor T ma .ThevalueslistedinTable1for r , k ,and C areallforchrysotile,and L isapproximatedasthe sumofthenetenthalpychangeofthebreakdownreactions plusthelatentheatsofpureforsterite(73.2kJ/mol)and enstatite(142kJ/mol)[ Spera ,2000].Consistentwiththe procedureusedforthecasesofgabbroandgranite,the contactindentationstrengthisapproximatedas s c  y fo 2 s fo +( y en 2 +2 y fo y en ) s en ,where y fo and y en arethevolume fractionsofforsteriteandenstatiteina1:1molarratio, s fo  9.94GPa,and s en  6.24GPa. [ 20 ]Thefluidviscosityvalues h ma and h m s inTable1are estimatesfortheaveragemeltcompositionatatmospheric pressure,andthetemperatures T ma and T m s withnoadded water.Forthesilicaterocks,theseestimatesarebasedonthe empiricalfitsreportedby HuiandZhang [2007]tocom- piledexperimentalviscositydata.Thefittingparameters summarizedinTable2werecalculatedassumingthatthe meltcompositionandsolidcompositionareidentical.As dolomiteisnotasilicaterock,theempiricalfitsof Huiand Zhang [2007]couldnotbeusedtocalculateitsmelt viscosity.Instead,weaveragedthepredictionsofempirical exponentialfitstotheviscosityofpureCaCO 3 andMgCO 3 melts[ Dobsonetal. ,1996].Plotsofmeltviscosityasa functionoftemperaturearegiveninFigure3foranhydrous conditionsatatmosphericpressure.Inthepresenceofwater, themeltingtemperaturecanbereducedandthemelt viscosityatagiventemperaturedecreasessignificantly. Thesetwoeffectstendtocanceleachothertosomeextent insofarastheappropriatevaluesof h m s areconcerned. However,lackofknowledgeabouttheavailabilityofwater thatcaninfluencethemeltingprocessisrecognizedasone ofthebiggestsourcesofuncertaintyinchoosingappropriate parametervalues. [ 21 ]Theeffectofconfiningpressure s c onthemelt viscosity h ispoorlyconstrained.Thegenerallyhighvis- cositiesofsilicatemeltsarefacilitatedinpartbythe formationofpolymerchains.Changesin s c produce changesinthedegreeofpolymerizationsothat h can decreasesignificantlyathigh s c .Forsomecompositions h increaseswith s c instead,andforothercompositions h reachesamaximumatintermediate s c (see,e.g.,discussion by Tinkeretal. [2004,andreferencestherein]).(Dissolved waterisexpectedtobeimportantformodifyingthesensi- tivityof h to s c aswell.)Empiricaldatatoconstrainthese changesaresparse,andthevaluesof h m s giveninTable1 arecalculatedfromempiricalfunctionsthatarecalibratedto dataacquiredatatmosphericpressure.Sufficientdatado existtoestimatetheapproximatemagnitudeoftheeffectof pressureon h forthecaseofdryalbite.Experimentsat 1350
Candarangeofconfiningpressuresupto2.4GPa suggestthat h dropsbyaboutoneorderofmagnitudeas s c isincreasedbythisamountfromatmosphericpressure [ Kushiro ,1976].Suchadropin h m s wouldcausean increasetothecalculatedvalueof V h byapproximatelya factorofthree.Theempiricaldataforcarbonatitemeltsused toapproximatetheviscosityofdolomitewerecollected underarangeofconditionsandshowlittleapparent dependenceonconfiningpressure. [ 22 ]Table3summarizestheimportantdimensionless parametersthatcontroltheexpectedfrictionalbehavior, calculatedusingthenominalparametervaluesfromTable 1.Calculationswereperformedforambienttemperaturesof 20
Ctorepresentconditionstypicaloflaboratoryrock frictionexperiments;210
Ctorepresentconditionstypical ofmidseismogenicdepthsattheonsetofearthquakeslip; and700
Ctorepresentconditionsfollowingprolongedslip andsignificantfrictionalheatingoftherocknearthesliding surface.Ineachcase,theweakeningtemperatureisassumed tocorrespondtotheonsetofbulkmeltingat s c sothat T w = T m s .Atagivenbackgroundtemperatureandthe correspondingtabulatedStefannumbers S ,andvaluesof V h / V w and V / V w ,Figure2canbeusedtogaugewhetherthe effectivefrictioncoefficientisbestapproximatedby m s or m l (i.e.,equation(6)or(7)).OnFigure2,starsfor T =20
C, plusesfor T =210
C,andsquaresfor T =700
