Rock Mech Rock Eng (2012) 45:607–617

DOI 10.1007/s00603-012-0221-6

ORIGINAL PAPER

Evaluation of Methods for Determining Crack Initiation

in Compression Tests on Low-Porosity Rocks
Mohsen Nicksiar • C. D. Martin

Received: 15 August 2011 / Accepted: 10 January 2012 / Published online: 25 January 2012
Ó Springer-Verlag 2012

Abstract Laboratory testing of rocks is traditionally 1 Introduction

carried out to determine the peak strength using the ISRM
Suggested Methods or other suitable standards. However, it Griffith (1921) used a constant tensile stress loading system
is well known that in low-porosity crystalline rocks there to support his energy theory for rupture of brittle solids. In
are at least three distinct stages of compressive loading that these experiments, as in any direct tension test for brittle
can be readily identified if the stress–strain response is solids the tensile stress required to initiate a crack is
monitored during the loading process: (1) crack initiation, essentially the same as the ultimate tensile strength. Hence
(2) unstable crack growth, i.e., crack coalescence and (3) in tension, crack initiation is synonymous with rupture
peak strength. Crack initiation is noted as the first stage of (Bieniawski 1967a). Griffith (1924), using a two-dimen-
stress-induced damage in low-porosity rocks, yet the sug- sional approach, applied his energy concepts to the rupture
gested guidelines of the ISRM for compression tests make of rock in compression and suggested that the rupture of
no mention of crack initiation. In addition, recent research rock in unconfined compression was eight times its tensile
suggests that crack initiation can be used as an estimate for strength. In 1963, Murrell extended Griffith’s theory to
the in situ spalling strength, commonly observed around triaxial stress conditions, which gave a ratio of uniaxial
underground excavations in massive to moderately jointed compressive strength (UCS) to tensile strength of 12.
brittle rocks. Various methods have been proposed for Laboratory tests have shown that the UCS is approximately
identifying crack initiation in laboratory tests. These 15–20 times the tensile strength for most rocks and there-
methods are evaluated using ten samples of Äspö Diorite fore the Griffith based criteria are seldom used to predict
and the results are compared with a simplified method, the rupture strength of rocks in compression. Nonetheless,
lateral strain response. Statistically, all methods give researchers have clearly identified that crack initiation that
acceptable crack-initiation values. It is proposed that the is routinely observed in the laboratory testing is a charac-
ISRM Suggested Methods be revised to include procedures teristic of common low-porosity crystalline rocks (Brace
suitable for establishing the crack-initiation stress. et al. 1966; Stacey 1981; Martin and Chandler 1994).
In Cook 1963, Cook studied the initiation of cracks
Keywords Crack initiation  Lateral strain response  around Witwatersrand gold mines using seismic events. He
Uniaxial compressive strength  Spalling found out that the events were associated with the forma-
tion of cracks associated with the mining front where the
compressive stress concentrations were the greatest. Fair-
hurst and Cook (1966) evaluated the formation of thin
slabbing, observed around deep South African tunnels and
M. Nicksiar (&)  C. D. Martin
commonly referred to as spalling. They concluded that the
Department of Civil and Environmental Engineering,
University of Alberta, Edmonton, AB T6G 2W2, Canada formation of the slabs could be explained using the
e-mail: nicksiar@ualberta.ca extension of Griffith cracks in a compressive stress field
C. D. Martin growing parallel to the maximum compressive stress. This
e-mail: derek.martin@ualberta.ca slabbing process was also examined by Stacey (1981) using

123
608 M. Nicksiar, C. D. Martin

an extensional strain criterion developed from laboratory 250 Lateral strain Volumetric strain Axial strain

compression tests.
The performance of square tunnels in South Africa was 200

Axial stress (MPa)

evaluated by Hoek and Brown (1980) who found that when
the far-field maximum stress magnitude exceeded 0.15 of 150

