Identifying crack initiation stress threshold in brittle...

J. Mt. Sci. (2018) 15(6): 1371-1382 e-mail: [email protected] http://jms.imde.ac.cn https://doi.org/10.1007/s11629-018-4847-z

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Abstract: As an estimate for the in-situ spalling strength around massive underground excavations to moderately jointed brittle rocks, crack initiation stress marks the initiation of rock micro fracturing. It is crucial to accurately identify crack initiation stress level by proper method. In this study, confined compression tests of sandstone samples are used to examine the validity/applicability of proposed axial strain stiffness method. The results show that by highlighting the minuscule changes in stress-strain curve, the axial strain stiffness curve provided further insight into rock failure process and revealed five stages: (a) irregular fluctuation, (b) nearly horizontal regular fluctuation, (c) irregular fluctuation gradually decreasing to zero, (d) extreme fluctuation, and (e) near zero, which mainly correspond to five stages of stress–strain curve. The ratio of crack-initiation stress to peak strength determined using this approach is 0.44–0.51, similar to the ranges previously reported by other researchers. In this method, the key is to

accurately detect the end point of the stage (b), “nearly horizontal regular fluctuation” characterized by a sudden change in axial strain stiffness curve, and the sudden change signifies crack initiation in rock sample. Finally, the research indicates that the axial strain stiffness curve can provide a mean to identify the crack-initiation stress thresholds in brittle rocks. Keywords: Brittle rock fracture; Rock failure; Sandstone; Crack initiation stress; Axial strain-stiffness; Triaxial test

Introduction

Crack-initiation (CI) stress of a rock under compression is the stress level at which the lateral and volumetric strain curves become non-linear. CI also marks the initiation of micro-fracturing in the rock (e.g., Brace et al. 1966; Bieniawski 1967; Lajtai 1974; Stacey 1981; Martin and Chandler 1994; Eberhardt et al. 1998). Martin (1993) reported that,

Received: 16 January 2018 Revised: 27 March 2018 Accepted: 08 May 2018

Identifying crack initiation stress threshold in brittle rocks

using axial strain stiffness characteristics

GAO Mei-ben1,2 http://orcid.org/0000-0002-1676-5783; e-mail: [email protected]

LI Tian-bin1,2* http://orcid.org/0000-0003-2167-586X; e-mail: [email protected]

MENG Lu-bo1,2 http://orcid.org/0000-0001-6647-9243; e-mail: [email protected]

MA Chun-chi1,2 http://orcid.org/0000-0002-5852-8022; e-mail: [email protected]

XING Hui-lin3 http://orcid.org/0000-0003-0863-9485; e-mail: [email protected]

* Corresponding author

1 State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China

2 College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu 610059, China

3 Centre for Geoscience Computing, School of Earth Sciences, The University of Queensland, St Lucia, QLD 4072, Australia

Citation: Gao MB, Li TB, Meng LB, et al. (2018) Identifying crack initiation stress threshold in brittle rocks using axial strain stiffness characteristics. Journal of Mountain Science 15(6). https://doi.org/10.1007/s11629-018-4847-z

© Science Press, Institute of Mountain Hazards and Environment, CAS and Springer-Verlag GmbH Germany, part of Springer Nature 2018

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unlike peak strength, the CI is essentially independent of the loading conditions. Recent researches suggest that CI can be used as an estimate for in-situ spalling strength, which is commonly observed in hard brittle rocks around underground openings (Martin 1997; Diederichs 2007; Anderson and Martin 2009; Martin and Christiansson 2009; Rojat et al. 2009; Chen et al. 2014, 2017). CI may also be used as a lower-bound estimate for the long-term strength of crystalline rocks (Damjanac and Fairhurst 2010) and has been related to the Kaiser effect, which is useful for establishing in-situ stress (Eberhardt et al. 1998; Cai et al. 2004). Therefore, it is significant to identify CI threshold in laboratory compression tests.

