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Identifying crack initiation and propagation
thresholds in brittle rock
E. Eberhardt, D. Stead, B. Stimpson, and R.S. Read
Abstract: Recent work at the Underground Research Laboratory of Atomic Energy of Canada Limited in Pinawa, Manitoba,
has shown that high compressive stresses near the tunnel face significantly contribute to the loss of strength, and eventual
failure of the rock, through stress-induced brittle fracturing. A program of laboratory testing has been undertaken to
investigate the effects of brittle fracture on the progressive degradation of rock mass strength. The work carried out in this
study involves a detailed analysis of the crack initiation and propagation thresholds, two key components in the brittle-fracture
process. This paper describes new techniques developed to enhance existing strain gauge and acoustic emission
methodologies with respect to the detection of these thresholds and their effects on the degradation of material strength.
Key words: tunnel, rock failure, brittle fracture, crack initiation, crack propagation.
Résumé : Des travaux récents au «Underground Research Laboratory» de l’AECL à Pinawa, Manitoba, ont démontré que les
fortes contraintes de compression près de la face du tunnel contribuent de façon significative à la perte de résistance, et
éventuellement à la rupture de la roche, par suite de fractures fragiles induites par les contraintes. Un programme d’essais en
laboratoire a été entrepris pour étudier les effets de la fracture fragile sur la dégradation progressive de la résistance de masse
de la roche. Le travail réalisé dans cette étude comprend une analyse détaillée de l’initiation de la fissure et des seuils de
propagation, deux composantes clés dans le processus de la fracture fragile. Cet article décrit de nouvelles techniques qui ont
été développées pour valoriser les méthodologies existantes de jauges de contraintes et d’émission acoustique pour la
détection de ces seuils et de leurs effets sur la dégradation de la résistance du matériau.
Mots clés : tunnel, rupture de la roche, rupture fragile, initiation de fissures, propagation de fissures.
[Traduit par la rédaction]
Introduction
The excavation of an underground opening in a stressed rock
mass results in the deformation of the near-field rock due to a
redistribution of stresses, resulting in induced stress concentra-
tions. This stress redistribution increases strain energy in zones
of increased compression. If the resulting imbalance in the
energy of the system is severe enough, it can result in the
progressive degradation of the rock mass strength through
fracturing. Thus it is important to establish the thresholds as-
sociated with microscale and macroscale fracturing in the
in situ rock mass.
The deformation and fracture characteristics of brittle rock
have been studied by numerous researchers over the past
30 years (Brace 1964; Bieniawski 1967a; Wawersik and
Fairhurst 1970; Lajtai and Lajtai 1974; Martin and Chandler
1994). The general consensus of these studies has been that the
failure process can be broken down into a number of stages
based largely upon the stress–strain characteristics displayed
through axial and lateral deformation measurements recorded
during uniaxial and triaxial laboratory tests. Based on the
stress–strain behaviour of a loaded material (Fig. 1), Brace
(1964) and Bieniawski (1967a) defined these stages as being
(1) crack closure, (2) linear elastic deformation, (3) crack in-
itiation and stable crack growth, (4) critical energy release and
unstable crack growth, and (5) failure and postpeak behaviour.
Crack closure occurs during the initial stages of loading
(σ<σcc in Fig. 1, where σis the total axial stress and σcc is
the stress at crack closure) when preexisting cracks orientated
at an angle to the applied load close. During crack closure, the
stress–strain response is nonlinear, exhibiting an increase in
axial stiffness (i.e., deformation modulus). The extent of this
nonlinear region is dependent on the initial crack density and
geometrical characteristics of the crack population. Once the
majority of preexisting cracks have closed, linear elastic de-
formation takes place. The elastic constants (Young’s modu-
lus, Poisson’s ratio) of the rock are calculated from this linear
portion of the stress–strain curve.
Crack initiation (σci) represents the stress level where mi-
crofracturing begins and is marked as the point where the lat-
eral and volumetric strain curves depart from linearity. Crack
propagation can be considered as being either stable or unsta-
ble. Under stable conditions, crack growth can be stopped by
controlling the applied load. Unstable crack growth occurs at
the point of reversal in the volumetric strain curve and is also
known as the point of critical energy release or crack damage
Can. Geotech. J. 35: 222–233 (1998)
Received July 24, 1998. Accepted December 2, 1997.
