英文翻译
建筑工程02
译文：
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；Warburton 1987；Goodman 1995）；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 & Strack 1979； Cundall 1988）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 & Brown 1980；
Hoek 1983）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
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C 1和内摩擦角ϕ。同样，用接触平面代替节理（图示平面中用m 来表示（本文中张应力采用正值计算）直线表示）。强度特性采用接触面上任意点的应力向量 (σ,τ)表示：
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（de Buhan等 1989；de Buhan & Salenc 1990；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 & Maghous 1996; 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 ξ1ξ2 ，并引入下列应力变量：
宏观破坏条件可简化为：
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α 为节理的方向，分别用σn 和τn 表示这个面上的正应力和切应力，用(σ11, σ22 , σ12) 表示条件（3），推求出一组许可应力(σn,τn ),然后求解出倾角α 。当α ≥ϕ 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：
1. 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Hj
2，宏观各向异性也是相当明显的，因为例如强度包络线绘制图。图2
是依赖于小面取向的。征曲线的一般概念，因此应被丢弃，而且各向
异性凝聚力和摩擦角的概念，作为暂定由积（1960），或MC Lamore和Gray （1967 ）介绍了。
也可以这样的各向异性可以通过标准的基础上，利用各向异性张量的概念的经典莫尔 - 库仑条件的扩展装置，适当地描述（伯勒尔和Sawczuk 1977;诺瓦1980 ; Allirot和Bochler1981 ） 。
适用于节理岩体开挖的稳定性
的封闭形式的表达式（3）的宏观失效的条件下获得的，使得它然后可以执行建立在这样一种材料，例如图中所示的挖掘任何结构的失败设计。 3 ，
其中，h 和β分别表示挖掘高度和倾斜角，分别为。因为没有付费施加到结构中， γ的构成材料的具体重量显然将构成对system.Assessing 这种结构的稳定性将达到评估的最大可能高度h +以外，将发生故障的唯一负载参数。这个问题的标准量纲分析表明，该临界高度可能是
放的形式
其中，θ =共同的方向和管开挖的稳定性K + =无量纲系数。这个因素上界估计会由现在的方式来确定
收率设计运动学方法，使用了2种故障机制中所示。
4 ，旋转失效机理[图4（a ） ]。
第一类在分析中考虑的失效机理是那些通常采用的均匀各向同性土壤或岩石边坡的直接换位。在这样的机制均质节理岩体的体积是绕一个点Ω与角速度ω 。分离这是保持不动的结构的其余部分本卷的曲线是速度跳跃线。因为它是角度的日志螺旋的弧命令聚焦Ω的速度间断在任何点这条线的倾斜角度WM 相对于切线在同一点上。
由外部力量和开发的最大抵抗工作所做的工作
这样的机制可以被写为（参见Chen 和Liu 1990; Maghous等人，1998 ） 其中weand 信号wME =无量纲的功能，并且μ1和μ2 =指定的角度旋转中心Ω.Since 产量设计的运动方式的位置指出一个
必要条件结构是稳定的写入
它遵循从方程。 （5 ）和（6 ），其从该第一类机制的派生最好上界估计通过最小化得到的相对于μ1和μ2
可数值确定。
分段刚性块失效机理[图。图4（b ） ]。
第二类故障机制涉及均质材料的两个转换块。它是由五个角的参数定义。为了避免任何误解，应当指定该块的术语在这里并不指岩石基质的团块中的初始结构，而仅仅意味着，在该屈服设计运动学方法，匀浆楔框架节理岩体被赋予一个（虚拟）刚体运动。的上界运动的实施办法，使得这种使用第二类的故障机制，导致了以下的结果。
其中U 代表下层块的速度的常态。因此，对于K +下上界估计 ：
结果与直接计算比较
约束优化已经计算数值的参数如下设置：
从均质化的方法得到的结果然后可以与来自直接计算而得相比，采用UDEC 计算机软件（ Hart 等人，1988）。因为后者能处理，其中指定的每个单独的关节的位置的情况下，已经执行了一系列的计算不同的规则间隔的关节数n ，倾斜相同的角θ = 10 °的水平，并相交的面的挖掘，作为描绘图。 5，本
稳定系数的相应估计已经被暗算n 个相同的数字。可以观察到，这些数值估计减少与交叉接头向下通过均质化方法产生的估计数。同质化和直接的方法之间观察到的差异，可以被看作是一个'' 大小'' 或'' 规模效应'' 这是不包括在经典同质化模型。克服后者的这种限制，同时还可以利用均质化的概念作为设计目的的计算节省时间的替代的一个可能的方法，可以诉诸于裂隙介质的描述，作为一个Cosserat 介质或微极连续体，由毕奥（ 1967）主张，例如; Besdo （1985）; Adhikary和Dyskin （ 1997年）; 和Sulem 和Mulhaus （ 1997）分层或块结构。本文的第二部分致力于将这样的模式来描述节理岩体介质的故障性质。
英语原文：
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 Cm and the friction angle m .