Over the past five decades, various methods for determining CI have been proposed (e.g., Brace et al. 1966; Bieniawski 1967; Lajtai 1974; Stacey 1981; Martin and Chandler 1994; Eberhardt et al. 1998; Diederichs et al. 2004). These methods are mostly based on the strain–stress data and acoustic emission (AE) techniques. They provide simple and feasible means for identifying CI values of rocks. However, the use of strain–stress data in previous studies has been somewhat limited to data sampling, computing and storage capabilities. Among these methods, volumetric strain (Brace et al. 1966), lateral strain (Lajtai 1974; Stacey 1981) and instantaneous Poisson’s ratio (Diederichs 2007) methods introduces considerable error and subjectivity into the analysis while picking CI from tangential lines, especially when the quality of the stress–strain curve is non-linear. The crack volumetric strain method (Martin and Chandler 1994) is less subjective and attempts to find CI by plotting crack volumetric strain against axial strain, though it is limited by elastic constants (i.e. Young’s modulus, E, and Poisson’s ratio, v) and is especially sensitive to Poisson’s ratio (Eberhardt et al. 1998). A more objective method has been proposed, known as the lateral strain response (LSR) method. This method involves fitting a parabola to find the stress associated with maximum strain difference (Nicksiar and Martin 2012). This method has been conducted to investigate CI stresses in igneous, metamorphic, and sedimentary rocks in compression tests (Nicksiar and Martin 2013). However, the LSR method relies on the accurate determination of

crack-damage stress (σcd) and on fitting a polynomial to find maximum LSR.

Owing to its sensitivity to rock cracking and damage, AE monitoring is another common tool used to identify the initiation and propagation of cracks in rocks under compression (Scholz 1968; Eberhardt et al. 1998). Eberhardt et al. (1998) used real-time AE ring-down counts to identify the CI of Lac du Bonnet granite during uniaxial compression. AE techniques are more objective, but the insignificance of AE activity in CI stages has made it difficult to differentiate between background noise and cracking-source acoustic events (Nicksiar and Martin 2012). Recently, via experimental and numerical methods, Haeri et al. (2014, 2015) had systematically studied microcrack initiation stress and direction in brittle materials, especially in wing cracks. Comprehensive reviews of various methods have been done in Nicksiar and Martin (2012) and Zhao et al. (2015).

In brittle rock, the failure process is a consequence of micro- and macro-scale fracturing that occurs in several stages, which are crack closure, elastic deformation, CI, crack growth, crack coalescence, failure and post-peak deformation (Bieniawski 1967; Lockner 1993; Thompson et al. 2006). Strain gauge measurements have provided an insight into delineating the stages of crack development in rock. Microcrack initiation, involves micro-scale changes, which may not be observable in strain–stress curve. The first derivative of the curve highlights changes in the slope and rate of change of the slope, which are helpful in identifying the stages of rock failure. For example, the value of the first derivative of the axial strain–stress curve would remain within a certain range during elastic deformation, before an irregular or sudden change of rate at microcrack initiation. This makes it possible to identify CI stress using the first derivative of the stress–strain curve.

The axial strain stiffness is determined by the first derivative of the stress–strain curve. In this paper, we propose a simple method to identify CI stress level in brittle rocks based on axial strain–stiffness characteristics. We then examine the feasibility and validity of this method in an experimental study of the brittle failure of sandstone samples in confined compression tests. Finally, the results are compared with the

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previously reported results of Nicksiar and Martin (2013). In comparison with the methods above, this method is simple, has a wider application (for both uniaxial tests and tri-axial tests), lower setup requirements (unlike AE method, which needs complex and expensive experimental setups) without dependence on parameters (unlike the method of crack volumetric strain, which is sensitive to E and v).

1 Concepts and Method

1.1 Assumptions

To identify the rock crack initiation (CI) stress level by using the strain–stiffness stiffness curves, the following assumptions and definitions have been proposed:

(a) The axial strain stiffness is defined as the ratio of the axial stress difference and axial strain difference, K=Δσ/Δε. It is the first derivative of the axial stress–strain curve.

