A simple mehod o esimae ensile srengh and Hoek-Brown srengh parameer m i of brile rocks M. Cai Mansour Group of Companies, 2502 Elm Sree, Sudbury, Onario, Canada, P3E 4R6 ABSTRACT: A pracical simple mehod is proposed o esimae he ensile srengh ( ) of srong brile rocks from he srengh raio of R= c / = 8 c /, where c is he uniaxial compressive srengh and is he crack iniiaion sress in an uniaxial compression es. 8 is he Griffih srengh raio and he erm c / accouns for he difference of crack growh or propagaion in ension and compression in uniaxial compression ess. can be reliably obained from volumeric srain measuremen or acousic emission (AE) monioring. Wih he srengh raio R deermined, he ensile srengh can be indirecly obained from he resul of uniaxial compression ess. In addiion, a pracical esimae of he Hoek-Brown srengh parameer m i is presened and i is suggesed ha m i = 12 c / can be applied o srong, brile rocks in high confinemen zone and m i = 8 c / can be applied o low confinemen o ension zones. I is found ha he prediced ensile srenghs and Hoek-Brown srengh parameer m i using his mehod are in good agreemen wih es daa. The rock srengh parameers like and m i, which require spealy ess such as direc ensile (or Brazilian) and riaxial compression ess for heir deerminaion, can be reasonably esimaed from uniaxial compression ess. 1 INTRODUCTION 1.1 Rock ensile srengh The ensile srengh is generally low compared o is compressive srengh. Direc ensile srengh ess of rocks are no rouinely conduced because of he difficuly in spemen preparaion. Indirec mehods, such as bending and Brazilian ess, are ofen used o obain he ensile srengh of rocks. When ensile srengh es daa are no available, he general approach o esimae rock ensile srengh makes use of he correlaion beween uniaxial compressive srengh ( c ) and ensile srengh ( ) and applies he generally agreed relaionship of c = R, where R 10 (Sheorey 1997). Esimaion of rock ensile srengh using = c /10 is only a firs sep bu definiely no he bes approach. The es daase compiled by Sheorey (1997), alhough limied in number, shows a large variaion of he srengh raio (R = c / ), from 2.7 o 39 wih an average of 14.7. Vuukuri e al. (1974) saed ha he srengh raio of mos rocks varies from 10 o 50. Brook (1993) noiced ha he srengh raio R depends on rock ype. However, he srengh raios of sandsone compiled by Sheorey (1997) range from 7 o 39 wih an average of 14.9. Wih such a large range of daa variaion and lack of direc correlaion of ensile srengh wih rock ype, i presens a significan challenge for engineers who wan o infer a srengh raio from he daabase. 1.2 Hoek-Brown srengh parameer m i The Hoek-Brown failure crierion (Hoek & Brown 1980a, Hoek & Brown 1997, Hoek e al. 2002) is widely used o describe he srengh of rocks and joined rock masses. To use his failure crierion, a few parameers such as c, s, m i, a, and GSI (Geological Srengh Index), are required. For inac rock srengh, s = 1, a = 0.5, GSI = 100, only c and m i are needed. For joined rock masses, c, m i, and GSI are needed. c can be obained from uniaxial compression ess. The m i values are believed o vary wih rock ype, and i is recommended ha hese values be deermined from a series of riaxial ess (Hoek & Brown 1980a). Quie ofen, riaxial ess PAPER 4063 1
are no rouinely conduced for mos projecs and engineers are forced o deermine he m i values empirically. Hoek (2007) provided some m i values as shown in Table 1. Possible daa ranges are shown by a variaion range value immediaely following he suggesed m i value. m i values range from 4 o 33 for some commonly encounered rocks and an impression ha m i depends only on rock ype can be seen from he able bu his is no rue. The m i value depends on many facors such as mineral conen, foliaion, and grain size (exure). A rend which can be easily seen from he able is ha he m i values are high for coarse grained rocks, moderae for medium grained rocks, and low for fine grained rocks. For sandsones, he m i values can vary beween 13 and 21, and for slaes, beween 3 and 11, according o Table 1. Such a large variaion range is expeced