Friction of Rocks and Stability of Rock Slopes

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Eleventh Rank& Lecture

FRICTION

JAEGER, J. C. (1971). GCotechnique 21, No. 2, 97-134.

OF ROCKS AND STABILITY OF ROCK SLOPES

J. C. JAEGER*

SYNOPSIS

The similarities and differences between soil and rock mechanics are discussed with particular reference to the stability of slopes. The effects of constraints and of the stiffness of the system apply- ing stress are of greater importance in rock mechanics. The criteria for failure of rocks are mostly empirical and lead to linear or power laws. Similar laws might be expected to hold for friction. While the Coulomb law is in general adequate for soils, it appears that the frictional behaviour of rocks is described better by a non-linear law and if a Coulomb law is used factors of safety are sensitive to the value adopted for the cohesion. The various methods for measuring friction are described and their limitations discussed. The process of wear and the area of contact between sliding surfaces are considered. It appears that in some cases residual values of friction are attained after small amounts of sliding. Gouge is built up during sliding and its behaviour appears to be time-dependent. At present, numerical values for friction and other parameters of jointed systems are uncertain and so simple formulae are still useful. A number of formulae for factors of safety for sliding on one or two plane surfaces are given. For the case of rock with closely spaced joints the use of soil mechanics theory for circular and other surfaces of sliding is reasonable. In this case, values of friction obtained from single joint surfaces or crushed rock are conservative since the interlocking of rock elements may cause a substantial increase in strength.

Les similitudes et les differences entre la mecanique des sols et la mecanique des roches sont etudiees, avec reference particuliere a la stabilite des pentes. Les effets des contraintes et de la rigidite du systeme appliquant les contraintes ont plus d’importance dans la mecanique des roches. Les criteres pour la rupture des roches sont pour la plu- part empiriques, et suggkrent des lois lineaires ou de puissance. On peut supposer des lois similaires pour le frottement. Alors que la loi de Coulomb est generalement correcte en ce qui concerne les sols, il semble que le frottement des roches serait plus correctement diifini par une loi non lineaire, et si l’on applique une loi de Coulomb les facteurs de securite sont affect&s par la valeur adopt&e pour la cohesion. Les differentes methodes de mesure du frottement sont decrites, et leur champ d’application est etudie. On tient compte Cgalement du pro- cessus d’usure et de la superficie de contact entre les surfaces en glissement. 11 semble que dans certains cas on obtienne des valeurs r6siduelles de frottement apres des glissements de faible importance. Des cavites se produisent pendant le glissement et leur comportement semble dependre d’un facteur temps. Actuellement, les valeurs numeriques de glissement et les autres parambtres des systemes joints ne sont pas certains, et une formule aussi simple rend encore de grands services. On donne un certain nombre de formules pour les facteurs de securite du glissement pour une ou deux surfaces planes. Pour les roches avec des joints rapproches on peut utiliser la theorie de mecanique des sols pour les surfaces de glissement circulaires et autres. Dans ce cas, les valeurs de frottement obtenues pour les surfaces a joint unique ou les roches CcrasCes sont plutot en de& de la &rite puisque l’enchevetrement des roches peut resulter en unimportant accroissement de la solidite.

INTRODUCTION

Most of the previous Rankine Lectures have dealt with topics in Soil Mechanics but Rankine by no means confined his attention to soils. In fact, much of his work is devoted to that historical branch of rock mechanics: the study of masonry structures. The past two decades have seen a stabilizing of the subject of soil mechanics and a rapid development of rock mechanics so that the two are now sometimes, as in Australia, joined into one geomechanics society.

It is therefore desirable to examine the similarities and differences between the two subjects and this is done with particular reference to a topic of great importance to both: the stability of slopes.

* Department of Geophysics and Geochemistry, Australian National University, Canberra. 1 97


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98 J.C.JAEGER

Soil mechanics is essentially a form of continuum mechanics in which the continuum is composed of small grains with the region between them filled with pore fluid. The grains may usually be taken to be rigid and the constitutive equations for the material involve their relative movement and the variation of pore pressure.

Rock mechanics is derived originally from another continuum: the elastic material. Much of the early work in it consisted of elastic analysis of stresses around openings and tunnels, and this is still needed and used extensively. However, the distinctive feature of rock mechanics, as emphasized in the classical works of Talobre (1957) and Mtiller (1963), is that a rock mass is not a mathematically uniform body but is broken up by a network of joints and faults into blocks on a scale of from inches to feet. To emphasize the fact that this is no longer a continuum, Trollope (1968) describes it as a discontinuum. Essentially, the position as set out by John (1969) is that the whole body of rock, described as a rock system, is broken up by joints into rock elements which are of such a size that they contain no joints or planes of weakness and may be regarded as elastic bodies subject to brittle failure under appropriate conditions. The term joint is used to cover all discontinuities; these might technically be joints, faults, bedding planes or other surfaces of weakness. The properties of the rock system are determined by the properties of the rock elements and of the joints as well as by the geometry of the system. The properties of the rock elements may be studied by conventional laboratory testing although it is not altogether clear how relevant these methods are. Under conditions in which all stresses are compressive and normal stresses on all joints are high it is realistic to treat the rock system as approximated to by an elastic continuum with the properties of the elements. It is only when larger displacements, either by sliding on joint surfaces or by passing over the top of the stress-strain curve (Fig. 5), are in question that a new formulation needs to be introduced. Several attempts have been made to measure the properties of joints in the laboratory or field, the ultimate aim of which is to provide a quantitative description of the mechanical properties of the joints. If the mechanical properties of the elements and the joints are known as well as the geometry of the system, the mechanical properties of the system could, in principle, be determined by putting the problem on a computer. Beginnings have been made in this direction, e.g. by Goodman et al. (1968), Zien- kiewicz et al. (1970). However, it seems that in most cases neither the mechanical laws nor the numerical values entering into them are sufficiently well known at present, so that there is still scope for the use of simple models and mathematical formulae, particularly in feasibility studies.

A further complication arises from the fact that any displacements in the rock system may change the relative position of rock elements and result in high localized stresses on them, which may cause individual failures. These failures may be of indirect tensile type or involve crushing. The stresses in the rock system may redistribute themselves in characteristic fashion after localized failures.

A large range of stresses is involved in the various branches of rock mechanics. In deep mining the stresses may be very high while in rock slopes they are relatively small. In the abutments of arch dams they may be of the order of 50 bars (the bar and kbar are units frequently used in geophysical literature; 1 bar = 14.5 lb/sq. in. = 1.02 kg/cm2).

The obvious similarity between soil and rock mechanics comes from the fact that both deal with discontinua and that the primary mechanism of distortion is by sliding between the elements. Among the differences may be listed: the jointing in rock systems may be compara- tively regular, the ratio of the size of elements to the size of the system is much smaller for soils, failure of individual elements of soils is less important, the elements of a rock system are frequently interlocked, the influence of the water pressure is more important and more complicated in soils, and large stress gradients may occur in rock elements.

Compression is reckoned positive throughout this Paper.


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FRICTION OF ROCKS AND STABILITY OF ROCK SLOPES 99

MECHANICAL PROPERTIES OF ROCK ELEMENTS

The mechanical testing of soils is in a highly developed state. The principal tests, triaxial and direct shear, are conducted under conditions comparable to those which occur in practice and more exotic tests to study failure criteria are easily carried out.

By contrast, the position in rock mechanics is chaotic. The measurements which have been made have to a large extent been determined by historical accident and the requirements of different disciplines. Of the various groups concerned, the geophysicists have been concerned with behaviour at relatively high confining pressures, usually in the range of l-20 kbars; the petrologists are interested in the deformation of minerals in this pressure range; workers in fracture mechanics are interested in precise measurements under well defined conditions, usually in a lower pressure range; engineering observations have to be made on specified rocks and are frequently confined to a determination of uniaxial compressive strength and modulus. The situation is further complicated by the fact that the effects of constraints and of the stiffness of the testing machine are of great importance in engineering rock mechanics. Also, although many different systems of stressing have been used to study the failure of ideal rock specimens under polyaxial conditions of stress, it is not clear how relevant these are to the failure of rock elements in practice.

Uniaxial compression has always been, and remains, the most common test; a review paper by Hawkes and Mellor (1970) contains over a hundred pages. Nevertheless, the effects of machine stiffness and constraint at the platens, clearly appreciated by Tredgold (l&42), are not fully understood and the general process of failure appears to be one of great complexity (Wawersik, 1968; Wawersik and Fairhurst, 1970).

Triaxial compression and extension have been increasingly used by geophysicists and engineers, following the pioneer work of von K&-man (1911) and the U.S. Bureau of Reclama- tion (1953), to specify the behaviour of rock under combined stresses and form the basic information for the study of criteria of failure. However, it is not clear how far this system conforms to that applied to rock elements since unrestricted lateral dilation of the specimen is allowed.

Direct tension has always been used, but sparingly because of the difficulty of avoiding bending stresses. For this reason the indirect tensile (Brazilian) test has been increasingly used. In the simplest form of this (Fig. l(a)) a cylinder of radius R and length L has line loads W applied to it at opposite ends of a diameter. On the classical theory of elasticity there is a uniform tensile stress WlrrRL across this diameter so that if W is the load at failure, the tensile strength T, is given by

T,, = W/?IRL . . . . . . . . (1)

Values of T, determined in this way are in reasonable agreement with those obtained in direct tension. Further, if the load is distributed over a small region, the size of this region does not

0 (b) (Cl (d) 0

Fig. 1. Indirect tensile failures; cylinder: T,, = W/xRL; sphere : 2’0 = kW/dP, k =OG!SO*~


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100 J. C. JAEGER

have much influence on T, and nor does the shape of the specimen (Fig. 1 (b)). The same remarks apply to spheres and cubes loaded along diameters. If the test is carefully conducted in a stiff testing machine, the fracture is accurately plane (cf. Fig. 2 for a sphere), the shattering frequently found with soft machines being a subsidiary effect. If a cylinder is loaded by several line loads (Figs l(c) and 1 (d)) failure takes place approximately along the dotted curves of maximum tensile stress or strain (Jaeger, 1967). Measurements of this type are probably much more relevant to rock mechanics than those of direct tension since slight move- ments of a rock mass will cause localized stresses to be set up on the individual rock elements which may well then fail in indirect tension (cf. Fig. 25 and the section of the effect of constraints). The effect is related to the axial cleavage fracture of Gramberg (1962).