(b) The linear elastic stage in the rock deformation and failure process is indicated by a linear section of the axial stress–strain curve (red line in Figure 1) and can be expressed in terms of a linear equation (y = kx+b).

(c) Rock deformation and failure is a continuous process and the first derivative of the axial stress–strain can highlight the changing features of the axial stress-strain curve. In then linear elastic stage, the rock sample can be considered a homogeneous, and isotropic material, and the rate of stress change of stress is nearly constant. Microcrack closure /initiation causes rate

irregularities/ fluctuations in the axial strain–stiffness curve.

1.2 Rock deformation-failure and its axial strain stiffness

Several characteristic stress levels have been identified by laboratory tests (Brace et al. 1964; Bieniawski 1967) and are important in understanding the failure process of brittle rocks (Cai et al. 2004). In this study, we use the stress–strain curve of a rock as well as the axial strain stiffness to investigate the relationship between characteristic stress levels and axial strain stiffness as follows:

(a) Crack-closure strength σcc Crack closure occurs during the initial stage of

loading when existing cracks that are oriented at an angle to the applied load close. During crack closure, as pre-existing pores, flaws or microcracks are compressed, the mechanical properties of the rock improve. This results in a non-linear stress–strain response and is indicated by an increase in the axial stiffness. In the meantime, there may be similarly large fluctuations in the axial strain–stiffness curve due to the crack closure. Once the majority of the pre-existing cracks have closed, linear elastic deformation takes place and the axial stiffness curve should be nearly horizontal. The corresponding stress is the crack-closure strength (σcc).

(b) Linear elastic deformation-CI stress σci In this stage, the inner pores, flaws or

microcracks are further compressed. The stress level is between crack-closure stress and CI stress (σcc<σ<σci) and is not high enough to initiate new

Figure 1 Identifying crack-initiation stress level using axial strain stiffness, and stress-strain curves characteristics.

σ1

σ3

Lateral strain gauge

Axial strain gauge

εvol

120 Axial stress (MPa) σf

σcd

σciσcc σ

r

Axial strain stiffnessM N

y=kx+bQ

80

40 εaxial

CI

120

80

40

0

-40

-80

Axi

al s

trai

n st

iffn

ess,

K (

GP

a)

0 Strain (%)-0.002 0.002 0.004

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cracks. The rock stress–strain curve has a positive gradient and axial strain–stiffness curve is a horizontal line. At the end of this linear elastic stage, CI begins with micro-scale changes, highlighted by the slope change in axial strain–stiffness curve (irregular point N in Figure 1). Then, the axial strain–stiffness curve departs from horizontal.

(c) Stable crack growth-crack-damage stress σcd In this stage, the stress level is beyond the CI

stress and internal cracks gradually begin to expand. New cracks may appear as the applied load increases. The mechanical properties of the rock gradually deteriorate and the non-elastic deformation occurs. Crack growth can be stopped by controlling the applied load. The stress–strain curve increases slowly and the axial strain stiffness gradually decreases. This stage ends at an inflection in the volumetric strain curve, which Martin (1993) refers to as the crack-damage stress threshold.

(d) Unstable crack growth- peak strength σf Bieniawski (1967) defines unstable crack

propagation as the condition when the relation between the applied stress and crack length diminishes and other parameters such as crack propagation velocity take control of the propagation process. In this stage, existing microcracks propagate, evolve, coalesce and form into a crack network within the rock. The crack can be observed as elastic energy releases and the volume expands. Even if the applied load does not increase, the fracturing process can continue. At the end of this stage, the sample has reached its peak strength.

Peak strength lies at the end point of this stage in the stress–strain curve and its value is highest in the stress–strain curve. As defined earlier, axial strain stiffness is obtained by taking the first derivative of the stress–strain curve. At the curve maximum, the value of the first derivative will be zero. Therefore, when the stress–strain curve reaches its peak, the axial stiffness value will be zero, shown in Figure 1 (point Q). Sharp points in the axial strain–stiffness curve indicate rupturing.

