外文翻譯---均質各向異性裂隙巖體的破壞特性

1、<p>　　Failure Properties of Fractured Rock Masses as Anisotropic</p><p>　　Homogenized Media</p><p>　　Introduction</p><p>　　It is commonly acknowledged that rock masses always d

2、isplay 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

3、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 structu

4、res inv</p><p>　　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

5、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-</p><p>

6、　　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 wo

7、rks 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 bodie

8、s. In this context, attention is primarily focused on the form</p><p>　　Since the previously mentioned direct approach is becoming highly complex, and then numerically untractable, as soon as a very large nu

9、mber 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 t

10、he 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 </p><p>　　The objective of the present paper is to derive a ri

11、gorous 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 orthogona

12、l 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 homoge

13、nization</p><p>　　Problem Statement and Principle of Homogenization Approach</p><p>　　The problem under consideration is that of a foundation (bridge pier or abutment) resting upon a fractured b

14、edrock (Fig. 1), whose bearing </p><p>　　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

15、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.</p><p>　　Likewis

16、e, the joints will be modeled as plane interfaces (represented by lines in the ?gure’s plane). Their strength properties are described by means of a condition involving the stress vector of components (σ, τ) acting at an

17、y point of those interfaces</p><p>　　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),

18、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

19、 the structure.</p><p>　　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

20、 jointed rock mass, insurmountable dif?culties 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 lo

21、wer than those of the rock matrix, the implementation of a kinematic approach would require the us</p><p>　　failure. Indeed, such a direct approach which is applied in most classical design methods, is becom

22、ing 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.</p><

23、p>　　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 su

24、ch 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).</p><p>

25、　　Macroscopic Failure Condition for Jointed Rock Mass</p><p>　　The formulation of the macroscopic failure condition of a jointed rock mass may be obtained from the solution of an auxiliary yield design bound

26、ary-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

27、 joints under plane strain conditions. Referring to an orthonormal frame Owhose axes are placed along the joints directions, and introdu</p><p>　　such a macroscopic failure condition simply becomes</p>

28、<p>　　where it will be assumed that </p><p>　　A convenient representation of the macroscopic criterion is to draw the strength envelope relating to an oriented facet of the homogenized material, whose

29、 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

30、 a the set of admissible stresses ( , ) deduced from conditions (3) expressed in terms of (, , ). The corresponding domain has</p><p>　　Two comments are worth being made:</p><p>　　1. The decreas

31、e 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

32、 as soon as condition is ful?lled.</p><p>　　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 n

33、otion 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).</p><p>　　Nor

34、 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 Boc

35、hler1981).</p><p>　　Application to Stability of Jointed Rock Excavation</p><p>　　The closed-form expression (3) obtained for the macroscopic failure condition, makes it then possible to perform

36、the failure design of any structure built in such a material, such as the excavation shown in Fig. 3, </p><p>　　where h and β denote the excavation height and the slope angle, respectively. Since no surcharg

37、e is applied to the structure, the speci?c 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 m

38、aximum 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</p><p>　　where θ=joint orientation and K+=non

39、dimensional 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 Fi

40、g. 4.</p><p>　　Rotational Failure Mechanism [Fig. 4(a)]</p><p>　　The ?rst class of failure mechanisms considered in the analysis is a direct transposition of those usually employed for homogeneo

41、us 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

42、which is kept motionless is a velocity jump line. Since it is an arc of the log spiral of angle and focus Ω the velocity discontinuity at any point of </p><p>　　The work done by the external forces and the m

43、aximum resisting work developed in such a mechanism may be written as (see Chen and Liu 1990; Maghous et al. 1998)</p><p>　　where and =dimensionless functions, and μ1 and μ2=angles specifying the position of

44、 the center of rotation Ω.Since the kinematic approach of yield design states that a necessary condition for the structure to be stable writes</p><p>　　it follows from Eqs. (5) and (6) that the best upper-bo

45、und estimate derived from this ?rst class of mechanism is obtained by minimization with respect to μ1 and μ2</p><p>　　which may be determined numerically.</p><p>　　Piecewise Rigid-Block Failure

46、Mechanism [Fig. 4(b)]</p><p>　　The second class of failure mechanisms involves two translating blocks of homogenized material. It is de?ned by ?ve angular parameters. In order to avoid any misinterpretation,

47、 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 kinematic approach, a wedge of ho

48、mogenized jointed rock mass is given a (virtual) rigid-body motion.</p><p>　　The implementation of the upper-bound kinematic approach,making use of of this second class of failure mechanism, leads to the fol

49、lowing results.</p><p>　　where U represents the norm of the velocity of the lower block. Hence, the following upper-bound estimate for K+:</p><p>　　Results and Comparison with Direct Calculation

50、</p><p>　　The optimal bound has been computed numerically for the following set of parameters:</p><p>　　The result obtained from the homogenization approach can then be compared with that derive

51、d 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 speci?ed, a series of calculations has been performe

52、d 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 </p><p>　　corresponding

53、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 homo

54、genization 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</p><p>　　homogenization

55、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 descri

56、ption 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 strati?ed or block structures.