Direct shear tests have been used relatively little for solid isotropic rocks but are of great importance for the study of anisotropic rocks, joints and friction. Essentially, in all cases the specimen S is held in two boxes, B and B’ (Fig. 3), of which one, B, is made to slide over the other by force T while it is pressed down by force N. Shear failure occurs on the plane PP’. The constraints applied to the block, and therefore the detailed stress distribution in it, vary greatly with individual arrangements. For example, N may be applied through a ball- jointed link or through rollers as shown in Fig. 3(a). Results obtained vary greatly with the stiffness and the constraints. In principle, the system has five degrees of freedom : horizontal and vertical (dilation) translation and rotation about a vertical axis and two horizontal axes. Equipment in use varies from large, specially built machines (Krsmanovic, 1967; Hoek and Pentz, 1969) through fittings built for testing machines, to commercial shear boxes built for use in soil mechanics.

Ideally the stress system is that of Fig. 3(b) in which the principal stresses are

01 = $Jd+[ 2+&J;]l’2 7d

>

. . . . . . (2) (T2 = i&d - [Ti + *cry

and the principal axes are inclined to PP’ at an angle 0 given by

tan 28 = -27,/a, . . . . . . . * (3)

where rd and Us are the shear and normal stresses over the area. This system occurs in torsion combined with axial compression which has been used in careful laboratory experiments (Baker, 1915; Handin et al., 1960, 1967).

This stress system is only a crude approximation to conditions in the shear box of Fig. 3(a) ; it should not lead to a shear fracture in the plane PP’ on any of the conventional theories of failure. These lead to a possibility of extension fracturing in a plane perpendicular to CS~ or to shear failure; they appear in torsion (Handin et al., 1960), and in secondary failures on sliding surfaces (Fig. 4). They also occur in discussion of secondary faulting (McKinstry, 1953; Lajtai, 1969). It appears subsequently that, for sliding surfaces, microfractures in these directions can be seen (cf. Fig. 27) so that it is possible that initial cracking occurs in these directions which combine to give a final major fracture in the direction of 7d.

These simple tests are those most directly relevant to the behaviour of rock elements. There are, of course, many others which have been devised to study the behaviour of rock under various systems of combined stresses.

The most important development of the past decade has been the full realization that any test involves a composite system consisting of the specimen and the testing machine, so that energy is stored in both and may be released suddenly in the specimen at failure. This has three repercussions: the machine may not be able to follow the stress-strain curve, the energy released may cause subsidiary fractures and it is not possible to stop the test during failure so that its microscopic behaviour may be observed. These considerations have led to the development of first stiff, and subsequently servo-controlled, testing machines which are beginning a


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new era in rock mechanics. The development of these was due in the first instance to workers in concrete (Turner and Barnard, 1962).

The significance of machine stiffness may be seen from Fig. 5(a) which represents an idealized load-displacement curve for uniaxial or triaxial compression. Frequently such a curve contains a region OA which is concave upwards, corresponding to closing of internal cracks; a moderately straight region AB; a region BC which is concave downwards with a maximum at C, in which internal cracks grow stably and the specimen is dilating; and a falling region CD in which the specimen, while technically failed, can still support a considerable load. Suppose that at a point P in CD a small displacement PQ=dx takes place, then the load which the specimen can support will fall by QR= (dF/&) Ax. If the stiffness of the testing machine is A, then the load applied by the machine will fall by /\Ax. Thus, if

h > IdF/dxj . . . . . . . . . (4)

the specimen can still support the load and the machine will follow the stress-strain curve. On the other hand, if h < JdF/dzJ the specimen will be overloaded and will fail and the machine will register a line PS whose slope is h. If h is small (a soft machine) failure will take place at C. Alternatively, a discussion may be given based on the energy stored in the machine and the specimen. Wawersik (1968) and Wawersik and Fairhurst (1970) have shown that for some rocks the curve CD (Fig. 5(a)) has a vertical tangent so that the curve cannot be followed even by a very stiff machine and so it is necessary to extract strain energy from the specimen to prevent violent failure. This effect has led to the development of servo-controlled machines.

Although this argument has been stated for compression, it applies in all cases. Fig. 5(b) shows the effect in a shear box experiment in which the machine is not able to follow the load- displacement curve in the dotted regions.

Apart from the value of knowing the complete stress-strain curve OABCD it is of great value to be able to stop the experiment in the region of CD to study changes in the micro- structure of the specimen. In fact, with the development of the servo-controlled machine and the scanning electron microscope, an era is beginning in which the process of failure can be observed instead of speculated about.

These concepts of stiff and soft loading systems carry over from the laboratory to the practical scale. Thus in rock slopes the loading system (gravity) is soft, while in much underground work it is stiff and the descending portion CD of the stress-strain curve is achieved.

FRACTURE CRITERIA

Frictional processes involve brittle fracture and in turn criteria for brittle fracture of rocks have been influenced by the concept of friction. The same empirical laws are frequently used in both cases. The various criteria for brittle fracture of rocks are discussed from this point of view for two-dimensional cases only. Essentially, a criterion for failure consists of a relation between principal stresses (TV and ug at failure. Such criteria may be derived in various ways : phenomenologically, empirically, mathematically and mechanistically.

The simplest and most important of the phenomenological criteria is that of Coulomb (1773), who suggested as a direct generalization of the laws of friction that shear failure takes place in that plane for which 171 -pun first reaches a constant S,, which may be described as the intrinsic shear strength (adherence) of the material. Here un and T are the normal and shear stresses across the plane and CL= tan 4 is a constant called the coefficient of internal friction. In terms of principal stresses this leads to a linear relation between u1 and u3 at failure, namely

u1= C,+qu, . . . . . . . . (5)


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102 J. C. JAEGER

i

(b) (4 (4

Fig. 6. Simple criteria for failure: (a) Coulomb, (b) Poncelet, (c) GrifBth, (d) power law

where C, is the uniaxial compressive strength and

C, = 2S0 tan C( 1

q=tan2G: , -*.-... (6)

7-r 4 u=-j+g

I

. . . . . (7) tancc = p+d(p2+l) = set++ tan+

In the discussion it is assumed that (T, > 0 and this may be shown to imply that equation (5) holds only for cI < C,/2. The criterion is shown in Fig. 6(a). The Mohr envelope corresponding to equation (5) is

7 =&+pLa, . . . . . . . (8)

The plane of failure is inclined at an angle

77 v 4 2 --“=;r-2 . . . . . . . .

to the crl direction. Equations (4)-(g) also apply for the general case (To > g2 > cr3 in which case the plane of failure passes through the o2 direction.

A second phenomenological law which goes back at least to Poncelet is that of maximum tensile strain. Suppose that failure occurs when Em = -A, where A is a constant. Then on elastic theory the uniaxial tensile and compressive strengths, T, and C,, are

-AE = -To = -VC, . . . . . . (10)

For triaxial conditions

as-v(u,+u,) = -AE

or using equation (10) l..rl = C,+[(1-v)/v]a, . . . . . . . (11)

This is a linear relation of the type of equation (5) (Fig. 6(b)) with

tan + = +(l-2~)/2/[~(1 -v)]

so = &C,z/[V/(l -V)] > * * * . . .


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Clearly v need not be the elastic Poisson’s ratio but simply a constant relating C,, and T,. This theory has been revived from another point of view by Trollope (1963). It has the advantage of explaining simply the longitudinal cracking observed in uniaxial compression. The history of strain theories of failure is discussed briefly by Timoshenko (1953). It is interesting that Todhunter and Pearson (1886) regard maximum stretch as the correct theory of rupture and remark that ‘Lame, Clebsch and innumerable English writers have fallen into the error of taking the maximum stress’.

Of the theories based on simple mechanical models, that of Griffith (1924) has been by far the most exploited. This is based on the assumption that tensile fracture is initiated at stress concentrations at the end of hypothetical flat elliptic Griffith cracks in the material. It leads to the result

(?--Q = C&1+%) ifa,+3u3 >O . . . * (13)

u’3 = -tc, ifa1+3u3 < 0 . . . . (14)

and to the parabolic Mohr envelope

72 = +&(a+&) . . . . . . . . (15)

The curves of equations (13) and (14) are shown in Fig. 6(c). The criterion is valuable as covering the transition from negative to positive values of us. The parabolic form of equation (13) does not fit the experimental results, but it was pointed out by McClintock and Walsh (1962) that it remains valid only so long as cracks remain open. For conditions in which compressive stresses are so high that cracks are closed, they assume that sliding can take place on their surfaces with coefficient of friction p and give an approximate theory which leads to the CouIomb relation, equation (5). Their modified Griffith theory consists of equations (13) and (14) for low values of u1 and a transition region tending asymptotically to the linear form of equation (5) shown dotted in Fig. 6(c). Griffith theory has proved extraordinarily useful as a mathematical model for studying the effect of cracks on rock, for example, on its elastic properties (Walsh, 1965) but it is essentially only a mathematical model; on the microscopic scale rocks consist of an aggregate of anisotropic crystals of different mechanical properties and it is these and their grain boundaries which determine the microscopic behaviour.

When experimental results are plotted as in Fig. 6 it is found that, while for some rocks they are approximately linear, for most rocks the curves are slightly concave downwards. This has led to a number of empirical attempts to represent them by a power law. Hobbs (1966) suggested that

u1-u3 = C,+FuS . . . . . . . . (16)

but states (Hobbs, 1970) that

u1 = C,+Fu$ . . . . . . . . (17)

fits his later observations better. Hoek (1968) proposed the law

(T,-T,,&/& = F(u&)f . . . . . . . (18)

where 7, = +(ul - uJ and urn = $(ul + uJ. In equations (16)-(18), F and f are constants and f<l.

Alternatively, laws of the form

T=T,,+FuJ~ . . . . . . . . * (19)

where 7 and u,, are the calculated shear and normal stresses across a plane of failure or are derived from the Mohr envelope have been proposed (Murrell, 1965).


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104 J. C. JAEGER

Although equations (16)-(19) give a reasonable representation of the results, they have not been adequately studied for low values of c3 and IS,, and it is in this region that they are least reliable since the curves have a vertical tangent. They are also not mathematically related; for example, the Mohr envelopes of equations (16)-(18) do not take the form of equation (19).

