1 INTRODUCTION

The mechanical properties of rocks are known to vary with sample size (e.g. Durelli & Parks 1962, Lundborg 1967, Lo & Roy 1969, Einstein et al. 1970, Brace 1981, Jackson & Lau 1990, Newman & Bennett 1990, Frantziskonis et al. 1991, Cunha 1993a, Andreev 1995, Ghosh et al. 1995, Goodchild & Wade 2002). This behavior is important because most rock engineering projects require the know-ledge of the rock media response at a relatively large scale. However, the direct measurement of rock properties at a representative scale is often difficult to conduct due to technical or economical con-straints.

The influence of scale manifests itself in various ways (e.g. Cunha 1990, 1993b, Bandis 1990, Hud-son & Harrison 1997). Figure 1 presents schemati-cally how the uniaxial compressive strength σc of rock media typically varies with size. On Figure 1, the unit bock is defined as the maximum size of in-tact rock units in situ (between geological disconti-nuities). For intact rock samples (larger than a few grain diameters dg), Figure 1 shows that a volume increase typically tends to increase the number and length of defects, thus increasing their potential for reducing material strength. At the intact rock scale, macroscopic defects are usually avoided when se-lecting laboratory test samples, so the dominant weakening flaws are micro-cracks and pores (at least for relatively isotropic rocks). These flaws reduce the intact rock strength until it reaches a minimum value (σL) when a sufficiently large size (dL) is

reached. At the rock mass (field) scale, new geologi-cal discontinuities (such as fractures and joints) are introduced, and these often constitute the controlling factors for the failure strength. A larger volume also tends to lead asymptotically to a minimum value of the rock mass strength.

dN

σN

rock mass

rock

unit block

scale corresponding to anisotropic behavior

σL

dLdg dS

σS

Figure 1. Schematic representation of the scale effect on the strength of rock media; σN is the nominal uniaxial compressive strength corresponding to nominal size dN (adapted from Aubertin et al. 1999, 2000).

Standard tests on intact rocks are normally con-

ducted on samples much smaller than the unit block size, which itself is often larger than dL. The corres-ponding minimum strength of intact rock σL (reached at dL) is of importance as it represents the

Influence of scale on the uniaxial compressive strength of brittle rock

L. Li, M. Aubertin, R. Simon & D. Deng Department of Civil, Geological and Mining Engineering, École Polytechnique de Montréal, Montreal, Quebec, Canada

D. Labrie CANMET Mining and Mineral Sciences Laboratories, Natural Resources Canada, Ottawa, Ontario, Canada

ABSTRACT: The prediction of failure conditions has always been a critical issue in rock engineering. One of the factors that makes this a complex problem is the influence of scale (size) on the response of rocks and rock masses. A variety of approaches have been developed and used to estimate the strength of rock media at different scales. These include a scaling function proposed by the authors, which is partly based on the statis-tical variability of uniaxial compressive strength of intact rock samples. In this paper, this approach is revi-sited. A simple method is presented for determining the required parameters for predictive applications, using standard laboratory test results. The validity of the proposed approach is evaluated using test results recently obtained by the authors and with some data taken from the literature.

typical media strength at an intermediate scale (ap-plicable to a small orepass or a venting shaft, for in-stance). The value of σL can also serve to establish the upper bound for the rock mass strength (Auber-tin et al. 2000). The authors have proposed a simple method for estimating the scale dependent strength of intact rocks up to dL, based on standard laboratory tests (Aubertin et al. 2001). This approach is used here, with new laboratory tests results, to illustrate how to estimate both the small (σS) and large scale (σL) strength of intact rocks. These new results have been obtained on hard rock samples collected in a Canadian underground mine located in Abitibi, Quebec, Canada (Labrie et al. 2006, Simon et al. 2006). The approach is also validated using addi-tional data taken from the literature.

