Int. J. Rock Mech. Min. Sci. Vol. 9, pp. 757-766. Pergainon Press 1972. Printed in Great Britain

FRACTURE-SURFACE ENERGY OF ROCKS

M. FRIEDMAN, J. HANDIN and G. ALANI

Center for Tectonophysics, Texas A&M University, College Station, Texas

(Received i 8 December 1971 )

Abstract--Effective fracture-surface energies 7~n, as determined from stable tensile-fracture tests on a variety of rock types, are of the order of 104 ergs/cm 2 in contrast to values for con- stituent crystals which are 102 103 ergs/cm 2. The rnajor source of discrepancy has been ascribed to improperly accounting for the entire new surface created by the propagation of fractures through granular media. One aspect of our continuing investigation of residual elastic strain in rocks has been the assessment of the contribution of such strain to the process of fracturing. Although the work is incomplete, it has confirmed this reason for these order-of-magnitude differences in single-crystal and rock values. From X-ray diffraction studies of changes in residual strain adjacent to tensile fractures in sandstone and also from thin-section studies of stained specimens, we conclude that intergranular displacements take place in a region as much as 5 mm wide on each side of the fiactures. When the total surface area of the grains within this region is equated to the total effective area of fracture created, 7<.n is reduced to the same order as that for single crystals.

INTRODUCTION

PUBLISHED values f'or the effective fracture-surface energies, 7o~-, determined f rom stable tensi le-fracture tests on granite, marble, l imestone, sandstone and quartzi te are one to two orders of magni tude larger than similarly determined values for the const i tuent single crystals of these rocks (Table 1). This result has been suspect because in most rocks fracture p ropaga tes both across grains and along grain boundar ies , and the surface energies of grain boundar ies are supposed to be less than those of the intact grains. That is the bounda ry is a region of compromise structure between two or three neighbor ing grains that is an a l ready strained region. I t has been suggested that the large values associated with rocks are merely appa ren t because much energy is " abso rbed by various inelastic and dissipative processes occurr ing near the extending edge of the crack" ([1], p. 311). We doub t that much inelastic de format ion occurs in very bri t t le quartzose sandstones and granite. More- over, s imilar dissipative processes such as sonic and thermal losses p robab ly occur also in the single crystals.

In general 7'e~ is calculated f rom the external energy U needed to p ropaga t e a stable tensile fracture through the rock and from the area of the newly created fracture surface as follows :

U ~-- 2A~,e~ ( 1 )

where A is the area of fracture surface in half of the symmetr ica l ly broken specimen [2]. The effective fracture-surface energy has also been called " the apparen t surface energy for rocks" [1], and " the specific surface energy for single crysta ls" [3].

The area under the fo rce-d i sp lacement record for stable fracture (Fig. 1) is an accurate measure o f U, p rov ided that any energy losses in the form of heat and sound are small. The

757

758 M. FRIEDMAN, J. HANDIN AND C. ALANI

largest source of error is in fact in the measurement of the true area of the fracture. surface.

This area has been taken by different workers [I, 4. 51 as (a) the nominal area ;L:, if it were perfectly plane; (b) the nominal surface corrected for visible side cracking (61 and (c) the nominal area corrected by an optically determined factor to account for topography (see below). At best these provide an area some IO-100 times too small to yield realistic values

of ye,, (Tables I and 2).

Here we show that our calculations of yCR from the stable fracture of bent, notched beams is somewhat smaller than those published previously for the same rocks. However,

our values too are still one or two orders of magnitude larger than those for the pertinent single crystals. Through X-ray studies of residual elastic strain before and after passage of a

TABLE 1. FRACTURE-SURFACE ENERGIES FOR SELECTED MAT~RIAIS*

Material

Effective fracture- surface energy

(IO4 ergs cm-‘) Source

Data from notched beams Firebrick

Chamotte High-alumina Basic Silica

Plate glass

Chehnsford granite Danby marble Indiana (Salem) limestone

Data from Gilman cleavage technique Calcite (1011) Calcite (1011) Quartz (1011) Quartz (1011) Quartz (1010) Carthage limestone Lueders limestone Indiana limestone Arizona sandstone (Coconino?) Lueders limestone Carthage limestone Tennessee sandstone

3.0 4.9 4.2 3.0 0.34 0.54 0.53 0.50 0.55 0.52 5.0 * 0.5t 5.0 :‘- 1.0t 2.0 11.0:

