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H.H. Einstein

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Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 48th U.S. Rock Mechanics/Geomechanics Symposium, June 1–4, 2014

Paper Number: ARMA-2014-7727

Abstract

Abstract Hydrocarbon extraction from unconventional oil and gas reservoirs requires more accurate ways to describe fracture processes in shale. Fracture initiation, propagation and coalescence has been studied in many rock-like materials [1,2] and natural rocks [1,2,3] . However, shale is typically heterogeneous and anisotropic with naturally formed bedding planes. Natural bedding planes can be weak zones where fractures can initiate and propagate along [1, 2, 3]. A series of unconfined compression tests were conducted on Opalinus clay shale with two pre-existing flaws and various bedding plane orientations (0, 30, 60, and 90 degrees with respect to the horizontal). High speed and high resolution imagery were use to capture fracture initiation, propagation and coalescence between the flaw pairs. Distinct coalescence and cracking patterns were observed when compared to previously tested rocks. As the bedding angle increased, fractures initiating at the flaw tips tended to propagate more frequently along the bedding planes. Coalescence of cracks between flaws trended from direct-combined (tensile and shear cracks) to indirect as the bedding plane angle increased. Tensile crack initiation and coalescence stresses showed a characteristic U-shape profile with a minimum at 60 degree bedding plane orientation and a maximum at 0 degree bedding plane orientation.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 46th U.S. Rock Mechanics/Geomechanics Symposium, June 24–27, 2012

Paper Number: ARMA-2012-421

Abstract

ABSTRACT: The fracture process zone (FPZ), a zone of weakened material surrounding the tip of a propagating crack, is common to many brittle materials, and is likely related to brittle material damage mechanisms. This study follows recent investigations of Carrara Marble and asks whether microstructure size, such as the size of marble grains, leads to an different extent of damage for a brittle material. Existing work has used acoustic emissions or laser interferometry and optical microscopy to answer this question, and found a positive relationship between grain size and size of the FPZ. Our study uses nanoindentation to probe the nanomechanical properties of the FPZ for two marbles of varying grain size, and attempts to relate mechanical properties of the FPZ to grain size. The marbles are from Carrara, Italy (typical grain size 300 µm), and Danby, Vermont (typical grain size 520 µm). Grids of nanoindentations were placed within the FPZ regions of Danby and Carrara marble specimens. Both marbles exhibited lower nanomechanical properties near the crack tip and near the area of future wing-crack formation, i.e. the FPZ. However, the Danby (large microstructure) marble exhibited this trend over a larger distance, and thus provides nanomechanical support for the increase of the FPZ with grain size. 1. INTRODUCTION Fracture mechanisms at the micro- and nanoscales govern many important geomechanical processes, such as drilling in rock [1]. An understanding of rock fracture at these small scales is thus critical to oil-, construction-, and other industries. The fracture process zone (FPZ) is a zone of damaged material surrounding the tip of a propagating crack [2,3]. This damage zone manifests itself in many brittle materials during fracture, and thus provides an opportunity to study fracture mechanisms. The FPZ of marble is the subject of this rock fracture study.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 46th U.S. Rock Mechanics/Geomechanics Symposium, June 24–27, 2012

Paper Number: ARMA-2012-593

Abstract

ABSTRACT: The study of crack initiation and propagation is important for the understanding of rock mass behavior, which affects many rock engineering problems. Such studies can be done experimentally in the laboratory or in the field, or numerically. Here, a numerical study is presented, in which the stress and strain fields around a flaw tip were analyzed using the finite element code, ABAQUS, to better understand the processes involved in crack initiation and propagation. Double-flaw geometries were modeled with ABAQUS with the intent of identifying the differences between stress and strain fields around the flaw tip, relating the stress and strain fields to crack initiation and propagation, and comparing numerical results with those of tests performed on gypsum and marble specimens. Both stepped and coplanar flaw geometries were studied, as well as different stages of crack propagation were modeled based upon laboratory results. For the stepped flaws, both stress and strain field analyses correctly explain wing and shear crack initiation and propagation in gypsum and marble. Furthermore, the two analyses are also capable of describing reasonably well tensile and shear coalescence in gypsum and marble, respectively. For the coplanar flaws, it was found that the stress field analysis is capable of explaining wing crack initiation and propagation observed in tests on gypsum and marble. It is also capable of explaining shear coalescence observed in gypsum, but it is not capable of describing the indirect coalescence observed in marble. The strain field analysis is not only capable of satisfactorily explain what the stress field analysis explains, but it also correctly describes the indirect coalescence that occurs in marble specimens. 1. INTRODUCTION The study of crack initiation and propagation is important for the understanding of rock mass behavior which affects many rock engineering problems.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 45th U.S. Rock Mechanics / Geomechanics Symposium, June 26–29, 2011

