StudyonthetransmissionandreflectionofstresswavesYexueLia,b,ZhemingZhuc,n,BixiongLic,JianhuiDenga,HeCollegeofWaterResource&Hydropower,SichuanUniversity,Chengdu610065,DepartmentofCivilEngineering,XiangfanUniversity,Xiangfan,Hubei441053,CollegeofArchitectureandEnvironment,SichuanUniversity,Chengdu61005,：Inordertoinvestigatethetransmissionandreflectionofstresswavesacrossjoints,afractaldamagejointmodelisdevelopedbasedonfractaldamagetheory,andtheyticalsolutionforthecoefficientsoftransmissionandreflectionofstresswavesacrossjointsisderivedfromthefractaldamagejointmodel.Thefractalgeometricalcharacteristicsofjointsurfacesareinvestigatedbyusinglaserprofilometertoscanthejointsurfaces.ThedynamicexperimentsbyusingSplitHopkinsonPressureBar(SHPB)forrockspecimenswithsinglejointareconductedtoconfirmtheyticalsolutionofthecoefficientsoftransmissionandreflection.TheSHPBexperimentalresultsofthecoefficientsagreewellwiththeyticalsolution.:StresswaveTransmissionandreflectionJointFractaldimensionDamageAsiswellknown,rocksusuallycontaindefectsordiscontinuities,suchasjoints,cracks,poresandfaults.Thesediscontinuitiesweakenrockmaterialstrengthandstabilityastheyaresubjectedtodynamicloads,suchasblastsandearthquakes,therefore,thestabilityofgeotechnicalstructures,suchastunnels,subways,dams,etc.,wouldbecontrolledbythesediscontinuities.Thedesignersofsuchgeotechnicalstructuresarerequiredtohavetheknowledgeofwavepropagationacrossjoints,andtherefore,thecorrespondingstudyontransmissionandreflectionofwavesacrossjointsissignificant.Currently,alargenumberoftheoretical,empiricalandnumericalmodelsconcerningthegeometricalandmechanicalpropertiesofjointshavebeenestablishedtostudythejointeffectsonwavepropagationandattenuation[1–8].However,ourunderstandingoftheprocessoftransmissionandreflectionofstresswavescrossingjointsisfarfromcomplete,asjointsarestillcomplexforus.Whenstresswavesencounterjoints,theywillsufferpartialreflectionandtransmission.Thecharacteristicsofreflectionandtransmissiondependonmanyfactors,suchas,jointspacing,jointorientation[9],jointwidth,jointroughness,jointstiffness,fillingmaterialinsidejoints[10,11],liquidsaturation,etc.Fortheissueofwavepropagationacrossjoints,CaiandZhao[4]haveinvestigatedtheeffectsofmultipleparallelfracturesonapparentattenuationofstresswaveinrockmasses.Fanetal.[6]havepresentedthestudyonwavepropagationandwaveattenuationinjointedrockmassesbyusingdiscreteelementmethod(DEM).GulyayevandIvanchenko[8]haveinvestigatedtheproblemaboutdynamicinctionofdiscontinuouswaveswithbetweenanisotropicelasticmedia.Perak-Nolte[12]andZhaoandCai[13]haveinvestigatedstresswavepropagationacrosslineardeformationjoints,andobtainedthecoefficientsofstresswavetransmissionandreflection.Wangetal.[14]havederivedthecoefficientsoftransmissionandreflectionofstresswavecrossingnon-linearlydeformationaljointsbyusingtherockdamagemechanicstheory[15]andstrainequivalenthypothesis,andtheresultofthecoefficientsagreedwellwiththenumericalresultspresentedbyZhaoandCai[13].Kahraman[16]hassimulatedjointsurfaceroughnessandinvestigatedtherelationbetweentheroughnessofjointsurfacesandtheparticlevelocityofstresswaves.Thesurfacesofnaturaljointsdevelopedduringalongtimeofgeologicaldigenesisarenotsmooth,andactuallytheyareveryrough.AcoefficientofjointroughnessJRC(jointroughnesscoefficient),proposedbyBamford[17]in1978,somehowcanexpresstheeffectofjointroughnessonstresswavepropagation.However,JRCisanempiricalparameter,anditcanonlyexpresstheroughnessofone-dimensionalcurveandcannotbeemployedtodescribethenaturaljointroughnesspreciselybecausethenaturaljointscontainthree-dimensionalroughsurfaces,inwhichthedimensionshouldbegreaterthan2andlessthan3.Asstresswavescrossjoints,itisverydifficulttodeterminetheanglebecausethejointsurfacesusuallyconsistofvarioussubsurfaceswithdifferentnormaldirections.Underthisscenario,itisnecessarytouseanotherparameter,fractaldimension,todescribethejointroughness.Fractaldimensionscanbeappliedtodescribingthejointroughness[18–20]anditwillbeemployedinthisstudy.Inthispaper,anattemptismadetoobtaintheyticalsolutionforthecoefficientsoftransmissionandreflectionofstresswavesacrossjoints.Thefractaltheory[21–23]combiningwithdamagemechanicstheorywillbeappliedtocharacteringthejointroughness.Thefractalgeometricalcharacteristicsofjointsurfacesareinvestigatedbyusinglaserprofilometer.Finally,byusingSplitHopkinsonPressureBar(SHPB)testingsystem,experimentalinvestigationsforthetransmissionandreflectionofstresswavesacrossasinglerockjointareconducted.TheoreticalWhenstresswavesencounterjoints,itisverydifficulttodeterminetheanglebecausethejointsurfacesusuallyareveryrough,andtherefore,theparametersofamplitudeandorientationofthetransmittedandreflectedwavescannotbedetermined.Inthefollowing,wewillinvestigatethefactualityofjointsurfacesbyfractaltheoryandderivethecoefficientsoftransmissionandreflectionofstresswavesacrossjoints.JointBasedonthefactthattheevolutionofrockdamaginghasthecharacteristicoffractal,XieandJu[18]havedefinedafractaldamagevariableω(?，ζ)thatnotonlycanexpressthedamageintrinsicmechanismtativelyintwo-dimensionalEuclideanspaces,butalsocanbeeasilyadoptedinmacro-scaledamagingysisbyusingthetheoryofdamageandfractalω(?，ζ)=1?(1?ω0)ζ(??de) where?isthedimensionofthecomplementofthedamagearea,ζisthemeasurementdeistheEuclideandimension,in2Dcaseitequals2,andin3Dcaseitis3,andω0isanominaldamagevariableandisdefinedas 00ω0S-00