for rocks bu a he same ime, i presens a major challenge for engineers o choose a reasonably accurae m i value for a paricular rock. I can be shown ha if he compressive o ensile srengh raio R is high (e.g. R 8), m i can be esimaed from his srengh raio, i.e., m i c R Hence, if he ensile srengh and uniaxial compressive srengh are available, he srengh raio R could be used as a good esimae for m i. In oher words, because c can be easily obained and if can be accuraely deermined, he m i parameer can be reasonably esimaed. In his sudy, we will examine he relaionship beween he uniaxial compressive srengh and he ensile srengh of rocks from a differen perspecive, by applying he Griffih s heory and using he characerisic sresses observed in he uniaxial compression es. A simple mehod o esimae and m i using uniaxial compression es daa is presened. Table 1 Updaed values of he consan m i for inac rock, by rock group. Noe ha values in parenhesis are esimaes (Hoek 2007) Rock Texure Class Group ype Coarse Medium Fine Very fine Conglomerae* Silsone 7±2 Claysone 4±2 Clasic (22±3) Sandsone 17±4 Greywacke Shales (6±2) Brecas (19±5) (18±3) Marls (7±2) SEDIMENTARY METAMOR- PHIC IGNEOUS Organic Chalk 7±2 Non- Crysalline Spariic Limesone (10±2) Limesone(9±2) (9±3) Micriic Dolomies Clasic Carbonaes Limesone (12±3) Evapories Gypsum 8±2 Anhydrie 12±2 Hornfels(19±4) Non Foliaed Marble 9±3 Measandsones Quarzies 20±3 26±6 Slighly foliaed Migmaie (29±3) Amphibolies 26±6 Foliaed* Gneiss 28±5 Schiss 12±3 Phyllies (7±3) Slaes 7±4 Pluonic Ligh Granie 32±3 Diorie 25±5 Granodiorie (29±3) Dark Gabbro 27±3 Dolerie (16±5) Norie 20±5 Hypabyssal Porphyries (20±5) Diabase (15±5) Peridoie (25±5) Volcanic Rhyolie (25±5) Dae (25±3) Obsidian Lava Andesie 25±5 Basal (25±5) (19±3) Pyroclasic (19±3) Agglomerae Breca (19±5) Tuff (13±5) * Conglomeraes and brecas may presen a wide range of m i values depending on he naure of he cemening maerial and he degree of cemenaion, so hey may range from values similar o sandsone o values used for fine grained sedimens. ** These values are for inac rock spemens esed normal o bedding or foliaion. The value of m i will be significanly differen if failure occurs along a weakness plane. (1) PAPER 4063 2
2 BACKGROUND 2.1 Griffih s heory Griffih (1924) proposed ha he failure of brile maerials is governed by he iniial presence of microcracks. Under uniaxial ension, he ensile srengh prediced by he Griffih s heory is E' c (2) / where E = E for plane sress problems and E ' E /(1 ) for plane srain problems and E is he Young s modulus, is Poisson s raio, is he spefic surface energy, c is he half crack lengh, and is a numerical consan ( 2/ ). Under uniaxial and biaxial compression, neglecing he influence of fricion on he cracks when closed, and assuming ha ellipical cracks will propagae from he poins of maximum ensile sress concenraion, a sress crierion is obained as (Griffih 1924) 2 8 1 3 1 0 0 3 3 if 3 0 1 if 3 0 1 where 1 and 3 are he wo prinpal sresses. I should be noed ha Griffih s original heory was inended for ensile srengh and Eq. (3) was exended based on assumpions of a parabolic Mohr failure envelope. The primary assumpion of his crierion is ha macroscopic failure is idenical o he iniiaion of cracking from he longes, mos criically oriened crack. I can be seen from Eq. (3) ha he Griffih sress crierion predics a srengh raio of R G = c / = 8. The original Griffih s heory was exended by Murrell (1963) ino hree dimensions o consider he riaxial sae of sress near he ip of fla ellipsoidal cracks. The crierion is wrien as 2 2 2 0 3 3 1 3 2 3 3 1 24 1 2 3 (4) Murrell s failure crierion predics ha he rock compressive srengh equals 12 imes he ensile srengh (i.e. Murrell s srengh raio is R M = 12). The Griffih srengh raios are smaller han generally observed for rocks bu is of he correc order. The reason why he Griffih crieria fail o predic he observed srengh raio of rocks is because he heories are only dealing wih he iniiaion of failure (crack iniiaion) whereas he srengh observaion refer o he final or macroscopic failure (Paerson & Wong 2005). As will be discussed in he nex secion, under ensile condiions, he crack iniiaion is ofen equivalen o oal failure. Under compressive condiions, crack iniiaion happens a a sress level normally much lower han he peak sress. Crack growh under compressive loading is sable and addiional load is required o increase he crack lengh. 