The case of planar anisotropy is of great importance in practical rock mechanics. Such rocks are usually tested either in direct shear parallel to the plane of weakness or triaxially with the plane of weakness orientated so that failure takes place in it. It is assumed that such measurements give S, and ,LL corresponding to the plane of weakness and that failure under any system of stresses is given by the theory discussed for failure of jointed rock. However, anisotropic rocks probably exhibit a continuous variation of both S, and p with direction. This problem was studied graphically by Casagrande and Carrillo (1953) for soils and Jaeger (1960) showed that for the case of constant p the simple theory of Coulomb could be extended to the case of rock whose intrinsic shear strength in a plane inclined at /3 to u1 is given by

s = S, -s, cos Z(w -/3) . . . . . . . (20)

so that S varies from a minimum of S, -.S, in the direction p= w to a maximum of S, +S, in the perpendicular direction. In this case the relation between (Jo and u3 is

(a,+S1 cot +)z-(7m+b)2 case? 4 = .Sz cot2 + (30s~ (2w+$) . . (21)

where o,,, = :p++; 7, = CT1 u’3

b = S, sin (2~ + #I) cos 4

This may be used in various ways, one simple one being that uniaxial compression normal to the plane of weakness gives a relation between S, and Sz so that, if S, -S, and $ are determined by direct shear, all the constants can be found.

SLIDING OF ROCK SURFACES

Many different types of measurement on the sliding of rock surfaces have been made under different conditions and for different purposes. There is a lack of any systematic study and so these tend to be lumped together and figures quoted without real appreciation of the complexity of the situation. From the point of view of practical rock mechanics it is desirable to have numbers describing the behaviour of a joint and these numbers may be found either by in situ measurements, which are slow and costly, or by laboratory measurements on much smaller samples which may not be representative.

The methods which have been used to study the sliding of rocks may be classified as follows.

Use of small sliders on larger surfaces

This is the classical method of studying friction (Fig. 7(a)) and a few observations on minerals are reported by Bowden and Tabor (1950). Usually the normal loads N are low and the system, both in N and T, is soft. This method has been used by Horn and Deere (1962) and Byerlee (1967a). In a modified form it has been used at higher normal loads by Jaeger and Cook (1970). A unique characteristic of the method is that although the same surface of the slider is always engaged it is continually moving on to new material and thus the method is valuable for studying the mechanism of wear.

Use of triaxial a@aratus

Triaxial apparatus has been extensively used and is probably the simplest method of studying drill core. A cylindrical sample with a joint or saw cut inclined at a suitable angle


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FRICTION OF ROCKS AXD STABILITY OF ROCK SLOPES 105

0 (b)

2T

J

Cd) (e)

Fig. 7. Systems for measuring friction

01 to the cylinder is jacketed and tested in an ordinary triaxial pot (Fig. 7(b)). If the axial and confining stresses are err and a3 the normal and tangential stresses across the plane are

un = a3+(al-a,) sin2cc . . . . . . * (22)

T= (a,-ug)sinacosa . . . . . . . (23)

The method seems to have been used first for testing the bond between mortar and aggregate by the U.S. Bureau of Reclamation (1954). A pioneer series of experiments using a variety of surfaces and angles was made by Jaeger (1959). The method is particularly useful for studying the residual sliding on surfaces of shear fracture which have been produced in intact rock in the apparatus, the angle a being measured subsequently and the surface being verified as suitable, and it has been used in this way by Lane and Heck (1964), Murrell(1965) and Hobbs (1970).

In practice, it follows from equation (22) that the normal a, is at least of the order of two or three times c3 so that the method is not particularly suitable for low normal stresses, say a,, < 10 bars. On the other hand, it is suitable for high normal stresses and has been used under these conditions by Byerlee (1967b) and Wiebols et al. (1968). Specimens with diameters as large as 6 in. may be used without difficulty (Jaeger, 1970).

The main objection to the method arises from its geometry. If one spherical seat is used, as is often done, this will rotate when sliding begins and conditions across the sliding surface become uncertain (Jaeger, 1959; Hobbs, 1970). Some workers, e.g. Byerlee (1967b) and Wie- bols et al. (1968), use no spherical seat, so that additional normal stresses determined by the transverse rigidity of the system are introduced. Rosengren (1968) made a careful study of the possibility of improving the system by using either two spherical seats or lubricated pads (Fig. 7(c)) to permit easy sliding at the ends of the specimen. This latter method proved successful and a correction can readily be applied for friction at the pads. With this system there is no constraint on lateral motion of the specimens.

One difficulty with the triaxial system, even with proper end conditions, is that only limited amounts of displacement are possible. A second difficulty is that dilation is not easily measured.

The effect of water is easily studied by using a pair of fine tubes leading from the platens along saw cuts in the sides of the specimen to the ends of the joint.

Conventiolzal shear box

Shear boxes (Fig. 3(a)) either built for use in soil mechanics or specially constructed (KrsmanoviC, 1967; Hoek and Pentz, 1969; Edvokimov and Sapegin, 1967; Mathews, 1970)


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106 J. C. JAEGER

have been extensively used for testing joints. The usual practice is to drill along the joint, setting the core in the shear boxes with the joint plane between its two halves. Dilation is easily measured by this system but the constraints on the system are uncertain. Some workers (e.g. Lecher, 1968) maintain the normal displacement constant.

Double shear

Hoskins et al. (1968), Jaeger and Rosengren (1969) and Dieterich (1971) used a simple double shear apparatus on accurately prepared specimens (Fig. 7(d)). The tangential load T was applied by a testing machine, a jack or a proving ring and the normal load N by flat or ordinary jacks or a proving ring. Specimens were usually 3 in. or 12 in. square. Rosengren (1968) made a careful comparison of results obtained by this method and those obtained by triaxial testing. Even with this relatively rigid system the constraint is not perfect and there may be a tendency for the side blocks to rotate slightly about a horizontal axis.

Rotation

In this system hollow cylinders are pressed together with axial load N and torque M is applied to them (Fig. 7(e)). An apparatus of this type has been built by the Mining Research Laboratory, Johannesburg. This system has great advantages, notably that indefinitely large amounts of sliding may be obtained without disturbing the surfaces. Also, it is easy to study the effect of water by introducing it inside the hollow cylinder.

Large scale in situ tests

Large scale in situ tests are being used increasingly for engineering testing (Rocha, 1964; Wallace et al., 1969; Serafim and Guerreiro, 1968). Essentially a block containing a joint of interest is exposed by quarrying, and normal and transverse loads N and T are applied to it in some manner by appropriate jacks (Fig. 7(f)). 0 ccasionally only one inclined jack is used. Sizes of up to 6 ft square or more have been used. The method has the advantage that it operates on a joint of interest and over a reasonably large area. One disadvantage is that the stresses may vary considerably over the area as Ruiz et al. (1968) show from a finite element analysis. Obvious disadvantages are the time and cost involved and the fact that relatively few can be made; this emphasizes the importance of obtaining comparisons on the same material of in situ and laboratory tests as has been done by the ComitC National Francais (1967) and Bernaix (1969).

Many different types of surface have been tested by various of these methods.

Cut and ground or polished surfaces. These have two properties : flatness and roughness. The roughness is usually measured by one of the commercial instruments and quoted in microinches or microns. Unless special precautions are taken, the departure from flatness will be much greater than this. As a result, relatively small surfaces of about 0.1 in. square are in approximate contact over the whole of their area and larger surfaces of about the order of 1 in. square may be in approximate contact only at their higher points. The extent of the initial contact when rock surfaces are pressed together may be judged from the sliding of rock on metal. Fig. 8 shows the early stages of sliding of rock with a surface roughness of 100 pin. on mild steel. The distance of sliding was 0.04 in., which sets a scale for the number of initial contacts.

Laboratory extension fractures. A number of workers (e.g. Byerlee, 1967b) have used extension fractures produced in the laboratory and subsequently replaced together. Such surfaces are highly interlocked.

Laboratory shear fractures. In many cases, after solid specimens have failed by shear fracture in triaxial or direct shear apparatus, testing is continued to study sliding on the fractured sur-


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Fig. 10. Load-displacement curves 2 for ground surfaces ; area : 9 sq. in. ; normal load: 1325 lb MO. I Sandstone

]1 Bowal trachyte

m Carrara marble

I

0 0.15 05

Displacement: in.

075

face (Jaeger, 1959; Maurer, 1965; Hobbs, 1970). The surfaces are frequently very irregular (Fig. 9) and contain a considerable amount of detrital material from the original failure. The amount varies greatly with the conditions under which the original failure took place.

Natural o$en joints. These may be obtained in large numbers from drill core. They can usually be mated together before testing so that there is macroscopic contact over most of their area.

Natural filled joints. Frequently open joints have been filled with materials such as calcite and testing of these gives a peak value corresponding to failure of the filling material followed by sliding on the failure surface. Natural shear faults are usually wider and clay filled. Their properties may be determined by testing the clay but with large scale tests on such joints great increases of strength can occur caused by interlock of rock material on opposite sides of the fault (Romero, 1968).

Artificial &?led joints. Cut surfaces with coatings of plaster have occasionally been used to simulate filled joints (Jaeger, 1959).

ArtiJicially constructed surfaces. Several workers (Patton, 1966; Lajtai, 1969; Barton, 1971) have used shaped surfaces of plaster or similar materials to demonstrate the effects of surface roughness. Although such experiments are instructive, the frictional behaviour of different rock types is very different (cf. Fig. lo), and so the behaviour of artificial materials may be different again.

The range in normal stresses used by various workers has been wide. Those interested in geophysical applications (Byerlee, 1967b; Raleigh and Paterson, 1965) use stresses up to 15 kbars confining pressure. Most workers in rock mechanics have used stresses comparable with those in underground rock mechanics, say 500-10 000 Ib/sq. in. Lower values are needed in rock slope studies and for some foundation work.

Given a particular apparatus and rock type, various methods of testing are possible. The simplest and most usual is to set the normal load at a prescribed value and measure the variation of shear force and dilation with displacement.

Even with ground surfaces it is usually found that there is a considerable variation of shear force with displacement during sliding. Fig. 10 shows curves of tangential force against displacement for three rock types in the apparatus of Fig. 7(d), the surfaces being approximately flat and of roughness 100 p in. In all cases, a constant value of tangential force is approached fairly slowly. The reason for this behaviour is discussed later.


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J. C. JAEGER

S S

(a) (b)

Fig. 11. (a) Typical curve of shear stress against displacement for (b) idealized curve

a natural joint,

For natural joints behaviour is much less regular (Fig. II), the tangential force usually rising to a peak and falling sharply. The peak is usually followed by considerable fluctuations (cf. Krsmanovic, 1967) but the tangential force again tends to a final residual value. For tightly interlocked joints, the fall in tangential force after the peak may be so rapid that the machine is unable to follow it (cf. Fig. 5(b)).

In describing results such as those of Figs 13 and 14 it is sometimes convenient to refer to the tangential and normal forces on the specimen. Alternatively, these may be divided by the area of the specimen to give the shear and normal stresses G- and u,,. These are macroscopic stresses and not those over the areas of real contact.

