Intrinsic shear strength of a brittle, anisotropic rock — III

Int. J. Rock Mech. Mitt. Sci. & Geomech. Abstr. Vol. 11, pp. 439-451. Pergamon Press 1974. Printed in Great Britain

Intrinsic Shear Strength of a Brittle, Anisotropic Rock III Textural Interpretation of Failure PETER B. ATTEWELL* MICHAEL R. SANDFORDI"

After considering two theories which accommodate the concept of crack cooperation towards terminal shear failure, it is concluded that the simple Griflith theory seems adequately to describe intrinsic strength anisotropy. The theory is examined by allocating the orientation distribution of clay mineral basal planes (determined by X-ray texture-goniometric methods) to an assumed equivalent distribution of pre-existing cracks in the rock and to an assumed equivalent distribution of crack lengths.

1. I N T R O D U C T I O N Any attempt to quantify the process of rock failure on a theoretical basis and to satisfy, for example, the experimental observations outlined earlier in Part I of this work [1] must recognize that the basic mechanism of the failure process will be conditioned by the internal structure of the rock material. Information about this structure and its properties may be obtained directly through operation of the universal stage on a petrological microscope (if the constituent crystals to be observed are large enough to permit this), by optical data processing methods [2, 3], by X-ray textural analysis as described in Part II of this work E4], or indirectly by measurements of elasticity, permeability, or other physical properties [5]. Direct observation generally reveals grains or small crystals of rock minerals, cemented together in places, but otherwise separated by cavities. These latter are totally or partially interconnected--confirmed by the fact that rocks possess some primary or intrinsic permeability. Mechanically, therefore, rock is inhomogenous on a microscopic intercrystalline basis and clearly it is difficult to model at this level. However, laboratory experiments are usually on such a scale that the rock may be considered homogenous, but not necessarily isotropic. It is now well known that intact rocks can exhibit highly non-linear stress--strain behaviour and a study of such behaviour, coupled with information on the variation of resistivity and microseismic activity with applied stress, lends insight into the bulk properties of rock and its failure mechanism. Figure 1 shows a set of generalized curves for these quantities, the curves being conveniently divided into four regions, each of which can be

interpreted by cross-reference between the various graphed parameters. Interpretation of the regions indicates that stress-strain non-linearity and hysteresis are a function of the cracks and pores in the material, that failure is initiated at stresses below the ultimate loadbearing failure stress (some 30 to 60 per cent of the failure stress--this effect has also been observed in rocks under tension by Wawersik [6]), and that before failure there is dilation, the increase in volume being due mainly to an increase in lateral strain. Any theoretical discussion of rock properties should include these points either as a basis or as a prediction.

2. FRACTURE THEORIES AND CRITERIA Existing theories and criteria for failure can be assessed with respect to the picture of rock behaviour just considered and to the observed dependence on experimental conditions. All the criteria embody some degree of empiricism and should probably be evaluated in such an order that takes into account the depth of treatment, beginning with Coulomb-Navier and progressing on a more genetic basis through Griffith to the closed crack modifications of McClintock and Walsh 1-29]. The three-dimensional character of any viable criterion is emphasized by the influence of the intermediate principal stress, not only for intrinsic failure but also for mass failure [7]. All the formulae may be expressed in the form: z = f(o)

(I)

f(a) oc(a) r

(2)

and in most cases

* Engineering Geology Laboratories, University of Durham, Dur- where F varies between (}5 and 1. Franklin [8] has also ham, England. t Present address: Central Electricity Generating Board, Park suggested an expression of this latter form to describe a Street, London, England. non-linear failure envelope in the z, tr plane.

439

440

Peter B. Attewell and Michael R. Sandford Maximum

(0)

~ tr°'n/ s'tArX',°l

Region

2

Region Region I

/

/

Young's Modutus IncreaAing Poisson's S~uS~Sl Young's R a t i o Modulus Zncreasing Constant Poissows ~roinmetric Ratio Constant

Number of

Volumetric

(b)

stress-100%

strain

events

, I00 ~" .~

--

5

I

0

o N

Resistivity

Fig, 1. (a) Generalized stress-strain curves for rock in compression (after Bieniawski [20]). (b) Generalized resistivity and microfracturing curves for rock in compression.

On reduction, such a basic similarity between theories is, however, a measure of the form of real rock data and not necessarily of the success of the theoretical predictions. For instance, the experimental data on rocks before failure imply that Griffith theory is only a means to an end and not a description of rock fracture. Failure initiation begins before fracture in both compression and tension and Griffith suggests that both events should be coincident. Furthermore, failure initiation leads, in compression, to a distribution of S-shaped cracks far removed from the simple ellipsoids treated by previous workers. Thus. it would seem that Griffith's theory is valid only as a theory for failure initiation and its applicability to fracture would appear to be coincidental. Barron [9] has observed that the coefficients of friction required to satisfy the modified Griffith theory are greater than the coefficients of friction for the same rocks measured directly. He has used this point, coupled with the fact that the form of the Griffith theory is satisfactory for failure prediction, to postulate an imaginary effective crack (the parameters of which are found from experiment) in order to describe fracture. He leaves the actual crack model, with parameters measured directly, to describe failure initiation. By this method he is able to relate the occurrence of failure initiation and fracture to the applied stress field, though his results, being so dependent on experimental data. a r e still little more predictive than, say, the Coulomb Navier expression. Any work based on a single detail, as for example Griffith's work was based on a single crack, might seem to be of limited application in the ticld of rock mechanics where the material to be dealt with is so complex and interaction effects so difficult to stud), either experimentally or theoretically. In order to tackle such complexity. statistical approaches have been adopted by a number of workers. In the field of pure statistics. Weibulrs weakest link theory was perhaps the earliest. He modelled a general

solid by a distribution of strengths within the solid and worked out the probability that the solid would fail under a given stress. The main result to come out of the theory was the prediction of a size effect, diminishing as the sample volume became very much bigger than the flaw volume; that is, as the number of discrete flaws became large. In intact rocks with no visible flaws, there is no observable size effect in normal laboratory sampies, implying that the flaws governing failure are very small. The disadvantage of Weibull's theory is that there is no physical basis for the assumption of the form of the flaw distribution. Weibols and Cook [10] have suggested that the process of failure may be controlled by the total amount of strain energy stored by the deformed cracks within the rock and in particular that fracture occurs when this strain energy reaches a maximum. Their procedure for calculating the strain energy of the crack system takes full account of the effect of the stress system on the ability of a closed crack to slide and the theory correctly predicts the form of the strength dependence on the magnitude of the interntediate principal stress. The difficulties with the theory are that the coefficient of crack interfacial friction required to fit the strength data does not seem to match with directly-determined values and that the crack distribution upon which the calculation is based is not necessarily the distribution obtaining at fracture. The physical principles of the theory relate more to failure initiation, or to the initiation of unstable crack growth, than to fracture. The concept of a rock as a mass of, perhaps, interacting cracks has also been used as the basis of an approach by Brady [11,12] who used the stress field applied to his model rock to determine the density of open cracks, the density of closed and sliding cracks and, by Griffith's theory, the density of cracks that are in the process of failing. He then used the results on dilatancy repot.ted by Brace et al. [13] to develop a semi-empirical stressstrain relationship for the model. He claims that stress-

