Int. J. Rock Mech. Min. &'i. & Geomech. Abstr. Vol. 31, No. 6, pp.

643~59, 1994
Copyright © 1994 Elsevier Science Ltd
Pergamon 0148-9062(94)00012-3 Printed in Great Britain. All rights reserved
0148-9062/94 $7.00 + 0.00

The Progressive Fracture of Lac du

Bonnet Granite
C. D. MARTINt
N. A. CHANDLERt

The strength of intact rock is made up of two components: the intrinsic

strength, or cohesion; and the frictional strength. It is generally assumed that
cohesion and friction are mobilized at the same displacements such that both
components can be relied on simultaneously. Damage testing of samples of Lac
du Bonnet granite has shown that as friction is mobilized in the sample,
cohesion is reduced. This progressive loss of intrinsic strength and mobilization
of friction is modelled using the Griffith locus based on a sliding-crack model.
There appears to be a maximum cohesion that can be relied on for engineering
purposes, and this strength is less than half of the unconfined compressive
strength measured in the laboratory.

INTRODUCTION LABORATORY DEFINED PARAMETERS

The progressive failure of clays and soft rocks is a Testing procedures for determining the compressive
well-known phenomenon involving a loss of cohesive deformational behaviour of rock samples are given by
strength [1]. In brittle rocks, however, progressive failure ISRM [2]. These include recording the axial (Eaxia~)and
is not generally recognized. It is obvious though that the lateral (ElateraI) strains in a sample as it is loaded with or
strength loss when progressing from an intact rock without a fixed confining stress. Richart et al. [3], in
through to a jointed rock mass must also be related to 1928, first noted that volumetric strain, in addition to the
a loss in cohesion, or intrinsic strength. axial and lateral strains, was also an important measure-
In many engineering applications, particularly at ment in compression testing, and Cook [4] proved that
shallow depths, jointed rock masses are the norm, and the volumetric strain of a sample measured by surface
the rock-mass strength, after sufficient deformation, is strain gauges was a pervasive volumetric property of the
essentially frictional resistance. However, in the mining, rock and not a superficial phenomenon. For a cylindrical
petroleum and nuclear industries, excavations are being sample subjected to axial loading, with or without a
made at ever increasing depths where large volumes of confining stress, and under small strains, the volumetric
essentially intact rock, i.e. tightly interlocked rock mass, strain (Ev or ~ ) is given by:
are being encountered. In these situations stresses are AV
generally much higher, and the cohesion component of Ev : V '~ Eaxial + 2Elateral" (1)
the failure envelope plays a major role in determining the
stability around underground openings. Hence by plotting the axial, lateral and the calculated
The strength of intact rock is made up of two com- volumetric strains versus the applied axial stress, the
ponents, cohesion and friction. For engineering purposes path of a rock sample to failure can be followed. An
it is often assumed that these components of intact example of axial, lateral and volumetric strain versus
strength are mobilized at the same displacements such axial stress curves for Lac du Bonnet granite in uniaxial
that both components can be relied on simultaneously. compression is given in Fig. 1.
The strength of intact rock is determined using labora- The failure of brittle rocks has been investigated by
tory triaxial tests, and the sum of the cohesion and many researchers [5-14]. These researchers showed that
friction components is obtained from these tests. In an the stress-strain curves for a brittle material can be
effort to understand the progressive failure of intact divided into five regions (Fig. 1). The initial region of the
rock, a series of damage-controlled tests has been carried stress-strain curves in Fig. 1 represents the closure of
out on samples of Lac du Bonnet granite. existing microcracks in the sample and may or may not
be present, depending on the initial crack density and
t A E C L Research, Whiteshell Laboratories, Pinawa, Manitoba, crack geometry. Once the existing cracks are closed, then
C a n a d a ROE 1L0. the rock is presumed to be a linear, homogeneous,
643
 644 M A R T I N and (.'IIANDLt-.IC PROGRESSIVE F R A C I I , ! R E ()F LAC DL BONNI l ( i R A N t F f

elastic, material (Region I1). The elastic properties of a The elastic volumetric strains are subtracted from the
rock sample can be determined from this portion of the total measured volumetric strains to determine the volu-
stress-strain curves. metric strains caused by the axial cracking (Fig. I}. a<,
The onset of dilation marks the beginning of Region is the axial stress at which dilation .just begins on the
llI. Brace et al, [6] found that dilation begins at a stress crack-volume plot, as shown in Fig. 1. Researchers
level of about 30 50% of the peak strength. It is worth [7,5, 15] have found that the cracking associated with
noting that this dilation is only registered on the lateral axial stresses slightly above a<, does not result in reduced
strain gauge and must therefore reflect the growth of rock strength. Therefore these random stable axial
axial cracks, i.e. cracks parallel to the direction of the cracks are not considered damaging to the rock strength
maximum applied load. Hence this stress level will be in laboratory tests.
referred to as the crack-initiation stress (%). These The axial stress level where the total volumetric strain
cracks are generally thought of as stable cracks since an reversal occurs marks the beginning of Region IV and
increase in load is required to cause further cracking. represents the onset of unstable crack growth, as defined
Crack initiation is difficult to identify from the labora- by Bienawaski [7]. It generally occurs at axial stress level
tory stress-strain curves, particularly if the sample between 70 and 85% of the short-term peak strength. It
already contains a high density of microcracks, The is at this stress [eve[ that the axial strain departs from
crack-initiation stress is best determined using a plot linearity (Fig. 1). The dominant mechanism resulting in
of crack volumetric strain versus axial strain. Crack such an increase in axial strains is sliding along inclined
volumetric strain is calculated as follows. First the elastic surfaces. Hallbauer et al. [I 3] pointed out that this region
volumetric strains are calculated using the elastic con-
is characterized by the most significant structural
stants (E, v) from the linear portion of stress-strain
changes to the sample, with the density of microcracks
curves in Region I1 by
increasing by about sevenfold. This stress level has
1 - 2v particular significance in the concrete industry as it is
AI" ,. 1/....
. ~ -. . E (o~ - o~) (2)
used to establish the long-term strength of concrete

Axial
Stress MPa)

(~f(Peak)_-~
v - - - . . . .