In the method of testing described, individual samples are tested at a fixed value of Q’, giving a curve of 7 against displacement S shown in Fig. 11 (b). From this the residual value T,,,, the peak value T,,,~~ and the displacement AS at which it occurs can be read off as well as the shear stress TV at any specified displacement S (KrsmanoviC and Langof, 1964; KrsmanoviC, 1967). From a series of tests of this type on different samples at different values of a,, any of the quantities of interest, -r,,*, rres, r,,,/AS, may be plotted against CT,. The disadvantage of this method is that specimens from the same joint may vary considerably so that a wide scatter in results may be found.

An alternative method, sometimes called the incremental method, uses only one specimen. In this, after the peak has been passed and conditions are stabilized to some extent, the normal stress is changed by specified amounts and the corresponding shear stresses are measured. Examples of measurements of this type in a triaxial apparatus on two graphite coated joints of different roughness are shown in Fig. 12. This method allows a plot of 7 against a,, to be made for sliding of one pair of surfaces at some stage of its development, usually chosen as near as possible to residual conditions. Some workers (e.g. Jaeger, 1959) raise the normal stress; others (e.g. Ruiz et al., 1968) lower it. These procedures are not the same since the nature of the sliding surface at any time depends on the normal stresses to which it has been subjected and some rocks show considerable hysteresis (cf. Fig. 13).

As a further variant of this method, CT,, may be varied continuously and 7 plotted against c,, on an X-Y recorder. Examples of this using the apparatus of Fig. 7(d) on ground surfaces are shown in Fig. 13 and in the rotating apparatus of Fig. 7(e) in Fig. 14. These give a direct indication of the amount of hysteresis involved.

Plots showing the values of T at fixed values of u,, are shown in Figs 15 and 16. Fig. 15 is for ground surfaces in the apparatus of Fig. 7(d) obtained by the incremental method. Fig. 16 shows the results of residual sliding on a number of different natural joints. These results are discussed later.


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2oooa

P

z

f loacoc .- s 5

0

Smooth joint

3om

2550

2ocQ

1500

IolM

0.3 0.6

Rough joint Fig. 12 (left). Load-displace-

ment curves for graphite 25w

r coated joints in six-inch core tested triaxiallg. Num- bers on curves are confIning pressures in lb/sq. in.

I Carran marble

1 Bownl trachyte

0.3

Displacement: in.

(a) (b)

0.6

0' 100 200

Q,,: Iblsq. in.

Fig. 13. Shear stress against normal stress for ground surfaces, showing hysteresis

Karoo shale. dry

400

0;: Ib/sq. in.

800

800

I Witwatersrand quartzlte, dry

4ou 8M)

9 : lb/rq. in.

Q

4000 -

Karoo shale, wet

2000

u” : Ib/sq. in.

(b)

Witwaterrrand quartzite. wet

c

0 2000 4000

en: Ib/rq. in.

Cd)

Fig. 14. Shear stress against normal stress in rotating friction apparatus


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110 J. C. JAEGER

Fig. 15 (left). Shear stress against normal stress for ground surfaces in the apparatus of Fig. 7(d)

400

CT, :Ib/rq. in.

600 800

u,, :Ib/sq. in.

0 (b)

Fig. 16 (below). Shear stress against normal stress: (a) residual sliding of natural joints in andesite under various conditions, (b) residual sliding of sheer fractures in mica schist

Figure 14 also includes some preliminary results on the effects of wetting from the Mining Research Laboratory of the Chamber of Mines of South Africa. Wetting has not been studied adequately but it is clear that surface wetness (apart from the effects of pore pressure) may have a considerable effect.

LAWS OF FRICTION

If -T is the shear stress corresponding to normal stress c,, under any defined set of conditions (e.g. residual) the variable coefficient of friction may be defined as

II* = T/U, . . . . . . . . . (24)

It may be obtained from results such as those of Figs 13-16. Amonton’s so-called second law states that CL* is a constant independent of u,. For metals, a simple explanation for this exists since, if A c is the actual area of contact per unit macroscopic area (cf. Fig. S), it may be expected that u,,= YA, and ~=SA,where Y and S are theyield strength and the shear strength respectively of the metal. A more refined discussion by Bowden and Tabor (1950) involving work-hardening changes this slightly. They introduce a ploughing term which might be specially relevant to sliding of rock on metal (Fig. 8) and leads to a law of form CL* = a + bdu,. Archard (1958) has discussed the case of many contacts, some of which may be elastic, and deduced that

r = Ku:: cL* = ,Q’-1 >

. . . . . . . . (25)

where 2/3<m<l.


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None of these discussions is applicable to brittle materials such as most rocks. Byerlee (lS67a) has developed a model based on tensile failures in asperities but at this stage it seems simpler to regard the process as one of shear failure under normal load so that criteria similar to those for the mechanical properties of rock elements would apply.

In the case of soil mechanics, the Coulomb law

7 = c+pan =c+Untan# . . . . . . - (26)

with the appropriate modifications for pore pressure, in general fits experimental observations well, c and $ being constants. The symbol TV is reserved for a constant called the coefficient of friction in the Coulomb relation (26). The variable coefficient of friction (equation (24)) in this case is

II* = /L+c/u, . . . . . . . . (27)

Jaeger (1959) suggested that the Coulomb law (equation (26)) also held for sliding of rocks with a cohesion c of about 50-100 lb/sq. in. This is suggested by the work of Hoskins et al. (1966) which led to results such as those of Fig. 15. Jaeger and Rosengren (1969) quote values of c for natural joints ranging from 0 to 200 Ib/sq. in. with a mean of 100 Ib/sq. in. Similar results by other workers are mentioned later. These relatively high values of c have a pro- found effect on slope stability calculations.

Results such as those of Fig. 16 suggest that the curve of T against u,, may be concave downwards and representable by the power of law of equation (25) or equation (19). For example, the points of Fig. 16(a) scatter around

7 = 1*2a;*9 . . . . . . . . . (28)

and those of Fig. 16(b) scatter around

7 = 5.2U9.7 n . . . . . . . . . (29)

units being lb/sq. in. The use of this law has been advocated in particular by Murrell (1965), Maurer (1965) and Hobbs (1970). Alternatively, a law of the form

7=79+Ku:: . . . . . . . . * (30)

may be assumed which is better written in dimensionless form

(T/C,) = (TO/C,)+K(Un/Co)m . . . . . . (31)

where Co is a suitable constant with the dimensions of stress, such as the uniaxial compressive strength of the material. If a law of the form of equation (25) or equation (30) holds and measurements are made in a range u1 < u, < ~9, the least squares linear fit to these will be of the Coulomb form of equation (26) with c increasing and p decreasing as ul increases. For example, in an extreme case Byerlee (1967b) found c =SOOO Ib/sq. in. for a ground granite surface with cr,, > 30 000 lb/sq. in.

A third proposal involves the effect of surface irregularities. Suppose that sliding is taking place up a slope inclined at 0 to the direction of 7 (Fig. 17). Then a simple calculation shows that in the absence of cohesion

7 = U,tan(ei-4) . . . . . . . * (32)

so that the effect is to increase the angle of friction by 19. If in any testy is the mean normal displacement corresponding to tangential displacement x

tan 0 = dy/& . . . . . . . . (33)

so that 0 can be determined and the effect corrected for.


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112 J. C. JAEGER

(a)

0;.

(b)

Fig. 17. Effects of surface inclination on friction

Bowden and Tabor (1950), and in the present context Patton (1966), suggested that at low normal stresses the effect might be one of sliding up asperities with zero cohesion, but that at higher normal stresses there might be shearing of asperities which would give an effect similar to cohesion. The simplest representation of this is the bi-linear law (Fig. 17)

on < c/b0 -PI) u?l > c/b0 ---I%) > * . . .

in which p. depends on the slope of the asperities, c depends on o, and /*.i is frequently regarded as a constant of the material.

None of these proposals has real theoretical or experimental status so that any attempt to fit experimental points to one of them may lead to misleading results. Equations (25) and (30) have the vertical tangent at the origin and in equation (26) the high value of c is frequently doubtful. Experiments at low values of (J,, seem to indicate a linear variation as in equation (34) and this is also suggested by curves such as those in Figs 13 and 14. However, the abrupt change in slope at c/(~~--J is unrealistic and some form of rounding off is needed. With the increasing use of computers some form of smoothing experimental data is needed and one which is probably adequate as corresponding with equation (34) but having continuous curvature is

7 = a{l-exp(-bo,)}+pa, . . . . . . . (35)

For example, the points of Fig. 16(a) may be represented reasonably well by

7 = 270{1-exp (-0~0015a,)}+0~41~, . . . . * (36)

On the basis of laboratory experiments Barton (1971) has proposed the law for peak strength

7 = us tan (20 log,, (Co/a,) +30)

which avoids the vertical tangent of equations (25) and (30) at the origin. At present there is no consistent practice in the reporting of results. Goldstein et al. (1966)

use equation (34). Murrell (1965), Hobbs (1970), M aurer (1965, 1966), KrsmanoviC and Popovic (1966) and Comes and Fournier (1968) use the power law. Edvokimov and Sapegin (1967), Serafim and Guerreiro (1963) and Rodrigues (1968) use the Coulomb form.

This discussion highlights the difficulty of providing accurate quantitative information for design purposes from a small number of laboratory measurements.

The phenomenon of stick-slip is an interaction between the testing system and the rock surface during sliding. It appears in most testing and it is not clear what is its significance or


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I Y

(b) (Cl

Fig. 18. Stick-slip: (a) and (b) simple mechanical model, (c) detail of regular stick- slip on smooth granite

what are the conditions for its occurrence. On a large scale some writers (Byerlee and Brace, 1968; Dieterich, 1971) regard it as providing a mechanism for earthquakes.

