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GSADataRepository2016005 Brodskyetal.

DocumentationofFaults

TableDR1.Scannedfaultsandroughnessprocessing

Name Scanner Location Lithology Sense Slip Processing

Cascia# LiDAR 42.719°N13.002°E

Carbonate Normal 50m Smallpatchesb+Taper5%d

GubbioUpper#

LiDAR 43.344°N12.597°E

Carbonate Normal 50‐100m Smallpatchesb+Taper5%d

GubbioLower#

LiDAR 43.344°N12.597°E

Carbonate Normal 200m Smallpatchesb+Taper5%d

MonteCoscerno# LiDAR

42.692°N12.887°E Carbonate Normal 250m Smallpatchesb+Taper5%d

MonteMaggio# LiDAR

42.762°N12.941°E Carbonate Normal 650m Smallpatchesb+Taper5%d

ValCasana# LiDAR 42.718°N12.857°E

Carbonate Normal 150m Smallpatchesb+Taper5%d

VenereLarge# LiDAR

41.971°N13.664°E Carbonate Normal >20m Smallpatchesb+Taper5%d

VenereSmall#

LiDAR41.971°N13.664°E

Carbonate Normal 4m Smallpatchesb+Taper5%d

WestFucino#

LiDAR41.940°N13.362°E

Carbonate Normal ~80m Smallpatchesb+Taper5%d

Vasquezrocks#

LiDAR34.483°N118.316°W

Sandstone Normal 10±5cm Smallpatchesb+Taper5%d

Yeelim# LiDAR 31.223°N35.354°E

Carbonate Normal 50‐80m Smallpatchesb+Taper5%d

SplitMountain#

LiDAR 33.014°N116.112°W

Sandstone Strike‐slip

30±15cm Smallpatchesb+Taper5%d

MeccaHills# LiDAR33.605°N115.918°W Carbonate

Strike‐slip 20±10cm Smallpatchesb+Taper5%d

FlowersPit# LiDAR42.077°N121.856°W Andesite Normal

100‐300m Smallpatchesb+Taper5%d

ChimneyRock# LiDAR

39.227°N110.514°W Carbonate Normal 8m Smallpatchesb+Taper5%d

LakeMead# LiDAR36.062°N114.831°W

Dacite Normal500‐1000m

Smallpatchesb+Taper5%d

CoronaHeights*

LiDAR+LPa

37.765°N122.437°E

ChertStrike‐slip

Severalmto>1km

Smallpatchesb+Largepatchesc+Taper3%d

Vuache‐Sillingy*

LiDAR+LPa

45.920°N6.049°E

CarbonateStrike‐slip

10‐30m Smallpatchesb+Taper3%d

DixieValley*LiDAR+LPa

39.947°N117.945°E

Rhyolites NormalSeveralmto>3‐6km

Smallpatchesb+Taper3%d

Bolu* LiDAR40.685°N31.568°E Carbonate

Strike‐slip

20m‐85km Smallpatchesb+Taper3%d

Klamath## LiDAR42.135°N121.678°W

Basalt+andesite

Normal 50‐300m Largepatchesc+Taper3%d

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Arkitsa§ LiDAR38.733°N23.000°E

Carbonate Normal>300‐400m

Largepatchesc+Taper3%d

Sources:#Brodskyetal.(2011),*Candelaetal.(2012),##Sagyetal.(2007),§ResorandMeer(2009)Notes:aLaserProfilometer,bSmallcleanfaultpatchesfreeofunwantedobjectsareselectedfromtheoriginalcloudofpoints,cLargecleanfaultpatchesareobtainedbyremovinglocallynon‐faultingfeaturesfromtheoriginalcloudofpoints,dDuringtheprocessingofindividualprofilesforcomputingtheFourierspectraweeitherapplyacosinetaperof3%or5%.

Scale‐dependentroughnessinspatialmaps

FigureDR1demonstratesthatthescale‐dependentroughnessisafeatureofindividualprofilesoftherawdatainthespatialdomain.TheparameterizationintheFourierdomainusefullycapturesthesamephenomenonshowninFigureDR1.

FigureDR1.Examplecrosssectionalprofilesatdifferentmagnificationsshowingthescaledependenceoftheaspectratio,H/L.Profilesaretakenfromaground‐basedLiDARscanoftheCoronaHeightsfaultsurface.Panelsshowsectionsthroughthefaultatdifferentmagnifications(Notethescalebarisdifferentineachpanel).

