Mechanics of Fracture Rocks

7/30/2019 Mechanics of Fracture Rocks

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Ranjith, P.G., Siew Foong, P., Hefny, A.M., Zhao, J. Strength and mechanics of fractured rocks under triaxial loadings.

ISRM 2003 Technology Roadmap for Rock Mechanics, South African Institute of Mining and Metallurgy, 2003.

Strength and mechanics of fractured rocks under triaxial

loadings

P.G. RANJITH F.P. SIEW A.M. HEFNY J. ZHAO

School of Civil and Environmental Engineering, Nanyang Technological University, Singapore

Fracture initiation and propagation, being one of the most intensive subjects in rock mechanics;

contribute significantly to the deformation of jointed rock masses. An experimental study on

fractured granitic rock was carried out to determine the peak strength of specimen under triaxial

loading conditions. The commonly observed failure mode for rock specimens in this study was

found to be the shearing of the plane of weakness. The minimum peak strength of fractured rock is

observed when the joint orientation is approximately 70o to the horizontal axis. Based onexperimental test data, an empirical expression was developed to accommodate the effect of joint

orientation and joint trace length in estimating the peak strength of fractured rocks. The predictedvalues of peak strength using the proposed equation well agree with the experimental results

carried out on singly fractured specimens under triaxial loading conditions. The threshold stress

values of crack initiation and propagation depend on joint geometrical parameters, their degree of

interconnectivity, as well as surrounding stresses on the fracture plane.

Introduction

In Singapore, due to the scarcity of land, underground

space has been utilized for the development of storage

facilities and deep sewerage system in rocks. Thus, acomprehensive understanding in the mechanism and the

behaviour of rock structure is required as the ultimateobjective is to control rock displacement into and around

the underground excavations. In addition, reliable

estimates of the shear strength and deformation

characteristics of a rock mass are required for analysis of

slopes and foundation.

This research program presents the effects of stresses on

the deformation characteristics of a single rock fracture

subjected to triaxial loading states. Furthermore, the study

investigates the effects of joint orientation and joint length

on the stress-strain behaviour of jointed rocks as well asthe threshold stress values of crack initiation and

propagation of fractured specimens.

Effects of joint orientation on the compressive

strength of rocks

The overall strength and permeability of rock a mass and

the stability of engineering structures are greatly

influenced by joint orientations (Ranjith, 2000).Higher the

interconnectivity of fractures which in turn lower the shear

strength, the greater will be the risk of failure of a rockmass. Therefore, it is fundamentally important to study the

influence of joint orientations on stress-strain

characteristics of rocks under different loading conditions.

The peak strength developed by transversely isotropic

rocks in triaxial compression vary with the orientation of

the plane of anisotropy, foliation plane or plane of

weakness, with respect to the principal stress direction

(Donath 1972, McLamore Gray, 1967). Figure 1 shows

measured variations in axial stress with the angle of

inclination () of the major principal stress to the plane ofweakness.

Brady and Brown (1994) introduced an instructive analysisof a case in which the rock contained a well-defined,

parallel plane of weakness whose normal was inclined at

an angle (Figure 2) to the major principal stressdirection.

Each plane of weakness has a limiting shear strength (s)defined by Coulombs criterion

wnw cs tan+= [1]

where, w = friction angle of the plane of weaknesscw= shear strength parameter of the plane of

weakness

n= normal stress

The stress transformation equations give the normal (n)

and shear () stresses on the weakness plane as:

)-()(n 2cos2

1

2

13131 ++= [2]

)( 2sin2

131 = [3]

1, 3 =major and minor principal stresses

Substituting for n into Equation [1], putting s=, andrearranging, Equation [4] gives the criterion for slip on the

plane of weakness.

)-()(c)-(

w

wws2sincottan1

tan2 331 += [4]


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928 ISRM 2003 TECHNOLOGY ROADMAP FOR ROCK MECHANICS

Figure 1: Variation of peak strength with the angle of

inclination of the major principal stress to the plane of

weakness for the confining pressure (Donath 1972,

McLamore Gray, 1967).

Figure 2:Variation of peak strength at constant confining

pressure with the angle, .

The principal stress difference required to produce slip

tends to infinity as 90 and as w. Between these

values of, slip on the plane of weakness is possible. Bydifferentiation, it is found that the minimum strength

occurs when

24

w += [5]

The variation of peak strength with the angle predicted

by this theory is illustrated in Figure 2.

In view of the deficiency of the original Hoek-Brown

criterion when applied to jointed rock mass, Hoek et al.