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 O ξ1ξ2whose 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 σn and τnthe 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 (σn , τn) deduced from conditions (3) expressed in terms of (σ11, σ22 , σ12). The corresponding domain has been drawn in Fig. 2 in the
particular case whereα≤ϕm .
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 conditionHj
2. The macroscopic anisotropy is also quite apparent, since for instance the strength envelope drawn in Fig. 2 is dependent on the facet orientation a.
The usual notion of intrinsic curve should therefore be discarded, but also
the concepts of anisotropic cohesion and friction angle as tentatively introduced 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 Excavation
The closed-form expression (3) obtained for the macroscopic failure condition, makes it then possible to perform the failure design of any structure built in such a material, such as the excavation shown in Fig. 3,
where h and β denote the excavation height and the slope angle,
respectively. Since no surcharge is applied to the structure, the specific weight γ of the constituent material will obviously constitute the sole
loading parameter of the system.Assessing the stability of this structure will amount to evaluating the maximum possible height h+ beyond which failure will occur. A standard dimensional analysis of this problem shows that this critical height may be
put in the form
where θ=joint orientation 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 design kinematic approach, using two kinds of failure mechanisms shown in Fig. 4.
Rotational Failure Mechanism [Fig. 4(a)]
The first class of failure mechanisms considered in the analysis is a direct transposition of those usually employed for homogeneous and isotropic soil or rock slopes. In such a mechanism a volume of homogenized jointed rock mass is rotating about a point Ω with an angular velocity ω. The curve separating this volume from the rest of the structure which is kept
motionless is a velocity jump line. Since it is an arc of the log spiral of angle mand focus Ω the velocity discontinuity at any point of this line is inclined at angle 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 mechanism may be written as (see Chen and Liu 1990; Maghous et al. 1998)
where weand wme=dimensionless functions, and μ1 and μ2=angles specifying the position of the center of rotation Ω.Since the kinematic approach of yield design states that a
necessary condition for the structure to be stable writes
it follows from Eqs. (5) and (6) that the best upper-bound estimate
derived from this first class of mechanism is obtained by minimization with respect to μ1 and μ2
which may be determined numerically.
Piecewise Rigid-Block Failure Mechanism [Fig. 4(b)]
The second class of failure mechanisms involves two translating blocks of homogenized material. It is defined by five angular parameters. In order to avoid any misinterpretation, it should be specified 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 kinematic approach, a wedge of homogenized jointed rock mass is given a (virtual) rigid-body motion.
The implementation of the upper-bound kinematic approach,making use of of this
second class of failure mechanism, leads to the following results.
where U represents the norm of the velocity of the lower block. Hence, the following upper-bound estimate for K+
:
Results and Comparison with Direct Calculation
The optimal bound has been computed numerically for the following set of parameters:
The result obtained from the homogenization approach can then be compared with that derived from a direct calculation, using the UDEC computer software (Hart et al. 1988). Since the latter can handle situations where the position of each individual joint is specified, a series of
calculations has been performed varying the number n of regularly spaced joints, inclined at the same angleθ=10° with the horizontal, and intersecting the facing of the excavation, as sketched in Fig. 5. The
corresponding estimates of the stability factor have been plotted against n in the same figure. It can be observed that these numerical estimates decrease with the number of intersecting joints down to the estimate produced by the homogenization approach. The observed discrepancy between homogenization and direct approaches, could be regarded as a „„size‟‟ or „„scale effect‟‟ which is not included in the classical
homogenization model. A possible way to overcome such a limitation of the latter, while still taking advantage of the homogenization concept as a computational time-saving alternative for design purposes, could be to resort to a description of the fractured rock medium as a Cosserat or micropolar continuum, as advocated for instance by Biot (1967); Besdo(1985);
Adhikary and Dyskin (1997); and Sulem and Mulhaus (1997) for stratified or block structures. The second part of this paper is devoted to applying
such a model to describing the failure properties of jointed rock media.