2.2 Microfracuring, damage, and failure of brile rocks Direc ension ess can be conduced o observe he crack iniiaion and propagaion process in ension and o deermine he ensile srengh of rocks. Because ensile fracuring under ensile loading is unsable, observaion of he fracuring process is difficul. Despie he difficuly, numerous direc ension ess have been conduced and meaningful resuls obained. I is seen from he es resuls ha he ensile failure process of rock, which includes iniiaion, growh, and propagaion of micro-fracures, happens when he axial sress is close o he peak srengh. For example, he crack iniiaion and propagaion sresses of norie under uniaxial ension are 95% and 97% of he peak ensile srengh, respecively (Bieniawski 1967). Hence, under overall ensile loading, he sress levels of crack iniiaion ( ( ) ) and crack damage ( ( cd ), which here is defined as he sress level a which crack coalescence occurs) are very close o he peak srengh ( ) so ha we can wrie cd 2 (3) ( ) ( ) (5) Several characerisic sress levels have been idenified from laboraory uniaxial compression ess. In he sress-srain relaion shown in Figure 1, cc is he crack closure sress level, is PAPER 4063 3
he crack iniiaion sress level, and cd is he crack damage sress level which is close o he long-erm rock srengh (Bieniawski 1967, Cai e al. 2004, Marin 1993, Marin 1997). AE (acousic emission) raes sar a and increase drasically when cd is reached. The maximum AE rae is usually reached near he peak srengh c. The hree sress levels, i.e.,, cd, and c, represen imporan sages in he developmen of he macroscopic failure process of inac rocks. Microscopic observaions indicae ha newly generaed cracks are ensile in naure, generaed by exension srain, and mosly aligned in he same direcion as he maximum compressive sress. Afer he crack iniiaion, he propagaion of he microcracks (ofen called as Griffih cracks) is a sable process, which means ha he cracks only exend by limied amouns in response o given incremens in sress. Crack iniiaion sars a sress levels of approximaely 1/3 o 2/3 imes he peak uniaxial compressive srengh for mos brile rocks (Brace e al. 1966, Bieniawski 1967, Marin 1997, Cai e al. 2004). In laboraory ess on inac rocks, he crack iniiaion sress or hreshold is defined by he onse of sable crack growh or dilaancy, which can be idenified from he sress volumeric srain curve as he poin of he deparure of he volumeric srain observed a a given mean sress from ha observed in hydrosaic loading o he corresponding pressure (Bieniawski 1967). As can be seen in Figure 1, boh he volumeric srain and he crack volumeric srain plos can be used o idenify he crack iniiaion sress level. can also be defined as he poin where he volumeric srain sars o deviae from he sraigh line in he elasic deformaion sage (Sage II) or he crack volumeric srain deviaes from zero, as shown in Figure 1. AE sars o appear sysemaically when he sress level is above. AE monioring is an alernaive ool o deermine he crack iniiaion sress bu a clear boundary beween background noise and rue crack iniiaion is difficul o be idenified someimes. The volumeric srain or he crack volumeric srain plo provides a more objecive approach for deermining he crack iniiaion sress, espeally under uniaxial compressive loading condiion. / c is an indicaor of rock heerogeneiy and exure. A low / c value indicaes high rock heerogeneiy (Marin 1993, Cai e al. 2004). For heerogeneous rocks such as coarse granie, he crack iniiaion sress level could be as low as 0.3. For fine grain dolomie, he crack iniiaion sress level could be as high as 0.6 (Cai e al. 2004). In he following discussion, is aken, along wih c and Griffih s heory, o link he ensile srengh o he uniaxial compressive srengh of rocks. Figure 1. Sress-srain diagram of a granie showing he sages of crack developmen (afer (Marin 1993)). PAPER 4063 4