the UCS, spalling was observed around the underground

opening. Martin et al. (1999) converted the Hoek and 100

Brown criterion to a maximum tangential stress criterion

50
and found that when the maximum tangential stress
exceeded approximately 0.4 of the UCS, slabbing was
0
observed. This criterion was applied by Rojat et al. (2009) 0.1 0.0 0.1 0.2 0.3
to the Lötschberg tunnel, Switzerland to analyse the stress- Strain (%)
induced problems associated with tunnel boring machine-
Fig. 1 Typical stress–strain response recorded in a uniaxial com-
excavated tunnel at the depth of over 1,000 m. They con- pressive test
cluded that in all cases the in situ spalling strength was
significantly lower than the laboratory UCS and hence
using the peak laboratory strength is problematic for
evaluating the spalling strength for tunnel design. More displays four important inflections: (1) crack closure,
recently, Martin and Christiansson (2009) and Andersson observed in the axial strain as the starting point of linearity;
et al. (2009) have suggested that crack initiation from (2) crack initiation, observed in the lateral strain as the
laboratory uniaxial compressive tests could be used as an inflection point in lateral strain; (3) start of unstable crack
estimate for the in situ spalling strength. Hence, having a growth, observed in the volumetric strain as the maximum
reliable method for determining crack initiation in labora- volumetric strain; and (4) peak, observed in the axial strain as
tory tests might benefit the engineering community when the point of maximum stress (see Fig. 1). Crack closure may
estimating the spalling strength. or may not be present in the stress–strain response as it
The ISRM makes no mention of crack initiation in its simply depends on the volume of cracks that exist in the
‘‘Suggested Methods for Determining the Uniaxial Com- samples being tested. At stress magnitudes above crack
pressive Strength and Deformability of Rock Materials’’. initiation it has been shown by many researchers that stress–
Nonetheless, various authors have proposed different strain response is dominated by the initiation and growth of
methods for identifying crack initiation (e.g., Martin and cracks (Brace et al. 1966; Bieniawski 1967a; Lajtai 1974;
Chandler 1994; Brace et al. 1966; Bieniawski 1967a; Lajtai Hallbauer et al. 1973). Acoustic emission (AE) monitoring
1974; Stacey 1981; Eberhardt et al. 1998; Diederichs et al. techniques used by Lockner et al. (1991) to locate the sources
2004). The purpose of this paper was to review and eval- of the cracking within confined cylinders of granite showed
uate the various methods that have been proposed to that initially the cracks were distributed uniformly
identify crack initiation, using ten laboratory tests of Äspö throughout the specimen. However, as the peak stress was
granodiorite. The results are compared with those obtained approached the cracks coalesced on the boundary of the
from a simplified methodology. sample, eventually propagating across the sample as a dis-
crete zone of AE activity as the load was maintained. Moore
and Lockner (1995) investigated the crack density in the
vicinity of this AE zone and concluded that the crack density
2 Stress–Strain in Laboratory Compression Tests in this zone was an order of magnitude greater than that found
in undeformed samples. Thompson et al. (2006) performed
The ISRM Suggested Methods (Brown 1981) for ‘‘Deter- three triaxial compression experiments of Westerly granite
mining Uniaxial Compressive Strength and Deformability where the load was applied in such a way as to maintain a
of Rock Materials’’ suggests measuring and plotting the constant AE-rate. The results indicate that these findings
axial stress versus axial (eax) and lateral (elat) strains were valid for both slow and fast loading rate. They also
response as illustrated in Fig. 1. Also plotted in Fig. 1 is clearly showed that the change from stable crack initiation
the calculated volumetric strain (DV/V) given by and growth to unstable crack coalescence occurs abruptly
DV near peak strength that in its final stage may lead to formation
 eax þ 2elat : ð1Þ
V of a fault (Fig. 2). These findings help explain the change in
Since the early work of Brace et al. (1966) and Bieniawski the volumetric strain response from contraction to dilation
(1967a) it is now recognized that the stress–strain response in observed in the stress–strain response, at the onset of
both unconfined and confined tests for low-porosity rocks unstable crack growth (see Fig. 1). Research over the past

123
Determining Crack Initiation in Low-Porosity Rocks 609

AE coalesence 0.14

0.12

Volumetric strain (%)

0.10

0.08

0.06

0.04

0.02

0.00
0 50 100 150 200 250
Axial stress (MPa)