whereS0isaninitialareaofazone(thenominalareainEuclideanspace),Sisthedamagedareainthezone,?istheno-damagedareainthezone,andtheirrelationisS0=S+?.UsingEsq.(1)and(2),onecanobtainthedamagevariableω(?,ζ),andthecorrespondingtheoreticalandexperimentalstudyhasbeenpresentedinreferences[18].However,forthevariableω0expressedinEq.(2),onlythedamageoccurredinsurfaceneshasbeenconsidered,andthedamagealongdepthisignored.ThiswillsomehowcauseerrorasthedamageactuallyvariesalongFig.1showsadamagedzonewithtwodifferentdamagedepths.ItcanbeseenthatifoneusesEq.(2)tocalculateω0,forthetwocases,theresultswillbethesame.However,becausethedamagedepthsaredifferent,thedamageextentsshouldbedifferenttoo.Inordertoaccurayandeffectivelydescriberockdamage,itisnecessarytousetheratioofvolumestorecetheratioofareasinEq.(2),thatisω=V=1-?

WhereV0istheinitialvolumeofazone(thenominalvolumeinEuclideanspace),Visthedamagedvolumeinthezone,?istheno-damagedvolumeinthezone,andtheirrelationisd?3).SubstitutingEq.(3)intoEq.(1),thefractaldamagevariablecanbewrittenas(note:incases,de=ω?，)=1(?