Paper Number: ARMA-11-449

Abstract

ABSTRACT This research study investigates the cracking processes associated with inclusion pairs of varying shape, orientation and inclusion materials. Specifically, this study summarizes a series of uniaxial compression tests on gypsum specimens with varying inclusion pair configurations. The inclusions consisted of differing materials, of contrasting Young’s Modulus (higher and lower than the matrix), shapes (hexagon, diamond, ellipse), and relative pair orientations (bridging angle). Similar cracking sequences were seen in the newly introduced inclusion pairs as in previous studies. Slightly increased debonding (usually corresponding to increased interface shearing) occurred as inclusion pairs with inclined interfaces were introduced. Coalescence behavior trended from indirect or no coalescence, to direct shear coalescence, to combined direct tensile-shear coalescence as the inclusion bridging angle was increased, similar to past studies on circular and square inclusion pairs and flaw pairs. Also, the coalescence related to inclusion interface inclination and bridging angles resembled the actual coalescence of flaw pairs with similar inclination and bridging angles. 1. INTRODUCTION The cracking processes in a brittle material consisting of a matrix with inclusions are important mechanisms for both natural materials (rocks) as well as synthetic composite materials (e.g. concrete). There have been many past studies regarding the cracking processes in brittle materials, which contain pre-existing cracks (called flaws) both analytically [1, 2], as well as experimentally [2, 3, 4]. Also, the cracking processes in brittle materials, which contain inclusions have been studied both analytically [6, 7, 8] and experimentally [7, 9, 10, 11, 12]. Only recently have experiments been performed with the technology capable of capturing high speed imagery to fully describe the crack propagation and coalescence behavior in a brittle material. The majority of the previous research performed on brittle materials with inclusions investigated the fracturing patterns associated with circular or rectangular (square) inclusions. The present research was conducted to develop a more detailed description of the coalescence patterns of uniaxially loaded gypsum specimens with inclusion pairs of varying shape, stiffness and orientation. Emphasis was placed on the coalescence behavior associated with the effects of varying these inclusion pair configurations. 2. PREVIOUS STUDIES 2.1 Flaw Coalescence Studies Amongst the many experimental studies regarding flaws in brittle materials, experimental work done by Wong and Einstein [5] is particularly significant because it incorporated the use of a high speed camera to follow crack propagation and coalescence. One of the most important contributions of Wong and Einstein’s study was a proposed set of coalescence categories for different co-planar and stepped flaw pairs (Figure 2.1). These coalescence patterns will later serve as a basis for comparing the coalescence patterns seen in brittle materials containing inclusions pairs. 2.2 Inclusion Coalescence Studies Extending on the macro-scale flaw testing techniques used in the Massachusetts Institute of Technology (MIT) Rock Mechanics Laboratory, brittle material with inclusions was investigated with high speed imagery by Janeiro and Einstein [12]. That study tested 1” single square, circle, diamond, and hexagon inclusion shapes as well as 1/2" circular and square inclusions with varying inclusion material stiffness (Figure 2.2).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 44th U.S. Rock Mechanics Symposium and 5th U.S.-Canada Rock Mechanics Symposium, June 27–30, 2010