Fig. 3 Volumetric strain method proposed by Brace et al. (1966) to

establish crack initiation

Pre-peak Peak Post-peak by Brace et al. (1966). Brace et al. (1966) found that the
onset of dilatancy when normalized to its peak values
Fig. 2 Incremental distribution of acoustic emission activity mea- varied from an average of 0.45 for granite, 0.5 for marble
sured by Thompson et al. (2006) during the confined testing of
Westerly granite and 0.55 for aplite. Bieniawski (1967b) conducted similar
experiments on norite and quartzite. Using photographic
imaging and the volumetric strain method he concluded
40 years has clearly shown that behaviour of low-porosity that fracture/crack initiation in uniaxial compression is not
rocks in compression is linked to the initiation and growth of affected by specimen shape, loading platens or loading
cracks. In the next section we review the methods that are machine and that the mechanism of fracture in compres-
used to establish the stress magnitude associated with crack sion is essentially the same in uniaxial and triaxial
initiation. compression.
Martin and Chandler (1994) noted that crack initiation is
difficult to identify from the axial-stress volumetric–strain
3 Methods for Determining Crack Initiation curve, particularly if the specimen already contains a high
in Compression density of cracks. They proposed that crack initiation could
be determined using a plot of crack volumetric strain ver-
The methods that researchers have used to establish the sus axial strain. Crack volumetric strain (DV/V)cr is cal-
load associated with the onset of crack initiation during culated by subtracting the elastic volumetric strain (DV/V)el
laboratory compression loading have relied primarily on from the calculated volumetric strains (DV/V).
the measured strains. The methods utilized either the vol-    
DV DV DV
umetric strain or the lateral strain and have been modified ¼  ; ð2Þ
V cr V V el
by various researchers and at times augmented by acoustic
emission techniques. These methods are reviewed below, where
and a new method that utilizes the lateral strain is intro-  
DV 2m  1
duced. It is assumed that the methods used to measure the ¼ ðr1 þ 2r3 Þ: ð3Þ
V el E
lateral strain are accurate and reliable.
The elastic volumetric strains are calculated using the
3.1 Volumetric Strain Methods elastic constants (E, m) from the linear portion of stress–
strain curves in Fig. 1. As shown in Fig. 4 the method is
One of the earliest studies that utilized volumetric strain less subjective than the previous method and can be readily
to establish the onset of dilatancy in compression was programmed. One of the criticisms of the method is that the
carried out by Brace et al. (1966). They examined the crack initiation stress is influenced by the elastic constants,
stress–strain response of granite, marble and aplite mea- and therefore extra care must be exercised when deter-
sured with strain gauges. They noted that the onset of mining those constants (Eberhardt et al. 1998). The method
dilatancy could be established using volumetric strain by is also more difficult to use when there are a significant
examining when the volumetric strain deviated from the volume of cracks prior to testing. These cracks influence
early linear portion. Figure 3 illustrated the approach used the determination of Poisson’s ratio and according to

123
610 M. Nicksiar, C. D. Martin

0.10 250
Crack volumetric strain (%)

200
0.05

Axial stress (MPa)

150
0.00
100

- 0.05
50

- 0.10 0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00
Axial strain (%) Lateral stain (%)

Fig. 4 Crack volumetric strain method proposed by Martin and Fig. 5 Lateral strain method proposed by Lajtai (1974) to establish
Chandler (1994) to establish crack initiation crack initiation

Eberhardt et al. (1998) this uncertainty can significantly 0.0010

Ratio (lateral strain/Axial stress)

affect the crack initiation values.
0.0008

3.2 Lateral Strain Methods

0.0006

It is well known that the lateral strain is more sensitive than

0.0004
the axial strain to the growth of cracks in the region of the
stress–strain response before the onset of unstable crack 0.0002
growth. Consequently, several researchers examined
methods to establish the crack-initiation stress based on the 0.0000
0 50 100 150 200 250
lateral strains. Lajtai (1974) suggested that the axial strains
remained linear from crack closure through to the onset of Axial stress (MPa)

unstable crack growth in Fig. 1. Hence Lajtai (1974) Fig. 6 Ratio of the lateral strain to axial stress using a data increment
applied the methodology used by Brace et al. (1966) for the of 25 to establish crack initiation. Tangent line represented as dashed
volumetric strain method to the lateral strain. Lajtai (1974) lines
proposed that the crack-initiation stress could be estab-
lished by defining the onset where the lateral strains tests could be determined using plots of lateral strain versus
deviated from linearity (Fig. 5). This approach is also axial strain (Fig. 7). In essence Stacey was also indirectly
subjective if the stress-strain response deviates from the defining the crack-initiation stress, although the crack-ini-
typical stress–strain response due to intense pre-existing tiation stress was not used by Stacey to assess tunnel sta-
cracks. bility. Inspection of Fig. 7 shows that the lateral strain
Because the lateral strain more clearly defines the onset versus axial strain is nonlinear for essentially its entire
of cracking, changes in the ratio of the lateral strain to axial length. Andersson et al. (2009) also noted that applying
stress may also indicate the onset of cracking. This can be Stacey’s extensional strain approach was problematic
easily programmed and can take advantage of the large because of the nonlinearity. This issue may simply be
number of data points that are collected during a com- related to the impact of modern day data acquisition. It is
pression test. Given that a test may contain 1,000 data now common to acquire many hundreds data points during
points, the ratio of the lateral strain to axial stress can be the loading of a test sample while inspection of Stacey’s
determined over various increments to assess the sensitiv- original figure shows that the interpretation was made with
ity of the crack initiation to the chosen increment. Figure 6 only tens of data points. Hence, as illustrated in Fig. 7 this
shows the ratio for a data increment of 25 (tangent line increase in data frequency makes inflection points more
represented as dashed line). The data in Fig. 6 have also difficult to detect.
been smoothed using a moving median technique. Diederichs (2007) examined crack initiation using a
Stacey (1981) observed that stress-induced failure discrete element program and proposed that the change in
observed around South African gold mines could be esti- Poisson’s ratio should be suitable indicator for establishing
mated using an extensional strain criterion. Stacey (1981) the stress magnitudes associated with crack initiation.
suggested that the extensional strain criterion in laboratory Diederichs suggested that plotting the Poisson’s ratio

