均质各向异性裂隙岩体的破坏特性

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侦11十小")上2§=(仃队―心"人2⑵suchamacroscopicfailureconditionsimplybecomes'山？十户＜（-Q十\21I*in以因）\o"CurrentDocument"PWggE（-^点勺加旺响whereitwillbeassumedthat〃阳=匚矿而褊叫=q也叫广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=层四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.叽=U\玷心…；3皿)呼皿=。。混卬思…；孔皿)(8)whereUrepresentsthenormofthevelocityofthelowerblock.Hence,thefollowingupper-boundestimateforK+:K-W理=min|不当⑼ResultsandComparisonwithDirectCalculationTheoptimalboundhasbeencomputednumericallyforthefollowingsetofparameters:9=75。，0=10°,=0.1,钿=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.均质各向异性裂隙岩体的破坏特性概述由于岩体表面的裂隙或节理大小与倾向不同，人们通常把岩体看做是非连续的。尽管裂隙或节理表现出的力学性质要远远低于岩体本身，但是它们在岩体结构性质方面起着重要的作用，岩体本身的变形和破坏模式也主要是由这些节理所决定的。从地质力学工程角度而言，在涉及到节理岩体结构的设计方法中，软弱表面是一个很重要的考虑因素。解决这种问题最简单的方法就是把岩体看作是许多完整岩块的集合，这些岩块之间有很多相交的节理面。这种方法在过去的几十年中被设计者们广泛采用，其中比较著名的是“块体理论”，该理论试图从几何学和运动学的角度用来判别潜在的不稳定岩块（Goodman&石根华1985；Warburton1987；Goodman1995）；另外一种广泛使用的方法是特殊单元法，它是由Cundall及其合作者（Cundall&Strack1979；Cundall1988）提出来的，其目的是用来求解显式有限差分数值问题，计算刚性块体或柔性块体的位移。本文的重点是阐述如何利用公式来描述实际的节理模型。既然直接求解的方法很复杂，数值分析方法也很难驾驭，同时由于涉及到了数目如此之多的块体，所以寻求利用均质化的方法是一个明智的选择。事实上，这个概念早在Hoek-Brown准则（Hoek&Brown1980；Hoek1983）得出的一个经验公式中就有所涉及，它来自于宏观上的一个直觉，被一个规则的表面节理网络所分割的岩体，可以看做是一个均质的连续体，由于节理倾向的不同，这样的一个均质材料显示出了各向异性的性质。本文的目的就是:从节理和岩体各自准则出发，推求出一个严格准确的公式，来描述作为均匀介质的节理岩体的破坏准则。先考查特殊情况，从两组相互正交的节理着手，得到一个封闭的表达式，清楚的证明了强度的各向异性。我们进行了一项试验：把利用均质化方法得到的结果和以前普遍使用的准则得到的结果以及基于计算机编程的特殊单元法（DEM）得到的结果进行了对比，结果表明：对于密集裂隙的岩体，结果基本一致；对于节理数目较少的岩体，存在一个尺寸效应（或者称为比例效应）。本文的第二部分就是在保证均质化方法优点的前提下，致力于提出一个新的方法来解决这种尺寸效应，基于应力和应力耦合的宏观破坏条件，提出利用微极模型或者Cosserat连续模型来描述节理岩体；最后将会

用一个简单的例子来演示如何应用这个模型来解决比例效应的问题。问题的陈述和均质化方法的原理考虑这样一个问题：一个基础（桥墩或者其邻接处）建立在一个有裂隙的岩床上（Fig.1），岩床的承载能力通过岩基和节理交界面的强度Fig.1,裂隙岩体基础的承载能力估算出来。岩基的破坏条件使用传统的莫尔-库伦条件，可以用粘聚力C1和内摩擦角©来表示（本文中张应力采用正值计算）。同样，用接触平面代替节理（图示平面中用直线表示）。强度特性采用接触面上任意点的应力向量（。其）表示：日（\丁尸）=|t|+otanipy—C/^0（1）根据屈服设计（或极限分析）推断，如果沿着应力边界条件，岩体应力分布满足平衡方程和结构任意点的强度要求，那么在一个给定的竖向荷载Q（沿着OZ轴方向）作用下，上部结构仍然安全。这个问题可以归结为求解破坏发生处的极限承载力Q+，或者是多大外力作用下结构能确保稳定。由于节理岩体强度的各向异性，若试图使用上述直接推求的方法，难度就会增大很多。比如，由于节理强度特性远远低于岩基，从运动学角度出发的方法要求考虑到破坏机理，这就牵涉到了节理上的速度突跃，而节理处将会是首先发生破坏的区域。这种应用在大多数传统设计中的直接方法，随着节理密度的增加越来越复杂。确切地说，这是因为相比较结构的长度（如基础宽B）而言，典型节理间距L变得更小，加大了问题的难度。在这种情况下，对节理岩体使用均质化方法和宏观等效连续的相关概念来处理可能就会比较妥当。关于这个理论的更多细节，在有关于加固岩土力学的文章中可以查到（deBuhan等1989；deBuhan&Salenc1990；Bernaud等1995）。节理岩体的宏观破坏条件节理岩体的宏观破坏条件公式可以从对节理岩体典型晶胞单元的辅助屈服设计边值问题中得到（Bekaert&Maghous1996;Maghous等1998）。现在可以精确地表示平面应变条件下，两组相互正交节理的特殊情况，建立沿节理方向的正交坐标系o§&2，并引入下列应力变量：TOC\o"1-5"\h\zP侦11十小"）上2§=（仃M一心"人2⑵宏观破坏条件可简化为：Q十血血甲m5「pr边士加f（抽）其中，假定宏观准则的一种简便表示方法是画出均质材料倾向面上的强度包络线，其单位法线n的倾角a为节理的方向，分别用气和Tn表示这个面上的正应力和切应力，用（a『a22,aJ表示条件（3），推求出一组许可应力（气气），然后求解出倾角a。当a>*m时，相应的区域表示如图2所示，并对此做出两个注解如下：

Fir.2.均匀介质平回日勺强茂包络技从图2中可以清楚的看出，节理的存在导致了岩体强度的降低。通常当HHm时，强度包络线和岩基破坏条件相一致，其前半部分被两个正交的半条线切去。宏观各向异性很显著。比如，图2中的强度包络线决定于方位角a。应该抛弃固有曲线和各向异性粘聚力与摩擦角的概念，其中后一个概念是由Jaeger（I960）或McLamore&Gray（1967）所引入的。通过莫尔-库伦条件进行扩展