Brittleness of rock and stability assessment in hard rock tunneling

Tunnelling and Underground Space Technology 18(2003) 35–48

0886-7798/03/$ - see front matter� 2003 Elsevier Science Ltd. All rights reserved.PII: S0886-7798Ž02.00100-1

Brittleness of rock and stability assessment in hard rock tunneling

Vahid Hajiabdolmajid*, Peter Kaiser

Geomechanics Research Centre, Laurentian University, Sudbury,, P3E 6B5, Canada

Received 2 August 2002; received in revised form 10 November 2002; accepted 21 November 2002

Abstract

Brittleness is a characteristic of many geomaterials in which the pre-existing heterogeneities among the mechanical andgeometrical properties of the constituent materials,(e.g. grains cementing materials and voids) and loading conditions promotenon-homogeneous distribution of the stresses inside the failing mass and eventually along the potential failure plane. This studyrelates the brittleness of failing hard rocks and tunnels to a strain-dependent brittleness index(I ) which characterizes the entireB´

failure process of rock(pre- to post-peak), and accounts for the involved mechanisms in inducing inelastic strains(damage)inside the failing rock. The strain-dependent brittleness of rock dictates the mobilized strength around underground excavations,affects their short- and long-term stability, and determines the shape of breakout(failed or inelastic) zone. The ground-supportpressure interaction mechanism is also affected by rock brittleness. Brittleness of rock is a time-(loading rate) and size-(geometry) dependent property.� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Brittleness; Brittle failure; Slabbing; Hard rock; Tunneling

1. Introduction

The main concern in construction of undergroundopenings in mining and civil engineering in general isground control, i.e. control of the rock failure processesleading to displacements of rock surrounding the exca-vations. Unlike many other engineering domains, wherethe behavior of structures(built from artificial materials)under load can be analytically predicted with greatcertainty, rock engineering still has to rely to someextent on empirical methods; lack of control over mate-rial properties and loading conditions that are encoun-tered around underground openings being two of manyreasons. At depths where stresses often exceed the stresslevel associated with the initiation of micro-cracks(crack initiation stress level) inside rock, additionaluncertainties are introduced by the behavior of rockduring the brittle fracturing process,(i.e. brittleness,defined later in the paper) and by the manner in whichthe transition process from a continuum(intact tomoderately fractured rock) to a discontinuum(spalling

*Corresponding author. Tel.:q1-705-675-1151x5095; fax: 1-q705-675-4838.

E-mail addresses: [email protected](V. Hajiabdolmajid),[email protected](V. Hajiabdolmajid).

or slabbing rock) takes place(Hajiabdolmajid et al.,2002b; Kaiser et al., 2000).In the design of openings in hard rocks, the theory of

elasticity has often been used. At depth, however, thecreation of a fractured,(i.e. inelastic or failed) zonearound openings in any types of rock, ‘hard or weak’ isalmost inevitable (especially in mining operations).Although the rock beyond the fractured zone maybehave elastically, it is the fractured(inelastic) zoneimmediately surrounding the excavation that really dic-tates the ground support requirements to ensure exca-vation stability. From a ground support perspective, thedepth and lateral extent of this fractured zone(breakoutzone) and its behavior in response to the far-field stressesand support pressure needs to be understood. The shapeof the fractured(or failed) zones and the magnitude ofthe induced displacements control the selection of sup-port, its dimensions and characteristics. Thus, the engi-neer must determine the stress level, theexcavation-induced stress, rock strength, the circum-stances(process) under which rock fails and the behav-ior of the broken rock in loading the support whilefailing.This paper first introduces a strain-dependent brittle-

ness index(I ) and a constitutive model called cohe-B´

36 V. Hajiabdolmajid, P. Kaiser / Tunnelling and Underground Space Technology 18 (2003) 35–48

Fig. 1. Damage mechanism in brittle failure process of cohesive geomaterials leading to the strain-dependent mobilization of strength components.

sion weakening-frictional strengthening(CWFS) model,which captures the failure mechanisms involved in thetransition from continuum to discontinuum during brittlefailure of rocks. The significance of the brittleness(asdefined in this study) in stability assessment of under-ground excavations is then demonstrated and the notionsof size and time effects in rock strength degradation arediscussed. Finally, some rock characteristics influencingthe rock’s brittleness are introduced which may be usedto classify the rock’s brittleness(I ).B´

1.1. Case study

Between 1990 and 1995 Atomic Energy of CanadaLimited (AECL) carried out the Mine-by Experiment.This experiment involved the excavation of a 3.5-m-diameter circular test tunnel in homogeneous massivegranite. The primary objective of the experiment was toinvestigate the brittle failure process. To achieve thisobjective, the tunnel was excavated in 0.5–1-m incre-ments using a line-drilling technique and monitored withstate-of-the-art displacement, strain and micro-seismicinstrumentation, coupled with extensive video documen-tation. The test tunnel was excavated at the 420-level inintact and very sparsely fractured Lac du Bonnet granite.One of the objectives of the Mine-by Experiment wasto assess the predictive capability of numerical modelsin capturing the extent and shape of the fractured zone.As part of this work, the in-situ stress magnitudes in thevicinity of the tunnel were determined to bes s60"3,1

s s45"4, s s11"2 MPa (Martin, 1997). The fol-2 3

lowing sections demonstrate the implication of theconcept of a strain-dependent brittleness index(I ) toB´

analyze the stability of deep underground excavationsin hard rocks using the observations made on theprogressive fracturing process around the Mine-bytunnel.