123
Determining Crack Initiation in Low-Porosity Rocks 611

0.4

105
0.3
Axial strain (%)

AE Count
0.2 104

0.1
103

0.0
0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 50 100 150 200
Axial stress (MPa)
Lateral strain (%)
Fig. 9 Acoustic emission count method proposed by Eberhardt et al.
Fig. 7 Extensional strain method proposed by Stacey (1981) to
(1998) to establish crack initiation. Tangent line represented as
establish crack initiation
dashed lines

versus the log of the axial stress should be suitable with 3.4 Proposed Lateral Strain Response (LSR) Method
stress magnitude-associated crack initiation (Fig. 8—tan-
gent line represented as dashed line). It is clear from the approach of Diederichs (2007) and the
ratio of lateral strain to axial strain used by Stacey (1981)
3.3 Acoustic Emission Method that a methodology utilizing the LSR should be used to
establish the stress magnitude associated with crack ini-
Eberhardt et al. (1998) have used several techniques to tiation. As discussed previously, beyond the onset of
detect crack initiation for Lac du Bonnet granite such as unstable crack growth the lateral strain increases signifi-
stress–strain data, the moving point regression technique cantly (see Fig. 1). Therefore, the LSR from zero stress to
and acoustic emission (AE). Acoustic emission is a low- the onset of unstable crack growth is examined for
energy seismic event which is generated by inelastic changes as the axial stress is applied. To detect changes in
deformation such as grain dislocation or crack initiation the LSR, the loading response is compared with a linear
(Hardy 1981). These three techniques can be used together reference line response taken from the onset of unstable
to find a more reliable result (Fig. 9—tangent line repre- crack growth to zero stress (Fig. 10a). The LSR method
sented as dashed line). However, the insignificant AE simply evaluates the difference between the measured
activity in crack initiation stages has made it difficult to loading response and the linear reference line. This dif-
differentiate between the background noise and the crack- ference is plotted as a function of axial stress and the
ing-source acoustic events. maximum difference is taken as the onset of crack initi-
ation (Fig. 10b). The methodology can be summarized as
follows:
1. Determine onset of unstable crack growth where total
0.35
Average incremetnal Poisson’s ratio

volumetric strain reversal occurs (see Fig. 1).

2. Determine the linear lateral strain reference line.
0.30
3. Find the change in lateral strain (DLSR) between the
loading and linear reference line.
0.25
4. Plot the axial stress versus change in lateral strain
(DLSR).
0.20 Crack
Initiation 5. Determine the maximum change in lateral strain
difference and the associated axial stress.
0.15
This methodology is amenable to programming in
0.10 Mathematica or Matlab and has been used by the authors to
50 70 100 150 200
examine the stress–strain response in a variety of rock
Log Axial stress (MPa)
types. A polynomial equation can be used to establish a
Fig. 8 Poisson’s ratio method proposed by Diederichs (2007) used to best fit to the data which facilitates finding the maximum
establish crack initiation. Tangent line represented as dashed lines LSR value and the associated crack-initiation stress. An

123
612 M. Nicksiar, C. D. Martin

a
250 Unstable crack
growth
Axial stress (MPa)

200
Lateral strain
150 Reference
line Circumferential
100 deformation
ΔLSR

50

0
−0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0
Lateral strain (%)

b
0.012 LVDT-Axial
Lateral strain difference (%)

deformation
0.01 0 50mm
0.008 Crack
initiation
Fig. 11 Example of the measurement system; LVDTs and chain used
0.006
to measure the axial and circumferential deformation, respectively.
0.004 Photo provided by SKB

