Mapping the state of fracture around cavities

Engineering Geology 31 (1991) 277-289

277

Elsevier Science Publishers B.V., Amsterdam

Mapping the state of fracture around cavities Emery Z. Lajtai, Bruce J. Carter and E.J. Scott Duncan Department of Geological Engineering, The University of Manitoba, 314 Engineering Building, Winnipeg, Man. R3T 2N2, Canada (Received J a n u a r y 14, 1991; revised version accepted J u n e 12, 1991)

ABSTRACT Lajtai, E.Z., Carter, B.J. and Scott Duncan, E.J., 1991. Mapping the state of fracture around cavities. Eng. Geol., 31: 277-289. The state of stress around cavities is heterogeneous. Consequently, the state of fracture may also vary from point to point. Under compressive loading, the rock surrounding the cavity may be in one of four possible states: pre-fracture (pre-microfracture initiation), microfracture propagation (initiation to the onset of dilatancy), dilatancy (onset of dilatancy to failure) and post-failure. These four states are separated by the crack initiation stress, the crack damage stress and the failure stress. The variation of the three fracture events with confining pressure is examined with reference to three intact rocks: a brittle granite, a semi-brittle limestone and a ductile salt rock. The maximum principal stress (al) at crack initiation, crack damage (onset of dilatancy), yielding and failure are established as a function of the minimum principal stress (a3). For a single intact rock, all four fracture events can be represented using one function (the Rocker function) with a single factor differentiating the various fracture states. The proposed fracture criteria, based on the experimental data, are combined with the existing state of stress to prepare a fracture map around an elliptical cavity located in intact Lac du Bonnet granite. The state of stability is expressed through a newly defined stability factor, the unconfinedstrength ratio (USR), which is similar in principle to the traditional safety factor. In contrast to the conventional safety factor in rock mechanics (SF = strength/a1), which is limited to the a 3 < tensile strength space, the USR is defined for any point in the a a - a l space of the stability diagram.

INTRODUCTION

In a compressive stress field, the fracture of intact rock during the strain-hardening phase of deformation is a stable evolutionary process which may be marked by four fracture events: crack initiation, onset of dilatancy (reversal of the volumetric strain rate), yielding and loss of (peak) strength or failure. The stress levels associated with these evolutionary steps depend on confining pressure, time, temperature, stress gradient and scale effects. The analysis in this paper is restricted to intact rock. The scale effects with respect to the rock mass are just too difficult to be considered in a single paper. The confining pressure dependence of rock strength (failure) is expressed through phenomenological theories known as failure criteria. The most often used criteria 0013-7952/91/$03.50

O 1991 Elsevier Science Publishers B.V. All rights reserved.

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for rock strength are the C o u l o m b - M o h r law with or without a tension cut-off (Paul, 1961), the Hock and Brown square root parabola (Hoek and Brown, 1980), and the Johnston three-parameter nonlinear criterion (Johnston, 1985). The influence of confining pressure on the fracture events before failure is less well known. In this paper, experimental data on all the fracture events are presented for the three rocks and appropriate fracture criteria are formulated for them in terms of a similar, threeparameter function. The various fracture criteria can be used to map out the state of fracture around openings contained within intact rock. As an example, the stress distribution around an elliptical cavity of arbitrary orientation (Bray, 1987) is combined with the criteria to show the state of fracture in the surrounding rock. The tool to do this is the unconfined strength ratio (USR), a stability function which is similar in concept to the safety factor. The stability analysis has been coded in True BASIC. The executable version of the Code, ELOP, can be obtained from the authors for a nominal charge covering postage and reproduction. EXPERIMENTAL DATA The granite of the study comes from the Cold Spring quarry located just outside of Lac du Bonnet, Manitoba. The limestone is from Garson, Manitoba and the salt rock was obtained from the Lanigan mine of the Potash Corporation of Saskatchewan. The geological and some of the mechanical properties of all three have been described earlier (Carter et al., 1991). Although the three rocks are widely different in their response to increasing stress, the process of fracture is similar in all three of them. The pre-failure fracture process consists of the initiation and the propagation of tensile microcracks that extend along the maximum principal stress trajectory (perpendicular to the minimum principal stress). Shear fracture that may occur in the post-failure phase is not considered in this paper. The stress-strain curves for each rock (Figs.l-3), constructed from electronic strain gauge measurements, identify the most important events in the evolution of fracture: crack initiation (C;), crack damage (onset of dilatancy, Ca), and yielding (potash only). The highest point on the stress-strain curves represents failure of the intact rock (peak strength). The variation of these values with confining pressure was evaluated through Brazilian tension tests and triaxial compression tests using confining pressures from 0 to 40 MPa for the limestone and granite, and 0 to 20 MPa for the much weaker potash rock (Figs.4-6). The specimen diameter for the Brazilian and the triaxial tests was 54 mm for the salt rock and the limestone and 30 mm for the granite. The length of the specimen was twice the diameter for the compression tests and about 25 mm for the Brazilian tension tests. The brittle rocks were loaded in a hydraulic loading system at an average loading rate of about 1 MPa/s. For the potash, a gear-driven, constant displacement rate machine was used to obtain an average strain rate of 1.75 g~/s. Crack initiation stress