H/L ~ 0.06

H/L ~ 0.08

H/L ~ 0.3

2 m

0.2 m

0.02 m

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FourierTransformComputation

Fromeachfaultpatch(LiDARorprofilometer),hundredstothousandsofprofilesareextractedintheslipdirection.Mostfaultpatcheshavemorethan500profilesintheslipdirectionandallpatcheshaveatleast100.Thefourstepsintheproceduretocomputethespectrumofeachprofileareasfollows:(1)Eachprofileisdetrendedbysubtractingthebest‐fittrend.(2)Eithera3%or5%cosinetaperisappliedtoeachroughprofiletoensurethattherearenostepfunctionsattheendofthefinitewindow.(3)ThediscreteFouriertransformiscalculated,andthepowerspectrumisequaltothesquareoftheamplitudesofthecoefficients.(4)Thepowerspectrumisnormalizedbytheprofilelengthtoobtainthepowerspectraldensity.ThemeanFourierspectrumofeachfaultpatchisthencomputedbyaveragingthespectraoftheprofilesandrestrictingtheresultstowell‐resolvedwavelengthsthataremorethanafactorof2lessthantheprofilelength.Finallywesmooththespectrainfrequencyspacebybinningeachaveragedspectruminto20logarithmically‐spacedintervalsinthewell‐resolvedfrequencyspaceandaveragingthepowerspectraldensitywithinthebin.Logarithmicbinningprovidesaconstantdensityofdatapointsinthelogarithmicrepresentationandthereforeavoidsgivingmoreweighttosmallerscalesinsubsequentfittingprocedures.

MaximumShearStraininHertzianContacts

FigureDR2.Cartoonofelasticinteractionbetweentwospheres.Dashedlinesshowundeformedoutlines.

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TheelasticdeformationbetweentwoidenticalsphereswasfirstmodeledbyHertz(Hertz,1881).WeusethesolutiontoillustratethegeneralformoftheelasticstressfieldbetweencontactingasperitiesstartingfromthegeneralresultssummarizedbyJohnson,1985,Appendix3.IftwoelasticspheresofradiusRareincontactasshowninFigureDR2andthecontactbetweenthemhasnoshearstress,thesolutionfortheradiusaofthecontactareais

a=(3WR/4E’)1/3 (A1)

whereWistheloadingforceonthesphereandE’isthemodifiedYoung’smodulus(2(1‐2)/E)‐1andEistheYoung’smodulusandisthePoissonratio.Theapproachdistancebetweenthetwospheresisrelatedtothecontactradiusby

=a2/R (A2)

Slightlymanipulatingeq.A1yields

a/R=¾W/(E’a2) (A3)

SinceW/a2isbydefinitionaveragenormalstressonthecontact,thencombiningeq.(A2)and(A3)yields.

/a=¾/E’. (A4)

Completesolutionsshowthatthemaximumshearstresswithintheasperityis0.31themaximumnormalstress,andthemaximumnormalstressis3/2(Johnson,1985,Appendix3).ForcompleteflatteningofacontactofheightHandlengthL,=Handa=L/2.Theshearstressisrelatedtotheshearstrainby=2GwhereGistheshearmodulus.Therefore,

H/L=/0.62G/E’ (A5)

andtheshearstrainisproportionaltotheaspectratioasexpected.Themaximumshearstrainoccursintheinterioroftheasperityandthereforetheasperityfailsratherthanthecontactsurface.Includingfrictionattheinterfacedoessignificantlyaffecttheinternalstressfieldofthecontactingspheres(Johnson,1985;Sect.5.4).

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Similarsolutionsarerecoveredformorecomplexgeometries.Forasinusoidal,single‐wavelengthsurface,thenormalstressrequiredforcompleteflatteningoftheasperityis21/2E’H/L,whichimpliesasimilarscalingwithstrainasin(A5)(Johnsonetal.,1985).Multi‐scalemodelsalsopreservetheproportionalitybetweencalculatedstresses,strainsandaspectratio(KrithivasanandJackson,2007;Jacksonetal.,2012).

Theroleofsurfaceslopeindeterminingsurfacefailureduringshearinghaslongbeenrecognizedintribology.Forinstance,theplasticyieldcriteriaforanindentordependsonthecotangentoftheapexangle,whichistheaspectratioH/L(Johnsonetal.,1985;Sect.6.1).Asaresult,wearmechanismsarepredictedbyusingtheplasticityindex(H/L)E’/HwhereHisthehardness(Mikic,1974;Johnson,1985).Therelated

approachpresentedherefocusesontheinverseproblemofdeterminingstrengthofamaterialgiventheobservedsurfaceroughness.

References

Hertz,H.,1881,Ontheelasticcontactofsolids.J.ReineAngew.Math92,156–171.

Jackson,R.L.,Ghaednia,H.,Elkady,Y.A.,Bhavnani,S.H.,andKnight,R.W.,2012,Aclosed‐formmultiscalethermalcontactresistancemodel:IEEETransactionsonComponents,PackagingandManufacturingTechnology,v.2,no.7,p.1158–1171,doi:10.1109/TCPMT.2012.2193584.

Johnson,K.L.,1985,ContactMechanics:CambridgeUniversityPress,Cambridge.

Johnson,K.L.,Greenwood,J.A.,andHigginson,J.G.,1985,Thecontactofelasticregularwavysurfaces:InternationalJournalofMechanicalSciences,v.27,no.6,p.383–396,doi:10.1016/0020‐7403(85)90029‐3.

Krithivasan,V.,andJackson,R.L.,2007,Ananalysisofthree‐dimensionalelasto‐plasticsinusoidalcontact:TribologyLetters,v.27,no.1,p.31–43,doi:10.1007/s11249‐007‐9200‐6.