0.002

0 and 73% (±6) Plagioclase (Janson et al. 2007). The

0 50 100 150 200 potassium feldspars (K-feldspar) are sparsely distributed as
Axial stress (MPa)
large crystals, often reaching 10 mm. Figure 12 shows a
Fig. 10 Example of the methodology used to establish the crack- typical example of Äspö Diorite that was used for labora-
initiation stress using the lateral strain response (LSR). Unstable crack tory testing.
growth is defined in Fig. 1. a Illustration of the LSR methodology and A total of ten specimens were tested in uniaxial com-
b example of the LSR result
pression tests by SP Technical Research Institute of
Sweden. The specimens were prepared according to
ASTM 4543-01 and stored from 21 to 24 days in water
attractive feature of the proposed method is that it does not prior to carrying out the uniaxial compression testing in a
require the subjective interpretation of the user. servo-controlled testing machine (Glamheden et al. 2010).
The axial load was determined by a load cell with max-
imum capacity of 1.5 MN. The axial and lateral defor-
4 Crack-Initiation Stress for Äspö Diorite mation measurements of the specimens were conducted
with miniature LVDTs with relative error of 0.6% in
The various methods described previously and capable of 1 mm for axial deformation and 1.3% in 3 mm for the
determining the crack-initiation stress were applied to radial deformation measurement. The loading rate was set
laboratory test results obtained used modern testing to a circumferential strain rate of -0.025%/min and
equipment illustrated in Fig. 11. The purpose was to increased after reaching the post-failure region (Glamhe-
establish if the various strain-based methods provided den et al. 2010). While the complete stress–strain
similar results. response into the post peak region was obtained, only the
The Swedish Nuclear Fuel and Waste Management Co. stress–strain results up to the peak strength are discussed
owns and operates the Äspö Hard Rock Laboratory (HRL) here.
located near Oskarshamn in Southern Sweden. Äspö HRL Table 1 lists the Young’s modulus (E), Poisson’s ratio
has been in operation since 1995 and has excavated tunnel (m) and the UCS for each specimen and the stress–strain
access to a depth of 450 m. The rock type encountered at curves of each test are presented in Appendix. The mean,
the Äspö HRL is called Äspö Diorite. Äspö Diorite is grey standard deviation (SD) and coefficient of variation (CoV)
to reddish grey, medium-grained low (&0.4%)-porosity expressed as percentage of each parameter are also pro-
igneous rock. The Äspö Diorite has a density of 2,740 kg/ vided. The UCS for the ten samples ranged from 171 to
m3 and consists of 19% (±4.5) quartz, 8% (±4) K-feldspar 242 MPa, with a mean value of 227 MPa, a standard

123
Determining Crack Initiation in Low-Porosity Rocks 613

Before UCS test After UCS test deviation of 31.3 MPa and a coefficient of variation of
13.8%.
The crack initiation (CI) stress values were determined
for each of the ten specimens using the six methods
described previously. Again the mean, SD and CoV were
determined and these results are also summarized in
Table 1. While the mean CI values from the six methods
only ranged from 105 to 111 MPa, the CoV ranged from
16.5 to 22.4%. Inspection of the results in Table 1 shows
that regardless of the method used to determine the crack-
initiation stress, the results appear surprisingly consistent.
A statistical methodology referred to as the analysis of
variance (ANOVA), was carried out in order to evaluate if
the mean values from the six CI methods statistically dif-
fer. ANOVA is a statistical method to test the variation in
an experimental outcome when there are more than two
groups. In our case we are testing if the results from the six
methods (groups) are all alike or not. One approach is to
compare the means obtained from each method, using the
F ratio in ANOVA, which is the ratio of the variation
between the methods to the variation within the method. In
the ANOVA F test, when the calculated F ratio is less than
the critical F ratio, there is no statistical difference in the
results. A detailed discussion of the ANOVA methodology
Fig. 12 Sample of Äspö Diorite used to compare the crack-initiation is beyond the scope of this paper and interested readers are
stress using various methods. The specimen is mostly composed of
referred to Walpole et al. (2002). For our dataset (6
Plagioclase, Oligoclase (orange) and Anorthite (dark brown). Quartz
grains are rarely observable as light-coloured mineral while K-feld- methods with 10 samples) F critical is 2.39, while the
spars are not obvious in the specimen (color figure online) F ratio is 0.26. The ANOVA results indicate that none of