Careusedto Figure3. Meltviscositiesasafunctionoftemperatureat 1barpressure,calculatedfromempiricalfitsusingthe parameterssummarizedinTable2.Foreachcomposition, themeltonsettemperatureat1barisshownwithared asterisk,andthemeltonsettemperatureat s c isshownwith abluesquare. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 6of14 B11308 representthevelocityratioslistedinTable3.Withthe exceptionofthealbite,atthesetemperaturesandthe nominalslidingrateof V =1m/seachofthesliding systemsconsiderediswellwithintheregimewhere m s � m l .Asnotedbefore,equation(6)predictsthat m  m s isnot sensitivetothemeltviscosityinthisregime.Thisisbecause t a = h V / h m andattimeswhen h m  4  a th q  , h m  h so t a isafunctionof V and q ,butnot h .Bycontrast,attimes when h m 4  a th q  , h m   h  so t a increaseswith h . Similarbehaviortothisispredictedbythemodelfor pervasivefrictionalmeltingby Sironoetal. [2006].Velocity ratiosforthealbitesystemplotclosetothetransitionregion where m s  m l ,asisdemonstratedbythepredictionsshown inFigure1. 4.ApplicationtoExperimentalSlidingSystems [ 23 ]Topredicttheevolutionoffrictionalbehaviorina typicalexperimentalsetting,wemustfirstdeterminethe evolutionofbackgroundtemperature.Weassumethatthe contactlifetime q isasmallfractionofthetimescaleover whichthebackgroundtemperaturechangessothateach contact‘‘sees’’aconstantbackgroundvalueof T ,butthat T evolvesoverthecourseoftheexperiment.Foraneffective heatsource V ( t ) t ( t )= V ( t ) m ( t ) s n , CarslawandJaeger [1948]givethetemperatureonthesymmetryplaneattime t as Tt  T 0  s n 2 k  a th p   t 0 Vt  u  m t  u   u  d u
 8  where T 0 = T ( t =0)istheinitialambienttemperature.For constantslidingspeedswith V V w thetemperature evolutionis T ( t )= T 0 + V m 0 s n  a th t  /( k  p  ).When V � V w , theevolutionof m canbeapproximatedfromequations(6) and(7)as m  max( m s , m l ). [ 24 ]Figure4showsthechangesin m and T thatare predictedwhendifferentmineralsurfacesarewarmedfrom aninitialtemperatureof T 0 =20
Cbyfrictionalslidingata constantvelocityof V =0.36m/swith s n =5MPa.These parametervalueswerechosentocorrespondwiththeupper endofconfiningstressesandslidingratesattainedin experimentsusingtheseslidingsystemsbyD.L.Goldsby andT.E.Tullis(manuscriptinpreparation,2008).Calcu- lationswereperformediteratively,withequation(8)evalu- atedfirstforaconstantvalueof m andtheresulting approximatetemperaturehistoryusedtoobtainabetter approximationfor m ( t )thatwasinturnusedtocalculate anupdatedestimatefor T ( t ).Thisprocesswasrepeateduntil furtherchangesin T werebelowathresholdtolerance. [ 25 ]Ataconstantsliprate,changestothesliding resistancewithincreasingslipdistancearedrivenbytem- peraturechangesinthevicinityoftheslidingsurface.As notedinTable3,atanambienttemperatureof20
C,the weakeningvelocityofdolomiteis V w  1.1m/s,which exceedsthemodeledslidingrateof V =0.36m/s.Thehigh valueof V w fordolomiteincomparisontotheothermineral systemsislargelyduetotherelativelylowvalueof t c for thismineralsystem.AsshowninFigure4,theeffective frictioncoefficientfordolomiteisexpectedtoremainatits Table2. RheologicalParametersUsedtoCalculateSilicateMelt Viscosityat1barPressureFromlog h = A + B / T +exp( C + D / T ) a System AB (K) CD (K) Quartz  6.831.81  10 4 02.16  10 3 Albite  7.311.79  10 4  0.381.29  10 3 Gabbro  10.41.99  10 4  23.72.13  10 4 Granite  7.601.83  10 4  1.592.74  10 3 Dolomite b  7.555.02  10 3  9.957.15  10 3 Serpentinite  13.52.28  10 4  63.36.74  10 4
MeltviscosityisinPas.Compositionaldatausedtoobtainthefitting parametersusingthemethodof HuiandZhang [2007]arefromthe followingsources:albite(D.L.Goldsby,personalcommunication,2007); gabbro[ HiroseandShimamoto ,2005];Westerlygranite[ Spray ,1993].As notedinthetext,serpentiniteisassumedtobreakdowntoforsterite, enstatite,andwater.Weassumethewaterescapestheasperitycontactsprior tomeltingandusetheoxidesformedfromananhydrousforsterite-enstatite 1:1molarcompositiontocalculatethemeltviscosity.  ForDolomite,theviscositywascalculatedfromtheaverageoffitsto dataforMgCO 3 andCaCO 3 meltsas h  0.5[exp( A )exp( B / T )+exp( C ) exp( D / T )][ Dobsonetal. ,1996]. Table3. DimensionlessParametersforVariousSlidingSystems,CalculatedUsingtheNominalParameterValuesFromTable1With T w = T m s and h = h m s PropertyQuartzAlbiteGabbroGraniteDolomiteSerpentinite ParameterValuesatT=20
C V w (m/s)0.250.0390.140.171.10.14 S 0.0730.190.190.0900.260.51 V h / V w 8.2  10 2 1.5  10 2 3.0  10 2 2.3  10 2 2.3  10 3 9.1  10 4 V / V w a 3.9267.35.80.93 b 7.0 ParameterValuesatT=210
C V w (m/s)0.220.0310.100.140.750.12 S 0.0780.210.220.100.310.56 V h / V w 8.8  10 2 1.7  10 2 3.5  10 2 2.6  10 2 2.8  10 3 1.0  10 5 V / V w a 4.5339.87.31.38.6 ParameterValuesatT=700
C V w (m/s)0.140.0140.0350.0660.180.063 S 0.0960.320.380.140.640.76 V h / V w 1.1  10 3 2.5  10 2 6.0  10 2 3.8  10 2 5.7  10 3 1.4  10 5 V / V w a 6.97228155.716