2 PHYSICAL PROCESSES AND SCALING FORMULATION

The scale-dependent failure strength of brittle and semi-brittle materials has been linked mainly to the energy field and energy dissipation around cracks, and to statistical effects due to random strength and defect distribution in the media (e.g. Hawkes & Mellor 1970, Jaeger & Cook 1979, Ingraffea 1987, Carter 1992, Bazant 1993, Tsoutrelix & Exadaktylos 1993, Bazant & Planas 1998, Fox et al. 2001, Han-son & Ingraffea 2003, Castelli et al. 2003, Lobo-Guerrero & Vallejo 2006). The magnitude of the size effects depends on the deformation processes involved, which in turn depend on the stress state. The influence of scale is generally more pronounced in the brittle field where microcrack initiation and propagation dominate. The size effect tends to de-crease under a semi-brittle behavior, where micro-cracking is coupled with intragranular dislocation motion and other ductile processes (Aubertin et al. 1998). The size dependency almost entirely disap-pears in the fully plastic regime of inelastic beha-vior, where isovolumetric deformation processes dominate. Accordingly, the influence of sample size can be reduced substantially by applying a large confining pressure, as the latter tends to limit crack-ing and promotes more ductile deformation mechan-isms (Aubertin et al. 1998, 1999, 2000). It has also been observed that the influence of scale is more pronounced in tension than in compression, at least for brittle materials.

With brittle rocks, experimental evidence indi-cates that significant size effects cannot be neg-lected. This applies to the uniaxial compressive strength, which is significantly reduced when the sample volume is increased (e.g. Pratt et al. 1972). This observation may be related to the fact that rock strength tends to decrease with the number and size of defects (e.g. Wong & Chau 1996, 1998), each of which can increase in larger specimens.

As shown in the schematic representation of Figure 1, the sample strength decreases until it reaches a minimum value σL at size dL. Beyond dL, the effect of scale practically disappears, unless new types of defects are introduced (such as joint sets in a rock mass). The large size intact rock strength, σL, can be much lower than the failure strength at the usual specimen scale. Likewise, specimen-scale strength is less than the strength measured on even smaller samples, which can increase to the maxi-mum strength, σS, corresponding to the smallest size dS which still shows an isotropic behavior (and hence is not controlled by the influence of grain boundaries).

Many functions exist to describe, and sometimes predict, the above mentioned size effect (Evans & Pomeroy 1958, Jaeger & Cook 1979, Hoek & Brown 1980, Bazant 1993, Tsoutrelis & Exadaktylos 1993). For instance, various researchers have shown that the progressive decrease of uniaxial compres-sive strength can be empirically related to the in-creasing diameter dN (in mm) of the sample using a power-law function. Such expressions can be written as (Hoek & Brown 1980, Cunha 1990):

β

⎟⎟⎠

⎞⎜⎜⎝

⎛σ=σ

Nc50N

50d

(1)

where σc50 represents the uniaxial compressive strength obtained on standard size samples (with a diameter of 50 mm). In this equation, the exponent β was fixed at 0.18 by Hoek & Brown (1980), while Cunha (1990) proposed a value of 0.22. As shown in Figure 2, this equation describes quite well typical results obtained on relatively small scale samples. However, Equation 1 incorrectly predicts that the strength decreases indefinitely with an increasing size. This equation also indicates that the strength can become extremely large when the sample size is reduced much below the standard size. To overcome these limitations, the authors have proposed a power law function bounded by realistic values for the small and large scale strengths. This function can be expressed as follows for the nominal strength (Aubertin et al. 2002):

( )λ

−−

σ−σ+σ=σSL

NLLSLN

dddd (2a)

or in a normalized form (for σc50 corresponding to d50 = 50 mm):

λ

−−

σσ−σ

+σσ

=σσ

SL

NL

c50

LS

c50

L

c50

N

dddd (2b)