0,023 0.035 0.041 0.050 0.103 3.8 1 .9 4.2

12.7 1 71, 3.5!1 8.811

NAKAYAMA [2] ., . . ,* ,. ,. *. . . ,,

FOR&AN-RNI and MOAVENZADEH [8] FOROOTAN-RAD and MOAVENZADEH 181 HOAGLAND et al. (41

GILMAN [7] SANTHANAN and GUPTA [IS] BRACE and WALSH (315

3. ,9

&Kttis and B(;RTLETT [5]

*. 9, ,, I,

PER~NS and KRE&~ I1 I

* MOAVENZADEH er al. [6] and FOR~CYTAN-RAD and MOA~EN~ADEH [8] have amply demonstrated that yd is influenced by heating and exposure to certain sutfactants. Values listed here are for untreated &‘mens fractured at room temperature and atmospheric pressure.

t Range of values at 50 per cent notch depth. : Range of values for ditrcren: fracture dir-ions relative to b&iing, Indiana limestone. Data for c&it+-

cemented Bcrea sandstone same as for the limestone. 5 Bruce and WALSH ([3], Table 2) s~mmarizc measured and calculated values of yes for sin& crystab of

eighteen different minerals. II These average values are for atmospheric pressure. Authors also present yd data to 200 bars -kg

pressure where y.,, is 21.9. 15.8 and 20.1 ( x l(r ergs/cm’) for Luedm limestone, mhW& k’KStme. and Tennessee sandstone, rtspectively.

FRACTURE-SURFACE ENERGY OF ROCKS

TABLE 2. DATA FROM NOTCHED-BEAM STABLE FRACTURE EXPERIMENTS

759

Nominal Corrected* U Ten Specimen no. area (2A) area (2A) ( x 10 4 ergs) ( × 10 '~ ergs/cm 2)

kueders limestone: LI0 4.1 5-5 6'7 1.2 Ell 4.1 6-8 6"2 0.9 El6 3.5 5 0 5.9 l .2 [,17 3.5 5.1 6.2 1.2 L12 3.4 4-8 4.3 0.9

Indiana limestone: W-5 4'5 7-4 13'5 l-8 W 6 4.6 7-2 11"6 1.6 W 8 3.4 5.1 8"4 1.7 W 7 3.3 5.2 11.1 2.2 W 9 2.7 4.2 6.5 1.6

Coconino sandstone : 1 5? 5 0 6-6 14.4 2-2 ( 0 . 2 2 ) + + II 10 5-3 7.0 18.1 2 6 (0-39) Ill 4 4-8 6-4 16-1 2.5 (0-23) IV 10 5-2 6.9 180 2.6 (035t

Tennessee sandstone: 1 2.6 4.7 17-9 3 8

Chilhowee quartzite: A 6§ 3-1 3.7 21-2 5 . 7 0 . 1 ) , + B 6 3-5 4-2 26-0 6.2 (0.7) C 3 2 8 3.4 10-5 3 l (1.2) D 5 3'4 4 0 196 4 9 (0-6)

* Fracture-surface areas underlined are corrected by optically determined ratios of actual vs nominal lengths established for each specimen. Areas not underlined are corrected by average ratios from five specimens.

t Data are averages from 5, 10, 4 and 10 separate beams for four orientations in the bedding plane: 1, II, III and IV respectively. No anisotropy with respect to Ten exists in the bedding ([10], 13th Quarterly Technical Report, Table 5).

++ Values in parentheses are standard deviations in corresponding average 7e~. § Data are averages from 6, 6, 3 and 5 separate beams for four orientations in the bedding plane:

A, B, C and D respectively, y~rr is significantly less for induced fractures of orientation C, which is parallel to the orientation of long grain boundaries.

nearby tensile f rac ture , we reason tha t gra in b o u n d a r y d i sp l acemen t s mus t occur wi thin a

reg ion at least 4 . 5 m m wide on each side o f the f rac ture . This o b s e r v a t i o n is con f i rmed by

studies o f th in sect ions cut across the ma in tensi le f rac tu re in quar tz i t e spec imens s ta ined

with si lver ni trate. W h e n the to ta l surface a rea o f the gra ins wi th in this d i s tu rbed reg ion is

equa ted to 2A [equa t ion (1)], Yerr is r educed to the o rde r o f 10 2 e rgs / cm 2, i.e. to that

m e a s u r e d for single crystals (Table 1).