..~ ~ = 8 0 % O'f .... -~ I

Can be used Unstable Cracking it.. .... - - ~ !

Stable Crack Growth [HX~ G.---~ / !

....... 'X~ 100 I- ..... c, ~ ! '1
C r a c k Initiation ~ - - -1 - "--4u~* u f - - - ~ l r ! ;
I Elastic Region II~N.~ / ,~11 I i
-= ; ! ',
_/.20 I 1 I t Ii I I I t I , 1
-0.16 -0.12 -0.08 -0.04 0 0.1 i 0.2 i 0.3 0.4
Lateral Strain *'/o 0 •2 V i Axial
' Strain' % iI I
,-- / I I I
I I I I
/n,L I IL-.~ X '
o'1 ~l"" / I/~ I ~etalsured--~ k I,

-j,i
0 0.2 0.10,3 ~ 0.4
Axial Strain %
Fig. 1. Stress strain diagram obtained from a single uniaxial compressEontest for Lac du Bonnet granite showing the defini.tion
of crack initiation (%), crack damage (a,a) and peak strength. Note only the axial and lateral strains are measured. The
volumetric strain and crack volumetric strain are calculated.
 MARTIN and CHANDLER: PROGRESSIVEFRACTURE OF LAC DU BONNET GRANITE 645

1 • . . . . , . . . . , . . . . • . . . . , . . . .

0.9 One Day

0.7 [---~D.~m--

e '" ......
0.8 °c

. ............... :?- ..........

~_ 0.7

NF
0.6

Samples tested wot at 200C

NF ,, Sample did not fail
l . . . . i , , , , l * , , i I I I " " i . . . .

0 10 20 30 40 50
Time (days)
Fig. 2. The strength of unconfinedsamples of Lac du Bonnet granite subjectedto long-termconstant load. trc is the standard
short-term unconfinedcompressivestrength.

[16, 17, 18]. Lajtai et al. [19] found that the unstable crack material parameters and which are a function of the
stress for unconfined samples of Lac du Bonnet granite particular loading conditions used in the uniaxial test.
from the Cold Spring Quarry occurred at 70% of the Hudson et al. [22] concluded that the peak strength of
short-term peak strength. Schmidtke and Lajtai [20] did a sample was a function of the boundary conditions of
extensive long-term testing of Lac du Bonnet granite the test, and hence not an inherent material property.
from Cold Spring Quarry. Their results were reanalysed Glucklich and Cohen [23, 24] used stored strain energy
by the authors and have been replotted in Fig. 2. Figure to explain peak-strength scale effects, which are com-
2 shows that for loads above ~0.70 of the peak strength monly observed. They point out that, during the stage of
(trc), failure occurs almost immediately. Thus the in- stable crack growth, there is equilibrium between the
crease in load above the unstable crack stress is a external load and the crack length. This was also
temporary strain-hardening effect that cannot be relied confirmed by Hoek and Bieniawski [5]. Both the loads
on for permanent loading conditions. Hence, we will and the stable crack lengths increase up to the critical
refer to this stress level as the crack-damage stress (acd) moment at which the strain-energy release rate equals or
since loads above this stress level result in damage to the exceeds that of energy absorption. At this moment crack
material which cannot be tolerated under a permanent propagation becomes unstable, and the material reaches
load. its peak strength. For heterogeneous materials such as
The peak strength of the material (af) marks the rock, the propagating crack will most likely encounter
beginning of post-peak behaviour, Region V, and is material that is stronger or weaker (an area of pre-exist-
almost universally used to establish the failure strength ing stable cracks) than the mean strength. In either case,
envelope. An example of the complete axial stress-strain after the propagating crack advances through the softer
curves for Lac du Bonnet granite is shown in Fig. 7. or harder material, there is an excess of energy released
Beyond the peak, the axial stress versus axial strain that is converted to kinetic energy and is available to do
shows a rapid decrease, which is interrupted by one or work against the remaining uncracked material. It is here
more short strengthening interludes, marked by steps in that the volume of the sample and the stiffness of the
the descending axial stress curve. Lockner et al. [21] testing machine play a critical role because the stored
reported that during the first portion of the post-peak energy in the total system dictates the energy release rate.
axial stress versus axial strain descent, the loci of the In essence, Glucklich and Cohen are pointing out that
seismic events indicated the development of a major a properly conducted compressive-strength test would
inclined shear fracture. balance the stored strain energy in the sample and
Thus far, three characteristic stress levels have been loading frame with the fracture surface energy required
identified in the laboratory stress-strain curves (see for fracture growth, i.e. there would be no kinetic energy
Fig. 1): the crack-initiation stress (tr,i), caused by stable available to propagate the crack. In reality this is
tensile cracking; the crack-damage stress (a~d), caused by very difficult to do because in compression testing two
crack sliding; and the peak strength (at). In order to modes of cracking are developing simultaneously,
better understand material behaviour, it is important to the axial crack and the sliding crack. One approach
establish which of these stress levels are characteristic to this problem is to reduce the loading rate such that
646 MARTIN and CHANDLER: PROGRESSIVEFRACTURE OF LAC DU BONNET GRANITE
. . . , . . . . , . , . . . , . , . . . . , - . _

200
~ k Strength
Q.
150
,J 4
ta~ .......... ' ........ ; .... ................... '
t ~

i i Mean Ocd = 133.9 MPa

1O0
C

50

. . . . | . . . . | , j . . . . . . .
Mean with One Standard Deviation
, . . . . • . . . . . . . . .