57、 The second part of this paper is devoted to </p><p>　　均質各向異性裂隙巖體的破壞特性</p><p><b>　　概述 </b></p><p>　　由于巖體表面的裂隙或節(jié)理大小與傾向不同，人們通常把巖體看做是非連續(xù)的。盡管裂隙或節(jié)理表現(xiàn)出的力學性質要遠遠低于巖體本身，但是它們在巖體結構

58、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(jié)理所決定的。從地質力學工程角度而言，在涉及到節(jié)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。 </p><p>　　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(jié)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(wěn)定巖塊（G

59、oodman & 石根華 1985；Warburton 1987；Goodman 1995）；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 & Strack 1979； Cundall 1988）提出來的，其目的是用來求解顯式有限差分數(shù)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(jié)理模型。</p><p>　　既然直接求解的

60、方法很復雜，數(shù)值分析方法也很難駕馭，同時由于涉及到了數(shù)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 & Brown 1980；Hoek 1983）得出的一個經(jīng)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(guī)則的表面節(jié)理網(wǎng)絡所分割的巖體，可以看做是一個均質的連續(xù)體，由于節(jié)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。</p><

61、;p>　　本文的目的就是：從節(jié)理和巖體各自準則出發(fā)，推求出一個嚴格準確的公式，來描述作為均勻介質的節(jié)理巖體的破壞準則。先考查特殊情況，從兩組相互正交的節(jié)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(jié)理數(shù)目較少的巖體，存在一個尺寸效應（

62、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(yōu)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(xù)模型來描述節(jié)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。</p><p>　　問題的陳述和均質化方法的原理 </p><p>　　考慮這樣一個問題：一個基礎（橋墩或者其鄰接處）

63、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(jié)理交界面的強度</p><p>　　估算出來。巖基的破壞條件使用傳統(tǒng)的莫爾-庫倫條件，可以用粘聚力C 1和內(nèi)摩擦角? m 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(jié)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 (σ,τ)表示： </p><p>　　根據(jù)屈服設計（或極限分析）推斷，如果沿著

64、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。 </p><p>　　這個問題可以歸結為求解破壞發(fā)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(wěn)定。由于節(jié)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(jié)理強度特性遠遠低于巖基，從運動學角度出發(fā)的方法要求考慮到破壞機理，這就牽涉到了節(jié)

65、理上的速度突躍，而節(jié)理處將會是首先發(fā)生破壞的區(qū)域。 </p><p>　　這種應用在大多數(shù)傳統(tǒng)設計中的直接方法，隨著節(jié)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(jié)理間距L變得更小，加大了問題的難度。在這種情況下，對節(jié)理巖體使用均質化方法和宏觀等效連續(xù)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(jié)，在有關于加固巖土力學的文章中可以查到（de Buhan等 1989；

66、de Buhan & Salenc 1990；Bernaud等 1995）。</p><p>　　節(jié)理巖體的宏觀破壞條件 </p><p>　　節(jié)理巖體的宏觀破壞條件公式可以從對節(jié)理巖體典型晶胞單元的輔助屈服設計邊值問題中得到（Bekaert & Maghous 1996; Maghous等 1998）。現(xiàn)在可以精確地表示平面應變條件下，兩組相互正交節(jié)理的特殊情況，建立沿節(jié)

67、理方向的正交坐標系O ，并引入下列應力變量： </p><p>　　宏觀破壞條件可簡化為： </p><p><b>　　其中，假定</b></p><p>　　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α 為節(jié)理的方向，分別用σn 和τn 表示這個面上的正應力和切應力，用(, , ) 表示條件（3），推

68、求出一組許可應力(σn,τn ),然后求解出傾角α 。當α ≥? m 時，相應的區(qū)域表示如圖2所示，并對此做出兩個注解如下： </p><p>　　1. 從圖2中可以清楚的看出，節(jié)理的存在導致了巖體強度的降低。通常當時，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。 </p><p>　　2. 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α 。應該拋棄固