(e) Failure and post-peak behaviour-residual strength σr

After peak strength, the stress level always drops dramatically but never down to zero. With an increase in the strain, the stress maintains a

constant value. Displacements are mainly caused by sliding blocks.

The residual strength can be obtained from the stress–strain curve and also from the axial stiffness curve. In the former, residual strength is the constant remaining stress. In the latter, when the axial stiffness remains at zero, its corresponding stress is the residual strength.

As cracks close, the stiffness gradually increases. In linear elastic deformation, stiffness maintains a constant value. At the peak, the stiffness value is zero, and for the post-peak stage, the value maintains at zero. Meanwhile, with crack closure and initiation, minuscule but undetectable changes occur in the stress–strain curve that are observable in the axial strain–stiffness curve. The axial strain–stiffness curve is schematically shown in Figure 1.

1.3 Proposed axial strain-stiffness method

The following steps describe how the CI stress threshold can be determined:

(a) Plot the axial strain–stiffness curve The axial strain-stiffness value can be

calculated from (εaxial, σ) data using the following equation:

K=(σn+1-σn)/(εn+1-εn) (1)

where K is the axial strain stiffness (GPa), σ is the axial stress (MPa), and ε is the percentage of axial strain.

(b) Investigate the axial strain–stiffness curve characteristics

Identify the horizontal zone and point N on the axial strain–stiffness curve (Figure 2). The horizontal zone represents the linear elastic deformation. The crack initiates at the end of linear elastic deformation and small undetectable changes occur in stress–strain curve, which are highlighted, but can be identified in the axial strain–stiffness curve (point N in Figure 2). The stress level coresponding to point N in the axial strain–stiffness curve is the stress level at which cracks initiate.

(c) Determine the strain and stiffness of point N (εaxial, K)

The coordinates of point N on the axial strain–stiffness curve indicate the strain and instantaneous stiffness associated with the CI

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stress level. (d) Determine the CI stress The stress level corresponding to the strain at

point N on the stress–strain curve (εaxial, σ) indicates the crack-initiation stress level (CI).

The parameters of the axial strain stiffness method used to identify the CI stress are shown in Figure 2.

2 Sample Description

The original microstructure and mineral composition are important for the proper understanding of micro crack initiation. The sandstone used in this study was fresh and/or slightly weathered, however weathering effects were not dominant and did not affect the specimen strength.

The mineralogy of the samples was

determined on randomly oriented powder specimens with X-ray powder diffraction analysis (Table 1). The rocks investigated were generally composed of fine grains of quartz, calcite, feldspar, chlorite and illite. The sandstone samples did not have distinct, macroscopic heterogeneities (e.g. veins) at the specimen scale. The average grain size was 0.4 mm.

3 Testing and Verification

Verification tests were performed using sandstone samples to check the validity/applicability of the proposed axial strain stiffness method for determining CI in brittle rock. Three cylindrical specimens (diameter, 50 mm; length, approximately 100 mm) were prepared. Sample ends were carefully polished to minimise the end effects.

Figure 2 Example of the methodology used to determine the crack initiation stress using the axial strain stiffness curve.

Table 1 Mineralogical profile of the investigated sandstone

Profile Sample No. 1-3 Sample No. 1-4 Sample No. 1-5 Average grain size

Fine (0.4 mm)

Fine(0.4 mm)

Fine (0.4 mm)

Mineral composition

Illite:10%;Chlorite:11%; Quartz:51%;Feldspar:12%; Calcite:16%

Illite:9%; Chlorite:12%;Quartz:52%; Feldspar:13%; Calcite:14%

Illite:10%; Chlorite:11%;Quartz:52%; Feldspar:13%; Calcite:14%

Photomicrograph of thin section

N (0.004868, 216.9538)