3 ESTIMATE OF TENSILE STRENGTH AND HOEK-BROWN PARAMETER m i OF ROCKS FROM UNIAXIAL COMPRESSION TESTS 3.1 Tensile srengh of srong brile rocks In he ligh of he observaions discussed in he previous secions, i is eviden ha when compared o is ensile srengh, a rock ends o have a much higher compressive srengh because microcrack iniiaion (ensile in naure), growh and propagaion in compression is sable and addiional load is required o cause furher crack growh. In conras o sable crack propagaion in compression, microcrack growh and propagaion in a pure ensile sress field is unsable. In ension, he crack iniiaion sress is very close o he peak ensile srengh. In compression, he crack iniiaion sress is lower han he peak uniaxial compressive srengh so ha a modificaion of he Griffih sress crierion assumpion is necessary in order o predic he correc srengh raio ( c / ). Crack iniiaion in ensile loading ofen means ensile fracure is imminen. On he oher hand, crack iniiaion in compression is only linked o a lower sress as compared o he peak compressive srengh. When boh crack iniiaion and peak srengh are considered he same (e.g. Griffih s approach), i leads o a c / raio of 8. However, addiional loading is required o bring he sress level from o c in compression. The raio of c / indirecly reflecs he gap beween crack iniiaion and peak srengh in compression. Hence, i is proposed o esimae he ensile srengh of srong brile rocks from he crack iniiaion sress ( ) and he uniaxial compressive srengh ( c ) using he srengh raio from he following equaion R R G c c 8 Here, he consan R G = 8 is adoped from he Griffih sress crierion (see Eq. (3)). As saed before, he Griffih sress crierion applies o crack iniiaion only so ha / = 8 (i.e., c = holds rue when applying Griffih sress crierion). In realiy, and c are differen. In our proposed approach, c / is used o reflec he difference beween he crack iniiaion sress and he peak srengh in compression. For coarse grained rocks such as coarse granie and diorie, / c is usually beween 0.3 and 0.4. Hence, he srengh raio R would usually be beween 20 and 27 according o Eq. (6). For medium grained dolerie and sandsone, / c is usually beween 0.4 and 0.5 and he srengh raio R will be in he range of 16 o 20. For fine grained silsone, limesone, and sandsone, / c is usually beween 0.5 and 0.7 so ha R is beween 11 and 16. These srengh raios are close o hose obained from laboraory ess. I follows ha he ensile srengh of he rock can be obained from c R 8 (6) (7) 3.2 Hoek-Brown parameer m i of srong brile rocks For srong brile rocks, he Hoek-Brown parameer m i can be approximaed by R, i.e., c m i R 8 The range of values of / c for coarse, medium, and fine grained brile rocks are 0.3 0.4, 0.4 0.5, 0.5 0.6; accordingly, m i values prediced by he proposed mehod are 20 27, 16 20, and 11 16, respecively, generally in good agreemen wih mos daa lised in Table 1. The overall good agreemen of he resuls from our mehod wih hose obained from experimens and suggesed by Hoek is by no means acdenal. Our proposed mehod, based on he assumpions regarding he failure envelopes in ension and compression according o Griffih s work, reveals an imporan linkage among, c,. However, deailed analyses of some published daa revealed ha he esimaed m i (by using m i R ) can deviae from he esed or he suggesed Hoek-Brown parameer m i in Table 1. (8) PAPER 4063 5
This can be aribued o how his parameer is obained. Sricly speaking, he Hoek-Brown parameer m i has o be obained from a series of riaxial compression ess. Alhough i has been shown ha m i is roughly he srengh raio R (see Eq. (1)), exising es daa were mosly in he compression zone. The uniaxial compressive and ensile srenghs from he fied Hoek-Brown envelope may no be exacly he same as he esed ones. In general, he error in uniaxial compressive srengh esimaion by riaxial daa fiing is small as compared o ha for he ensile srengh. Tes daa by Carer e al. (1991) show ha for poash rock and limesone (sof rocks), he fied Hoek-Brown curves overesimae he ensile srengh of he rocks, bu for