Engineering Fracture Mechanics Vol. 37, No. 1, pp. 59-74, 1990 0013-7944/90 $3.00+ 0.00 Printed in Great Britain. Pergamon Press plc.

CRITERIA FOR BRITTLE FRACTURE IN COMPRESSION

E. Z. LAJTAI, B. J. CARTER and M. L. AYARI

Departments of Civil and Geological Engineering, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

Abstract-Fracture nucleation and propagation in the compressive stress field of the geological and the mining environment is considered with the purpose of formulating an empirical, but general fracture criterion that is in agreement with experimental evidence. Present fracture criteria are inadequate for compressive loading. The boundary stress-based theories ignore the effect of the stress gradient while the critical stress intensity concept of fracture mechanics neglects the normal stress that acts parallel with the direction of fracture propagation. A new, empirical crack resistance (CR) function is defined based on experimental data and then combined with an ‘averaged’ state of stress in front of the cracktip to formulate a ‘crack driver’ (CD) function. The crack driver is analogous to the safety factor, but with values greater than unity representing the fractured state. The crack driver concept is implemented to predict the nucleation and propagation of fracture in a compressive environment. The evolution of the failure process around underground openings is then described, with special reference to the primary, the remote and the slabbing types of fracture of rock mechanics and mining terminology.

INTRODUCTION

FRACTURE in the compressive stress field of underground workings is of great practical concern. In the high stress field of the mining or the tunnelling environment, the failure of the intact rock component of the rock mass is a common occurrence. It may manifest itself through seemingly harmless microcracking, quiet stable macroscopic spalling, or through the frightening violence of rock bursts. Although fracturing at all scales has been experienced through the centuries, the success of developing a theory to explain and more importantly, to predict fracture of rock seems to have eluded the rock mechanics community. The search for a comprehensive theory of fracture has a long history. Failure of materials is a common engineering problem, thus research in this area is carried on by several fields of engineering science. Rock mechanics in particular has been borrowing extensively from continuum mechanics, material science, structural geology and more recently from fracture mechanics.

Traditionally, the approach to fracture in rock mechanics has been through stress-based, strength theories. This method can now be contrasted with the fracture mechanics technique which uses the stress-intensity factor formulation. Assumptions are made in both approaches, either to simplify the mathematics or to make the formulation suit the area of application. The purpose of this paper is to show that for fracture in a compressive stress field, both approaches are handicapped by these assumptions.

STARTING WITH THE GRIFFITH THEORY

Practicing engineers in the rock environment are always baffled by the emphasis fracture mechanics place on microscopic flaws. In their view, the rock mass has plenty of macro- and megascopic discontinuities which influence rock response, without having to worry about features that cannot be seen with the naked eye. Rock engineers prefer to work therefore, at the more ‘practical scale’ of the rock mass and the rock supports, and with the stress field that surrounds them. They are also aware of the fact that the response of the rock mass to stress is just too complicated to be reliably modeled through the use of conventional analogs. For the rock mechanic, the physical and the numerical models serve only as guides to rock response and are never expected to be reliable predictors of material behaviour. Therefore, the stress-based safety factor approach (e.g. [1]) is considered to be quite adequate in a design that ultimately must call on engineering judgement. This approach has been deemed to work at the ‘practical’ level, but it is of limited use to the theoretician. The latter must work from fundamental principles, which in this case unavoidably leads back to the work of Griffith[2].

59

60 E. Z. LAJTAI et al.

The Griffith theory, or at least its basic premise that fracture starts from flaws, is fundamental to all disciplines investigating brittle fracture. The original Griffith theory is of course stress-based; it uses the Inglis[3] solution for the distribution of stress around an elliptical tunnel. On the other hand, the discipline of fracture mechanics has changed to the theory of the mathematical crack, which seems to have originated with Westergaard[4].

The Griffith, and the Modified Griffith theories[5] of rock mechanics and the discipline of fracture mechanics, both model the crack starting flaw as a ‘flat ellipse’. Besides the obvious exclusion of starting flaws of other shapes, the introduction of the flat flaw concept has had one disquieting consequence. Flat cracks do not disturb the distribution of the crack-parallel normal stress (the axial compressive stress), and hence this component of the stress tensor is neglected in fracture mechanics theory. In an evaluation of the Griffith theory in compression through physical modelling, Lajtai[6] has shown that this assumption leads to erroneous results in the case where the starting flaws do not close under stress. This includes a variety of equidimensional flaws in rock, such as pores, vugs, and other voids and possibly elliptical cavities that have aspect ratios higher than about 0.01. The problem is just as serious for the mathematical slit of fracture mechanics. The three independent modes of fracture correspond to loading by the normal stress component acting perpendicular to the slit (the opening mode) and the two shear components directed parallel to the slit (the sliding and tearing modes). Neglecting the other two, crack parallel, normal stress components creates some degree of ‘academic’ discomfort. By putting a normal stress component of the stress tensor to nil, an originally all-compressive stress field is drastically altered; one of the principal stresses becomes either zero (uniaxial compression) or in the more general case, tensile. Using the theory of the flat crack, one must however accept the lack of contribution from normal stress directed parallel to the slit. In addition to the ‘discomfort’, there is a more compelling reason for not neglecting at least one of the normal stress components. Observation of fractures in laboratory specimens or in the tectonically deformed rocks of nature suggests rather clearly, that the dominant mode of fracture in compression, for all kinds of rock types, is consistently in the direction of the far-field maximum principal stress (compression considered positive in this paper). In experiments, such cracks are seen to extend in response to the increasing maximum principal (compressive) stress. In compression, it is one of the two normal stress components, neglected by fracture mechanics, which may control the fracture process! In uniaxial compression, the ‘neglected’ normal stress, the only non-zero far-field stress component, is in fact the sole cause of fracture. The introduction of a fourth mode of cracking (the axial compression mode?), would be required to account for the dominant mode of fracture in the compressive stress enironment of underground activities.