E. Eberhardt.1Department of Geological Sciences, University
of Saskatchewan, Saskatoon, SK S7N 5E2, Canada.
D. Stead. Camborne School of Mines, University of Exeter,
Redruth, Cornwall TR15 3SE, England.
B. Stimpson. Department of Civil and Geological Engineering,
The University of Manitoba, Winnipeg, MB R3T 5V6, Canada.
R.S. Read.2Whiteshell Laboratories, Atomic Energy of Canada
Limited, Pinawa, MB R0E 1L0, Canada.
1Present address: Engineering Geology, ETH H`nggerberg,
8093 Zhrich, Switzerland.
2Present address: Klohn–Crippen Consultants Ltd., Calgary,
AB T2E 7H7, Canada.
222
© 1998 NRC Canada
stress threshold σcd (Martin 1993). Bieniawski (1967a) defines
unstable crack propagation as the condition which occurs
when the relationship between the applied stress and the crack
length ceases to exist and other parameters, such as the crack
growth velocity, take control of the propagation process. Un-
der such conditions, crack growth would continue even if the
applied load were kept constant.
Unstable crack growth continues to the point where the nu-
merous microcracks have coalesced and the rock can no longer
support an increase in load. Martin (1993) notes that the peak
strength of granite (including the uniaxial compressive strength
in unconfined tests) is not a unique material property but is de-
pendent on loading conditions such as the loading rate. The crack
initiation and crack damage stresses were found to be more char-
acteristic, essentially independent of loading conditions.
Work at the Atomic Energy of Canada Limited (AECL)
Underground Research Laboratory (URL) has concentrated on
using the crack initiation (σci) and crack damage (σcd) stress
thresholds to better quantify rock damage. The detection of
these thresholds, however, has proven difficult especially with
respect to crack initiation. A series of uniaxial tests were sub-
sequently performed to refine existing analysis techniques for
determining the crack initiation stress threshold of intact rock.
These techniques involve the use of stress–strain data and
acoustic emission monitoring. New methods of data analy-
sis were also introduced to help substantiate the interpreta-
tion of the data with respect to crack initiation and growth.
These methods include the application of a moving point
regression technique to the stress–strain data and an exami-
nation of the changes in the acoustic event properties with
loading. Testing was performed on cylindrical samples of Lac
du Bonnet granite obtained from the URL 130 Level (130 m
below ground surface). This paper examines the effects of
crack initiation and damage in the degradation of material
strength, emphasizing the underlying mechanisms and char-
acterization methods.
Detection of crack development in brittle
rock
A number of techniques have been developed to detect and
study crack growth in brittle materials. The most common of
these involves the use of electric resistance strain gauges to
measure slight changes in sample deformation that can be re-
lated to the closing and opening of cracks (Brace et al. 1966;
Bieniawski 1967b). To a lesser extent, acoustic emission
monitoring has been used to correlate the number of acoustic
events to various strain gauge responses (Scholz 1968; Ohnaka
and Mogi 1982; Khair 1984). Other techniques have involved
the use of photoelastics, optical diffraction patterns, scanning
electron microscopes, laser speckle interferometry, ultrasonic
probing, and electrical resistivity.
Stress–strain data
Strain gauge measurements have provided the most insight
into delineating the stages of crack development in rock.
The use of strain gauges in past studies, however, has been
somewhat limited by data sampling, computing, and storage
Fig. 1. Stress–strain diagram showing the elements of crack development (after Martin 1993). Note that only the axial (εaxial) and lateral (εlateral)
strains are measured; the volumetric strain and crack volume are calculated. σaxial, axial stress; σucs, peak strength; ∆V, change in volume; V,
initial volume.
Eberhardt et al. 223
© 1998 NRC Canada
capabilities. Recent work by the Rock Mechanics Research
Group at the University of Saskatchewan has been directed
towards using more powerful computers with larger data stor-
age capabilities in conjunction with faster data logging sys-
tems. These capabilities have allowed for tests to be conducted
in which the sampling rate has been increased five to ten times
that allowable with older testing systems (i.e., capable of five
measurements per second). Thus more data points along the
axial and lateral stress–strain curves can be collected and ex-
amined for indications of crack growth. In essence, higher
resolution of sample deformation relating to crack initiation
and growth is achieved.