granie (srong and brile rocks), he curve fiing underesimaes he ensile srengh (Table 2). Tes daa by Alber and Heiland (2001) on limesone show ha he ensile srengh inferred from he fied curve in he compression zone consisenly overesimaes he rue ensile srengh. The saemen expressed by Eq. (1) is rue only if he fied c and are used (columns 5 and 6 in Table 2), no he es values (columns 2 and 3 in Table 2). Table 2 Tes daa and fiing resul summary (daa from (Carer e al. 1991)) (1) (2) Tesed c (MPa) (3) Tesed (MPa) (4) R (es daa) (5) H-B c (MPa) (6) H-B (MPa) Poash rock 25 2 12.5 25.5 6 4 Limesone 52 4 13 57 6.5 8.6 Lac du Bonne granie 226 13 17.4 249 8.5 29 (7) H-B m i As was concluded by Carer e al. (1991), fiing riaxial es daa using he Hoek-Brown failure crierion is very good for hard, srong crysalline rocks. The fi is espeally good a high confining pressure, bu much less so in he low confining pressure and ension zone. Hence, we consider ha he m i value is no a consan bu confining pressure dependen. As shown in Figure 2, wo differen m i values, (m i ) for he ension zone and (m i ) c for he compression zone, can be used in he Hoek-Brown failure crierion. When a rock is subjeced o a high confining sress, microcrack iniiaion from mos preexising defecs in he rock will follow he sress sae defined by 3D Griffih ellipsoidal cracks (Murrell 1963). I is observed ha he esimaed m i from Eq. (8) provides a m i value for he ension o low confinemen zones (e.g. 3 < 5 MPa, for rocks near he underground excavaion boundary, which are of a concern for excavaion sabiliy), since all cracks are assumed o follow 2D Griffih s sress sae. Under a high confining sress sae, ellipsoidal microcrack iniiaion and propagaion will dominae and hence Murrell s srengh raio R M, insead of Griffih s srengh raio R G, is considered more appropriae for he esimaion of m i. I is proposed o esimae he m i value for he high confinemen zone from m i m i c R M c c 12 where he subscrip c in (m i ) c indicaes he m i value for he high compression zone. Hence, he m i value for he high compression zone is 1.5 imes larger han he value (Eq.(8)) for he ension zone (and low confinemen zone) for srong brile rocks. I can be seen from a few examples below and in he nex secion ha he m i values obained from Eq. (9) are close o he riaxial es values of srong crysalline rocks. Tes daa in he ension and low confinemen zone in Figure 2 are from Johnson e al. (1987) for Weserly granie. The Hoek-Brown failure envelope in he compression zone in Figure 2 was based on c = 214 MPa, m i = 26.7, obained from es daa by Heard e al. (1974) (ed in (Hoek & Brown 1980b)). The Hoek-Brown failure envelope ha fis well o he es daa by Johnson e al. (1987) is defined by c = 190 MPa, m i = 14. Clearly, his m i value is smaller han he one obained from he riaxial daa in he compression zone. If he Hoek-Brown envelope, defined by c = 214 MPa, m i = 26.7, is exended o he ension zone (curve A-B in Figure 2), we find ha he ensile srengh of he rock is underesimaed ( = 8 MPa). If we fix c = 214 MPa and change m i o fi he es daa by Johnson e al. (1987), hen, m i = 16 resuls (as indicaed by he blue line (curve B-D) in Figure 2), which is very close o 17.8, a value obained by dividing (m i ) c (= 26.7) by 1.5. (9) PAPER 4063 6
In summary, he m i values esimaed using mi 8 c /, based on 2D Griffih s crack model, need o be considered as he lower bound, applicable o ension and low confinemen zones for srong brile rocks. Since m i is radiionally deermined and applied for riaxial sress saes in compression o a confinemen range of 0.5 o 1.0 of c, he upper bound esimaion using m 12 / provides a beer mach o he riaxial es daa. i c Figure 2. m i values for ension and compression zones for he Weserly granie. (m i ) and (m i ) c are m i values for ension ( 3 < 0) and compression ( 3 > 0) zones, respecively. 