The term is used in two different contexts. First, it is used for occasional cases in which the machine cannot follow the load displacement curve (cf. Fig. 5(b)) ; this is more properly described as irregular stick-slip (Rabinowicz, 1959). Stick-slip may be described as a relaxa- tion oscillation whose period is determined by the frictional properties of the system which arises when the system is driven at a constant speed. An example of a load-displacement curve for ground granite surfaces of roughness 35 p in. is shown in Fig. 18(c). The simplest mechanical model which illustrates such behaviour is that in which the coefficient of dynamic friction CL’ is constant and less than that of static friction ,LL. Suppose that a mass M is pressed against such a surface by force W and driven through a spring of stiffness k whose other end is moved at constant speed V (Fig. 18(a)). If the mass is at rest, it will slip when the extension of the spring is x1 and the applied force is F, such that

F, = pW = kx, . . . . . . . . (37)

Suppose that slipping ceases when the extension of the spring is x2 so that the force is F,=p2. If V is very small compared to the velocity of slip, the work done against friction during sliding is p’W(x, -x2) and equating this to the loss of energy stored in the spring which is k(xf -x$/2 gives

It follows that

k(x,+x,) = 2$W . . . . . . . . (38)

F, = kx, = (Z/L’-~)W . . . . . . * (39)

F,-F, = k(x,-x,) = 2(p-p’)W . . . . . (40)

It appears that F oscillates between F1 and F, (Fig. 18(b)) and on this simple model p and p’ can be determined. From equation (40), measurement of x1 -x2 gives k.

Stick-up is a complex phenomenon. It is related to surface roughness, speed of sliding V or time between slips, wetness of surface and the amount of gouge present.


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J. C. JAEGER

Fig. 19 (left). Contour lines (units of O*OOl in.) for a graphite coated joint surface of area 5 in. by 6 in. (after Mathews, 1970)

0.5 I I.5 Displacement : in.

Fig. 20. Shear force and dilation for the joint of Fig. 19 in a shear box with normal load of 6500 lb

SLIDING AND WEAR

The frictional force during sliding varies continuously with displacement and this variation is associated with wear of the surface. It is important to understand this process, both macroscopically and microscopically.

On the scale of laboratory specimens, Fig. 19 shows surface contour lines for a graphite coated joint surface (Mathews, 1970) and Fig. 20 shows the variation of shear force and mean normal displacement y (dilation) with displacement x from an initial position in which the two surfaces of the joint are mated as closely as possible. Clearly, there is uphill sliding in the sense of Fig. 17. Knowing y, or in practice the normal displacements of the four corners of the shear box, the positions of the two surfaces after any amount of sliding can be calculated. Fig. 21 shows one cross-section after one inch of sliding; it is clear that for the most part contact has been lost except in a few regions where intense shearing and removal of material has taken place. Ultimately this sheared material will fill the hollows and the end result will be a surface of gouge material.

It is reasonable to assume that the behaviour of larger surfaces is similar to that of Fig. 21, scaled up by a factor of 10 or 100.

To indicate the way in which a residual surface of sliding is approached, Fig. 22 shows the behaviour of a pair of mated extension fractures in Bowral trachytel in the apparatus of Fig. 7(d). The fractures were induced in the material by transverse loading in Brazilian fashion. They were approximately parallel but each surface had an irregularity of about 0.1 in. and a grain size of about 0.03 in. The surfaces were mated together and tested at a normal load of 1350 lb, their area being 5.2 sq. in. The first load-displacement curve is marked 1 in Fig. 22 and shows a marked peak. After about 0.4 in. displacement the surfaces are remated (detrital material being brushed off) and the procedure is repeated. This gives curve 2 in which the initial peak has diminished markedly in size. The seventh run is shown in curve 7 on which incremental loading and unloading at loads of 2050,271O and 3460 lb have been superimposed. In the fifteenth run, curve 15, all trace of the initial peak has disappeared. At this stage, the surface has been ground down to a number of moderately rounded lands, occupying about 80% of its area, with hollows of original surface between them (Fig. 23). Sliding will be on only a small fraction of the area of lands. Thus, although there has been an

1 Bowral trachyte is a commercial building stone properly described as a microsyenite and consisting of interlocking tabular crystals of orthoclase of grain size 0.5-l mm which constitute 80% of its volume. The other major mineral is aegerine-au&e which has been extensively altered to siderite and calcite.


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FRICTION OF ROCKS ASD STABILITY OF ROCK SLOPES 115

enormous change in the nature of the surface, a residual value seems to be approached after only one inch of sliding. Values of I” from all runs agree around the rather high value of 0.9.

As an example of a slightly different type, Fig. 24 shows typical load-displacement curves for a block of Bowral trachyte with ground plane-parallel surfaces forced between a pair of rough surfaces formed by extension fractures. The normal load is again 1350 lb and the area 6.3 sq. in. Curves 1,6, 18 represent the first, sixth and eighteenth runs, each of about 0.5 in., the surfaces being replaced in the same relative position after each run. The behaviour of the surfaces is now completely different: the highest spots of the tension-cracked surfaces are worn down and score the originally flat surface. The same regions of the tension-cracked surfaces are always in contact with the opposite surface and the area of these contacts grows steadily. After 15 runs it is of the order of 1 sq. in. so that surface wear is far from complete. The load-displacement curves tend to rise and show a great deal of short period fluctuation. Similar results have been obtained by Jaeger and Cook (1970) for sliding of gravel on plane surfaces.

Another case arises in the sliding of approximately plane surfaces on one another. The protuberances in contact may be expected to be of large radius of curvature with small surface roughness superimposed on them. The gradual increase of frictional force with displacement often observed (Fig. 10, curve II) is probably attributable to growth of the gross areas of contact. This effect is particularly marked in marble (Fig. 10, curve III). In this case, there may be plastic deformation below the contacts; also, gouge material is produced profusely and may dilate so much so that rubbing blocks may fail by indirect tension caused by local stress at the contacts (Fig. 25).

It has been suggested that the frictional force may arise from shearing of rock under high normal stresses at localized zones of contact (cf. Fig. 21). By analogy with the linear laws for fracture criteria the shear strength S at actual normal load u,, might be expected to be given by

s = &+&a,, . . . . . . . . (41)

where c, and pa are constants of the rock. There is experimental evidence for this (Jaeger, 1962). If the actual area of contact of two sliding blocks of rock is A,, the tangential and normal stresses on them will be

T = c,&+~naAa N = u,,A, >

. . , . . . .

and if the total area of the blocks is A so that the conventional stresses are T=T/A and on = a,,,/A , equations (42) become

T = c,(A./A)+pau,, . . . . . . . . (43)

which is of the Coulomb form of equation (26) in which the cohesion c is proportional to A,/A. This is supported by results of Byerlee (1967b) who tested granite cylinders, 0.66 in. in diameter in triaxial apparatus at confining pressures in the range 15 000-225 000 lb/sq. in. He used two classes of surface-ground surface and mated extension fractures-and in both cases the surfaces were pressed together with hydrostatic pressure of 15 000 lb/sq. in. before tangential stress was applied. Both sets of surfaces obeyed the Coulomb law

7 = 9000+0~6u, . . . . . . . * (44)

in the range 30 000~ u, ~225 000 lb/sq. in. Under these conditions the surfaces of the extension fractures must have been perfectly interlocked and so the whole of the material in this plane would have been completely sheared. It is concluded that in both cases A, = A and the value of 9000 lb/sq. in. is that of c, for intact material. For an ordinary contact at low normal stresses A,/A might be of the order of 0.01 giving a cohesion of 90 lb/sq. in., which is of


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116 J. C. JAEGER

the order of those values already reported. Results of Jaeger and Cook (1970) on sliding of nearly spherical trachyte contacts on trachyte with varying areas of contact and high normal stresses may be interpreted in the same way with c~= 10 000 Ib/sq. in. and pe=0*32; at low normal loads the values are c = 10 and p = 0.48, suggesting that the area of contact is small.

If the alternative form, equation (25), is assumed for complete contact, the result for fractional area of contact A,/A is

7 = K(A,/A)l-“u~ . . . . . . . . (45)

so that the exponent remains the same but the constant changes. It seems possible that equation (43) may be applicable to the practical testing of rock

joints. Figs 21 and 23 and an elaborate series of laboratory measurements by Barton (1971) suggest that at any stage of sliding the fractional area contact A,/A is relatively small. If the rock is tested at normal stress crna=(A/A,)a,, values of c, and pa are determined and the behaviour under normal stress a,, follows from equation (43), i.e. using (A,/A)c, for the cohesion. If this reasoning is applied to rock slope studies, samples of joints taken from drill core represent portions of the surface which will eventually be taking stress and they should be tested at high normal stresses; e.g. if a,=50 lb/sq. in. and A/A,=100 they should be tested at u,, = 5000 lb/sq. in. and the cohesion so obtained divided by 100 for use in actual slope stability calculations. This reduces the anomalously high values of the factor of safety such as those obtained from equation (63) in Table 1. The difficulty is that A,/A is unknown but it is possible that it could be estimated by photogrammetric measurements or by comparison of values of the cohesion estimated from slides and measured on corresponding surfaces.

200

wn:Ib/sq. in.

-

Fig. 28. Sliding of a thick layer of finely powdered Bowral trachyte

Fig. 29 (right). Variation of coemcient of friction with time of stick (after Dieterich, 1971)

295 bars normal stress

0.82 1 I 100 IO’ IO2 10’ IO’ 10s 106

Duration of stick: s


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FRICTION OF ROCKS ATU’D STABILITY OF ROCK SLOPES 117

Fig. 30. Variation of T with q, for various materials; lengths of vertical bars indicate the effect of time of stick (after Dieterich, 1971)

The mechanism of wear is still obscure. Two different processes probably occur: the fracture of protuberances in the manner suggested by Byerlee (1967b), and fracture in a relatively plane surface caused by normal and tangential loads on it in the manner of Fig. 3(b). In the Author’s experience with small sliders on plane surfaces a network of small cracks appears in the surfaces around the contacts. These can be seen in Figs 25 and 26. Fig. 26 shows the sliding of an initially spherical surface of Bowral trachyte on Carrara marble; the ploughing of the harder material through the softer with the build-up of fragments round the contact is shown clearly, and the fact that some fragments are detached from the harder material.

In many cases the process of break-up can be studied either on individual surfaces or by impregnating the region between two sliding surfaces with resin without separating them (Gay, 1970). Fig. 27 shows one example in which fractures occur in directions suggested by Fig. 3(b). The amount of detrital material produced during sliding varies greatly with the nature of the materials and surfaces. It tends to aggregate, forming accretion steps in the lee of protuberances and developing slickensides (Gay, 1970; Norris and Barron, 1969). Such structures are of geological interest.