Calculatedfor V =1m/s.  Note V w � V sonoweakeningexpected. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 7of14 B11308 basevalueof m 0 =0.56,untilaslipdistanceofapproxi- mately4m,bywhichtimethebackgroundtemperaturehas risenabove500
C,and V w hasdroppedbelow V sothat flashweakeningbegins.Bycontrast,alltheothermineral systemsconsideredherehave V w 0.36m/sevenat20
C, sotheybeginwith m m 0 .Withincreasesintemperaturethe effectsofflashweakeningreducethemodeled m further,as showninFigure4a).Inadditiontotheslipspeed,thedegree ofweakeningisdependentonseveralkeypropertiesofthe mineralsystem,includingtheweakeningvelocity V w ,and theStefannumber S .Once V V w becomessufficiently largeathighertemperatures,theviscosity-dependentchar- acteristicvelocity V h alsoplaysarole.Asshownbya comparisonofthetrendsinFigure4a)withtheparameters ofTable3,thevalueof V w providesagoodindicationofthe predictedrelativedegreeofweakening.Dolomitehasthe highest V w attheinitialtemperatureandremainsstrongest throughoutthecalculatedsliphistory.Thisisfollowednext byquartz,thengranite,gabbroandserpentiniteatalmostthe same m andsimilarinitialvaluesof V w .Finally,albitehas thelowest m andalsothelowestvalueof V w atagiven temperature. [ 26 ]ExperimentalresultsreportedbyD.L.Goldsbyand T.E.Tullis(manuscriptinpreparation,2008)(seealso Figure4of Beeleretal. [2008]andFigure2bof Hiroseand Shimamoto [2005])suggestweakeningvelocitiesforgab- bro,graniteandquartzoforder0.1m/satambientlabora- torytemperatures.Thisisinreasonableagreementwithour predictionsof V w  0.14,0.17,and0.25m/srespectively fortheseslidingsystems(seeTable3).Usingequation(2) toextrapolatetheseexperimentalresultstoatypicalcoseis- micsliprateof1m/s,D.L.GoldsbyandT.E.Tullis (manuscriptinpreparation,2008)predictfrictioncoeffi- cientsofapproximately0.2forthesethreemineralsystems andnotethatsuchalowfrictioncoefficientissufficientto satisfyheatflowconstraintsontheSanAndreasfault.Here, wecalculate m  m s  0.26forgabbro,0.23forgranite,and 0.25forquartzat V =1m/sandanambienttemperatureof 20
C.Therewasnoobservedweakeninginslidingexperi- mentswithdolomitetoslipspeedsof0.36m/s(D.L. GoldsbyandT.E.Tullis,manuscriptinpreparation,2008). Thisisinagreementwithourpredictionof V w  1.1m/s atanambienttemperatureof20
C.Bycontrast,weakening ofalbitewasnotobservedbyGoldsbyandTullisuntil V  0.3m/s,whereaswepredictasignificantlylower V w  0.04m/ssothat m wouldapproach0.11at V  1m/s. Asdiscussedfurtherbelow,additionalaffectssuchasthe productionofgougeontheslipsurfacemightreducethe rateofslipatasperitycontacts.Theuncertaintyinparameter choicesdiscussedabovemayalsoproducesignificant modelingerrorsthatexplainthediscrepancieswithexper- imentalobservations.ExperimentalresultsbyGoldsbyand Tullisforserpentiniterevealasignificantlymorecompli- catedfrictionalevolutionthanfortheotherslidingsystems theyinvestigated.Theyreportaslowonsetofweakening beginningatjust10–20mm/s,followedbymorerapid weakeningstartingatapproximately0.1m/sandabrupt strengtheningbeginningatapproximately0.2m/s.By design,ourmodelisonlycapableofpredictingtheoutcome fromasinglephasechangeprocess,inthiscasewith weakeningbeginningat V w  0.14m/sandleadingto m  0.26at V  1m/swhen T =20
C.Itseemsclearthat additionalphysicaleffectsmustberesponsibleforthebe- haviorobservedintheserpentiniteexperimentsofD.L. GoldsbyandT.E.Tullis(manuscriptinpreparation,2008). [ 27 ]Acomparisonbetweenpredictedfrictioncoefficients andtheexperimentalresultsof HiroseandBystricky [2007] foraserpentiniteslidingsystemareshowninFigure5.The experimentswereconductedusingcylindricalcoresamples, withonesideheldfixedandtheotherrotatedathighspeeds asresistancetoshearwasrecorded.Ineachofthefour modeledruns,thepredictionshavemuchlessstructurethan theexperimentaldata,yettheoveralltrendsagreereason- ablywell.Someofthevariationsinmeasuredfrictional resistanceareundoubtedlycausedbyadditionalphysical effectsthatwerenotmodeledhere.Asnotedby Hiroseand Bystricky [2007],theseincludevariationsinslipspeedwith radialdistance,fractureandshorteningoflaboratorysam- Figure4. (a)Changeineffectivefrictioncoefficientwithtimeastheslidingsurfacewarmsdueto frictionalheating.(b)Evolutionofbackgroundtemperature T .ParametersfromTable1wereusedwith appliednormalstress s n =5MPa,slidingrate V =0.36m/s,andaninitialtemperatureof T 0 =20
C. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 8of14 B11308 ples,andpossiblyeventhermalpressurizationofthewater releasedbydehydrationreactions.Indeed,calculationsby Rice [2007]suggestthatwateradsorbedtomineralsurfaces atambienthumiditymightbeexpectedtoundergosignif- icantthermalpressurizationduringlaboratoryexperiments. Forexample,aporepressureincreaseof0.5MPawouldbe sufficienttodropthemodeledeffectivefrictioncoefficients forthethreelowercurvestoapproximately0.32,0.19,and 0.15,arguablymoreinlinewiththemedianvaluesobtained experimentally. [ 28 ]Weemphasizethatnoadjustableparameterswere usedtoobtainthemodelresultswedescribe.Ourprimary goalisnottoobtainanexactfitwithexperimentalresults, butrathertoprovideasimplephysicallybasedmodelthat makesuseofpropertiesmeasuredindependentlyyetis capableofreproducingtheessentialtrendsoffrictional experiments.Weanticipatethatfutureexperimentalefforts, aimedbothatmeasuringfrictionalresistanceandatfurther constrainingtherelevantphysicalproperties,willprovide morestringenttestsofthiselementarytheoryandthe importanceofadditionalphysicaleffectsthathavenotbeen included. 5.Discussion [ 29 ]Thesimplifiedmodelofasperitymeltingpresented herepredictsfrictionalbehaviorthatisinreasonableagree- mentwithavailableexperimentalresults.Becauseofthe uncertaintyinchoosingappropriatemodelparameters,the adoptionofseveralsimplifyingapproximations,andpoten- tialcomplicatingeffectssuchassolid-statephasetrans- formations,somedifferencesbetweenpredictedand experimentalfrictioncoefficients m areinevitable.Two- parameterfitsoftheformgiveninequation(2)thatusea constantweakenedcoefficient m w andempiricalonset slidingrate V w alsofitexperimentaldataquitewell.Itis reasonabletoaskwhethertheadditionalcomplications involvedinpredictingfrictionalbehaviorfromthefunda- mentalpropertiesofthesystemcomponentsisworththe extraeffortinvolved.Ifthegoalistoextrapolateknown experimentalresultstopredictfrictionalbehaviorunder otherconditions,empiricalfitsmaywellprovideabetter guidethanfirst-principlescalculationsthatfailtomatch preciselyobservationsintheoverlappingrangeofslip rates V ,normalstresses s n ,andambienttemperatures T . Nevertheless,approximationsfor m oftheformgivenby equations(6)and(7)dosuggestimportantfeaturestothe systembehaviorthatarenotencapsulatedinprevious treatments.Theyalsoprovideabasisforgaugingthe importanceofotherphysicaleffectsthatmaycomplement orcounteracttheeffectsofasperitymelting. [ 30 ]Fortheslidingsystemsconsideredhere,undertyp- icalexperimentalandcoseismicconditionsweexpect m  m s ,asdescribedbyequation(6).Inthisparameterregime theratioof m tothelow-speedfrictioncoefficient m 0 dependsonlyontheweakeningvelocity V w  ( r C ) 2 ( T w  T ) 2 pa th /( t c 2 D a )andtheStefannumber S
L /[ C ( T w  T )],whichmeasurestherelativeimportanceoftheheat requiredtomeltasperitiesincomparisontothatrequiredto raisethetemperaturefromambientlevels.Crucially,the valueof m s isindependentofthemeltviscosity h ,whichis oftenoneofthemoredifficultphysicalpropertiesto ascertain.Thethermalpropertiesanddensitiesofmost mineralsystemsarewelldeterminedandtheambient temperatureisalsoknownunderexperimentalconditions andcanbeestimatedasafunctionofdepthduringseismic slip.Theweakeningtemperature T w andtheproductof asperitysize D a withthesquareofthecontactstrength t c 2 arethephysicalquantitiesthatareleastwellknownfor assessingappropriatevaluesof V w and S .Asanalternative toseekingthesepropertiesindependently,onestrategyfor determiningtheirvaluesistofitempiricaldatausing equation(6)withdifferentvaluesof V w and S andsoinfer T w and t c 2 D a .Closerexaminationofequation(6)indicates that m s / m 0 isinverselyproportionalto V / V w atsmall S ,which alsofitswellwiththepredictionsofequation(2),butthe dependenceof m s / m 0 on V / V w weakensas S increases. Brownetal. [2007]report m  V  0.36 inexperimentswith diabase;thisisconsistentwithourpredictionsfor m s at valuesof S approachingunity. [ 31 ]Atthehighestslidingratesweexpect m  m l ,as describedbyequation(7).Inthisparameterregimetheratio m / m 0 dependson V , V w andthevelocityscale V h
