Rock Mech. Rock Engng. (1998) 31 (4), 237±250Rock Mechanicsand Rock Engineering

: Springer-Verlag 1998Printed in Austria

Microscopic Fracture Processes in a Granite

By

E. Z. Lajtai

Department of Civil and Geological Engineering, The University of Manitoba, Winnipeg,Canada

Summary

The deformation of a competent, brittle, granitic rock is thought to have two main com-ponents: elastic and brittle deformation, the latter caused by axial microcracking. Dynamicfatigue testing of Lac du Bonnet granite would, however, suggest the presence of a thirdmechanism, compaction. Compaction is not the same as elastic crack closure; compactionentails permanent damage along grain boundaries that are under high compression. Duringcompaction, the axial sti¨ness (elastic modulus) of the rock increases and the permanentcrack volume becomes negative (compression). Compaction is active at all stress levels, butit is most noticeable at low stress where its presence is not masked by dilation caused byaxial microcracking.

1. Introduction

The important role played by pre-existing microcracks (¯aws) in nucleating axialmicrocracks has been well recognized since Gri½th had formulated his theory onbrittle fracture (Gri½th, 1924). For rock fracture, a theory based on material ¯awsis particularly attractive since rocks carry discontinuities of many types and size.Suitably oriented microcracks, or Gri½th ¯aws, act as stress concentrators, whichgenerate both tensile and compressive stress concentrations. The tensile stressconcentrations are thought to be the source of the axial tensile cracks appearing inthe uniaxial, or even the triaxial compression tests of brittle rocks. The potentialrole that the compressive stress concentrations (that can be orders of magnitudegreater than the tensile stress concentrations) play, has largely been ignored(Lajtai, 1971).

According to present concepts, the pre-failure deformation of brittle rocks hastwo major components: elastic deformation and deformation produced throughaxial microcracking. There are also number of less signi®cant processes: pre-existing crack closure, internal crack sliding and apparent plasticity. Crack closureis a low stress, elastic phenomenon, and apparent plasticity, usually due to thebuckling of rock slabs bounded by axial fractures, appears only at high stress,

close to failure. An additional mechanism, internal crack sliding, has been sug-gested on theoretical grounds (Walsh, 1965), but physical evidence for pre-failureshear displacement in rock is scarce.

The beginning and progress of a particular deformational mechanism ismirrored by the shape of the stress-strain diagram (Bieniawski, 1967), throughchanges in the position of stress markers and in the slope of stress-strain curves(Fig. 1). In addition to elastic deformation, the nonlinear, low stress part of theaxial stress-strain curve displays pre-existing crack closure (cc), and just beforefailure apparent plasticity (y) may appear. The axial strain curve should not bein¯uenced by the axial microcracking, because axial sti¨ness is una¨ected by thepresence of stress free axial discontinuities. Between the crack closure and the yieldstress, the axial stress-strain curve behaves elastically and its slope (the axial sti¨-ness) should de®ne the elastic modulus (E ). The lateral strain curve is normallyuna¨ected by crack closure, but it is strongly in¯uenced by the dilation of the axialmicrocracks. Elastic conditions prevail at low stress, before the crack initiationpoint (ci). In the elastic zone, the slope of the lateral strain curve de®nes the lateralsti¨ness. The ratio of the elastic modulus to the lateral sti¨ness yields the Poisson'sratio (v). From the axial strain (ea). and the lateral strain (e1) curves one may de-rive two additional relationships: the volumetric strain (D) and the crack volu-metric strain (ecv). The ®rst is computed using the in®nitesimal strain assumption:

D � 2el � ea:

The crack volumetric strain is the excess of the measured volumetric strain (D)over the elastic volumetric strain:

ecv � Dÿ s�1� 2u�E

:

Fig. 1. Stress strain diagram for Lac du Bonnet granite showing the axial, lateral, volumetric and crackvolumetric strain. The axial curve reveals the crack closure (cc), the yield (y) and the peak stress ( p).The lateral strain curve displays the crack initiation stress (ci). The volumetric strain curve shows thevolumetric strain rate reversal (onset of dilatancy) stress (vrr). The crack volumetric strain has its own

reversal stress (cvrr) marking the point, above which axial cracking dominates crack closure

238 E. Z. Lajtai

For compression, s must be entered negative. The crack volumetric strain for Lacdu Bonnet granite is initially negative (compression), goes through a rate reversalpoint (cvrr), and normally ends with a positive value (dilation). The general shapeof the crack volumetric strain curve re¯ects the interplay between two microscopicprocesses, crack closure and axial crack propagation.