Moving point regression technique
Improvements can also be made in the way strain gauge data
are analyzed. Stress–strain data analysis has traditionally con-
centrated on picking noticeable slope changes in the plotted
stress–strain curves (axial, lateral, and calculated volumetric)
which may then be correlated to several of the theoretical
stages in crack development. However, a high degree of error
and subjectivity is incorporated into this analysis procedure
when one considers the combined use of poor data resolution
and the manual picking of points. Bearing in mind that certain
inflections, some of which may be undetectable to the unaided
eye, in the stress–strain curves are of key interest, a moving
point regression technique, which uses the first derivative of
the curves to highlight any slope or rate changes in the curves,
was developed.
The moving point regression technique uses a “sliding win-
dow” approach to move through an x,ydata set, fitting a
straight line over a user-defined interval. The slope at each
point is calculated over the interval and recorded, the process
being repeated at successive points (Fig. 2). When plotted
against the parameter of interest, inflections in the original x,
ydata curve are highlighted. For example, using an axial stress
versus axial strain curve, the technique produces a moving
point average of the changes in the Young’s modulus through-
out loading (Fig. 3). This is referred to as the average axial
stiffness, therefore avoiding problems in terminology when
calculating the slope outside the range of linear elastic behav-
iour. In regards to the sensitivity of the method to the user-
defined regression interval, it was found that the general shape
of the stiffness curve remained the same with increasing inter-
val sizes, but small-scale fluctuations in the measured defor-
mation response were filtered out when extremely large
regression intervals were used. Analysis results indicate that
the size of the regression interval should be approximately 5%
of the total number of x,ydata pairs.
Acoustic emission response in rock
Acoustic emission (AE) in polycrystalline rock originates as a
result of dislocations, grain boundary movement, or initiation
and propagation of fractures through and between mineral
grains. The sudden release of stored elastic strain energy ac-
companying these processes generates an elastic stress wave
which travels from the point of origin within the material to a
boundary where it is observed as an acoustic event (Hardy
1977). AE techniques have been used with some success in
identifying microfracturing in brittle materials. Scholz (1968)
found that characteristic AE patterns in rock correlate closely
with stress–strain behaviour. However, most of the success in
correlating AE activity to microfracturing has involved the
later stages of crack development. This is due to the fact that
the majority of AE events occur just prior to failure. The lack
of significant AE activity in the initial stages of loading makes
it more difficult to distinguish background noise from fracture-
related acoustic events. A balance must be struck between set-
ting event threshold limits high enough to filter out the
majority of the background noise, yet low enough to pick up
the beginning of the microfracturing process.
Characteristics of an acoustic event
In addition to recording the number of acoustic events and
correlating this number to the measured deformation response
in the rock, it is also possible to record certain properties of the
AE waveforms. The signal waveform of an acoustic event is
affected by the characteristics of the source, the path taken
from the source to the sensor, the sensor characteristics, and
the recording system. Generally, these waveforms are complex
and using them to characterize the source can be difficult. Due
to these complexities, AE waveform analysis can range from
simple parameter measurements to more complex pattern rec-
ognition. However, relatively little work has been done in the
area of waveform analysis with respect to rock mechanics and
the progressive degradation–failure process in rock.
The event threshold serves as a reference for several of the
simple waveform parameters (Fig. 4). These AE event proper-
ties are defined in Table 1. The characteristics of an acoustic
event may also be used to approximate the release of kinetic
energy through the AE event. The true energy is directly pro-
portional to the area under the acoustic emission waveform
which in turn can be measured by digitizing and integrating
the waveform signal. However, this can be both difficult and
time consuming. As a simplification, the event energy can be
approximated as the square of the peak amplitude (Spanner
et al. 1987; Lockner et al. 1991) or the square of the peak
amplitude multiplied by the event duration (Beattie 1983;
Mansurov 1994). The resulting values are actually more
Fig. 2. Moving point regression technique.
Can. Geotech. J. Vol. 35, 1998
224
© 1998 NRC Canada
representative of the event power (the units are given in dB
and dB⋅s, respectively), but are commonly referred to as en-
ergy calculations in the literature due to their approximately
linear relationship with energy. This type of “energy” analysis
helps in accentuating AE events with abnormally large ampli-
tudes or durations.
Laboratory testing of Lac du Bonnet
granite
Laboratory uniaxial testing was performed on 20 samples of
pink Lac du Bonnet granite from the 130 Level of the URL.