Table 1 Comparison of results from different methods available for determining the crack-initiation stress of Äspö Diorite
Sample ID E (GPa) m UCS (MPa) Crack initiation stress (MPa)
Bracea Lajtai Stacey Martinb Diederichs LSR Mean SD CoV CI
UCS
(1966) (1974) (1981) (1993) (2007) (MPa) (MPa) (%)

46G02-02 79 0.18 171 74 74 74 71 70 78 74 2.4 3.3 0.46

46G02-03 80 0.25 238 118 122 118 115 111 118 117 3.1 2.7 0.50
46G02-04 77 0.25 242 122 115 116 117 100 119 115 6.5 5.7 0.49
46G05-01 74 0.28 203 115 119 117 97 110 101 107 9.1 8.5 0.50
48G02-01 77 0.29 224 108 112 114 96 98 116 108 7.2 6.7 0.52
48G02-02 75 0.31 224 107 113 105 117 115 115 112 4.2 3.7 0.51
51G01-01 72 0.27 218 109 117 117 102 103 108 109 5.5 5.1 0.50
54G01-02 73 0.29 294 157 118 118 150 146 164 142 19.7 13.9 0.56
54G02-01 78 0.26 237 118 132 114 116 113 113 117 6.2 5.3 0.48
54G06-01 72 0.24 218 76 73 74 71 88 76 76 5.1 6.7 0.35
Mean 76 0.26 227 110 110 107 105 105 111 0.49
SD 2.9 0.04 31.3 23.5 19.8 17.6 23.6 19.7 24.5 0.05
CoV (%) 3.8 13.8 13.8 21.3 18.1 16.5 22.4 18.7 22.1 11.3
a
Brace et al. (1966)
b
Martin and Chandler (1994)

123
614 M. Nicksiar, C. D. Martin

200 Read (2004) compiled the experience obtained while

excavating variously shaped tunnels in Lac du Bonnet granite
Crack initiation stress (MPa)

APSE at AECL’s Underground Research Laboratory. Read con-

150 0.55 UCS
cluded that regardless of the tunnel shape or tunnel direction,
the compressive stress at the in situ initiation of spalling on
100
the tunnel wall was about 50–60% of the uniaxial compres-
0.49 UCS sion strength. Martin et al. (1999) examined spalling in tun-
50 nels in rock types with UCS that ranged from 36 to 350 MPa.
They concluded that the initiation of spalling occurred when
the maximum tangential boundary stress exceeded approxi-
0
0 50 100 150 200 250 300 mately 0.40 ± 0.1 of the UCS. Hence, the experience from
Uniaxial compressive strength (MPa) carefully controlled in situ experiments and observations
Fig. 13 Relation between the uniaxial compressive strength (UCS)
made while tunnelling all suggest that the spalling strength
and the crack-initiation stress for Äspö Diorite measured by LSR cannot be estimated from the peak UCS. While more in situ
method. Also shown is the stress associated with onset of in situ experiments are needed in different rock types, there is suf-
cracking (APSE) for Äspö Diorite determined by Andersson et al. ficient evidence to warrant using the crack-initiation stress in
(2009)
laboratory uniaxial compressive as an estimate for the in situ
spalling strength.
the six methods have a significant statistical advantage over
the other. Hence choosing a particular method to establish
the crack-initiation stress is one of convenience. 6 Conclusions

Five strain based-methods were reviewed for establishing

the onset of cracking in laboratory compression tests on
5 Discussion low-porosity rocks. These methods utilized the laboratory
locally measured axial and/or lateral strains. Identifying the
One of the notable findings from the evaluation discussed stress level associated with crack initiation using these five
in the previous section is that regardless of the method, the methods relied on user judgement. A new method was
crack-initiation stress is consistently lower than the peak introduced called the LSR, which relies only on the LSR
UCS. Figure 13 shows the UCS versus the crack-initiation and removes the user judgement. A statistical evaluation of
stress determined using the LSR method. Also shown in the results from the application of all six methods to ten
Fig. 13 is the linear least squares fit to the ten data points samples of Äspö Diorite showed that any of the strain
which suggest that the crack-initiation stress for Äspö methods provided statistically accurate results.
Diorite occurs at approximately 0.49 of the UCS. Crack initiation in Äspö Diorite begins at stress levels well
Andersson et al. (2009) conducted an in situ experiment below the peak strength. Comparison of the stress magnitudes
(APSE) to investigate the onset of spalling (cracking) in a required to initiate spalling in Äspö Diorite in large-scale in
fractured Äspö Diorite rock mass. They used mechanical situ experiments showed that the laboratory crack-initiation
and thermal loading to gradually increase the boundary stress provided an estimate value for estimating the onset of
stresses around 1.8-m-diameter mechanically excavated spalling. Given the importance of establishing laboratory
holes until spalling occurred. Spalling was recorded using testing procedures that can be used for estimating in situ
displacement and acoustic emission monitoring. Andersson strength, it is proposed that the ISRM Suggested Methods
et al. (2009) concluded that spalling initiated when the develop standardized procedures for establishing crack ini-
stress magnitudes ranged between 114 and 133 MPa with a tiation from laboratory stress–strain data.
mean value of 124.6 MPa. The ten samples of Äspö Diorite
used in the previous section were taken from a borehole Acknowledgments We would like to acknowledge the financial
drilled parallel and in close proximity to APSE experiment. contribution of Swedish Nuclear Fuel and Waste Management
Company through the DECOVALEX Project. The authors would like
When compared with the laboratory mean UCS to thank Lars Jacobsson (SP Sweden) for providing the stress–strain
(227 MPa), the in situ spalling strength can be expressed as data for Äspö Diorite.
0.55 of the UCS. This value and the range are also shown
in Fig. 13. As concluded by Andersson et al. (2009) and
shown in Fig. 13 it would appear that the crack initiation Appendix
stress determined from unconfined laboratory compression
tests provides an estimate for the in situ spalling strength. See Fig. 14.