Paper Number: ARMA-10-309

Abstract

ABSTRACT: This study explores the interaction between crack initiation and nanomechanical properties in the crack tip process zone (zone of microcracking at the tip of a propagating crack) of a brittle material. Samples of Carrara marble with pre-existing cracks (“flaws”) were loaded in a uniaxial testing machine until the process zone appeared at the tips of the pre-existing cracks in the form of “white patching”. Two techniques were then used to obtain nanomechanical properties of the process zone and relate them to macroscale crack initiation: digital photography, to visually assess the macrostructure and crack formation, and nanoindentation, to yield nanomechanical properties and assess nano/microheterogeneities. Nanoindentation testing was comprised of lines and grids of single nanoindentations located both near and far from the process zone. The purpose of nanoindentation testing is to investigate the underlying trend in nanomechanical property change between intact and process zone marble. Analysis of nanoindentation testing results showed a decrease of both modulus and hardness (a) near grain boundaries in intact material, and (b) with closeness to the process zone. Ultimately, the study confirms that the crack tip process zone manifests itself as an area of reduced nanoindentation hardness and nanoindentation modulus in marble. 1. INTRODUCTION The study of geomaterial cracking at its most fundamental scale, nano and microscales, is critical to predicting crack propagation. With this in mind, this study explores the interaction between crack initiation and nanomechanical properties in the crack tip "process zone" of Carrara marble. Nanomechanical properties of material in the fracture process zone (FPZ), i.e. the material within the zone of microcracking around the crack tip, are compared with nanomechanical properties of the "intact material". The differences in nanomechanical properties between intact and process zone material reveal the existence of nanoscale damage in the FPZ that may well contribute to the fracture propagation. The literature contains extensive information on the three defining aspects of this investigation - process zone, nanomechanical properties, and geomaterials - but this study represents the first occasion to bring these aspects together in such a way. Many theories exist regarding the process zone, but the experimental investigation of this theoretical region in rock is a relatively recent development. In Linear Elastic Fracture Mechanics (LEFM) the process zone is defined as the core region of plastic-behaving material. Rice investigated the process zone region from a theoretical perspective to ultimately configure the J-Integral, which expresses the energy release rate. However, the integral does not consider the interplay between nano/micromechanical properties and the energy release rate or fracture energy [2]. The Dugdale model for the size of the process zone, or “inelastic zone,” has been applied even at the fault scale (several kilometers), but does not consider the variability of properties in the region [3]. Application of the closely related Cowie and Scholz model to existing faults reveals a logarithmically decreasing microfracture density within and moving away from the “cohesion zone” (a low-strength area surrounding a displaced fault), and a finite scale-independent limiting microfracture density [4].

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 42nd U.S. Rock Mechanics Symposium (USRMS), June 29–July 2, 2008

Paper Number: ARMA-08-162

Abstract

ABSTRACT: Uniaxial compression tests were conducted on prismatic Barre Granite specimens with two pre-cut, straight, open flaws. Using a high-speed video system, crack initiation, propagation, and coalescence were observed. Coalescence patterns for the granite specimens fit into a previous framework (developed for Molded Gypsum and Carrara Marble) except for one new coalescence pattern. The crack initiation stress and the maximum stress were measured for each specimen, and these results are interpreted, also. 1. INTRODUCTION Crack coalescence, which is the linkage of pre-existing flaws, is a common phenomenon in nature. The current study examined coalescence in Barre Granite as an extension of previous work presented by the MIT rock mechanics group [1]. A high-speed video system was used to record the coalescence of two artificial flaws in Barre Granite (the term flaw will be used in this paper to refer to an artificially made, pre-existing crack). This observation method made it possible to distinguish between shear and tensile cracks during their formation and propagation as well as record the sequence of cracks during coalescence. The observed coalescence behavior was complex, but also fit into the framework proposed earlier for Molded Gypsum and Carrara Marble [2]. 2. EXPERIMENTS 2.1. Specimen Preparation While this study used Barre Granite, several other materials have been used in the past being part of the continuing research of the MIT rock mechanics group. For nearly forty years, the group has studied the behavior of discontinuous (jointed) geo-materials [3]. In this context, a specific study began 17 years ago to experimentally investigate the effects of material type, geometric parameters, and loading conditions (Table 1) on fracture initiation, propagation, and coalescence [1, 4, 5, 6]. Bobet continued his work at Purdue University [7]. Several other groups have also researched crack coalescence in geo-materials [e.g. - 8, 9, 10]. To enable one to make comparisons between this study and previous work, similar procedures for specimen preparation were followed. Prismatic specimens of Barre Granite specimens with dimensions ~152 mm x~76 mm x~25 mm (6" x 3" x 1") were prepared. North Barre Granite, Inc. cut slabs ~25 mm (1") thick with a diamond saw. The other two dimensions (6" and 3") were cut with an OMAX waterjet, which cuts with a high-pressure mixture of water and a garnet abrasive. Two open, straight flaws 12.7 mm (0.5") long were cut in each specimen with the waterjet. The flaws were cut with different geometric relationships. Ligament length (L) was always equal to flaw length. The flaw inclination angle was varied (ß= 0°, 30°, 45°, 60°, and 75°) for two values of bridging angle (a= 0°?and 60°). As a consequence, flaw pairs were either coplanar (a= 0°) or left stepping (a= 60°) and could be nonoverlapping, partially overlapping, or completely overlapping. Three specimens were prepared and tested for each flaw-pair geometry. The two geometries summarized in Table 2 were geometries identical to those tested by Wong for Molded Gypsum and Carrara Marble [1]. Table 1. Parameters tested by the MIT rock mechanics group (both specimen parameters and loading conditions). Refer to Figure 1 for explanation of terms marked with an asterisk.(available in full paper)