PREVIOUS WORK

Prev ious d e t e r m i n a t i o n s o f ;~e~ in a var ie ty o f rocks and minera l s have focused on (a)

expe r imen t a l des ign to ach ieve s table f rac ture [1, 2, 4 -7 ] ; (b) effects o f spec imen size and

p r epa ra t i on , inc lud ing o r i en t a t i on re la t ive to fabr ic and t h e r m a l and chemica l t r ea tmen t s

760 M. FRIEDMAN, J. HANDIN AND G. ALANI

[2, 6, 8, 9] and (c) the influence of deformation rate and of confining pressure [1 ]. In Table 1 are listed representative values for a variety of rocks and for quartz and calcite single crystals. Although this list is not meant to be complete, it does show the disparity between values of 7oft for rocks and single crystals.

METHODS

Stable fracture

The energy considerations involved in the propagation of a stable tensile fracture from a notched or partially fractured specimen and the distinction between catastrophic, semi- stable and stable fracture (Fig. 1) are given by NAKAYAMA [2] and by MOAVENZADEH et al.

< o

/ / DEFLECTION

for FIG. 1. Typical load-deflection curves notched beams showing unstable or catastrophic fracture (A), semi-stable fracture (B) and stable fracture (C) [6].

BLOCK \ \ \ \ \ \

~°i~, ~-4- -~. -1 ._1

Fro. 2. Schematic diagram of the paralle|-beam loading device.

F R A C F U R E - S U R F A C E E N E R G Y OF R O C K S 761

[6]. To achieve stable fracturing one must be able to propagate the fracture at will, and this requires a testing machine of very high stiffness. One can effectively stiffen the machine in several ways. The simplest and cheapest method suitable to our application is the parallel loading of a thick steel beam and the notched rock beam to be tested (Fig. 2).

In our device the force is applied by a 100-ton hydraulic press through the thrust block to the four-point loaded steel beam, the dimensions of which are so chosen that the force required to deflect it is about 50 times larger than the force needed to deflect the rock beam a similar amount. The force on the three-point loaded, notched, rock beam is measured by tile gauge located between the two beams. The mid-point deflection is determined by a transducer that measures the relative displacement between the force-gauge rod and one of the end rods supporting the rock test-beam. For our purposes the error of this measurement relative to the true deflection of the top surface of the beam is negligible, and the accuracies of both force and deflection measurements are better than l per cent. A displacement rate of 4 10 -4 cm/sec is used in our experiments.

Guided by the work of others, we have held our specimen size and type of notch constant. Our specimens are 1.4 cm thick, 2.5 cm wide and are long enough to be supported and loaded in the parallel-beam device along two outside line loads spaced 10.2 cm apart (see Fig. 2). The notches are produced by a saw cut (0.9-ram blade) that yields a width of I • 5 mm and a U-shaped profile. We have evaluated the effect of notch depth in beams of Lueders and Indiana limestone and Coconino sandstone. We find, as did FOROOTAN-RAD and MOAVENZADEU [8], that both U and yeJ, [equation (l)] decrease markedly with increasing notch depth until the depth exceeds about 35 per cent of the beam thickness and then remain essentially constant for greater depths. Moreover, fracture at shallow notch depths is catastrophic, becomes semi-stable with increasing depth and is stable only when the depth exceeds about 25 per cent of the thickness [10].* Our values for yen" (Table 2) are for notch depths of between 39 and 65 per cent. Areas under the force-deflection curves (Fig. 3) are measured with a polar-planimeter.

25

20 t

~15

%

ao~o J

I I I I I I I

---____q I [ I l i t 5 I0 15 210 25 30 2,5 DEFLECTION (x 10-3cm) 4 0

FIG. 3. Bending load vs deflection for tensile fracture of Tennessee sandstone. Shape of curve is typical of stable fracture. Area under curve is external energy required to fracture the rock.

* 13th Quarter ly Technical Report , pp. 2-220.