(1992) modified the criterion to account for the effect of

fractures on the strength of rocks. The modified criterion

conforms to the strength prediction given by the original

criterion, for different stress conditions, and predicts a

tensile strength of zero for a rock mass. The modified

criterion is expressed in the following form:a

c

bc

m

+=

331 [6]

where, mb and a are the constants for fractured rock.

McLamore and Gray (1967) suggested a genetic

classification of anisotropy based on the shape of the

anisotropy curve between compressive strength and joint

orientation angle as shown in Figure 3. The planar type of

anisotropy (Figure 3a, Figure 3b) is the result of cleavage

whereas the linear type of anisotropy or the bedding plane

type of anisotropy (Figure 3c) is due to the weakness of

rock along the bedding planes.

Fracture initiation

According to Eberhardt et al. (1997), the point where

majority of fractures began to initiate is defined as thecrack initiation threshold. With increasing loads, further

cracking is observed to initiate intragranularly within the

stronger quartz grains for the case of granitic rocks. This

point is identified as the secondary cracking threshold.

Brace et al. (1966:3948) and Latjai and Latjai (1974)

define the point where the axial stress versus lateral strain

curve departs from linearity as the initiation of the

microcracking process which is referred to as the crackinitiation stress threshold. This point represents the stress

at which a significant number of critically orientated

cracks initiate and propagate in the direction of major

principal stress, (1).

Noting the difficulty in using lateral strain data, especially

in damaged samples, several researchers including Martin

and Chandler (1994) suggested using the calculated crackvolumetric strain to identify crack initiation. In this

respect, crack initiation can be defined as the stress level at

which dilation begins in the crack volume.

Figure 3: Classification of anisotropy (McLamore and

Gray, 1967).

Test program

The rock specimens of cylindrical shape with

approximately 47.5 mm in diameter and 97 mm in heightwere used in the testing program. Two types of pink Bukit

Timah granitic specimens found in

Singapore were used for the experimental study: (a) intact

and (b) a single fracture. Six numbers of intact specimens

and twenty-seven numbers of fractured specimens with a

single fracture and different joint orientations (i.e. =60-

90 from the horizontal axis) were experimentally testedusing a high pressure triaxial testing apparatus. Prior totesting, all visible surface fractures in samples were

mapped relative to some co-ordinates axes. The confining

pressures applied to the fractured specimens during the

testing were 5, 20, 30 and 40MPa, for all the joint

orientations. A schematic diagram of the triaxial testing

facility used for this study is shown in Figure 4.


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Table 1: Peak strength of singly fractured granitic rocks: experimental and theoretical values.

Theoretical values Experimental valuesFailure

criterion

Confining

pressure, 3(MPa)

Joint

orientation

(o)

Trace length,x

(m) Peak strength

1 (MPa)

Failure

load Pf

(kN)

Peak strength

1 (MPa)

Failure

load Pf

(kN)

60 0.080 148.76 263.61 145.83 259.50

70 0.058 146.81 260.15 116.79 206.95

80 0.100 221.14 391.86 186.49 330.465

90 0.096 1827.09 3237.61 235.56 410.41

60 0.079 222.96 395.08 238.34 422.33

70 0.066 220.20 390.20 148.34 262.86

80 0.115 325.14 576.15 286.33 507.3820

90 0.095 2592.38 4593.71 311.47 556.58

60 0.080 272.41 482.71 262.53 465.20

70 0.075 269.12 476.88 207.40 367.50

80 0.100 394.45 698.96 306.40 542.94

30

90 0.100 3102.38 5497.43 400.10 697.10

60 0.097 321.86 570.33 351.79 623.37

70 0.077 318.03 563.55 327.73 578.29

80 0.098 463.76 821.78 475.22 842.09

Modified

Mohr-

Coulomb

Equation

[4]

40

90 0.100 3612.38 6401.15 507.63 873.20

60 0.080 145.83 259.50

70 0.058 116.79 206.95

80 0.100 186.49 330.465

90 0.096

130.53 231.30

235.56 410.41

60 0.079 238.34 422.33

70 0.066 148.34 262.86

80 0.115 286.33 507.3820

90 0.095

223.93 396.80

311.47 556.58

60 0.080 262.53 465.20

70 0.075 207.40 367.50

80 0.100 306.40 542.9430

90 0.100

265.02 469.62

400.10 697.10

60 0.097 351.79 623.37

70 0.077 327.73 578.29

80 0.098 475.22 842.09

Modified

Hoek-

Brown

Equation

[6]

40

90 0.100

299.92 531.45

507.63 873.20

In Table 1, the modified Mohr-Coulomb criterion is not applicable for = 90 because the function cot is not

defined at = 90. Therefore, theoretical values estimated by modified Mohr Coulomb theory at = 90 shouldnot be compared with the experimental values.