英文翻译
建筑工程02
译文：
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；Warburton 1987；Goodman 1995）；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 & Strack 1979； Cundall 1988）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 & Brown 1980；
Hoek 1983）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
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C 1和内摩擦角ϕ。同样，用接触平面代替节理（图示平面中用m 来表示（本文中张应力采用正值计算）直线表示）。强度特性采用接触面上任意点的应力向量 (σ,τ)表示：
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（de Buhan等 1989；de Buhan & Salenc 1990；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 & Maghous 1996; 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 ξ1ξ2 ，并引入下列应力变量：
宏观破坏条件可简化为：
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α 为节理的方向，分别用σn 和τn 表示这个面上的正应力和切应力，用(σ11, σ22 , σ12) 表示条件（3），推求出一组许可应力(σn,τn ),然后求解出倾角α 。当α ≥ϕ 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：
1. 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Hj
2，宏观各向异性也是相当明显的，因为例如强度包络线绘制图。图2
是依赖于小面取向的。征曲线的一般概念，因此应被丢弃，而且各向
异性凝聚力和摩擦角的概念，作为暂定由积（1960），或MC Lamore和Gray （1967 ）介绍了。
也可以这样的各向异性可以通过标准的基础上，利用各向异性张量的概念的经典莫尔 - 库仑条件的扩展装置，适当地描述（伯勒尔和Sawczuk 1977;诺瓦1980 ; Allirot和Bochler1981 ） 。
适用于节理岩体开挖的稳定性
的封闭形式的表达式（3）的宏观失效的条件下获得的，使得它然后可以执行建立在这样一种材料，例如图中所示的挖掘任何结构的失败设计。 3 ，
其中，h 和β分别表示挖掘高度和倾斜角，分别为。因为没有付费施加到结构中， γ的构成材料的具体重量显然将构成对system.Assessing 这种结构的稳定性将达到评估的最大可能高度h +以外，将发生故障的唯一负载参数。这个问题的标准量纲分析表明，该临界高度可能是
放的形式
其中，θ =共同的方向和管开挖的稳定性K + =无量纲系数。这个因素上界估计会由现在的方式来确定
收率设计运动学方法，使用了2种故障机制中所示。
4 ，旋转失效机理[图4（a ） ]。
第一类在分析中考虑的失效机理是那些通常采用的均匀各向同性土壤或岩石边坡的直接换位。在这样的机制均质节理岩体的体积是绕一个点Ω与角速度ω 。分离这是保持不动的结构的其余部分本卷的曲线是速度跳跃线。因为它是角度的日志螺旋的弧命令聚焦Ω的速度间断在任何点这条线的倾斜角度WM 相对于切线在同一点上。
由外部力量和开发的最大抵抗工作所做的工作
这样的机制可以被写为（参见Chen 和Liu 1990; Maghous等人，1998 ） 其中weand 信号wME =无量纲的功能，并且μ1和μ2 =指定的角度旋转中心Ω.Since 产量设计的运动方式的位置指出一个
必要条件结构是稳定的写入
它遵循从方程。 （5 ）和（6 ），其从该第一类机制的派生最好上界估计通过最小化得到的相对于μ1和μ2
可数值确定。
分段刚性块失效机理[图。图4（b ） ]。
第二类故障机制涉及均质材料的两个转换块。它是由五个角的参数定义。为了避免任何误解，应当指定该块的术语在这里并不指岩石基质的团块中的初始结构，而仅仅意味着，在该屈服设计运动学方法，匀浆楔框架节理岩体被赋予一个（虚拟）刚体运动。的上界运动的实施办法，使得这种使用第二类的故障机制，导致了以下的结果。
其中U 代表下层块的速度的常态。因此，对于K +下上界估计 ：
结果与直接计算比较
约束优化已经计算数值的参数如下设置：
从均质化的方法得到的结果然后可以与来自直接计算而得相比，采用UDEC 计算机软件（ Hart 等人，1988）。因为后者能处理，其中指定的每个单独的关节的位置的情况下，已经执行了一系列的计算不同的规则间隔的关节数n ，倾斜相同的角θ = 10 °的水平，并相交的面的挖掘，作为描绘图。 5，本
稳定系数的相应估计已经被暗算n 个相同的数字。可以观察到，这些数值估计减少与交叉接头向下通过均质化方法产生的估计数。同质化和直接的方法之间观察到的差异，可以被看作是一个'' 大小'' 或'' 规模效应'' 这是不包括在经典同质化模型。克服后者的这种限制，同时还可以利用均质化的概念作为设计目的的计算节省时间的替代的一个可能的方法，可以诉诸于裂隙介质的描述，作为一个Cosserat 介质或微极连续体，由毕奥（ 1967）主张，例如; Besdo （1985）; Adhikary和Dyskin （ 1997年）; 和Sulem 和Mulhaus （ 1997）分层或块结构。本文的第二部分致力于将这样的模式来描述节理岩体介质的故障性质。
英语原文：
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 Cm and the friction angle m .