2. Brittle failure process of cohesive geomaterials

Unlike failure of ductile materials where shear slipsurfaces form in such a manner that continuity of

material contact is maintained, brittle failure is a processwhereby continuity is disrupted to create slabs or blocksthat need to be separated and follow kinematicallyfeasible failure modes with localized(point) contacts.From a mechanistic point of view, what happens

during the brittle failure process of rock is the reduction(or destruction) of the strength derived from bondsbetween grains(cohesive strength, Fig. 1a). The fric-tional strength component is gradually mobilized as thedisintegrated blocks readjust or deform by shearing atnewly created surfaces(Fig. 1). Fig. 1a illustrates thateven when a uniform confinement(s ) is applied at the3

boundaries, due to the heterogeneous nature of failingrock material the local confinement(or effective con-finements ) is highly variable and may even be tensile.n¯In Fig. 1b, ´ and ´ are the plastic strain(damage)p p

c f

levels necessary for cohesion loss and frictional strength-ening, respectively. In the following sections it is dem-onstrated that these two plastic strain limits´ and ´p p

c f

can be used to characterize the failure process of rock(stress–strain curve) and the brittleness of a failingrock.It should, however, be emphasized that the terms

‘plastic strain’ and ‘plasticity’ in this context representsonly ‘inelastic strain’ and ‘lack of elasticity’, respective-ly, which are due to various mechanisms inside thefailing rock mass in low confinement environments,(i.e.brittle field). These terms should be distinguished fromthe similar terms used in metal plasticity or in failureof rocks at very high confinement(andyor temperature)levels,(i.e. ductile field) in which the involved mecha-nism is mainly due to dislocations along slip planes.During excavation of the Mine-by tunnel at the 420

level of URL (Atomic Energy of Canada Limited,AECL) in intact and very sparsely fractured Lac duBonnet granite a progressive brittle fracturing processresulted in the development of v-shaped notches, typicalof borehole breakouts, in the regions of the compressivestress concentration in the roof and floor(see Fig. 5).The maximum induced stress at the tunnel wall was

37V. Hajiabdolmajid, P. Kaiser / Tunnelling and Underground Space Technology 18 (2003) 35–48

Fig. 2. CWFS model:(a) illustration of cohesion-loss and frictional strengthening as a function of plastic strain,´ (b) evolution of the yieldp

envelopes due to frictional strengthening(friction hardening) in brittle failure of Lac du Bonnet granite. Note, 160 MPa difference between themobilized strengths withI sy1 condition andI s1.5 in situ.B´ B´

calculated to be considerably lower(s s150 MPa)induced

than the measured short-term laboratory unconfinedcompressive strength(UCSs224 MPa) of cylindricalsamples taken from the surrounding rock. Varioushypotheses have been suggested to explain this brittlefailure process of rock(Martin, 1997; Potyondy andCundall, 1998).Hajiabdolmajid(2001) attributed the low mobilized

in-situ strength around the Mine-by tunnel to the brittle-ness of the failing rock in situ which is a result of astrain-dependent mobilization of the strength compo-nents(cohesive and frictional), causing the rock to failin-situ at much lower stress levels than in the laboratorycompression tests. It was found that in situ(roof), thenon-simultaneous mobilization of the cohesive and fric-tional strength components could be represented usingtwo bilinear functions illustrated in Fig. 2a with andp

c

´ . The plastic strain limits(´ and ´ ) are the inputp p pf c f

parameters of a strain-dependent Mohr–Coulomb failurecriterion called the CWFS model, defined by Eq.(1).

p p¯ ¯f s sf c,´ qf s ,´ tanf (1)Ž . Ž . Ž .n

In Eq. (1), ´ is a strain(or stress) path-dependentp

measure of plastic shear strain called the effective plasticstrain and expressed by Eq.(2). The effective plasticstrain, ´ , represents the accumulated inelastic strainp

(damage), inflicted by the process of micro cracking

and used to determine the weakeningystrengtheningparameters,(i.e. ´ and ´ ) in the CWFS model. Thep p

c f

parameter´ is calculated from the principal plasticp

strain increments and is essentially a measure of theplastic shear strain(Hill, 1950; Vermeer and de Borst,1984).

2p p p p p p p¯ s (d´ d´ qd´ d´ qd´ d´ )dt (2)1 1 2 2 3 3|y3

where, d , d´ and d are the increments of principalp p p1 2 3

plastic strains.In this model, the yield envelope expands as plastic

strain increases. At the early stage of failure, the mobi-lized strength is mainly due to the cohesive strengthcomponent and as inelastic strains accumulate, the fric-tional strength progressively mobilizes and replaces thecohesive strength component(Fig. 2b). This type ofbehavior results in a series of linear failure envelopesexpanding by plastic straining(damage). The evolutionof the yield envelopes in brittle failure of Lac du Bonnetgranite around the Mine-by tunnel is illustrated in Fig.2b.