)

Accordingtothedefinitionofdamagevariables,thejointstiffness?canbeexpressedintermsofthedamagevariableω(?，ζ),as?=K0(1?ω(?，ζ)) SubstitutingEq.(4)intoEq.(5),thejointstiffnessofdamagedrockjointscanberewritten?= Fig.2showstworockspecimens;theheightforthespecimenwithjointisH,andtheheightforthespecimenwithoutjointisH-?,where?isthejointwidth.Underuniaxialcompression,thediscementsofbothspecimens(δjforthespecimenwithjoint,andδforthespecimenwithoutjoint)canbeeasilymeasured,andthedifference,δj-δ,isthejointdiscement,thus,therelationofloadsPversusjointdiscements,δj-δ,canbeobtained,andthejointnormalstiffnesscanbeCoefficientsofwavetransmissionandAsstresswavesencounterjoints,theywillsufferpartialreflectionandtransmission.Inthispaper,byusingthefractaldamagetheoryandcombiningwiththeformeryticalsolutionofwavepropagationacrossstraightjoints,thecoefficientsofwavetransmissionandreflectionacrossjointswillbederived.Fig.3showsaroughjointanditscorrespondingequivalentstraightjoint;thejointstiffnessofthestraightjointissupposedtobethesameasthatoftheroughjoint.ThepropagationofPwavesandSwavesareindependent,buttheyaregenerallymutuallyrelativewheneitherPwaveorSwavetransmittedandreflectedonjoints.Exceptforthecasethatawaveisnormallyprojectedontheinterface,thatis,normalincidence,twokindsofwavescouldgothroughuncoupledsolution.Forstresswavesobliquelytojoints,inordertomeetboundarycondition,whetherPwavesorSwavesaresuretosimultaneouslyreflectPwavesandSwaves,andtransmitPwavesandSwaves.ConsideringthecasethatonlyanePwaveonthestraightjointinterface,whichwillproducessimultaneouslyareflectedPwaveandSwave,andatransmittedPwaveandSwaveasshowninFig.3.Foreachofthesefivewaves,thereisacorrespondingwavepropagationequation,thus,therearetotallyfiveequationswhichcanbeexpressedasfollows[24]:u0=A0u1=A1expu2=A2exp[i(Kx2x+ky2y?ωt)]u3=A3exp[i(Kx3x+ky3y?ωt)]u4=A4exp

where{u0，u1，u2，u3，u4}arethediscementsinducedbythe Pwave,reflectedSwave,transmittedPwaveandtransmittedSwave,respectively,{A0,A1,A2,A3,A4}arethecorrespondingwaveamplitudes,respectively,andωandkaretheangularfrequencyandwavevector,respectively.ThediscementcomponentsinzoneIandzoneIIshowninFig.3canbeexpressed(ux)1=u0cosα1?u1cosα1+u2sinyy

)=010

sin

+

sin

+

(ux)2=u3cosα2?u4siny{(u)=usinα+ucosy

1whereα1，α2，β1andβ2aretheanglesshowninFig.3,1

2thediscementcomponentsalongxandyaxialdirection,respectively,thesubscripts1and2inEsq.(8)and(9)representzoneIandzoneII,respectively.Supposingthedeformationsinducedbythestresswavesareintheelasticrange,then,therelationbetweendiscementandstresscanbewrittenas2 =(λ+2μ)?ux+λ =μ?uy+ ( WhereμisshearmodulusAccordingtothediscementdiscontinuitymodelproposedbyPerak-Nolte[12],thestressesonthecommonboundaryofzoneIandzoneIIsatisfythefollowing(σxx)1= 11

=

2

(ux)1?(ux)2=???? y(uy1

?

=???

Where?andKyarethejointnormalstiffnessandshearstiffness,respectively.Thecoefficientsoftransmissionandreflectionaredefinedas =A1，

=

=A3，

=

WhereFR1andFR2arethereflectioncoefficientsofthereflectedPwaveandSwave,respectively,andFT1andFT2denotethetransmissioncoefficientsofthetransmittedPwaveandSwave,respectively.Asstresswavespropagateinanelasticbody,therelationoftheparameterscanbeexpressedasPλ+2μ= PSμ= Sλ=ρ(?2? Whereλ=

,νisthePoisson’sratio,CpisthePwavespeed,CsistheSwaveμistheshearmodulusandρisthedensity.FromEsp.(7)to(19),weobtainthefollowing

s1sin2α1?FR1s1sin2α1?FR2?s1cos

?2[FT1s2sin2α2+FT2?s2cos

ρ

(?p1?sin2α1s1)(1+FR1)?FR2?s1sin2β?2[FT1(?p2?sin2α2s2)