Intrinsic Shear Strength of Anisotropic Rock--IlI dependent strain and fracture are related by a critical volumetric strain criterion; that is, the material fails when a maximum value of volumetric strain is attained within the rock. The strain criterion is equivalent to one in which the total densities of deformable, and Griffithextendable, cracks reach a critical value representative of the number of microcracks necessary to join and create a failure plane. His argument thus leads him to an integration involving a crack distribution. He assumes that the final distribution is proportional to the initial distribution and is thus able to reduce his integral to 'known' functions. For a random distribution function this theory shows an apparently linear relationship between tr I and tr3 with no dependence on tr2 when tr3 is compressive. Brady also shows that a curved relationship between trl and tra, of the correct form, may be produced by assuming that the rock contains a number of different phases and therefore cracks with a range of values for interfacial friction. Such a material may be approximated by a homogenous material with an equivalent coefficient of friction which varies with applied pressure. Presumably the form of the pressure variation may be changed to accommodate any experimental data. Brady's theory is empirical in its approach and the results are not conclusive but it remains as a major attempt so far to relate cracking to stress-strain behaviour and stress-strain behaviour to failure. The theories of both Brady and Weibols and of Cook are able to relate rock properties to something that is, at least in principle, measurable by means other than strength tests. That "something' is the crack distribution and its determination may be regarded as a step towards a fuller theoretical description of rock behaviour.

3. ROCK INTRINSIC ANISOTROPY Many metamorphic and argillaceous rocks are banded, foliated or laminated as a result of the tectonic or sedimentological conditions under which they were formed. Such visible evidence of a defined directionality within the rock is usually accompanied by a preferred orientation of some of the constituent mineral crystals (particularly the inequant ones) and it is necessary to distinguish conceptually between orientation density and spatial density. The latter more directly controls the possible linkages of microscopic 'Griffith' discontinuities. In contrast to fine-grained, laminated sedimentary rocks in which there are compositional and structural variations resulting from cyclic sedimentational behaviour, low-grade metamorphic rocks can be considered anisotropic but homogenous on the scale of the experimental work reported in Part I. The presence of banding or foliation and its accompanying crystallite texture indicates that the bulk physical properties of the intact rock may exhibit some degree ofanisotropy due either to the crystallite distribution directly or to a non-random crack distribution. Brace [14] related variations in 'intact" (no cracks) elasticity to

441

fabric izl quartzite and Attewell [15] showed that a directional concordance existed between the axes of fabric

symmetry and intrinsic elasticity in Pem'hyn slate. Rodrigues [16] and Pinto [17] have obtained experimental data on the variation of Young's modulus with direction in particular rocks. Anisotropy may also be induced by an applied stress field. Tocher [18] observed a wave velocity anisotropy in a rock under a non-hydrostatic stress field and Nur [19], reporting a similar effect, suggests that the stress field may preferentially close cracks, so creating a non-random crack distribution which will then account for the observed anisotropic response. Nur was able to predict velocity anisotropy from a given distribution function but found that the inverse problem--that of determining a unique crack field given a distribution of velocities---could not be solved. Observations of a directional variation in Young's modulus under a non-hydrostatic stress field were explained by Brace [5] using similar arguments. An equivalent directional variation of strength might also be expected in rocks possessing distinctive foliation, fissility and cleavage, and indeed such variation has been proved experimentally for cleaved rocks such as slate and fissile mudstone. In other, less visible anisotropic rocks, directional variation is insufficiently intense to merit separate study as it is considered merely as a factor affecting the scatter of experimental results (Bieniawski [20]; Bernaix [21] ). Casagrande and Carillo [22] were perhaps the first to study the effect quantitatively and Jaeger [23] considered a rock where the shear strength S at an angle ~ (usually defining a shear failure plane) varies according to the expression: , S = $1 + S2 cos 2(at - fl).

(3)

As in the earlier paper of this work, fl is the angle contained by the anisotropic feature plane and the axis of major principal stress. Donath [33] expanded Jaeger's results to obtain a failure criterion of the form: 01 = a - b cos 2(30 ° - fl)

(4)

which fitted his experimental strength data on Martinsburg slate quite well. Walsh and Brace [24] and Hoek [25], using an expanded Griffith theory and a non-random distribution of crack lengths, considered a distribution of long cracks, the planes of which were all mutually parallel, and managed to obtain a good fit to Donath's [33] results. For a crack system of particular length, Walsh and Brace have suggested the equation: C -- Co [(1 + I/2) 1/2 - # ] -]- 2/t2fl 2sin fl cosfl(1 - / a t a n fl)

(5)

for the strength of that system, where Co is the strength of the equivalent random array of the cracks and/~ is the coefficient of crack interfacial friction. The theory has the same data-fitting power as any Griffith-theory derivative and by allowing Co to become a function of angle --that is, by stating that any cracks usually become

442

Peter B. Attewell and Michael R. Sandford

shorter as their planar orientation departs from bedding or cleavage--then almost any form of strength anisotropy may be accounted for. Barron [9], for instance, has applied a multiple crack system to a sandstone with some success and Rodrigues [16] has formulated an empirical relationship to satisfy observed strength variations within a cleavage plane itself. A proper test for a theory such as that of Walsh and Brace would consist of a comparison between the crack distribution suggested by fitting a suitable distribution function to experimentally-derived strength data and a distribution measured by some independent means. Such a comparison has not been performed hitherto, and although Walsh and Brace suggest that their slate results are qualitatively correct and the theory is consistent with the long crack concept of Morlier [34], it is doubtful if, were it possible to measure a distribution, the result would be positive owing to the supposed rapid changes in crack formation undergone by a failing rock. The theories of Weibols and Cook and of Brady may be applied to the anisotropic strength behaviour of rock. Both theories require prior knowledge of a crack distribution, and the introduction of a non-random distribution should produce a variation in strength with direction. The next step is to compare the results of these two theories when applied to Penrhyn slate, the intrinsic anisotropic characteristics of which are reasonably wellknown in both strength [1] and textural [4] detail.