3.3 Applicaion examples 3.3.1 Medium o coarse grained granie from he Mine-by unnel The Lac du Bonne granie from he Mine-by unnel in Canada is among he mos horoughly invesigaed rocks in he rock mechanics communiy, boh in he laboraory and in siu. The mechanical properies of he medium o coarse grained pink granie are: elasic modulus 655 GPa, Poisson s raio 0.250.05, uniaxial compressive srengh 21320 MPa, ensile srengh (obained from Brazilian es) 8.91 MPa, crack iniiaion sress 70 o 80 MPa (Marin 1993, Read & Marin 1996, Diederichs 1999). According o Eq. (6), he srengh raio R is found o vary beween 21 and 24. For comparison, he average srengh raio from he es daa is R = 213 / 8.9 24. If only he uniaxial compression es is conduced, we can esimae he ensile srengh as / 8 = (70 o 80) / 8 = 8.75 o 10 MPa wih an average of 9.4 MPa, he Hoek- Brown srengh parameer m i as 34 and 23 for he compression and ension zones, respecively. The m i value for he compression zone is very close o he riaxial compression es daa lised in Marin and Simpson (1994). The m i values from he es daa for he medium o coarse grained granie are 30.8 from he shallow ground and 34.8 from he 240 m level. PAPER 4063 7
3.3.2 Medium grained granie of Lac du Bonne The microscopic fracure process in he Lac du Bonne granie was invesigaed by Lajai (1998) using cyclic loading. The es samples were obained from he Cold Spring Quarry near Lac du Bonne, Manioba, Canada. The mechanical properies of he granie are: elasic modulus 70 GPa, Poisson s raio 0.21, uniaxial compressive srengh 225 MPa, ensile srengh (from Brazilian es) 13.5 MPa. The crack iniiaion sress of he granie sudied by Lajai is 100 MPa (Lajai 1998). According o Eq. (6), he srengh raio is R = 8 225 / 100 = 18, which is very close o he es resul of 225 / 13.5 17. Accordingly, he ensile srengh and he Hoek-Brown srengh parameers m i for his rock can be esimaed as 12.5 MPa, and 27 (compression zone), and 18 (ension zone), respecively from he uniaxial compression es. 412 riaxial ess of he granie a 13 confining pressures (he maximum confining pressure was 40 MPa) were conduced by Carer e al. (1991). The Hoek-Brown parameer m i from he es daa was 29. The esimae of (m i ) c = 27 is very close o he es daa. 3.3.3 Tes daa from a mine sie Laboraory ess were carried ou a he Geomechanics Research Cener of Laurenian Universiy in Canada, using rock samples from a mine sie in Canada and he es resuls are summarized in Table 3. In each uniaxial compressive srengh es, wo srain gauges were insalled o measure he axial and laeral srains. Brazilian ess were used o obain he ensile srengh of he rocks. Two Brazilian es samples were cu righ adjacen o he sample used for he uniaxial compressive srengh es o ensure he comparabiliy of he obained o c. Loading direcions of he wo Brazilian samples were perpendicular o each oher. The ensile srengh ha is corresponding o a paricular UCS sample is obained by aking he average of he wo Brazilian es resuls. In general, he ensile srenghs in he wo orhogonal direcions are very close bu in one case (Sample-11), here is a large difference in he ensile srenghs of he wo Brazilian ess. I is believed ha his is due o he influence of weak planes in he sample. In he uniaxial compression ess, he crack iniiaion sresses are idenified from he volumeric srain plos. Figure 3 shows he sress-srain relaionships for Sample-2. The peak UCS srengh is 83 MPa. No crack closure sress can be idenified from he plo. The crack iniiaion sress is idenified from he volumeric srain plo as 42 MPa. The esimaed m i parameers and ensile srengh are 16 (ension zone), 24 (compression zone), and 5.25 MPa, respecively. The average ensile srengh from he Brazilian es is (6.6+4.8)/2 = 5.7 MPa. In his case, he error in ensile srengh esimaion by he proposed mehod is 7.9%. Figure 3. Sress-sain relaionship of rock Sample - 2. PAPER 4063 8