3. Brittleness of failing rock

The concept of the brittleness of a material or structureis not a new one and it is often defined as the lack of


Fig. 3. Schematic representations of various stress–strain curves withequal peak and residual strength levels and different rate for strengthloss rates.

ductility, (e.g. Hetenyi, 1966), and ductility is defined´as the ability of a material to endure a large inelasticdeformation without loss of its bearing capacity. How-ever, the quantification of brittleness has long beenfuzzy, as Andreev(1995) lists 20 different definitionsand relationships used to quantify the rock brittleness.The main reason is that the apparent brittleness of astructure or a rock mass depends simultaneously on thematerial properties, the geometry and size and on theloading conditions. This makes it very difficult to finda proper definition and relationship, which incorporatesall the above-mentioned influences. However, it can besaid that any measure of brittleness should be vanish-ingly small for the elastic-perfectly behavior and infi-nitely large for the elastic-brittle failure.In the geotechnical engineering field, brittleness is

normally understood as a material condition character-ized by its reduced ability to carry load as the strainincreases. In this context, brittleness is a characteristicof many geomaterials in which their natural heteroge-neities and loading conditions promote non-homogene-ous distributions of the stresses inside the mass, causinglocal failures and progressively form the failure plane(shear band). Thus, brittleness affects the developmentof frequently observed progressive failure process inthese materials. However, this general understanding ofbrittleness concerns only the post-peak behavior of thematerial, as Bishop(1967) introduced a brittleness index(I ) expressed by Eq.(3) in terms of peak(t ) andB f

residual strength(t ) (Bishop, 1967).r

t ytf rI s . (3)Btf

However, the post-peak behavior of material has itsorigin in the pre-peak range, thus brittleness shouldencompass a broader sense and include the whole failureprocess(pre- to post-peak stage). The brittleness indexshould not only represent the severity of strength lossafter failure but also the ability of a material to withstandinelastic deformations before failure. In other words, thepeak and residual strength levels are not sufficient tocharacterize the susceptibility of a material to brittlefracturing and progressive failure, the strength loss rateand the stress(strain) path followed between these twostrength levels must also be considered. The brittlenessindex defined by Eq.(3) does not determine the mobi-lized strength during various stages of failure(pre- topost-peak) during any particular stress path. Fig. 3demonstrates one of the difficulties in defining a stress-based definition for rock brittleness. In Fig. 3, the peakand residual strength levels are the same in all thestress–strain curves, however, the implication of variousprocesses and paths in pre- and post-peak stages willlead to different stress redistributions regimes.

Therefore, it is necessary to introduce a measure ofbrittleness that depends on strain, a measure that estab-lishes the rate of strengthening and weakening duringthe pre- and post-peak stages. Using plasticity theoryand the concept of strain-dependent strength mobiliza-tion (Fig. 1), Hajiabdolmajid(2001) introduced a strain-dependent brittleness index(I ) expressed by Eq.(4)B´

(Hajiabdolmajid, 2001).

p p´ y´f cI s . (4)B´ p´c

This definition of brittleness separates the mobiliza-tion and loss of strength components(c ™c andf™i r i

f ) during the strain-dependent process ofr

micro-cracking inside rock(as it is demonstrated in Fig.1) by considering the rate at which the strength com-ponents are lost or mobilized as functions of inelasticstrain (damage). From the physical point of view, thebrittleness index as defined by Eq.(4) reflects howpresent the tension(andyor shear) induced cracks arein the failure process, and how constrained(or free),these cracks are to propagate. Since, the plastic strainlimits, (i.e. ´ and ´ ) are determined using a stressp p

c f

path-dependent measure of plastic shear strain(´ ), thusp

the stress path dependency of the induced damage,(i.e.mobilization and loss of strength or brittleness) isconsidered in this relationship. Fig. 4 demonstrates thesensitivity of the mobilized strength and the stress–strain curve(pre- to post-peak stages) to the strain-dependency of the mobilization of the strengthcomponents using various plastic strain limits(´ andp

c

´ in the insert in Fig. 4). The results in Fig. 4 will bepf

explained in Section 6.The back analysis of the brittle failure of the Lac du

Bonnet granite around the Mine-by tunnel demonstratedthat the frictional strength mobilizes to its full capacitywith a slower rate than the cohesive strength stabilizes


Fig. 4. The unifying definition for brittleness ‘(I )’ considers theB´

pre-peak non-linearity, mobilized strength and post-peak ductility.Note the sensitivity of the mobilized strength and stress–strain curveto the plastic strain limits, and strain-dependent functions(non-linearin these tests) for cohesion loss and friction mobilization.

