均質(zhì)各向異性裂隙巖體的破壞特性

FailurePropertiesofFracturedRockMassesasAnisotropic

HomogenizedMediaIntroductionItiscommonlyacknowledgedthatrockmassesalwaysdisplaydiscontinuoussurfacesofvarioussizesandorientations,usuallyreferredtoasfracturesorjoints.Sincethelatterhavemuchpoorermechanicalcharacteristicsthantherockmaterial,theyplayadecisiveroleintheoverallbehaviorofrockstructures,whosedeformationaswellasfailurepatternsaremainlygovernedbythoseofthejoints.Itfollowsthat,fromageomechanicalengineeringstandpoint,designmethodsofstructuresinvolvingjointedrockmasses,mustabsolutelyaccountforsuch''weakness''surfacesintheiranalysis.Themoststraightforwardwayofdealingwiththissituationistotreatthejointedrockmassasanassemblageofpiecesofintactrockmaterialinmutualinteractionthroughtheseparatingjointinterfaces.Manydesign-orientedmethodsrelatingtothiskindofapproachhavebeendevelopedinthepastdecades,amongthem,thewell-known''blocktheory,''whichattemptstoidentifypotei-tiallyunstablelumpsofrockfromgeometricalandkinematicalconsiderations(GoodmanandShi1985;Warburton1987;Goodman1995).Oneshouldalsoquotethewidelyuseddistinctelementmethod,originatingfromtheworksofCundallandcoauthors(CundallandStrack1979;Cundall1988),whichmakesuseofanexplicitfinite-differencenumericalschemeforcomputingthedisplacementsoftheblocksconsideredasrigidordeformablebodies.Inthiscontext,attentionisprimarilyfocusedontheformulationofrealisticmodelsfordescribingthejointbehavior.Sincethepreviouslymentioneddirectapproachisbecominghighlycomplex,andthennumericallyuntractable,assoonasaverylargenumberofblocksisinvolved,itseemsadvisabletolookforalternativemethodssuchasthosederivedfromtheconceptofhomogenization.Actually,suchaconceptisalreadypartiallyconveyedinanempiricalfashionbythefamousHoekandBrown'scriterion(HoekandBrown1980;Hoek1983).Itstemsfromtheintuitiveideathatfromamacroscopicpointofview,arockmassintersectedbyaregularnetworkofjointsurfaces,maybeperceivedasahomogeneouscontinuum.Furthermore,owingtotheexistenceofjointpreferentialorientations,oneshouldexpectsuchahomogenizedmaterialtoexhibitanisotropicproperties.Theobjectiveofthepresentpaperistoderivearigorousformulationforthefailurecriterionofajointedrockmassasahomogenizedmedium,fromtheknowledgeofthejointsandrockmaterialrespectivecriteria.Intheparticularsituationwheretwomutuallyorthogonaljointsetsareconsidered,aclosed-formexpressionisobtained,givingclearevidenceoftherelatedstrengthanisotropy.Acomparisonisperformedonanillustrativeexamplebetweentheresultsproducedbythehomogenizationmethod,makinguseofthepreviouslydeterminedcriterion,andthoseobtainedbymeansofacomputercodebasedonthedistinctelementmethod.Itisshownthat,whilebothmethodsleadtoalmostidenticalresultsforadenselyfracturedrockmass,a''size''or''scaleeffect''isobservedinthecaseofalimitednumberofjoints.Thesecondpartofthepaperisthendevotedtoproposingamethodwhichattemptstocapturesuchascaleeffect,whilestilltakingadvantageofahomogenizationtechnique.ThisisachievedbyresortingtoamicropolarorCosseratcontinuumdescriptionofthefracturedrockmass,throughthederivationofageneralizedmacroscopicfailureconditionexpressedintermsofstressesandcouplestresses.Theimplementationofthismodelisfinallyillustratedonasimpleexample,showinghowitmayactuallyaccountforsuchascaleeffect.ProblemStatementandPrincipleofHomogenizationApproachTheproblemunderconsiderationisthatofafoundation(bridgepierorabutment)restinguponafracturedbedrock(Fig.1),whosebearingFig.1.Bearingcapacityoffbiindationonfracturedrockmasscapacityneedstobeevaluatedfromtheknowledgeofthestrengthcapacitiesoftherockmatrixandthejointinterfaces.ThefailureconditionoftheformerwillbeexpressedthroughtheclassicalMohr-CoulombconditionexpressedbymeansofthecohesionCandthefrictionangle.Notethattensilestresseswillbecountedpositivethroughoutthepaper.Likewise,thejointswillbemodeledasplaneinterfaces(representedbylinesinthefigure'splane).Theirstrengthpropertiesaredescribedbymeansofaconditioninvolvingthestressvectorofcomponents(o,t)actingatanypointofthoseinterfacesFjo,t=t+gtan%.—G.<0(1)Accordingtotheyielddesign(orlimitanalysis)reasoning,theabovestructurewillremainsafeunderagivenverticalloadQ(forceperunitlengthalongtheOzaxis),ifonecanexhibitthroughouttherockmassastressdistributionwhichsatisfiestheequilibriumequationsalongwiththestressboundaryconditions,whilecomplyingwiththestrengthrequirementexpressedatanypointofthestructure.ThisproblemamountstoevaluatingtheultimateloadQ+beyondwhichfailurewilloccur,orequivalentlywithinwhichitsstabilityisensured.Duetothestrongheterogeneityofthejointedrockmass,insurmountabledifcultiesarelikelytoarisewhentryingtoimplementtheabovereasoningdirectly.Asregards,forinstance,thecasewherethestrengthpropertiesofthejointsareconsiderablylowerthanthoseoftherockmatrix,theimplementationofakinematicapproachwouldrequiretheuseoffailuremechanismsinvolvingvelocityjumpsacrossthejoints,sincethelatterwouldconstitutepreferentialzonesfortheoccurrenceoffailure.Indeed,suchadirectapproachwhichisappliedinmostclassicaldesignmethods,isbecomingrapidlycomplexasthedensityofjointsincreases,thatisasthetypicaljointspacinglisbecomingsmallincomparisonwithacharacteristiclengthofthestructuresuchasthefoundationwidthB.Insuchasituation,theuseofanalternativeapproachbasedontheideaofhomogenizationandrelatedconceptofmacroscopicequivalentcontinuumforthejointedrockmass,maybeappropriatefordealingwithsuchaproblem.Moredetailsaboutthistheory,appliedinthecontextofreinforcedsoilandrockmechanics,willbefoundin(deBuhanetal.1989;deBuhanandSalenc,on1990;Bernaudetal.1995).MacroscopicFailureConditionforJointedRockMassTheformulationofthemacroscopicfailureconditionofajointedrockmassmaybeobtainedfromthesolutionofanauxiliaryyielddesignboundary-valueproblemattachedtoaunitrepresentativecellofjointedrock(BekaertandMaghous1996;Maghousetal.1998).Itwillnowbeexplicitlyformulatedintheparticularsituationoftwomutuallyorthogonalsetsofjointsunderplanestrainconditions.ReferringtoanorthonormalframeO&&whoseaxesareplacedalongthejointsdirections,andintroducingthefollowingchangeofstressvariables:TOC\o"1-5"\h\zP偵11十小")上2§=(仃隊(duì)―心"人2⑵suchamacroscopicfailureconditionsimplybecomes'山？十戶＜（-Q十\21I*in以因）\o"CurrentDocument"PWggE（-^點(diǎn)勺加旺響whereitwillbeassumedthat〃陽=匚礦而褊叫=q也叫廣Aconvenientrepresentationofthemacroscopiccriterionistodrawthestrengthenveloperelatingtoanorientedfacetofthehomogenizedmaterial,whoseunitnormalnIisinclinedbyanangleawithrespecttothejointdirection.Denotingbycandtthenormalandshearcomponentsofthestressvectoractinguponsuchafacet,itis

possibletodetermineforanyvalueofathesetofadmissiblestresses(cnnFig.2.Strengthenvelopeattachedtofacetofhomogenizedmaterial"("tan啊/#J"Jdeducedfromconditions(3)expressedpossibletodetermineforanyvalueofathesetofadmissiblestresses(cFig.2.Strengthenvelopeattachedtofacetofhomogenizedmaterial"("tan啊/#J"JTwocommentsareworthbeingmade:ThedecreaseinstrengthofarockmaterialduetothepresenceofjointsisclearlyillustratedbyFig.2.Theusualstrengthenvelopecorrespondingtotherockmatrixfailureconditionis''truncated''bytwoorthogonalsemilinesassoonasconditionH,<Hisfulfilled.Themacroscopicanisotropyisalsoquiteapparent,sinceforinstancethestrengthenvelopedrawninFig.2isdependentonthefacetorientationa.Theusualnotionofintrinsiccurveshouldthereforebediscarded,butalsotheconceptsofanisotropiccohesionandfrictionangleastentativelyintroducedbyJaeger(I960),orMcLamoreandGray(1967).NorcansuchananisotropybeproperlydescribedbymeansofcriteriabasedonanextensionoftheclassicalMohr-Coulombconditionusingtheconceptofanisotropytensor(BoehlerandSawczuk1977;Nova1980;AllirotandBochler1981).ApplicationtoStabilityofJointedRockExcavationTheclosed-formexpression(3)obtainedforthemacroscopicfailurecondition,makesitthenpossibletoperformthefailuredesignofanystructurebuiltinsuchamaterial,suchastheexcavationshowninFig.3,