762 M. F R I E D M A N J. H A N D I N , A N D G. ALANI

X-ray measurements

The residual strains locked in the quartz grains of sandstones are a source of potential energy. That these strains are elastic, that is they can relax virtually instantaneously, has been demonstrated earlier by X-ray diffraction studies of d spacings before and after the diffracting grains are freed from their nearest-neighbor constraints by etching with HF (FRIEDMAN [1 l ]). The facts that the strains are potentially recoverable and that they appear to be a factor in controlling the orientations of both tensile and shear fractures [12] lead to the proposition that the stored strains may affect the energy balance at the tip of a fracture, and hence the propagat ion of the fracture itself. In order to evaluate this idea, it is necessary to determine if the residual strains are in fact relaxed by inducing a fracture through the rock, and if so, to measure the volume of material involved in the relaxation in order to establish quantitatively how much of the available stored elastic energy is released.

The experimental data consist of X-ray measurements of the d spacing for ~3254) in quartz before and after fracture for a total of 20 areas of irradiation distributed over three notched bars of Elliot Lake quartzite and of Tennessee sandstone. The procedure is as follows:

1. Thin bars of rock ranging in size f r o m 5 × 25 x 3 8 m m t o 8 , 30 2 4 0 m m a r e cut from the starting blocks. The top surface is polished with 600-mesh grit in preparation for X-ray study.

2. The bottom surface is then notched with a fine-diamond saw blade in order to control the propagation of a tensile fracture perpendicular to the top surface as in step 4 below.

3. The strain before fracture is measured in a number of 3 x 9-ram areas o f irradiation at known distances from the expected trace of the induced fracture (Table 3, columns l-3).

TABLE 3. X-RAY STRAIN RELAXATION DATA

1 2 3 4 5 6 7 8 9 10 Rock Area of Distance * Stored Std dev. Av. peak Stored Std dev. A v peak Change and irradiation to fracture strain ( • 10 - 6 ) intensity strain ( × 10 -6) intensity in

specimen (ram) before Nt before after /N÷ after strain fracture fracture fracture fracture ( × 10-~,

I ~ 10-°1 (cps) ( 10 -6 ) ~cns~

Elliot Lake quartzite [

3

Tennessee sandstone I

A 0 '5 80~ 10/5 15"9 100 20/3 18.5 20§ B 2-0 80 0/5 14"5 100 17/2 14.4 20§

C 2.5 80 12/3 15.8 0 I8/4 15.5 80 D 1-3 --100~ 20/12 t5 .2 --10 2018 14.3 90 E 3"0 180 2218 21.5 120 12/4 16"2 60

F 3.5 --130 32/30 31.4 --90 28110 30.4 40

A 1"5 80 15/6 31.3 20 16/5 27.7 60] B 1.5 80 26/6 30.4 10 1015 24"2 70 C 1"0 100 20/5 21.9 70 10/5 2 l -4 30 D 1 '0 130 1215 26.8 50 20/5 25.t 80

A 11.0 100 32/7 21-1 60 1315 22.9 40 B 5.0 200 20/6 19"6 200 20/5 21-6 0 C 10'0 50 20/6 26-8 10 1015 32-~ 40 D 3.0 360 1516 30'3 290 6/5 38-0 70]] E 1.0 220 1416 17.6 140 2415 14.6 80H F 0.5 10 38/6 20.2 10 7/5 25-2 0

G 53"5 120 0/4 18.4 120 8/3 17-8 0 FI 14.5 --20 34•7 61.1 --20 2018 44-1 0 I 4.5 60 2616 24-1 0 8/5 22.8 601~ K 27.5 100 1616 18.5 100 20•5 25.8 0

i Distance to fracture sur£ac¢ is measured from the mi'd.p.oint of ~ 3 m m side o f the 3 x 9 mm area o f irradiation. N is the number o f r c p ~ ~ t s of f /~ ti l t ' faction .profile i n ¢~ch area. Expansions are calculated as pos~ive and compressive strains as ne~t ive .

d~In two cases the strains increased in magnitude after fracture; however~ the increase is small compared to the standard eviations,and therefore suspect.

11 Changes in strain that are large relative to standard deviations; mean of eight values is 75 x 10 -6.

FRACTURE-SURFACE ENERGY OF ROCKS 763

Relative to the top surface of the bar, the diffracting (hkil) planes strike parallel to the strike of the fracture and dip at 45 ° on both sides of the fracture. The normal strain components, en, across these (hkil) planes, therefore, are oriented at 45 ° to the top surface of the bar and they lie ill the plane containing the perpendiculars to the top surface and to the fracture surface. The X-ray measurements are repeated from 3 to 30 times for each area; the slab is removed from the sample holder and then replaced between each measurement of the diffracting peak (Table 3, column 5). The peak 20 value and the intensity of the peak height are recorded for each step scan of the diffraction profile. Strains are calculated from the d spacing as follows:

d(observed) -- d(strain free) e, = (2)

dstrain free

The average strain value from the reproducibility tests is then assigned to the given area of irradiation along with the corresponding standard deviation and peak intensity (Table 3. columns 4, 5 and 6, respectively).