FT2?s2sin2β2]= (cosα1?FR1cosα1+FR2sinβ1)?(FT1cosα2?FT2sinβ2)?(?p1?sin2α1s1)(1+FR2)?FR2?s1sinx???? x(sinα1+FR1sinα1+FR2cosβ1)?(FT1sinα2+FT2cosβ2) s1sin2α1?FR1s1sin2α1?FR2?s2cos (i?ρ

Where?arewaveangularfrequency,andthesubscripts1and2intheaboverepresentzoneIandzoneII,respectively.Fornormalincidence,i.e.thePwaveisperpendiculartothejointsurfaces,onecanhaveα1=α2=β1=β2=0.Iftherockspropertiesbothsidesofthejointarethesame,thenthedensityofrockmassandthevelocityofstresswaveatbothsidesareidentical,thus,fromEsq.(20)to(23),weobtainthecoefficientsofthereflectedPwaveandthetransmittedPwaveas

= ，

=2?/?Z

)

)Where:Zisthewaveimpedance.SubstitutingEq.(6)intoEq.(24),weobtaintheyticalsolutionofthecoefficientsoftransmissionandreflectionasstresswavescrossjoints ?(??3)

])/(?Z)

2K0(?)ζ(??3)= (?Z) = ? ? V√1+4((K0[(VV0

ExperimentInordertovalidatethetheoreticalsolution,i.e.Eq.(25),dynamicexperimentalstudyforrockspecimenswithasinglejointbyusingSplitHopkinsPressureBar(SHPB)isconducted,andbyusingalaserprofilometer,thejointsurfaceshavebeenscanned,andthejointsurfacefractaldimensionshavebeenmeasured,andtheresultshavebeenemployedtocalculatethecoefficientsoftransmissionandreflectionofwavesacrossjoints.SHPBtestingSplitHopkinsPressureBariswidelyusedforcharacterizingdynamicresponseofengineeringmaterials.TheSHPBsystemusedinthisstudyisshowninFig.4.ThelengthsofthebarandtransmissionbaroftheSHPBarethesame,2.0m,andandthefrictionbetweentheendsofthespecimensandtheinputandoutputbars,weappliedasinglejointedspecimenwith12mminlengthand30mmindiameter.Theimpactwavesweregeneratedandpropagatedalongtheaxialdirectionsofthespecimencylinders,andperpendicularlyprojectedtothejointsurfaces,whichislocatedinthemiddleofthe12mmlongspecimen.Theobjectiveofthisexperimentalstudyistoinvestigatetheeffectofjointsurfaceconfigurationonwavereflectionandtransmission,soastovalidatethetheoreticalresultsofthecoefficientsofwavereflectionandtransmission.Inordertominimizethesideeffectoflargesticdeformationandadditionalcracking,theimpactstrikerspeedshouldbecontrolledwithinacertainrange,suchthatthesticityandcrackinginthespecimeninducedbytheimpactwerenegligible,thus,noirreversibleenergywilldissipateexceptforthediscementofjointsurfaceduringthewavepropagation.AccordingtothepreliminarySHPBtestsusingintactrockswithdifferentimpactspeed,theimpactspeedof6.8m/swasadoptedfinallyintheSHPBtests.Thewave,reflectedwaveandtransmittedwavearecollectedbysuperdynamicapparatus.Stressgaugeswereemployedinrecordingthestresswaves,andweresituatedinthemiddleoftheandtransmissionbars,respectively(seeFig.4).Inordertorecordthestresswavescontinuouslyandaccurayandtoavoidthein?uenceofpulsesreflectedfromtheandthecontactedends,thedistancebetweenthestraingaugesandtheendsofthebarsislargerthanthelengthofthestrikerbarwhichis200mminlength.SHPBspecimenpreparationandtestingRockcoreswithadiameterof30mmwerefirstdrilledfrommarbleblocks.Toavoidthewereselectedtomakerockspecimens.Bythree-pointbendingmethod,theselectedrockcoreswerefracturedintotwoparts,andthesetwopartswereadheredtogetherandbycuttingthetwoendsofthecore,a12mmlongspecimenwithajointinthemiddle,showninFig.5,wasmade.Theendsofthespecimenshavebeengroundverycarefullysuchthatthefrictionlessconditioncanbesatisfiedbeforeweinstalledthespecimenbetweenthesteelbars.Becausethespecimenlength12mmisveryshortascomparedtothebar,thestrainsonthetwoendsareapproximaythesameinaveryshorttimeinterval,whichcanmeetthehypothesisofstrainuniformity.Becausethelengthofthespecimenssatisfiestherequirementofh=√3νr,whereristheradiusofthespecimencylindersandnisthedynamicPoisson’sratiooftherock,stresswavespropagatingalongthespecimencanbeconsideredasone-dimensionalstresswave[25–27].Therefore,intheseteststhestraininsidethespecimenwasone-dimensionalanduniformelasticstrain.Basedontheelasticwavetheory,onecanexpressthehistoriesofstressσ(t),strainrateε?andstrainεwithinthespecimenasσ(t)=EA[ε(t)+ε(t)+ε ∫ε(t)= t[ε(t)?ε(t)?ε∫