THE PHYSICAL EVIDENCE FOR FRACTURE IN ROCKS

The flat starting crack, the flat ellipse from Inglis, or the mathematical slit of fracture mechanics, is the centre piece of all theories of fracture. Ironically, the published literature is devoid of convincing visual evidence for fractures starting from such pre-existing features. Certainly, there have been many notable demonstrations of the so-called wing cracks propagating from modeled starting cracks[7,8,9, lo], but to the authors’ knowledge no natural equivalents have yet been documented. Over the years, the authors have examined many cracks in microscopic thin sections of Lac du Bonnet granite, that in total length would add up to about three-quarters of a meter. Only three cases have been identified where the analogy to the Griffith model is striking. One of these, where the grain boundary around a biotite flake assumes the appearance of the Griffith crack, is shown in Fig. 1. In the other two cases, the wing cracks of the Griffith configuration nucleated at inclined grain boundaries between grains of quartz. The similarity however is not perfect. The initial curvature as the extending fracture is expected to swing into the direction of the maximum principal (compressive) stress is either missing, or perhaps is only observable at a smaller physical scale. At the microscopic scale of investigations, the whole length of the stress-induced cracks is approximately parallel with the far-field compressive stress direction. In fact, the inspection of test specimens, that were stressed in triaxial compression to just below their strength, suggests that over 70 per cent of the cracks propagate within ten degrees of this direction (Fig. 2). When fractures depart from this trend, they do it mostly for obvious microstructural reasons (Fig. 3).

Brittle fracture in compression 61

Fig. 1. A natural ‘Griffith Crack’. A biotite grain, or the boundary around it nucleates two long wing cracks. Lac du Bonnet granite, 40 x magnification in transmitted light microscopy. Loading in triaxial

compression with the maximum principal stress being vertical. Fig. 2. Tensile fractures nucleating at various points in the rock, subjected to a compressive stress field, and extending parallel with the maximum principal (compressive) stress trajectory. Lac du Bonnet granite, 40 x magnification in transmitted light microscopy. Loading in triaxial compression with the maximum

principal stress being vertical.

Fig. 3. The strong compressive-stress parallel trend is sometimes disrupted by microstructural effects, in this case the weak cleavage in biotite deflects the fractures. Furthermore, fractures are not distributed uniformly in the rock, but rather display a certain amount of ‘crack-bunching’. Lac du Bonnet granite, 40 x magnification in transmitted light microscopy. Loading in triaxial compression with the maximum

principal stress being vertical. Fig. 4. ‘Step-out Fractures’ in Lanigan potash. The primary fracture creates stress concentrations off its tip which generates the ‘step-out’ fractures. Lanigan potash, 4x magnification in reflected light microscopy. The mother crack is a ‘remote’ fracture that formed off the perimeter of a 66 mm cylindrical cavity located in a 230mm high, 2SOmm wide and 120mm deep block of potash loaded in uniaxial

compression.

62 E. Z. LAJTAI et al.

Fig. 5. ‘Remote Fractures’ forming around a cylindrical cavity in a Lanigan potash block loaded in uniaxial compression (vertical). Block dimensions are as described in Fig. 4. Remote fractures form about 80 per cent of the total in this potash and also in similarly tested granite. Only 20 per cent of the cracks are nucleated at the tensile stress concentrations of the perimeter. Slabbing at the compressive stress concentrations are not obvious at this scale. A few are observable under the microscope. Note the ‘en-echelon’ development at the top left and the incipient buckling instability between the top two cracks.

Brittle fracture in compression 63

There is no doubt however that grain boundaries in polycrystalline rocks are favoured as crack starters. For Lac du Bonnet granite, about 85 per cent of the cracks start at a grain boundary while only 15 per cent nucleate inside a crystal. The most effective grain boundaries in this case are those that separate quartz grains. Although boundaries of all orientations seem to start new cracks, a statistical distribution suggests that the most effective ones are those that run in the 20-40 degree band measured from the compression direction. The 30 degree direction, of course, is the direction of the ‘critical crack orientation’ of the Griffith theory. With the mentioned exceptions, only a few of the grain-boundary nucleated cracks conform to the typical Griffith configuration of wing cracks. More commonly, several cracks are seen to start at the same grain boundary suggesting that the full length cannot be modeled as a single starting crack; in fact, there may be several potential sites at the same grain boundary which might become active independently from each other. This observation has been reported by others as well[l 1] suggesting that the starting flaws could be located at the submicroscopic rather than microscopic scale.