The pink granite is medium to coarse grained, with an average
grain size between 3 and 4 mm. Based on work by Jackson and
Lau (1990), 61 mm diameter cores were chosen to minimize
size effects. Jackson and Lau found that samples of Lac du
Bonnet granite with smaller diameters were more sensitive to
sample disturbance, thereby influencing the observed me-
chanical behaviour of the rock samples during testing.
Samples were prepared for testing according to American
Society for Testing and Materials standards (designation
ASTM D4543-85) with length to diameter ratios of approxi-
mately 2.25. Considerable care was taken in minimizing the
influence of end effects on strain gauge and AE transducer
readings. This entailed the use of a specially constructed frame
that allowed for the sample ends to be highly polished. Meas-
urements of end surface flatness and perpendicularity five
times lower than those recommended by ASTM standards
were attained. Each sample was instrumented with six Micro-
Measurement electric resistance precision strain gauges (three
axial and three lateral at 60° intervals, 12.7 mm in length, with
a 5% strain limit) to record sample deformation and four
175 kHz piezoelectric transducers to record acoustic emis-
sions. Strain gauges were epoxied directly to the cleaned sam-
ple surface to ensure a solid bond, whereas the AE transducers
were mounted onto wave guides, which were in turn epoxied
to the sample surface. Acoustic emissions were recorded with
an AET 5500 logging system using a gain of 40 dB and a
threshold value of 0.1 V.
Prior to uniaxial testing, P- and S-wave traveltimes were
recorded for each sample (Table 2). Results from these meas-
urements indicate a significant increase (40–50%) in both P-
and S-wave velocities when compared with previously tested
granite and granodiorite samples from the 420 Level of the
URL (Eberhardt et al. 1996). With maximum in situ stresses
increasing from approximately 15 MPa at 130 m depth to
60 MPa at 420 m depth (Martin 1990), considerable damage
through stress-induced microcracking would be expected in
samples from the 420 Level. Static elastic properties were de-
termined from the stress–strain data following ASTM stand-
ards (designation D3148-93) and include Poisson’s ratio,
average Young’s modulus, and secant Young’s modulus (Ta-
ble 3). The difference between the average and secant modulus
Fig. 3. Moving point regression analysis of axial stress–strain data showing the changes in axial stiffness throughout loading for a 130 Level
pink granite. Eavg, average Young’s modulus.
Fig. 4. Definition of simple acoustic emission waveform
parameters.
Eberhardt et al. 225
© 1998 NRC Canada
(which would include any nonlinearity due to preexisting
cracking) for the 130 Level samples is relatively small
(5 GPa), indicating that initial sample damage is relatively
low.
Interpretation of crack development in Lac
du Bonnet granite
It is generally accepted that the first fractures in a uniaxially
loaded brittle material are tensile microcracks (Lajtai and La-
jtai 1974). The growth of these cracks has been shown to occur
in the direction of the major principal stress (σ1), where cracks
not aligned with σ1grow along a curved path to align them-
selves with σ1(Fig. 5). This phenomenon has been observed in
a number of materials including glass (Brace and Bombolakis
1963; Hoek and Bieniawski 1965), hard plastics (Nemat-
Nasser and Horii 1982; Cannon et al. 1990), plaster (Lajtai
1971), ice (Schulson et al. 1991), and rock (Wawersik and
Fairhurst 1970; Peng and Johnson 1972; Huang et al. 1993).
The opening of crack faces parallel to the applied load and the
closure of crack faces perpendicular to the load cause certain
changes in the relative lateral and axial deformations, respec-
tively. These changes appear as inflections in the stress–strain
curves which, in turn, can be used to identify the different
stages of rock deformation and failure.
Test results were analyzed to define the stages of rock
deformation and failure as defined by Brace (1964) and
Bieniawski (1967a). The following discussion highlights some
of the key observations. Results from this analysis are summa-
rized in Table 4.
Crack closure
The crack closure stress level (σcc in Fig. 1) indicates the load
at which a significant number of preexisting cracks have
closed and near-linear elastic behaviour begins. This point is
approximated by determining the point on the stress–strain
curve where the initial axial strain appears to change from
nonlinear to linear behaviour. Crack closure stresses (σcc) were
picked for each test using the moving point regression analysis
(Fig. 3). As was expected, a rapid increase in axial stiffness
was observed before values levelled off and behaved in a rela-
tively linear manner. This pattern and the corresponding values
were consistent for each test (Table 4).