l0 ε?(t)=C[ε(t)?ε(t)?ε whereA0andl0aretheinitialareaandinitiallengthofthespecimen,respectively,EandAaretheYoung’smodulusandtheareaofthebar,respectively,Cisthelongitudinalstresswavevelocityofthebar.Thetiesindicatingbythesubscripti,randtrefertothe,reflectedandtransmittedwave,respectively.ByusingtheaboveSHPBtestingsystemandthespecimens,thecurvesofstrainversustimeforthe,reflectedandtransmittedwavescanbemeasuredandtheresultscanbecollectedautomatically.Fig.6showsthreetypicalcurvesofstrainversustimeforthespecimenNo.10,No.13andforthespecimenwithoutjoint.Thecorrespondingcoefficientsoftransmissionandreflectionofstresswavesacrossthejointscanbecalculated.Fig.6showsthatbothamplitudesofthereflectedandtransmittedwavesarelessthanthoseofthewaves.TheamplitudesoftransmittedwavesinFig.6aregreaterthanthoseofthereflectedwaves,whichindicatesthatmostoftheenergyofthewavehastransmittedthejointsandonlylessenergyhasreflectedbacktothebar.Forthespecimenwithoutjoint,theamplitudeofitstransmittedwaveislargerthanthoseforthespecimenswithjoints,andtheamplitudeofitsreflectedwaveislessthanthoseforthespecimenswithjoints.Thisistobeexpected,asjointscanblockwavepropagation.FractalitymeasurementofjointAlarge-scalelaserprofilometerisappliedtoscanningtherockspecimensurfacesbeforeSHPBtests.Thescanningcapacityofthelaserprofilometerscopesupto30mmwitharesolutionof7mmandaminimumscanningincrementof7.5mm.Anadvancedsurfacemethod,non-contactsurfacemeasurementmethod,whichcanavoidcuttingordamagingthetargetsurfacebytheprobeduringscanningtest,wasemployed.Intotal,thirteenjointsurfacesofrockspecimensweretestedbyusingthelaserprofilometer,andthethree-dimensionalcoordinatesoftheroughsurfaceswerecollected.Thefractaldimensionsoftheroughsurfacesarecalculatedbyapplyingthefractalprojectivecoveringmethod[20–22],andthespecimensurfaceprofilesaswellasthecorrespondingbi-logarithmrelationofthecoveringboxnumberversustheboxscaleforfourtypicalmarblespecimensbeforeSHPBtestsareshowninFig.7,whereDisthedimensionofthespecimensurfaces.Fig.7showsthattheerrorofthelinearfitbetweenthelogarithmofcoveringboxscaleandthelogarithmofthenumberofboxesusedforcoveringthejointsissmall.ForspecimenNo.6,theerroris4.3%whichistheumerrorinallbi-logarithmplots,whereasforspecimenNo.13,theerrorisonly0.99%.Fromthefourjointconfigurationsandthecorrespondingbi-logarithmplotsshowninFig.7,itcanbeseenthatthefractaldimensionincreaseswiththeroughnessofthejointsurfaces,thatis,therougherthejointsurface,thehigherthefractaldimension.TheoreticalandexperimentalresultsofthecoefficientsoftransmissionandEq.(25)canbeemployedtocalculatethecoefficientsoftransmissionandreflectionofwavesacrossjoints.BeforeusingEq.(25),theinitialvolumeV0ofazoneandtheno-damagedvolume?ofthezonehavetobedetermined.FromFig.1,V0canbewrittenasV0=πr2(hmax? Wherehmaxandhminarethe umandminimumheightofthejointsurfacescannedbythelaserprofilometer,respectively,risthespecimenradius.Theno-damagedvolume?canbewrittenas?=∑m