50 1O0 150 200 250 300 350

Diameter (mm)
Fig. 3. The effect of sample diameter on peak strength, crack-damage stress and crack-initiationstress of 53 samples of Lac
du Bonnet granite from the 240 Level of the Underground Research Laboratory.

the fracture surface has time to grow and increase unconfined testing (Fig. 5). This also concurs with the
the sample volume to minimize the effect of heterogen- previously mentioned notion that the failure mechanism
eity on the fracture process. A similar approach is for god is one of sliding. It would appear that with
used to reduce strain-hardening effects in more ductile the appropriate boundary conditions, i.e. loading rate,
materials [25]. loading-frame stiffness and sample volume, the peak
The long-term test data of Schmidtke and Lajtai [20] strength of a sample of Lac du Bonnet granite would
(see Fig. 2) also suggest that the peak load above the
crack-damage stress is only sustained by the rock for
Axial
a short duration and cannot be relied on for the long
Stressl(MPa) Ocd (Peak)~x, a
term. This leaves only aci and acd as possible material
parameters that should therefore be independent of 1201 /
sample volume.
To determine the effect of scale on acd and aci, the °°[ /
stress-strain curves were analysed for 53 samples, with
diameters ranging from 33 to 300 mm diameter. The °I /
results are summarized in Fig. 3, and the peak strength
is shown for comparison. The peak strength, as ex-
pected, shows a modest reduction in strength for larger
samples, but both Oci and acd appear to be unaffected by
sample volume. Note that except for the largest sample -0.1 -0.05 0.05 0.1 0.1~ 0.2 0.25 0.3
diameter tested, the data suggest that the peak strength I
Lateral Strain (%) Axial Strain (%)
is trending towards the a~d strength, i.e. about 70% of I

the peak strength. This result is in keeping with that of I

Hoek and Brown [26], who showed the unconfined 0.15

compressive strength reducing to about 80% of the peak
strength of a small sample, as the sample diameter ._c .__., 0.1
increased from l0 to 200 mm.
In an effort to minimize the potential influence of 8
/ o.os
uncontrolled strain energy on the test results, one final
series of tests was carried out that attempted to combine _=
the effects of scale and a slow loading rate. Four
200-mm-diameter samples were tested at the loading rate c~ -0.05
of 0.00075 MPa/sec, which is 1000 times slower then the . . . . . . . . . . . . . . . . . . . . . . , , m

normal loading rate. In two samples, failure occurred 0.05 0.1 0.15 0.2 0.25 0.3
at the stress level generally associated with a~, and
Axial Strain (%)
those samples did not display the normal volumetric
strain reversal (Fig. 4). Interestingly, the samples that Fig. 4. The combined effect of sample diameter (200mm) and slow
loading (0.00075MPa/sec) on peak strength, crack-damagestress and
were subjected to slow loading rates all developed crack-initiation stress. In this sample the crack-damage stress is the
characteristic shear planes not generally associated with peak strength.
 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LAC DU BONNET GRANITE 647

trolled, servo-hydraulic compression machine, consisting

of a 2.22 MN rated load cell, load frame, hydraulic
power supply, triaxial cell, confining pressure subsystem,
test controller, test processor and DEC micro PDP 11/73
computer. The triaxial cell is equipped with three
linear variable differential transformers (LVDT) for
the measurement of axial strain and a circumferential
extensometer to measure the lateral strain.
The confining pressure and the axial stress were
initially increased from zero to the required confining
stress at the rate of 0.75 MPa/sec. The axial stress was
then increased using axial strain-rate control at a rate
approximating 0.75 MPa/sec. The instrumentation was
scanned every 3 sec. Up to approx. 75% of the expected
peak strength, the load-unload cycles were carried out
at 40 MPa increments. As the peak strength of the
sample was approached, special care was taken to
prevent rapid failure in order to continue the test into the
post-failure region. After the axial stress reached ~ 75%
of the expected peak strength, the load-unload cycles
were performed at 0.063 mm increments of circumferen-
tial deformation using axial-strain control. A test took
about 8 hr to complete, and a typical result is shown in
Fig. 6.
The initial concern was whether the testing method
influenced the results. Figure 7 compares the results from
an unconfined damage-controlled test with a traditional
unconfined post-failure test without the damage incre-
Fig. 5. The failure surface, developed in a sample subjected to ments. Figure 7 illustrates that the general shape of the
0.00075 MPa/sec loading rate, is inclined 23~ with respect to the
direction of loading. Note the short axial cracks that form adjacent to stress-strain plot, particularly in the prepeak region, is
the failure surface. unaffected by the testing method. This was also found
true for the confined tests.

be reduced to about 80% of the standard uniaxial

TEST RESULTS
strength (at), similar to the failure loads of Schmidtke
and Lajtai [20]. The purpose of the testing was to determine the effect
Having established that acd is the true peak strength of of damage on the stress levels associated with crack
a rock in a monotonically loaded uniaxial compression initiation and crack damage. The volumetric strain
test and that ~ca and ac~ are scale-independent parameters encompasses both the damage in the lateral and axial
with completely different modes of origin, the next step direction and can be related to crack-initiation and
is to determine the effect of increasing crack damage in crack-damage stress (see Fig. 1). In a given test, a
a specimen on these two parameters. damage increment (i), i.e. a load-unload cycle, will
produce permanent volumetric damage (E uP). A damage
parameter (co) is therefore defined as the cumulated
DAMAGE-CONTROLLED TESTING
permanent volumetric strain (Fig. 8)
The Lac du Bonnet granite is medium to coarse
grained and composed of approx. 30% K-feldspar, 30% co = ~ (E P),%. (3)
plagioclase, 30% quartz and 10% mafic minerals, mainly i=1