4. ANALYSIS O F P E N R H Y N SLATE

(a) Specification of the problem Penrhyn slate (Greenschist facies) was formed in what is now North Wales during the Caledonian orogeny through processes of dynamic metamorphism, the character of the distribution of its constituent crystals seeming to reflect the symmetry and relative amplitudes of the constraints that were imposed during metamorphism. The 'hard blue' slate studied here is fine-grained and exhibits a well-defined cleavage. Donath's cosine formula (equation 4) can be generalized to include any minimum angle and then fitted to the experimental strength curves for Penrhyn Slate. Two such fits are shown in Fig. 2 and these indicate that the data are rather less symmetrical about the minimum than is Donath's curve, although it should also be noted that the minima of the fitted curves do not coincide with the data minima. However. from the general degree of consistency of fit and from the general appearance and self-consistency of the results and trends, it would seem justifiable to use the strength data presented in Part I of this work for a general study of the characteristics of rock failure theories. The best available information on structural anisotropy derives from the X-ray texture diagrams defining the clay mineral basal plane sub-fabrics and discussed in Part II of this work. The X-ray data only provide information on the orientation density distribution of crystal

70

i

I

i

,

60

50

% t~4 ~ t"

40

3C

7. 2C

I

I0

b0

I

I

15

30

,

|

45

,

I

I

60

75

90

~,* Fig. 2. Data from P e n r h y n slate compared with the modified form of Donath's equation tr~ = a + bcos2(~t - fl). Upper curve at tr 3 = 10,000 lb/in 2, ,r~ = 6 3 , 0 0 0 + 4 0 , 5 0 0 c o s 2 ( 4 7 - fl). Lower curve at tr3 = 0, tx~ = 25,300 + 21,000 cos 2(47.7 - fl). Marked points represent experimental data.

planes {hkil} but say nothing about spatial density distribution. There is no information on crystal size or on crack size, and this means that any attempt to develop a crack distribution profile for the material must be based on an interpretation of the X-ray texture in the light of some additional facts or hypotheses. In this present case, the facts must be the available strength data and the hypotheses must be the failure theories to be tested. It must then be assumed that a combination of interpretation and theory which matches the data should be close to the real situation. Such an assumption is open to some criticism, but in the absence of any proven method of obtaining a crack distribution directly, it must stand as a basis for this analysis. One assumption which can be made independently of any failure theory is that the anisotropy is sufficiently described by the character of the clay mineral basal plane sub-fabrics. This assumes that the intrinsic structure of the rock, as manifest for example in strength experimentation, is dominated by the illite and chlorite crystals having a platy morphology, and it ignores any contribution of a more isotropic nature from the more

O n g~ ,..-Z ,<

t-"

v

3

Cos

(~,~)

Cos

(;~o~)

Fig. 3. Plot of the function o,'(r/,;'.)n(rl, 2) used in the Weibols a n d Cook integration showing its generally parabolic shape.

Intrinsic Shear Strength of Anisotropic Rock~III equant-habit minerals in the rock. There is no fundamental difference---other than one of degree of anisotropy---between the c-axis sub-fabrics of illite and chlorite and either could be used for further analysis.

(b) Application of Weibols and Cook theory Weibols and Cook advanced a strain energy theory of failure which they restricted to an isotropic distribution of cracks. The argument is that the energy stored within the material prior to failure is compounded from elastic deformation and more specifically from the shear distortion of closed microcracks. This latter problem can be summarized in terms of the normal stress a~ across a closed crack, zc the shear stress along it, and the friction coefficient,/~c. Sliding will take place along the crack when

443

length a~ a function of orientation. The assumptions leading to this are as follows: first, that the crystallites in the rock which dominate the anisotropy do not vary in size; second, that the crystallites are spatially independent; third, that the cracks of anisotropic significance are located at and parallel to the basal planes of the platy crystallites; fourth, that the cracks are spatially independent. These assumptions permit the X-ray textural data to specify the number of crystallites directly and the number of cracks indirectly. Justification, or otherwise, for these assumptions will arise from the quality of the results. The form of the function:

(1 I)

f = co(r/,).)N(t/,2)

is shown in Fig. 3 and, being smooth and quasi-parabolic, lends itself to a simple Simpson's rule integration [z~l >/a~o= (6) procedure. In fact, two such procedures, one for r/and and there will be an effective shear stress for movement one for 2, were interleaved to perform the double inof tegration. Each integration was performed repeatedly, with decreasing step length, until a convergence criterion Z'eff = Iz l (7) had been satisfied. Execution time for the program was Under these conditions, the crack will store recoverable markedly dependent upon the accuracy required and, as shear strain energy co, where a suitable compromise between the two factors, the convergence criteria were set such that the overall accuracy co = (8) of the result was 1 per cent or better. k is a constant, its value being dependent on the elastic Values of W were derived for various combinations of constants of the surrounding material. fl, tan ~b, at, a3, an example of the relationship between The total distortional energy for a large number of at and W for two input values of cr3 and for a range cracks will be the sum of all co, and may be written: offl being shown in Fig. 4. Logically, such a set of curves could be used to produce a solution for Wmx by selecting values of the shear strain energy parameter W until the W kf so(r/,2) N(r/, 2) dr/d2 (9) values of a~ generated by the curves converge to concordr .ib dance with the experimental values of at. However, where N(r/,2)* is the number of cracks having poles observation of the Fig. 4 relationships indicates that the orientated r/(latitude dip), 2 (azimuth), the integral being intrinsic strength of the material decreases progressively taken over those areas on the sphere of projection (r/,2) with increasing fl up to a 60 ° cleavage inclination and where movement can occur (tar > 0). The constant k is this, of course, conflicts with the experimental results assumed to be independent of direction, although for an which reverse this sequence from a minimum strength at anisotropic matrix this will not be so. fl = 30 °. It is concluded that this particular strain energy If it is proposed that failure occurs when W reaches concept does not stand up to the test of experimental some limit Win,x, then knowing Win,x and for a given tan ~b~, the amplitude of the stress difference at failure observation but there are grounds for suggesting that requires the solution of the equation: 9O

W = W.,a~

.

(lO)

and knowledge of the distribution N(q,2). Since W and Wmax need not be known absolutely, values of k and N proportional to the real values only need be used. For numerical solutions, therefore, k has been entered as unity and N has been taken unmodified as the normalized distribution function obtained directly from the pole figure analysis. The use of the above distribution function implies that the cracks in the material vary in number but not in * These symbols are used to generalize the discussion. In terms of the specific P e n r h y n slate sub-fabrics presented in Part 1I, and taking account of the symmetry a r g u m e n t s a n d evidence on the genetic history of the slate. 2 ~ 0 and r/ = "/.