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 31st U.S. Symposium on Rock Mechanics (USRMS), June 18–20, 1990

Paper Number: ARMA-90-0261

Abstract

ABSTRACT INTRODUCTION Rock fracture parameters such as size, attitude, spacing and shape vary in space in a way that can be practically described only in probabilistic terms. Stochastic fracture geometry models have been developed by Baecher, et al. (1977), Veneziano, (1979), Dershowitz, (1985), Long et al. (1985, 1987), Hestir et al., (1987) and La Pointe and Hudson (1985) among others. These models are not completely satisfactory because: -They do not account for spatial nonhomogeneities such as fracture clustering (An exception is the parent/daughter model of Long et al., 1987). -The models are only loosely tied to the geologic genesis of the fractures. In particular, most models assume independence among fracture sets. From a physical viewpoint, this assumption is often incorrect. -Only in a few cases have the models been validated using actual fracture data. A way to address these concerns is proposed here. The main features of our model are that fracture sets are described in a hierarchical order and dependencies among fractures of the same set or of different sets are accounted for. The sequential generation and correlation of fracture sets correspond to what happens in nature. Equally important as the modelling principle is the availability of statistical procedures to estimate parameters and validate the model. In its present form, the model is two-dimensional, i.e. it can be used to describe fracture trace patterns on outcrops. Some comments will be made at the end of the paper how to extend the hierarchical model. The model will be presented by first introducing the basic ideas and then developing the details, the latter simultaneously with showing an application to actual data.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 22nd U.S. Symposium on Rock Mechanics (USRMS), June 29–July 2, 1981