biotite. The average grain size of the medium-grained It is useful to plot the peak stress, acd and ~,, versus
granite is about 3 4 ram. Six post-failure uniaxial com- the damage parameter co. The collection of these values
pression tests and thirty-one post-failure triaxial com- of peak stress, acd and ec~, for any one test will be referred
pression tests were conducted on the 63-mm-diameter to as the peak (co) locus, the Oca locus and the % locus.
grey samples. The samples were obtained from the 420
Level of AECL's Underground Research Laboratory. Crack initiation and crack damage
The testing was carried out by C A N M E T (Canada The crack-initiation stress occurs when the load first
Centre for Mineral and Energy Technology) [27], exceeds about 0.2-0.4 of the peak strength. Initially, in
and specimens were prepared in accordance with the the early stages of the test, the crack-initiation stress
methods suggested by the International Society for Rock appears to increase slightly, however, as damage ac-
Mechanics [2]. The post-failure tests were conducted cumulates the slope of the crack initiation locus appears
using an MTS 815 Rock Test System, a computer-con- to level off. One could speculate that the initial increase
648 MARTIN and CHANDLER: PROGRESSIVE F R A C T U R E OF LAC DU BONNET GRANITE

200 1 1 I
I ' ' ' I ' ' ' I ' ' ' I ' ' '

MB124205
~3 = 2 MPa

150

A
w
o,.

"" 100
w
2

50

0
0 0.2 0.4 0.6 0.8 1

Axial Strain (%)

Fig. 6. Example of the repeated loading and unloading used in a damage-controlled test.

in the aa locus is related to less critical cracks requiring independent of the damage accumulated in the sample
more load to reach crack initiation. Given the difficulty (Fig. 9).
in determining the aci stress in the early stages of the The crack-damage stress occurs at about 0.8 of the
test, it may be that the initial a¢~ slope is within the peak strength. However, unlike the crack-initiation
error of the analysis. Also, this phenomenon was not stress, the crack-damage stress reduces significantly in
observed in all the test results. Thus it is reasonable to the early stages of the test and reaches a threshold as the
conclude that the crack-initiation locus remains fairly damage accumulates in the sample (Fig. 9). This
constant with each damage increment and is therefore phenomenon is seen at all confining stress levels and is

Lac du Bonnet Granite

150 420 Level URL
03 = 0 MPa

125 Damage-Controlled
Test

~v 100

-~ 75

50

Standard
25 Post-failure
Test

0 . . . . • . . . . • . . . . i

0.2 0.4 0.6

Axial Strain (%)
Fig. 7. Comparison of an unconfined damage-controlled test and a standard unconfined post-failure test.
 M A R T I N and C H A N D L E R : P R O G R E S S I V E F R A C T U R E O F LAC D U B O N N E T G R A N I T E 649

175
Peak(m)~9 8 7 t 6 "/ Damage
5 Increment
4
150
"''"- / "N
"7. 2 "N A :,
• Io #°

125 " "<; "• "\ -I

O. , , ! /
/" :,; , ,,
--.
C/)
100
C/)
=
N 75
._ ' ( /,' ,"
,t , , •: ,'/ , • / .,.//I
50
, ;: ?.'
25 •) oo) S

•"34, oo S't°° o d)

.' o..- , ..--" Load-Unload

• Cycle
0
-0.2 -0.1 0 0.1 0.2
Damage (P.~) Volumetric strain (%)
Fig. 8. D a m a g e is defined as the permanent volumetric strain resulting from a single damage increment.

quite consistent from test to test (Fig. 10). Similar treated as a separate test, and the modulus and Poisson's
observations [28,29,30] have been made during the ratio are computed for the part of the stress-strain curve
cyclic testing of other brittle rocks. It should be noted that lies between the crack-closure stress and the crack-
that the drop in trca is smaller at higher confining stresses initiation stress. A plot of Young's modulus and Pois-
and that the threshold value of aca corresponds approxi- son's ratio versus damage is compared with the
mately to tr~ when the sample is unconfined. As the crack-damage locus in Fig. 12. As the sample is subjected
confining stress is increased, the threshold value of acd is to increasing damage, a gradual reduction in stiffness is
greater than tr¢i (Fig. 11). indicated (Fig. 12). In the post-peak region of the test in
which the peak (co) stress dropped from about 150 to
Deformation constants 56 MPa ( ~ 3 5 % of maximum value), the modulus de-
Young's modulus and Poisson's ratio can be deter- creased from 50 to 24 GPa ( ~ 50% of maximum value).
mined for each damage increment. Each increment is At a confining pressure greater than 20MPa, the

200 • . . . . . . . . . , . . . . o . . . . • . . . . • - .

Lac du Bonnet Granite

iv ---q~ Pea " URL 420 Level
175 / ~ _ Sample MB 124205
150

~125

.= 100

7s

• ~(~cd Threshold

2s , ~ r - ~ . _ ,=,,=,..~t1.s.,~_4.,,'
Oci ~i)
0 i . . . . a . . . . a . . . . | . . . . , . 0

0 0.5 1 1.5 2 2.5

Damage to
Fig. 9. Example o f the crack-initiation stress and the crack-damage stress as a function of damage. Note that at low confining
stresses, the crack-damage stress is essentially the same magnitude as the crack-initiation stress.
650 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LAC DU BONNET GRANITt:

200 • . . . . , . . . . , .