In these equations, dL is taken as the reference size (diameter in laboratory tests) at which the minimum uniaxial compressive strength σL is reached; dS is the size corresponding to the maximum isotropic

strength σS. In Equation 2 (a and b), ⟨ ⟩ are the MacCauley brackets (⟨x⟩ = (x + |x|)/2 ≥ 0); here, their use means that the corresponding term disappears when size dN becomes larger than dL). The exponent λ controls the non linearity of the relationship, and its value may depend on material characteristics and in some cases, on the sample geometry. Various re-sults analyzed by the authors indicate that this expo-nent value is fairly uniform for different types of rock, so it can be taken as a constant (λ = 20) for most practical applications. Equation 2 implies that the size effect function of intact rocks tends toward two limiting strength values (i.e. σL ≤ σN ≤ σS) when the scale increases from a few grains to the large unit block size. Figure 3 shows schematically the general shape of this function, with the influence of parameters λ (Fig. 3a), dL (Fig. 3b), and σL (Fig. 3c).

Figure 2 introduced above also shows the appli-cation of Equation 2 to a few test results taken from the literature. It can be seen that the proposed equa-tion also describes these experimental results quite well. In this case, the values of dS and dL (and of σS and σL) are not known before hand. These have been estimated based on the limited number of results (obtained on different types of rocks) given in Fig-ure 2. In the following section, a more specific ap-proach is proposed to establish the value of the pa-rameters to be used in Equation 2.

Figure 2. Application of two size functions (Equations 1 and 2) to test results obtained on a variety of rocks (data taken from Cunha 1990).

3 PARAMETER DETERMINATION

The application of Equation 2 requires that the value of five parameters be estimated: σS, σL, dS, dL and λ. These could be determined directly through a tar-geted testing campaign, but this is not a commonly used approach in rock engineering. For practical ap-plications, a simple and pragmatic approach is pro-posed (in the absence of relevant specific data).

a)

Log d N

σN

σS

σL

d L

λ = λ1

d 50d S

λ = λ3

λ = λ2

λ1 > λ2 > λ3

σc50

b) Log d N

σN

σS

σL

d L = d L1

d 50d S

d L = d L2

d L = d L3

d L1 > d L2 > d L3

σc50

c)

Log d N

σN

σS

d Ld 50d S

σL = σL2

σL1 > σL2 > σ L3

σL = σL3

σL = σL1

σc50

Figures 3. Influence of parameters λ (a), dL (b), and σL (c) on the shape of the scaling function (Equation 2) describing the strength of intact rocks.

First, the following simple predictive equation

has been developed for estimating the large scale strength of intact rocks (Aubertin et al. 2001):

σcL = x1( c50σ + x2 S0) (3)

where σcL is the large scale uniaxial compressive strength of rock, c50σ is the average observed mean value of the uniaxial compressive strength on stan-dard size specimens (50 mm), S0 is the correspond-ing standard deviation of the test results (when a minimum of about 10 results are available), and x1 and x2 are two statistically obtained parameters (see

0.5

1

1.5

0 1 2 3 4 5d N / d 50

σN / σc50Eq. (2)β = 0.22β = 0.18NoriteMarbleBasaltGraniteLimestoneQuartz dioriteBasalt-andesite lavaGrabbro

λ = 20d L/d 50 = 40d S/d 50 = 0.1

σL/σc50 = 0.7

σS/σc50 = 1.2

σL/σc50

Eq. (1)

details in Aubertin et al. 2001). This equation was applied by Li et al. (2001) for the analyses of the URL tunnel, with x1 ≅ 0.08 and x2 ≅ 5 to 6. It will be used below with new test results and Equation 2.