An explanation for the parallel growth of fractures in compression is found in the concept of fracture occurring from a multitude of crack starters that more or less act independently. Fracture nucleation from a number of sites is possible, because the process of propagation in compression is stable; the compressive stress must continuously increase to drive the process. In response to the rising stress, the less favourable sites become critical as well and new fractures are nucleated. Although the most striking feature of brittle fracture in compression is the propagation of a set of cracks parallel with the compression direction, there are a number of additional features of this process that may be just as important.

Ultimate failure of brittle rocks is often thought to occur through some type of coalescence of a suitably positioned set of parallel cracks. In fact, there is no physical evidence for this in pre-failure specimens. The cracks propagating in the compression direction show little inclination to leave the maximum principal stress trajectory. There are however other types of interference among the parallel cracks. One type follows from the ‘shadowing effect’ of adjacent cracks. Horii and Nemat-Nasser[8] demonstrated the sudden and explosive growth of fractures between two adjacent cracks. This effect should be even more important when a confining pressure, acting perpendicular to the crack, is present as well. A normal stress acting laterally would retard crack extension, but its effect is diminished in the stress shadow formed by two proximally growing and presumably dilating cracks. The consequence of this is the formation of several closely spaced fractures. This could explain why cracks are not uniformly distributed in the tested rock specimens, but tend to crowd in certain parts. This ‘crack bunching’ is a common feature of the fracture pattern observed in Lac du Bonnet granite (Fig. 3).

Two proximally growing cracks are not even necessary to generate additional cracks in the immediate neighbourhood. A new type of crack interaction, in the form of a more direct mother-daughter relationship was observed while testing a block of Lanigan potash, a salt rock from Saskatchewan. Stabilizing, compression parallel cracks were seen to generate offset, or ‘step-out’ fractures (Fig. 4). This process was repeated with the daughter crack, giving rise to a set of fractures. Fracture sets of this type are notably unstable. In response to increasing compression, the rock bridge between the end of the mother and the starting point of the daughter crack was observed to become the site of additional cracks. Eventually, the thin slabs of rock between the newly formed cracks showed signs of buckling instability (top left corner of Fig. 5).

There is also evidence for the existence of ‘step-out’ fractures in nature where they are usually called ‘en-echelon’ fractures[ 121. Tertiary mafic dikes on the Colorado Plateau[ 131 provide one good example. Closely spaced joints near the dike tips were interpreted to have formed in response to the ‘tensile maxima’ located on either side and to the front of the tip of the dike plane. The dike itself is interpreted to have intruded into a pre-existing joint.

The ‘step-out’ fractures of the Lanigan potash experiment relate to a class of experimentally observed fractures, known as secondary or remote fractures. Remote fractures have been identified in connection with model tests of fracture around cavities[l4, 15, 161 and the criterion for nucleation has now been established[l7]. With few exceptions[l6, 181, researchers in fracture mechanics treat only the fractures that nucleate at the tensile stress concentration of the flaw perimeter. These are the primary fractures of the terminology used here. In addition to primary fractures, the physical model tests display a large number of fractures that do not nucleate at the

64 E. Z. LAJTAI et al.

cavity boundary; they are ‘remote’. In fact, in the uniaxial compression of physical models containing cavities, (e.g. the potash model of Fig. 5 and similar models of Lac du Bonnet granite and Beebe anorthosite), remote fractures dominated and primary fractures 4 : 1. The significance of ‘remote’ fractures with respect to microscopic cavities or flaws however has not yet been established.

No matter where and how they start, cracks growing in a compressive stress field seem to know where they want to go, and they do this in a most systematic way, not only in the laboratory, but also in nature. Joints, the tensile fractures that formed in the tectonic stress field of the geological environment[l9-211, occur almost always in sets with individual fractures displaying consistent parallelism. The law that controls the process is quite simple; like the fractures of the laboratory, most joints can be shown to line up with the maximum principal paleo-stress direction.

MODELLING FRACTURE IN COMPRESSION

The exact size and shape of the starting flaw in rocks, at which the fractures of a specific type (primary or remote) originate, are still not clear, but this may be unimportant. In all probability, the crack starters have a variety of sizes, orientations and shapes, with most nucleation sites occurring along grain boundaries. The initial path of the stress-induced crack is probably a function of the geometry of the starting crack, but this influence is short-lived and probably restricted to the submicroscopic scale. What is clear, is that observable fractures in compression extend in their own plane and follow the direction of the maximum compressive principal stress. Fracture propagation is stable; the driving stress must be raised continuously to increase the crack length.

Crack propagation in response to the in-plane normal stress is not directly amenable within the framework of the three modes of loading of fracture mechanics. In an indirect way, a starting crack inclined to the compressive load can simulate the process of fracture in the direction of the compressive load through the mixed-load effect and this is the usual approach taken by researchers in fracture mechanics.

Most authors take the average grain size in rock as being the ‘suitable’ critical crack length. However, attempts to find the critical crack length from independently measured tensile strength and fracture toughness data often lead to strange results. Lac du Bonnet granite for example may have the odd plagioclase phenocryst that is as large as 40 mm in length, but the average grain size is only about 5 mm. Using a penny shaped crack, 14 MPa for tensile strength (Brazilian test) and 2.45 MPaJrn for K,, (double torsion test), the critical crack length computes to 48 mm. This of course is greater than even the largest grain in Lac du Bonnet granite. In fact, it is greater than some of the test specimens used to determine the tensile strength. When this exercise is repeated with a limestone, the popular Tyndalstone, the results are equally puzzling.

