BRITTLE FRACTURE OF SPHERICAL HOLLOW SPECIMENS BY UNIFORM PRESSURE I. L. Zel'manov, L. S. Kozachenko, and O~ S. Kolkov

The most widely used strength characteristics of solid rock in place is the crushing strength, which is determined in relatively simple static experiments on axial compression of cylindrical or prismatic specimens. However, the strength so determined is not a physical characteristic of the medium. The fracture mechanism may differ according to the material of the specimens and the conditions on the loading planes; for example, we may have splitting along the planes of maximal shear in some cases, and severance along planes parallel to the specimen axis in others. Therefore, in our opinion, it hardly seems expedient or justifiable to use this simple characteristic in problems on fracture in states of stress differing from those arising under axial compression. For a number of problems, for example, in the investigation of fracture during camouflet blasting of a concentrated charge, the stability of spherical cavities in the ground, etc., spherical symmetry is characteristic. In this connection it seems expedient to investigate fracture during hydrostatic loading of spherical hollow specimens of brittle materials. The state of stress of such specimens is given by elasticity theory; fracture must begin at the cavity surface and must be determined by the shear strength, which may be regarded as a physical characteristic of the medium. As media simulating a rock in the laboratory, it is natural to use substances without cracks or inhomogeneities, thus limiting the range of phenomena investigated. We selected three such substances: rosin, sulfur, and potash alum. The specimens were obtained from a melt of the initial substances without significant superheating above the melting point (rosin IO0~ sulfur I15-120~ alum 92~ Immediately before filling, the molds for casting half-specimens were placed on a cooled (to --5~ massive metal hearth, but the exposed surface was left to cool freely in air. Thus gradual upward cooling was realized; this is insignificant if we bear in mind that the substances used undergo marked shrinkage during crystallization, and that uniform cooling on all sides could induce undesirable initial stresses in the specimens. The results of the experiments confirmed the desirability of this procedure. The rosin specimens were spheres with outer diameter b = 60 mm and a center hole with diameters a = 12, 18, and 24 mm, glued from two halves. We also performed experiments on cubelets with edge length b = 50 mm and a center hole with diameters ~ = 12, 18, and 24mm. The sulfur specimens were of two types: spheres of two glued half-spheres with external diameters of 18 and 60 mm and hemispherical ones with outer diameter 60 mm. In the experiments the half-spheres werelapped to a flat steel disk with contact on vacuum grease; this ensured equivalence to the case of spherically symmetrical compression. The alum specimens were two glued half-spheres with outer diameter b = 18 mm and inner diameter a = 4, 6, and 8 ~n. Figure 1 shows the apparatus used for the experiments on rosin specimens. The cylinder 1 containing the specimen 2 is equipped with a pressure inlet and outlets for control and release of pressure; it is filled with water, the pressure of which can be smoothly raised to i000 atm. The cylinder has a Plexiglas window 3 on the side of both bases, permitting the specimen to be photographed at different loading stages. Anticipating somewhat, we may remark that fracture of rosin is stagewise and that the visible contours of the crushing zone in the successive stages can be photographed in transmitted light. The distortion of the size of the crushing zone as a result of the difference between the refractive indices of water and rosin was allowed for by means of a preliminary experiment. O. Yu. Shmidt Institute of Physics of the Earth, Moscow. Translated from FizikoTekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 3, pp. 65-71, May-June, 1979. Original article submitted April 18, 1978. 0038-5581/79/1503-0245507.50