+ +

?

j=1[4(hi,j+

Wherehistheheightofatargetpointscannedbythelaserprofilometer,mandnarethenumbersofthetargetpointsscannedalongxandyaxis,respectively,andhmin=min(hi,j+hi+1,j+hi,j+1+hi+1,j+1)FromEqs.(27)And(28),onecanhave?

∑m∑n + + +

?

Table1showsrockparametersmeasuredinthisexperimentalstudy.Thevelocityofstresswavewasmeasuredbyanultrasonoscope.SubstitutingtheparametersinTable1andEq.(29)intoandthecorrespondingtheoreticalresultsarepresentedinTable2.FromtheSHPBtestingresultsofthecurvesofstrainversustimeasshowninFig.6,onecaneasilyobtainthecoefficientsoftransmissionandreflectionofstresswavesacrossjoints,andtheSHPBtestingresultsarealsopresentedinTable2forcomparison.Itcanbeseenthatforthetransmissioncoefficients,theaveragepercentdifferencesbetweenthetheoreticalandexperimentalcoefficientsisonly1.14%,andforthereflectioncoefficients,itis5.42%.ThisindicatesthatthecoefficientsoftransmissionandreflectionpredictedbyEq.(25)arereliable,andaccordinglyEq.(25)iseffective.Fig.8showsthecurvesofthetheoreticalandtheSHPBtestingresultsforthecoefficientsofreflectionandtransmissionasstresswavescrossjoints.Itcanbeseenthatwiththeincreaseofthefractaldimensionofjointsurfaces,thetransmissioncoefficientsofstresswavedecrease,andthecorrespondingreflectioncoefficientsincrease.Thisiseasytounderstandbecausethelargerthefractaldimensionis(ortherougherthejointsurfaceis),thelargerthejointreflectionis.FromFig.8,itcanbeseenthatgenerally,thetheoreticalresultsofthecoefficientsoftransmissionandreflectionagreewellwiththeSHPBtestingresults,andtheerrorbetweenthemisverysmall,whichindicatesthatthefractaldamagemodelofjointspresentedinthispaperiseffective,andtheyticalsolutionforthecoefficientsoftransmissionandreflectionofwavesacrossjointsisreliable.Afractaldamagejointmodelhasbeendeveloped,andtheyticalsolutionEq.(25)forthecoefficientsoftransmissionandreflectionofstresswavesacrossjointshasbeenestablishedbyusingthefractaldamagejointmodel.Itisshownthatasthefractaldimensionofjointsurfacesincreases(orastheroughnessofjointsurfacesincreases),thetransmissioncoefficientsdecrease,whereas,thereflectioncoefficientsincrease.ASHPBtestingsystemhasbeenemployedtomeasurethecoefficientsoftransmissionandreflectionforrockspecimenswithasinglejoint.ItisshownthattheSHPBtestingresultsagreewellwiththetheoreticalresultspresentedinthispaper,whichindicatesthatthefractaldamagemodelofjointsdevelopedinthisPaskaramoorthyR,DattaSK,ShahAH.Effectofinterfacelayersonscatteringofelasticwaves.JApplMech1988;55:871–8.ChevalierY,LouzarM,MauginGA.Surfacewavecharacterizationoftheinterfacebetweentwoanisotropicmedia.JAcoustSocAm1991;90:3218–27.ShindoY,NiwaN.Scatteringofantineshearwavesinafiber-rein dcompositemediumwithinterfaciallayers.ActaMech1996;117:181–90.CaiJG,ZhaoJ.Effectsofmultipleparallelfracturesonapparentattenuationofstresswavesinrockmasses.IntJRockMechMinSci2000;37:661–82.KingM.Elasticwavepropagationandpermeabilityforrockswithmultipleparallelfractures.IntJRockMechMinSci2002;39:1033–43.FanSC,JiaoYY,ZhaoJ.Onmodelingof boundaryforwavepropagationinjointedrockmassesusingdiscreteelementmethod.ComputGeotech2004;31:57–66.KrasnovaT,JanssonP,BostromA.Ultrasonicwavepropagationinananisotropiccladdingwithawavyinterface.WaveMotion2005;41:163–77.GulyayevVI,IvanchenkoGM.Discontinuouswavein ctionwithinterfacesbetweenanisotropicelasticmedia.IntJSolidsStruct2006;43:74–90.GongQM,JiaoYY,ZhaoJ.NumericalmodellingoftheeffectsofjointspacingonrockfragmentationbyTBMcutters.TunnellingUndergroundSpaceTechnol2006;21:46–55.ZhuZ,ntyB,XieH.Numericalinvestigationofblasting-inducedcrackinitiationandpropagationinrocks.IntJRockMechMinSci2007;44:412–24.ZhuZ,XieH,ntyB.Numericalinvestigationofblasting-induceddamageincylindricalrocks.IntJRockMechMinSci2008;45:111–21.Pyrak-NolteLJ.Seismicvisibilityoffractures.Ph.D.Thesis,UnivCalif,Berkeley;ZhaoJ,CaiJG.TransmissionofelasticP-waveacrosssinglefractureswithanonlinearnormaldeformationalbehavior.RockMechRockEng2001;34:3–22.WangWH,LiXB,ZuoYJ.Effectofnon-linearlynormaldeformationaljointonelasticp-wavepropagation.ChinJRockMechEng2006;25:1218–25.[in DougillJW,LauJC,BurtNJ.MechanicsinEnging.ASCE,EMDKahramanS.TheeffectsoffractureroughnessonP-wavevelocity.EngGeolBamfordWE.Suggestedmethodsforthetatived ptionofdiscontinuitiesinrockmasses.IntJRockMechMinSci1978;15:319–69.XieH,JuY.Astudyondamagemechanicstheoryinfractionaldimensionalspace.ActaMech1999;31:300–10[in JuY,SudakL,XieH.Studyonstresswavepropagationinfracturedrockswithfractaljointsurfaces.IntJSolidsStruct2007;44:4256–71.XieH,WangJA.Directfractalmeasurementoffracturesurfaces.IntJSolidsStructMandelbrotBB.HowlongisthecoastlineofBritain?Statisticalself-similarityandfractaldimensionScience1967:155–636.ZhouHW,XieH.Directestimationofthefractaldimensionsofafracturesurfaceofrock.SurfRevLettMandelbrotBB.TheFractalGeometryofNature.New man;Pyrak-NolteLJ,MyerLR,CookNGW.Transmissionofseismicwavesacrosssinglenaturalfractures.JGeophysRes1990;95(6):8617–38.DaviesEDH,HunterSC.ThedynamicscompressiontestingofsolidsbythemethodsofthesplitHopkinsonpressurebar.JMechPhysSolids1963;11:155–79.LifshitzJM,LeberH.DataprocessinginthesplitHopkinsonpressurebartests.IntJImpactEng LindholmUS.SomeexperimentswiththesplitHopkinsonpressurebar.JMechPhysSolids