On the other hand, the small scale strength σS may appear easier to obtain experimentally, but again, it is seldom determined by direct measure-ments. Because of this, the authors have also devel-oped a simple statistical approach to estimate this parameter from laboratory tests conducted on stan-dard size samples. The approach assumes that for a low enough probability (which depends on the num-ber of tests available, n), the statistical upper bound value of the strength may correspond to the maxi-mum strength value. This is deemed to provide a representative value for the small scale uniaxial compressive strength of the rock to be used in Equa-tion 2. The estimate is based on the following ex-pression:

0,2/

ccS 1S

nt

−+σ=σ να (4)

This equation represents a particular form of the up-per bound value expression, which can be found in many statistical textbooks (e.g. Spiegel 1961, Bun-tinas & Funk 2005). This equation is applicable to a small random population (sample) having a normal distribution. In Equation 4, the value of tα/2,ν and σcS (i.e. σS in uniaxial compression) depends upon the selected confidence level (1-α) (or cumulative prob-ability) and on the number of degrees of freedom ν (= 1-n). Here, cσ is the mean uniaxial compressive strength obtained from a minimum number of labor-atory tests (n.b. smaller samples can be used if stan-dard size samples are not available), and S0 is the corresponding standard deviation. The statistical Students’ law parameter tα/2,ν is defined from the following equation (e.g. Spiegel 1961, Buntinas & Funk 2005):

α−=∫ ρα

α−1d)(

2

2

/

/

t

tvt (5)

where t is the Student distribution function. The other 3 parameters (dS, dL and λ) have been

found to take values that are almost the same for various types of rocks (having a small grain size). Hence, these will become constants in the following applications of Equation 2.

4 LABORATORY TESTS 4.1 New results As part of a research project aimed at evaluating the strength of hard rocks with respect to scale, several parallel holes of different calibers were cored with a diamond drill (Labrie et al. 2006). The coring was done in 2005 at the CANMET Experimental Mine

near Val-d’Or, Quebec. Four calibers were used for the holes (AQ – 29.8 mm, BQ – 36.3 mm, NQ – 47.4 mm, and HQ – 63.3 mm) which were between 6 and 9 m in length. The drilling was performed at three locations in the mine, at depths of 40 m, 70 m, and 130 m. Rock cores were recovered, prepared, and tested at the rock mechanics laboratory of the École Polytechnique de Montréal (Simon et al. 2006). The uniaxial compression testing program was conducted on cores from each hole. The main results obtained on n standard size samples are given in Table 1.

Calculation results indicate that the minimum large scale uniaxial compressive strength of the rock σL (at a depth of 40 m) is equal to 23.1 MPa, based on Equation 3 with x1 = 0.08 and x2 = 5.5. This sig-nificant strength reduction (from a mean uniaxial compressive strength value cσ of 104.8 MPa) is in accordance with laboratory test results obtained on these same rock samples (see below) and also with stiffness measurements made in situ with a dilato-meter (Labrie et al. 2006, Simon et al. 2006). Then, using Equations 4 and 5 with tα/2,ν = 4.781, corres-ponding to a confidence level of 99.9% (chosen as a function of the small number of samples tested), one obtains a small scale strength σS of 158.2 MPa. These small and large scale strengths correspond to sizes that have been fixed at dS = 5 mm and dL = 2000 mm, respectively. As mentioned above, the analysis of these and various other results indicates that the λ value can be taken as a constant (λ = 20), as most back calculations lead to a value close to this preset exponent.

Table 1. Uniaxial compressive strength test results obtained on standard size (NQ) samples at the three depths; the large and small scale strengths are also given (the latter is based on a confidence level of 99.9%; experimental data taken from Si-mon et al. 2006). __________________________________________________ Location n cσ † S0

† σL † tα/2,ν σS

† __________________________________________________ at 40 m 10 104.8 33.5 23.1 4.781 158.2 at 70 m 10 168.2 37.6 30.0 4.781 228.2 at 130 m 8 133.4 33.1 25.2 5.408 201.1 __________________________________________________ † in MPa

Figure 4 shows that the proposed Equation 2 with

the above mentioned parameters represents the expe-rimental results well for the other sample sizes tested (AQ, BQ, HQ). The same procedure was ap-plied to the testing results obtained for rock cores sampled at a depth of 70 m and 130 m. The esti-mated parameters are also given in Table 1. The ap-plication of Equation 2 to the laboratory test results is presented in Figures 5 and 6. Again, a good agreement is obtained for both cases between the es-timated and the experimental results. This consti-tutes a partial (limited) validation of the approach introduced above for defining the scaling function.