Paper Number: ARMA-81-0361

Abstract

ABSTRACT 1. INTRODUCTION Uncertainty in predicting geological conditions often leads to postulating worst conditions and thus to conservatism in design and construction. Savings are possible by adapting design and construction methods to the conditions actually encountered during excavation. Specifically, among the excavation and support methods that can be technically used for a given combination of geological parameters, only one will be the most economical. Usually, however, the cost of changing construction method is such that adapting to different geologic conditions is only economical if these conditions persist over long segments of the tunnel. It follows that in defining a set of design-construction options prior to construction and in adaptively selecting them in the course of the excavation, one has to take into account the variability of geologic conditions. This can be done by probabilistically describing the geologic conditions, and by then selecting excavation-support methods using the tools of decision theory under uncertainty. Information that becomes available during construction should of course be used to update the probabilistic description of the geologic condition ahead of the tunnel face. A simple model that accomplishes these objectives is proposed here. Its viability is demonstrated through a case study analysis of the Seabrook Power Station discharge tunnel. Prior to this example, modeling assumptions are stated and a few results from mathematical analysis of the model are given here. 2. PROBABILISTIC GEOLOGICAL PREDICTION An attractive feature of the probabilistic method for geological prediction described in this section is that it makes more complete use of information already available, without requiring new or more sophisticated exploration programs. Therefore, the approach is implementable under standard practices and procedures. It is based on a Markov-process representation of the spatial variation of geologic parameters. The form of this process makes it possible to easily reduce parameter uncertainty as more information becomes available during preliminary exploration and subsequently during tunnel excavation. Mathematical analysis readily provides initial and updated geological predictions which form the key data for decision making. 2.1 Modeling Assumptions Tunneling operations and financial planning depend on such parameters as rock type, faulting, degree of jointing and permeability. Some parameters are discrete. For example, "rock type" X 1 may be Schist (X 1 =1) , Metaquartzite (X 1 =2) , and Diorite (X 1 =3). Other parameters are continuous but can be conveniently discretized. For example, "degree of jointing" X 2 may be classified as not severe (X 2 =1) or severe (X 2 =2). Discretization corresponds to accepted practice and greatly simplifies the analytical model. From a mathematical point of view, the generic parameter X i can be regarded as a scalar random function of distance from the portal of the tunnel (Fig. 1). Simultaneous characterization of the vector Random process X (l)=[X 1 (l) .... X n (l)] T can be either through the joint characteristics of its components (through the distribution of X(l) for each l, the joint distribution of x(l 1 ) and x(l 2 ) for each l 1 and l 2 ' and so on) or through the marginal characteristics of one component and the conditional characteristics of the other components.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 20th U.S. Symposium on Rock Mechanics (USRMS), June 4–6, 1979

Paper Number: ARMA-79-0233

Abstract

ABSTRACT ABSTRACT A numerical method, the displacement discontinuity method, has been reformulated and computerized to provide a practically applicable method for comprehensive analysis of rock masses. In particular, stiffness relationships have been incorporated to describe comprehensively the history-related behavior of rock discontinuities. The method has been successfully applied in the analysis of fracture propagation through loaded wedge of intact rock. INTRODUCTION The behavior of rock masses is usually governed by the geometry and behavior of discontinuities. The effect of discontinuities can only be fully understood and analyzed if their history is taken into consideration. By determining and analytically describing the behavior of individual discontinuities from their creation to the present state it becomes possible to better predict their subsequent behavior. This paper describes a model that is capable of providing such an analytical description and then illustrates the model in an application. DISPLACEMENT DISCONTINUITY METHOD (DDM) Various numerical techniques are available to model the behavior of a discontinuous rock mass. The method chosen and described here is the Displacement Discontinuity Method (DDM), which is an influence function technique, similar to boundary element or boundary (J) integral methods. Such methods have been developed and applied by various researchers(Ronved and Fraser (1958), Hackett (1959), Berry (1960), Bilby (1960), Weertman (1964), Massonet (1965), Rizzo (1967), Cruse (1969), Benjumea and Sikarskie (1972), Diest et al (1973), Thompson (1974), Crouch (1976), Lachat and Wilson (1976), Brady and Bray (1978),Roberds (1979), Among others). The DDM incorporates the ability to adequately model rock masses as a quasicontinua similar to finite element techniques, without either the high cost or imposition of often artificial boundaries inherent in finite element modelling. Analysis by DDM essentially entails the representation of a problem as surfaces of displacement discontinuity within an infinite and otherwise continuous body (Fig. 1a).

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 20th U.S. Symposium on Rock Mechanics (USRMS), June 4–6, 1979