The primary goal for the research described in this paper was to ®nd the e¨ectof axial crack density on the two deformational parameters, the axial and the lat-eral sti¨ness. To do this, the testing of specimens with di¨erent crack densities wasrequired. The dynamic fatigue test o¨ers a convenient and economic way to sys-tematically change the density of axial microcracks. In the dynamic fatigue test,the load is cycled between two limits (Fig. 2). In this case, the lower limit is at zerostress and the upper limit is set at a compressive stress (the fatigue load), which is afraction of the uniaxial compressive strength. If the fatigue load is high enough toinduce axial microcracking, successive loading cycles should keep on increasingthe crack density. Crack density can be de®ned in many ways. In microscopicinvestigations, crack density is usually expressed as the number of cracks crossinga unit length. This quantity cannot be derived from the analysis of the stress-straindiagram. The stress-strain diagram however will yield a similar parameter, thepermanent crack volume. The permanent crack volume (pcv) created during aloading cycle is the di¨erence in the volumetric (or crack volumetric strain) mea-sured at the beginning and the end of the loading cycle (Fig. 3). Since at zero stress,there is no elastic volumetric strain, the volumetric and the crack volumetric straincurves merge. The volume of the cracks created during the dynamic fatigue test isrecorded by a cycle to cycle shift in the hysteresis loops of the volumetric strain.The shift is to the left (compression) at low and to the right (dilation) at high

Fig. 2. Stress-strain diagram of a loading cycle (2nd at 185 MPa) showing the axial, the lateral and thevolumetric strain. During the test, the uniaxial compressive stress varied between zero and the maxi-mum value of 185 MPa. After a few loading cycles, the specimen failed at this load. The axial sti¨ness isthe slope of the axial strain curve taken above the crack closure stress (cc). The lateral sti¨ness, is theslope of the relatively straight section of the lateral strain curve below the crack initiation point (ci)

Microscopic Fracture Processes in a Granite 239

stress. Shifts in the position of stress markers and changes in the sti¨ness parame-ters, caused by microscopic fracture processed during a loading cycle, are re¯ectedin the stress-strain diagram of the following loading cycle. Hence, by analyzing thecycles of a dynamic fatigue test, the trend of stress markers and sti¨ness parame-ters and their relationship to the permanent crack volume can be established.

2. Experiments

The granite rock specimens came from the Cold Spring Quarry near Lac duBonnet, Manitoba. The mechanical properties of Lac du Bonnet granite are typi-cal of many granites (elastic modulus: 70 GPa, Poisson's ratio: 0.21, uniaxial com-pressive strength: 225 MPa; Brazilian tensile strength: 13.5 MPa). Each cylindricaltest specimen, 31 mm in diameter and 64 mm long, was equipped with three axialand three lateral strain gauges, that were attached to the middle third of the spec-imen to avoid end e¨ects (Fig. 4). Loading was between steel platens with a ball-jointed loading head at one end of the testing arrangement. Testing was conductedin an MTS testing frame in the load control mode, producing a loading rate ofabout one MPa per second. The data acquisition system was set to read the load,the elapsed time and the output of the six strain gauges at 2-second intervals.

Being aware of the strong in¯uence water moisture has on rock properties,three air-dried and three wet specimens have been prepared. For the wet tests, thestrain gauge installation was water proofed. The specimens were kept submergedin water for a minimum of one day prior to testing and the drift of the straingauges was recorded for the duration. From the six tests, specimen No. 812 (dry)and specimen No. 801 (wet) produced the most useful data. They both failed in theend, but only after several well-de®ned loading cycles and without prematurestrain-gauge failure. In the other tests, failure occurred while the stress was incre-

Fig. 3. Volumetric and crack volume strain during the last (11th) complete loading cycle. The specimenfailed on the next load application. The start and the end of volumetric or the crack volume strain curveat zero stress de®ne the initial and the ®nal crack volume. The di¨erence is the permanent crack volume( pcv) created during the loading cycle. The wide hysteresis loop is caused by rapid crack growth onapproaching failure. Compare this with the more usual narrower loop of the 2nd cycle shown in Fig. 2

240 E. Z. Lajtai

mented to the next fatigue load. The data coming from these tests were howeversimilar to those for No. 812 and No. 801.