40

/9.

o

3o_ 4 5 ¢=01 75

£- 30

=I ®,~ 9

_

~o

x 20~

i,oL / ....

, ....... 6000

,,=7000. . . . . . . .

W - f w . N 017 dX, orbitrory

1,o

units

Fig. 4. Plot of Weibols and Cook strain energy against az for various values of [/. The upper curves correspond to a aa value of 6000 Ib/in 2 and the lower curves to a a~ value of 2000 lb/in e. Poisson's ratio v is taken as 0-4.

444

Peter B. Attewell and Michael R. Sandford

this failure may be a property of the distribution adopted rather than of the theory itself. It may be noted that the families of curves in Fig?4 tend to converge with decreasing W, so implying by extrapolation an elimination of strength anisotropy for small W. Griffith theory would produce the same result when applied to a crack distribution where the crack length was constant. This limit, which is at a lower level than the minimum strength values arising from the experimental work, increases with increasing interfacial friction and tends to the minimum experimental strength value at fl = 30 ° when
Brady's theory, embodying analysis and empiricism, is based on some aspects of the pre-failure stress-strain behaviour of rock and in this sense it has, intuitively, an a priori relevance to an integrated crack extension concept. It is developed from a prediction of the permanent strains generated by cracks undergoing failure and the failure criterion makes use of the predicted strains. Numerical predictions rely quite heavily on experimentallydetermined strain and failure measurements. Although this theory is quite capable of handling very general stress systems, it will be necessary only to consider compressive systems in which 0.2 = 03 in order to try to achieve compatibility with the experimental data. As in the Weibols and Cook theory, a closed microcrack situation only will be considered. As a rock is strained, Fig. 1 shows that a level of applied stress is reached at which microcracking begins and induced, permanent, lateral strain becomes just noticeable even though the applied axial stress remains elastic. This situation can lead to two assumptions: first, that permanent lateral strain is caused by microcrack extension and, second, that microcracking does not contribute to permanent axial strain. Thus, if the change in strain ¢; for a single microfracturing event is dEi Brady writes: (dEl) ~-- 0

(12)

(permanent axial strain)

and (&3) = :}Go

- ~,,1

(0.( _ 0.1)"

(15)

where G and m are constants, 0.{ is the macroscopic failure strength of the material, and 0(2, q) is an unknown function defining the crack orientation. In general, the formula states that strains become larger for increasing al, but, in particular, that they tend to infinity when failure occurs. This form of expression is not unreasonable for strains and stresses close to failure but the proven existence of a finite post-failure strength in a number of different rocks studied in stiff testing machines suggests either that Brady's proposed form is incorrect or that the development of a failure plane immediately prior to failure allows the operation of a mechanism other than that of differential crack extension. However, since the failure plane does not begin to form until some 98 per cent of the failure stress has been applied, its presence may be temporarily and conveniently ignored and Brady's results taken as they stand. From the incremental strain contributed by each crack, the average microcracking strain may be written as: d0.1 (16) (dE3)=

½A2 P (0.( -- 0.1) m

where A2

= G1

"

(40.,q)P(2,tl, c ) d 2 d r 1

(17)

=0

and p = p,.|

P(2,~l,c)d2dqdc.

.2Cl

(18)

P(2, q, c) is the crack extension probability, p is the density ofmicrocracks able to extend and pe = N e / V = the density of flaws within the specimen. Thus, r/l and r/2, cl and c2 are the crack dimension limits defined by the Griffith criterion: It[ - #ca,,c > 2 ( K / c ) 1:2

(19)

with K a constant. Since (d~l) ~ 0, the total average microcrack strain is given by: da,

(dE 2 ) ~ (dE 3

(13)

for an axially-symmetric (about El axis) microcrack distribution. These permanent strains will be functions both of crack orientation (and possibly length, although Brady does not suggest this) and applied stress, and for a general crack distribution where )~,r/are the polar coordinates of the crack-normal with respect to the at axis, Brady's formula becomes: (dE2)

,

d0.l

4G g (z, tl) (0.{ _ iT1),,

(14)

-(dr)

"~ A p ( o f

_ 0.1) m,

(20)

Now Paulding's work on dilatancy (Brace et al. [13]) suggests to Brady that a critical volumetric strain criterion may be operative. Accordingly, he takes the total volumetric work done over the microcrack strains:

Wv =

vfa.tdeiit

(21)

where V is the volume of the specimen, and he states a criterion such that W v = Wm~x. This is equivalent to the postulate in the Weibols and Cook theory (equation 10).

Intrinsic Shear Strength of Anisotropic Rock--III The total volumetric strain is given by:

Id~il --

¢

f~

da~ Ap ( a { - a l ) m"

(22)

At this stage, Brady states, without proof, that the above work criterion is equivalent to the density criterion p = C, where C is a material constant. Either form of the criterion, he states, expresses the assumption that total failure occurs when there is a sufficient number of microcracks available so that the probability of their joining up to form a macroscopic fracture surface is large. The constant crack density form of the criterion requires for its operation the evaluation of the integral in equation 18. For P(c, ~/,2) = P(r/, 2), the same distribution function as used for the Weibols and Cook theory may again be adopted. Using this same distribution function (that is, one in which the crack length is assumed constant and the number of cracks is taken to vary) produces much the same shape plot when applied to texturally-interpreted crack densities with much the same negative results as for the previous evaluation. The same comments apply here as were made at the end of the Weibols and Cook analysis. The apparent failure of these two related theories to satisfy a quantifiable strength anisotropy from a unique microcrack orientation density distribution must be assumed to be due to some characteristic of that distribution. It is now proposed to consider an approach which should go further towards resolving this problem.

(d) Modified approach to failure interpretation Since the actual strength anisotropy of the slate, based on a particular crack distribution, cannot be interpreted in terms of the volume interaction theories independently proposed by Weibols and Cook and by Brady, and since the formulation of the distribution function might be claimed to be based on rather weaker assumptions than those embodied in the two theories, then it would seem reasonable to formulate an alternative distribution function but with the provision that the function is directly related to the microstructural evidence as supplied from the texture-goniometer. The work of Walsh and Brace. and of Hoek provides the starting point for the formulation of such a new crack distribution function. These workers used crack distributions such that all cracks had the same length and were all aligned parallel to the bedding or cleavage plane as the case may be and they used their results to postulate that a good fit with experiment could be obtained over a wide range of angle fl by applying Griffith's theory to a system of cracks the lengths of which decrease as their orientations depart from the bedding or cleavage plane. The implication is that a crack length distribution could be derived from the experimental data, and Barron performed such a derivation for a sandstone.