Lac du Bonnet Granffe

175 URL 420 Level
Peak(w) 03= 0
150

a. 125

¢O

e 100
u)
._.R 75

50

25

0 • . . . . • . . . . ' . . . . • . . . . ' . . . . • - i i i ! i i i i ! i

0 0.5 1 1.5 2 2.5 3 3,5

Damage o)
Fig. 10. Unconfined example of the crack-damage stress and the peak stress as a function of damage. Note the repeatability
between tests.

reduction in the modulus in the post-peak regime was and after the initial large drop in the post-peak strength,
considerably less. In all cases, the strength reduced faster the ratio remains relatively high, ranging from 0.6 to 0.9.
than the modulus. The locus of Poisson's ratio clearly establishes that two
In the early portion of testing, i.e. before acd is phases of axial crack growth occurs. The first phase
reached, Poisson's ratio is about 0.14 and increases to occurs in the pre-peak portion of the test when the ratio
about 0.2 at the maximum a~d (Fig. 12). As the peak (o9) is increasing quite rapidly, indicating significant axial
stress level exceeds the initial aoa and starts its descent to crack growth. The second phase occurs when the sample
the post-peak strength, Poisson's ratio increases quite enters into the post-peak region and the first significant
sharply to about 0.9. It is obvious that above 0.5 this strength drop occurs. This phase is indicative of when
ratio is only relating lateral strains to axial strains and the sample has developed a major shear fracture as
is not an elastic constant. As the a~d threshold is reached identified by Lockner et al. [21].

400 • . . . . • - . . , . . .

Lac du Bonnet Granite

URL 420 Level
350
Sample MB 122382
(~3 = 15 MPa
300

#. 2so

w= 200

. 150

100

50 OQ O0,~....___O.ei
• S • • tN

0 | n - - • • . . . . • . . . . • • • • i . . . . J . . . . • . . . . t .

0.5 1 1.5 2 2.5 3 3.5

Damage (o
Fig. 11. Confined example of the crack-initiation stress and the crack-damage stress as a function of damage. Note that at
higher confining stresses, the crack-damage stress is considerably higher than the crack-initiation stress.
 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LAC DU BONNET GRANITE 651

200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Lae du Bonnet Granite

175 [- URL 420 Level
/-- Peak(Co) Sample MB1-21.065

~" 125

100

75

5o

25

0 I , i - • • • - t i I i i i • * - - t i I i i i i i . . . . . . . . . .

0 0.5 1 1.5 2 2.5 3 3.5

Damage m

100 ................................... , 1.0

#

t¢m', ~ , - - Poisson's ratio _~

, "~ , * ~ / .r

j , =e
/- Youn. mo u,u t o
) ,0 r

eQ
0 0
0 0.5 1 1.5 2 2.5 3 3.5
Damage
Fig. 12. Modulus and Poisson's ratio as a function of damage.

CRACK DAMAGE LOCUS removed from the system, the excess energy will be
converted to kinetic energy. It is generally not possible
The theoretical limit of crack growth for brittle ma-
to follow the unloading path AB since most systems,
terials has been evaluated by Berry [31,32] for tensile
even a stiff testing apparatus, have a finite unloading
crack extension, and by Cook [33] for shear crack
stiffness, represented by AC in Fig. 13. Thus a crack
extension. Cook [33] refers to this limit for crack growth
starting at o A will propagate dynamicallyt. Berry [31]
as the Griftith locus, and it follows the general form
noted that the excess strain energy represented by the
ABCD, given in Fig. 13. According to Berry, the Griffith
shaded area ABC will cause the crack growth to acceler-
locus can be interpreted in the following way. The
ate, hence the crack will continue to extend even as the
portion AB, during the early stages of crack extension,
stress drops below trc corresponding to point C on the
indicates a rapid loss in strength with no increase in axial
failure locus. Below trc the crack will finally stabilize
strain. Unless the strain energy released from the elasti-
when the excess strain energy ABC is equal to the strain
cally strained regions around the propagating crack is
energy CDE. The area ABC = CDE, and hence the
energy CDE, is the surface energy required to create
longer cracks. The longer crack is now represented by
tin a servo-controlled laboratory test the energy that causes the
dynamic propagation is controlled. Hence, it is possible to follow OD, with a reduced modulus Ec+dc. These cracks are
this stage of crack growth. now loaded to a subcritical stress level trE, and hence will
652 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LAC DL] BONNET GRANITI,

not extend until the stress level is increased to oil. Thus number of cracks per unit volume (see Fig. 13). Using
the Griffith locus has two key elements: the stiffness of Ihe approach of Cook the critical axial strain ~,,. at
the initial material which controls the position of OA, which sliding occurs, is given b)
and the crack properties that control the shape and
(r~ - 2vo.~ + 2 ( H,~ + H"~
position of BCD (see Fig. 13). to, (4)
Thus far the crack damage locus has been plotted +Ti :::.,i \ 2 1"
versus the damage parameter oJ, which is defined by the where
volumetric strain. Although o~d is defined by the volu-
metric strain reversal, it also corresponds to the onset of W, = ~ ( 1 - - v ) G c2
nonlinearity in the axial stress versus axial strain plot, as
shown in Fig. 1. Earlier it was suggested that this n (T - / 1 o , )
non-linear response is a direct result of sliding along Wf= ~(1 - V)laa° G c2
crack surfaces angled with respect to the direction of the
maximum load. Thus it is instructive to replot the test 8~G
results to see the relationship between the crack damage 7I" (1 - - Y)(T - - ]./O'n) 2
locus and the peak (a)) stress versus axial strain.
and where v is the Poisson's ratio, G is the modulus of
Figure 14 shows a typical plot of the crack-damage locus
rigidity,/~ is the friction across crack faces, ~ is the shear
and peak (co) stress locus versus axial strain. The
stress in the direction of the crack slip, on is the normal
volumetric strain versus axial strain is also shown to
stress acting on the surface of the crack, and c¢ is the
illustrate how the plot is generated. Note that the full
fracture surface energy.
reduction in er~dto its threshold level occurs by damage
For the conditions of triaxial compression and assum-
increment 10, which takes place before the peak strength
ing that the crack is parallel to the direction of the
of the sample is reached. This was not evident from the intermediate compressive stress
previously presented plots. The shape of the damage
locus on an axial stress versus axial strain plot is similar O'] -{- 0" 3 G I -- 0" 3
an - cos 20
for other confining stresses. 2 2
It has been suggested that the crack-damage stress is
o I- G~ .
defined by sliding since the axial strain registers perma- - sm 20 (5)
2
nent damage. This would imply that the crack-damage
locus is the locus of strength required to initiate crack where 0 is the angle between the critical crack surface
sliding or simply the Griffith locus, as defined by Cook and the direction of the maximum applied stress, al.
[33]. Only a brief summary of Cook's original work is Hence E~ can be obtained provided the crack density
provided, and the interested reader is referred to Cook's n and the fracture surface energy ~ are known. The other
paper for the derivation of the formulas that follow. For parameters are available from standard laboratory tests
the confined case, consider a single elliptical crack of and the fracture surface energy can be equated to the
length 2c, inclined at some angle 0 to the direction of the strain energy release rate ~ by ~ = 2~. The strain energy
applied stress a~, in a specimen where n represents the release rate is one of the most important parameters,