Fig.3.Stabilityanalysisofjointedrockexcavationwherehandpdenotetheexcavationheightandtheslopeangle,respectively.Sincenosurchargeisappliedtothestructure,thespecificweightyoftheconstituentmaterialwillobviouslyconstitutethesoleloadingparameterofthesystem.Assessingthestabilityofthisstructurewillamounttoevaluatingthemaximumpossibleheighth+beyondwhichfailurewilloccur.Astandarddimensionalanalysisofthisproblemshowsthatthiscriticalheightmaybeputintheform布一二?”"*"而押J(4)where0=jointorientationandK+=nondimensionalfactorgoverningthestabilityoftheexcavation.Upper-boundestimatesofthisfactorwillnowbedeterminedbymeansoftheyielddesignkinematicapproach,usingtwokindsoffailuremechanismsshowninFig.4.Fia.4.FailuremechanismsusedinkinematicannroachRotationalFailureMechanism[Fig.4(a)]Thefirstclassoffailuremechanismsconsideredintheanalysisisadirecttranspositionofthoseusuallyemployedforhomogeneousandisotropicsoilorrockslopes.InsuchamechanismavolumeofhomogenizedjointedrockmassisrotatingaboutapointQwithanangularvelocityro.Thecurveseparatingthisvolumefromtherestofthestructurewhichiskeptmotionlessisavelocityjumpline.SinceitisanarcofthelogspiralofangleandfocusQthevelocitydiscontinuityatanypointofthislineisinclinedatanglewmwithrespecttothetangentatthesamepoint.Theworkdonebytheexternalforcesandthemaximumresistingworkdevelopedinsuchamechanismmaybewrittenas(seeChenandLiu1990;Maghousetal.1998)川L=層四i巾2)附在"nJ倒;，私押了部；卬】：蛀|⑴wherewandw=dimensionlessfunctions,and四]and^2=anglesspecifyingthepositionofthecenterofrotationQ.Sincethekinematicapproachofyielddesignstatesthatanecessaryconditionforthestructuretobestablewrites(6)itfollowsfromEqs.(5)and(6)thatthebestupper-boundestimatederivedfromthisfirstclassofmechanismisobtainedbyminimizationwithrespectto四1and^2K-WK件minim](7)whichmaybedeterminednumerically.PiecewiseRigid-BlockFailureMechanism[Fig.4(b)]Thesecondclassoffailuremechanismsinvolvestwotranslatingblocksofhomogenizedmaterial.Itisdefinedbyfiveangularparameters.Inordertoavoidanymisinterpretation,itshouldbespecifiedthattheterminologyofblockdoesnotreferheretothelumpsofrockmatrixintheinitialstructure,butmerelymeansthat,intheframeworkoftheyielddesignkinematicapproach,awedgeofhomogenizedjointedrockmassisgivena(virtual)rigid-bodymotion.Theimplementationoftheupper-boundkinematicapproach,makinguseofofthissecondclassoffailuremechanism,leadstothefollowingresults.嘰=U\玷心…；3皿)呼皿=。?