445

The present problem is, however, the inverse of this. Strength must be related to a known rock property (a derivative of the X-ray texture). Tbe problem is simplified in the sense that it is known from the earlier work that a variation in crack length is required such that the longest cracks lie parallel to the bedding, and the form of the experimental data suggests that this variation should be smooth with the shortest cracks lying perpendicular to the bedding. If, for example, it were to be assumed that a crack was associated with each clay mineral face, the length of the crack being proportional to size of the face, then such a continuous size variation with orientation might at first be thought capable of expression most simply through a convolution of the orientation distribution functions generated by (0001) (long cracks) and (1120~ (0110) (short cracks). However, considerations of crystallographic symmetry and plane multiplicities suggest that the functions generated by the basal plane distributions should adequately reflect the necessary relative crack length distribution with departure from bedding or cleavage. In his work on wave attenuation in rocks, Walsh [26] observed that his results suggested the existence of one crack for every constituent grain in the rock. For the type of cracks that Walsh considered, that is, those that were just closed under very small normal compressive pressures, he thought that a crack-to-grain ratio of almost unity was unlikely. The work of Brace et al. on electrical resistivity of rock implies the presence of a large volume of contiguous pores within (most) rock and Walsh suggests that a real rock should have a looser structure than his own model, with the wave attentuation caused by larger displacements on fewer cracks. The corollary is--although Walsh does not say so explicitly-the concept of a rock having larger cracks than existed in his model, with each crack encompassing more than one grain. This means that it is probably unrealistic to assume that each grain contributes one crack; it must be conceded that larger, continuous cracks owe their genesis to--or are associated in a more general sense with--adjacent individual grains, and if this phenomenon is in some way dependent on orientation, then crack lengths may indeed vary with orientation. Although elongate crack-shaped cavities can be associated only with sub-parallel grains, the concept of an orientation distribution also indicates the possibility of a series of elongate cavities having many orientations and all joining-up to create a large crack which may be far from planar. However, while acknowledging the possible internal complexities of natural polymineralic aggregates, it is both necessary and reasonable to erect and test a model of somewhat reduced complexity. Murrell and Brace et al. have suggested that all cracks which are going to close and operate subsequently in a frictional manner will close under a reasonably low applied stress. Brady has also said that it is the closed cracks which will be correctly orientated to initiate failure. The first point implies that cracks that are almost closed will close and the second point shows that it is

446

Peter B. Attewell and Michael R. Sandford

only necessary to look at closed cracks of one orientation at a time. The two points taken together further imply that subsequent discussion may be directed solely towards sub-parallel grains or crystallites; that is, in interpreting X-ray textural information as a means of formulating a crack length distribution, the amplitude data at each discrete orientation may be considered in isolation from the data at all other orientations. Given this assumption, it should be possible to relate the numbers of grains in a given orientation to the probability of their forming clusters of a given size and thence to an estimate of a crack distribution function. Mack [27, 28] studied an appropriate two-dimensional form of this problem and produced a formula for the expected number of a k-aggregate in a 'random' distribution of points, where a k-aggregate is defined as k points which may be covered by a window of known size. With respect to the present problem, a cluster of grains or crystallites may be said to have formed if the centres of those grains fall within a given area, and Mack's work may then be adopted directly. There are, however, a number of problems associated with the application of this concept. For cracks considered to be related to crystallite faces, it is possible to imagine two overlapping crystallites so producing an effective crack length of less than two unit crack lengths. A window which included all reasonable pairs of cracks would also include all overlapping cracks and it would be necessary to perform at least two calculations, with different window sizes, to take the overlap effect into account. Such a calculation would not, in itself, be excessively demanding, but there is a further problem which may be much more difficult to overcome: there is no simple way of distinguishing a k-aggregate that is also part of a k + 1 aggregate. In particular, it may be seen that a 3-aggregate may contain 1, 2 or 3-off 2-aggregates depending on the sizes of the windows used to view the aggregates. For the case of clusters of two or three, it should be possible to take account of this effect, but the complexity of the problem increases with the size of the cluster being considered and rapidly becomes unmanageable. These problems, coupled with the complexity of Mack's original expression, mean that it will not be possible to produce a complete crack distribution function even though Mack's approach is apparently suited to the problem. For the purposes of Griffith's theory it is not necessary to access the complete distribution. All that is required is a knowledge of the length of the longest crack cluster in any orientation. In theory, this is only limited by the size of the sample and the total number of discrete unit cracks in the sample having that particular orientation, but clearly these limits are rarely reached, especially when samples tend to be chosen so that they appear to be homogenous, with no major flaws! A more practical limit should be obtained if it is assumed that the number of k-aggregates, which are not part of (k + l)-aggregates, reduces the population available for (k + D-aggregates.

However, this approach still requires the complete distribution function and, as has already been noted, this is not available. We are left, therefore, with the indication that the maximum crack length will be some function of the total number of unit cracks and, for the rest of this discussion, it will be assumed that this function is linear. This means that the assumed maximum crack length will be directly proportional to the (0001) distribution function as processed from the X-ray texture-goniometer output. Having thus obtained a form of crack distribution, Griffith theory can be applied and predicted strength curves obtained. Re-writing equation (19) and taking c from the unmodified, normalized X-ray texture, the criterion then becomes: Izl -

uca~ = B/x/1

(23)

with I being the normalized diffracted X-ray intensity value. The values of p~ and B have to be determined by experiment. The process of finding values for #, and B and using these values to predict anisotropic strengths was carried out in two stages. The first stage involved the use of a direct application of the formula of Walsh and Brace to the simple distribution of cracks, all having the same length and all being mutually parallel. Parameters for this formula (Co, the unconfined compressive strength for a random array of the same crack and #c the coefficient of crack interfacial friction) were chosen such that the theory produced a least squares fit with experimental data at/3 = 30 °. In this case, #c was found to be 0"5--of. range of(}8-1 (McClintock and Walsh, [29]); 0-5 to 1.5 (Hock and Bieniawski, [30] }--for various rock types. This value of (}5 is determined largely by the location of the minimum strength condition with respect to/3 and since this location is well defined by experimentation it is possible to attribute a quite reasonable degree of confidence to the value. In Fig. 5 a number of curves of %ff(= ]z~] - ~a,~) as a function of angle /3 are drawn for #~ = 0.5 and for various values of al and a3. These curves may be used with a plot of B/x/c vs/3 to determine strength levels and, conversely, strengths may be used with the Zerf curves to generate B/x/c values. For the. simple crack distribution considered so far, there is only a single B/v/c value at zero angle and, for these conditions, the value of B/x/c is 2.9 in the units used. This value is required to correspond to the minimum B/x/c for any more general function that may be used to describe these distribution data and, since B is constant, the function must also satisfy the maximum value of c. In the more general function chosen, the crack length c is equivalent to I, the normalized X-ray diffraction intensity, and it is known that the suggested value of lmax is 8 as taken from the smoothed pole figure. From this, the value of B is 8"2. The second stage of the analysis is actually to plot the values of B/x/c = B/x/l and to use the Lfr curves to determine the angles/3 for which failure can take place under a given stress system. This is done by superimpos-