For specimen 801, the lower fatigue load boundary was set at zero; the upperranged between 120 and 185 MPa uniaxial compression (Table 1). For a speci®cfatigue load, the load was cycled for about a working day to produce 40 to 60cycles. Each morning, the maximum load was incremented and the procedure re-peated. The wet specimen No. 801, underwent 312 cycles starting with a fatigueload of 120 and ®nished with failure at 185 MPa (Table 1). In a simple uniaxialcompression test, the peak stress would normally be about 225 MPa.

3. Analysis

The data acquisition system produced about 150 measurements per cycle, su½-cient to produce smooth stress-strain curves. Although, the stress-strain curves for

Fig. 4. Test specimen and the loading condition. The specimen was instrumented with three axial andthree lateral gauges and compressed between hardened-steel loading platens. The compressive load was

transmitted to the upper platen through a ball-jointed cylinder head

Table 1. The test history of specimens No. 812 and No. 801

Dry specimen No. 812 Wet specimen No. 801

Fatigue load (MPa) Number of cycles Fatigue load (MPa) Number of cycles

140 24 120 77160 13 140 54180 40 160 64200 44 170 50220 16 180 56

185 12

Microscopic Fracture Processes in a Granite 241

Lac du Bonnet granite are rarely linear in any stress interval, it is possible tomanually ®t a straight line to both the axial and the lateral strain curves above thecc (crack closure) and below the ci (crack initiation) stress markers, respectively(Fig. 2). The low stress, curving part of the lateral strain curve was ignored inthe ®tting procedure. However, the actual computational procedure to determinethe axial and lateral sti¨ness was more involved. First, the momentary axial andlateral sti¨ness was found by running a moving-point linear regression procedurefor the whole length of the stress-strain curves. From plots of the momentarysti¨ness parameter against the axial stress, through inspection a ``relatively con-stant sti¨ness interval'' was identi®ed (Fig. 5). The axial and the lateral sti¨nessquoted in this paper is the average of the momentary sti¨ness of this stress inter-val. This way, the constant sti¨ness interval for the axial strain may not corre-spond to the constant sti¨ness interval of the lateral strain. Finding a commonstress interval for both was inconvenient, mainly because the common stress in-terval, between the crack closure and the crack initiation stress, narrows at highstress (Fig. 6).

3.1 The Crack Closure and the Crack Initiation Stress

The crack closure stress should mark the point where all the pre-existing cracksclose and the material enters the elastic stage. The axial sti¨ness is measured abovethis stress, along the linear portion of the axial strain curve. Crack closure isthought to be an elastic process and the compressive strain accompanying itshould be recoverable. Most of the crack closure strain is expected to come frompre-existing cracks that are oriented at high angle to the compressive load. Somedegree of closure, however, is possible for all cracks except those that have a load-

Fig. 5. Determining the axial and the lateral sti¨ness from the strain measurements. The momentarysti¨ness curves, shown in the diagram, were found through the running of a moving-point regressionprocedure on the strain data. The lateral strain curve is always non-linear; the axial sti¨ness becomesclose to linear above the crack closure stress. The quoted value of the two sti¨ness parameters is the

average of the momentary sti¨ness between the arrows

242 E. Z. Lajtai

parallel orientation. Because during the compression test, the axial microcrackspropagate parallel to the compressive stress, they should have no e¨ect on theshape of the axial strain curve, and consequently on the crack closure stress.Nevertheless, the crack closure stress did change in the dynamic fatigue tests ofLac du Bonnet granite (Fig. 6).