Since there appears to be no acceptable way to measure the size, shape and orientation of the ‘critical’ starting crack, the selection of a ‘suitable’ flaw becomes rather arbitrary. In fact, in the case of the elliptical void, one is faced with the task of supplying as many as three parameters: size, shape and orientation. This certainly allows for a good deal of flexibility in fitting a numerical model to the experimental data, but inspires little confidence in the outcome.

Much has been made of the point in rock mechanics that the Griffith type of theories are ‘mechanistic’ since they are based on the mechanism of fracture. In fact, the ‘mechanistic theories’ are only different from the empirical or ‘phenomenological’ theories in that they carry the investigation to a finer, physical scale. Essentially, the Griffith or Modified Griffith theories are based on a phenomenological theory, the maximum stress theory. The sophistication comes from the mathematical exercise involved in finding the maximum value of the tensile principal stress on the flaw boundary. In the end, the theory is developed by equating this stress to the tensile strength. This approach may be adequate at the megascopic scale of underground openings where the stress gradients are negligible, but for microscopic cavities, the influence of the stress gradient is overwhelming. Data from physical models rarely agree with the predictions of Griffith-based theories[22, 231.

Concentrating on the shape, size and orientation of starting flaws distracts from the real cause of fracture, the stress concentration. It is difficult, if not impossible, to build a theory for rocks around a physical flaw that cannot adequately be identified. On the other hand, there is little

Brittle fracture in compression 65

argument about the concept that fractures are nucleated from stress concentrations. Formulating a fracture criterion that is based on the stress concentration rather than on its possible causes, offers a way out of the dilemma of selecting a ‘suitable flaw’.

There are two important parameters associated with a stress concentration, the maximum value of the relevant stress or stresses and the rate at which the stress diminishes away from the maximum value, in short, the stress gradient. Similarly, material response to stress could be formulated as some combination of the strength of the material in a homogeneous stress field (as usually determined in the laboratory) and the material’s ability to redistribute stresses in a high stress gradient environment. This would suggest the need for at least four parameters, two for the ‘crack driver’ (maximum value and gradient) and two for the ‘crack resistance’ (strength in homogeneous stress and a parameter that reflects the position of the material in the brittle to plastic scale of material response to stress). In fact, the relevant stress and gradient parameters can only be defined if the stress distribution itself is known in detail.

There are a number of ways to implement this simple concept of fracture. In this paper, two different implementations are introduced, both in the general area of fracture from macroscopic cavities. One is a closed-form solution for primary fracture from a single cylindrical or elliptical void, and the other, a hybrid numerical/theoretical, general solution designed to search for fracture under a general state of stress. The latter is required to analyse ‘remote’ and ‘step-out’ fractures and fractures associated with compressive stress concentrations (slabbing).

A CLOSED-FORM CRITERION FOR PRIMARY FRACTURE PROPAGATION

Physical model tests have convincingly demonstrated that fractures associated with cavities in brittle rock loaded in compression are mostly tensile and fall into one of three possible categories: primary fractures (radial for a cylindrical cavity) growing out of the tensile stress concentration on the cavity perimeter, compressional fractures (peripheral) forming at the compressive stress concentration of the perimeter (spalling or slabbing of mining terminology) and remote fractures that form off the perimeter. The latter may unite with the slabbing fractures at high stress[l7]. The nucleation, exact position, or sequence of fracture formation is strongly stress dependent; in particular the ratio of the in-plane, far-field principal stresses governs the resulting fracture pattern. In uniaxial loading situations, all three fracture types are present; with all around nearly hydrostatic compression, only the slabbing mode is in evidence[24].

Primary fracture is the easiest to treat as it occurs at a well defined point and it starts in an area where the compressive principal stress (radial stress) is either zero or negligibly small. Since

the latter is the stress parallel to the crack, this case can be (and has been) analysed using fracture mechanics principles[25]. The state of stress around a circular, or elliptical void with vertical and horizontal axes parallel with the load directions, can be defined from the solution given in Terzaghi and Richart[26] who in turn identified the sources as Inglis[3], Neuber[28] and Edwards[29]. Although this represents a much simpler geometry than the inclined sliding crack, the outcome of the analysis is very similar for both geometries. This solution was used earlier for fracture nucleation and was offered as an explanation for the size effect observed in the testing of physical models[22].

When both the far-field, maximum compressive principal stress and the major axis of the elliptical void are vertical, then the largest tension at the tip is:

Here P, and P2 are the vertical and horizontal, far-field in-plane principal stresses acting along the major (a) and minor (h) axes of the ellipse respectively. The stress gradient for the same stress field, with respect to the primary fracture path and at the same location is:

G = (P,/h)(2 + 34h) - (PJb)[4(a/b)* + 3(a/b)].