~2
°3
l
Unloading path of a
OAt......................... , ~ . / rigid system
¢R
.e / J'i ' / / / Unloadingpath of a
l /[ I~(~ / / non-rigidsystem
:i Excess E n e r g y
°c ............... Grifllth Locus of
or, .......... D Crack Growth
0= ...... O ~ . de;

Fracture Surface
Energy
0 "
Axial Strain
Fig. 13. Illustration of the Griffith locus.
 M A R T I N and CHANDLER: PROGRESSIVE F R A C T U R E OF LAC DU BONNET G R A N I T E 653

200 '''' I' ''' I '' ' '1 ''' ' I ....
.~.~=="-- Peak
I' '' '

0"3 = 2 MPa ,'"+-v~,

+~'~ Strength

A 150 • ~

m
ca. ,.+" Pea ~-

100 "

0"cd Locus +~-

+" I I
,,,, I, ,,I, I,I, ,,I .... I .... I ....
0 ....... I, ,I ...............
"~t 0.4~ I I ' I ' ' '
/
E Damage --
~ Increment3~"
, 0"
c#
8 o E-

[-- ~ / . ' ~ / ~ ~ ~ ~ ~./--Damage

~ ' ~ , ~ ~lncrementlO .
g -0.2

-0.4 ''
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Axial Strain (%)
Fig. 14. Crack-damage locus and peak strength versus axial strain. The volumetric strain for each damage increment is also
shown. Note that the major drop in the crack-damage locus occurs before the peak strength is reached.

with regard to fracture, and is defined as the amount test were added to the calculated results in order to
of energy release per unit increase in crack surface compare with the measured strains. The parameters
area. Rice [34] proposed a method for determining used in equation (4) are shown in Fig. 15. It should
fie, the strain energy release rate at failure, for shear be noted that in Fig. 15, the initial positive slope of
faulting. Using Rice's approach, Kemeny and Cook [35] the locus is somewhat less than that of the measured
calculated a ffc value of 1.05 J/m 2 for the creation of values at the higher confining stresses and the dis-
a single shear fault for Westerly granite. Since ff~ is crepancy increases with confining pressure. This occurs
considered to be a material property, the value because no correction for the increasing stiffness (E) of
determined by Kemeny and Cook will be used as a the samples with confining stress was made. Because of
starting point in evaluating the model for Lac du Bonnet the high density of microcracks in the samples, Young's
granite. modulus increases from about 50GPa for the un-
To calibrate the model for the crack density n, the confined samples to about 60GPa for the 30 MPa
critical axial strains were fitted to the data from an confining stress. Even without this correction, the agree-
unconfined test to estimate a value for n. All other input ment between the measured and the predicted loci is
parameters were taken from laboratory test results and quite good.
used to predict the critical strain at confining stresses of According to Cook, the critical condition for the
2, 15 and 30 MPa. The predicted values were compared initiation of failure occurs when
with the measured crack-damage locus at these confining
stresses (Fig. 15). It should be noted that the non-linear 7r (T - ~Uan)2
(1 -- v) G c/> 4~. (6)
strains that occurred in the initial seating phase of the
RMMS 3 I / 6 ~ F
654 MARTIN and CHANDLER: PROGRESSIVEFRACTURE OF LAC DU BONNET GRANITE

400 . . . . . . . . . . - . . . . . . . . . . .

350 G= 20 G P a
q = 1 J/m 2 0~
300 = Tan(46.5°)

Q.
n = 1200
250 O= 25 ° O3 = 30 MPa
1/1
9 200

~ 150
/ I~BI=,- L • O3= 15 MPa
100

50 ~ll~=... "03 = 2 MPa

, , , , i , , , , I , l i l I i - - • • . . . .

0.2 0.4 0.6 0.8

Axial Strain (%)

Fig. 15. Crack-damage locus at various confining stresses.