447

Intrinsic Shear Strength of An!sotropic Rock--III

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70

. 80

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ing the zefr curve on the B/~/c curve such that, for a given at, the two curves just touch. The angle through which the "Ceff curve must be displaced to achieve this result is the angle at which the particular al raises a shear failure condition. The B/x/c curve is shown in Fig. 6 and a plot of strength against angle for four values of as is shown in Fig. 7. Experimental values for as = 0 are also shown in Fig. 7 and by reference back to Part I it can be seen 60

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that the fit of theory to experimental data is less good for low values of ft. Such incompatibilities arise because the theoretical values are almost symmetrical about the minimum when the experimental values clearly are not. Figure 8 plotted in the (at - a3)/(~3 plane expresses the quality of match between the variable crack length theory and the experimental data for several fl angles. It can be seen that the quality of fit decreases with a greater departure of fl from 30° and is unacceptable for fl < 30 °. It should also be noted that the theory predicts a linear relationship between at and aa at failure and that the slope of this line does not vary significantly with ft.

5. DISCUSSION The theory applied in this work is a Griffith theory and as such produces a straight line plot between (a~ a3) and a3, with the slope being dependent on the coefficient of crack interfacial friction pc. There is, however, one less degree of freedom in fitting the Griffith theory to anisotropic data in as much as the value o f / ~ must also define the angle flmi, at which the minimum strength is observed such that 2flmi, = tan- 1/~/-t. For this reason, it seems acceptable to suppose that the value of /~ obtained, giving, as it does, such a good fit to both slope and minimum, will represent quite closely the average value of the real coefficient of friction actually operating in the rock. Walsh [26] quotes values of ~ obtained experimentally for both smooth and roughened crystal surfaces• The figure of 0.5 fits into his range of values for various crystal surfaces ro]/ghencd by sliding across one another, the situation appropriate to the prc-macroscopic shear failure and one which imposes a frictional control on the final failure. Finding a value of Co appropriate to the experimental data has no theoretical significance unless it can be used to provide some indication of crack size. This operation, however, demands some knowledge of the elastic constants of the intrinsic material of the rock, and that information is not readily available.

448

Peter B. Attewell and Michael R. Sandford 50 o -

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The fact that the application of Griffith's theory to the simple, adopted crack length distribution produces an anisotropic strength distribution is not, in itself, remarkable, but the result is, nevertheless, a useful one in that the strength distribution so formulated bears a close resemblance to that obtained by experiment, In particular, there is quite a close fit between theory and experiment for angles fl > 30 ° and in this the crack distribution postulated from the evidence of clay mineral basal plane orientation density distribution improves on the simplest distribution where all cracks are parallel. The crack length distribution derived from the X-ray data is not necessarily--and, rather, is unlikely to be--the true distribution, but the very fact that such a distribution, obtained independently of the data that it was hoped to fit, does produce such results argues against simple coincidence. The main difficulty in allowing the texture-based theory to describe the experimental data lies in the discrepancy between the two for fl < 30 °. As mentioned earlier, this discrepancy arises because the theoretical curve is symmetrical about its minimum, a symmetry which is not shared by the experimental points. The symmetry is built in to the two components which constitute the final curve: the crack distribution is naturally symmetrical and an asymmetric distribution would be most unlikely; the ~effcurve is almost symmetrical about its maximum, but what asymmetry exists does not become obvious until some distance below each peak and it can therefore have little real effect. With respect to the experimental data, the asymmetry is similar to that noted by Donath in his Martinsburg slate results.

Donath, in fact, only noticed a serious departure from his symmetrical curve in the data for fl = 0 and in this instance claimed, reasonably, that the constraining effect of the sample end plates affected the result, with ~t following fl closely for fl angles up to 30 °. This same effect has been noted by Attewell and Sandford in Part I. In Fig. 9, the failure plane inclination ~t with respect to tr~ is plotted against fl, the sample orientation, for all Penrhyn slate samples at all angles and confining pressures (the latter without distinction). This plot shows a trend towards control of the failure plane angle by sample orientation and it seems significant that the data for fl = 15° and fl = 30 ° show much less scatter than that for the other angles. These relationships may be generalized to the assumption that ~ = fl for fl ~< 30 °, and with this assumption some new results may be obtained from the theory presented in the last section. The expected form of interaction between the effective shear stress and the crack lengths is shown in Fig. 10. 50

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449

Intrinsic Shear Strength of Anisotropic R o c k - - I I I

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Fig. 10. Showing graphical solution of the equation: Lrr = BIv'c. The two curves are moved relative to each other until they just touch. Displacements a and fl represent the orientation ofcq with respect to the failing crack and the cleavage plane respectively.

Where the L . . curve just touches the B/x/c curve defines the failure point. The point of contact would also be expected to define angle ~t, as shown. Usually, this angle would lie quite close to the expected angle of 30 °, but a different situation arises if it is known that there is some other constraint imposed on ~t. In particular, the case where ~< = fl is shown in Fig. 11, such a situation clearly requiringa higher value ofcr~ than would normally be predicted. A new set of values for tTl(ct = fl for fl ~< 30 °) when ~3 = 0 is plotted in Fig. 12 and produce the required upturn of the curve for the low fl angles. In a similar manner, a plot of(al - O'a)against Ga for fl = 15 ° in Fig. 13 n~ay be compared with Fig. 8a which shows the same plot before this modification. Since the improvement is significant, the relationships between 0t and/7 can now be re-computed in Fig. 14 on the basis of the above assumption. It can be seen that the predicted angles ~t above fl = 30 ° are larger than the data suggest but that the general form of the data is preserved. However, for angles ~ = fl close to zero, the %, curve vanishes and the theoretical strength of the rock becomes infinite. Thus, the proposed modification still leaves an unknown gap in the interpretation of strength anisotropy. A further--although perhaps related--deficiency is rooted in the reason why the shear plane should be constrained to follow the cleavage for a range of angles