The crack initiation stress (C~) is marked on the stress-strain diagram by the point where the initial linearity of the lateral and volumetric strain curve ends: that is at

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strain (%) Fig.1. Stress-strain curves from a triaxial test of Lac du Bonnet granite. This is a very strong, brittle rock that has no yield point. C~ and Cd=the initiation and the damage points, respectively. The crack initiation point is located at the deflection point of the lateral strain curve and the damage point at the point where the volumetric strain rate changes from compression to dilation. The highest stress on the stress-strain curves represents the peak strength of the rock and marks the point of failure.

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microstrain Fig.2. Stress-strain curves from a uniaxial compression test of Tyndallstone, a semi-brittle limestone. This rock may or may not show a yield point. The points of crack initiation and crack damage are well marked on the lateral and volumetric stress-strain curves, respectively. C~ and Cd as in Fig.1.

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strain (%) Fig.3. Stress-strain curves from a uniaxial compression test of the ductile Lanigan potash rock displaying all stages of fracture: initiation, yielding, damage and failure. Ci and Cd as in Fig.l.

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MAPPING FRACTURESTATE

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Fig.5. Triaxial fracture parameters for Tyndallstone. The crack initiation and the crack damage data were derived from incremented-load, cyclic tests. Failure data from other non-cyclic tests are included as well.

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Fig.6. Triaxial fracture parameters for Lanigan potash rock, a salt rock from Saskatchewan. This is the only one o f the three rocks considered here, that has a yield point.

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the first deflection point. Since the al-parallel forming cracks do not change the axial stiffness, the stress-axial strain curve is not expected to reflect the initiation of the first crack. Locating the crack initiation point on stress-strain curves is not always a simple operation. The concept of crack initiation is based on the idea of homogenous rock. Most rocks, however, are heterogeneous at any scale, and therefore may exhibit several crack initiation points. In the extreme, the lateral and volumetric strain may start curving from a very low stress. Through biasing the location of strain gauges on the granite samples, crack initiation in the feldspars can be demonstrated to occur at around 25% (Fig.7) and in quartz at 60% of peak strength. Low-stress fracture initiation at grain boundaries may further complicate the interpretation. A coarse-grained version of potash rock from Rocanville, Saskatchewan, was instrumented to reflect the deformation at grain boundaries (Fig.8). The corresponding volumetric curves show that deformation may be significantly different at grain boundaries from that within crystals. The eventual result is that the crack initiation data (Figs.4-6) are more widely dispersed than the test data representing the peak strength of intact rock. Crack damage stress

The crack initiation point is usually expected to signal the first damage to rock, in the form of a newly-formed axial tensile crack. However, the crack initiation point may not be that significant: the dilation of pre-existing axial cracks could produce the same increase in the lateral strain as a newly-formed axial crack. The 180 /

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l a t e r a l s t r a i n (iJe) Fig.7. Crack initiation at 55 MPa in the feldspar and at 130 MPa in the quartz crystals of Lac du Bonnet granite. The strain data were obtained by biasing the placement of small strain gauges.