Table 1Parameters of the CWFS model for in situ brittle failure of Lac duBonnet granite

Cohesion Friction angle Dilation angle

Initial 50 MPa 08 308Residual 15 MPa 488 308´ , I s1.5p

B´¯ 0.2% 0.5%(Roof) 0%´ , I sy0.4p

B´¯ 0.2% 0.12%(Floor) 0%

at its residual value(´ )´ in Fig. 1b and Fig. 2)p pc f

(Hajiabdolmajid, 2001). The plastic strain limits atwhich the cohesive strength reaches a residual value(´ ), and the frictional strength fully mobilizes(´ ) arep p

c f

two material properties in the CWFS model and theywere obtained by using the laboratory damage-controlledtests results on Lac du Bonnet granite reported by Martin(1997), and by back analyzing the breakout zone aroundthe Mine-by tunnel(Hajiabdolmajid, 2001).The pioneering work of Schmertmann and Osterberg

(1960) showed that even in cohesive soils, these twostrength components(cohesive and frictional) might notnecessarily be mobilized simultaneously. They showedthat the cohesive component of strengthc(´) wasmobilized early in their compression tests, while thefrictional components9(´)tanf required more strainingto reach full mobilization.

4. Brittleness of rock and stability of undergroundexcavations

The well-documented Mine-by Experiment(URL,Pinawa, Manitoba) was used to verify the ability of theCWFS model in predicting the shape of the breakoutzones. The details of these back-analyses are presentedby Hajiabdolmajid (2001), Hajiabdolmajid et al.(2002a,b). Here, some insights into the excavationresponses regarding the rock’s brittleness are presented.The plastic strain limits together with the initial and

ultimate strength parameters listed in Table 1 were usedin the CWFS model to simulate the brittle failure ofLac du Bonnet granite, near the Mine-by tunnel.

FLAC (Itasca, 1995) was used to simulate the brittle2d

fracturing process using the CWFS model, and to deter-mine the brittleness associated with this process. FLACcodes use an explicit time-marching scheme to find thesolution to a problem. The equations of motions aresolved to derive new velocities and displacements fromforces and stresses. Velocities are then used to calculatestrain rates, from which new stresses can be foundthrough a constitutive equation.Fig. 5a illustrates the simulated shapes of the failed

zones around the Mine-by tunnel using the CWFSmodel. The plastic strain limits and´ which simulatep p

c f

the shape of the failed zone in the roof(Fig. 2a andFig. 5a) correspond to the brittleness index(I ) equalB´

to 1.5. Fig. 5b demonstrates the mechanism of failurearrest by showing the induced damage(plastic strain)contours and the progressive frictional strengtheningwhich leads to the failure arrest at the breakout tip. Thedamage induced by the cracking process around open-ings in hard rocks weakens the rock by reducing itscohesional strength component. The largest reductionsoccur at the wall of the opening, where the damage isthe greatest. This is schematically shown for a circulartunnel subjected to high vertical stress,s in Fig. 6.1

Outside this damaged zone, the cohesional componentof strength can be considered almost unaffected fromthe stability point of view(Fig. 6). In Fig. 6, the amountof induced damage(D , represents the induced damagei

or plastic strain) decreases significantly from the bound-ary of the tunnel towards the breakout zone tip. At thetunnel wall, the coalescence and propagation of cracksare not kinematically constrained and the frictionalstrength does not significantly contribute to the mobi-lized strength(Fig. 5b and Fig. 6, i.e. low mobilizedstrength, in situ failure envelope in Fig. 2b). However,when the breakout zone is deepened and reaches acertain geometry, the propagation of micro-cracks form-ing macro-cracks by coalescence is increasingly con-strained which in the presence of higher confinementfinally leads to the arrest of failure.Thus, cohesion loss and frictional strengthening rates

(´ and ´ ) which are the characteristics of rock typesp pc f

and the loading conditions determine how deep theprocess of brittle fracturing will propagate. In otherwords, the brittleness(I , i.e. the ease of crack coales-B´

cence and propagation) in the presence of favorable in


Fig. 5. (a) Prediction of the notch formation around the Mine-by tunnel in the roof and floor using the CWFS model,(b) inelastic strain contoursinside the breakout zone and the mechanism of failure arrest.

Fig. 6. Mobilization of the strength components in the slabbing zoneand stabilization of the new geometry(notch zone) as a result of thedecrease in the induced damage and progressive frictional strength-ening towards the notch tip.D , represents the induced damage byi

brittle fracturing.

situ stress conditions determines the depth of failure.Thus, the brittleness(as it is defined in this study)influences the final shape of an underground excavationwhere stress level is high enough to bring the rockmaterials close to or beyond the crack initiation stresslevel (s )crack initiation stress level). Thus, theinduced

manner in which the strength is mobilized and lost,(i.e.I ) in any particular rock type needs to be consideredB´

in the stability assessments and numerical simulationsof the brittle failure around underground excavations.In practical rock engineering, the reduced ability of

rock to withstand stresses in situ is often attributed tothe effect of size and time. The following sections

demonstrate that the size and time effects can be viewedas factors influencing the rock’s brittleness (I ) in situB´

relative to the brittleness(I ) of cylindrical samplesB´

tested under laboratory conditions. The bilinear envelope(heavy line) in Fig. 2b represents the stress levelsassociated with the spalling initiation and failure arrestat the breakout zone tip, in the roof. This bilinearenvelope demonstrates that spalling at the tunnel walloccurs at the stress level equal to 100 MPa, which ismuch lower than the long-term(165 MPa) and short-term (224 MPa) UCS (unconfined compressivestrength). If both strength components were completelyand simultaneously mobilized(I sy1), the failure ofB´

Lac du Bonnet granite by brittle fracturing could onlyoccur when a stress level of 260 MPa is exceeded.These results imply that the brittleness(I ) of failingB´

granite in laboratory tests is lower than in situ(Hajiab-dolmajid, 2001).Two different material behaviors(I ) are chosen toB´

simulate the shape of the failed zones in the roof andin the floor (Fig. 5a). A lower brittleness had to bechosen in order to simulate the failed zone in the floor;the reason is related to the effect of muck pressure,which toughens the rock in this region. This is explainedlater in the paper.