Fig. 12. Plot of ~rl vs. cleavage plane angle/7 for zero confining pressure. The full line is the theoretical curve as modified by shear plane entrainment. The circles are the experimental data points.

fl < 30 ° but not for a similar range fl > 30 °. It is possible, however, to imagine a causative mechanism, a description of which can be attempted. A number of workers have shown that a crack will propagate in the sigmoidal form drawn in Fig. 15 such that the extensions tend to become parallel to the al axis. An initially propagating crack will be selectively aligned at about 30 ° to t~l, but the new extensions will serve to produce a new effective crack at a smaller angle to tTt. Thus, the crack would seem to rotate either away from or towards the cleavage plane according to whether fl is greater or less than 30 ° (see Fig. 16). In this way, the propagating cracks may become constrained towards the plane of weakness, the cleavage plane, and the distribution will become depleted for crack orientations relatively remote from the cleavage plane. This constraint into the cleavage becomes more difficult for fl approaching zero degrees and there must be some gradual modification into yet another failure mode as, for example, the effective tensile stress mode of Brown and Trollope [31 ]. Having obtained a theory which enables the anisotropic strength variations of a natural material to be predicted with a reasonable degree of accuracy, it is then necessary to examine the success of that theory in the light of the pre-failure behaviour of the rock. The fact that the proposed theory relies solely on a Griffith criterion immediately suggests that the pre-failure behaviour of the material has been ignored. This is 50

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Fig. 13. Plot of(el - ~3) vs. ~a for fl = 15°. The full line represents the theoretical relationship modified for shear plane entrainment. The circles arc the experimental data points.

450

Peter B. Attewell and Michael R. Sandford

4O 30 ao

20 I0

....

"

/ Fig. 15. Approximate form of a failed crack (after Brace et al. [13]).

becomes modified by the surrounding activity. Morlier also provides part of the answer when he suggests that 0 I0 20 30 40 50 60 70 80 initial microcracking may be due to cracks in weaker B° material propagating until halted by, for example, a Fig. 14. Modified ~ - B relationship. stronger grain, so causing fracture hardening. The existence of either of these postulated phenomena contravenes none of the observed pre-failure occurrences but, not necessarily true. An examination of rock failure equally, neither necessarily implies co-operation in the theories has shown that only the Weibols and Cook and sense envisaged by Weibols and Cook or by Brady. the Brady theories make any attempt to take into However, a form of co-operation is taking place in that account a cooperation of cracks in generating and conthe pre-failure activity is setting up the required condiditioning the failure, but it is easily seen that these tions for failure. theories provide little or no improvement over the GrifIf cracks in rocks are formed by associations of subfith one. In particular, the constraints on the crack fricparallel crystal faces it must also be stated that such tion coefficient Pc are the same in all theories. Examinacracks may take all orientations and may join together tion of Fig. 4 shows that the strain energy function, for to produce larger cracks which are not flat but which constant crack length, produces the same slope for the bend and twist in space in an undefined manner. It is o 1 vs. o"3 line independently of the total strain energy and now possible to envisage such distorted cracks as having there is no reason to suppose that the change in the form weak points, for example, at crystallite edges, and that of the distribution would change that. Similarly, the under stress these weaknesses permit some form of microshape of the tr~ vs. ~ variation would remain symmetrifracture. Such microfractures will propagate only to cal as in the simple case, although the detailed shape the surface of the crystallites concerned and any excess would depend on the exact form of the B/\/c curve used energy will appear as additional strain on the whole and, as has been shown, this is open to some manipulacrack and as microseismic activity. This means that the tion. Thus, the two co-operative theories would produce surface of such a distorted crack will become ground and very much the same results as does the simple Griffith splintered until it becomes fiat, and the orientation of the theory. resulting flat crack will be just that orientation along However, where the strain energy and strain volume which maximum resolved shear stress operates. Only the criteria do depart from the Griffith theory is in their abicrack spatial density distribution, not the overall texture lity to predict the angle of the failure plane. Neither of the rock, will be altered by this operation and, in fact, theory contains any mechanism whereby the expected it will be the crystallite structure of the rock which will failure plane may be derived and it would be difficult to control the formation of the new crack distribution. This modify either theory to take into account the crack modification of the crack distribution will only take entrainment mechanism which has been proposed for place for those orientations along which the shear stress < 30°. Such a modification is not impossible and it is large, but within that limitation the crystallite texture may well be feasible to implement the appropriate inof the rock will specify the final crack distribution and tegration over a dynamic crack population by the use of not the initial one. In this way, terminal, catastrophic faila digital computer, but having done so. the answer ure will not occur until a well-defined crack-like cavity, would turn out to be a series of applications of Griffith's terminated by some stronger region of rock material, has theory and the basis of the work. either strain energy or formed, whereupon the Griffith criterion and its closed strain volume, would become lost in detail. Indeed, such crack derivatives comes into operation with all that that theories become irrelevant when such a detailed model implies. of the structure becomes available. It would seem that. even tbr anisotropic data. the GrifCleavage / / p l a n e / fith theory still holds good despite a large quantity of 30 ° data on pre-failure rock behaviour which suggests that for ~ > ~7 /~// //C leavage a more complex co-operative theory is required. The plane question is: if the Griffith theory is applicable to rock failure rather than failure initiation, then what is happening between failure initiation and terminal failure? Murrell provides the clue. s n o w i n g new I Murrell [32-] points out that the microcrack events effective direction which occur prior to failure contribute to the formation of a range of new crack distributions and that any crack Fig. 16. Showing how a failed crack may become constrained into the cleavage plane for angles ~ < 30 . may propagate more than once as the stress distribution 0

Intrinsic Shear Strength of Anisotropic Rock--III

45i

6. CONCLUSIONS

5. Brace W, F. Relation of©lastic properties of rocks to fabric. J. Geo-

It has been shown that the simple Griftith theory may be applied with some success to the failure interpretation of an intrinsically anisotropic rock provided that the crack lengths used by the theory are taken as the crack lengths obtaining in the rock immediately prior to ultimate failure. In the case of Penrhyn slate considered, the required distribution of crack lengths has been taken from the unmodified data of an illite basal plane pole figure. It would appear from this that the plate-like illite and chlorite crystals constitute a major factor in determining the manner in which large cracks form in the material. It has also been shown that the lack of symmetry in the experimental relationships between strength and cleavage orientation may be explained by accepting the observed phenomenon of ultimate failure plane entrainment into the cleavage plane for cleavage plane orientations of less than 30 ° with respect to the major principal stress axis. There are a number of still-unresolved imponderables in the problem but, given the assumptions outlined in the text. the importance of the present results can be regarded as threefold. First, they tend to show that the use of intrinsically anisotropic material can perhaps provide a greater insight into the mechanism of rock failure under compression than can the equivalent use of isotropic material. Second, they suggest that the Gritfith theory seems still to describe rock failure behaviour rather better than do many others. Third, they indicate that it is possible to relate quantitatively an independently-measured property--in this case, an important element of the rock texture measured by X-ray methods - - t o experimentally--derived strength data and to find quite reasonable agreement between them.