123
Determining Crack Initiation in Low-Porosity Rocks 615

Specimen ID: 46G02-02 Specimen ID: 46G05-01

Lateral Strain Volumetric Strain Axial Strain Lateral Strain Volumetric Strain Axial Strain
180
250
160
Axial stress (MPa)

140

Axial stress (MPa)

200
120

100 150
80
100
60

40
50
20

-0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.10 0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.30
Strain (%) Strain (%)

Specimen ID: 46G02-03 Specimen ID: 48G01-01

Lateral Strain Volumetric Strain Axial Strain Lateral Strain Volumetric Strain Axial Strain
250 250
Axial stress (MPa)

200 200
Axial stress (MPa)
150 150

100 100

50 50

-0.15 -0.10 -0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 -0.15 -0.10 -0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Strain (%) Strain (%)

Specimen ID: 46G02-04 Specimen ID: 48G02-02

Lateral Strain Volumetric Strain Axial Strain 250 Lateral Strain Volumetric Strain Axial Strain
250

200
Axial stress (MPa)

200
Axial stress (MPa)

150
150

100
100

50
50

-0.15 -0.10 -0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
-0.15 -0.10 -0.05 0.0 0.05 0.10 0.15 0.30 0.25 0.30 0.35
Strain (%) Strain (%)

Fig. 14 Äspö Diorite stress–strain curves in unconfined compressive strength test

123
616 M. Nicksiar, C. D. Martin

Specimen ID: 51G01-01 Specimen ID: 54G02-01

Lateral Strain Volumetric Strain Axial Strain Lateral Strain Volumetric Strain Axial Strain
250 250

200 200

Axial stress (MPa)

Axial stress (MPa)

150 150

100 100

50 50

−0.2 −0.1 0.0 0.1 0.2 0.3 −0.10 −0.05 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35
Strain (%) Strain (%)

Specimen ID: 54G01-02 Specimen ID: 54G06-01

Lateral Strain Volumetric Strain Axial Strain Lateral Strain Volumetric Strain Axial Strain

300 250

250
Axial stress (MPa)

Axial stress (MPa)

200

200
150
150
100
100

50
50

−0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4
Strain (%) Strain (%)