4.2 Data taken from the literature Figures 7 to 9 show the application of the proposed procedure to other rocks using data taken from the literature. The experimental data and parameters are given in Table 2. These results were retained be-cause they provided information on the response of rocks from the small to the large scale.

As can be seen in Figures 7 to 9, the scaling func-tion proposed here, with the parameters obtained from the proposed determinative approach, appears to well represent the experimental results measured on these types of rock, even if these are quite differ-ent in nature.

Table 2. Uniaxial compressive strength of laboratory test sam-ples (with a level of confidence of 99.9%). __________________________________________________ Fig. n dN (cm) cσ † S0

† σL † tα/2,ν σS

† __________________________________________________ 7 10 2.5 41.9 5.9 6.0 4.781 51.3 8 11 4.6 29.6 16.8 10.4 4.587 56.0 9 11 2.5 58.0 11.9 9.9 4.587 75.3 __________________________________________________ † in MPa.

Figure 4. Representation of the scaling function (arithmetic and semi-log scales) for the rock sampled at the 40 m level (see Table 1).

Figure 5. Representation of the scaling function(arithmetic and semi-log scales) for the rock sampled at the 70 m level (see Table 1).

Figure 6. Representation of the scaling function (arithmetic and semi-log scales) for the rock sampled at the 130 m level (see Table 1).

5 DISCUSSION

The results shown above tend to indicate, among other observations, that the large scale strength of rocks at size dL can often be significantly lower than the value obtained on standard size specimens. The magnitude of the strength reduction is compatible with other results ensuing from direct measurements conducted on relatively large scale samples (e.g. Jahns 1966, Bieniawski 1968, Pratt et al. 1972, Herget & Unrug 1974, Singh 1981, Swolfs 1983, Natau et al. 1983, 1995). The low strength values obtained here are also in line with the reduction ob-served on the in situ modulus at a larger scale (Simon et al. 2006, Labrie et al. 2006). The modulus is known to be correlated to the uniaxial strength of rocks and other geomaterials (e.g. Peck 1976).

The relatively low rock strength at a large scale, deduced from the analysis presented here, may help to explain some of the observations made around underground openings, which indicated that some failure had been observed close to the excavation boundaries where the stress state was much lower than the lower bound strength value expected from commonly used analyses (e.g. Martin 1997, Li et al. 2001). It should be recalled here that the phenome-non of strength reduction associated with an increas-ing scale is limited mostly to the region close to the opening boundary, as the larger mean (confining) stress encountered deeper in the rock mass tends to reduce the influence of size on the material strength (Aubertin et al. 2000, 2001). The rock at depth with-in a confined rock mass may thus remain very strong at a large size. It is only when creating new open-ings, within regions of lower mean stress (which tends to promote the above described scale effect), that this scaling reduction is observed.

0

50

100

150

200

0 500 1000 1500 2000d N (mm)

σN (MPa)

0

50

100

150

200

1 100 10000

Eq. 2dataestimated

0

50

100

150

200

250

0 500 1000 1500 2000d N (mm)

σN (MPa)

050

100150200250

1 100 10000

Eq. 2dataestimated

0

50

100

150

200

250

0 500 1000 1500 2000d N (mm)

σN (MPa)

050

100150200250

1 100 10000

Eq. 2

data

estimated

Figure 7. Representation of the scaling function (with a close up) for a coal (data taken from Bieniawski 1968; see Table 2).