Figure 4. Sress-sain relaionship of rock Sample - 5. Table 3 Tes resul summary and ensile srengh and m i value predicion for a group of rocks from mine sie in Canada (m i) 6 c Prediced # Rock c / c Srengh Brazilian Brazilian Average Error in ype 1 (MPa) (MPa) raio R 5 (MPa) (a) (MPa) 3 (b) (MPa) 4 (MPa) predicion (%) 1 sumx 70 33 0.47 17.0 25.5 4.13 5.7 3.2 4.45 7.3 2 sumx 83 42 0.51 15.8 23.7 5.25 6.6 4.8 5.7 7.9 3 sumx 91 53 0.58 13.7 20.6 6.63 6.2 7.4 6.8 2.6 4 sumx 115 43.5 0.38 21.1 31.7 5.44 4.9 5.4 5.15 5.6 5 sch 65 52 0.80 10.0 15.0 6.50 8.6 8.9 8.75 25.7 6 sch 69 54 0.78 10.2 15.3 6.75 9.3 10.5 9.9 31.8 7 sch 73 51 0.70 11.5 17.2 6.38 9.4 8 8.7 26.7 8 qe 152 78 0.51 15.6 23.4 9.75 10.1 12.7 11.4 14.5 9 prd 67 47 0.70 11.4 17.1 5.88 7.7 4.7 6.2 5.2 10 prd 155 75 0.48 16.5 24.8 9.38 9.6 7.1 8.35 12.3 11 peg 169 82 0.49 16.5 24.7 10.25 8.6 15.1 11.85 13.5 12 masu 124 84 0.68 11.8 17.7 10.50 13 10.7 11.85 11.4 13 amp 110 64 0.58 13.8 20.6 8.00 10.5 7.3 8.9 10.1 Noe: 1. sumx sulphide mix; sch schis; qe quarzie; prd peridoie; peg pegmaie; masu massive sulphide; amp amphibolies. 2. Deermined from volumeric srain plos. 3. The Brazilian samples are cu righ adjacen o he UCS sample. 4. The loading direcion in Sample b is perpendicular o he loading direcion in Sample a. 5. (m i ) for ension zone ( c / ). 6. (m i) c for compression zone (12 c / ). For some rock ypes, he iniial sage for crack closure is characerized by a prolonged concave porion in he axial sress axial srain plo (Figure 4). We need o firs idenify he linear elasic response from he axial sress axial srain plo by drawing line a-a. This will define he crack closure sress. Line b-b is hen drawn saring from he crack closure sress on he volumeric srain plo. The deviaion poin ha marks he crack iniiaion sress is 52 MPa. The peak UCS srengh of he sample is 65 MPa. The esimaed m i parameer and ensile srengh are 10 (ension zone), 15 (compression zone), and 6.5 MPa, respecively. The average ensile srengh from he Brazilian es is (8.6+8.9)/2 = 8.75 MPa. The error in ensile srengh esimaion by he proposed mehod is 25.7%, which is slighly high bu accepable for ensile srengh esimae of schis. For comparison, he m i values suggesed in Table 1 is 124 for schis. The esimaed ensile srengh and m i values are lised in Table 3 for oher esed samples. For mos cases, he error in ensile srengh esimae is wihin 15%, in line wih he uncerainy PAPER 4063 9
range found by Brace e al. (1966) for deermining. The error is only found o be large for schis which has higher crack closure and crack iniiaion sresses ( / c = 0.7 o 0.8). The exisence of weak seams in schis is also responsible for he large error in ensile srengh predicion. 3.4 Discussion I is noed ha he srengh raios R for boh he pink granies and gray granies (granodiories) from he URL are very close o he es daa bu he m i values esimaed by (m i ) R are smaller han he values suggesed in Table 1. In fac, hey are even smaller han he lower limis suggesed for granies (29) and granodiories (26). m i values in Table 1 are obained from riaxial ess which usually apply very high confinemen sresses and are hence are applicable o he high confinemen zone. The srengh raio R canno be used o esimae accuraely he m i value for he srengh envelope in he high confinemen zone. Esimaion of m i R can only be used in he ension zone (and low confining pressure zone). On he conrary, he m i values esimaed by (m i ) c 12 c / are very close o he es daa and he values suggesed in Table 1. I indirecly proves ha he logic of using 3D Griffih crack model under high confinemen condiion is correc. Hence, mi 12 c / is a good esimae for he Hoek-Brown parameer m i for srong crysalline rocks under high confining sress condiions. Daa from Bell and Jermy (2000) also suppor his noion. The average uniaxial compressive srengh of he doleries is 169.2 MPa (120 samples) and he average ensile srengh is 12.4 MPa. The srengh raio is 13.6. The m i value for he ension zone is 13.6 according o (m i ) R. However, he average m i value from 35 riaxial ess is 18.3, higher han wha is obained from (m i ) R. If we apply he mehodology described here, he average m i value for high confinemen zone would be 1.5 13.6 20, a beer approximaion of he es value. Figure 5 presen he Hoek-Brown srengh envelopes of srong brile rocks wih m i = 20 and 30. The lower value of 20 defines he ensile zone and he low confinemen zone. The higher value of 30, which is 1.5 imes of he lower value of 20, is applicable o he high confinemen condiion. We can see ha here mus be a ransiion zone in he figure ha changes