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某某某大学毕业设计（论文）Failure Properties of Fractured Rock Masses as AnisotropicHomogenized MediaIntroductionIt 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 nite-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 Browns 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 nally illustrated on a simple example, showing how it may actually account 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 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 gures plane). Their strength properties are described by means of a condition involving the stress vector of components (, ) acting at any point of those interfacesAccording 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 satises 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 difculties 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 offailure. 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 MassThe 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 becomeswhere 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 fullled.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 ExcavationThe 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 denote the excavation height and the slope angle, respectively. Since no surcharge is applied to the structure, the specic weight 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 formwhere =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 rst 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 with an angular velocity . 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 and focus 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 and =dimensionless functions, and 1 and 2=angles specifying the position of the center of rotation .Since the kinematic approach of yield design states that a necessary condition for the structure to be stable writesit follows from Eqs. (5) and (6) that the best upper-bound estimate derived from this rst class of mechanism is obtained by minimization with respect to 1 and 2which 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 dened by ve angular parameters. In order to avoid any misinterpretation, it should be specied 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 CalculationThe 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 specied, a series of calculations has been performed varying the number n of regularly spaced joints, inclined at the same angle=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 gure. 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 classicalhomogenization 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 stratied 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.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 m 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 (,)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 ，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 为节理的方向，分别用n 和n 表示这个面上的正应力和切应力，用(, , ) 表示条件（3），推求出一组许可应力(n,n ),然后求解出倾角 。当 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。 2. 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 。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（1960）或Mc Lamore & Gray（1967）所引入的。通过莫尔-库伦条件进行扩展，利用各向异性张量的方法来描述各向异性也是不妥当的 (Boehler & Sawczuk 1977； Nova 1980；Allirot & Bochler 1981) 。在节理岩体开挖稳定性中的应用 式（3）的封闭形式是从宏观破坏条件中得到的，该式可以用来对此种材料的结构体进行破坏设计，如图3所示的开挖，h 和 分别表示开挖高度和边坡角。由于结构上没有其他荷载，材料比重 就成为系统唯一的加载参数。该结构的稳定性评价需要在破坏发生的部位算出最大可能高度 h+，通过标准量纲分析表明，这个临界高度表示为： 其中为节理方位角，K +为表示开挖部位稳定性的一个无量纲因子，该因子的上界估计值可以分别使用图4所示的两种类型的破坏机制，通过屈服设计的运动学方法来确定。转动破坏机理 Fig. 4(a) 第一种类型的破坏机制，通常把分析对象直接转换为均匀各向同性的岩坡（或土坡）。若采用这种破坏机制，各向同性的节理岩体围绕点产生角速度为的旋转，把静止的部分和运动的部分分开的曲线即为速度突跃线，在这条角度为 m 、圆心为的滑弧上的任意一点上，速度都不是连续的，速度方向与该点处的切线成倾角 m 。在这种破坏机制下，外力所做的功和最大抵抗功可以表示为下列形式（Chen & Liu 1990；Maghous 等 。w e 和w mr为无量纲函数，1 和2 为滑移体的圆心角，由于屈服设计状态的动力学方法是结构稳固的一个必要条件，故有： 联立（5）式和（6）式，取1 和2 的最小值进行计算，可以得到第一种类型破坏机制的最佳上界估计： 分段刚性块体破坏机理Fig. 4(b) 第二种类型的破坏机制涉及到了两种均匀材料块体的转换，由五个角度参数定义。为了避免误解，应该具体指出，“块体”并不是指代初始状态下的岩基块体，在屈服设计运动学方法的框架下，它代表的不仅仅这个意思，一块均质节理岩体的运动可以近似看做是刚体运动。 对于第二种类型的破坏机制，运用上界运动学方法，可以得出以下结果：（在Frard (2000)的文中可以找到详细的计算过程） 其中U 表示下盘块体的速度（如图4-b所示）。因此，K +上界估计值为： 计算结果以及与直接计算结果之间的对比 经过计算，选定最优上界参数值为： 屈服时有 利用UDEC软件（Hart等 1988）对均质化方法的计算结果和直接计算方法的结果进行对比发现，当每一个节理的位置都已知时，利用后一种方法就可以求解这个问题，当节理间隙很规则，且倾角保持在与水平成10 的方向切割开挖平面时，随着节理数n的变化，计算出一系列的结果，点绘于图5中，与n相应的稳定性因子的估计值也在图5中表示出来，容易看出，随着分割节理数目的降低，这些估计值的大小降低到了均质化方法的估计值。均质化方法和直接计算法的差异可以看成是由于“尺寸效应”而引起的，而均质化方法并没有尺寸效应的问题。为了设计上计算的省时高效，克服直接计算法的局限，同时要运用均质化的概念，考虑对裂隙岩体介质采用一种新的描述方法Cosserat 或者是微极连续，Biot (1967)， Besdo (1985)，Adhikary & Dyskin (1997)，以及 Sulem & Mulhaus (1997) 对于分层岩体或者是块体结构都有所描述。本文的第二部分就是致力于应用这个模型来描述节理岩体介质的破坏特性。
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