Paper Number: ARMA-79-0317

Abstract

ABSTRACT: Rock slope stability analysis based on a single statistically representative orientation for each joint set can lead to erroneous conclusions regarding the stability of a slope. The fact that joint orientations are random variables must be considered in slope reliability analysis consisting of probabilistic kinetic and kinematic analysis. This paper presents a numerical procedure to establish the probability of kinematic instability of a 2-joint rock wedge. The method is a first step in a slope reliability analysis and, can be used, by itself, for preliminary slope geometry optimization. 1. INTRODUCTION Stability analyses for rock slopes based on limit equilibrium approaches usually employ a two stage procedure. In the first stage one identifies bodies that can potentially move--the so called kinematic analysis. In the second stage one assesses the stability (or instability) of those bodies by examining force equilibrium--the so called kinetic analysis. In rock slopes the potential failure bodies are usually (but not necessarily) bounded by discontinuities i.e. joints, foliation surfaces, bedding planes or faults. The movement of the failure body may occur in any several modes: translational sliding, rotational sliding, toppling, falling or a combination of modes. If a limit equilibrium approach for stability is used, the kinematic analysis must be performed for each of the failure bodies and failure modes. This paper treats the kinematic analysis for a wedge bounded by two surfaces (Figure 1) within the slope and two free surfaces defining the slope, i.e., the classic 2-plane rock wedge is considered. However, the concept and approach discussed here can be applied to any of the other failure bodies. The two surfaces of the wedge within the slope consist--as indicated above--of discontinuities.* The usual procedure is either to use specific discontinuities or to determine statistically representative discontinuities to define the wedge.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the 19th U.S. Symposium on Rock Mechanics (USRMS), May 1–3, 1978

Paper Number: ARMA-78-0090

Abstract

ABSTRACT: The Probabilistic Model for Shearing Resistance of Jointed Rock is a step in the development of complete methods for reliability analysis of rock slopes. The model derives the probability distribution of the strength of a discontinuous rock mass taking uncertainty on the joint pattern into consideration. The distributions of joint spacing and length from field surveys, the shear resistance parameters of intact rock and rock discontinuities, and the in-situ stress field are input data to the model. The paper describes the two main parts of the method, the stochastic model of joint geometries and the mechanical model. The results of actual computations are then presented and discussed to show the practical applicabilities of the method and to examine the sensitivity to various input parameters. The main qualitative, conclusion is that the probability distribution of apparent persistence is insensitive to variations of the stress field and of the joint orientation relative to the stress field. Apparent persistence depends strongly however on the distribution parameters of joint spacing and length together with cohesion of intact rock. 1. INTRODUCTION Probabilistic approaches to rock slope stability analysis have become increasingly common because they provide a rational incorporation of the uncertainty of parameters affecting slope stability. Risk analyses and Bayesian updating techniques used in site exploration require the use of probabilities of failure rather than the traditional factor of safety. Complete probabilistic modeling is theoretically possible but may be somewhat premature because the assumed mechanisms and associated parametric relations may introduce inaccuracies that have a greater effect than the uncertainty of individual parameters. In this paper a limited topic, the effect of discontinuity geometry on slope stability, will be treated. This partial model can later be incorporated into a complete probabilistic slope stability analysis.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 18th U.S. Symposium on Rock Mechanics (USRMS), June 22–24, 1977

Paper Number: ARMA-77-0400

Abstract

ABSTRACT ABSTRACT Statistical description of rock mass properties is essential for two reasons: 1 -Analyses in rock engineering require statistical descriptions to take the distributive character of properties into account and 2-Field sampling requires statistical descriptions to develop sampling plans and to draw inferences from data. For both purposes, it is essential to know appropriate distributions of rock mass properties. Based on the evaluation of a large number of joint data and taking previous work into account, it was determined that the best fitting distribution for joint "length" is lognormal and for joint spacing exponential. Based on these conclusions, a model was developed for inferring joint set parameters and for estimating the intensity of jointing (joint surface area per volume) from outcrop data. Intensity of jointing is an indicator of persistence. 1. INTRODUCTION Accuracy in predicting rock mass behavior depends on adequate characterization of in situ rock properties. The performance of natural and cut slopes, of underground openings and of foundations on rock depends on geometric characteristics of discontinuities (attitude, spacing and persistence), on the resistance characteristics of the discontinuities and the intact rock, and on the effect of water. Difficulties arise because parameters describing these characteristics are not unique, but distributed. This distributive character affects both analyses in which rock mass parameters are used and sampling plans by which parameters are obtained. Analysis should incorporate uncertainty associated with parameter distribution. Traditionally, this has been done using factors of safety. Factors of- Safety do not relate parameter distributions with the probability of failure or excessive deformation. Such predictions can only be made using probabilistic approaches. Field sampling is in part a statistical problem. Thus statistical techniques should be used both to develop sampling plans and to draw inferences from collected data. As probabilistic analysis in rock engineering becomes more common, statistical sampling will become absolutely necessary. Statistical analyses depend on assumptions, of which an important one is the distributional form of the properties sampled. Most uses of statistics in sampling and analysis are based on standard assumptions, the most common of which is Normality. To the extent these assumptions reflect reality, inferences are accurate. However, the assumptions themselves are seldom tested. This paper concentrates on determining appropriate distributions of joint spacing and joint surface or "length" (surface or length are related to persistence), Extensive joint data from two rock excavation sites were analyzed and various analytical forms tested. In Section 2 the literature on distributional properties of jointing is summarized, and in Section 3 the present data is reviewed in light of that previous work. In Section 4, a model of joint geometry is presented and used to draw inferences from field data. In Section 5 these inferences are related to estimates of persistence. 2. LITERATURE Since 1970 attempts have been made to determine appropriate distributional forms of joint length and spacing. These are summarized in Table 1. Priest and Hudson (lg76) have shown that combinations of evenly spaced, clustered and randomly spaced discontinuities will yield an exponential distribution of spacing. However, large predominances of evenly spaced joints lead to normal distributions.

