设计英文翻译

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) where̅isthedimensionofthecomplementofthedamagearea,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,andtheirrelationisd¼3).SubstitutingEq.(3)intoEq.(1),thefractaldamagevariablecanbewrittenas(note:incases,de=ω̅，)=1(̅

)

Accordingtothedefinitionofdamagevariables,thejointstiffness̅canbeexpressedintermsofthedamagevariableω(̅，ζ),as̅=K0(1−ω(̅，ζ)) SubstitutingEq.(4)intoEq.(5),thejointstiffnessofdamagedrockjointscanberewritten̅= 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

−

=̅̅̅

Where̅andKyarethejointnormalstiffnessandshearstiffness,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∁s1sinx̅̅̅̅ x(sinα1+FR1sinα1+FR2cosβ1)−(FT1sinα2+FT2cosβ2) s1sin2α1−FR1s1sin2α1−FR2∁s2cos (iͥρ

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

= ，

=2̅/ϕZ

)

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

])/(ϕZ)

2K0(̅)ζ(̅−3)= (ϕZ) = ̅ ̅ 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).Inordertorecordthestresswavescontinuouslyandaccurayandtoavoidtheinfluenceofpulsesreflectedfromtheandthecontactedends,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),strainrateε̇andstrainεwithinthespecimenasσ(t)=EA[ε(t)+ε(t)+ε ∫ε(t)= t[ε(t)−ε(t)−ε∫

l0 ε̇(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-damagedvolume̅ofthezonehavetobedetermined.FromFig.1,V0canbewrittenasV0=πr2(hmax− Wherehmaxandhminarethe umandminimumheightofthejointsurfacescannedbythelaserprofilometer,respectively,risthespecimenradius.Theno-damagedvolume̅canbewrittenas̅=∑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),onecanhave̅

∑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,(.大学水利水电学院,中国60065;2.襄樊学院土木工程系,中国襄樊44053;3.大学建筑与环境学院,中国60065):为探讨应力波穿越不规则节理时的透反射规律,基于分形损伤理论建立分形损伤节理模型，在此基础上，根据分形损伤节理模型，推导应力波穿越节理时透反射系数,SHPB节理岩石冲击动力学试验解与试验结果进行对比分析.SHPB试验的结果和应力波穿越分形节理的解析构面面由于承受了动力负荷使得岩体材料的强度和稳定性都有所减小。这些动力负荷包如今,为了研究节理面对应力波的和能量耗散[–8]的影响建立了大量的关于节理方向[9],[0,]以及是否干性节理等等。关于波在节理中的课题，Cai和Zhao[4]探讨了多分形节理面对岩层中应力波衰减的影响。Fanetal.[6]曾提出过使用离散单元法(DEM)对在节理发育的岩体中波的和衰态相互作用的相关问题。Pyrak−Nolte[2]、Zhao和Cai[3]研究过线性变形节理中应力波的特性，并且推导出应力波的透反射系数。Wangetal.[4]借鉴岩石损伤力学思想，基值角度与Zhao和Cai[3]的研究结论完全一致。Kahraman[6]模拟节理面的粗糙性，并探Bamford于978年节理粗糙度系数JRC(jointroughnesscoefficient)可以描述节理粗糙度对应力波的JRC只是一个经验参数，且仅能用来描述曲线的粗糙度对于空间曲面的粗形下,有必要引入另外一个参数,分形维数,粗糙度8–20]，并且应用于此项研究之中。本文将尝试推导应力波穿越节理时透反射系数的解析解。结合分形理论[2–23]和损伤SHPB试验设备对应力波穿越单一岩石节理的透反射规律进行试验研究。当应力波穿越节理面，由于界面面一般比较粗糙所以很难判断其入射角度。而且，应力波的振幅参数和透反射方向的理论推导也十分在下文中，利用分形理论依据材料损伤演化具有分形性质的基本事实Xie和Ju[8]结合损伤力学原理与分形理论，给出了一种兼顾损伤细观特征描述和宏观损伤力学分析需要的分形损伤变量ω(̅，ζ)，0ω(̅，ζ)=1−(1−ω （0：̅3；ω0为表观损伤变量，可表示为 ω0S

式中：S0为承载面的初始截面积，即二维欧氏空间中各区域的表观面积，S为损伤后的有效截面积，̅是无损伤的有效截面积，并且有S0=S+̅的关系。ω(̅,æ)，已经[18]。可是等式(2)中的变量ω0，只有考虑到假设损伤发生在表面并且损伤的深度忽略不计。但是当损伤却是发生在不同的深处时必将一些错误。图显示了含两种不同损伤深度的损伤区域。很明显如果应用等式(2)计算ω0，对于下ω=V-̅

式中：V0V̅是无损伤体积，并且有V0V+̅的关系。把式(3)代入式()中，分形损伤变量可以表示为（3维计算中，de=3）̅ω(̅，ζ)ù(̅，ζ)̅=K0(1−ù(̅，ζ))

̅=

̅

K02显示了两种岩石样本，H表示有节理样本的高度，H-∆表示无节理样本的高度，∆δj-δP，2.2.3表示一个粗糙节理和它的等效平直节理；即平直节理的节理刚度等效于此粗糙节2种可以分别解耦地处理外，在斜入射情况下，为满足给定的边界条件，则不论入射波u0=A0exp[i(Kx0x+ky0y−ωt)]u1=A1expu2=A2exp[i(Kx2x+ky2y−ωt)]u3=A3exp[i(Kx3x+ky3y−ωt)]u4=A4exp

射横波质点位移；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)中的下标，2表示区、2 =(ë+2ì)∂ux+λ

xy=μ(∂x

(式中μ是切变模数。.依据L.J.Pyrak-Nolte[2]线弹性位移不连续模型可知，在节理处的，2区内应力、位移应满足应力连续、位移间断的假定，即(σxx)1= (11

=

(

−

=̅̅̅

(y(uy1

−

=̅̅̅

(式中：和Ky =A1， =

( =A3， =

( 式中FR1为反射纵波的反射系数，FR2为反射横波的反射系数，FT1为透射纵波的透射系数，FT2为透射横波的透射系数。当应力波在弹性介质中时，应力波波速与弹Pλ+2μ= (PSμ= (Sλ=ρ(∁2− ( λ= CP(7)到式(9)可以推导出下列

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∁̅̅̅