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―心"人2⑵suchamacroscopicfailureconditionsimplybecomes'山？十戶＜（-Q十\21I*in以因）\o"CurrentDocument"PWggE（-^點勺加旺響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=。。混卬思…；孔皿)(8)whereUrepresentsthenormofthevelocityofthelowerblock.Hence,thefollowingupper-boundestimateforK+:K-W理=min|不當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理大小與傾向不同，人們通常把巖體看做是非連續的。盡管裂隙或節理表現出的力學性質要遠遠低于巖體本身，但是它們在巖體結構性質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理所決定的。從地質力學工程角度而言，在涉及到節理巖體結構的設計方法中，軟弱表面是一個很重要的考慮因素。解決這種問題最簡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理面。這種方法在過去的幾十年中被設計者們廣泛采用，其中比較著名的是“塊體理論”，該理論試圖從幾何學和運動學的角度用來判別潛在的不穩定巖塊（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數值問題，計算剛性塊體或柔性塊體的位移。本文的重點是闡述如何利用公式來描述實際的節理模型。既然直接求解的方法很復雜，數值分析方法也很難駕馭，同時由于涉及到了數目如此之多的塊體，所以尋求利用均質化的方法是一個明智的選擇。事實上，這個概念早在Hoek-Brown準則（Hoek&Brown1980；Hoek1983）得出的一個經驗公式中就有所涉及，它來自于宏觀上的一個直覺，被一個規則的表面節理網絡所分割的巖體，可以看做是一個均質的連續體，由于節理傾向的不同，這樣的一個均質材料顯示出了各向異性的性質。本文的目的就是:從節理和巖體各自準則出發，推求出一個嚴格準確的公式，來描述作為均勻介質的節理巖體的破壞準則。先考查特殊情況，從兩組相互正交的節理著手，得到一個封閉的表達式，清楚的證明了強度的各向異性。我們進行了一項試驗：把利用均質化方法得到的結果和以前普遍使用的準則得到的結果以及基于計算機編程的特殊單元法（DEM）得到的結果進行了對比，結果表明：對于密集裂隙的巖體，結果基本一致；對于節理數目較少的巖體，存在一個尺寸效應（或者稱為比例效應）。本文的第二部分就是在保證均質化方法優點的前提下，致力于提出一個新的方法來解決這種尺寸效應，基于應力和應力耦合的宏觀破壞條件，提出利用微極模型或者Cosserat連續模型來描述節理巖體；最后將會

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

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