-200

-100

0

100

200

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

Axi

al s

trai

n st

iffn

ess,

K (

GP

a)

Axial strain (%)

Micro-crack ruptures (crack initiates)

The crack initiation strain

Horizontal zone

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3.1 Testing procedure

We conducted the confined compressive strength tests using a stiff loading frame with a closed-loop servo-control system (MTS815) at the Rock Mechanics Laboratory, State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology. The maximum axial load capacity of the frame is 2000 kN. The test machine was in load-control mode with lateral displacement measurement limits. Axial and circumferential strain gauges were mounted onto the specimen at half the specimen height to eliminate the influence of end effects on strain measurements (Figure 3). Linear variable differential transformers and extensometers connected to a chain wrapped around the specimens were used for the axial and

circumferential deformation measurements, respectively.

The loading procedure in these experiments followed was in accordance with the guidelines proposed by the International Society of Rock Mechanics. During the initial loading phase, the confined pressure was 30 MPa at 0.005 mm/s and axial was used at the same rate (0.005 mm/s) until the peak load was reached. The confined pressure was kept constant throughout.

3.2 Test results

(a) Stress–strain curves The complete stress–strain curves of the three

samples are shown in Figure 4. The sample axial stress–strain curves under the confined (30 MPa) pressure can be divided into the following four stages: (1) the linear elastic, (2) CI and stable growth, (3) crack unstable growth and (4) failure and residual-presenting as observed in typical

Figure 3 MTS815 rock mechanics testing machine.

Figure 4 Stress–strain curves of sandstones.

0

70

140

210

280

-0.02 -0.01 0.00 0.01 0.02

Axi

al s

tres

s (M

Pa)

Strain (%)

(a) Sample 1-3

εradial

εvol

εaxial

0

70

140

210

280

-0.020 -0.010 0.000 0.010 0.020

Axi

al s

tres

s (M

Pa)

Strain (%)

(b) Sample 1-4

εradial εaxial

εvol

0

70

140

210

280

-0.03 -0.02 -0.01 0.00 0.01 0.02

Axi

al s

tres

s (M

Pa)

Strain (%)

(c) Sample 1-5

εradial

εvol

εaxial

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brittle failure of rocks. (1) There is no obvious crack-closure stage in

axial stress–strain curves, which indicates that the sandstone did not have a significant number of microscopic cracks and defects. As shown in Table 1, the samples do not have distinct microscopic heterogeneities.

(2) In the linear elastic stage, the stress–strain curves increase rapidly with an increase in the stress. The stress–strain curves maintain a straight line as the pressure increases, producing a convex shape. However, it is difficult to identify the end of the linear elastic stage.

(3) After failure and into the residual stage, the stress–strain curves drop as stress is quickly reduced and the change in strain also decreases, indicating a brittle failure. Sandstone’s high cohesion and the bond strength between inhomogeneous particles resulting in a non-elastic deformation stage are indicated by the convex part of the curve.

(4) The three axial stress–strain curves of the samples are in accordance with Type II curve compression testing. According to Wawersik and Fairhurst (1970), Type II corresponds to an unstable failure, which often occurs in hard brittle rocks. Additional external loading is not necessary to promote failure. Instead, the sudden outburst of accumulated strain energy can cause rock failure.

(b) Physical-mechanical parameters The complete stress–strain curves were

measured, enabling the calculation of the samples’ crack-damage stress, peak strength, elastic modulus and Poisson’s ratio. The average values for these measures are listed in Table 2.

3.3 Axial strain–stiffness curves

First derivatives reflect change rates; the rate of change (axial strain stiffness) of axial stress–strain curves can be calculated by data obtained from tests.

Combining the test data and Equation (1), the axial strain stiffness of the samples can be calculated (Figure 5).