 pr L a th  h S 2   =( T w  T )  p kC  h L   .Themeltviscos- ity h hasanimportantcontrollingeffectunderconditions when m  m l .Inthesimplifiedmodelpresentedherewe havetreatedthemeltlayerasisothermalsothat h  h m s isa constantandallheatthatisnotconductedawaythroughthe solidgoestoextendingthemeltfilmthickness.Inreality, thetemperatureonthesymmetryplaneat y =0isexpected toexceedthatatthephaseboundarywhere y = h m ,and variationsin h withtemperatureareexpected,asshownin Figure3.InAppendixAanalternativemodelisdescribed thatapproximatestheeffectsofvariationsinmeltviscosity withtemperaturebyevaluatingchangesintheaverage viscosity h avg ofthemeltlayerasitgrowsandapproximat- ingtheresistancetoshearingatthemoltenasperitycontacts as t a  h avg V / h m .AsshowninFigure6,thefrictional Figure5. Comparisonbetweenpredictedeffectivefriction coefficientasafunctionofslipdistancefortheserpentinite systemattheslipratesandconfiningstressesgiveninthe legendandthelabeledexperimentaldataof Hiroseand Bystricky [2007]. B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 9of14 B11308 evolutionpredictedbythisalternativemodelisinreason- ablygoodagreementwiththatproducedusingtheisother- malmelttreatmentofthemodeldescribedinsection2. Becauseforceequilibriumrequiresthatstrainratesbe higherwhere h islower,andbecausetheeffectsofan overpredictedviscosityandanoverpredictedfilmthickness partlyoffseteachother,thetemperature-dependentviscosity doesnotaffectthetotalshearresistanceasmuchasone mightotherwiseexpect,atleastforthekinematictreatment consideredhere.Inexperimentsthatspanasufficientrange ofslidingvelocitiessothatthefrictionalbehaviortransitions fromaregimeinwhich m  m s tooneinwhich m  m l , empiricalfitsto m / m 0 atthehighest V usingtheformgivenby equation(7)mightbeusedtoinfer V h andsodetermine h . [ 32 ]When m  m l ,Figure1showsstrengtheningwith increasesin V afteraminimumin m / m 0 isreached.Weak- eningwithincreasedslipratehasbeensuggestedasa mechanismforcausingshearlocalizationingranularmedia [e.g., Riceetal. ,2005].Atfacevalue,thepotentialfor asperitymeltingtoproducestrengtheningwithincreased slipratesuggestsamechanismforbroadeningtheshear zoneduringlaterstagesofseismicorexperimentalslip.The kinematictreatmentpresentedheredoesnotaccountforthe variationsinslipratethatareexpectedundermorerealistic conditions.Itisreasonabletoexpectthattheambient temperature T oftheshearzoneshouldtypicallyincrease withslip.Assumingthat m l canbeusedtoapproximatethe frictionalbehaviorwithchangesin T evenas V changes,we differentiateequation(7)tofindthat  m l  T   2 m 0 T w  T V w V 1  1  2 V 2 h V 2 V V w  1    1  2        0   9  Thisindicatesthat m l alwaysdecreasesas T risesandhints thatthepositivedependenceof m l on V mayonlycausethe shearzonetobroadenunderarestrictedsetofcircum- stances.Accuratepredictivemodelsfortheshearzone thicknessanditsevolutioningranularmediaareneeded. [ 33 ]Significantvelocitystrengtheninghasbeenobserved duringexperimentsonserpentenite(D.L.GoldsbyandT. E.Tullis,manuscriptinpreparation,2008)anddiabase [ Brownetal. ,2007].Intheformercase,theincreasein m with V wasmuchmoreabruptthanpredictedbyourmodel forasperitymelting.Inthelattercase,evidenceofmelt quenchingand‘‘welding’’wasimplicatedinthestrength- eningbehavior,muchasinferredduringexperimentson gabbroby HiroseandShimamoto [2005].Additionalphys- icalmechanismsbeyondthosetreatedinourmodelare requiredtoexplaintheseobservations. Noda [2008]givesa detailedandinformativeanalysisofsomeoftheseissues. Recentexperimentalandtheoreticaleffortstotrackstrength evolutiontoconditionsinwhichthroughgoingmeltlayers coattheslipplanealsoshowpromiseinthisdirection[e.g., DiToroetal. ,2006a,2006b; Nielsenetal. ,2008; Sironoet al. ,2006].Thetreatmentof Sironoetal. [2006]shares manysimilaritiestothecurrentmodelandpredictsthatmelt viscosityshouldhaveonlyaminoreffectontheeffective frictionalbehaviorevenwhentheentireslidingsurfaceis coatedwithamoltenlayer.Anotabledifferencewiththe predictionsofourmodelisthatwhereaswefindthat m  m s isindependentofviscosity h becausemeltthickness h is proportionalto h and t  h / h , Sironoetal. [2006]findthat forthemineralsystemstheyconsiderthetemperaturesof thethroughgoingmeltlayersattainsuchlevelsthat h reachesnearlythesamevalueineachcase. [ 34 ]Wehaveconsideredslidingalongasingleplaneand havenotaccountedfordistributedshearthroughfinite gougelayers.Granularshearzonescanbepreexisting,as inthecaseforearthquakesthatoccuralongmaturefaults. Theycanalsobeformedduringslip,asinthecaseof slidingexperiments,whicharetypicallyconductedusing rocksamplesthatareinitiallyintact,butfoundtobecoated bywearproductsafterexperimentalruns.Ifshearis uniformlydistributedover N graindiametersatagiven instantintime,thentherelativeslipvelocitybetween adjacentgrainsisexpectedtobereducedtoavalue