The lateral sti¨ness is measured on that part of the lateral strain curve whichlies below the crack initiation stress. De®ning the crack initiation point for Lac duBonnet granite has always been problematic. Microstructural studies (Lajtai,1991), have indicated that Lac du Bonnet granite may have as many as three crackinitiation points (Fig. 7). The two low stress ``initiation points'' (20±60 MParange) are responsible for the low stress curvature of the lateral strain curve, andthey are thought to be associated with recoverable dilation along cleavage infeldspars (60% of granite) and possibly in biotite (10% or less). In the granitestructure, the stronger quartz (30%) envelopes both the feldspar and the biotitecrystals. This quartz framework gives the high strength to granites. During uni-axial compression, this framework is ®rst disrupted at a uniaxial stress of about100 MPa. Since this is the stress level at which stress-induced structural changesbecome noticeable in Lac du Bonnet granite, the high-stress crack initiation pointat 100 MP is the true crack initiation point in Lac du Bonnet granite. The trend ofthis point seems to follow that of the crack closure stress, the crack initiationmoves to higher stress as the fatigue load of the dynamic fatigue test is increased.In the high stress series at 170, 180 and 185 MPa, both crack closure and crackinitiation occur at almost the same stress, around 160 MPa.

3.2 The Axial and the Lateral Sti¨ness

In general both the axial and the lateral sti¨ness seem to decrease as the number ofloading cycles and the fatigue load increase (Fig. 8). For the scaling used in the

Fig. 6. Crack initiation and crack closure data for the whole stress history of specimen 801. The maxi-mum stress applied for a particular range of cycles is shown at the top. Both the crack initiation and thecrack closure points move to higher stress and practically merge at high stress. Note the almost in-

stantaneous response to an increase in the fatigue load

Microscopic Fracture Processes in a Granite 243

®gure, the axial sti¨ness seems to change but little until the maximum load israised to 170 MPa. In fact, the axial sti¨ness does display a very de®nite trendbefore this. When using a ®ner scale, the axial sti¨ness can be shown to increase atlow stress from 69 to 71 MPa (Fig. 9). For the lateral sti¨ness, the trend is alwaysnegative. Starting with the 120 MPa series, the lateral sti¨ness clearly decreases,with the rate become greater with increasing fatigue load.

Fig. 7. Non-linear lateral strain curve for Lac du Bonnet granite. When strain gauge locations areselected to report strain from the feldspar and quartz crystals separately, as many as three crack initi-ation points can be identi®ed. The arrows show the location of these. The two low-stress points are re-lated to recoverable dilation, possibly along the cleavage of feldspar and possibly biotite. Permanent

damage occurs when the quartz framework of granite is fractured around 100 MPa

Fig. 8. Cycle to cycle changes in the axial and the lateral sti¨ness for specimen 801. The lateral sti¨nessdecreases right from the beginning, with the rate of drop accelerating with stress. The axial sti¨ness

changes less, but just as de®nitely as shown in Fig. 9

244 E. Z. Lajtai

3.3 Permanent Crack Volume

The di¨erence between the permanent volumetric strain at the beginning and theend of a loading cycle yields the crack volume created during the cycle. When thefatigue load is greater than the crack initiation stress, one would expect the per-manent crack volume to increase, re¯ecting the nucleation of new axial cracks andthe extension of those created previously. If in addition to elasticity, axial micro-cracking was the only strain producing mechanism, the permanent crack volumecould only increase. The actual situation, however, is more complex. In fact, thepermanent crack volume strain remained negative until the fatigue load was raised

Fig. 9. Cycle to cycle changes in the axial sti¨ness for specimen 801. The axial sti¨ness increasesinitially from 69 to 71 GPa. With the rise in the fatigue load to 170 MPa, the sti¨ness starts to dropgradually to 67 MPa. The early rise is attributed to the compaction, probably at grain boundaries

Fig. 10. Cycle to cycle trend of the permanent crack volume for specimen 801. Initially, the permanentcrack volume is negative suggesting compaction rather than dilation at low stress. The contribution ofaxial microcracking however can be seen as early as in the 140 MPa series where the permanent crack

volume begins to increase. The rate of increase in crack volume accelerates at high stress

Microscopic Fracture Processes in a Granite 245

to 180 MPa (Fig. 10). The rate of change in the permanent crack volume howeverturned positive (dilatant) at a stress lower than this, at the beginning of the 140MPa series. Negative crack volume indicates a reduction in void space. This pro-cess of compaction is the most obvious during the low stress series. Compactionmay, however, take place at all stress levels, but at high stress the volume of newlycreated axial cracks is large enough to mask its presence.

