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Failure Properties of Fractured Rock Masses as Anisotropic
Homogenized Media
Introduction
It is commonly acknowledged that rock masses always display discontinuous surfaces of various sizes and orientations, usually referred to as fractures or joints. Since the latter have much poorer mechanical characteristics than the rock material, they play a decisive role in the overall behavior of rock structures,whose deformation as well as failure patterns are mainly governed by those of the joints. It follows that, from a geomechanical engineering standpoint, design methods of structures involving jointed rock masses, must absolutely account for such ‘‘weakness’’ surfaces in their analysis.
The most straightforward way of dealing with this situation is to treat the jointed rock mass as an assemblage of pieces of intact rock material in mutual interaction through the separating joint interfaces. Many design-oriented methods relating to this kind of approach have been developed in the past decades, among them,the well-known ‘‘block theory,’’ which attempts to identify poten-
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 distinct element method, originating from the works of Cundall and coauthors (Cundall and Strack 1979; Cundall 1988), which makes use of an explicit finite-difference numerical scheme for computing the displacements of the blocks considered as rigid or deformable bodies. In this context, attention is primarily focused on the formulation of realistic models for describing the joint behavior.
Since the previously mentioned direct approach is becoming highly complex, and then numerically untractable, as soon as a very large number of blocks is involved, it seems advisable to look for alternative methods such as those derived from the concept 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 intuitive idea that from a macroscopic point of view, a rock mass intersected 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 homogenized material to exhibit anisotropic properties.
The objective of the present paper is to derive a rigorous formulation for the failure criterion of a jointed rock mass as a homogenized medium, from the knowledge of the joints and rock material respective criteria. In the particular situation where twomutually orthogonal joint sets are considered, a closed-form expression is obtained, giving clear evidence of the related strength anisotropy. A comparison is performed on an illustrative example between 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 shown that, while both methods lead to almost identical results for a densely fractured rock mass, a ‘‘size’’ or ‘‘scale effect’’ is observed in the case of a limited number of joints. The second part of the paper is then devoted to proposing a method which attempts to capture such a scale effect, while still taking advantage of a homogenization technique. 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 finally illustrated on a simple example, showing how it may actually account for such a scale effect.
Problem Statement and Principle of Homogenization Approach
The problem under consideration is that of a foundation (bridge pier or abutment) resting upon a fractured bedrock (Fig. 1), whose bearing

capacity needs to be evaluated from the knowledge of the strength capacities of the rock matrix and the joint interfaces. The failure condition of the former will be expressed through the classical Mohr-Coulomb condition expressed by means of the cohesion and the friction angle . Note that tensile stresses will be counted positive throughout the paper.
Likewise, the joints will be modeled as plane interfaces (represented by lines in the figure’s plane). Their strength properties are described by means of a condition involving the stress vector of components (σ, τ) acting at any point of those interfaces

According to the yield design (or limit analysis) reasoning, the above structure will remain safe under a given vertical load Q(force per unit length along the Oz axis), if one can exhibit throughout the rock mass a stress distribution which satisfies 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 jointed rock mass, insurmountable difficulties are likely to arise when trying to implement the above reasoning directly. As regards, for instance, the case where the strength properties of the joints are considerably lower than those of the rock matrix, the implementation of a kinematic approach would require the use of failure mechanisms involving velocity jumps across the joints, since the latter would constitute preferential zones for the occurrence of
failure. Indeed, such a direct approach which is applied in most classical design methods, is becoming rapidly complex as the density of joints increases, that is as the typical joint spacing l is becoming small in comparison with a characteristic length of the structure such as the foundation width B.
In such a situation, the use of an alternative approach based on the idea of homogenization and related concept of macroscopic equivalent continuum for the jointed rock mass, may be appropriate for dealing with such a problem. More details about this theory, applied in the context of reinforced soil and rock mechanics, will be found in (de Buhan et al. 1989; de Buhan and Salenc ,on 1990; Bernaud et al. 1995).
Macroscopic Failure Condition for Jointed Rock Mass
The formulation of the macroscopic failure condition of a jointed rock mass may be obtained from the solution of an auxiliary yield design boundary-value problem attached to a unit representative cell of jointed 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 joints under plane strain conditions. Referring to an orthonormal frame Owhose axes are placed along the joints directions, and introducing the following change of stress variables:

such a macroscopic failure condition simply becomes
where it will be assumed that

A convenient representation of the macroscopic criterion is to draw the strength envelope relating to an oriented facet of the homogenized material, whose unit normal n I is inclined by an angle a with respect to the joint direction. Denoting by and the normal and shear components of the stress vector acting upon such a facet, it is possible to determine for any value of a the set of admissible stresses ( , ) deduced from conditions (3) expressed in terms of (, , ). The corresponding domain has been drawn in Fig. 2 in the particular case where .

Two comments are worth being made:
1. The decrease in strength of a rock material due to the presence of joints is clearly illustrated by Fig. 2. The usual strength envelope corresponding to the rock matrix failure condition is ‘‘truncated’’ by two orthogonal semilines as soon as condition is fulfilled.
2. The macroscopic anisotropy is also quite appa