comparableto V / N .Becauseparticlesizesinshearzones areoftensubmicroninscale, N canbelargeenoughthat V / N V w evenforsubmillimeter-scaleshearzonesatthehighest slipspeedsattainableduringearthquakesandlaboratory experiments.Similarconclusionsarereachedforthecase wheretherelativesliprateisnotuniform,butinsteadhasa Gaussiandistributionacrosstheshearzone[e.g., Andrews , 2002]. Rice [2006]suggestedthatevenwhenshearis distributedthroughafinitezone,therelativesliprate betweenadjacentparticlesatagiveninstantintimemight actuallyapproach V .Thiscorrespondstotherateexpected forsliponaplane,withtheideabeingthattheslipplane Figure6. Comparisonbetweenthefrictionalbehavior predictedbythemodelforisothermalmelting(dashed)from section2(seealsoFigure1)andthatpredictedbythe similaritysolutionfromequation(A5).Parameterswere chosenfordryalbiteandtheinitialtemperature T 0 =210
C. Thedot-dashedlineiscalculatedusing h m s ,whichisthe highestviscosityinthemeltandoccursatthefilm boundaries.Thedottedlineshowstheapproximationfrom equation(A4),whichisvalidinthelimitoflow h .Thesolid curvegivesthepredictionsofequation(A5)withtheviscosity assignedthespatiallyaveragedvalue h avg forthecalculated temperatureprofileacrossthefilm(usingtheviscosity parameterizationsummarizedinTable2). B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 10of14 B11308 itselfmovesthroughafinitezonetoaccommodatethe effectsofgeometricalirregularitiesandsoproduceashear zonewithanapparentfinitewidth.Thelargerstrength reductionimpliedbythisassumptionhasbeencompared tothatpredictedforuniformshearwith m approximatedby equation(2),where V isreplacedby V / N asappropriate [ Rempel ,2006].Becausethetemperatureevolutionissen- sitivetothethicknessoftheregioninwhichheatis dissipated,thesecomparisonsattesttotheparamountim- portanceofshearzonethicknesstomechanismsforstrength evolutionthatarethermalinorigin.Forexample,the strengtheningobservedbyD.L.GoldsbyandT.E.Tullis (manuscriptinpreparation,2008)intheirexperimentson serpentinitemightbeattributedtoreductionsin m that accompanydecreasesin V / N asgougeisproduced;the largevalueof V w inferredfromexperimentsonthealbite systemmightbeattributedtosimilareffects. [ 35 ]Duringhigh-speedfrictionexperimentsthatwere conductedby HiroseandShimamoto [2005]macroscopic quantitiesofmeltwereejectedfromtheslidingsurface[see also DiToroetal. ,2006a,2006b; Nielsenetal. ,2008]. Similarbehaviormightmodifytheevolutionofmeltfilm thicknesses h m atasperityjunctionssothattheythicken moregraduallythanpredictedhere.Thiswouldtendto increase m andmightbepartiallyresponsibleforthe strengtheningbehaviorobservedby Brownetal. [2007]. Inadditiontoreducing h m belowwhatitwouldotherwise be,meltejectionmayalsocausetheeffectiveareaofcontact toincreasesothat s c decreasesduringthecontactlifetime. Sincemeltonsettemperatureistypicallyreducedby decreasesinconfiningpressure,thiscouldsignificantly complicatethestrengthevolutionbyflashmelting.Without firmexperimentalevidencetoguideus,weviewmodel extensionsthataccountforsucheffectsaspremature. [ 36 ]Themeltfilmswemodelalongasperityjunctionsare typicallynanometerstoperhapsmicronsinthickness,with thepreciseevolutionof h m sensitivetothemeltviscosity h . Itisreasonabletoquestionwhetherthemeltviscosities themselvesmightbealteredfromtheirbulkvaluesbythe effectsofconfinement.Experimentswithotherliquidsys- tems[e.g., Granick ,1991]suggestthatviscositiesdecrease toapproachtheirbulkvaluesforfilmsthatareonlyafew molecularlayersinthickness.Forsilicatemeltsthatare significantlyinfluencedbytheformationofpolymerchains, thechainlengthisthemorerelevantlengthscale.Weare notawareofanyexperimentaldatathatteststheeffectsof confinementinthesesystems.However,onecaneasily envisionscenariosinwhichconfinementwithinathinfilm mightcausetheaveragelengthofpolymerchainsto decrease,andsoproduceareductionineffectiveviscosity atlowvaluesof h m .Insofarastheeffectivefriction coefficientisnotsensitivetothemeltviscositywhen m  m s ,sucheffectsmayonlybeofacademicinterestiftheydo notaltertheoverallfrictionalbehavior,butjustinfluence theevolutionof h m .Forphysicalintuition,usingthe parameterslistedinTable1thefilmthicknessesrequired forviscousshearresistance t  h m s V / h m  t c when V =1 m/sareapproximately0.002,0.2,0.006,and0.02 m mfor quartz,albite,gabbro,andgraniterespectively.Fordolomite