Failure of rocks under tensile conditions

Int J. Rock Meth Mm Set Vol 4, pp. 219-227 Pergamon Press Ltd 1o67 Printed m Great Britain

FAILURE OF ROCKS UNDER TENSILE CONDITIONS

J. C. JAEGER Department of Geophysics, Austrahan National University, Canberra, Austraha

(Received 15 August 1966)

Abstract--This paper examines, theoretically and experimentally, a number of systems m which extension failures occur under applied compressive loads. These are (i) compression of a circular cyhnder along two diameters or by three symmetrically placed line loads (u) diametral compression of spheres and cubes (iii) bending of circular plates. In all these cases, one principal stress is tensile over a portion of the body and failure occurs in this region, m many cases the calculated maximum tensile stresses in the region are of the order of the umaxial tensile strength of the material. For circular plates, as for other cases of mhomogeneous stresses, they are higher. Many of these systems are related immediately to the crushing of single particles or aggregates of particles

1. INTRODUCTION

FAmURE of rocks under conditions in which one or more of the prinopal stresses is tensile has usually been studied in an endeavour to find simple methods of measuring tensile strength, for example, BERENBAUM and BROI~IE [1] consider bending of beams, diametral compression of cylinders (the 'Brazilian' test) and diametral compression of squares from this point of view. The general question is much wider than this, since what appear to be extension fractures occur very commonly in rocks under practical conditions. Most surfaces produced by spalling or slabbing are of this type, i.e. they show no evidence of dis- placement in the plane of fracture. Typical examples are longitudinal fracture m uniaxial compression (GRIGGS and HANDIN, [2]; FAIRHURST and Cook [3] and disking during dia- mond drilling (JAEGER and COOK [4]; OBERT and STEPHENSON [5]. Whatever the primary causes of these may be, they are probably associated with tensde stresses produced in some way and it is therefore of interest to study the behaviour of simple systems under stresses, some of which may be tensile.

The systems to be studied here are (i) the generalization of the Brazihan test to compression of cylinders by three or four line loads, (ii) diametral compression of spheres, (ill) diametral compression of squares or cubes, (iv) bending of orcular plates. Apart from their interest m the present context, they arise in connexion with the crushing of single particles or aggregates of particles.

When possible, the relevant mathematical theory is given and experiments have been made on three rock types previously used by JAEGER and HOSK]NS [6, 7]. These are: Bowral trachyte, a fine-grained igneous rock; Gosford sandstone, a fine-grained weakly cemented quartz sandstone; and Carrara marble. Some work has also been done on a lithographic limestone, presumed to be Solenhofen.

2. CYLINDER COMPRESSED SYMMETRICALLY BY FOUR EQUAL LINE LOADS

This case occurs if a cylinder is compressed between vee blocks or in the compression of a number of cylinders packed together in contact. Suppose that the loads are W per unit length and that the acute angle between their directions is co as in Fig. l(a).

219

220 ,, c. JAEGER

The stresses have been ca lcula ted f rom HONDROS's [8] formulae for the case m which the loads W are not concent ra ted but d i s t r ibu ted over an angle o f ct --= 15 °. The difference between the stresses for a --- 0 and ct ~ 15 ° is small except near the loads.

F igure l (b) shows the na ture o f the stresses for oJ - - 90 °. Wi th in the curve 1, both pr incipal stresses in the p lane o f the disk are compress ive; on the curve I one o f them vanishes; in the

W

W W

(a) {b) {c)

Fzo. 1. (a) Cyhnder compressed in two diametral planes inclined at co. (b) oJ = 90 °. Curve i, limit of tensile field; Curve II, locus of maximum tensile stress. (c) ~o = 60 °. Curve 1, limit of tensile field; Curve II, locus

of maximum tensile stress.

region outs ide curve I, one pr incipal stress is tensile and this has its greatest value on curve II. F igure l(c) is s imilar ly d rawn for the case oJ = 60 °. In this case the tensile region jus t reaches the axis o f the cylinder.

The var ia t ion o f the m a x i m u m tensile stress ,~ a t ta ined in the cyl inder is given in Table 1 toge ther with the rad ius r a t which it occurs, R being the radius o f the cylinder, oJ ---- 0 corres- p o n d s to the o rd ina ry Brazi l ian test with loads 2 W i n which case the m a x i m u m tensile stress is

~rt = 2 W / r r R . (1)

TABLE 1. VALUES AND POSITION OF 1"HE MAXIMUM TENSILE STRESS ATTAINED IN THE SYSTEM OF FZG. 1 (a)

0 30 ° 45 ° 60 ° 90 °

RtrdW 0"622 0"452 0'253 0'112 0"039 r/R 0 0 0 0"5 0"85

A number o f tests o f this type has been made on mater ia ls ment ioned in Sect ion 1. N o significant differences were observed between loads at fai lure for vee blocks with and with- ou t ro l ler bear ings at their surfaces so tha t the la t te r were used for simplici ty. Specimens were 2 in. d i amete r × 1 in. long.