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Fig.8. The effect of grain boundaries on crack initiation and volumetric strain rate reversal (crack damage). These events occur at differentstress levelsin the crystalsand at the site of grain boundaries. Lac du Bonnet granite is a good example of this. The feldspar grains are the first to show dilation, presumably along one or two of the three cleavage systems. If the rock is unloaded at this stage, no permanent strain is suffered. On reloading, the new stress-strain diagram cannot be distinguished from the original. From a practical point of view, the stress at which the rock suffers permanent damage is more important than the crack initiation stress. The stress at the point of permanent damage is the crack damage stress (Ca). There are several ways to evaluate the crack damage stress. A good measure of damage is the permanent strain. This can be evaluated through cyclic loading where the axial load is incremented for each loading cycle. The Tyndallstone criteria (Fig.5) were developed in this way. Another method involves the Brazilian testing of cylinders which were previously loaded in compression. The presence of axial cracks should be reflected by a decrease in tensile strength. As can be seen from Fig.9, the trend of both parameters suggests that the permanent damage stress is close to the stress at which dilatancy begins.

Yield stress Yielding in a material is marked by the first deflection point on the stress-axial strain curve. A brittle material, such as the Lac du Bonnet granite, usually reaches peak strength without displaying a yield point (Fig.l). This feature may change at higher confining pressures, but for the relatively low confining pressures used no yield points could be identified. Tyndallstone, which is less brittle than granite, occasionally shows a yield point. Only the potash rock displayed a yield point in a consistent manner (Fig.6). Yielding for this rock takes place before the onset of

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Uniaxial Compressive Strength (MPa) Fig.9. Determining the stress at which permanent damage occurs in Lac du Bonnet granite. The volumetric strain rate reversal is a convenient marker for this. This is the level at which the permanent strain starts increasing at a rapid rate. The tensile strength of the specimens that had earlier been subjected to compression is another indicator of damage. The tensile strength, however, starts to decrease at somewhat higher stress.

dilatancy. The yield stress increases with confining pressure. If the data were displayed as deviatoric stress versus confining pressure space, the trend o f the yield point with increasing confining pressure would essentially be flat. Peak stress

The highest point on the stress-strain curve establishes the strength o f a material. In the compressive field, the strength o f the three rocks increases with confining pressure. F o r the tensile field, the relationship is not well defined. C o m m o n l y , the only data available for guiding the strength curve t h r o u g h the tensile field comes from the Brazilian test. Positioning the Brazilian test data in a 3 - a l space is problematic. Equivalence between the Brazilian tensile strength and the uniaxial tensile strength is often assumed, but there are, in fact, substantial differences in the m o d e o f testing and in the stress state imposed by the two techniques. In this paper, the Brazilian test is plotted along the alfi73 = - 3 locus rather than on the a t = 0 axis, since this is the appropriate stress ratio at the center o f the Brazilian disc. FRACTURE CRITERIA In order to describe the functional relationship between the m a x i m u m principal stress at the various fracture events and the confining pressure, one could use any

MAPPING FRACTURE STATE

285

of the existing failure criteria. In rock mechanics the most popular criterion is the Hock and Brown parabola (Hock and Brown, 1980). This is a form of the y2 =f(x) type of parabola that has been shown to fit the triaxial strength of competent crystalline rocks tested at moderately high to high confining pressures. It is less satisfactory in the low confining pressure and the tensile fields of al-o.3 space. In fact, no two-parameter parabola gives an adequate fit to data covering both the tensile and the compressive stress fields. Three-parameter functions, such as the one proposed by Johnston (1985), can describe any form of strength relationship, but require difficult and time-consuming, computer-aided fitting procedures. A good fit in the tension and low compression region is essential, since most of the fracture processes around underground openings occur under a condition of low confining pressure. For the purpose of developing a simpler but still adequate fracture criterion for the tensile and low confining pressure region, a functional form was selected that anchors the function at two points, the uniaxial tensile and the uniaxial compressive strength. A third parameter, R, is then used to shape the curve beyond these two points. R is a fractional exponent, with 0.5 (square root) being an approximate value. For the present use, an additional, optional scaling factor, A, is attached:

The uniaxial tensile strength (To) is a negative quantity, o-1:, the maximum principal stress at failure (peak strength), is a measure of strength at confining pressure o-3, and Co is designated as the strength in uniaxial compression. This functional form, the Rocker function (Carter et al., 1991), can be adapted to formulate criteria for all the fracture parameters of this study by simply dimensioning the scaling factor, A as shown in Figs.4-6. One disadvantage of this form is that it forces all three (initiation, damage and peak strength) curves to unite at the uniaxial tensile strength, implying that all three events take place at the point of failure. It has long been assumed that the points of fracture initiation and peak strength coincide in brittle rocks. Although this is not strictly true, the presently available data on tension does not allow, nor justify, a more complex formulation.

Safety factor Once the functional form of the fracture parameters in a l - a a space is established, the state of stress (63, o-1) can be compared with the available resistance to formulate a safety factor. Hock and Brown (1980), for example, define the safety factor in terms of the maximum principal stress: Safety Factor = o.1: O.1

which is valid as long as a3 > To. When aa < To, al: is no longer defined. This would be the case with any criteria that is based on the o.2 =f(a3) type of parabola. A

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possible solution of the 0.3 < To condition is to express the safety factor with respect to the minimum principal stress: Safety Factor = To 0.3

For many applications, the double-criteria approach may be adequate. It has, however, one disadvantage; it produces an artificial discontinuity in the safety factor distribution by causing sudden shifts in the value of the safety factor at the point where the fracture criteria change. One can avoid this only through a redefinition of the safety factor using a functional form that is valid at every point of 0.3-0.1 space.

Unconfined strength ratio The role of the safety factor is to measure the state of stability; its value should reflect the severity of the state of stress when compared with the available resistance. There is no particular reason why its value should be measured with respect to either the ordinate (0.1) or the abscissa (0.3) of the coordinate system. In order to remain close to the generally accepted conncept of the safety factor (SF) (safe when SF > 1, unsafe when S F < 1), but at the same time gain the advantage of a single function formulation, the unconfined strength ratio (USR) is proposed. For the state of stress, expressed through 0.3 and 0.1, one may define an equivalent strength function that represents the SF = 1 condition (the strength line includes the state of stress). Since a single point does not fully define a strength parabola, the additional characteristic of the equivalent strength function is that it maintains the actual tensile to compressive strength ratio (To~Co). For three-parameter formulations, the fractional exponent (R of the Rocker function) is kept the same as well (Fig.10). USR then becomes the ratio between the actual uniaxial compressive strength and the equivalent uniaxial compressive strength (Coe). Because the tensile to compressive strength ratio is kept constant, an expression in terms of the two tensile strengths is equally valid: USR-

Co Coe

-

To Toe

For the two-parameter parabolic criteria, such as the Hoek and Brown function, USR can be expressed explicitly: USR -

2Cos - - m 0 . 3 + [ ( m 0 . 3 ) 2 + 4 s ( 0 . 1 - - 0-3)211/2

where the constants m and s are those given in Hoek and Brown (1980). Similarly, for the Rocker function at R = 0.5: USR

To -

For R not equal to 0.5, USR cannot be expressed explicitly. Taking R = 0 . 5 will

287

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03 (MPa) Fig.10. Determining USR at a point P(a3, 01). The equivalent strength curve through the point SF= I has the same tensile to compressive strength ratio and the same functional form as the strength itself. USR is the ratio of the actual to the equivalent unconfined compressive (or tensile) strength.

always give an adequate fit in the tension field, that is, between the anchor points of To and Co, but slight errors may result at higher confining pressure. If a general solution is required for any fractional exponent, an iterative solution can be used. A computer program, "USR", was written in TRUE BASIC for the IBM PC to accomplish this task. The interested reader is invited to write for a free copy. The parameter s in the Hock and Brown formulation of rock mass strength is intended to be a scaling parameter (s< 1). By including this fractional factor in the definition of strength, the strength of the intact rock (s= 1) can be scaled to a level that is thought to be more appropriate for the rock mass. This scaling factor is not included in the Rocker function for the simple reason that a similar effect can always be achieved by scaling down the strength parameters, Co and To. MAPPING FRACTURE AROUND AN ELLIPTICAL CAVITY