After considerable sliding has taken place, the surfaces of sliding may consist almost entirely of detrital material and then the behaviour of such material is important. Fig. 28 shows the variation of shear stress against normal stress for sliding in a thin uniform layer of finely powdered Bowral trachyte. The curve is approximately linear and ~=064 is of the same order as that for solid material. The presence of stick-up is noteworthy and also the fact that it is regular for decreasing normal loads and irregular for increasing loads. Dieterich (1971) associates stick-slip with the presence of gouge and has noted an important time dependence. Fig. 29 shows his measured variation of coefficient of friction with time of slip, and Fig. 30 is a summary of his results for various rocks. He attributes the increase of coefficient with time to compaction of gouge. These are probably the first indications of time dependence


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118 J. C. JAEGER

and creep in frictional phenomena and since many fault surfaces are covered with gouge they are of considerable importance in rock mechanics. They have to be taken in conjunction and the suggestion of Miiller and Malina (1968) that distortion in a sliding mass will cause a build- up of localized stresses.

SPECIFICATION OF PROPERTIES OF JOINTS OF ROCK SYSTEMS

Certain quantities have to be recorded by field measurements. These are directions of joint planes, surface regularity and roughness, continuity, spacing, aperture (width of open spaces) and filling. In addition to these, some estimate of the quality of the rock elements and water pressure in the joints is needed.

Field measurements may be made either on surface exposures, in underground workings or in drill holes or from drill core. These provide different types of sampling. The systematic logging of drill core orientated with some device such as that manufactured by Craelius is the simplest methodof obtainingdetailed statisticalinformation. Comparison of information from underground measurements and from suitably orientated drill holes shows good agreement.

The systematic plotting of the poles of joint planes on the stereographic or equal area projection (Mtiller, 1963; John, 1962; Panet, 1967; McMahon, 1968) is standard practice.

The regularity of the surface is the hardest quantity to estimate. Rengers (1970) described methods for detailed measurement of it. Jennings (1968) distinguished between roughness, which is local, and waviness on a scale of feet which he measured by the maximum departure from planarity over 24 in. In fact more detailed specification is needed. In dealing with ground surfaces and possibly with natural slickensides irregularities of the order of 10m4 in. are significant ; with interlocked crystals in extension fractures and joints the scale may be of the order of 10-l in. ; the grosser irregularities of the joint of Fig. 19 are on a scale of 1 in. and on a larger scale this might represent waviness. Like roughness the term asperity is used in many senses; in the past, Bowden and Tabor (1950) and Byerlee (1967b) used the term to imply a local, relatively sharp point, but Goodman et al. (1968) define it as an irregularity in a joint surface.

Continuity is perhaps the most difficult quantity to specify. It is usually estimated from surface observations, e.g. whether a joint can be observed on both walls of an underground excavation, but an alternative method consists of observing the number of joints which ter- minate in bore core. For a given joint surface the continuity u is defined so that a fraction a of the surface area is open or filled and a fraction 1 -a consists of solid rock. The cohesion c for the whole surface is then estimated to be

c = UC,+ (1 --a)~, . . . . . . , . (46)

where c, is the cohesion of the joint and c, that of the solid material. Since c,>>c, this leads to high values of c unless 1 --a! is small.

Spacing of joints is probably most simply studied from drill holes or drill core. It appears that the distribution of distances between joints is very skew and suggests a Poisson distribution (Snow, 1968, 1970). It follows that it can be misleading to speak of an average joint spacing.

Aperture and filling can readily be determined from drill core. Frictional properties for peak or residual sliding may be determined as already described.

They should be adequate for simple problems of sliding. For computer calculations involving joint elements, further information is needed (Zienkiewicz et al., 1970). When movement is occurring on a joint the relationship between normal and transverse displacements and the corresponding stress is given adequately by curves such as those of Figs 11,20 and 22. Before sliding occurs there will be some displacement, mainly elastic, across the joint given by the early part of these curves but they are not usually recorded with sufficient accuracy for this


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J. C. JAEGER

20

IO

1

I

0 a

(a)

Fig. 32. (a) The function cosec u set (a++) which controls sliding on a plane inclined at a to 0, : case for ri2 =30D. (gj Polar pldt of Fig. 36(a). (c) Stereographic projection of boundary between regions in which slip is possible and impossible; case for ~=0*67 and a,laI=O*l

(47) very large, failure will take place through the solid ignoring the joint and for a solid material satisfying the Coulomb criterion, equation (5), the condition for this would be

cr1-u3 = C0+(~-l)a3 . . . . . . . (48)

This graphical representation of equations (47) and (48) is convenient for laboratory observations (Jaeger, 1959; Donath, 1961). The simple result of equation (47) has been extended to the case of stresses in three dimensions by Jaeger and Rosengren (1969) who plotted the boundaries between regions in the stereographic projection in which slip is possible or impossible. An example is shown in Fig. 32(c). It shows that in some cases the results are sensitive to the value of vs.

For the representation of field data it is preferable to plot the results of equations (47) and (48) in polar co-ordinates (Fig. 32(b)) as is done by John (1962, 1969) and Mtiller and Pacher (1965) and, in a slightly different representation, by Bray (1967). The advantage of this system is that it makes the representation of the effects of several joint systems much easier.

All these results refer to the sliding of a joint or joints under a specified system of stresses. They can be profoundly modified by constraints and the stiffness of the system through which the stresses are applied. This question is discussed later.

ELEMENTARY TWO-DIMENSIONAL THEORY FOR STABILITY OF ROCK SLOPES

The two-dimensional theory of stability of earth slopes has been in a highly developed state for many years. Until recently it has been based almost entirely on the Coulomb criterion

7 = c+/-Lcr, = c+a,tan# . . . . . . - (49)

for drained slopes with appropriate modifications for the case of pore pressure. So firmly entrenched in the literature is this that most results have been expressed in terms of the dimensionless quantity

c/yH . . . . . . . . . * (50)

which is sometimes called the stability number, where y is the unit weight of the material and H is the vertical height of the slope.

The first approach dates back to Coulomb and considers plane sliding of an undistorted wedge of the material. Subsequently it became clear that most slides were on nearly circular


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arcs (Fellenius, 1927). In this case again the mass is undistorted and an exact solution of the statical problem was given by Taylor (1937), which has remained fundamental up to the present time. With the development of computer facilities the method of slices has been developed to take into account non-circular surfaces of sliding and other laws of friction than the Coulomb one (Bishop, 1955; Bishop and Morgenstern, 1960).

For rock slopes there are three simple situations which may be used to give an appraisal of a situation. These are as follows.

(a) A definite plane of sliding is suggested by a favourably orientated set of joints or bedding planes.

(b) In highly anisotropic rocks there is a definite plane of weakness but there may also be little change in shear strength of the rock for angles of up to 20” from this plane (cf. equation (20)).

(c) The rock may be so broken up by close irregular joints that it may be treated to a first approximation as a soil.

If, as in many excavated slopes, a detailed knowledge of the joint patterns is available, a more fundamental study involving cataclastic movement of the individual blocks can be attempted, but for feasibility studies with limited information this is not possible.

The fundamental difficulty for rock slopes is that the law of friction for low values of the normal stress is not clearly known and attempts to measure the parameters in it by laboratory experiments on joint surfaces are liable to considerable scatter. In particular, if the Coulomb law of equation (49) is used, there is great uncertainty about the value of c and this is exacer- bated if attempts to allow for joint continuity by formulae such as equation (46) are made.

Until the parameters can be better determined there seems little point in making elaborate computer calculations except from the point of view of computer simulation, and it seems preferable first to consider simple systems for which simple formulae are available. If the results of such calculations, based on laboratory measurements of rock properties, are in reasonable agreement with observations of rock slides it is possible to regard calculations with greater confidence. This is essentially the philosophy of Hoek (1970).

Only drained slopes are considered here to keep the discussion confined as far as possible to the simplest frictional problems.

The simplest problem is the sliding of a wedge of unit thickness down a plane at CL, the inclination of the slope OA being i and its vertical height H. The law of friction for the surface OB is written in the form of equation (24).

For a factor of safety F with respect to strength (Bishop, 1955) the total shear strength (T,P* available on the surface must be F times that necessary to maintain equilibrium, i.e.

F = (~,,/+L* . . . . . . . . (51)

From the geometry of Fig. 33, it follows that the weight W of the slice OAB is

W = &yH2(cot CL- cot ;) = &yH2 sin (;-a) cosec i cosec a . . (52)

Also the length OB = I = H cosec a, so that

!F B

a u, = (W/Z) cos cz = +yH sin (i -CX) cosec i cos C( . (53)

I 7 = (W/I)sincr . . . . . . . - (54)

The assumption which is made in this elementary treatment is that the block OAB behaves perfectly

W rigidly, the stress being uniformly distributed over the Fig. 33. Simple plane sliding surface OB.


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122 J. C. JAEGER

The values of u,, given by equation (53) are extremely low; e.g. if i=55’, a=45’, y= 160 lb/cu. ft, H=700 ft then u,=58 lb/sq. in. For this reason the form of the law of friction at low normal stresses is vitally important.

Using equations (53) and (54), equation (51) gives

F=p*coto: . . . . . . . * (55)

This is the most convenient formula for F, where a,, is calculated from equation (53) and substituted in CL*. However, formulae can easily be written out explicitly. Thus for the Coulomb law of equation (49), p* =p+c/o, and equation (55) becomes

F = Clcota+ 2c sin i

yHsin (i-ol)sincr ’ . ’ ’ ’

Similarly, the power law of equation (25) gives

F = koz-l cot ct . . . . . . . . . . . * (57)

= k[$yH sin (i-a) cos a cosec ilm-l cot a . . . , - w

To compare results given by equations (56) and (58), suppose that a Coulomb law is fitted to values given by the power law at two points, g,, and uuo, then the constants in the Coulomb law are

(59)

where b= (am- l)/(a- 1) and, denoting the value of F obtained from equation (56) with these constants by F,, this is given by

F, = ko;-l bcota+ 2( 1 - b)uo sin a

yHsin(i--a)sincr > . ’ ’ ’

As an example, consider the case k = 5.2, m = O-7 stresses being in lb/sq. in., i= 55”, CL =45”, y = 170 lb/ft, H = 700 ft for which equation (58) gives F = 1.5. Putting a = 2 and b = 0.62 in equation (60) this becomes

F c = 3.22 u;“.3 +0.0319 o.7 00 . . . . . . (61)

If u. = 100, F, = 1.61; if u. = 200, F, = 1.96; if crO = 300, F, = 2.31. This shows clearly the large effects on the factor of safety which may arise from changes in the conditions of measurement.