Figure 8. Representation of the scaling function (with a close up) for a coal (data taken from Singh 1981; see Table 2).

Figure 9. Representation of the scaling function (arithmetic and semi-log scales) for Cedar City quartz diorite (data taken from Pratt et al. 1972; see Table 2); the equivalent diameter was used with Equation 2 for these prism samples, based on dN = 2[V/(πh)]1/2 (V and h are the volume and height of prism sam-ples, respectively).

There are nonetheless a number of uncertainties that exist when trying to predict the large (and small) scale strength based on results obtained from standard laboratory specimens. For instance, in the proposed approach, the margin of error on the mean and on the standard deviation (Aubertin et al. 2001), which depends on the available number of tests n, could play a major role. For the approach to be va-lid, the margin of error must be within a certain range, which requires that a minimum number of tests be conducted on the same sample set (generally 10 tests or more; e.g. Gill et al. 2003). A minimum number of results is required to help insure that the data reflect the natural variability of rock properties. This is important here because the proposed ap-proach assumes that small and large scale strengths can be statistically related to the weak and strong portions of the strength distribution curve of a given rock. Hence, the proposed methodology could not be strictly applied to results shown in Figure 2, because too few tests are available for each type of rock. This is why in this case, a simple regression was used to obtain the relevant parameters. This may ex-plain why the large scale strength appears to be much higher when compared to more complete data sets.

On the other hand, it should also be noted that Equation 3, used here to obtain σcL for the different rock types, is somewhat empirical, although it was deduced initially from a representative statistical analysis of the data (Aubertin et al. 2001) which in-cluded the evaluation of the margin of error for both the mean and the standard deviation. At this point however, it is not practical to base the estimate on the use of these two margins because these are much too dependent on the number of test results, espe-cially when n is small (as is the case here). This means that a few additional (or missing) tests may be sufficient to tip the balance toward an unrealistic value for σcL. Hence, it was deemed more realistic for the general scaling function to assess the value of σcL with Equation 3 (which is more robust), instead of using the more complete formulation introduced previously. More work is underway to validate the proposed semi-empirical relationship, and to assess its general validity from a more complete statistical evaluation. It should be noted nonetheless that ob-taining a precise estimate will always remain diffi-cult for a testing program with a small number of samples.

More work is warranted in this area because the large scale strength of intact rock is of particular in-terest in many applications (Aubertin et al. 2000, 2001).

0

10

20

30

40

50

60

0 50 100 150 200d N (cm)

σN (MPa)

0

20

40

60

0 10 20 30 40 50

Eq. 2exp. dataestimated

0

10

20

30

40

50

60

70

0 50 100 150 200d N (cm)

σN (MPa)

0

20

40

60

0 10 20 30 40 50

Eq. 2exp. dataestimated

0

20

40

60

80

100

0 50 100 150 200d N (cm)

σN (MPa)

020406080

100

0.1 1 10 100 1000

Eq. 2estimatedexp. data

6 CONCLUSION

In this paper, a scaling function proposed by the au-thors is revised and the procedure to estimate the model parameters is defined. These parameters in-clude the small and large scale sizes (dS and dL) that correspond to the limiting (maximum and minimum) strength values of intact rocks. These sizes can be given constant values of 5 mm and 2 m, respective-ly. The procedure also includes an estimate of the small and large scale strengths based on common la-boratory tests. A good agreement has been obtained between the predictions obtained from the proposed scaling function and the experimental results. The proposed scaling function and corresponding para-meter estimation procedure can thus be considered a useful tool for rock engineering applications.

ACKNOWLEDGEMENTS

The work has been financed in part by IRSST and by the Industrial NSERC Polytechnique-UQAT Chair on Environment and Mine Wastes Manage-ment (http://www.polymtl.ca/enviro-geremi/). These contributions are gratefully acknowledged. The au-thors also thank Dr. John Molson for his help in im-proving the manuscript.

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