1,(.大學水利水電學院,中國60065;2.襄樊學院土木工程系,中國襄樊44053;3.大學建筑與環境學院,中國60065):為探討應力波穿越不規則節理時的透反射規律,基于分形損傷理論建立分形損傷節理模型，在此基礎上，根據分形損傷節理模型，推導應力波穿越節理時透反射系數,SHPB節理巖石沖擊動力學試驗解與試驗結果進行對比分析.SHPB試驗的結果和應力波穿越分形節理的解析構面面由于承受了動力負荷使得巖體材料的強度和穩定性都有所減小。這些動力負荷包如今,為了研究節理面對應力波的和能量耗散[–8]的影響建立了大量的關于節理方向[9],[0,]以及是否干性節理等等。關于波在節理中的課題，Cai和Zhao[4]探討了多分形節理面對巖層中應力波衰減的影響。Fanetal.[6]曾提出過使用離散單元法(DEM)對在節理發育的巖體中波的和衰態相互作用的相關問題。Pyrak?Nolte[2]、Zhao和Cai[3]研究過線性變形節理中應力波的特性，并且推導出應力波的透反射系數。Wangetal.[4]借鑒巖石損傷力學思想，基值角度與Zhao和Cai[3]的研究結論完全一致。Kahraman[6]模擬節理面的粗糙性，并探Bamford于978年節理粗糙度系數JRC(jointroughnesscoefficient)可以描述節理粗糙度對應力波的JRC只是一個經驗參數，且僅能用來描述曲線的粗糙度對于空間曲面的粗形下,有必要引入另外一個參數,分形維數,粗糙度8–20]，并且應用于此項研究之中。本文將嘗試推導應力波穿越節理時透反射系數的解析解。結合分形理論[2–23]和損傷SHPB試驗設備對應力波穿越單一巖石節理的透反射規律進行試驗研究。當應力波穿越節理面，由于界面面一般比較粗糙所以很難判斷其入射角度。而且，應力波的振幅參數和透反射方向的理論推導也十分在下文中，利用分形理論依據材料損傷演化具有分形性質的基本事實Xie和Ju[8]結合損傷力學原理與分形理論，給出了一種兼顧損傷細觀特征描述和宏觀損傷力學分析需要的分形損傷變量ω(?，ζ)，0ω(?，ζ)=1?(1?ω （0：?3；ω0為表觀損傷變量，可表示為 ω0S