4. A tensile fracture is then propagated through the bar by placing the notched side of the beam in tension.

5. The known irradiated areas on each bar are then restudied after fracture to determine the nature of the d spacing (strain) change.

Measurement qf fracture area

The nominal area of the fracture surface induced with the parallel-beam device is simply the product of the mean lengths of long and short sides. The true area is always larger simply because the surface is irregular, that is the fracture has propagated around as well as through grains. This is shown convincingly in the scanning and electron photomicrographs published by HOAGLAND et al. [4]. In order to obtain a more accurate measure of the true area, we measured lengths along the irregular surfaces as viewed on edge at 100 x magnifica- tion with the aid of a calibrated eyepiece-micrometer. Surface irregularities down to less than 0.01 mm were recorded. The cumulative lengths along the irregular surfaces are div- ided by the nominal lengths to obtain a ratio by which the nominal area is corrected[10]*. Corrected values range from 120 to 170 per cent of the nominal areas (Table 2).

Obsert:ations of side fractures

Microscopic study of our unstained specimens in thin sections or polished surfaces cut perpendicular to the tensile fractures reveals a negligible amount of side fracturing adjacent to the major breaks in both sandstone and limestone. Side fractures are much more con- spicuous in the specimens of MOAVENZADEH et al. [6]. When the specimens are stained with silver nitrate solution, exposed to a strong light source, and then thin-sectioned [13] micro- fractures primarily along grain boundaries become conspicuous (Fig. 4).

E X P E R I M E N T A L R E S U L T S

Fracture-surface energy

Data from our notched-beam tests on Lueders and Indiana limestone, Coconino and Tennessee sandstone and Chilhowee quartzite are listed in Table 2. Use of fracture-surface

* 13th Quarterly Technical Report, p. 6, Table 2.

764 M. FRIEDMAN, J. HANDIN AND G. ALANI

areas corrected optically for topography yield values for ~'¢~r that range from 0.9 to 6.2 x 104 ergs/cm 2. These are somewhat smaller than corresponding values determined by others who have used nominal areas, but are still much larger than values for calcite and quartz by a factor of 100 (compare Tables 1 and 2).

Relaxation of residual strain

Data from X-ray studies of the relaxation of residual elastic strains caused by tensile fracturing of Elliot Lake quartzite and Tennessee sandstone are listed in Table 3. These are significantly different types of rock. The former is poorly sorted and it contains about 40 per cent quartzite rock fragments, 20 per cent feldspar grains, 10 per cent quartz grains and 30 per cent very fine-grained quartz--epidote cement. Grain-contact types range from point to sutured surface. Tensile fractures in this rock cut across grains and also go around them; relief on the fracture surface averages about one framework grain diameter about (1.0 mm). The Tennessee sandstone is moderately well sorted and bimodal with layers of larger grains (about 0.3 mm) that alternate with layers of finer ones (about 0-1 mm). Quartz grains and silica overgrowths (cement) make up more than 90 per cent of the rock. Grain contacts are mostly long and concavo-convex; no sutured contacts are observed. Fractures in this rock go primarily around the grains; and relief on the fracture surface is as much as 1.0 mm.

Table 3 shows the following facts:

(a) Close agreement in peak heights of diffraction profiles before and after fracture (columns 6 and 9) indicates that essentially the same set of diffracting grains are being measured in each area before and after fracture. The small changes in peak height are not systematic.

(b) Strains relax from initial states that are both expansions and compressions. In only two of the twenty areas do the strains tend to increase, and even this is doubtful statistically (see Table 3, footnote §).

(c) Relaxations occur in both the quartzite and the sandstone. Tiffs suggests that relaxation is independent of whether the fracture cuts through grains or goes around them.

(d) The magnitude of the relaxation in nine of the areas (column 10, footnote IL) are large compared to the standard deviation (columns 5 and 8). For these areas the average change in strain is 70 x 10 -6.