Substituting equation (5) into equation (6) provides Thus equation (7) can be rewritten as
the strength of the specimen in a general M o h r -
Coulomb form:
cTz(1 - v) { 1+# +#
O'1~ + o3 :-- (8)
N/1 + / ~ 2 - # ~/1 + # ' - #
crc (1 - v) 1+
at/> sin 0 cos 0 (1 - / ~ tan 0 ) + a3 i----p- t--~n-0 " (7) It was stated previously that the crack-damage loci
were similar in shape regardless of confining stress.
Note that the critical angle of 0 will be related to /~ Figure 16 shows typical examples of the crack-damage
in order that ( ~ - / w , ) in equation (6) is a maximum locus for some of the confining stresses tested. Not all of
when the results are shown in this figure for clarity reasons,
however all of the results are shown in Fig. 17. The
1 tl condition for sliding, according to equation (6) is a linear
0critical: ~ tan -.
# relationship in at-a3 space. The crack-damage threshold

500 , , ' I ' ' ' I ' ' ' I ' ' ' I ' ' '

400

A
¢1 300
I1.

6" 200

100

l . . c,-,.---'-~O 3 ,, 2
I I I I I I I I I I I I = t = I i = I
0.20.4 0.6 0.8 1
Axial Strain ( % )
Fig. 16. Crack-damage locus at various confining stresses.
 MARTIN and CHANDLER: PROGRESSIVEFRACTURE OF LAC DU BONNET GRANITE 655

700 . . . t . . . . . . . . . . . . . . . . . . . . . . . . . .

600 Peak
m=35, s=1 . _~-
o c = 157 MPa
500

~- 400

IO 300

200 ~ _jL.,¢.' ' ° ~

: _.,.,~o..,IL"~,,,~_ Crack
100 ~-~° Damage
~,L44"" Threshold
0 n , , , , I , - . . . . . . . ' . . . . t . . . . . . . . o . . . .

0 30 40 10 50 60 20 70
~3 (MPa)
Fig. 17. The peak strength and crack-damage threshold failure envelopes.

values also follow a linear relationship in o1-o"3 space, are combined in the strength value obtained from any
which gives a friction angle of 47.6 °. This friction value one test. The Griffith locus can be used to determine the
is close to the residual friction angle of 45 ° reported by relationship between cohesion and friction during the
Gyenge e t al. [36] and 42-43 ° reported by Lajtai and failure process.
Gadi [37] for Lac du Bonnet granite (Fig. 18). The peak The shear criterion given in equation (8) can be
strength is also shown in Fig. 17 with a Hoek-Brown reduced to
failure envelope fitted to the data.
a1=2 cTz~-~ v) tan 4 5 + ~ +a3tan 2 45+ (9)
DISCUSSION
by substituting # = tan ~b, where ~b is the friction angle.
Coulomb (1796) [38] postulated that the shear The shear strength of a frictional material is also given
strength of rock and of soil is made up of two com- by the well-known Mohr-Coulomb criterion
ponents--a constant cohesion, and a normal stress-
dependent frictional component. Schmertmann and Os- ~j=2S0tan 45+~ +0.3tan 2 4 5 + (10)
terberg [39] showed that for clays these components are
not mobilized at the same displacements. However, for where So is the empirical cohesion intercept or intrinsic
rock engineering design purposes, it is generally assumed strength. It is interesting to note that the two shear
that these components are mobilized at the same dis- criteria in equations (9) and (10) are identical. In
placements such that both components can be relied on equation (9) the empirical cohesion of equation (10) is
simultaneously in rock engineering design. The strength expressed in terms of fracture surface energy and crack
of intact rock is determined in the laboratory using length. More importantly, an examination of equations
triaxial tests, and the cohesion and friction components (9) and (10) reveals that the fracture surface energy and

120 . . . , - . . , . . , . . , . . . , . .

100
A N N
ta. 8o
m

60
¢/)

~ 40

20 Lac du BonnetGranite

• .

60 20 80 100 40 120
Normal Stress (MPa)
Fig. 18. Basic friction for Lac du Bonnet granite, after Gyengeet al. [36],
656 MARTIN and CHANDLER: PROGRESSIVEFRACTURE OF LAC DU BONNEt(iRANITt!

400
Sample MB122382
350 G 3 = 15 MPa

300
t~
13. 250
O3
o3
.e 200
C::eSFr~nticLoss
~,/, wi ,t7 obilized Friction

150

100

50

0 " • ' ' ' . . . . • . . . . . . . . . . . . . i

0.2 0.4 0.6 0.8 1

Axial Strain (%)
Fig. 19. Mobilization of friction and cohesion as a function of axial strain.

crack length only apply to the cohesive component of the in crack length. Figure 19 presents an example of this
material and that the frictional strength is not dependent progressive fracturing, illustrating the loss in cohesion
on these parameters. and the mobilization of friction, and Fig. 20 illustrates
The interpretation of a standard set of uniaxial test the concept in terms of a Mohr stress diagram. Ulti-
results, using equation (10) would imply that mately, in Fig. 19, at large displacements, the peak stress
and the crack damage stress should be equal.
(11) This concept of cohesion loss can only be explained in
this manner if non-elastic deformations are required to
mobilize friction and if the frictional component is made
Thus the cohesion can be equated to the strength just as
up of a residual component (~bb) and a roughness
sliding starts, which for our tests results is the crack-
of interlocking component (~bi), such that the total
damage stress. For an unconfined test, at the instant
frictional residence can be expressed as ~bb+~. The inter-
when sliding starts, the cohesion becomes
locking decreases from a maximum as damage accumu-
2S0 = O'cd.
lates and as friction is mobilized, the residual friction
and the minimum cohesion must be approaches (see
Once sliding initiates, i.e. friction is being mobilized, the Fig. 21). For Fig. 21, the value of 4~ was calculated using
strength of the sample starts to increase above the
crack-damage stress. However, we have seen from our ~b=2tan L(crl~ -~
test results that as the sample is subjected to increasing ~cd / 2"
damage, only a small amount of damage to the sample Furthermore in Fig. 21, the damage has been normalized
is necessary to bring the crack-damage stress to the with respect to the value of e) at the end of the test and
threshold value, suggesting that cohesion must also the strength has been normalized to the peak strength.
decrease. The total strength of the sample has not Figure 21 illustrates that the peak friction angle
changed, therefore the frictional component of equation (4,b + ~= 63 °) is only reached when most of the cohesion
(11) must be increasing as the cohesion is decreasing. In is lost. With increasing damage, the friction angle gradu-
equation (9) the drop in cohesion is related to an increase ally decreases to about 42 ° . This friction value is similar