5. Size effect

The effect of size in stability assessments of under-ground excavations has been investigated by many, and


Fig. 7. Rock strength:(a) in experiments with increasing sample size(normalized with the strength of 50-mm diameter sample) after Hoekand Brown(1980) and Wagner(1987); and(b) in the CWFS modelswith increasing brittleness index(normalized with the strength ofI sy1 model) wafter Hajiabdolmajid(2001)x.B´

strength reduction factors are often applied, e.g. it isnormally assumed that in-situ strength should be consid-ered as a function of sample size, e.g. UCSsUCS =(50yd) (Hoek and Brown, 1980).0.18

50 mm( )

While in a larger volume, discontinuities weaken therock, at depths and in massive rocks, the weakeningeffects of discontinuities are minimal(Hoek et al.,1995). Thus, the observed in-situ strength should berelated to other factors.For instance, the Mine-by tunnel was excavated in

massive, joint free Lac du Bonnet granite. Therefore,the reduced mobilized strength in Fig. 2b(in situ vs.UCS) cannot be explained with the general understand-ing of size effect. However, if the size effect is viewedas a result of the differences in the failure mechanismsleading to different material brittleness(I ) and lowerB´

mobilized strength, then it follows that larger excava-tions walls fail differently than smaller ones. In otherwords, large blocks of rocks near excavation walls failin a more brittle manner than smaller blocks or labora-tory samples. This can be explained by the fact thataround large excavation walls the coalescence and prop-agation of micro-cracks forming macro-cracks and visi-ble fractures is less kinematically constrained andmobilize much less frictional strength component, whencompared to the laboratory compression tests on smallcylindrical samples.Fig. 7a compares the mobilized strengths in compres-

sion tests carried out on various sample sizes normalizedwith the strength of the 50-mm diameter cylindricalsample reported by the extended relationship of Hoekand Brown(1980) and by the relationship obtained inexperimental works on massive rock reported by Wagner(1987). Fig. 7b illustrates the results of the compressiontests models using the CWFS model with various brit-tleness indices. The increase in the brittleness indexresults in a lower mobilized strength. It is interestingthat similar trends are observed in both experiments andthe CWFS models. This interpretation of size(andgeometry) effect is consistent with the experimentsreported by Hudson et al.(1971) who concluded thatthe larger samples tended to indicate lower mobilizedstrength and post-peak ductility when compared withthe smaller samples. The fact that in intact rocks, thesize effect is more pronounced in brittle rocks alsodemonstrates that the difference in the involved failuremechanisms(I ) is the origin of the observed sizeB´

effect in these materials.The brittleness associated with the spalling in the roof

of the Mine-by(I s1.5) limits the mobilized strengthB´

to only 100 MPa. However, in the laboratory conditionsthe lower brittleness,(i.e. I fy1 or ´ f0) allows thep

B´ f

mobilized strength to be almost two times higher(short-term UCSs224 MPa and in the CWFS model, UCSs260 MPa forI fy1 or ´ f0). Fast mobilization ofp

B´ f

the frictional strength component(or slow cohesion loss

rate) in laboratory condition,(i.e. small sample size) isrepresented byI F0, i.e. 0F´ F´ , and in situ con-p p

B´ f c

ditions (or large blocks of rock in laboratory tests) arecharacterized byI )0, (i.e. ´ )´ ).p p

B´ f c

6. Time effect

Another notion used to explain the reduction of themobilized strength in situ is the effect of time or loadingrate. It has long been recognized that a decreasingloading rate lowers the mobilized strength of rocks(Brace et al., 1966; Kaiser and Morgenstern, 1981).Various mechanisms have been identified as the causefor this time effect observations. For example, stresscorrosion is a time-dependent cracking phenomenon inrock that depends on load, temperature and moisture(Atkinson and Meredith, 1987; Okui and Horii, 1997;Potyondy and Cundall, 1998).Fig. 8 illustrates the effect of loading rate on the

mobilized strength in compression tests of Westerlygranite reported by Brace et al.(1966). In Lac du


Fig. 8. Effect of loading rate on the mobilized strength in compression tests on Westerly Granitewafter Brace et al.(1966)x.

Fig. 9. Mobilized strength and post-peak ductility in unconfined compression tests:(a) in the CWFS model by changing the rates of cohesionloss and frictional strengtheningwafter Hajiabdolmajid(2001)x; and(b) in experiments by reducing changing loading ratewafter Peng and Podnieks(1972)x.