The material response could be formulated in terms of the uniaxial tensile strength T, and a brittleness factor, n. The latter was originally defined as an absolute distance. It was derived from the concept of ‘stress-averaging’ which a material may do in an area where very steep stress gradients exist. In fact, the distance of 2d was assumed to represent that fraction of the crack path

66 E. Z. LAJTAI et al.

over which stress averaging would occur. The inclusion of d in a fracture criterion is necessary. Without it, a primary fracture, originating at the tip of an elliptical slit of finite width, would propagate indefinitely, i.e. in an unstable manner regardless of the nature of the stress field.

The fracture criterion may now be constructed using the concept that fracture nucleation is controlled by the average stress over a distance 2d, which for a linear gradient is the stress (r3 existing at a distance d in front of the crack tip. Accordingly, the controlling stress can be approximated and the fracture criterion becomes:

03d = ~rlI,X +dG=T.

Substitution for the stress and the gradient (G) yields a linear function in P, - Pz space with parameters T, d and aspect ratio a/b:

P, = T/[d/b(2 + 3a/b) - l] + Pz{d/b[4(a/b)* + 3(a/b)] - [1 + 2a/b]}/[d/b(2 + 3a/b) - 11.

Although originally proposed as a criterion for fracture nucleation, this long, but simple equation can simulate crack propagation in the direction of P,, the maximum, far-field principal stress. Since the stress distribution around the elliptical shape depends only on the aspect ratio and not on absolute size, fracture propagation is simulated by decreasing the b/a ratio. Figure 6 represents a case, where P,, the increasing vertical stress, is related to the length of the crack propagating parallel with it, while the width of the crack (b, the minor axis) and P, are kept constant. The trend of the curves is in agreement with experimental evidence. Except for the case of P, < - 0.2 MPa, primary fracture propagation is stable; after a short, initial extension at slightly increasing stress, P, must be substantially raised to drive the fracture. This phenomenon has been demonstrated for the inclined mathematical slit as well, both experimentally and theoretically while using the more sophisticated fracture mechanics formulation[25,29-311. Primary fracture propagation is very sensitive to the lateral far-field stress, Pz. Even a small lateral stress, can increase crack resistance substantially and hence inhibit the propagation of primary fractures. The fracture process associated with the propagation of this primary fracture may not however stop at this point. The propagation of the primary fracture shifts the critical stress concentration to a location which is off the tip of the perimeter. Alternatively, with increasing load, the compressive stress concentration of the perimeter may itself become large enough to cause slabbing. In both the remote and the compressive locations, however, the state of stress is more complex, and the above approach is no longer adequate.

Crack Extension in Compression

150

100

50

0

-S&?

4

I --

I-

‘m

I I

Numbering for

Confining Pressure

Crack Length (mm)

Fig. 6. The relationship between primary crack length nucleating at the perimeter of a cylindrical cavity and the maximum principal stress at constant minimum principal stress (confining pressure). The minimum principal stress must be negative to propagate the primary fracture indefinitely, in an unstable

manner.

Brittle fracture in compression 67

GENERAL CRITERION FOR FRACTURE

For fracture nucleation from points where the stress field is general, that is both U, and (TV are different from zero, the above approach is no longer tenable. In fact, the only good reason for including the previous implementation is because it yields a close-form solution in P, - Pz space, The case of slabbing at the compressive stress concentration of the elliptical voids has been treated essentially the same way[23]. At this time, it is more appropriate to develop a general criteria that will encompass the nucleation of fracture under any stress condition.

A general criterion for fracture should be based on the physical evidence on fracture propagation in both tension and compression and should preferably use parameters that are not too difficult to measure in the laboratory. By far the most important physical evidence is the stable propagation of tensile fractures along the maximum principal stress trajectory. The fractures extend in response to axial compression with the rate strongly influenced by the lateral principal stress. Therefore, both 0, and g3 must be included in the general criterion along with parameters representing the material’s resistance to fracture. In a stability analysis, the stress field is established from the boundary conditions using experimental or numerical techniques; the material properties are determined in the field or the laboratory.

The two most commonly performed tests in the rock mechanics laboratory are the uniaxial compression and the Brazilian tension tests. It is generally assumed that under tensile loading, the stresses at fracture nucleation and failure are identical. This is not strictly true, but the margin of error through this assumption is not too great. Cracks do start well before failure in uniaxial compression however and appropriate instrumentation is thus needed to detect the start of dilation; the mark of crack nucleation. For relatively homogeneous materials, like fine-grained limestones, the start of dilation in response to axial cracking is well marked on the stress-lateral strain or the stress-volumetric strain curves. For more heterogeneous materials, such as Lac du Bonnet granite which has several types of crack nucleation sites, there may be more than a single deflection point. The result is a stress-lateral strain curve that appears to be nonlinear from almost the beginning (Fig. 7). Some of the lower stress, deflection points may simply represent reversible dilation of the weak cleavage planes in feldspar which constitutes about 60 per cent of the granite. Perhaps a better measure of new fracture formation is given by the permanent lateral strain. This can be obtained through cycled loading involving progressively higher uniaxial loads. For this granite, the permanent strain increases substantially only after about 170 MPa or after about 75 per cent of

600

500

400

m

10TRI39M9 Lac du Bonnet Granite

Lateral Strain

Volumetric Strain

Confining Pressure = 39 MPa

N m t Y) Lo Ic m

Percent Strain

Fig. 7. A typical stress-strain diagram for Lac du Bonnet granite loaded in triaxial compression, Both extension and shortening are plotted as positive. Crack initiation is interpreted to occur at the arrows,

where the lateral and/or volumetric strain curves depart from linearity.