(e) Relaxation to within 4-20 x 10 -6 of zero occurs in 6 cases (column 7, Elliot Lake areas 2C and 2D, Tennessee sandstone areas 1A, 1B, 5C, and 7J).

(f) There is no systematic relation between the degree of relaxation (column 10) and distance from fracture or magnitude of initialstraln (columns 3 and 4, respectively). For example, zero strain change is recorded at distances of 0-5, 5.0, 14.5, 27.5 and 53.5 ram. Furthermore, several areas of high strain show no relaxation (e.g. Tennessee 5B, 7G and 7K). These facts suggest the strain release is hetero- geneous and probably dependent on the nature of local grain-to-grain constraints.

(g) Statistically significant relaxations (Table 3, column 10, footnote It) occur to a distance at least 4-5 mm from the induced fracture (Tennessee 7.1). Zero or only minor relaxations (of doubtful statistical significance) are found for the six areas located between 5.0 and 53.5 m m away.

(a)

(b) (c)

Fi(;. 4. Photomicrographs of stained fractures and grain boundaries in thin section cut parallel to foliation and perpendicular to the induced, stable, tensile fracture in Chilhowee quartzite.

(a) Stained grain boundaries and microfractures along grain boundaries (arrowed) in experimentally un- deformed rock. Grain elongation trends E-W. (b) Area immediately adjacent to the reduced macrofracturc. notched-beam test. Grain boundaries are more conspicuous than in (a) indicating grain boundar> dis- turbance near macrofracture. (c) Detail of grain boundaries adjacent to macrofracture (arrowed). Specimen,

stained with silver nitrate solution (see text). Scale lines for each photomicrograph are 0. I ram.

RM f.p. 764]

FRACTURE-SURFACE ENERGY OF ROCKS 765

DISCUSSION AND CONCLUSION

Even though we improved the technique for measuring the true area of the fracture surface, our data and that of other workers as well (Table 1) indicate that ~'e~ for rocks is of the order of 10* ergs/cm 2, whereas values for single crystals of quartz, calcite and feldspar lie between 10 2 and l0 3 ergs/cm 2 (Table 1). The fact that a rock has a larger apparent 7e~ than that of its constituent single crystals has been suspect because in the rock fracturing occurs both across and around grains. The surface energy of the grain-boundary region is less than that within the grain because intergranular boundaries, if chemically bonded, are regions of compromise structure and hence of strained bonds. However, this effect would tend to decrease, not to increase the surface-energy value for the aggregate.

We argue from X-ray data that small intergranular movements along grain boundaries must take place within narrow regions adjacent to tensile fractures in sandstone. When the total surface area of the affected grains within this region is equated to 2A [equation (1)], ~e~ is reduced to 2 /,, l02 ergs/cm 2 for sandstone.

We demonstrate that residual elastic strains stored in the quartz grains of sandstones relax all or in part within a region 4.5 mm wide on either side of an induced tensile fracture. This relaxation is not solely a function of distance from the newly made free surface. Recall that the strain data are obtained by recording the d spacing for (3234) in suitably oriented quartz grains within definite areas of irradiation before and after the fracture is propagated through the specimen. The diffracting atomic planes are inclined at 45 ° to the polished top surface of the sample, and they also lie at 45 ° to the induced fracture surface. It follows that proximity to the new free surface cannot of itself relax the strains, because previous to fracture the stored strains are in fact recorded within only 0.05 mm of the polished surface. This distance is determined from the absorption coefficient of SiO2 and the angle of incidence of the X-ray beam. Indeed, most of the diffracted energy comes from the first 0.02 ram. Since stored elastic strains exist at these very shallow depths, and since they are relaxed all or in part after the specimen is fractured, some other factor must be also significant. Recalling that the strains do relax when the boundaries of quartz grains are etched with HF [l 1], but that photoelastic model studies show remarkably little dis- turbance of prestrain by a newly formed free surface [14], we propose that small move- ments along grain boundaries occur within a narrow zone of an induced fracture. Inter- granular bonds are broken, and loss of constraint permits the relaxation of the stored strains in the grains within that zone. In other words fracturing has occurred in a zone of finite width and not just on a single surface as is usually supposed.