Bonnet granite the long-term and short-term UCS arereported as 165 and 224 MPa, respectively(Martin,1997).Here, the effect of time or loading rate is viewed as

a result of the changes that occur in the failure mecha-nism of rock,(i.e. I or brittleness). As explained byB´

Kaiser and Morgenstern(1981) a lower loading ratepromotes a more brittle behavior as loads are distributeddifferently between cracking elements. Reducing the

loading rate enhances the time effect by giving time tomore cracks to propagate andyor propagate more withless mobilized friction. A low loading rate increases thedelay between the cohesion loss and frictional strength-ening, (i.e. increases y´ and I ). As one can seep p

f c B´

from Fig. 8, an increase in loading rate expands theyield envelope, which can be interpreted as an increasedrate of frictional strengthening leading to higher mobi-lized strength(i.e. low brittleness and reduced).p

f


Fig. 10. Depths of failures in the roof and floor of the Mine-by tunnel,and in granodiorite sections of the Mine-by tunnel and their associatedbrittleness indices in the CWFS model.

Fig. 11. Geometric characteristics of the failed zone used in Eqs.(5)and(6).

Fig. 12. Simulation of the failed zones around the Mine-by tunnel ina three-dimensional model.

Fig. 9a illustrates the results of the simulation of theuniaxial compression tests using two non-linear func-tions for cohesion loss withc s0 and various rates forr

cohesion loss and frictional strengthening(´ and´ inp pc f

insert in Fig. 9a). In all these models, the materialproperties remain the same, however, both the mobilizedstrength (peak strength) and the entire stress–straincurve (pre- to post-peak) have been affected by chang-ing the strain limits (´ and ´ ). The most brittlep p

c f

behavior is associated with the lowest cohesion losslimit (small ´ ) and highest frictional strength limitp

c

(large ´ , i.e. low strengthening rate), leading to thepf

highest brittleness index and the lowest mobilizedstrength and post-peak ductility. Fig. 9b illustrates theuniaxial compression tests on Tuff reported by Peng andPodnieks(1972), which demonstrate that increasing theloading rate increases the mobilized strength and affectsthe strength drop rates after peak. The experimentalresults in Fig. 9b demonstrate that the mobilized strengthand post-peak behavior are affected by loading rate,which in fact determines the manner in which thestrength components are mobilized during failure pro-cess, (i.e. ´ and ´ ), this effect of loading rate isp p

c f

simulated in the compression tests in Fig. 9a. In otherwords, the lower the loading rate, the lower the tough-ness or the higher the brittleness(I ), and the lowerB´

the mobilized strength.

7. Effect of support pressure on rock brittleness

Fig. 10 illustrates the depths of failure around theMine-by tunnel for different sections associated withdifferent brittleness indices. It was found that the brittle-ness has a direct relation with the depth of failure,which is expressed by Eq.(5). A similar relationshipwas found for the lateral extent,(i.e. a in Fig. 11) ofbreakout zone as expressed by Eq.(6).

0.5DOFs25 1qI (5)Ž .B´

0.4as25 1qI (6)Ž .B´

It follows that the brittleness(I ) is a dominantB´

factor, often more than stress, in controlling the breakoutzone shape. This explains the failure of methods adoptedby many researchers in establishing stress related break-out prediction models.In mapping the failed zones around the Mine-by

tunnel, it was noticed that the breakout zone in the floorwas smaller than the one in the roof(Fig. 5a). Thedifference was explained by the effect of muck pressureandyor the effect of having two different stress pathsfollowed by the regions in the floor and in the roof(Martin, 1997). In order to clarify this uncertainty, athree-dimensional implication of the CWFS model wasused in FLAC (Itasca, 1997) to predict the formation3d

of the failed zones around the Mine-by tunnel in whichthe non-alignment of the tunnel axis with the axial stress(which is the cause for the stress path difference betweenthe roof and floor) is considered.


Fig. 13. Effect of support pressure in reducing the depth failure aroundthe Mine-by tunnel without considering the effect of support pressurein reducing the rock brittleness.

Fig. 14. Ground reaction curves and the effect of cohesion loss rate in the CWFS model.

Fig. 12 illustrates the formation of the failed zones inthe roof and floor in a three-dimensional model, whichare similar even though a slightly different stress pathhas been considered. Considering the results in Fig. 10and Fig. 12, one can conclude that the presence of muckhas lowered the rock brittleness(I ), resulting in anB´

increase in the mobilized strength in failure of rock inthe floor. These results can be used to argue that thesupport pressure tends to reduce the rock brittleness,which limits the propagation of the breakout zone.In hard brittle rocks the support pressure(confine-

ment) will act as a crack propagation inhibitor by

lowering the rate of cohesion loss(increase ) andyorpc

accelerating the frictional strengthening rate(decrease´ ), which leads to a higher mobilized strength in thep

f

process zone(active failing zone where micro-cracksare propagating). In other words, the presence of supportpressure around the openings in hard brittle rocks aidsthe rock to mobilize more of its strength capacity duringthe failure process. Indeed if the effect of supportpressure in lowering the rock brittleness is ignored, thenecessary support pressure which can affect the depthof failure around openings in hard brittle rock is muchhigher than the capacity of the conventional supportsystems. For instance, the support pressure necessary toreduce the depth of failure by 15%(the differencebetween the depths of failure in the roof and floor)around the Mine-by tunnel is approximately 20 MPa,which is out of reach of any types of support(Fig. 13).It is expected that in less brittle(ductile) rocks, supportpressure act as an additional strengthening factor, whichadds to the available strength in rock rather than a causefor exploitation of the rock mass strength as it does inhard brittle rocks.