68 E. Z. LAJTAI et al.

COMPDAM Lac du Bonnet Grani ts

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PERMANENT MICROSTRAIN

Fig. 8. The beginning of permanent structural damage is indicated when some of the strain is not recovered after unloading. This, being due to axial cracking, is reflected only in the lateral direction.

strength; there is no permanent axial strain since the axially growing cracks do not produce strain in this direction (Fig. 8). At stresses between the permanent crack damage limit and failure the only fracture process in brittle rocks is the growth of axial cracks; no shear fractures forming prior to failure have been documented in this, or for that matter, any other experimental study. The exact mechanism(s) which actually causes the ultimate failure in brittle rock is still not positively identified.

Clearly, establishing the start of cracking in compression requires some interpretation of stress-strain data, but it is a measurable quantity, designated here as CIC, (new-)crack initiation in uniaxial compression. Crack initiation in a Brazilian test takes place at the centre of the test disc where the state of stress is characterized by the relationship: 6, = -30,. Sub-critical cracking in the Brazilian test of Lac du Bonnet granite begins at about o3 = 12 MPa.

The Brazilian and uniaxial compression tests yield two points in P, and P, space. The simplest function connecting two points is a straight line (Fig. 9). This is of course the precise

form of relationship found earlier for primary fracture and previously for failure at the compressive stress concentration of elliptical flaws[23]. In fact, the linearity in P, and P2 space may simply be an expression of the principle of superposition, the manner in which the effect of an additional source of stress is evaluated. Linearity should be maintained as long as the point at which the crack nucleates is stationary for all states of stress. With respect to cavities, this is strictly true only for the ends of the axes where the primary and slabbing fractures originate; the remote fracture point shifts as the confining pressure changes. Still, linearity is a reasonable assumption even for remote fracture[l7]. Other types of considerations, based on the damage mechanics concept as established through acoustic emissions, suggest a linear form as we11[32].

The linear function can now be defined. Instead of the Brazilian stress parameters, it is more appropriate to define the straight line in terms of the intercepts at the Pz = 0 (CIT, crack initiation in uniaxial tension) and P, = 0 (CIC, crack initiation in uniaxial compression) locations. Accordingly, the crack resistance (CR) function becomes:

P, = CIC - (CIC/CIT)P2 = CR.

The negative sign is needed since CIT is negative; the crack resistance actually increases

with confining pressure. This formulation should be valid for both tensile and compressive loading.

Brittle fracture in compression 69

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Lac du Bonnet Granite

Triaxial Strengtf

Permanent Crack Damage 55 Crack Initiation

Minimum Principal Stress (MPa)

Fig. 9. Crack initiation, permanent crack damage and strength for Lac du Bonnet granite. Axial crack growth is interpreted to occur between the permanent damage and the strength curves. The strength curve is a second order polynomial fitted to 412 tests. The crack initiation curve is based on 10 tests. The position

of the Permanent Crack Damage line is approximate.

THE CRACK DRIVER

For the stress field of an underground opening, maps of the principal stresses and the crack resistance define all the parameters needed to assess stability. In fact, the separate maps would most likely confuse rather than educate the operating engineer. Traditionally, the stress and strength parameters are combined in a safety factor and this is then plotted on a separate map. This is possible in this case as well, where a safety factor at a point could be defined as:

Safety Factor = CR/a,

or alternatively in terms of c3 since the latter could be equally valid. The crack resistance should now be evaluated in terms of the ‘at-the-point’ stress parameters, rather than P, and Pz.

It is convenient to introduce the concept of the ‘crack driver’, or CD:

CD, = (a, - a,(CIC/CIT))/CIC

where p signifies that the stresses are those found at the point. This special form is designed so that fracture nucleation will coincide with the condition of CD = 1; values higher than unity signify a failure condition while the rock is stable at the point of reference when CD < 1.

The definition in this form ignores the stress gradient that may exist at the point of fracture nucleation. This may be acceptable at points where the stress gradient is negligibly small, as it usually is in the remote fracture position. The solution is, however, not acceptable where the stress gradient is high such as at the tensile and compressive stress concentrations occurring on the perimeter of cavities. It is possible to introduce the gradient effect in a number of ways, specifically through some type of stress averaging. Averaging would have to be performed over an absolute distance, or perhaps area, depending on the method in which this concept is implemented. In any case, the technique should always be calibrated against appropriate physical models. The end result is the replacement of the ‘at-the-point’ principal stresses with the averaged values or their ‘stress equivalents’:

CD = (a, - a,(CIC/CIT))/CIC.

In the next section, the general criterion for fracture is implemented for the case of fracture around cavities, again using the solution from[26].