Examination of the lateral stiffness curve (Fig. 6) over this
region reveals relatively high stiffness values when the load is
first applied to the sample (the lateral stiffness term, although
somewhat unconventional, represents the change in the lateral
strain rate with axial loading). Artificially high values of this
term during the initial stages of loading represent a point in the
load history where there is not a continuous transmission of
stresses due to the presence of open microcracks, therefore the
lateral and axial strain responses are not fully coupled. These
initially high values are followed by a marked reduction (35%)
during the first 25 MPa of loading. The initial stages of crack
closure appear to predominantly involve the simple movement
of preferentially aligned crack walls towards one another,
parallel to the direction of applied load. This would have a
significant effect on the axial strain but little effect on the
lateral strain, since the displacement would be in the axial
direction. With increasing load, values of lateral stiffness be-
gin to rapidly decrease, possibly signifying shear or sliding
movement between the faces of closing or closed cracks. This
behaviour has been observed in glass plates by Bieniawski
(1967b) who noted that the sliding deformation demonstrated
by single closed cracks continues even during linear elastic
behaviour.
Linear elastic behaviour
Figure 3 shows that after crack closure is reached, a period of
relatively linear axial strain occurs. The average Young’s
AE event property Description
Ring-down count The number of times a signal crosses a preset threshold datum; in general, large events require more cycles to ring
down to the threshold level and will produce more counts than a smaller event; provides a measure of the intensity of
the acoustic emission event
Peak amplitude Related to the intensity of the source in the material producing an AE event; measurements are generally recorded in
logarithmic units (decibels dB) to provide accurate measurement of both large and small signals
Event duration When an acoustic event first crosses the preset threshold, an event detector measures the time that the waveform
amplitude remains above the threshold, thereby giving the event duration
Rise time Measures the time it takes to reach the peak amplitude of an event; provides an account of the positive-changing AE
signal envelope
Table 1. Definition of acoustic emission (AE) event properties (as shown in Fig. 4).
Location rock type Density (g/cm3)Pwave (m/s) Swave (m/s)
130 Level pink granite 2.62 (±0.01) 4890 (±190) 3030 (±120)
420 Level grey granite 2.61 (±0.02) 3220 (±100) 2160 (±60)
Note: Standard deviation is given in parentheses.
Table 2. Average acoustic velocities of URL granites.
Property 130 m Level Pink Granite
Average modulus, Eavg (GPa) 66.1 (±2.5)
Secant modulus, ES(GPa) 61.0 (±2.9)
Poisson’s ratio, ν0.31 (±0.04)
Note: Standard deviation is given in parentheses.
Table 3. Average static elastic moduli for URL pink granite at
the 130 Level.
Can. Geotech. J. Vol. 35, 1998
226
© 1998 NRC Canada
modulus was taken as a least squares fit along this region. In
terms of lateral stiffness, linear behaviour is never truly
reached. Instead, the lateral stiffness continuously decreases
from values of approximately 300 GPa to values less than
20 GPa prior to failure (Fig. 6). This would seem to indicate
that a number of processes may be contributing towards the
gradual but continual loss of lateral stiffness in the specimens
tested. These may include sliding (shear) between faces of clos-
ing cracks, tensile opening of cracks during crack initiation, and
possibly further shear movement related to crack coales-
cence – columnar buckling during the latter stages of rock de-
formation. Following ASTM standards, Poisson’s ratio was
calculated using a least squares fit over the same interval as
that used in calculating the average Young’s modulus (i.e., lin-
ear region of the axial stress–strain curve). It will be shown
later that this may not be the most appropriate interval over
which to calculate Poisson’s ratio.
Crack initiation
The crack initiation stress threshold, as determined through
laboratory testing, has been defined as the point where the
lateral strain curve departs from linearity (Brace et al. 1966;
Bieniawski 1967a; Lajtai and Lajtai 1974). Examination of the
lateral strain curve (Fig. 6) reveals that the identification of this
point can be very subjective. This is clear from analysis of the
lateral stiffness curve which indicates that at no time is the
lateral stress–strain curve truly linear. Noting the difficulty in
using lateral strain gauge data, especially in highly damaged
samples, Martin (1993) suggested using the calculated crack
volumetric strain to identify crack initiation. For a cylindrical
sample loaded uniaxially, crack volume is determined by sub-
tracting the linear elastic component of the volumetric strain,
given by
[1] εVelastic =1−2ν
Eσaxial
where Eand νare the elastic constants and σaxial is the axial
stress level, from the volumetric strain calculated from the
measured axial (εaxial) and lateral (εlateral) strains, given by
[2] εV=ε
axial +2εlateral
The remaining volumetric strain is attributed to axial cracking,
i.e.,
[3] εVcrack =ε
V–ε
elastic
Martin (1993) defines crack initiation as the stress level at
which dilation (crack volume increase) begins in the crack
volume plot (Fig. 1).