4. Discussion

For competent, brittle rocks the expectation is that strain is produced through twomajor mechanisms: elastic deformation and axial microcracking. There are alsosuch minor contributors as the elastic closure of pre-existing cracks at low stressand perhaps some nonlinear behavior (plasticity), occurring close to the peakstress. Microscopic examination of both pre-failure and post-failure fracture pat-terns in Lac du Bonnet granite would suggest that internal shear displacements,whether due to crack sliding or plasticity, develop only after the peak stress ispassed, i.e. during the post-failure phase of the stress history. Others reached sim-ilar conclusions (e.g. Tapponnier and Brace, 1976 and references therein).

To the two major deformational mechanisms of elastic deformation and axialmicrocracking, one may now add a third one: compaction. Compaction is not thesame as elastic crack closure. Compaction is not an elastic process; it involvespermanent damage! Although this deformational process may be active at allstress levels, the permanent crack volume due to compaction is usually quite small,in the order of 100 me. The total amount of compaction may however be largerthan this, but its presence could be masked by the dilation caused through axialmicrocracking. Through compaction, the axial sti¨ness of Lac du Bonnet granite

Fig. 11. Orientation of microcracks with respect to the loading direction (vertical). The microscopicexamination of fractured Lac du Bonnet granite specimens shows that most of the stress-induced cracksfollow the loading direction. There is, however, a small increase in the distribution of microcracks at the

load-normal, orientation

246 E. Z. Lajtai

increases by about 2 GPa, from an initial 69 GPa to a peak value of 71 GPa(Fig. 9).

To this point, the evidence in support of compaction comes indirectly, from theinterpretation of the axial and the volumetric strain curves. In contrast, axialmicrocracks have been observed and documented by several researchers (Kranz,1983). Is there visual evidence for compaction? The author and his students havestudied and documented many microcracks in stressed Lac du Bonnet granite. It isclear that by far the majority of microcracks are oriented within a narrow rangeabout the compressive stress direction (Fig. 11). These are the axial microcracks.At the same time, there is a small increase at the 90� position, perpendicular to thecompression axis. In micrographs of fractured granite (Fig. 12), the fractures thatcould be responsible for the 90� increase follow low angle (load-normal) grainboundaries. Presumably, these load-normal fractures develop on unloading, at

Fig. 12. Micrograph of a laboratory-fractured Lac du Bonnet granite at 40� magni®cation. The darkmaterial is quartz; the light is feldspar. The loading direction was vertical. Note that most of the cracksrun parallel with the loading direction (marked by horizontal arrows), but there are a few cracks in theload-normal position as well (vertical arrows). The largest load-normal crack follows a grain boundary;

two similar but smaller cracks developed along the cleavage planes of feldspar

Microscopic Fracture Processes in a Granite 247

locations where the compaction damage was concentrated. Grain boundaries areneither continuous nor smooth enough to transmit stress without disturbance.They often contain minor cavities separated by rock bridges (Sprunt and Brace,1974). Under compression, some of the bridges concentrating the compressivestress may be crushed. On unloading, the elastically compressed and still unfrac-tured material around the damaged grain boundary rebounds, pulling the twosurfaces of the damaged grain boundary apart (Fig. 13).

The e¨ects of compaction are displayed best by the axial strain curve, throughthe shift in the crack closure stress to higher stress, and from the change in axial sti¨-ness. Axial microcracking, on the other hand, is best interpreted from the changein the shape of the lateral strain curve, in particular from the decreasing lateralsti¨ness.

All the examples given above refer to the behavior of Lac du Bonnet granite at100% humidity; the test specimen was soaked in water and tested while submergedin water. Dynamic fatigue tests were, however, conducted on air-dried specimensas well. As expected, the presence of moisture has a strong in¯uence on practicallyall the strength parameters. As an example, Fig. 14 compares the evolution of per-manent crack volume under both dry and humid conditions. In general, micro-cracking starts and progresses at lower stress levels at high humidity regardless ofthe testing procedure (Lajtai and Schmidtke, 1987).