70 E. Z. LAJTAI et al.

FRACTURE AROUND AN ELLIPTICAL CAVITY

The major advantage of the general criterion of fracture just developed is that it treats fracture nucleation at any point as long as CD can be defined at that point. The stress parameters that enter into CD, are the maximum and the minimum principal stresses. The fracture is assumed to extend perpendicular to g3. In this plane formulation, the assumption is that g3 is the ‘in-plane’ and cZ the ‘anti-plane’ principal stress. In certain instances, this may not be true. The experimental parameters, CIC and CIT must come from laboratory testing. In fact, a third experimental parameter is required as well, the absolute distance over which stress averaging occurs. In this implementation, it is referred to as DeX.

In the earlier treatment of primary fracture, the constant d was introduced as an absolute distance along the primary fracture path with stress averaging occurring over 2d. Since the direction of primary fracture propagation is predictable, this approach created no computational difficulty. At this location, the gradient is actually the steepest in the propagation direction. For fracture at the compressive stress concentration, the propagation direction is peripheral. The gradient in this direction is negligible, however fracture is still strongly affected by the other, the radial stress gradient[l7]. The problem can be overcome by introducing gradient effects in both the radial and the peripheral directions. On this basis, CD is computed twice, once using the gradient in the X and once in the Y direction, taking the lower value as the effective crack driver. Optionally, the stresses could be averaged over an area, DeX square. In finite-element formulation, an effective procedure is to use the average mesh stress, with the element size made equal to DeX. Either way, calibration of the numerical technique against a physical model must occur. For Lac du Bonnet granite, sufficient data for primary fracture nucleation from a BX sized borehole (36 mm diameter) have been collected. With the uniaxial tensile strength (extrapolated from the Brazilian test) being 15 MPa, and the value of the uniaxial compressive load (P,) at primary fracture nucleation measured as 16 MPa, the appropriate value of DeX was found to be 0.225 mm. Using 170 MPa for CIC, the permanent crack damage limit (Figs 8 and 9) rather than the crack initiation, all the experimental constants are now defined and the crack driver can be computed, provided the principal stresses are known.

The Terzaghi and Richart equations, along with a procedure to compute the crack driver, are now part of a computer program, ELOPE, written in True BASIC. The Program scans the area beyond and including the perimeter, at each point computing the principal stresses, the crack resistance, and the crack driver. It then sorts the crack driver data to locate the most critical point. Some of the results have been used to produce Figs 10 and 11. The current investigation was limited to fracture nucleation. A finite element procedure, incorporating the ‘crack driver’ concept, would be required in order to analyse both fracture nucleation and propagation, which is now in progress.

The theoretical solution itself is independent of absolute size; the input is the aspect ratio of the ellipse. One can however simulate crack propagation by decreasing the minor axis, making the ellipse increasingly more slender. Figure 10 is a plot of the crack initiation stress in uniaxial compression (CIC) for an elliptical cavity whose major axis is oriented parallel to the compression direction. For equidimensional shapes, down to an aspect ratio of about 0.5, fracture starts at the tip as expected. This is the usual primary fracture. For more slender shapes down to about 0.2, the nucleation point moves slightly away from the tip but remains on the perimeter. At an aspect ratio of about 0.2, the fracture no longer nucleates on the perimeter but moves into a remote position off the tip, by slightly more than the length of the minor axis. Once the initial fracture reaches a certain length it stabilizes and then gives birth to a daughter crack in an offset position.

There is general agreement that tensile fractures require a ‘low confining pressure’ environ- ment. In the potash mining environment, where the far-field stress field is probably close to hydrostatic, there is however plenty of evidence for the formation of tensile fractures. The influence of the confining stress on fracture around underground cavities is illustrated by the stability diagram of Fig. 11. The figure refers to a 1 cm long ellipse in Lac du Bonnet granite loaded as indicated. Three aspect ratios are shown: 1,0.46 and 0.05. The boundaries of the stability diagrams represent conditions where the crack driver equals one. Points inside represent stable conditions, while points outside indicate fracture. For the circular shape, the stable zone is symmetrical about the hydrostatic stress line; for the elliptical shape the stability zone, for obvious reasons, is no longer

Brittle fracture in compression 71

200

LDB Granite-Uniaxial Compression

CIT=lS MPa

M

ii 140

L

Is 120 Remote

On Tip

Aspect Ratio of El 1 ipse

Fig. 10. Crack initiation from a 1 cm long cavity in Lac du Bonnet granite. The aspect ratio is varied between a relatively flat (I : 1000) crack and a circular cavity with the major axis of the cavity oriented parallel with the uniaxial compressive stress. As the aspect ratio is decreased, the crack initiation point moves off the tip in two stages: off the tip but still on the perimeter and off the perimeter into a ‘remote’

position.

I I I I

Z-------Y :

7

‘NO-CAVITY’ Fracture

STRESS ALONG MINOR AXIS (MPo)

Fig. II. Stability chart (curves marking the locations where the crack driver is equal to one) for a 1 cm long cavity in Lac du Bonnet granite showing the location and the type of fracture forming under the given stress condition; P for primary, R for remote, S for slabbing and T for transitional types of fracture. Three different aspect ratios are considered: a circle, a 0.46 and 0.05 ellipse. The shaded area marks the regions within which the crack driver is greater than unity under homogeneous (no-cavity) stress conditions. In the stability chart, points falling inside the ‘closed-in’ curves represent unfractured

conditions.