8. Ground reaction curves

Ground reaction curves(GRC) are often used instability analyses of tunnels and for evaluation of theground-support interaction(Brown et al., 1983). Mostof the previously reported works in this field use closedform solutions or numerical methods with simple and


Table 2The CWFS model parameters in Fig. 14

Test ´pc ´p

f c

A 0.2 0.2 08B 0.2 0.2 308C 0.06 0.2 08D 0.06 0.2 308

Table 3The CWFS model parameters in Fig. 15

Test ´pc ´p

f c

A 0.01 0.0 08, 308B 0.01 0.2 08C 0.01 0.2 308

Fig. 15. Ground reaction curves and the effect of frictional strengthening rate on the observed displacements in the CWFS model.

often unrealistic material behaviors to calculate theground reaction curves for tunnels in hard brittle rocks.The CWFS model can be used to calculate the ground

reaction curves. Fig. 14 illustrates the GRC with thelisted parameters in Table 2, which demonstrates theeffect of cohesion loss rate on GRC. Fig. 15 illustratesthe GRC with the listed parameters in Table 3, whichdemonstrates the effect of frictional strengthening rateon GRC. As can be seen using the same materialproperties but different brittleness results in a completelydifferent GRC. This demonstrates the significance ofrock brittleness on the interpretation and prediction ofthe observed displacements around openings and deter-mination of the support system necessary for the exca-vation stability. In calculating the ground reaction curvesin Fig. 14 and Fig. 15, a constant dilation angle(cs308) was used. It should be mentioned that using astrain-dependent function for dilation angle,wi.e. c(´)xas it was used for the cohesive and frictional strengthcomponents, will affect the induced inelastic strains anddisplacements in the failing rocks around the tunnel walland different ground reaction curves will be produced.

9. Rock properties and brittleness

It is expected that different rocks possess differentstrain limits for cohesion loss and frictional strengthen-ing (different brittlenessI , i.e. ´ y´ ) in differentp p

B´ f c

loading conditions. Thus, creating the same opening inthe same far field stress environment inside differentrocks with different brittleness will lead to differentfailed zones. The Mine-by tunnel at the URL nicelyillustrates this where the tunnel passes from granite togranodiorite of the Lac du Bonnet formation. Fig. 16adepicts the distribution of gray granite and granodioriteon an unfolded perimeter map of the Mine-by tunnel.Fig. 16b illustrates the depth of the breakout zone incontour form (in the roof and floor) on an unfoldedperimeter map. Spalling(depth of notch) is almosteliminated where the tunnel passes through the grano-diorite. Fig. 16c illustrates the brittleness indices asso-ciated with the observed depths of failure. Thebrittleness indices associated with the observed extentangle(a) are also shown(by squares).Using the concept of brittleness index(I ) one canB´

explain the differences observed, between the grano-


Fig. 16. Perimeter maps of geology(a), the depth of the breakout(notch in meters), granite is shown as the light regions and granodiorite as thedark regions on the geology map(b) wafter Martin et al.(1997)x and(c) associated brittleness indices.

diorite and granite of Lac du Bonnet around the Mine-by tunnel, while the in situ, stresses and environmentalconditions can be considered constant along the lengthof tunnel(Fig. 16b). The stability in the granodiorite orthe slabbing in the granite can be related to differencesin their brittleness(I ). The two materials(granite andB´

granodiorite) are reported to have very similar strengthand deformational(laboratory) properties but differentgrains size distributions(granite is coarser than grano-diorite). In the granodiorite, the size of all grains isapproximately equal to(1 mm) and somewhat moreuniformly distributed. It is likely that the presence oflarger grains in granite contributes to a faster cohesionloss rate (with straining) andyor slower frictionalstrengthening,(i.e. higher brittleness) in situ. Therefore,the mobilized strength of the granodiorite in situ ishigher than the mobilized strength in situ in the granite(Fig. 10). This cannot be simulated and considered inthe models in which the failure initiation and arrest aresimulated using purely stress-based criteria.Several researchers have shown that the UCS of the

granitic and other rock types decreases with the increasein the mean grain size(Olsson, 1974; Hugman and