The loads W at fai lure for the three rocks (calculated f rom the machine loads assuming tha t f r ic t ion is negligible) are shown in Table 2. Each figure is the mean o f three.

FAILURE OF ROCKS UNDER TENSILE CONDITIONS 221

It appears that up to ~o = 45 ° the tensile stresses at failure are very nearly equal to the

Brazilian tensile strength of the material. For greater angles, there are still substantial ten-

sde stresses, but they are less than the Brazihan tensile strengths. Failure is invariably in almost straight lines jo ining the loading points. It therefore follows

very closely the hnes of maximum tensile stress in F~g. 1 (b) and (c) even it ff is not initiated

TABLE 2 NORMAL LOADS W(lb) AND MAXIMUM TENSILE STRESS ot (lb;'in ~) AT FAILURE IN THE SYSTEM OF F I G l(a)

Carrara marble W qt

Gosford sandstone W at

- 0 45 60 90"

1600 4080 4130 5160 1000 1030 460 200

850 2350 2900 3640 540 590 330 140

on these lines. The lines of fracture for Carrara marble with ~ ~ 60 ° are shown in F~g. 2(a) which was obtained by coating the surface of the cyhnder with penetrant dye after failure. In the case co ~ 90 ° the circular cylinder is frequently reduced to a square.

(a)

w

-0.5

i I

I I

r/a ~----~30 i]

(b)

FtG. 3. (a) Co-ordinate system for a cylinder compressed by three line loads. (b) Variat'on of stress with radius" Solid lines are for concentrated loads at the values 0 °, 30 °, 60 ° of 0. Broken lines are for 0 = 0 and

loads distributed over angles of 15 ° (Curve I) or 5 ° (Curve II).

3. THE CASE OF A CYLINDER WITH THREE LINE LOADS

Suppose that loads W and ½W sec ~ per uni t length are applied to a cylinder of radius R at angles 7r - - % 2% ~ -- ~ to each other as in Fig. 3(a). The problem is a special case of that in which the normal stress N applied to the cylinder is given by the cosine series

N = A o + ] ~ An cos nO (2)

222 J c. J,~rG~R

in which the te rm in cos0 is absent since the forces are to be in equil ibrium. Stresses are reck- oned posit ive when compressive. The stresses can most easily be calculated by the use of the complex variable as in J*EC;ER [9]. They are found to be, in polar co-ordinates, r. # Fig. 3(a)

J

% 4 ,r, : 2Ao-~-4"~' Aup" cos nO (3) Jt 2

a o - - - a r = 2 ( p 2 - 1 ) ~ nAnp *~-2cosnO (4) t t 2

- /

rr . = {pz __ 1 ) ~ nAnp "-2 sin nO (5) ~t - = o -

where

p : : r/R.

For the special case q~ - - 60 ° and concentrated line loads, the series can be summed and give

3 W ( I - - p6) aO -~- ar ~ ~rR[1 -- 2p 3 cos 30 + [,6] (6)

9 W p ( p z - 1)[(1 + ,06) cos 30 -- 2p 3]

~. - a,. : rrR[1 - - 2p 3 cos 30 + p612 (7)

9Wp(p z -- 1) (1 -- p6) sin 30 f r o = 2~rR[1 - - 2 p 3 c o s 3 0 + p~]e"

Some values of the least principal stress for this case ale shown in Fig. 3(b). In this, , r R u / W , where o is the least principal stress, is plotted against r / R for the values 0 °, 30 ° and 60 ° o f 0. I t appears tha t bo th principal stresses are compressive in the region 0 < r < 0.4R for all values o f 0. Fo r 0 = 0 ° or 120 ° the tensile principal stress increases to the Brazilian value W/TrR beneath the loading points.

As in HONDROS'S [8] extension o f the theory o f the Brazilian test, the effect o f finite width of the loading region can be taken into account. I f each load is distr ibuted over an angular width o f 2a, the stresses on the line 0 = 0 are found to be, for ~ -= 60 °,

~R(o# + ,~)/W :- 3 + p3 sin 3a t

a2 t a n - 1 1 -- p3 cos 3aJ (9)

3p(p 2 -- 1) sin 3a ~,R(,,o - ,,,)/w= iql-:-2p~cos 3~ + :}" (1o)

Both principal stresses become compressive immediately beneath the loaded region, but beyond this there is a region in which one o f them becomes tensile. The value o f the least principal stress on the line 0 = 0 is shown by the dotted curves in Fig. 3(b), Curve I being for 2a = 15 ° and Curve II for 2a = 5 °. It appears that for small values o f a , such as those

FAILURE OF ROCKS U N D E R TENSILE CONDITIONS 223

occurring with flat platens, tensile stresses approaching the Brazilian values WIreR are attained.