andserpentinitetheeffectiveviscositiesmustbeconsider- ablygreaterthanthevaluesof h m s listedinTable1inorder forrealisticfilmthicknesses(i.e.,greaterthanmolecular dimensions)tobeachievedwith t = t c .Withfurther experimentalcontrolsamoredetailedinvestigationofthe conditionsatmeltonsetmayleadtoanimprovedunder- standingoftheshearresistanceduringtheearlystagesof flashmeltingforthesesystems.Wenotehoweverthat Figure2andTable3indicatethattheeffectiveviscosity wouldneedtoincreasebyseveralordersofmagnitudeforit tochangetheinferencesmadehereandbecomeanimpor- tantfactorincontrollingtheeffectivefrictionalbehaviorof dolomiteorserpentinite. [ 37 ]Onthebasisofthesimplemodelpresentedhere, fieldevidencefortheeffectsofflashmeltingislikelytobe sparseatbest.Asnotedabove,meltfilmsgeneratedalong asperityjunctionsareexpectedtobeextremelythinand theirquenchedproductswouldbedifficulttodetect,par- ticularlyfollowingthelongresidencetimesrequiredfor exhumationfromseismogenicdepths.Typically,meltsare expectedtoquenchtoglasses,thatwouldbeexpectedto formpatchycoatingsonthefaultsurfaces.Inrarecircum- stances,meltsmightcrystallizeashigh-pressurepoly- morphsoftheinitialmineralphases.However,asperities areexpectedtobecooledandunloadedsimultaneously,so conditionsshouldfavortransformationbacktotheoriginal crystalstructures.Carefulexaminationofslipsurfaces followingcontrolledlaboratoryexperimentsmayyetyield evidencefortheproductsofflashmelting.However, comparisonsbetweenmodelpredictionsandobservedfric- tionalevolutionarelikelytobethemostcompellingtestof flash-weakeningmodels.Thepotentialforflashmeltingto contributetothedeterminationofshearzonewidthsuggests afutureresearchdirectionthatmightonedayproduce predictionsofflash-meltingcharacteristicsthataretestable withfieldevidence. 6.Conclusions [ 38 ]Laboratoryobservationsofflashweakeningare broadlyconsistentwiththepredictionsofsimplifiedmodels fortheeffectsoflocalizedmeltingandviscousheatingat asperitycontacts.Forestimatedvaluesofthephasebehav- iorandmechanicalpropertiesthatcontrolflashmelting undertypicalseismicandlaboratoryconditions,theevolu- tionofeffectivefrictioncoefficient m isexpectedtobe relativelyinsensitivetomeltviscosity.Instead,theweak- enedfrictioncoefficientiswellapproximatedbyequation (6),whichiscontrolledbytheratiooftheslipratetothe thresholdvaluerequiredforweakeningtobegin V / V w ,and theStefannumber S ,whichmeasurestheratiooflatentto sensibleheat.Atthehighestslipratesandatthehigher Stefannumbersthatcorrespondtoslipatambienttemper- aturesnearertomeltonset,equation(7)givesabetter approximationtotheexpectedbehavior,withaweak dependenceonmeltviscosityenteringthroughtheinfluence of V / V h on m . [ 39 ]Formanymineralsystems,thehighconfiningpres- suresexpectedofasperitycontactsaresufficienttopromote solid-statephasetransformations,buttheshortdurationof asperitycontactmaynotbesufficientforequilibriumtobe achieved.Confidenceinappropriateparameterchoicesis hamperedbysuchcomplications,togetherwithuncertain- B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 11of14 B11308 tiesinfactorsthatincludetheavailabilityofwatertoalter thephasebehaviorandmeltviscosity.Flashweakeningis expectedtocommonlyoccurinconcertwithotherimpor- tantprocessesincludingthethermalpressurizationofpore fluids,theproductionofgouge,andchangesinthethick- nessandlocationoftheshearzone.Quantificationofthese effectsisnecessarytoachieveamorecompleteunderstand- ingofthestrengthevolutionduringhigh-speedseismicand laboratoryshear. AppendixA:AnAlternativeModelforMeltFilm Growth [ 40 ]Themodelforflashmeltingdevelopedinsection2 makesuseofanapproximatetreatmentoftheheatflow fromthemeltfilm,whilethepossibilityofrheologically significanttemperatureincreasesinthefilminterioris neglected.Furtherinsightintothevalidityofthisisothermal modelisgainedbycomparingitspredictionswiththoseofa two-layer(melt/solid)modelformeltproductionthatis formulatedtobothallowfortemperatureincreaseswithin thefilmitselfandsolveforthechangingheatfluxoutofthe growingfilm. [ 41 ]Inthemeltfilm,energyconservationrequiresthat  T  q  a th  2 T  y 2  V 2 h avg r Ch 2 m
 A1  andbeyond y =± h m /2thetemperaturesatisfiesthediffusion equation.Theboundaryconditionsaretheheatflux condition(comparewithequation(4))  2 k  T  y     y  h  m  2  2 k  T  y     y  h  m  2  r L  h m  q