Friedman, 1979; Onodera and Kumara, 1980; Fredrichand Evans, 1990; Prikryl, 2001). Onodera and Kumara(1980) also showed that the percentage of boundaryrelated cracks(inter-granular cracks) decreases withincreasing grain size whereas the number of intra-granular cracks increases. This might be used to arguethat in fine-grained polycrystalline rocks the inter-gran-ular cracks are more abundant than in the coarse grainrocks. The propagation of inter-granular cracks(thereare more in fine-grained rocks) mobilizes more strengthdue to more involvement of the frictional strength andyor the slower cohesion loss rate. The propagation ofintra-granular cracks is more brittle and involves a lessfrictional strength contribution. This leads to the conclu-sion that the homogenous fine-grained rocks tend topossess lower brittleness(I is small), than heteroge-B´

neous grain sized rocks. More investigations should becarried out in this area.Mineralogical characteristics of rock also influence

the rock brittleness. In this regard, the presence of asmall percentage of flaky, soft and altered minerals tendsto increase the rock’s brittleness and lower the mobilizedstrength. Quartz is another important mineral in rock


Table 4Textural and mineralogical properties of Lac du Bonnet granite andgranodiorite

Rock type Origin Grain size Unsound Quartzyfeldsparminerals ratio

LDB granite IN F to C Non 0.66LDB granodiorite IN F Non 0.85

IN: igneous intrusive; F: fine grain size; C: coarse grain size.

construction, which has a primary effect on the strengthmostly due to the interlocking structure among its grains.The abundance of easily cleavable minerals (such asfeldspars) causes a reduction in the mobilized strengthand results in higher brittleness. Thus, the ratio of quartz-feldspar contents can be considered as an indicator. Areview of the previous studies and experiments andobservations of the brittle fracturing around deep under-ground openings in hard rocks was carried out byHajiabdolmajid(2002), which revealed that the follow-ing rock’s characteristics might be considered to deter-mine the brittleness of rocks (I ).B´

– Rock type: igneous(intrusive or effusive), metamor-phic, sedimentary;

– grain size: fine-medium-coarse;– ratio of compressive to tensile strength;– presence of voids(pre-existing cracus and pores);– ratio of quartz to feldspar contents; and– presence of flaky, soft and altered minerals.

Table 4 compares some of characteristics of graniteand granodiorite of Lac du Bonnet. The differences canbe observed in grain size, and in the ratio of QuartzyFeldspar contents.It is expected that the dominant factors, influencing

the rock brittleness vary among various rock types. Thisis due to the fact that various micro-mechanisms areinvolved in inducing micro-cracks in rocks, which even-tually lead to the failure of rock in macro-scale. Themechanisms such as elastic mismatch among grains andbetween grains and cementing material, sliding alongpre-existing cracks and concentration of stress aroundpores might be acting simultaneously during the failureprocess of a rock. The presence and dominance of anyof these or other mechanisms will result to differentbrittleness(I ) in any particular rock. The mechanisms,Be

which involve the initiation and propagation of cracksat the grain boundaries(frictional cracks), cleavage,foliation and soft inclusion, pores, point loading, graincrushing and local stiffness mismatch are different interms of the involved plastic straining during theirprocess and as a result the brittleness of the process,(i.e. ´ y´ ) differs in each mechanism. More investi-p p

f c

gations should be conducted to determine the brittlenessof various rock types by considering various involvedmicro-mechanisms during their failure process.

10. Conclusions

The significance of rock brittleness in stability assess-ments of underground excavations was demonstrated.Rock brittleness is an important property of rock andshould concern the behavior of materials in pre- to post-failure stages. It was demonstrated that rock brittlenessdictates the mobilized strength around undergroundexcavations and thus, determines the failed zone shapeand the ground-support interaction mechanisms. In mod-eling the brittle failure of rock around undergroundexcavations in hard rocks, the brittleness of rock cannotbe adequately modeled by assigning only the peak andresidual strengths of rock. The effects of stress path,strength loss rate after peak, and the strain dependencyof the strength mobilization(in pre- and post-peakstages) on the stress redistribution and the progress ofbreakout zone should be considered.In hard brittle rocks, the support pressure tends to

reduce the rock’s brittleness leading to a higher mobi-lized strength and results in a less progressed breakoutzone (failed or inelastic zone). Brittleness is a materialproperty with respect to the rock behavior under certainloading conditions, which means that rock’s brittlenessmight be viewed as a time- and size-(i.e. geometry)dependent property of rock. For instance, the brittlenessof a rock increases with decreasing loading rate (timedependent) and increasing the size of the sample (size-dependent). Different rocks possess different brittleness,which are the results of their unique textural andmineralogical characteristics. More investigations shouldbe conducted to classify the rock brittleness and relatethese properties to the strain-dependent brittleness index(I ) introduced in this study.B´

While in this study the concepts of non-simultaneousand strain-dependent mobilization of the strength com-ponents were used to explain and interpret the observa-tions made in failure of a hard rock such as granite.However, these concepts may be applied for a widerange of geomaterials in which the damage under com-pressive loading is primarily due to the tension-inducedmicro cracking.

Acknowledgments

The authors wish to thank Dr Derek Martin forproviding information about the rock mass surroundingthe Mine-by tunnel experiment and the laboratory testdata on the Lac du Bonnet granite and for helpfuldiscussions on the concept of cohesion loss-frictionmobilization. The partial support from NSERC(NationalSciences and Engineering Research Council) of Canadais gratefully acknowledged.

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Brittleness of rock and stability assessment in hard rock tunneling