Experiments were made on cylinders of Gosford sandstone and Carrara marble 2 in. m diameter and I in. thick. Flat platens were used. The loads Wat failure for various values of 7 are given in Table 3. ~ ---- 0 is the ordinary Brazilian test.

TABLE 3. LOADS W (lb) AT FAILURE IN THE SYSTEM OF FIG. 3(a)

~0 0 22½ ' 30" 45 o 60 ~

Carrara marble 3200 3400 3500 5300 5800 Gosford sandstone 1700 2400 2500 3350 3450

These loads increase with increase of % For line loads, the stresses below them would be expected to have the Brazilian values and failure would be expected to be initiated below the greatest load. In fact, in Carrara marble a crack is always observed below the greatest load which extends along this diameter as the load is increased and ultimately swings towards one of the other two loads, the final failure being as in Fig. 2(b). The same behaviour has been observed with lithographic limestone.

4. DIAMETRAL COMPRESSION OF SPHERES

It would be expected, by analogy with the cylindrical case, that a sphere subjected to concentrated forces If" at opposite ends of a diameter would show an extension failure in a plane or planes through this diameter. Formulae for stresses in this case have been given by STERNBERG and ROSENTHAL [10] and ABRAMIAN et al, [11]. Unfortunately the formulae are rather complicated and no numerical values are available. In contrast to the cylindrical case, Poisson's ratio occurs in the formulae.

I f I4 / is the load at failure for a sphere of radius R, it would be expected on dimensional grounds and by analogy with the cylindrical case that failure would take place when

k W / R ~ = To (11)

where To is the tensile strength and k is a numerical constant. Results for four rock types are given in Table 4, the values of k are computed using the values of To determined by the Brazilian test with fiat platens. Each figure is the mean of three. Tests were rapid, stress being increased at a rate of 500 Ib/min.

TABLE 4. DIAMETRAL COMPRESSION OF SPHERES

R W To k (in.) (lb) (lb/in 2)

Bowral trachyte 1-0 7520 1740 0.23 0.5 1930 0.23

Gosford sandstone 1.0 2160 540 0"25 0"5 580 0'23

Carrara marble i'0 2800 1265 0-45 0'5 800 0"40

L~thograph~c limestone 1.0 4200 1600 0-38

224 J.C. JAEGER

It seems that the law (11) is reasonably well obeyed. Carrara marble shows the greatest departure and this is probably due to effects of rate of straining.

For the sandstone, marble and limestone, the usual method of failure appears to be by an extension fracture in a single diametral plane. In a few cases there is some shattering and one of the hemispheres is divided into two parts, but this is believed to have occurred after the primary failure. In the harder trachyte, division into three lunes of angle,,: of approximately 120 ° occurs in some cases with considerable shattering.

These resultsmay be compared with the observations of GILVARRY and BEROSTROM [12], [13] on spheres of glass which are found to shatter into many pieces. These authors and DRUCKER [14] regard such shattering as presumptive evidence for the existence of internal cracks. Only in the trachyte, and only in some specimens, is there evidence of this sort of behaviour.

Spheres of rock may readily be loaded by contact with hardened steel spheres in various packing formations to produce complicated patterns of failure which generalize those described above for cylinders.

l v/ I I

2R

t w ( a l (bl

w i

I ' I _J I o i I

(c)

FiG. 4. (a) Square cylinder of side 2R loaded over a region of width 2b. (b) Circular cylinder of radius R loaded over a region of width 2ttR. (c) Circular plate of thickness d stressed over rings of radii a and b.

5. FAILURE OF SQUARE PLATES AND CUBES COMPRESSED ALONG A DIAMETER

The case of a square plate of side 2R compressed as m Fig. 4(a) by loads W per unit length distributed over width 2b of its surface has been used by BER~NB^UM and Baom~ [1 ] for measurement of tensile strength. They determine the tension in the central region photoelastically for various values of the ratio b/R. For example, they find the maximum tensile stress crt when b/R = ~ to be

,~t == 0-366 W/R. (12)

It might be expected that this stress would not differ greatly from that in a circular cylinder of radius R loaded over a region of the same width, i,e. 2a = 15 °, approximately, Fig. 4(b). This is given very nearly by the Brazilian formula

,~t -: W/R,,. (l 3)

GOODIER [15] has given a complete analysis for line loads applied to a rectangle and shows that for the case of a square the tensile stress across most of the loaded diameter is given very nearly by (13). For a long rectangle compressed across its shortest diameter in this way, the tensile stress at the centre is of the order of 0-85 W/~rR. JAEGER and HOSKINS [6]


have discussed the corresponding problem for a ring of thickness 2R pinched between forces W applied at its inner and outer surfaces and find a maximum tension of 0.63 W[zrR.