6. Wawersik W. R. Detailed analysis of rock failure. Ph.D. thesis, University of Minnesota (1968). 7. Attewell P. B. and Woodman J. P. Stability of discontinuous rock

phys. Res. 70, 5657-5667 (1965).

masses under polyaxial stress systems. In Stability of Rock Slopes, 13th Symposium on Rock Mechanics (E. J. Cording, Ed.) University of Illinois, Urbana, Aug. 30-Sept. 1, 1971, A.S.C.E., pp. 665-683

(1972). 8. Franklin J. A. A strength criterion for rock, Imperial College, London, Rock Mechanics Research Report (1968). 9. Barron K. Brittle fracture initiation in and ultimate failure of rocks--I-Ill. Int. J. Rock Mech. Min. Sci. 8, 541-575 (1971). lO. Weibols G. A. and Cook N. G. W. An energy criterion for the strength of rock in polyaxial compression. Int. J. Rock Mech. ~lin. Sci. 5, 529-549 (1968). 11. Brady B. T. A statistical theory of brittle fracture for rock materials--I: Brittle failure under homogenous axisymmetric states of stress. Int. J. Rock Mech. Min. Sci. 6, 21-42 (1969). Brady B. T. A statistical theory of brittle fracture for rock materials 12. --II: Brittle failure under homogenous triaxial states of stress. 'Int. J. Rock Mech. Min. ScL 6, 285--300 (1969). 13. Brace W. F., Paulding B. W. and Schoh C. Dilatancy in the fracture of crystalline rocks J. Geophys. Res. 71, 3939-3953 (1966). 14. Brace W. F. Brittle fracture of rocks. In Int. Conf. on State of Stress in the Earth's Crust, pp. 110-178, Santa Monica, California (W. R. 15. Judd, Ed.). Elsevier, New York (1963). Attewell P. B. Triaxial anisotropy of wave velocity and elastic moduli in slate and their axial concordance with fabric and tectonic symmetry. Int. J. Rock Mech. Min. Sci. 7, 193-207 (1970). 16. Rodrigues F. P. Anisotropy of rocks. Most probable surfaces of the ultimate stresses and of the moduli of elasticity. Proc. 2nd Congr.. Int. Soc. Rock Mech. Belgrade, 1, 133-142 (1970). 17. Pinto J. L. Deformability of schistous rocks. Proc. 2nd Congr., Int. Soc. Rock Mech. Belgrade, 1,491-496 (1970). 18. Tocher D. Anisotropy in rocks under simple compression. Trans. Am. Geophys. Union 38, 89-94 (1957). 19. Nur A. Effects of stress on velocity anisotropy in rocks with cracks. J. Geophys. Res. 76, 2022-2034 (1971). 20. Bieniawski Z. T. Mechanism of brittle fracture of rock--I: Theory of the fracture process. CSIR Report No. MEG 520, Pretoria, South Africa (1967). 21. Bernaix J. New laboratory methods of studying the mechanical properties of rocks. Int. d. Rock Mech. Min. Sci. 6, 43-90 (1969). 22. Casagrande A. and Carillo N. Shear failure of anisotropic materials, J. Boston Soc. Cir. Engrs 31,122-135 (1944). 23. Jaeger J. C. Shear failure of anisotropic rock. Geol. Mag. 97, 65-72

(1960). Acknowledgements--Dr. S. A. F. Murrell has kindly read through the script and commented on several points. The work was supported by a Natural Environment Research Council grant to one of the authors (P.B.A.) and a research studentship to M.R.S.

Received 5 April 1974.

REFERENCES 1. Attewell P. B. and Sandford M. R. Intrinsic shear strength of a brittle, anisotropic rock--I: Experimental and mechanical interpretation. Int. J. Rock Mech. Min. Sci. 11,423--430 (1974). 2. Pincus H. J. Optical processing of vectorial rock fabric data. Proc. 1st Congr. Int. Soc. of Rock Mechanics. Lisbon, v. 1, pp. 173-177 (1966). 3. Pincus H. J. Sensitivity of optical data processing to changes in rock fabric--I : Geometric patterns, int. J. Rock Mech. Min. Sci. 6, 259-268. Part II. 6, 269-276 (1969). 4. Attewell P. B. and Sandford M. R. Intrinsic shear strength of a brittle, anisotropic rock--lI: Textural data acquisition and processing. Int. J. Rock Mech. Min. Sci. 11,431-438 (1974).

24. Walsh J. B. and Brace W. F. A fracture criterion for brittle anisotropic rock. J. Geophys. Res. 69, 3449-3456 (1964). 25. Hock E. Fracture of anisotropic rock. J. S.A. Inst. Min. Metall. 64, 501-518 (1964). 26. Walsh J. B. Seismic wave attenuation in rock due to friction. J. Geophys. Res. 71, 1(3, 2591-2599 (1966). 27. Mack C. The expected number of aggregates in a random distribution of points. Proc. Camb. Phil. Soc. 46, 285-292 (1949). 28. Mack C. The effect of overlapping in bacterial counts of incubated colonies. Biometrika, 40, 220-222 (1953). 29. McClintock F. A. and Walsh J. B. Friction on Griflith cracks under pressure. Proc. 4th U.S. Natl. Congr. Appl. Mech. Berkeley, pp. 1015-1021 (1962). 30. Hock E. and Bieniawski Z. T. Brittle fracture propagation in rock under compression. Int. J. Fracture Mech. 1,137-155 (1965). 31. Brown E. T. and Trollope D. H. The failure of linear brittle materials under effective tensile stress. Rock Mech. Eno. Geol. 5, 229-241 (1967). 32. Murrell S. A. F. Micromechanical basis of the deformation and fracture of rocks. Proc. Civ. E ~ . Materials Conf. Southampton (M. Te'eni, Ed.) pp. 239-248 (1969). 33. Donath F. A. Experimental study of shear failure in anisotropic rock. Geol. Soc. Amer. Bull. 72, 985-990 (1961). 34. Morlier P. Sur le comportement des roches fragiles avant le rupture. Proc. Syrup. on Rock Fracture, Nancy, France, Sect. 1, Paper 4.