The main goal of the research was to de®ne the e¨ect of axial microcracking onthe axial and the lateral sti¨ness of granite under uniaxial compression. Plottingthe two sti¨ness parameters against the permanent crack volume may show this. Asexpected, the lateral sti¨ness decreases with increasing crack volume (Fig. 15).What was not expected, was that even a small increase in crack volume from the

Fig. 13. Interpretation of load-normal fractures. Grain boundaries are rarely continuous; they oftencontain gaps separated by rock bridges. In the hypothetical four-mineral medium, the compressiveloading crushes the rock bridges of the load-normal grain boundary between the two grains in the middle.On load-removal, the two undamaged crystals regain their initial shape. Because they are bounded tothe damaged crystals in the middle, The two damaged grains are pulled apart along the grain boundary

to form a load-normal fracture

248 E. Z. Lajtai

initial (intact) condition would cause a very large decrease in lateral sti¨ness. How-ever the rate of decrease decelerates so that, for large crack volumes, the decrease inlateral sti¨ness becomes, negligible. The axial sti¨ness shows a similar trend startingwith an initial 71 GPa, but soon dropping to and then stabilizing at 67 GPa.

Fig. 15. E¨ect of permanent crack volume on sti¨ness. Although a decrease in sti¨ness with increasingcrack volume would normally be expected, the actual trend of the relationship is somewhat surprising.Even for a very small increase in crack volume from the initial (intact) state, both the axial and thelateral sti¨ness drop steeply. The in¯uence is much less at large crack volume, to a degree that over

about 600 me, neither the axial nor the lateral sti¨ness is a¨ected much

Fig. 14. comparison of permanent crack volumes in air-dried and water-saturated granite. Althoughthe total amount of permanent crack volume at failure is not much di¨erent, the stress necessary toinduce cracking and failure is substantially less for the wet rock. The numbers on the graph indicate the

fatigue loads

Microscopic Fracture Processes in a Granite 249

5. Conclusion

During the dynamic fatigue test, the deformational behavior of an elastic brittlerock, such as Lac du Bonnet granite, is shown to be controlled through threemajor mechanisms of stress-induced deformation: elastic deformation, and twomicrofracture related processes, compaction and axial microcracking. The pres-ence of the compaction mechanism is supported by the low stress appearance ofnegative permanent crack volume accompanied by increasing axial sti¨ness, andthe shift in the crack closure point to higher stress. Compaction delays the ap-pearance of the expected dilation caused by axial microcracking. The amount ofcompaction is quite small, in the order of 100 me. With increasing stress, axialmicrocracking becomes the dominant strain-producing mechanism.

The axial and the lateral sti¨ness of the granite are very sensitive to the per-manent crack volume contained in it. Compaction increases the axial sti¨nesswhile axial microcracking causes a decrease in the lateral sti¨ness.

As expected, the presence of moisture in the testing environment has a strongin¯uence on practically all the deformational and strength parameters. In partic-ular, axial cracking starts at lower stress when the environment is humid ratherthan dry.

Acknowledgements

The author thanks the Natural Sciences and Engineering Research Council for its ®nancialsupport through the Research Grants Program. The aid of W. Grajewski and Fei Yin inpreparing the test specimens and arranging for the testing is gratefully acknowledged.

References

Bieniawski, Z. T. (1967): Mechanism of brittle fracture in rock. Int. J. Rock Mech. Min.Sci. 4, 395±430.

Gri½th, A. A. (1924): Theory of rupture. First Int. Cong. Exp. Mech., Delft, 55±63.

Kranz, R. L. (1983): Microcracks in rocks: A review. Tectonophysics 100, 449±480.

Lajtai, E. Z. (1971): A theoretical and experimental evaluation of the Gri½th theory ofbrittle fracture. Tectonophysics 11, 129±156.

Lajtai, E. Z., Schmidtke, R. H. (1987): The e¨ect of water on the time-dependent deforma-tion and fracture of a granite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 24, 247±255.

Lajtai, E. Z., Carter, B. J., Duncan, E. J. Scott (1991): Mapping the state of fracture aroundcavities. Engng. Geol. 31, 277±289.

Sprunt, E., Brace, W. F. (1974): Direct observations of microcavities in crystalline rocks.Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 11, 139±150.

Tapponier, P., Brace, W. F. (1976): Development of stress-induced microcracks in Westerlygranite. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 13, 103±112.

Walsh, J. B. (1965): The e¨ect of cracks on the uniaxial elastic compression of rocks. J.Geophys. Res. 2, 399±411.

Author's address: Dr. E. Z. Lajtai, Department of Civil and Geological Engineering,The University of Manitoba, Winnipeg, Manitoba, Canada, R3T 5V6.

250 E. Z. Lajtai: Microscopic Fracture Processes in a Granite