As a second example, consider the points of Fig. 16(a) which may be approximately represented by the power law of equation (28) or the exponential form of equation (36). The least squares line through the points for U, <: 1000 is

7 = 20+0.65 u, . . . . . . . - (62)

Table 1

H, ft

1000 100

I&. ft

160 160

a i

35” 45” 35” 75”

un, F from F from F from F from lb/sq. in. equation equation equation equation

(28) (36) (62) (63)

112 1.07 1.12 1.18 2.58 30 1.22 1.15 1.88 7.81


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124 J. C. JAEGER

F = pcotct-f- 2c sin i sin (a1 - @)

yHsin (i-/3) sin (fxl-a) sina . ’ . ’

Similarly the power law of equation (57) gives

F = k(+yH)m-‘[Sin (“-$?; ;ain;l-a)]m-lcot a . . .

(66)

(67)

Some method of stepping between joints of the same set such as this seems the most likely method of failure.

Tensile fractures normal to joints. It is sometimes postulated that fracture spreads between joints of one system by tensile failure normal to them, i.e. the situation of Fig. 34 with c(~= (rr/2) + a and tensile forces T, x PR acting across the face PR.

In this case for the Coulomb law

F = cL cot ,+2[c cos (8-a) +T, sin (B-41 sin i yH sin (i - /3) sin a

. . . (W

Thus if ,!I - a is of the order of 10” this corresponds to adding a quantity of the order of T,/5 to c, which is a considerable increase.

These cases comprise the simple order of magnitude results which can be stated for plane sliding in the presence of a well defined fault or joint system. They have the great advantage that results can be obtained with a slide rule in a few minutes and that the crude physical approximations to the method of sliding and the values of u,, are made clear.

However, even the simplest formula (equation (56)) contains three parameters-c/yH, a

and i-and so the results are difficult to display. Hoek (1971) has shown that this formula and a variety of cases, including the height H, of the water table and the depth 2, of a vertical tension crack, may be adequately represented in terms of two parameters

X = 22/((i--)(a-+[l -cz(H,/H)~])}

Y = (1 + bZ,/H)yH/c

where a =0 for a drained slope, a = O-1 for normal drawdown and a = 0.5 for a horizontal water table; b = 0 if there are no tension cracks; b = 1 for a dry tension crack and b = 3 for a water-filled tension crack. His chart is of the type shown in Fig. 35. Here and in equation (69) angles are in degrees.

In the other extreme case in which rock is so broken up that it may be treated as a soil to a first approximation, a plane surface of sliding may be postulated. However, it is known from soil mechanics theory and practice that circular surfaces of sliding are more usual and in general lead to lower factors of safety. The case of circular surfaces of sliding is of a different order of complication to that of plane sliding since the worst possible circle has to be determined and hence either eIaborate computer calculation or some form of design charts are necessary. Bishop and Morgenstern (1960) gave charts for this purpose and Hoek (1971) has given charts which give F as a function of X and Y defined by

X = i-+(1+2-aH,/H)

Y = [l+ (i-b)Z,/lOOHJyH/c > ’ ’ ’ * ’ . (69)

where H, is the height of the water table, 2, is the depth of any vertical tension crack and a=0 for a drained slope; a=0*3 for normal drawdown; a =0*5 for a horizontal water table; b=i for no tension crack; b=25 for a dry tension crack and b= 10 for a water-filled tension crack. Hoek’s (1971) chart is shown in Fig. 35. Both this and the charts of Bishop and


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Fig.


35. Simple circular sliding (after Hock, 1971)

60 .

Morgenstern (1960) involve the Coulomb law and the effect of assuming other laws is not clear. However, Hoek (1970) pointed out that any non-linear law may be regarded as the envelope of a family of Coulomb laws and if curves of H against i are plotted for these for any given value of F, their envelope will be the corresponding curve for the non-linear law.

In view of the difficulty in using laboratory measurements to provide parameters for design purposes attempts have been made to compare actual rock slides with design charts in an attempt to determine c. Coates et al. (1963) analysed over twenty rock slides in various types of rock, using an angle of friction of 37” (similar to that for intact rock) and various reasonable assumptions about tension cracks and water table; they found values of c ranging from 410 to 2620 lb/sq. ft. Again Coates et al. (1965) analysed fifteen slides in material which, when crushed and tested triaxially (cf. the section on closely fractured and interlocking material), gave $ = 34” and c between 450 and 750 lb/sq. ft. They concluded that the average value of c mobilized was 710 lb/sq. ft.

Hoek (1971) studied a number of stable and unstable slopes in Rio Tinto Espanola’s Atalaya open pit using his curves for plane sliding similar to those shown in Fig. 35. He found that, by choosing ‘field’ values of +=35” and c= 15 lb/sq. in., the unstable slopes had factors of safety clustering around 1, while those for the stable slopes clustered around 1.3. Laboratory studies on joints gave values of 4 varying from 38” at peak to 31” residual, while c varied from 150 lb/sq. in. at peak to zero residual. He takes a pessimistic view about the possibility of basing accurate rock slope design on laboratory figures, and this opinion may be reinforced by the preceding discussion. However, the position is that for feasibility studies, at least, only laboratory-type tests are available. A direct comparison needs to be made between direct shear tests on single joints of differing areas, triaxial tests on single joints of various areas, triaxial tests on crushed rock and triaxial tests on closely jointed core if such is


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126 J. C. JAEGER

available (cf. section on closely fractured and interlocking material). If such studies were made on slopes expected to fail in the course of excavation, an invaluable comparison between laboratory and field estimates would be obtained. One possible method of using such a comparison has been indicated in the discussion of equation (43).

SLIDING ON TWO PLANES

The simple two-dimensional theory of the previous section may be generalized to consider the sliding of tetrahedral blocks on a pair of surfaces. The theory of this has been worked out largely in connexion with the possibility of movement of actual, accurately delineated blocks in dam abutments. It is complicated by the geometrical possibilities of movement. Wittke (1965) used a vectorial treatment and was followed by Goodman and Taylor (1967). John (1968), Londe (1965) and Londe et al. (1969) use the stereographic projection. All these authors use the Coulomb law.

From the point of view of slope stability in which case both the directions of possible surfaces of sliding and their frictional properties may not be well known, it is important to be able to estimate, in order of magnitude, the improvement in the factor of safety for the present case over that of the previous section. The simplest treatment is that of Paulding (1970).

Suppose that the two planes of sliding are OP,Q and OP,Q (Fig. 36(a)), their line of inter- section OQ being inclined at CI to the horizontal. Take OQ as the z axis, OX horizontal and Oy to make a right-handed system. Viewed in the direction z0 the system appears as in Fig. 36(b), where & and & are the inclinations of the planes OP, and OP, to the vertical plane through Oz. Let W be the weight of the block under consideration. Let N, and N2 be the normal reactions on the planes OP, and OP, and let & and & be the variable coefficients of friction on these surfaces so that, if the factor of safety is F, the system would be in limiting equilibrium with frictional forces FmlN,pT, and FelN,& parallel to Oz acting in these two planes. ResoIving in the x, y and z directions gives

N,cos#,-N,cos#, = 0 . . . . . . . (70)

N,sin~,+N,sin&.--Wcosa = 0 . . (71)

Wsin ct-F-l(N1pT+N&) = 0 . . . . . . (72)

Hence

N

1

= w cos a cos qlz

sin (9% + A) N

2

= w,cos cl cos *I

sin (9% + ~4

F = (d cos $2 + CL: cos #I) cot O1

sin (Cl + 9J . . . . . .

(73)

(74)

Here the geometry of the system enters only through the quantities pt and & which in general are functions of the normal stresses across the two planes. These stresses are N,/A, and N,/A, where A, and A, are the areas of contact with the two planes. Equation (74) with F = 1 may be used to infer values of pf (assumed equal to &) from known slope failures. This has been done by Paulding (1970) and the large and scattered values he finds are probably a result of these assumptions.

If t~f and & are assumed to be constant, equation (74) gives F independent of the shape of the mass sliding. In all other cases this must be specified. To get a simple result the case of a flat topped slope of height H and inclination i is considered, where the direction of sliding


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Fig. 36. Sliding on two surfaces

OQ is in a plane perpendicular to P,P, (Fig. 36(c)). The dip 6, and strike wi of the plane OP,Q are related to the parameters a and I& previously used by

cos 6, = cos CL sin *I

tanw, = sin CL tan *I > * ’ ’ ’ ’ (75)

with similar results for 6, and w2. The weight W of the tetrahedron OP,P,Q is

W = &yH3(cot ~-cot i)2(tan &+ tan ti2) sin a . . . . (76)

and the area A, of the plane OP,Q is

A1 = @!Z2(cot u-cot i) sech . . . . (77)

Hence from equation (73)

i?I_1 - YZY sin (i-a) cos CC N,

A, - 3sini =& . . . . . (76)

If the Coulomb law is assumed

/.$ = PI+% = PI+ 3c, sin i

1 yHcoscfsin(i-a) . . . * P-9)

and from equation (74)

F = [

p1co~~2+p2cos~1+ 3(c1 c~~ao~o~~s~~s~~~~~ ;] cot CL cosec (#r +#2) . . (SO)

For the symmetrical wedge & = $2 with pL1 = p2 = p and c1 = c, = c this becomes

3c sin i yH cos ct sin (i - cz) I

cot CL cosec +& . . . . . (81)

This differs from the result of equation (56) for plane sliding by the factor cosec #1 and the replacing of 2c by 3c. The latter change corresponds to the change in shape of the body sliding from a prism to a tetrahedron.

Another case is that in which one plane is vertical, ti2= 0, and as before pcl =pa=p and c1 = c2 = c. Then equation (SO) gives

3c sin i

yH cos 01 sin (i-a) I cot acot *,/2 . . . . (82)


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128 J. C. JAEGER

If a power law and the symmetrical case are assumed so that PT = ~(2 = kot-l = k(N,/A,)m-l equation (74) gives

F = k cot CL cosec #l{+,H cos CC sin (i-a) cosec ;>‘“-l . . . (83)

which differs from equation (58) by the factor cosec & and the substitution of l/3 for l/2. Thus the factor of safety for downslope sliding of a right-angled tetrahedron $I =& = 7~14 is improved by a factor of the order of 42.

The result of equation (74) for the symmetrical case PT = & = p and & = (crZ = $ is easily generalized to the case of horizontal tectonic stresses. Suppose that the horizontal force on the block is P, then the factor of safety F is

F = bp* cot a cosec I/ . . . . . . (84)

where sin2 # + h’ cos2 (CI

b = 1/{1 +(K-1)2 cot= a co? I+} . . . . (W

K = (2P/W) set a tan $ . . . . . (86)

Horizontal tectonic stresses are frequently measured near the surface. Also, in curved pit slopes the effect of curvature is to introduce circumferential stresses. This effect has been discussed for the case of a Coulomb material by Jenike and Yen (1963).