In Figure 5, the features of axial strain stiffness are obvious and can be divided into five stages: (a) irregular fluctuation, (b) nearly horizontal regular fluctuation, (c) irregular fluctuation gradually decreasing to zero, (d) sudden sharp fluctuation and (e) near zero. These stages roughly correspond to the five stages of the stress–strain curve. The features and identifying procedure are detailed below:

(1) Stage a may correspond to crack-closure stage. The random distribution of pores, flaws and microcracks results in axial strain stiffness with irregular features. The axial strain stiffness fluctuates irregularly and is mainly greater than in Stage b. To some extent, there is a trend towards gradual increase. From Figure 4, it was difficult to determine crack closure, whereas highlighting the change rate in the axial strain–stiffness curve reveals the crack closure. For these intact sandstone samples (Table 1), there were few microcracks, making crack closure non-obvious in the stress–strain curve and occupying only a small area in the axial strain–stiffness curve.

(2) Stage b may correspond to the linear elastic stage. The axial strain stiffness maintains a nearly horizontal constant value with only a few irregularities. When rock is in the linear elastic stage, it can be considered a homogeneous and isotropic material and its rate of deformation is constant. At the end of this stage, the crack initiates, and minuscule undetectable changes occur in stress–strain curve, but result in sudden change in the axial strain stiffness due to the occurrence of microcrack ruptures (Figure 5).

(3) Stage c may correspond to the stable and unstable crack growth. The axial strain–stiffness curve gradually decreases to zero and irregular values occur frequently. In this stage, the stress is greater than the CI and microcracks develop. With micro crack initiation, development and coalescence, the properties of the rock gradually deteriorate. When a crack coalesces, irregular values may be seen in the axial strain–stiffness curve. When the sample reaches its peak strength, the corresponding axial stiffness value is zero.

(4) Stage d may correspond to the crack networking and initiation of block sliding but not

Table 2 Physical–mechanical parameters of sandstone samples from confined compression tests

Sample ID ρ(g/cm3) E (GPa) v σcd (MPa) σf (MPa)1-3 2.46 32.14 0.252 171.93 253.391-4 2.44 30.25 0.244 170.97 246.491-5 2.46 29.01 0.256 161.50 259.41

Average 2.45 30.47 0.250 168.13 253.10

Notes: ρ, Density; E, Elastic modulus; v, Poisson’s ratio; σcd, Crack damage stress; σf, Peak strength.

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the residual stage. The axial strain stiffness is mainly negative and fluctuates wildly. Peak stress represents the maximum bearing capacity of the rock sample and in that state, the sample is not sliding but forming a crack network (Figure 6). Stress begins to decrease immediately once it reaches its maximum value (Figure 4). There is a redistribution of stress between the peak stress and block sliding (Figure 6), which is transient and difficult to identify in a stress–strain curve. In

contrast, this process can be observed in the axial strain–stiffness curve. When extreme fluctuation occurs in the axial strain–stiffness curve, block sliding is initiated (Figure 6). This feature is obvious and may be related to the macro-sliding face.

(5) Stage e may correspond to the residual stage (in which the block stably slides). The axial strain–stiffness curve remains near zero at this stage.

Figure 5 The relation curves between axial strain stiffness and axial strain for sandstones.

-200

-100

0

100

200

0.000 0.004 0.008 0.012 0.016

Axi

al s

trai

n st

iffn

ess,

K

( G

Pa)

Axial strain (%)

(a) Sample 1-3

a b c d e


Peak strain

Block sliding

-200

-100

0

100

200

0.000 0.003 0.006 0.009 0.012 0.015

Axi

al s

trai

n st

iffn

ess,

K (

GP

a)

Axial strain (%)

(b) Sample 1-4

b c d e

Block sliding initiates

a


Peak strain

-200

-100

0

100

200

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016

Axi

al s

trai

n st

iffn

ess,

K (

GP

a)

Axial strain (%)

(c) Sample 1-5

a b c d e


Peak strain

Block sliding initiates

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The axial strain–stiffness curve provides an alternative way of observing features of rock failure. It can highlight any slope or rate changes in the stress–strain curve and may be used to identify crack or block sliding initiation.