This method is limited, however, because of its dependence
on the use of the elastic constants Eand ν. Although the
Young’s modulus Ecan be determined with a reasonably high
degree of confidence and consistency, the nonlinearity of the
lateral strain response complicates the measure of Poisson’s
ratio. The resulting value is the ratio of lateral to axial strain
magnitudes based on the linear elastic axial strain behaviour
and the “best approximation” of a straight line through a non-
linear region of lateral strain over the same stress interval.
Table 5 lists the respective Poisson’s ratio values calculated
over the same stress interval as the average Young’s modulus
(as per ASTM standards) and over the stress interval between
crack closure and crack initiation as determined using the mov-
ing point regression analysis. This variability introduces a
large degree of uncertainty into the crack volume calculation
used to determine crack initiation. Figure 7 demonstrates the
sensitivity of crack initiation values to changes in the Poisson’s
ratio using crack volume reversal as the indicator (for example,
a change of ±0.05 in the Poisson’s ratio results in a ±40%
change in the σci value). The crack volume stiffness plot shown
in Fig. 7 is calculated as the change in slope of the crack vol-
ume strain curve, the reversal of which is noted by the change
from a positive to a negative slope.
Using an approach that involved the combined use of the
moving point regression analysis and acoustic emission re-
sponse, it was found that that the crack initiation stress thresh-
old could be more accurately determined. From the strain
gauge data, it was found that, although the lateral strain is
nonlinear, rate changes do occur and can be correlated to the
growth of cracks in the sample. These rate changes are most
evident when analyzing the volumetric strain and stiffness
curves (Fig. 8). The volumetric stiffness curve is based on the
stress-dependent rate of change in the volumetric strain. Volu-
Fig. 5. Model of internal crack extension towards the major
principal stress. σt, major principal stress; τ, shear stress.
Property Stress threshold (MPa)
Crack closure, σcc 47.5 (±2.9)
Crack initiation, σci 81.5 (±3.7)
Crack coalescence, σcs 104.0 (±3.8)
Crack damage, σcd 157.3 (±9.9)
Peak strength, σucs 206.5 (±10.0)
Note: Standard deviation is given in parentheses.
Table 4. Average strength parameters for URL pink
granite.
Eberhardt et al. 227
© 1998 NRC Canada
metric strain is defined in eq. [2]. The volumetric stiffness is
calculated as the slope of the volumetric strain versus axial
stress curve. The rate at which the volumetric strain curve
changes is dependent on the rate of change in the measured
axial and lateral strain.
Examination of the volumetric stiffness curve indicated a
series of characteristic patterns (Fig. 8) that recur in each of
the uniaxial tests performed. During the initial stages of load-
ing, the axial strain controls the shape of the volumetric strain
curve. The nonlinear behaviour of the axial strain during crack
closure distinguishes itself as an irregular region along the
volumetric stiffness curve (Fig. 9). This irregular region is fol-
lowed by a linear region marked by a small break in slope
signifying a rate change. This break in slope represents the
transition from crack closure to near-linear elastic behaviour.
This linear region also marks the stress interval in which the
lateral strain achieves its most linear behaviour (i.e., the Pois-
son’s ratio should be calculated in this region as shown in
Table 5). The volumetric stiffness curve then makes a transi-
tion to a less regular region without any discernible break in
slope at approximately 80 MPa. Throughout this region the
axial strain rate maintains a near-constant level, therefore any
change can be attributed to a change in the lateral strain rate.
Changes in the lateral strain rate result in slight slope changes
in the volumetric strain curve. However, because the axial
strain rate still dominates in controlling the shape of the volu-
metric strain curve, no noticeable slope change occurs in the
volumetric stiffness curve. Although these changes in the lat-
eral strain rate do not contribute to the overall volumetric strain
enough to cause a major change in the slope of the volumetric
stiffness curve, they do contribute enough to cause irregulari-
ties in it. These changes in the lateral strain rate, and conse-
Fig. 7. Variability of crack volume strain reversal with Poisson’s ratio for a 130 Level pink granite.
