Failure Properties of Fractured Rock Masses as AnisotropicHomogenized MediaIntroductionIt is com monly ack no wledged that rock masses always display disc on ti nu ous surfaces of various sizes and orie ntatio ns, usually referred to as fractures or joi nts. Si nee the latter have much poorer mecha ni cal characteristics tha n the rock material, they play a decisive role in the overall behavior of rock structures,whose deformati on as well as failure patter ns are mai nly gover ned by those of the join ts. It follows that, from a geomecha ni cal engin eeri ng sta ndpo int, desig n methods of structures inv olvi ng joi nted rock masses, must absolutely acco unt for such‘‘ weakness' ' surfaeesahalysis.The most straightforward way of deali ng with this situati on is to treat the joi nted rock mass as an assemblageof pieces of in tact rock material in mutual in teracti on through the separat ing joint in terfaces. Many desig n-orie nted methods relat ing to this kind of approach have been developed in the past decades, among them,the wedlnown ‘‘ block theory, ' ' which attempts to ide ntify pote n- tially unstable lumps of rock from geometrical and kinematical considerations (Goodman and Shi 1985; Warburton 1987; Goodman 1995). One should also quote the widely used distinet element method, originating from the works of Cundall and coauthors (Cundall and Strack 1979; Cundall 1988), which makes use of an explicit ?nite-difference numerical scheme for computing the displacements of the blocks considered as rigid or deformable bodies. In this con text, atte nti on is primarily focused on the formulatio n of realistic models for describ ing the joint behavior.Since the previously men ti oned direct approach is beco ming highly complex, and the n numerically untractable, as soon as a very large number of blocks is involved, it seems advisable to look for alter native methods such as those derived from the con cept of homogenization. Actually, such a concept is already partially conveyed in an empirical fashion by the famous Hoek and Brown ' s criterion (Hoek and Brown 1980; Hoek 1983). It stems from the in tuitive idea that from a macroscopic point of view, a rock mass in tersected by a regular network of joint surfaces, may be perceived as a homogeneous continuum. Furthermore, owing to the existence of joint preferential orientations, one should expect such a homoge ni zed material to exhibit ani sotropic properties.The objective of the present paper is to derive a rigorous formulation for the failure criteri on of a joi nted rock mass as a homoge ni zed medium, from the kno wledge of the joi nts and rock material respective criteria. In the particular situation where twomutually orthogonal joint sets are con sidered, a closed-form expressi on is obta in ed, giving clear evide nce of the related stre ngth ani sotropy. A comparis on is performed on an illustrative example betwee n the results produced by the homogenization method,making use of the previously determined criterion, and those obtained by means of a computer code based on the distinct element method. It is show n that, while both methods lead to almost ide ntical results for a den sely fractured rock mass, a ‘‘ size ' ' or ‘‘ scale effect ' ' is observed in the case of ajoints. The second part of the paper is then devoted to proposing a method which attempts tocapture such a scale effect, while still tak ing adva ntage of a homoge ni zati on tech niq ue. This is achieved by resorting to a micropolar or Cosserat continuum description of the fractured rock mass,through the derivation of a generalized macroscopic failure condition expressed in terms of stresses and couple stresses. The implementation of this model is ?nally illustrated on a simple example, show ing how it may actually acco unt for such a scale effect.Problem Statement and Principle of Homogenization ApproachThe problem under consideration is that of a foundation (bridge pier or abutment) resting upon a fractured bedrock (Fig. 1), whose beari ngFig, 1, Bearing capacit>p of foundation on fractured rockmasscapacity needs to be evaluated from the knowledge of the strength capacities of the rock matrix and the joint in terfaces. The failure con diti on of the former will be expressed through the classical Mohr-Coulomb condition expressedby means of the cohesion C m and the friction angle m . Note that tensile stresses will be counted positive throughout the paper.Likewise, the joi nts will be modeled as pla ne in terfaces (represe nted by lines in the ?gure ' s plane). Their strength propertieiseadescribed by means of a condition involving the stress vector of comp onents ( c , T ) act ing at any point of those in terfacesF7(O-,T)=|T|+<J tan <f f—C r^0 (l)According to the yield design (or limit analysis) reasoning, the above structure will remai n safe un der a give n vertical load Q(force per un it le ngth along the Oz axis), if one can exhibit throughout the rock mass a stress distribution which satis?es the equilibrium equations along with the stress boundary conditions,while complying with the strength requirement expressed at any point of the structure.+This problem amounts to evaluating the ultimate load Q beyond which failure will occur, or equivalently within which its stability is ensured. Due to the strong heterogeneity of the joi nted rock mass, in surm oun table dif?culties are ilkely to arise whe n tryi ng to impleme nt the above reasoning directly. As regards, for instanee, the case where the strength properties of the joints are considerably lower than those of the rock matrix, the implementation of akinematic approach would require the use of failure mechanisms involving velocity jumps across the joi nts, since the latter would con stitute prefere ntial zones for the occurre nee offailure. In deed, such a direct approach which is applied in most classical desig n methods, is beco ming rapidly complex as the den sity of joi nts in creases, that is as the typical joi nt spaci ng l is beco ming small in comparis on with a characteristic len gth