Proceedings Papers

Publisher: American Rock Mechanics Association

Paper presented at the The 11th U.S. Symposium on Rock Mechanics (USRMS), June 16–19, 1969

Paper Number: ARMA-69-0083

Abstract

ABSTRACT The objective of the model studies described in this paper is to determine the effect of planar discontinuities on the strength and deformability of a rock mass. A model material was used because it satisfied concurrently three requirements that available natural-rock samples did not: (1) low strength, which was desirable for minimizing the loads that had to be applied in the laboratory; (2) ease of producing test specimens with geometrically regular joint patterns; and (3) uniformity of the test specimens. The model tests were performed on specimens made from a single modeling material and tested under triaxial stress conditions with s2--s3. Results are presented in this paper for tests on specimens that had (1) a single joint set inclined at several angles to the major principal direction, including 0° and 90°, and (2) two joint sets that were mutually perpendicular, one set perpendicular to and one set parallel to the major principal direction. Several different joint spacings were used in each series. Future tests will include a series with two sets of joints that make oblique angles with each other and with the major principal direction. The results of model tests and tests on cores of jointed rock by several investigators (e.g., MÜller and Pacher, 1 Moore, 2 Krsmanovic and Milic, 3 Hayashi 4 , Rosenblad 5 , Jaeger 6 , and Lane and Heck 7 ) have contributed significantly to the design of these experiments, which are believed to be the first in which a comprehensive study has been made of the combined effects under triaxial loading of joint spacing and joint orientation (Fig. 1). PROCEDURE Selection of Model Material A substantial investigation was made to select an appropriate modeling material. That investigation has been described in detail by Nelson and Hirschfeld, 8 and the most important aspects of it are reviewed in the following paragraph. Selection Criteria SIMILITUDE REQUIREMENTS --The strength of a jointed rock mass can be represented by tile function (Mathematical Equation)(Available in full paper) This function can be written in terms of dimensionless factors expressed in terms of the foregoing variables; for example (Mathematical Equation)(Available in full paper) Each of the dimensionless factors (¿-factors) must be the same for model and prototype ( (¿model=(¿prototype ) if the model is to fulfill the requirements of similitude. It was not the purpose of this study to model any particular prototype rock but rather to ensure that the model correspond so the general range of brittle rocks that are most commonly encountered in civil engineering. The first two ¿-factors (Mathematical Equation)(Available in full paper) were combined into a single ¿-factor (Mathematical Equation)(Available in full paper), which, together with the shape of the stress-strain curve, the modulus of deformation, and the failure mode, is an indicator of ?brittles.? Typical values of (Mathematical Equation)(Available in full paper) for brittle rocks are between 10 and 20, occasionally even higher. Values that are higher than 20 have usually been determined in investigations in which the tensile strength was determined by means of point-load tests, which tend to give lower tensile-strength values than carefully conducted tension tests, and correspondingly higher apparent values for the ratio (Mathematical Equation)(Available in full paper).