9 1980 Plenum Publishing Corporation

245

/ 2

Fig. 1

v

0

QQ

2

4HID

Fig. 2 For opaque alum and sulfur specimens, which are stronger, we used a cylinder permitting the pressure to be raised to 2500 atm. These substances did not exhibit stagewise fracture, and the moment of fracture was recorded from the sound emitted or from the closure of a contact placed in the cavity (in the case of hemispherical specimens) when it was filled with the crushed medium. In all the experiments the rate of pressure rise was 2-3 atm/sec. In addition to the experiments on uniform compression of hollow specimens, we subjected rosin and sulfur cylinders to axial crushing. Unfortunately, we were unable to obtain cylindrical alum specimens of uniform length. The rosin cylinders had a diameter D = 3 0 m m , the height ranged from 24 to 120 mm. The end faces of the cylinders were glued to steel washers with an epoxy compound to exclude uncertainty in the degree of friction between the end faces of the specimen and the surfaces of the press. These conditions on the cylinder end faces clearly give the upper strength at the given ratio H/D. Fracture of the rosin cylinders was "explosive" and accompanied by a loud noise. The specimen disintegrated into very fine dust and thin flakes measuring 2-3 mm. The prismatic columns approximately parallel to the cylinder axis, which appeared during fracture, were retained on the end faces. Figure 2 is a plot of the breaking stress [o] vs the ratio of the height of the specimen to the diameter; it will be seen that approximately at H/D = 1.5 the breaking stress remains constant and equal to about 300 kgf/cm 2. Tsvetkov et al. [I] give a value of 200 kgf/cm2; in our opinion, this is due to the difference in the conditions on the end faces in comparison with our experiments. The sulfur cylinders had a diameter and height of 18 and 36 mm, respectively. For seven specimens we obtained [o] = 760 • 50 kgf/cm 2. The fracture bore the same character as that for the rosin specimens. Figure 3 shows photographs of the successive stages of fracture of cubic and spherical hollow rosin specimens under uniform compression. For the cubes, a and b were 12 and 50 mm, respectively, for the spheres -- 12 and 60 ram, respectively. Fracture had the following character. Under an external pressure P~ we observed the first break at the inner surface of the cavity. However, the boundary of the fracture zone did not reach the specimen surface but halted (Fig. 3a, b). With a further increase in pressure, for a certain time the picture did not change up to a pressure P2, which induced the next fracture stage and an increase in the size of the fracture zone. Complete fracture with emergence on the surface took place in

246

Fig. 3. la) P = 0-440 kgf/cm2; 2a) P = 440-600; 3a) P = 600-650; ib) P = 0-440; 2b) P = 440-550; 3b) P = 550-650; 4b) P = 650-760 kgf/cm 2. 2-4 stages for the rosin specimens. Complete fracture for the cubes and spheres took place at pressures of 650 and 760 kgf/ cm 2, respectively. The frames corresponding to complete fracture are not given, because on contact of air in the inner cavity with the surrounding water at a high pressure the field was completely darkened. Certain rosin specimens were sawed up after the release of pressure on attainment of the different stages of fracture. In the stages before emergence of fracture on the surface, the fracture zone consisted of tightly compacted powder (completely opaque). The medium around the fracture zone, or more precisely the crushing zone, did not display cracks, and the inner cavity was retained without a visible change in diameter. After the final fracture of the specimen the cavity was completely filled with compacted powder with a density close to the initial value. Note that the final stage differed in the character of fracture from the preceding stages, namely, the crushing zone did not reach the surface, and the layer adjoining the latter was broken by individual cracks and consisted of fragments which had retained their transparency. Stagewise fracture was not observed in the experiments on sulfur and alum; after the initial crushing the cavity was completely filled with compacted powder. Clearly, the difference in the types of fracture of these substances and rosin is due to the fact that under pressure crushed rosin powder acquires a coherence sufficiently great for the crushed layer, together with the ambient unfractured medium, to withstand the increasing external pressure. During the fracture of sulfur specimens the sphere surface did not display visible signs of fracture, i.e., filling of the cavity was probably due to density redistribution within the specimen. The ejection of the crushed material of the specimen into the cavity and the tight filling of the latter resemble the phenomenon of a shock bump. It is possible that our hollow specimen crushing procedure provides a felicitous elementary model for investigating this destructive effect of great importance in mining. The difference between the experiments with cubes denoted by a cross and a circle (Fig. 4) is as follows: In the first case half-cubes with hemispherical indentations were glued over the surface, which remained free during crystallization; in the second case they were glued over the surface subjected to forced cooling. Fairly close agreement was observed, showing that the specimen preparation procedure was satisfactory. For all three substances the curves of P vs the ratio a/b are qualitatively similar and complete coincidence is obtainable by using certain constant factors for the pressure (0.21 for alum and 0.31 for sulfur). Knowing the breaking pressure and assuming that the specimen behaves elastically up to the moment of crushing (the justification of such an assumption for rosin was shown in [i, 2]), we can calculate the initial strength of the medium. Since there are no breaking stresses at the cavity surface, and the radial component of the stress ~i = 0, it makes sense to 247

.e fOOr

/OOC

~0

3A

.