；靺n思…；孔皿)(8)whereUrepresentsthenormofthevelocityofthelowerblock.Hence,thefollowingupper-boundestimateforK+:K-W理=min|不當(dāng)⑼ResultsandComparisonwithDirectCalculationTheoptimalboundhasbeencomputednumericallyforthefollowingsetofparameters:9=75。，0=10°,=0.1,鈿=35。，孔=20?！?wKu=min{K：,K；}=1.47Theresultobtainedfromthehomogenizationapproachcanthenbecomparedwiththatderivedfromadirectcalculation,usingtheUDECcomputersoftware(Hartetal.1988).Sincethelattercanhandlesituationswherethepositionofeachindividualjointisspecified,aseriesofcalculationshasbeenperformedvaryingthenumbernofregularlyspacedjoints,inclinedatthesameangle0=10°withthehorizontal,andintersectingthefacingoftheexcavation,assketchedinFig.5.TheFig.5,Estimatesforsrabiiityfactor:homogenizationversusdirectapproachcorrespondingestimatesofthestabilityfactorhavebeenplottedagainstninthesamefigure.Itcanbeobservedthatthesenumericalestimatesdecreasewiththenumberofintersectingjointsdowntotheestimateproducedbythehomogenizationapproach.Theobserveddiscrepancybetweenhomogenizationanddirectapproaches,couldberegardedasa''size''or''scaleeffect''whichisnotincludedintheclassicalhomogenizationmodel.Apossiblewaytoovercomesuchalimitationofthelatter,whilestilltakingadvantageofthehomogenizationconceptasacomputationaltime-savingalternativefordesignpurposes,couldbetoresorttoadescriptionofthefracturedrockmediumasaCosseratormicropolarcontinuum,asadvocatedforinstancebyBiot(1967);Besdo(1985);AdhikaryandDyskin(1997);andSulemandMulhaus(1997)forstratiedorblockstructures.Thesecondpartofthispaperisdevotedtoapplyingsuchamodeltodescribingthefailurepropertiesofjointedrockmedia.均質(zhì)各向異性裂隙巖體的破壞特性概述由于巖體表面的裂隙或節(jié)理大小與傾向不同，人們通常把巖體看做是非連續(xù)的。盡管裂隙或節(jié)理表現(xiàn)出的力學(xué)性質(zhì)要遠(yuǎn)遠(yuǎn)低于巖體本身，但是它們?cè)趲r體結(jié)構(gòu)性質(zhì)方面起著重要的作用，巖體本身的變形和破壞模式也主要是由這些節(jié)理所決定的。從地質(zhì)力學(xué)工程角度而言，在涉及到節(jié)理巖體結(jié)構(gòu)的設(shè)計(jì)方法中，軟弱表面是一個(gè)很重要的考慮因素。解決這種問題最簡(jiǎn)單的方法就是把巖體看作是許多完整巖塊的集合，這些巖塊之間有很多相交的節(jié)理面。這種方法在過去的幾十年中被設(shè)計(jì)者們廣泛采用，其中比較著名的是“塊體理論”，該理論試圖從幾何學(xué)和運(yùn)動(dòng)學(xué)的角度用來判別潛在的不穩(wěn)定巖塊（Goodman&石根華1985；Warburton1987；Goodman1995）；另外一種廣泛使用的方法是特殊單元法，它是由Cundall及其合作者（Cundall&Strack1979；Cundall1988）提出來的，其目的是用來求解顯式有限差分?jǐn)?shù)值問題，計(jì)算剛性塊體或柔性塊體的位移。本文的重點(diǎn)是闡述如何利用公式來描述實(shí)際的節(jié)理模型。既然直接求解的方法很復(fù)雜，數(shù)值分析方法也很難駕馭，同時(shí)由于涉及到了數(shù)目如此之多的塊體，所以尋求利用均質(zhì)化的方法是一個(gè)明智的選擇。事實(shí)上，這個(gè)概念早在Hoek-Brown準(zhǔn)則（Hoek&Brown1980；Hoek1983）得出的一個(gè)經(jīng)驗(yàn)公式中就有所涉及，它來自于宏觀上的一個(gè)直覺，被一個(gè)規(guī)則的表面節(jié)理網(wǎng)絡(luò)所分割的巖體，可以看做是一個(gè)均質(zhì)的連續(xù)體，由于節(jié)理傾向的不同，這樣的一個(gè)均質(zhì)材料顯示出了各向異性的性質(zhì)。