69、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（1960）或Mc Lamore & Gray（1967）所引入的。通過莫爾-庫倫條件進行擴展，利用各向異性張量的方法來描述各向異性也是不妥當?shù)?(Boehler & Sawczuk 1977； Nova 1980；Allirot & Bochler 1981) 。</p><p>　　在節(jié)理巖體開挖穩(wěn)定性中的應用 &l

70、t;/p><p>　　式（3）的封閉形式是從宏觀破壞條件中得到的，該式可以用來對此種材料的結構體進行破壞設計，如圖3所示的開挖，h 和β 分別表示開挖高度和邊坡</p><p>　　角。由于結構上沒有其他荷載，材料比重γ 就成為系統(tǒng)唯一的加載參數(shù)。該結構的穩(wěn)定性評價需要在破壞發(fā)生的部位算出最大可能高度 h+，通過標準量綱分析表明，這個臨界高度表示為： </p><p>

71、　　其中θ為節(jié)理方位角，K +為表示開挖部位穩(wěn)定性的一個無量綱因子，該因子的上界估計值可以分別使用圖4所示的兩種類型的破壞機制，通過屈服設計的運動學方法來確定。</p><p>　　轉動破壞機理 [Fig. 4(a)] </p><p>　　第一種類型的破壞機制，通常把分析對象直接轉換為均勻各向同性的巖坡（或土坡）。若采用這種破壞機制，各向同性的節(jié)理巖體圍繞點Ω產(chǎn)生角速度為ω的旋轉，把靜止

72、的部分和運動的部分分開的曲線即為速度突躍線，在這條角度為? m 、圓心為Ω的滑弧上的任意一點上，速度都不是連續(xù)的，速度方向與該點處的切線成傾角? m 。</p><p>　　在這種破壞機制下，外力所做的功和最大抵抗功可以表示為下列形式（Chen & Liu 1990；Maghous 等 。</p><p>　　w e 和w mr為無量綱函數(shù)，</p><p&g

73、t;　　μ1 和μ2 為滑移體的圓心角，由于屈服設計狀態(tài)的動力學方法是結構穩(wěn)固的一個必要條件，故有： </p><p>　　聯(lián)立（5）式和（6）式，取μ1 和μ2 的最小值進行計算，可以得到第一種類型破壞機制的最佳上界估計： </p><p>　　分段剛性塊體破壞機理[Fig. 4(b)] </p><p>　　第二種類型的破壞機制涉及到了兩種均勻材料塊體的轉換，

74、由五個角度參數(shù)定義。為了避免誤解，應該具體指出，“塊體”并不是指代初始狀態(tài)下的巖基塊體，在屈服設計運動學方法的框架下，它代表的不僅僅這個意思，一塊均質節(jié)理巖體的運動可以近似看做是剛體運動。 </p><p>　　對于第二種類型的破壞機制，運用上界運動學方法，可以得出以下結果：（在Fréard (2000)的文中可以找到詳細的計算過程） </p><p>　　其中U 表示下盤塊體的

75、速度（如圖4-b所示）。因此，K +</p><p><b>　　上界估計值為： </b></p><p>　　計算結果以及與直接計算結果之間的對比 </p><p>　　經(jīng)過計算，選定最優(yōu)上界參數(shù)值為： </p><p><b>　　屈服時有 </b></p><p>　

76、　利用UDEC軟件（Hart等 1988）對均質化方法的計算結果和直接計算方法的結果進行對比發(fā)現(xiàn)，當每一個節(jié)理的位置都已知時，利用后一種方法就可以求解這個問題，當節(jié)理間隙很規(guī)則，且傾角保持在與水平成10 °的方向切割開挖平面時，隨著節(jié)理數(shù)n的變化，計算出一系列的結果，點繪于圖5中，與n相應的穩(wěn)定性因子的估計值也在圖5中表示出來，容易看出，隨著分割節(jié)理數(shù)目的降低，這些估計值</p><p>　　的大小降低

77、到了均質化方法的估計值。均質化方法和直接計算法的差異可以看成是由于“尺寸效應”而引起的，而均質化方法并沒有尺寸效應的問題。為了設計上計算的省時高效，克服直接計算法的局限，同時要運用均質化的概念，考慮對裂隙巖體介質采用一種新的描述方法——Cosserat 或者是微極連續(xù)，Biot (1967)， Besdo (1985)，Adhikary & Dyskin (1997)，以及 Sulem & Mulhaus (1997)