~O

oj

o,4 a/b

o,f

Fig. 4

0,4 a/b Fig. 5

Fig. 4. Crushing pressure vs ratio a/b: x) alum spheres; A) sulfur spheres, b = 18 mm; m) sulfur spheres; m) sulfur hemispheres; 0) rosin spheres, b = 60 ram; o, +) rosin cubes, b = 50 mm. examine as the breahing stress either the compressive tangential stress [o~] of the associated shear stress [r] = 0.5[02]. Using the results of the Lame solution (see [2]), for a hollow sphere subjected to an e~=ternal pressure P we obtain an expression for the shear strength:

Ix] - - 0.75P/[ l-- (a/b) a].

(i)

Figure 5 gives the values of [T], calculated from (i). All three substances display a tendency for the strength to vary with a/b or the pressure P. Although this tendency is very weak it is systematic and must not be disregarded if we bear in mind that the range of pressure change within experiments on the same substance is small (15-25%). Clearly, henceforth it r.mkes sense to perform experiments with simultaneous loading of the specimens from within so as to reveal the dependence of the strength on the principal stresses. Direct comparison of the shear strengths obtained with the results obtained from crushing of cylinders is not meaningful owing to the difference in the states of stress and therefore a possible difference in the fracture mechanism. If, however, we assume, as is frequently the case, that the compressible breaking stress in the crushing experiments is double the value of the shear stress, then a comparison of our shear strengths [T]for sphericalspecimens with the values of [T'] =0.5[0] from the crushing experiments gives ratios [T]/[~'] of about 2.6 and 2.0 for sulfur and resin, respectively. Two facts must be noted. Firstly, the crushing strength and the strength determined on hollow specimens are different; secondly, the ratios [T]/[T'] themselves are different for rosin and sulfur, as would be expected. From this we can infer that it is inadmissible to use the results of experiments on uniaxial crushing in problems concerning crushing in spherical (and also apparently in cylindrical) geometry, even as a relative characteristic of different rocks. The fact that the crushing strength does not coincide with the strength obtained in explosive experiments was noted by Tsvethov et al. [2], but they attributed it to the influence of the loading rate, excluded in our experiments. Note also that the actual value of the shear strength of rosin, calculated from the elastic potential for explosive motion in [2], was 300 kgf/cm =, agreeing closely with our experiments on fracture of hollow spheres. It seems that the halting of the crushing zone front and the equilibrium observed up to the next crushing stage are understandable if we assume the following mechanism. As is known, the first crushing begins at the cavity surface when the maximum tangential stress becomes equal to the shear strength [~]. In this connection the crushing medium loses

248

all or part of its supporting power, inducing radial propagation of the crushing zone. With constant external pressure, as the radius of the crushing zone increases the elastic (outer) zone undergoes considerable deformations; in this connection the displacements are directed toward the center, inducing compression of the inner crushed zone. If we now postulate that the strength of the crushed layer is enhanced when the pressure on it increases, thele will come a moment at which the "supporting" radial stress at the boundary of the crushing zone is sufficient for fracture of the outer layer to cease (this, of course, presupposes the condition

I~-~[<~[~)

9 Using the solution to the problem of elastic equilibrium of a hollow

sphere under the influence of external and internal pressures [3] and replacing the internal pressure in it by a radial "supporting" stress o~ of reverse sign, we obtain the condition of halting of the crushing zone boundary: o,-

o;

o.Ts (o, + P) =

I--(R*/R2)8

= [$1,

(2)

where R2 is the internal radius of the specimen, R* is the radius of the crushing zone after its boundary has halted, and P is external pressure at which crushing began. The fact that we use the solution to the problem of equilibrium, not motion, signifies that we examine the moment at which wave processes ceased and the crushing zone boundary halted. From Eq.