本文的目的就是:從節(jié)理和巖體各自準(zhǔn)則出發(fā)，推求出一個(gè)嚴(yán)格準(zhǔn)確的公式，來描述作為均勻介質(zhì)的節(jié)理巖體的破壞準(zhǔn)則。先考查特殊情況，從兩組相互正交的節(jié)理著手，得到一個(gè)封閉的表達(dá)式，清楚的證明了強(qiáng)度的各向異性。我們進(jìn)行了一項(xiàng)試驗(yàn)：把利用均質(zhì)化方法得到的結(jié)果和以前普遍使用的準(zhǔn)則得到的結(jié)果以及基于計(jì)算機(jī)編程的特殊單元法（DEM）得到的結(jié)果進(jìn)行了對(duì)比，結(jié)果表明：對(duì)于密集裂隙的巖體，結(jié)果基本一致；對(duì)于節(jié)理數(shù)目較少的巖體，存在一個(gè)尺寸效應(yīng)（或者稱為比例效應(yīng)）。本文的第二部分就是在保證均質(zhì)化方法優(yōu)點(diǎn)的前提下，致力于提出一個(gè)新的方法來解決這種尺寸效應(yīng)，基于應(yīng)力和應(yīng)力耦合的宏觀破壞條件，提出利用微極模型或者Cosserat連續(xù)模型來描述節(jié)理巖體；最后將會(huì)

用一個(gè)簡(jiǎn)單的例子來演示如何應(yīng)用這個(gè)模型來解決比例效應(yīng)的問題。問題的陳述和均質(zhì)化方法的原理考慮這樣一個(gè)問題：一個(gè)基礎(chǔ)（橋墩或者其鄰接處）建立在一個(gè)有裂隙的巖床上（Fig.1），巖床的承載能力通過巖基和節(jié)理交界面的強(qiáng)度Fig.1,裂隙巖體基礎(chǔ)的承載能力估算出來。巖基的破壞條件使用傳統(tǒng)的莫爾-庫(kù)倫條件，可以用粘聚力C1和內(nèi)摩擦角?來表示（本文中張應(yīng)力采用正值計(jì)算）。同樣，用接觸平面代替節(jié)理（圖示平面中用直線表示）。強(qiáng)度特性采用接觸面上任意點(diǎn)的應(yīng)力向量（。其）表示：日（\丁尸）=|t|+otanipy—C/^0（1）根據(jù)屈服設(shè)計(jì)（或極限分析）推斷，如果沿著應(yīng)力邊界條件，巖體應(yīng)力分布滿足平衡方程和結(jié)構(gòu)任意點(diǎn)的強(qiáng)度要求，那么在一個(gè)給定的豎向荷載Q（沿著OZ軸方向）作用下，上部結(jié)構(gòu)仍然安全。這個(gè)問題可以歸結(jié)為求解破壞發(fā)生處的極限承載力Q+，或者是多大外力作用下結(jié)構(gòu)能確保穩(wěn)定。由于節(jié)理巖體強(qiáng)度的各向異性，若試圖使用上述直接推求的方法，難度就會(huì)增大很多。比如，由于節(jié)理強(qiáng)度特性遠(yuǎn)遠(yuǎn)低于巖基，從運(yùn)動(dòng)學(xué)角度出發(fā)的方法要求考慮到破壞機(jī)理，這就牽涉到了節(jié)理上的速度突躍，而節(jié)理處將會(huì)是首先發(fā)生破壞的區(qū)域。這種應(yīng)用在大多數(shù)傳統(tǒng)設(shè)計(jì)中的直接方法，隨著節(jié)理密度的增加越來越復(fù)雜。確切地說，這是因?yàn)橄啾容^結(jié)構(gòu)的長(zhǎng)度（如基礎(chǔ)寬B）而言，典型節(jié)理間距L變得更小，加大了問題的難度。在這種情況下，對(duì)節(jié)理巖體使用均質(zhì)化方法和宏觀等效連續(xù)的相關(guān)概念來處理可能就會(huì)比較妥當(dāng)。關(guān)于這個(gè)理論的更多細(xì)節(jié)，在有關(guān)于加固巖土力學(xué)的文章中可以查到（deBuhan等1989；deBuhan&Salenc1990；Bernaud等1995）。節(jié)理巖體的宏觀破壞條件節(jié)理巖體的宏觀破壞條件公式可以從對(duì)節(jié)理巖體典型晶胞單元的輔助屈服設(shè)計(jì)邊值問題中得到（Bekaert&Maghous1996;Maghous等1998）?，F(xiàn)在可以精確地表示平面應(yīng)變條件下，兩組相互正交節(jié)理的特殊情況，建立沿節(jié)理方向的正交坐標(biāo)系o§&2，并引入下列應(yīng)力變量：TOC\o"1-5"\h\zP偵11十小"）上2§=（仃M一心"人2⑵宏觀破壞條件可簡(jiǎn)化為：Q十血血甲m5「pr邊士加f（抽）其中，假定宏觀準(zhǔn)則的一種簡(jiǎn)便表示方法是畫出均質(zhì)材料傾向面上的強(qiáng)度包絡(luò)線，其單位法線n的傾角a為節(jié)理的方向，分別用氣和Tn表示這個(gè)面上的正應(yīng)力和切應(yīng)力，用（a『a22,aJ表示條件（3），推求出一組許可應(yīng)力（氣氣），然后求解出傾角a。當(dāng)a>*m時(shí)，相應(yīng)的區(qū)域表示如圖2所示，并對(duì)此做出兩個(gè)注解如下：

Fir.2.均勻介質(zhì)平回日勺強(qiáng)茂包絡(luò)技從圖2中可以清楚的看出，節(jié)理的存在導(dǎo)致了巖體強(qiáng)度的降低。通常當(dāng)HHm時(shí)，強(qiáng)度包絡(luò)線和巖基破壞條件相一致，其前半部分被兩個(gè)正交的半條線切去。宏觀各向異性很顯著。比如，圖2中的強(qiáng)度包絡(luò)線決定于方位角a。應(yīng)該拋棄固有曲線和各向異性粘聚力與摩擦角的概念，其中后一個(gè)概念是由Jaeger（I960）或McLamore&Gray（1967）所引入的。通過莫爾-庫(kù)倫條件進(jìn)行擴(kuò)展