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Table 2: Peak strength of granitic rocks with a single fracture: theoretical (Equation [11]) and experimental values.

Theoretical values Experimental valuesFailure

criterion

Confining

pressure, 3(MPa)

Joint

orientation

(o)

Trace

length,x

(m)

Peak strength

1 (MPa)

Failure

load Pf

(kN)

Peak strength

1 (MPa)

Failure

load Pf

(kN)

Degree of

accuracy

(%)

60 0.080 120.99 214.39 145.83 259.50 83.0

70 0.058 76.35 135.29 116.79 206.95 65.4

80 0.100 152.30 269.88 186.49 330.46 81.75

90 0.096 139.78 247.69 235.56 410.41 59.3

60 0.079 204.99 363.24 238.34 422.33 86.0

70 0.066 153.03 271.17 148.34 262.86 103.2

80 0.115 335.25* 594.06 286.33 507.38 117.1*

20

90 0.095 234.97 416.37 311.47 556.58 75.4

60 0.080 247.15 437.95 262.53 465.20 94.1

70 0.075 210.89 373.70 207.40 367.50 101.7

80 0.100 305.78 541.84 306.40 542.94 99.830

90 0.100 301.59 534.42 400.10 697.10 75.460 0.097 368.22 652.49 351.79 623.37 104.7

70 0.077 249.46 442.04 327.73 578.29 76.1

80 0.098 333.99 591.83 475.22 842.09 70.3

(Equation

[11])

developed

by authors

40

90 0.100 340.36 603.12 507.63 873.20 67.0

*overestimated theoretical values.

Figure 4: A schematic diagram of triaxial testing facility.

Test results and discussions

Effects of joint orientations on the compressive

strength of rock

According to Equation [4], modified Mohr-Coulomb

theory incorporates the effect of joint orientations but not

the joint trace length. Nonetheless, neither one of the

parameters are incorporated into Equation [6], which

depicted the modified Hoek-Brown criterion. Theoretical

peak strength values calculated using modified Mohr-

Coulomb criterion and modified Hoek-Brown criterion are

compared with the experimental values obtained from

triaxial testing on fractured granitic rocks, as given inTable 1. The theoretical values of peak strength and failure

load as predicted by the modified Mohr-Coulomb are

relatively consistent for joint orientations of 60 and 70.

However, the experimental values for 90 are well out the

predicted range. This is because in Equation [4], the

function cot in the denominator is converged to zero as

the angles approaches 90 and to infinity for the case of

=0. The modified Mohr-Coulomb would thereforesignificantly overestimate the predicted peak strength of

fractured rocks under such circumstances (i.e., =90 and0).

The theoretical critical joint orientation for granitic rock

with a single fracture at minimum peak strength is

calculated as 65 (Line AB-Figure 5) with w of 40.7(Equation [5]). Experimental results show that the critical

joint orientation, which gives minimum peak strength, is

found to be approximately 70 (Line CD-Figure 5). There

is a small discrepancy of 5 between the experimental andtheoretical value, and the reason for this is explained

below.

The shape of the curves as obtained in Figure 5 is now

compared to the genetic classification of anisotropy as

suggested by McLamore and Gray (1967:65). It can be

observed that the curves produced by the modified Mohr-

Coulomb criterion exhibit the cleavage or planar type of

anisotropy, which is further categorized as the U-type

(Figure 3a). However, for the curves obtained

experimentally, they are generally categorized under the

bedding plane type, which is of the shoulder-type (Figure

3c). A smooth shoulder formation on either end of the U-shape suggests a sliding mode failure of the specimen in

the U-shape region and splitting in the shoulder region.

Thus, the occurrence of the small discrepancy in critical

joint orientation between the experimental and theoretical


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result is due to the variation of the planar type that

influences the critical joint orientation.

Figure 5:Effects of joint orientations on peak strength of

rocks at different confining pressures.