(2) we obtain the value of the "supporting" 1~*

stress

31

(3)

However, this condition is inadequate. It is necessary that under the influence of the compressive stress o~ the inner layer does not fracture; we denote its new strength by [TI]. Assuming that as before the broken and eventually compacted m e d i u m behaves elastically (with new moduli of elasticity, of no significance here), by replacing o~ in Eq. (2) by O, P by oi, and [~] by [TI], we obtain the strength of the crushed medium

I --

(RI/R*)3

'

(4)

where RI is the cavity radius, which we assume to be unchanged. Similarly, by regarding the fractured m e d i u m as ideally plastic with yield condition

Io,--o=i=-2~,, (~.>0),

(5)

using the stress distribution over the radius at plastic equilibrium of the hollow sphere (see, e.g., [4]), we can determine the shear strength (T s = 0.5Os, where o s is the yield stress in the accepted sense):

"~

--

4 |n

R*/R~

(6)

Calculations of [T~] and ~s for elastic and plastic schemes were performed for three experiments with hollow rosin spheres with internal radius RI = 6 mm and external radius R2 = 30 mm. As the crushing zone radius we took the radius of a circle with an area equal to that bounded by the visible contour of the crushing zone (allowing for the change in size of the recording system). The initial data and the results of the calculations are given in Table i. The values of the fractured medium strength required for equilibrium are higher when the crushing zone radius and the external pressure are higher, but they are much lower than the initial strength. In general as R* approaches R2, from Eqs. (3) and (4) it follows that o~ § --P, and [TI] § [~]. For us, however, the m a x i m u m radius at w h i c h the crushing zone is halted is 15.2 mm. Clearly, fracture then takes place as a result of loss of stability due

249

TABLE 1

Expt. No. 1

I p, kgf/ [ I cm2 P*, mm 550 650 410 490 565

400 550 610

9.9 1 ]

[~1 333

222

9 9,2

310

8,3

302

15,2

I ] [

9,4 11,|

-(r*,

[x,]

~,

1 lg '5

3 61 66

181

98,5 205

164

228

lg 203

5

52 55

4,6 89

92

to asymmetry of the crushing zone. This explains why the final fracture of the specimen reduces to breakage into relatively large transparent fragments. In conclusion, we may r,~ntion that continuation of these experiments should apparently be concerned with the kinematics of fracture and the variation of the geometry of the specimens and methods of load application. LITERATURE CITED i.

2. 3. 4.

V. M. Tsvetkov, I. A. Sizov, and A. D. Polikarpov, "Behavior of a brittle-fractured medium during camouflet blasting," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 4 (1977). V. M. Tsvetkov, I. A. Sizov, and N. M. Syrnikov, "Fracture mechanism of a brittle medium during camouflet blasting," Fiz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 6 (1977). S. P. Timoshenko and J. Guider, Jr., Elasticity Theory [in Russian], Moscow (1975). L. M. Kachanov, Fundamentals of Plasticity Theory [in Russian], Moscow (1969).

DURABILITY OF ROCKS UNDER COMPRESSION Yu. A. Veksler

According to the thermofluctuationconceptof the strength of solids [i], the time to failure under uniaxial elongation by stresses o is determined from the equation

"r='~o exp u(o)/kO,

(1)

where u(o) is the energy of activation fracture, To is a constant, k is Boltzmann's constant, and 0 is the absolute temperature. Insufficient work has been done on the durability of solids under the conditions of a complex stressed state [2, 3]. In estimates of the strength of underground structures, we have to deal with a more complex stress distribution. To use Eq. (i) under these conditions we need to know how each component of the stress tensor or their combination determines the durability. The durability of a body can be calculated from the limiting deformation; this requires the use of the corresponding solution of the problem of creep theory. i. Investigations were performed on sandstone and siltstone specimens consisting mainly of irregular quartz grains bound together by carbonate clay cement. A very small amount of feldspar grains and mica flakes was also present in the specimens. Solid cylinders 32 mm in diameter and 42 ram in height were subjected to axial compression. During the tests we measured the axial deformation and the stress, which was kept constant. Tubular specimens Karaganda Polytechnic Institute. Translated from Fiziko-Tekhnicheskie Problemy Razrabotki Poleznykh Iskopaemykh, No. 3, pp. 71-76, May-June, 1979. Original article submitted March 14, 1978.

250

0038-5581/79/1503-0250507.50

9 1980 Plenum Publishing Corporation