Fig. 6. Plots of lateral strain and lateral stiffness against axial stress for a 130 Level pink granite.
Can. Geotech. J. Vol. 35, 1998
228
© 1998 NRC Canada
quently the volumetric stiffness curve, signify crack initiation
at approximately 80 MPa.
Correlation between the behaviour of the volumetric stiff-
ness curve and crack initiation can also be validated through
acoustic emission analysis. The average response from the four
AE transducers shows that the majority of activity occurs to-
wards the end of the test. Although AE activity occurs con-
tinuously throughout the test, the logarithmic plot in Fig. 10
shows that the beginning of significant AE activity is at ap-
proximately 80 MPa. This coincides with the crack initiation
stress threshold (σci) of 80 MPa as determined using the volu-
metric stiffness curve in Fig. 9. AE activity prior to this point
can be attributed to movement along crack faces during crack
closure, as recognized in the lateral strain rate and previously
discussed. It is also likely that small cracks may form during
low stresses in areas already weakened by stress-relief cracking.
Fig. 8. Plots of volumetric strain and volumetric stiffness against axial stress for a 130 Level pink granite.
Fig. 9. Breakdown and correlation of volumetric stiffness with the stages in the compressive failure process of rock for a 130 Level pink granite.
Method of calculation ν
ASTMa0.31 (±0.04)
σcc →σ
cib0.24 (±0.03)
Note: Standard deviation is given in
parentheses.
aCalculated according to ASTM standards.
bCalculated over the stress interval between
crack closure and crack initiation as
determined through the moving point
regression analysis.
Table 5. Average Poisson’s ratio νfor
URL pink granite at the 130 Level.
Eberhardt et al. 229
© 1998 NRC Canada
In addition to the acoustic emission response, the properties
of the acoustic events themselves are markedly different be-
fore and after crack initiation. Figures 11 and 12 contain plots
of the maximum event duration and ring down counts. From
these plots it can be seen that a marked increase in their re-
spective magnitudes occurs at approximately 80 MPa, coin-
ciding with crack initiation. Larger event durations and ring
down counts both signify larger acoustic events. Although
acoustic activity occurs prior to this point, the size of the events
is relatively small (in terms of event duration, these events are
70% shorter in duration than those occurring above 80 MPa).
This may indicate that the acoustic events generated through
crack closure are much smaller than those generated through
stress-induced tensile cracking. Calculations of the acoustic
event “energy” based on the peak amplitude and event duration
of the event waveform (herein referred to as the elastic impulse
energy so as not to confuse it with the true energy) also rein-
forces these observations. Plots of the elastic impulse energy
and its stress-dependent rate change (Fig. 13) show that the
size of the events in terms of energy dramatically increases
shortly after crack initiation begins.
Crack coalescence and crack damage
In defining the various stages of crack behaviour, stable crack
growth is interpreted as one stage bounded at the lower end by
the crack initiation stress threshold (σci) and at the upper end
by the crack damage stress threshold (σcd). Analysis of labora-
tory data, however, indicates that this region may consist of
two stages distinguished by a major change in the volumetric
strain rate. Analysis of both the axial and lateral stiffness
curves indicates that a large rate change occurs in strain well
before unstable crack propagation (i.e., σcd). During stable
crack growth, rate changes are believed to occur solely in
terms of lateral strain, since crack growth is predominantly in
Fig. 10. Plot of log acoustic emission event count vs. axial stress for a 130 Level pink granite.
Fig. 11. Plot of event duration vs. axial stress for a 130 Level pink granite.
Can. Geotech. J. Vol. 35, 1998
230
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the direction of uniaxial loading (i.e., σ1). Volumetric strain
reversal then occurs as a result of increases in the lateral strain
rate surpassing the constant axial strain rate as the dominant
component in the volumetric strain calculation. This results in
a reversal of the volumetric strain curve defining the crack
damage threshold. Analysis indicates, however, that the axial
stiffness is not constant but decreases well before σcd at
160 MPa (Fig. 3). Analysis confirms that the lateral strain rate
change does eventually exceed the rate change in axial strain
thereby causing the reversal in the volumetric strain curve.