 A2  thesymmetryrequirementthat  T /  y =0at y =0,thefar- fieldconditionthat T  T 0 atlarge y ,andthemelt equilibriuminterfacecondition T = T m at y =± h m /2.The symmetryvariableisdefinedas x
y /(2  a th q  )andthe evolutionoffilmthicknessissoughtintheform h m = 2 l  a th q  .Thetransformedgoverningequationinthemelt regioncanbeintegratedtowritethetemperaturegradientas d T d x  V 2 h avg k l 2 e  x 2  x 0 e t 2 d t 
where k
r C a th isthethermalconductivity,andtheterm inbracketsontherightisDawson’sintegral[ Abramowitz andStegun ,1964],whichsatisfiesthesymmetrycondition at x =0.Outsidethemeltregionthetemperatureprofile satisfies T ( x )= T 0 +( T w  T 0 )erfc( x )/erfc( l /2).Theheat fluxconditionat x = l /2describedbyequation(A2)canbe writtenas  2 erfc l 2  p 3  2 V 2 S l 2 V 2 h  l  2 0 e t 2 d t  S  p  l e l 2  4
 A3  whichissolvedfortheunknowninterfacecoordinate. Differencesinthermalpropertiesbetweenthesolidandmelt componentshavebeenneglectedforsimplicity.Takingthe limitingcaseofsmall l forillustrativepurposesgives l  p 3/2 /(4 S )( V / V h ) 2 .Thisissubstitutedintoequation(1)forthe effectivecontactstrengthusingthemodifiedweakening time  q w = pq w V 4 /(4 l 2 V h 4 S 2 )  4 q w / p 2 atwhich V h avg / h = t c forthismodel,tocalculate
t c andfindthattheeffective frictioncoefficientfor q �  q w is m  m 0 4 p  V w V  1   V w  p  V     A4  Moregenerally,forlarger l thismodelpredictsthat m  m 0  p V w V 3  l V 2 h S 1   p V w V 3  4 l V 2 h S    A5  [ 42 ]Thesourceterminequation(A1)isproportionalto theresistancetoshearinthemeltfilm t a = h  g .Force equilibriumrequiresthat  t a /  y =0sotheeffectsof gradientsinfluidviscositymustbecompensatedby correspondinggradientsinstrainrate  g .Weapproximate thestrainrateas  g  V / h m andregardthefluidviscosityas anaverageoverthemeltlayer h avg .Forthesimilarity solutionleadingtoequation(A5)tobevalid, h avg isfurther approximatedasanaverageoverthecontactlifetime,as wellasoverthefilmthicknesssothat h avg  2 l  l  2 0 h T  d x  Here h ( T )iscalculatedusingtheempiricaldependenceof meltviscosityontemperaturethatischaracterizedbythe parameterslistedinTable2.Toevaluatetheeffective frictioncoefficientfromequation(A5),wefirstobtainedan approximationfor l with h avg  h m s .Wethenusedthis resulttodeterminetheapproximatetemperatureprofile withinthefilmandobtainarevisedapproximationfor h avg thatweusedtoadjustthevalueof V h inequation(A3)so thatanewapproximatevalueof l couldbedetermined.The processwasrepeateduntilfurtherchangesto l were insignificant. [ 43 ]Comparisonsbetweentheeffectivefrictioncoeffi- cientspredictedbytheflash-meltingmodelsforthecaseof albitewith T 0 =210
CareshowninFigure6.Thesolid curveshowsthepredictionsofthesimilaritysolutionfrom equation(A5),calculatedwith V h determinedfortheaver- agemeltviscosity h avg .Thedashedcurvegivesthepre- dictionsoftheisothermalmodeldescribedinsection2.At lowslidingrates,thesimilaritysolutionisexpectedtobe lessreliablebecauseitdoesnotproperlycapturethe frictionalbehaviorpriortomeltonset.Athighslidingrates, theisothermalmodelsuffersbecauseitdoesnotaccountfor theenergythatgoesintochangingmelttemperatureinthe interiorofthefilm;nordoesitaccuratelytreatthedistrib- utednatureofviscousheating.Overall,however,the agreementbetweenthetwoapproachesisreasonablygood. Forreference,thepredictionsofthesmall l limitof equation(A4)areshownwiththedottedcurve,andthe predictionsofthesimilaritysolutionwith V h determined using h = h m s areshownwiththedot-dashedcurve. Temperatureincreasesinthefilminteriorcanproduce B11308 REMPELANDWEAVER:FLASHWEAKENINGANDMELTING 12of14 B11308 significantreductionsto h ,andthedot-dashedcurvecanbe thoughtofasacrudeupperboundon m .Bycontrast,inthe limitofsmall l describedbyequation(A4),thefluid viscosityisunimportantandthedottedcurveisalower boundonthepredictionsofthesimilaritysolutionfor m .In thislimit,differenceswiththepredictionsshownbythe dashedcurveareprimarilyduetoinaccuraciesinthe approximationoftheeffectsofheatflowfromthefilmin theisothermalmodel. [ 44 ] Acknowledgments. WethankPaulWallaceformanyhelpful discussionsandDavidGoldsbyforsharingunpublishedexperimental results.ThisworkwasimprovedbythecarefulreviewsofStefanNielsen andHiroyukiNodaandthecommentsofAssociateEditor,EricDunham. FundingcamefromNSFEAR-0711048. References Abramowitz,M.,andI.A.Stegun(1964), HandbookofMathematical Functions ,p.298,Natl.Bur.ofStand.,Washington,D.C. 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