CLOSELY FRACTURED AND INTERLOCKING MATERIAL

In many practical cases the rock mass is intersected by closely spaced joints and planes of weakness which divide it into small interlocked elements. If there are no joint systems especially favourable for sliding, it is reasonable as a first approximation to use soil mechanics theory to study the behaviour of the mass. The essential problem is that of determining the frictional parameters of the material for inserting in the soil mechanics equations.

The most obvious method is that of studying specimens of joint surfaces by any of the methods described (Hobbs, 1970). This ignores the effects of interlocking of rock elements and so represents a conservative approach.

Alternatively, specimens of the broken rock may be tested in one of the large triaxial pots used for testing aggregate for rock fill (Fumigalli, 1970; Coates et al., 1963). Terzaghi (1962) refers to measurements by Silvestri (1961) who found an angle of friction of +=65” for crushed aggregate. Jaeger and Cook (1969) quote a measurement by the Snowy Mountains Authority giving a value of 4 = 40” for crushed aggregate.The results of such tests depend greatly on the porosity of the aggregate. While such measurements make an accidental allowance for inter- Iocking of the aggregate, they probably give an underestimate. The Mohr envelopes are usually strongly curved so that a power law representation may be suitable.

Rosengren and Jaeger (1968) made a laboratory study of the effect of interlocking using coarse grained marble which had been heated to about 500°C. The thermal expansion of calcite is highly anisotropic and so the effect of heating is to fracture almost all grain boundaries so that the final material consists of a mass of interlocked crystals and has negligible tensile strength. Tested triaxially, the strength increases rapidly with confining pressure. The Mohr envelope is shown in Fig. 37(a). This is strongly curved and may be represented by 7 = 400O.~~, units being lb/sq. in. Failure at the lower confining pressures is always on a well defined shear plane.

Similar results were obtained by Jaeger (1970) who tested six-inch cores from closely jointed rock in which the distance between fractures was c-1 in. Again the Mohr envelope is approximately represented by a power law (Fig. 37(b)) and the material selects a definite


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Huted marble (W

I Pangunr rndcrite

0 : Ib/sq. in. 0 : Ib/rq. In.

Fig. 37. Mohr diagrams for closely fractured material

plane for failure. Comparison of these results with those from tests on individual joint surfaces indicates a considerable strengthening due to interlocking.

The results suggest that the most satisfactory method of studying rock of this type is by triaxially testing the largest cylinders possible. With the improvement of drilling techniques, the obtaining of such cylinders should be possible.

Many writers (e.g. Brown and Trollope, 1970; Einstein et al., 1970; Mtiller and Pacher, 1965; John, 1969) have done triaxial and biaxial compression tests on models composed of materials such as plaster or patternstone usually made up of cubical blocks orientated at various angles to the applied stresses. In general planes of shear failure running through the masses are obtained and the Mohr envelopes suggest a power law. These effects are determined partly by friction on the bounding surfaces and partly by failure of the rather weak blocks.

The most important practical cases in which these considerations may apply are in some slope stability problems and in the study of the behaviour of fractured rock around tunnels.

In slope stability problems, the parameters obtained in this way are used to determine factors of safety. The simplest case, that of a circular surface of sliding, may be estimated from Fig. 35 or from the tables of Bishop and Morgenstem (1960). Non-circular surfaces of sliding and arbitrary laws of friction may be studied by sophisticated computer techniques (Morgenstern and Price, 1965; Little and Price, 1958; Whitman and Bailey, 1967).

For the study of stresses in broken rock around drives and tunnels with applications to rock bolting, Morrison and Coates (1955) used the Coulomb theory of soil mechanics and Jaeger and Cook (1969) discussed the case in which solid rock obeyed a different Coulomb relation. Daeman et al. (1969), Horvath (1964) and Ewoldsen and McNiven (1969) also use Coulomb theory. Hobbs (1966, 1970) uses a power law.

EFFECT OF CONSTRAINTS

Most of the previous discussion has been based on the assumption that the constraints on the system are such as to permit sliding. However, after the peak has been passed, and possibly even before it, there are substantial displacements which affect the configuration of the system and hence the stresses on it in a manner determined by the external constraints.

This is a different effect from interlocking. Fig. 38 shows an experiment on two-dimensional sliding on an array of cubes of Bowral trachyte of side lengths 3 in. The corners of the

2


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130 J. C. JAEGER

1 N= 14000 tb

Fig. 38. Multiple sliding of square blocks: (a) system, (b) sy%m&rical sliding, (c) slightly asymmetrical sliding

(a) (b) 0

cubes were removed so that there would be no interlocking. PP is the platen of a testing machine and QQ is a block at right angles to it. The sides of the cubes are initially at 45” to these. Load T is applied through a block AB by the testing machine and load N by a line of jacks through a plate CD. The half cubes in contact with PP, QQ, AB and CD have metal sheets attached with epoxy resin and the contacts of these with the platens are lubricated with Molybond to reduce friction. The gross effect should be that of pure shear of the set of cubes.

If the system is carefully orientated with all faces parallel, symmetrical sliding takes place on all surfaces and a load-displacement curve such as Fig. 38(b) is obtained, the upper part of which corresponds to the surface AB moving inwards and the lower part to CD moving inwards. Both are consistent with a value of p = 0.65 similar to that obtained from sliding on a single surface. However, if asymmetry is present or develops, the plate CD can rotate and the blocks can rotate and replace surface contact by line contact. The load-displacement curve increases steeply (Fig. 33(c)) and violent stick-slip occurs. Ultimately the blocks may fracture. A similar effect has been observed by Jaeger and Rosengren (1969) for sliding on a pair of inclined surfaces. It appears that, apart from interaction of blocks, multiple sliding is sensitive to the applied constraints.

As another example of practical interest, consider the case of a rectangular pillar ABB’A’ (Fig. 39) containing a plane of weakness CD inclined at Q to its axis. Suppose that a vertical load P is applied to the pillar which is of area A and that sliding on CD is governed by a Coulomb relation with cohesion c and angle of friction $. Possibly lateral stresses u2 are applied to the wahs AA’ and BB’ by fill If there were no constraints at the pillar boundaries AB and A’B’, sliding would be governed by equation (47). However, it is more likely that the behaviour is governed by the constraints at AB and A’B’. For example, shear failure might take place across these surfaces (shown dotted) with Coulomb values S, and angle of internal friction &. In this case equation (47) generalizes to

(TV cos (a + $ + &) = [c cosec CC cos 4 + (S, + a, cot a) sin (a + C)] cos $, * (37)

This formula also arises as a correction for end effects in triaxial apparatus (Mathews, 1970; Rosengren, 1968). It appears that slip is impossible if Q > (7~/2) -$-&.

As an alternative possibility, the portion ABCD of the pillar may behave as a cantilever


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Fig. 39. (a) Faulted pillar with sliding at its base, (b) cantilever approximation, (c) load-displacement curves for two values of a

A +’ m-m

1 a D

--_

C

2octi

0

ri’

IOMM

so@

AL 45’

K tp B’ 0 0.1

x: In.

(a) 6) (4

so that, for a small relative lateral displacement x of the two portions of the pillar, the lateral force is kx where k is the stiffness of the system (Fig. 39(b)). Neglecting the cohesion c gives

P = kxtan(a++) . . . . . . . . (88)

This condition can be approximated in uniaxial testing in a heavy testing machine if the specimens are attached rigidly to the platens. For example, Fig. 39(c) shows load-displacement curves for rectangular blocks of sandstone 3 in. by 1 in. in cross-section and 4 in. long. For the sandstone (b =30” and the two curves are for ~r=45’ and a=50°. Their slopes both correspond to k of the order of 50 000 lb/in. Failure in both cases is approximately along the dotted diagonal shown in Fig. 39(b) and the higher value of 6000 lb/sq. in. is of the order of the uniaxial compressive strength, 9000 lb/sq. in. Horino (1968) reported similar results.

ACKNOWLEDGEMENTS

The Author is greatly indebted to many friends, notably Professor E. Hoek, Dr J. H. Dieterich, Mr K. E. Mathews, Dr N. G. W. Cook and Dr N. Gay for discussion, advance information about their work and permission to use figures from it. He also thanks Mr W. McIntyre for assistance with his experimental work.

REFERENCES

ARCHARD, J. F. (1958). Elastic deformation and the laws of friction. Proc. R. Sot., Series A 243, 190-205. BARTON, N. R. (1971). A model study of the behaviour of steep excavated rock slopes. Ph.D. thesis,

University of London. BERNAIX, J. (1969). New laboratory methods of studying the mechanical properties of rocks. Int. JnZ

Rock Mech. & Min. Sci. 6, 4%90. BISHOP, A. W. (1955). The use of the stability circle in the stability analysis of earth slopes. Ge’otechnique

5, No. 1, 7-17. BISHOP, A. W. & MORGENSTERN, N. R. (1960). Stability coefficients for earth slopes. Gtotechnique 10,

No. 4, 129-150. BBKER, R. (1915). Die Mechanik der bleibenden Formanderung in Kristallinisch aufgebauten Kijrpern.

Verb. dt Ing. Mitt. Forsch. 175, I-51. BOWDEN, F. P. & TABOR, D. (1950). The friction and lubrication of solids. Oxford: Clarendon. BRA;~J: W. (1967). A study of jointed and fractured rock. Rock. Mech. & Engng Geol. 5, 119-136, 197-

BROWN, E. T. & TROLLOPE, D. H. (1970). Strength of a model of jointed rock. J. Soil Mech. Fdns Div. Am. Sot. civ. Engrs 96, SM2, 685-704.

BYERLEE, J. D. (1967a). Theory of friction based on brittle fracture. J. appl. Phys. 38, 2928-2934. BYERLEE, J. D. (196713).

72, 3639-3648. Frictional characteristics of granite under high confining pressure. J. geophys. Res.


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VOTE OF THANKS

PROFESSOR HOEK said that he had been privileged to hear Professor Jaeger lecture on many occasions. Most of those who were concerned with the details of rock or soil engineering were so close to the problem that they seldom had time to sit back and examine the question of whether they were using the right concepts. It was always a valuable experience listening to Professor Jaeger because he had an extraordinary ability to see the fundamental principles involved in the problem, however complex, and to communicate these principles to others.

In presenting his Rankine Lecture, Professor Jaeger had examined many of the basic principles of rock mechanics and the large amount of material contained in his Paper would make it a valuable reference work for many years to come. Professor Hoek asked those present to join him in showing their appreciation for a very good lecture.

The vote of thanks was carried by acclamation.

Friction of Rocks and Stability of Rock Slopes

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