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 O ξ1ξ2whose 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 σn and τnthe 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 (σn , τn) deduced from conditions (3) expressed in terms of (σ11, σ22 , σ12). The corresponding domain has been drawn in Fig. 2 in the
particular case whereα≤ϕm .
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 conditionHj
2. The macroscopic anisotropy is also quite apparent, since for instance the strength envelope drawn in Fig. 2 is dependent on the facet orientation a.
The usual notion of intrinsic curve should therefore be discarded, but also
the concepts of anisotropic cohesion and friction angle as tentatively introduced 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 Excavation
The closed-form expression (3) obtained for the macroscopic failure condition, makes it then possible to perform the failure design of any structure built in such a material, such as the excavation shown in Fig. 3,
where h and β denote the excavation height and the slope angle,
respectively. Since no surcharge is applied to the structure, the specific weight γ of the constituent material will obviously constitute the sole
loading parameter of the system.Assessing the stability of this structure will amount to evaluating the maximum possible height h+ beyond which failure will occur. A standard dimensional analysis of this problem shows that this critical height may be
put in the form
where θ=joint orientation 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 design kinematic approach, using two kinds of failure mechanisms shown in Fig. 4.
Rotational Failure Mechanism [Fig. 4(a)]
The first class of failure mechanisms considered in the analysis is a direct transposition of those usually employed for homogeneous and isotropic soil or rock slopes. In such a mechanism a volume of homogenized jointed rock mass is rotating about a point Ω with an angular velocity ω. The curve separating this volume from the rest of the structure which is kept
motionless is a velocity jump line. Since it is an arc of the log spiral of angle mand focus Ω the velocity discontinuity at any point of this line is inclined at angle 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 mechanism may be written as (see Chen and Liu 1990; Maghous et al. 1998)
where weand wme=dimensionless functions, and μ1 and μ2=angles specifying the position of the center of rotation Ω.Since the kinematic approach of yield design states that a
necessary condition for the structure to be stable writes
it follows from Eqs. (5) and (6) that the best upper-bound estimate
derived from this first class of mechanism is obtained by minimization with respect to μ1 and μ2
which may be determined numerically.
Piecewise Rigid-Block Failure Mechanism [Fig. 4(b)]
The second class of failure mechanisms involves two translating blocks of homogenized material. It is defined by five angular parameters. In order to avoid any misinterpretation, it should be specified 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 kinematic approach, a wedge of homogenized jointed rock mass is given a (virtual) rigid-body motion.
The implementation of the upper-bound kinematic approach,making use of of this
second class of failure mechanism, leads to the following results.
where U represents the norm of the velocity of the lower block. Hence, the following upper-bound estimate for K+
:
Results and Comparison with Direct Calculation
The optimal bound has been computed numerically for the following set of parameters:
The result obtained from the homogenization approach can then be compared with that derived from a direct calculation, using the UDEC computer software (Hart et al. 1988). Since the latter can handle situations where the position of each individual joint is specified, a series of
calculations has been performed varying the number n of regularly spaced joints, inclined at the same angleθ=10° with the horizontal, and intersecting the facing of the excavation, as sketched in Fig. 5. The
corresponding estimates of the stability factor have been plotted against n in the same figure. It can be observed that these numerical estimates decrease with the number of intersecting joints down to the estimate produced by the homogenization approach. The observed discrepancy between homogenization and direct approaches, could be regarded as a „„size‟‟ or „„scale effect‟‟ which is not included in the classical
homogenization model. A possible way to overcome such a limitation of the latter, while still taking advantage of the homogenization concept as a computational time-saving alternative for design purposes, could be to resort to a description of the fractured rock medium as a Cosserat or micropolar continuum, as advocated for instance by Biot (1967); Besdo(1985);
Adhikary and Dyskin (1997); and Sulem and Mulhaus (1997) for stratified or block structures. The second part of this paper is devoted to applying
such a model to describing the failure properties of jointed rock media.