he m i = 20 srengh envelope o he m i = 30 srengh envelope gradually. This ransiion zone is illusraed in he figure in an area confined by lines 1 / 3 = 20 and 30, and Hoek-Brown srengh envelopes defined by m i = 20 and 30. I is seen ha for brile srong rocks wih c = 100 o 200 MPa, he confinemen which divides he low and high confinemen zones is beween 5 o 10 MPa. This observaion is in agreemen wih Paerson and Wong (2005), who sae ha ha below 10 MPa confining pressure, he sress maximum is sill assoaed wih proliferaion and growh of predominanly axial microcracks bu he macroscopic longiudinal slabbing ha ends o develop in uniaxial ess is suppressed. 1 / 3 = consan is called he spalling limi by Kaiser e al. (2000). Spalling limi of rocks defines he boundary over which he sress raio 1 / 3 is greaer han, he rock will fail predominanly in he mode of spalling and slabbing caused mainly by he ensile fracure propagaion under low confinemen condiion. The spalling limi depends on rock heerogeneiy, and for less heerogeneous inac rocks i can be higher han 30. In general, 1 / 3 = 20 o 30 is applicable o some brile rocks. When we consider he mechanism of crack propagaion under differen confinemen condiion, we observe ha he spalling limi for inac rocks can be explained as he ransiion from 2D Griffih crack propagaion under low confinemen condiion o 3D Griffih crack propagaion under high confinemen condiion. When he ransiion zone is considered, he srengh envelope becomes muli-segmened. This ype of muli-segmened srengh envelope had been suggesed by Kaiser e al. (2000) bu our explanaion of 2D o 3D Griffih s crack deformaion behavior ransiion in combinaion wih low and high m i values for low and high confinemen zones provides a new dimension in undersanding his ype of failure crierion. I should be noed ha he srengh envelope presened in Figure 5 may presen difficuly in numerical modeling because he disconinuiy on he gradien and he concave in he zone of ransiion, respecively. These problems should be addressed when one ries o employ hese srengh envelopes in numerical modeling. In addiion, he muli-segmened (Figure 5) srengh envelopes are only applicable o some paricular rocks and canno be generalized. PAPER 4063 10
Figure 5. m i value change and he spalling limi. 4 CONCLUSIONS The ensile srengh of srong brile rocks can be linked o he crack iniiaion sress and he peak srengh c in uniaxial compression ess using he srengh raio of R c / 8 c /. The original Griffih s heory predics well he crack iniiaion in boh ension and compression. However, crack growh in ension is unsable while sable in compression. The raio of c / is employed o accoun for he difference of crack growh or propagaion in uniaxial ension and compression. When he crack iniiaion sress is deermined by AE monioring or volumeric srain measuremen, he ensile srengh can be esimaed by / 8. The proposed mehod, which reveals an imporan linkage among, c, and, is validaed using laboraory es daa. The prediced ensile srenghs are in good agreemen wih he es values, wih error generally less han 15%. The Hoek-Brown srengh parameer m i depends on confining pressure. For pracical esimae of m i, i is found ha mi 12 c / can be applied o srong, brile rocks (e.g. crysalline rocks), applicable o high confinemen zone. For low confinemen o ension zone, espeally for he ension zone, mi 8 c / can be used. Rock ype canno be used direcly o define he srengh raio R and he m i value. Daa inferred from he daabases can only be used when here are no es daa available a he iniial design sage. Whenever possible, laboraory ess should be conduced o deermine he ensile srengh and he m i values more accuraely. The mehod suggesed in his paper provides an easy and ye accurae way o deermine hese wo imporan parameers for brile rocks from convenional uniaxial compression ess. 5 ACKNOWLEDGEMENTS The auhor would like o hank Mr. S.J. Kim, Mr. G. Maybee, and Mr. D. Marr for conducing he laboraory ess a he Geomechanics Research Cenre, MIRARCO Mining Innovaion, Laurenian Universiy, Canada. PAPER 4063 11
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