The modified Hoek-Brown criterion predicted a set of

peak strengths for the given confining pressures regardless

of joint orientations for any given granitic rock with a

single fracture (Table 1). Experimental values of peak

strength and failure load for joint orientation of 60 is

relatively consistent with theoretical values obtained from

the modified Hoek-Brown criterion but not with 70, 80

and 90. Moreover, the experimental values for the joint

orientation of 90 is observed to be consistent with the

theoretical values for intact granitic rocks obtained using

Hoek-Brown criterion. This is only applicable when the

joint oriented parallel to the line of the axial load as well as

the joint trace length distributes along the specimen from

top to the bottom of the specimen. Thus, it can be

concluded that singly fractured granitic rocks having a

joint orientation of 90 shows similar strength

characteristics as observed in intact rocks under same

boundary conditions. This is because with the fracture

starting from the top to the bottom of the specimen in astraight line, it is difficult to mobilize opening or closing

of the fracture or shearing the fracture along its plane of

weakness.

The inconsistency of the theoretical and experimental peak

strengths for joint orientation of 70 is probably due to the

fact that 70 has been the experimental critical joint

orientation, which gives minimum peak strength as

discussed in Figure 5. Thus, the rock specimens with

critical joint orientation would experience lower peak

strength as compared to the theoretical values because it

has a weak plane due to the critical joint orientation.

Therefore, it can be concluded that the peak strength at the

joint orientation of 70 for any given confining pressure isthe lowest (Figure 5).

Neither the original Hoek-Brown criterion nor the

modified Hoek-Brown criterion incorporates both joint

orientations and the joint trace length of rock fractures into

the relevant equations. The parameters in Equation [6] for

granitic rock with a single fracture are determined as, a =

0.35, c = 173.3MPa and mb = 13.8. With reference toFigure 5, it is observed that the effect of joint orientationin relation to the peak strength takes a form of sinusoidalcurve. Thus, the joint orientation of fracture can be

incorporated into Equation [6] as a function of sin. The

resulting equation is:

sin

)(48.71 35.0331

+= [7]

A back-analysis is used to obtain a relationship between

the variable k in Equation [8] and the trace length of

fracture,x.

k

sin

)(48.71 35.0331

+= [8]

For a given values of3= 5MPa, 1= 145.83MPa and =

60, k is estimated as 0.97. The variable (k) has beenplotted against the trace length of fractures in order to

develop a relationship between these two parameters

(Figure 6).

Figure 6:Variable kversus joint trace length.

The exponential function is chosen to represent the

variable (k) and trace length (x) due to the fact that the

function is able to produce the highest correlation with

respect to the relationship between kandx, as compared to

other functions such as polynomial (Figure 6). Taking an

average of the four exponential functions obtained with an

optimum factor of safety of 1.3, the following relationship

between kandxis developed as follows:x

ek377.18

1839.0= [9]

Equation [9] is modified to account for the effect of

different joint orientations and trace length of fractures as

given below:

sin

)(48.71 35.0331

+= (0.1839e18.377x) [10]

It can be seen from Table 2 that the modified Equation

[10] which includes the effect of joint orientations and

joint trace length, is able to estimate peak strength at

various joint orientations. In contrast, the modified Hoek-

Brown criterion is independent ofxand (Equation [6]).Using Equation [10], theoretical peak strength can now be

predicted and be compared with the experimental peak

strengths. The results are as depicted in Table 3.

The degree of accuracy in predicting the peak strength of

granitic rock with a single fracture using Equation [10],


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can reach an accuracy of as high as approximately 100.0%,

with the lowest precision of 59.3%. However, it must be

emphasized that Equation [10] is highly dependant on the

parameter of trace length provided in any test conducted.

The trace length measured in the tests shall be of fractures

that distribute along the diameter of the specimen. As

observed from Table 2, the accuracy in predicting strength

of rocks with joint orientation of 90 is relatively lowwhich is within the range of 59.3% to 75.4%. As discussed

in the previous section, a granitic rock with joint

orientation of 90 is inclined to the strength characteristicsof an intact rock.

The complete equation for the modified Hoek-Brown

criterion with inclusion of the effect of joint orientation

and joint trace length is given below:

)e1839.0(sin

)/( 18.37735.0

c3c31

xb

m += [11]

Crack closure

The pre-existing cracks and voids inside a rock mass close

up upon the initiation of stresses. In triaxial compression

test, the pre-existing cracks are forced to close in both

vertical and horizontal axes. The stress threshold for the

crack closure can be observed from the stress-strain curve

whereby the curve depicts a slight non-linearity at thebeginning of the curve. This phenomenon can be observed

from Figure 7 as shown below. For a given specimen

subjected to a confining pressure of 5MPa, crack closure is

observed at deviator stress of 12MPa. Also, the on-set of

crack closure can be determine with a higher degree of

accuracy from plot of crack volumetric strain versus axial

strain as suggested by Martin and Chandler (1994:644) asshown in Figure 8.