Fig. 14 continued

References near-face stress rotation. Int J Rock Mech Min Sci

41(5):785–812
Andersson C, Martin CD, Stille H (2009) The Äspö pillar stability Eberhardt E, Stead D, Stimpson B, Read R (1998) Identifying crack
experiment: part II—rock mass response to coupled excavation- initiation and propagation thresholds in brittle rocks. Can
induced and thermal-induced stresses. Int J Rock Mech Min Sci Geotech J 35(2):222–233
46(5):865–878 Fairhurst C, Cook NGW (1966) The phenomenon of rock splitting
Bieniawski ZT (1967a) Mechanism of brittle fracture of rock, part I— parallel to the direction of maximum compression in the
theory of the fracture process. Int J Rock Mech Min Sci neighbourhood of a surface. In: Proceedings of the 1st congress
Geomech Abstr 4(4):395–406 of the international society of rock mechanics, Lisbon,
Bieniawski ZT (1967b) Mechanism of brittle fracture of rock, part pp 687–692
II—experimental studies. Int J Rock Mech Min Sci Geomech Glamheden R, Fälth B, Jacobsson L, Harrström J, Berglund G,
Abstr 4(4):407–423 Bergkvist L (2010) Counterforce applied to prevent spalling.
Brace WF, Paulding B, Scholz C (1966) Dilatancy in the fracture of Technical Report TR-10-37, Swedish Nuclear Fuel and Waste
crystalline rocks. J Geophys Res 71:3939–3953 Management Co, Stockholm, Sweden
Brown ET (ed) (1981) Rock characterization, testing and monitoring, Griffith AA (1921) The phenomena of rupture and flow in solids.
ISRM suggested methods. Pergamon Press, Oxford Philos Trans R Soc Lond 221A:163–198
Cook NGW (1963) The basic mechanics of rockbursts. J South Afr Griffith AA (1924) Theory of rupture. In: Biezeno CB, M BJ (eds)
Inst Min Metall 63:71–81 Proceedings of the first international congress on applied
Diederichs MS (2007) The 2003 Canadian Geotechnical Colloquium: mechanics, Delft, Tech. Boekhandel en Drukkerij J Walter Jr,
mechanistic interpretation and practical application of damage Delft, pp 55–63
and spalling prediction criteria for deep tunnelling. Can Geotech Hallbauer DK, Wagner H, Cook NGW (1973) Some observations
J 44:1082–1116 concerning the microscopic and mechanical behaviour of
Diederichs MS, Kaiser P, Eberhardt E (2004) Damage initiation and quartzite specimens in stiff, triaxial compression tests. Int J
propagation in hard rock during tunnelling and the influence of Rock Mech Min Sci Geomech Abstr 10:713–726

123
Determining Crack Initiation in Low-Porosity Rocks 617

Hardy HR (1981) Applications of acoustic emission techniques to Moore DE, Lockner DA (1995) The role of microcracking in shear-
rock and rock structures: a state of the art review. In: Drnevich G fracture propagation in granite. J Struct Geol 17(1):95–114
(ed) Acoustic emission in geotechnical engineering practice, Murrell SAF (1963) A criterion for brittle fracture of rocks and
ASTM STP750, pp 4–92 concrete under triaxial stress, and the effect of pore pressure on
Hoek E, Brown ET (1980) Underground excavations in rock. The the criterion. In: Fairhurst C (ed) Proceedings of the 5th U.S.
Institution of Mining and Metallurgy, London symposium on rock mechanics, Pergamon Press, New York,
Janson T, Ljunggren B, Bergman T (2007) Modal analysis on rock pp 563–577
mechanical specimens. Specimen from borehole KLX03, Read RS (2004) 20 years of excavation response studies at AECL’s
KLX04, KQ0065G, KF0066A and KF0069A. Oskarshamn site Underground Research Laboratory. Int J Rock Mech Min Sci
investigation. SKB P-07-03, Swedish Nuclear Fuel and Waste 41(8):1251–1275
Management Co., Stockholm, Sweden Rojat F, Labiouse V, Kaiser PK, Descoeudres F (2009) Brittle rock
Lajtai EZ (1974) Brittle fracture in compression. Int J Fract Mech failure in Steg Lateral Adit of the Lötschberg Base Tunnel. Rock
10:525–536 Mech Rock Eng 42:341–359
Lockner DA, Byerlee JD, Kuksenko V, Ponomarev A, Sidorin A Stacey TR (1981) A simple extension strain criterion for fracture of
(1991) Quasi-static fault growth and shear fracture energy in brittle rock. Int J Rock Mech Min Sci Geomech Abstr
granite. Nature 350:39–42 18:469–474
Martin CD, Chandler NA (1994) The progressive fracture of Lac du Thompson BD, Young RP, Lockner DA (2006) Fracture in Westerly
Bonnet granite. Int J Rock Mech Min Sci Geomech Abstr granite under AE feedback and constant strain rate loading:
31(6):643–659 nucleation, quasi-static propagation, and the transition to unsta-
Martin CD, Christiansson R (2009) Estimating the potential for ble fracture propagation. Pure Appl Geophys 163(5):995–1019
spalling around a deep nuclear waste repository in crystalline Walpole RE, Myers RH, Myers RH, Ye K (2002) Probability &
rock. Int J Rock Mech Min Sci 46:219–228 statistics for engineers & scientists, 7th edn. Prentice Hall, Upper
Martin CD, Kaiser PK, McCreath DR (1999) Hoek–Brown param- Saddle River
eters for predicting the depth of brittle failure around tunnels.
Can Geotech J 36(1):136–151

123