These changes can be seen in the volumetric stiffness curve,
where large increases in the lateral strain rate combined with
changes in the axial strain rate cause large irregularities as the
volumetric strain curve approaches reversal (Fig. 14).
The unexpected departure from linear behaviour seen in the
axial strain response prior to the crack damage threshold may
be explained through the coalescence of propagating cracks in
a loaded sample. At the beginning of crack initiation, small
tensile cracks begin to grow parallel to the applied load. These
cracks originate from small flaws preferentially aligned to the
loading direction so as to induce cracking through high tensile
stress concentrations. The cracks are assumed to appear ran-
domly throughout the sample and, for the most part, are iso-
lated from one another. They have very little effect in
decreasing the overall competency of the rock. As the load is
increased additional cracks begin to grow as their individual
strengths are exceeded, incrementally contributing to the deg-
radation of the inherent rock strength (i.e., cohesion). This is
evident in the acoustic emission data which show increasing
bursts of AE activity leading up to σcd (Fig. 10).
As cracks increase, both in number and size, they eventu-
ally begin to interact with one another. Crack interaction then
becomes extremely complex as stress shadows overlap. This
has been demonstrated using a numerical modelling study of
the process (Eberhardt et al. 1998). Eventually cracks will be-
gin to step out and coalesce (i.e., develop en echelon (Lajtai
et al. 1994)). This coalescence of cracks would involve some
crack growth at oblique angles to the loading direction and
perhaps an element of shearing, thereby contributing to a
change in the axial strain rate. Thus, the changes seen in the
axial and volumetric stiffness curves may be attributed to a
stage of crack coalescence (σcs) prior to the crack damage
stress threshold.
Following crack coalescence, determination of the crack
damage stress threshold (σcd) is relatively straightforward. Al-
though a certain degree of subjectivity may be introduced by
picking the point of volumetric strain reversal directly off the
volumetric strain curve, the point stands out very clearly in a
plot of volumetric stiffness versus axial stress (Fig. 14). The
crack damage threshold could represent the intermediate-term
strength of the sample, since beyond this point failure will even-
tually occur through unstable cracking (Martin 1993). Values of
σcs and σcd for the tested samples are given in Table 4.
Conclusions
The deformation and fracture characteristics of brittle rock are
an important consideration in assessing its long-term strength.
The initiation, propagation, and interaction of stress-induced
cracks are closely linked but extremely complicated. Through
the combined use of strain gauge analysis and acoustic emis-
sion monitoring, techniques were developed to aid in the iden-
tification and characterization of mechanisms leading to brittle
failure.
The following observations were made with respect to
uniaxial testing performed on samples of pink Lac du Bonnet
granite:
(1) Crack closure involved both axial and lateral strain com-
ponents.
(2) Near-linear elastic behaviour was seen only in axial
strain measurements. Lateral strain followed a nonlinear trend
throughout the entire test, as depicted through continuously
decreasing values of lateral stiffness.
(3) The combined use of moving point regression analysis
(performed on the axial, lateral, and volumetric stress–strain
curves) and acoustic emission response (including the event
Fig. 12. Plot of ring down counts vs. axial stress for a 130 Level pink granite.
Eberhardt et al. 231
© 1998 NRC Canada
Fig. 13. Plot of cumulative elastic impulse energy vs. axial stress and the stress-dependent energy rate vs. axial stress for a 130 Level pink granite.
Fig. 14. Plot of average volumetric stiffness vs. axial stress, indicating the occurrence of major strain rate changes between crack initiation and
crack damage for a 130 Level pink granite. σcs, crack coalescence stress threshold.
Can. Geotech. J. Vol. 35, 1998
232
© 1998 NRC Canada
properties and energy calculations) provided the most accurate
and reliable means of identifying crack initiation (σci).
(4) Analysis of both the axial and lateral stiffness curves
indicates that a significant rate change in strain occurs prior to
the crack damage threshold, possibly marking the small-scale
coalescence of cracks.
Acknowledgments
Parts of the work have been supported by Atomic Energy of
Canada Limited and a Natural Sciences and Engineering Re-
search Council of Canada operating grant. The authors wish to
thank Dr. Derek Martin and Zig Szczepanik for their sugges-
tions and contributions to this work. Special thanks are ex-
tended to Dr. Emery Lajtai for his insights into the initial stages
of this work.
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