The region of crack closure for the specimen shown in

Figure 8 is in the range of 0 to 0.05% axial strain. During

the formation of crack closure, no energy is emitted

because the energy is consumed for the closing up of pre-existing cracks. The crack closure stops at axial stress

15MPa and at 0.05% axial strain as determined by Figure

7. Furthermore, the closure of microcracks is represented

by the decrease in crack volumetric strain at the beginning

of the curve (Figure 8).

Figure 7:Stress-strain curves for a granitic rock specimen

with a single fracture (=90) at confining pressure of5MPa.

Figure 8: Volumetric strain curve for a granitic rock

specimen with a single fracture (=90) at confiningpressure of 5MPa.

Crack initiation

Crack initiation starts when the deformation of rock

becomes irreversible in which depicts the beginning of the

mechanism of stable crack growth. Noting the difficulty

in identifying crack initiation with the point of departure

from the stress-strain curve, Martin and Chandler

(1994:644) suggested in using the calculated crack

volumetric strain curve to identify crack initiation.

However, this method is highly dependant on the Youngsmodulus and Poisson ratio taken within the region of

elastic deformation. For a specimen shown in Figure 7 and

Figure 8, the Youngs modulus (E) and Poissons ratio (v)

are 50.5GPa and 0.2, respectively. In this respect, crack

initiation can be defined as the stress level at which

dilation begins in the crack volumetric plot. From Figure 7

and Figure 8, it can be seen that crack initiation started at

0.29% axial strain which corresponds to a crack initiation

axial stress of 160MPa from the stress-strain curve.

Crack damage

The threshold stress of crack damage can be determined

from the volumetric strain curve at the point of reversal ofthe curve. For the specimen used in Figure 8, crack

damage threshold stress is determined at 0.36% axial strain

which corresponds to axial stress of 185MPa.

It can be deduced that the volumetric strain reversal

occurred when the relative increase in lateral strain rate

surpassed the axial strain rate and emerged as the

dominant component in the volumetric calculation. The

point of reversal is highly distinguishable in the plot of

crack volumetric strain versus axial strain curve as in

Figure 8.


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Conclusions

A series of laboratory tests on fractured rocks was carried

out to determine the effects of joint orientation and joint

trace length on the peak strength of rocks as well as to

estimate the threshold stress values for the crack closure,

crack initiation and crack damage. Findings of the study

show that critical joint orientation which yields the

minimum peak strength, for Bukit Timah granitic rocks

with a single fracture is found to be 70. Granitic rockspecimens with joint orientations of 80 and 90 exhibitsimilar strength characteristics as in intact rock thus their

peak strength deviates significantly from the theoretical

values predicted by modified Mohr-Coulomb and

modified Hoek-Brown. The joint orientation and trace

length are incorporated into the modified Hoek-Brown

criterion as given below:

)e1839.0(sin

)/( 18.37735.0

c3c31

xb

m +=

The predicted values of peak strength using the proposed

equation well agree with the experimental test results

carried out on singly fractured specimens under triaxialloading conditions.

Acknowledgements

The authors would like to thank the Geotechnical

Laboratory technical staffs, Nanyang Technological

University for their assistance in the laboratory testing.

Sincere gratitude goes out to a number of colleagues for

their invaluable contributions to this paper.

References

Brace W.F., Paulding, B.W., Jr, and Scholz, C (1966):

Dilatancy in the fracture of crystalline rocks. Journal ofGeophysical Research, Vol. 71 (16), pp. 3939-3953.

Brady B.H.T. and Brown, E.T. (1985): Rock Mechanics

for Underground Mining. 1st ed, Chapman & Hall,

London.

Donath F.A. (1972): Effects of cohesion and granularity

on deformation behaivour of anisotropic rock, in studies inMineralogy and Precambrian Geology (eds. B. R. Doe andD. K.Smith), Geol. Soc. Am. Memoir, 135, pp. 95-128

Eberhardt E., Stead, D., Stimpson, B. and Read, R.S.

(1998). Identifying crack initiation and propagation

thresholds in brittle rock. Can. Geotech. J., Vol. 35, pp.

222-233.

Hoek E., Wood, D. and Shah, S. (1992): A Modified

Hoek-Brown Failure Criterion for Jointed Rock Masses.

ISRM Symposium, Eurock 1992, British Geotechnical

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Lajtai E.Z. and Lajtai, V.N. (1974): The evolution of

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McLamore R. and Gray, K.E. (1967): The mechanical

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