of the structure such as the foundation width B.In such a situati on, the use of an alter native approach based on the idea of homoge ni zati on and related con cept of macroscopic equivale nt continuum for the join ted rock mass, may be appropriate for dealing with such a problem. More details about this theory, applied in the con text of rein forced soil and rock mecha ni cs, will be found in (de Buha n et al. 1989; de Buhan and Sale nc ,on 1990; Bernaud et al. 1995).Macroscopic Failure Condition for Jointed Rock MassThe formulati on of the macroscopic failure con diti on of a join ted rock mass may be obtained from the solution of an auxiliary yield design boundary-value problem attached to a unit represe ntative cell of joi nted rock (Bekaert and Maghous 1996; Maghous et al.1998). It will now be explicitly formulated in the particular situation of two mutually orthogonal sets of joi nts un der pla ne strain con diti ons. Referri ng to an orth ono rmal frame O 1 2 whose axes are placed along the joints direct ions, and in troduci ng the follow ing cha nge of stress variables:p = ((T]|+(T22)/\2, I?=((T22-O，I1)/ \2, 1= \2(T|2 (2) such a macroscopic failure con diti on simply becomes（如mwh ere it will be assumed that15 = 5tan<p;.A convenient represe ntati on of the macroscopic criteri on is to draw the stre ngth env elope relating to an oriented facet of the homogenized material, whose unit normal n I is inclined by an an gle a with respect to the joint direct ion. Deno ti ng by - n and n the no rmal and shear comp onents of the stress vector act ing upon such a facet, it is possible to determ ine for any value of a the set of admissible stresses , n) deduced from conditions (3) expressed in terms of ( J,、, -12). The corresponding domain has been drawn in Fig. 2 in the particular case where 乞mTwo comme nts are worth being made:1. The decrease in stre ngth of a rock material due to the prese nceof joints is clearly illustrated by Fig.2. The usual strength envelope corresponding to the rock matrix failure condition is ‘‘ truncatedy two orthogonal semilines as soon as condition H j ： H m is ful?lled.2. The macroscopic anisotropy is also quite apparent, since for instance the strength envelopedrawn in Fig. 2 is dependent on the facet orientation a. The usual notion of intrinsic curve should therefore be discarded, but also the con cepts of ani sotropic cohesi on and frictio n an gle as ten tatively in troduced by Jaeger (1960), or Mc Lamore and Gray (1967).Nor can such an anisotropy be properly described by means of criteria based on an extension of the classical Mohr-Coulomb condition using the concept of anisotropy tensor(Boehler and Sawczuk 1977; Nova 1980; Allirot and Bochler1981).Application to Stability of Jointed Rock ExcavationThe closed-form expressi on (3) obta ined for the macroscopic failure con diti on, makes it thenpossible to perform the failure design of any structure built in such a material, such as the excavati on show n in Fig. 3,where h and Bdenote the excavation height and the slope angle, respectively. Since noFig. 2. Strength envelope attached co facei of homogenized materialFig. 3. Stability analysis of jointed rock excavationsurcharge is applied to the structure, the speci?c weight y of the constituent material will obviously constitute the sole loading parameter of the system.Assessing the stability of this structure will amount to evaluati ng the maximum possible height F bey ond which failure will occur. A standard dimensional analysis of this problem shows that this critical height may be put in the formwhere 0 =jointorientation and K+=nondimensional factor governing the stability of the excavation. Upper-bound estimates of this factor will now be determined by means of the yield desig n kin ematic approach, using two kinds of failure mecha ni sms show n in Fig. 4.Rotational Failure Mechanism [Fig. 4(a)]The ?rst class of failure mecha ni sms con sidered in the an alysis is a direct tran spositi on of those usually employed for homogeneous and isotropic soil or rock slopes. In such a mecha nism a volume of homoge ni zed joi nted rock mass is rotati ng about a point Q angular velocity 3 . The curve separating thjisrvofrom the rest of the structure which iskept motionless is a velocity jump line. Since it is an arc of the log spiral of angle m andfocus Q the velocity disc on tin uity at any point of this line is in cli ned at an gle wm with respect to the tangent at the same point.The work done by the external forces and the maximum resisting work developed in such a mecha nism may be writte n as (see Chen and Liu 1990; Maghous et al. 1998)二丁3片片％(B；»i .42)_'' mr| P，「*申/A t，42 ) (')J mwhere W e and W me =dimensionless functions, and ® and %=angles specifying theposition of the center of rotation Q .Sinee the kinematic approach of yield design states tha n ecessary con diti on for the structure to be stable writes化w (6)it follows from Eqs. (5) and (6) that the best upper-bound estimatederived from this ?rstclass of mechanism is obtained by minimization with respect to 1 and 2 卩which may be determ ined nu merically.Piecewise Rigid-Block Failure Mechanism [Fig. 4(b)]The sec ond class of failure mecha ni sms invo Ives two tran slat ing blocks of homoge ni zed material. It is de?ned by ?ve an gular parameters .In order to avoid any misi nterpretati on, it should be speci?ed that the terminology of block does not refer here to the lumps of rock matrix in the initial structure but merely means that, in the framework of the yield design kin ematic approach, a wedge of homoge ni zed join ted rock mass is give n a (virtual) rigid-body moti on.The implementation of the upper-bound kinematic approach,making use of of this second class of failure mechanism, leads to the following results.叫=匕号沪叫(…；"gj 炉(…；山g)(8)where U representsthe norm of the velocity of the lower block. Hence, the following+upper-bo und estimate for K: Results and Comparison with Direct CalculationThe optimal bound has bee n computed nu merically for the follow ing set of parameters:0 = 75。