72 E. 2. LAJTAI et al.

symmetrical. The small diagrams intend to illustrate the position of the corresponding fractures. Although the diagram itself is self-explanatory, there are a few points that should be emphasized.

The proper primary fractures are quickly inhibited once a confining pressure is introduced, or the cavity flattened. Admittedly, this depends to a great extent on the value of DeX used in the analysis. If DeX was set close to zero, primary cracks would propagate much more easily. Alternatively, if the size of the opening was large, meters rather than a centimetre, the effect of the absolute distance, DeX, would again be negligible; primary cracks would survive the higher confining pressure, or the flatter shape. The influence of size on the strength parameters of rock is well recognized in rock mechanics. In contrast to stress based theories, the above introduced ‘crack driver’ approach to fracture analysis does account for the size effect.

In uniaxial compression, no failure in the compressive zone was noted (Fig. 10). Upon application of confining pressure, slabbing in this zone quickly becomes the most important fracture mechanism[33]. Physical tests suggest that slabbing is confined to the compression zone; the individual fractures do not propagate far out of this zone. They have however been observed to merge with the more distant remote fractures[l7].

The physical experiments, conducted so far were all in uniaxial compression, thus the relevance of remote fractures at high confining pressure must be established through theory. The nucleation of such fractures, even at very high confining pressure, is suggested by our analysis, but only in the case of relatively flat openings. For the 20: 1 shape, remote fractures are indicated under conditions that are very close to hydrostatic loading (Fig. 11). They seem to persist for the flatter ellipses, specifically for the 100: 1 shape. Ellipses more slender than this may close elastically, invalidating the analysis.

SUMMARY AND CONCLUSIONS

The purpose of this paper was to formulate a simple, empirical fracture criterion for compression that satisfies the following physical evidence:

-A fracture forming in a compressive stress field prior to failure of the rock as a whole (strength), is a tensile fracture. ‘Shear fractures’ do not form before failure and therefore do not cause or even contribute to the failure of brittle rocks. (The shear fractures produced in the laboratory and the large scale faults observed in the field, represent post-failure phenomena.)

-The fracture propagates perpendicular to the minimum principal stress, i.e. parallel to the maximum principal stress trajectory.

-Fracture formation in the compressive stress field occurs between the crack initiation point and the stress at failure (strength). The majority of the fractures however form at high stress, close to strength.

-The fractures propagate in a stable manner. The driving stress must continually be raised to cause further crack extension.

-An increasing driving stress nucleates additional fractures at some of the less critical stress concentrations.

-The driving stress is the averaged state of stress, with averaging occurring over an absolute distance or area. Hence, the driving stress reflects the stress gradient at the point of nucleation.

-The maximum principal stress aids, while the minimum principal stress, if compressive, hinders fracture propagation.

-Fractures from the crack nucleation sites propagate parallel to each other. -When a confining pressure is present, the stress shadow between proximally extending

fractures facilitates the nucleation and propagation of new fractures causing ‘crack-bunching’. -As an individual fracture stabilizes it gives rise to ‘step-out’, or secondary fractures that

nucleate at the stress concentrations set up by the original fracture, off and slightly ahead of the crack tip.

-The resistance to fracture is formulated here as a linear function in TV, - g3 space. Its parameters are derived from a set (at least two) of conventional rock mechanics experiments (the Brazilian and the uniaxial compression tests using strain-gauged specimens are adequate). In homogeneous rock, where the crack initiation point represents the formation of a ‘permanent’ fracture, the corresponding state of stress should provide the appropriate point for the definition

Brittle fracture in compression 13

of the crack resistance (CR) function. In more heterogeneous rocks, such as granite, the first deflection on the lateral or volumetric strain curves may only reflect recoverable dilation along grain boundaries or cleavage in feldspar. Permament crack damage occurs only when the quartz network surrounding the feldspar crystals is disrupted. Accordingly, the ‘permanent crack damage’ stress is a more appropriate parameter to define the crack resistance function.

-To represent the state of fracture in a heterogeneous stress field, the state of stress at nucleation, the stress gradient and the crack resistance are combined to form the ‘crack driver’ (CD). The crack driver is analogous to the ‘safety factor’ of the conventional representation of stability around underground openings, but with values greater than unity representing the fractured state.

-The inclusion of the maximum principal compressive stress in the fracture criterion allows modeling fracture at any point around the perimeter of an underground cavity. In fact, fractures may nucleate at the tensile stress concentration, (primary fracture), at the compressive stress concentration (slabbing fracture), and in positions off the perimeter (remote fracture). Fractures may also nucleate in intermediate positions.

Acknok~ledgements-This paper is the outcome of research into fracture development around underground cavities with special emphasis on potash mining. The Project is supported through the Research Partnership Program of the National Science and Engineering Research Council of Canada and it is a cooperative venture between the University of Manitoba (Departments of Civil and Geological Engineering and the Department of Geological Sciences), the Potash Corporation of Saskatchewan and Cominco Fertilizers Ltd. The support of NSERC and the industrial partners through their representatives in the research, David Mackintosh and Parvis Mottahed, is gratefully acknowledged. Special thanks are extended to Professor Shah for his constant guidance and to Professor Rajapakse for his help with the mathematics of underground cavities.

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(Received 20 June 1989)