The essential points are that the calculated tensile stresses in all these systems are much the same and experimental failure takes place at very similar loads.

The same results may be expected to apply in three dimensions. Table 5 compares the applied forces at fadure of spheres 2 in. in diameter with those for 2-in. cubes loaded along a diameter. In the latter case the load was applied from contact with a sphere of diameter 3 in.

TABLE 5 COMPARISON OF DIAMETRAL COMPRESSION OF CUBES AND SPHERES

W Sphere Cube (lb) (lb)

Gosford sandstone 2160 2770 Carrara marble 2800 2950

As would be expected, the values for cubes are of the same order as, but rather higher than those for spheres.

6. FAILURE OF CIRCULAR PLATES WITH AXIALLY SYMMETRICAL LOADING

The theory of the stresses in a circular plate of radius a and thickness d, Fig. 4(c), simply supported at its circumference r ---- a and uniformly loaded over a concentric region of radius b, is well known (MORLEY [16]; SOKOLINKOFF [17]. For a concentrated central load the stresses become infinite. The simplest system, both theoretically and experimentally, is the case in which total load W is applied on a circle r = b. In this case the maximum tensile stress is constant over the region 0 ~ r < b and is given by

(tXr)max : (%)m~x 3(1 + v ) W , a l - - v ( b 2 ) ) = 4~rd~--121n b + 1 q -v l - - a 2 "

Experiments were made on a number of materials with a variety of values of b, a, d. a was either 1.5 in. or 1 in; b either 0.5 in. or 0.25 in. and d either 0.125, 0-187 or 0.25 in. While there is probably some systematic difference between results obtained on different systems this is not large and all disks (about 16) of each material are lumped together in Table 6.

TABLE 6 TENSILE STRENGTHS (lb/ln 2) MEASURED BY BENDING OF CIRCULAR PLATES AND OTHER METHODS

I II III IV V VI

Bowral trachyte 3640 3500 3700 3650 1740 1990 Gosford sandstone 1140 1200 1100 1140 540 520 Carrara marble 2600 2500 2300 1710 1265 1000

I--Circular plates, II--Dlametral compression of 2:1 rings, III--Diametral tension of 2:1 rings, IV--Three point bending, V--Diametral compression of solid cyhnders with 15 ° contact. VI--Direct tension.

226 J, C. JAEGER

The difference between the extreme values and the mean given is less than l0 per cent. The value v = 0.25 was used in the reduction.

The values o f the calculated maximum stress at failure are shown, 'Fable t~, and compared with values given by JAEGER and HOSKINS [6] for other tests on the same materials. I t appears that the values derived f rom circular plates are o f the same order as those obtained with other systems involving inhomogeneous stresses and o f the order o f twice those obtained f rom direct tension. The same effect has been observed by DURELLI and PARgS [18]. The values for three-point bending in Table 6 were intended to give order ,,f magnitude only and were based on only a few specimens.

In the case o f Carrara marble (and of the rather coarse-grained Wombeyan marble which was also used) cracks can be observed to form before complete failure occurs. These cracks do not propagate and close on unloading. This behaviour makes all obselvat ions on marble a matter o f considerable uncertainty.

In all cases failure occurs inside the ring r =- b and perhaps gives an insight into behawour when two tensile principal stresses are equal. In no case do more than three cracks meet at a point, the mean angle between crack directions at a point being 120 °. In many cases a disk will separate into more than three pieces and there will be more than one point o f inter- section o f cracks in the region r < b. A histogram of angles between cracks is shown in Fig. 5(a). I t may be compared with Fig. 5(b) which is a histogram o f angles bounding col- umns in columnar jointing [19] this, again, probably corresponds to failure under equal principal tensile (thermal) stresses.

2 0 -

i o -

I 066 9o

(a) (b)

FIG. 5. (a) Histogram of angles of fracture in circular plates of various materials. (b) Histogram of angles m columnar jointing.

Note added in proof--Si~e the above was written, the stress distribution in a sphere compressed diametrally has been calculated by HrRAMArSU and OKA[20] who find 0.225, approximately, for the constant k in (11). This agrees very well with the results of Table 4, except for those on marble. The system of a tong cylinder which is compressed diametrally between point loads has been used by D'ANDREA et al.[21] who use a formula of type (11) with k = 0.24.

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Failure of rocks under tensile conditions