If the foregoing argument is valid, then the true area to be associated with equation (J) approaches the surface area of all the sand grains in the affected zone. This idea has been tested by a calculation for idealized spheres 0- I mm in diameter within the volume of rock defined by the specimen size and by the ±4 .5- ram wide zone normal to the fracture surface. For Coconino sandstone this volume is reduced by the porosity, 16 per cent. When this idealized grain-boundary area is used along with the measured external energy, U, the mean value of ~eff becomes 2 × I0 2 ergs/cm 2. This agrees favorably with values for single quartz crystals.

This argument is supported by thin-section studies of beams of Chilhowee quartzite stained with silver nitrate. Microfractures along grain boundaries are conspicuous in the regions adjacent to the main tensile fracture (Fig. 4). Further support comes from the l:act that regions adjacent to the major fractures have increased permeabilities as indicated by differential penetrations of epoxy resin ([4], Fig. 9).

ROCK 9/6--G

766 M. FRIEDMAN, J. HANDIN AND G. ALAN1

From these observations on quartzose rocks and by inferences for limestone it is reasonable to conclude that the true fracture surface energy of a well-indurated rock probably does not after all exceed that of its constituent single crystals. In very loosely consolidated aggregates where fracturing occurs entirely along grain boundaries, yea should be even less than that for the individual grains.

Acknowledgments-We are indebted to Dr M. S. Paterson (Australian National University, Canberra) for his helpful suggestions made during a recent visit to our laboratories.

This work was supported by the Advanced Research Projects Agency, Department of Defense (Contract DACA 73-68-C-ooo4) and was monitored by the Missouri River Division, U.S. Army Corps of Engineers.

1.

2.

3.

4.

5.

PERKINS T. K. and KRECH W. W. Effect of cleavage rate and stress level on apparent surface energies of rocks. Sot. Petrol. Enars J. 6. (4) 308-314 (1966).

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NAKAYAMA J. Direct measure~ment of fracture energies of brittle heterogeneous materials. J. Am. Cerum. Sot. 48, 583-587 (1965). BRACE W. F. and WALPH J. B. Some direct measurements of the surface energy of quartz and orthoclase. Am. Miner. 47, 111 l-l 122 (1962). HOAGLAND R. G., HALM G. T. and ROSENFIEW A. R. Injluence of Microstructure on Fracture Propaga- tion in Rocks, Semiannual Research Report, Battelle Columbus Laboratories, ARPA Contract No. HO210006 (1971). PERKINS T. K. and BARTLE~ L. E. Surface energies of rocks measured during cleavage. Sot. Pefrol Engrs J. 3, (4) 307 (1963).

6. MOAV~N~AD~H F., WILLIAMSON R. B. and W1.s.s~ A. E. Z. Rock Fracture Research, M.I.T., Department of Civil Engineering Research Report, R66-56. p. 85 (1966).

7. GILMAN J. J. Direct measurement of surface energies of crystals. J. appl. Phys. 31,2208-2218 (1960). 8. FOR~~TAN-RAD P. and MOAV~NWDEH F. Crack Initiation and Propagation in Rock, MIT. Department

of Civil Engineering Report, R68-29, p. 126 (1968). 9. SUMMERY D. A.. COR~~NE J. and CHEN LI-KING. A Comnarison of Methods Available for the Determina-

tion of Surface’Energy, Proceedings of the Twelfth Sym&xivm on Rock Mechanics (1970). 10. HANDIN J. Studies in Rock Fracture, Task I, U.S. Army Corps of Engineers, Contract No. DACA

73-68-C-0004, Quarterly Technical Reports, Nos 1-15 (1967-1971). 11. FRIEDMAN M. X-ray Analysis of Residual Elastic Strain in Quartzose Rocks, in Basic and Applied

Rock Mechanics (K. E. Grey, Ed.). pp. 573-595, Proceedings of the Tenth Symposium on Rock Mech- anics, Am. Inst. Min. Metall. Petrol. Engrs (1972).

12. FRIEDMAN M. and LOGAN J. M. The influence of residual elastic strain on the orientation of experi- mental fractures in three quartzose sandstones. J. geophys. Res. 75, 387405 (1970).

13. BALDRIDGE S. Private communication (1971). 14. GALLAGHER J. J. Phatomechanical Model Studies Relating to Fracture and Residual Elastic Strain in

Granuhr Aggregates, Ph.D. Dissertation, Texas A&M University (1971). IS. SANTHANAN A. T. and GUAVA Y. P. Cleavage surface energy of calcite. Int. J. Rock Mech. Min. Sci. 5,

253-259 (1968).