Smax 0o
Cohesion Loss J
As ¢ Mobilized ,~
Smobilize(

2Smax "1 I
TO
Cred ~f (Peak)
Fig. 20. Illustration of cohesion loss and mobilization of friction in terms of Mohr stress diagram.
 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LAC DU BONNET GRANITE 657

Lac du Bonnet Granite

URL 420 Level
Sample MB121065
Peak(~) %=0
0.8
e- bt

e-

0.6
"O
O
.N
~0.4
O
z ~qt~~-- (~cd
"• ° 0"o o.
0.2

0 0.2 0.4 0.6 0.8 1

Normalized Damage role)maX

• , o . . . . • • , ,

t
I
P
80 I 0.8
I
1

Z
O

"-"
"0" 60 0.6
®
I
N
O
=, O= 0 (I)b ,- a.
e-
:;...:. :..~ ~L :..~::......: ....... i...................... :.......................................~ m O 0 0 = = _ A _ . = ~ =
e,.
._o 40 0.4 o
0
"6 • ~ ..= / ~ Cohesion
0

20 0.2

0.2 0.4 0.6 0.8

Normalized Damage to/COmax
Fig. 21. Mobilization of friction and cohesion as a function of normalized damage.

to the residual friction angle of 42--45 ° for Lac du Bonnet As the face of the opening approaches and passes the
granite reported by [36, 37]. Thus for the test presented volume of rock that eventually becomes part of the
in Fig. 21, it appears that the residual friction is nearly tunnel surface, the principal stresses associated with this
reached. The peak friction angle of 63 °, although high, rock will change significantly in both magnitude and
is not unrealistic, e.g. Dusseault and Morgenstern [40] direction. Stress concentrations in excess of the crack-
reported that natural slopes of uncemented locked sands damage locus, occurring at any period in the rocks
have inclinations greater then 54 ° . Thus 63 ° does not loading history around the opening, will result in a
seem unreasonable for perfectly interlocked mineral localized increase of damage to the rock and a corre-
grains subjected to small displacements. sponding loss of cohesion. The degree of damage will be
highest at the surface of the opening where confinement
Application of the Griffith locus is zero and stress concentrations are greatest, and this
The Griffith crack-damage locus can be readily ap- damage will decrease with increasing distance into the
plied to the rock surrounding an underground opening. rock. From equations (9) or (10), it can be seen that
658 MARTIN and CHANDLER: PROGRESSIVE FRACTURE OF LA( DU BONN[:.I GRANITt

when o"3 = 0, the frictional c o m p o n e n t o f the rock's reduction is a function of the a c c u m u l a t e d d a m a g e .

strength plays essentially no role in determining the Hence the strength a r o u n d an unconfined u n d e r g r o u n d
strength a r o u n d an u n d e r g r o u n d opening. The strength opening, in a high stress environment, will be lowest at
is d e t e r m i n e d by the cohesion which is a function o f the tunnel surface. Thus back analysis o f the failed or
crack damage. Hence the strength a r o u n d an under- d a m a g e d openings to determine the in s i t u strength
g r o u n d opening, in a high stress environment, will be ~ o u l d suggest m u c h lower strength than that found from
lowest at the tunnel surface. F o r this class of p r o b l e m s routine l a b o r a t o r y testing, even if the rock mass was
back analysis o f the failed or d a m a g e d openings to fairly massive. This low-strength d a m a g e d material
d e t e r m i n e the in s i t u strength would suggest m u c h lower would be localized a r o u n d the opening, and the rock
strength than that f o u n d from routine l a b o r a t o r y testing, mass outside this d a m a g e d zone would still have the
even if the rock mass was fairly massive. F o r our test u n d a m a g e d strength. It is suggested that this may be one
results, the cohesion loss was greater than 50% o f o f the c o n t r i b u t i n g factors to restricting the depth o f
l a b o r a t o r y unconfined compressive strength. This low- b o r e h o l e - b r e a k o u t s , or failure zones, a r o u n d deep tun-
strength material would only be present in the local area nels and m a y also help explain an observed p h e n o m e n o n
a r o u n d the tunnel that has experienced the d a m a g e , and that the b a c k - a n a l y s e d strength a r o u n d tunnels with
the rock mass outside this d a m a g e d zone would still have stress-induced failures is a b o u t half the m e a s u r e d un-
the u n d a m a g e d strength. This m a y be one o f the con- confined compressive strength.
tributing factors to restricting the d e p t h o f borehole-
b r e a k o u t s , or failure zones a r o u n d deep tunnels and m a y A c k n o w l e d g e m e n t s T h i s work was supported by the Canadian
Nuclear Fuel Waste Management Program which is jointly funded by
also help explain an observed p h e n o m e n o n that the AECL and Ontario Hydro under lhe auspices of the Candu Owners
b a c k - a n a l y s e d strength a r o u n d tunnels with stress-in- Group.
duced failures is a b o u t h a l f the measured unconfined
Accepted[or puhlication 22 April 1994.
compressive strength [41,42,43, 44].

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Martin et al. (1994) investigated the failure of granite specimens under uniaxial and triaxial compression conditions
by damage controlled testing. it is demonstrated that the crack initiation stress is independent and crack damage
stress is dependent of the damage accumulated on the sample.