式中：S0為承載面的初始截面積，即二維歐氏空間中各區域的表觀面積，S為損傷后的有效截面積，?是無損傷的有效截面積，并且有S0=S+?的關系。ω(?,?)，已經[18]。可是等式(2)中的變量ω0，只有考慮到假設損傷發生在表面并且損傷的深度忽略不計。但是當損傷卻是發生在不同的深處時必將一些錯誤。圖顯示了含兩種不同損傷深度的損傷區域。很明顯如果應用等式(2)計算ω0，對于下ω=V-?

式中：V0V?是無損傷體積，并且有V0V+?的關系。把式(3)代入式()中，分形損傷變量可以表示為（3維計算中，de=3）?ω(?，ζ)ù(?，ζ)?=K0(1?ù(?，ζ))

?=

?

K02顯示了兩種巖石樣本，H表示有節理樣本的高度，H-?表示無節理樣本的高度，?δj-δP，2.2.3表示一個粗糙節理和它的等效平直節理；即平直節理的節理剛度等效于此粗糙節2種可以分別解耦地處理外，在斜入射情況下，為滿足給定的邊界條件，則不論入射波u0=A0exp[i(Kx0x+ky0y?ωt)]u1=A1expu2=A2exp[i(Kx2x+ky2y?ωt)]u3=A3exp[i(Kx3x+ky3y?ωt)]u4=A4exp

射橫波質點位移；A0，A，A2，A3，A4(ux)1=u0cosα1?u1cosα1+u2siny{(uy1

=u0sinα1+u1sinα1+u2cos (ux)2=u3cosα2?u4siny{(uy2

=u3sinα2+u1cos 1式中：α1，α2，β1和β23中所示的的角度。式中：(ux)1，(uy)1

x，y方向的位移分量。式(8)和式(9)中的下標，2表示區、2 =(?+2ì)?ux+λ

xy=μ(?x

(式中μ是切變模數。.依據L.J.Pyrak-Nolte[2]線彈性位移不連續模型可知，在節理處的，2區內應力、位移應滿足應力連續、位移間斷的假定，即(σxx)1= (11

=

(

?

=???

(y(uy1

?

=???

(式中：和Ky =A1， =

( =A3， =

( 式中FR1為反射縱波的反射系數，FR2為反射橫波的反射系數，FT1為透射縱波的透射系數，FT2為透射橫波的透射系數。當應力波在彈性介質中時，應力波波速與彈Pλ+2μ= (PSμ= (Sλ=ρ(?2? ( λ= CP(7)到式(9)可以推導出下列

s1sin2α1?FR1s1sin2α1?FR2?s1cos2β1 [FT1s2sin2α2+FT2?s2cos2β2]0

(?p1?sin2α1s1)(1+FR1)?FR2?s1sin2β1

[FT1(?p2?sin2α2s2) FT2?s2sin2β2]= (2

(cosα1?FR1cosα1+FR2sinβ1)?(FT1cosα2?FT2sinβ2)?(?p1?sin2α1s1)(1+FR2)?FR2?s1sin????