As an example of the use of the fracture criteria and the USR concept based on it, a map showing the different stages of fracture was constructed for an elliptical opening in intact rock. The general solution for the elastic stress distribution around an elliptical tunnel is detailed in Bray (1987) and this is now coded in TRUE BASIC (ELOP). The far-field principal stresses may take any orientation with respect to the axes of the elliptical cross section. ELOPfinds the stresses over a user-defined elliptical coordinate grid and combines them with user-supplied strength parameters to compute USR at every point. Using the distribution of USR, the four fracture zones representing pre-cracking, crack initiation to crack damage, crack damage to failure and post-failure can be delineated.

288

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For convenience, all fracture criteria are expressed in terms of the Rocker function. The rock used in the example is Lac du Bonnet granite which is an elastic, brittle rock. The crack initiation (A=0.44) and the crack damage (A=0.71) criteria are expressed as a fraction of strength, as shown in Fig.4. This way, all three criteria converge on the a3 = To point. The far-field stresses for the case solved (Fig.11) were selected to give a nice distribution of potential fracture zones. It must be noted that the fracture map is rather hypothetical since it is based on the pre-fracture, elastic stress distribution within intact rock. The stress distribution changes slightly as new microfractures form and changes considerably when yielding and/or fracture propagation occur. Fracture maps of this type however give an idea about the influence of shape, orientation and loading condition on overall stability. A more complex, numerical solution would be required to follow the true fracture evolution and to interpret the failure mechanism involved in the eventual collapse of an underground cavity. CONCLUSIONS

Rocks subjected to a compressive stress field exhibit several stages in their fracture history: crack initiation, crack damage (onset of dilatancy), yielding, peak strength and post-peak strain-softening. The points marking these events can be identified from the stress-strain curves of uniaxial or triaxial compression tests.

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Fig. 11. M a p p i n g the state of fracture around an elliptical cavity oriented at 30 ° with the m a x i m u m farfield stress. The fracture parameters are for Lac du Bonnet granite. Ci, Cd and F m a r k the areas where the state of stress leads to microfracture (crack initiation to onset of dilatancy), to crack damage (onset of dilatancy to failure) and to failure, respectively. The boundaries are based on the value of U S R which combines the state of stress with the strength parameters shown in Figs.1 and 4.

MAPPINGFRACTURESTATE

289

The stresses associated with crack initiation, crack damage, yielding and peak strength increase with increasing confining pressure. For a single rock type, the three points can be modeled through a single function, the Rocker function, by varying the scaling parameter, A. In order to indicate the local condition of fracture in rock, the stress state was compared to the various fracture strengths through a safety factor, the unconfined strength ratio (USR). USR is more convenient to use than the conventional safety factor since the underlying function is valid at any point of o l - o 3 space. ACKNOWLEDGEM ENTS

The experimental data used for this study could not have been made available without the financial assistance of Atomic Energy of Canada Limited, Cominco Fertilizers, the Potash Corporation of Saskatchewan and the Natural Sciences and Engineering Council of Canada. REFERENCES Bray, J.W., 1987. Some applications of elastic theory. In: E.T. Brown (Editor), Analytical and Computational Methods in Engineering Rock Mechanics. Allen and Unwin, London, pp.72-78. Carter, B.J., Duncan, E.J.S. and Lajtai, E.Z., 1991. Fitting strength criteria to intact rock. Geotechn. Geol. Eng. (in press). Hoek, E. and Brown, E.T., 1980. Underground Excavations in Rock. Inst. Min. Metall., London, p.527. Johnston, I.W., 1985. Strength of intact geomechanical materials. Geotechn. Eng., I 11: 730-749. Lajtai, E.Z., Carter, B.J. and Ayari, M.L., 1990. Criteria for brittle fracture in compression. Eng. Fracture Mech., 37: 59-74. Lajtai, E.Z., Duncan, E.J.S. and Carter, B.J., 1991. The effect of strain rate on rock strength. Rock Mech. Rock Eng. (in press). Paul, B., 1961. Modification of the Coulomb-Mohr theory of fracture. J. Appl. Mech., 28: 259-268.