3.4 CI stress

The details of identifying the CI, peak strength and residual strength using axial strain stiffness are shown in Figure 7.

According to the test data from Sample 1–3 in Figure 7, the crack-initiation strain is 0.004481 and the corresponding stress (CI) is 128.49 MPa. The strain to reach the peak strength is 0.011957 and the peak strength value is 253.38 MPa. The strain to reach the residual strength is 0.013130 and the residual strength value is 128.58 MPa. The strengths of samples 1–4 and 1–5 can be identified from test data in the same way. These values and methods for determining the stress parameters are shown in Table 3.

4 Discussion

4.1 Fluctuation of the axial strain–stiffness curve

The axial strain stiffness was calculated by investigating the compression test data, indicating the change rate of two consecutive data points.

Figure 6 The process between peak strength and block sliding.

Figure 7 Identifying the CI strengths (Sample ID 1-3).

σf : Crack

network formed Block sliding initiates σ

r: Block sliding

Table 3 Strength parameter values for each method

Sample ID Stress parameters

Method By axial strain stiffness curve

By axial stress strain curve

1-3 σci (MPa) 128.49 / σf (MPa) 253.38 253.39σr (MPa) 128.58 128.21



N0tes: σci, Crack initiation stress;σf, Peak strength;σr, residual strength.

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Minuscule changes between the two consecutive points can be observed in the axial strain–stiffness curve, especially when the axial strain–stiffness fluctuates.

During the closure stage, the random distribution of pores, flaws and microcracks in the sample results in irregular rock deformation features and causes irregular fluctuation in axial strain–stiffness curves.

The cracking process itself is a key factor that causes fluctuations in the axial stiffness curves. When the crack initiates, the stress–strain curve presents a nearly vertical spike up or down. Single crack ruptures or initiations cause only minuscule change in the stress–strain curve, which is difficult to observe. Since there is a nearly vertical change in the axial strain–stiffness curve, this is much easier to detect. This is also the reason why axial strain–stiffness curves fluctuate frequently from the stable crack growth stage to peak strength stage.

In the linear elastic deformation, the axial strain–stiffness curve’s fluctuation is probably related to system errors, equipment errors and other errors in the data acquisition process. However, although the axial strain–stiffness curves fluctuate, they stay within a certain horizontal zone that reflects linear elastic deformation.

4.2 Validation

(a) CI stress

The ratio of the peak strength to CI for the tested sandstone samples are shown in Table 4. In addition, Nicksiar and Martin (2012) summarised results obtained from different methods for determining the CI stress of Ӓspӧ Diorite, as shown in Table 5. According to Table 5, the ratio of CI stress to peak stress is in the range between 0.35 and 0.56.

In this study, the sandstone samples were subjected to a confined pressure of 30 MPa and the ratio of CI stress to peak strength was between 0.44 and 0.51, which is within the expected range, as reported by Nicksiar and Martin (2012). Therefore, we can contend the axial strain stiffness method to be a plausible approach for CI identification.

(b) Peak strength and residual strength It is widely recognised that the peak and

residual stresses can be accurately determined from stress–strain curves and test data. Therefore, in this paper, we discuss the accuracy with these strengths that can be determined by axial strain stiffness.

The peak strength is the largest value in the stress–strain data and is easily obtained from the

Table 4 Ratio of peak strength to CI for sandstones

Sample ID σci(MPa) σf (MPa) σci/σf

1-3 128.49 253.38 0.511-4 107.45 246.47 0.441-5 127.22 259.41 0.49

N0tes: σci, Crack initiation stress; σf, Peak strength.

Table 5 Comparison of results from different methods available for determining the CI of Ӓspӧ Diorite

(Nicksiar M and Martin CD 2012)

Sample ID E (GPa) v UCS

(MPa)

Crack initiation stress (MPa)Bracea

(1996) Lajtai(1974)

Stacey(1981)

Martinb

(1993) Diederichs(2007) LSR Mean

(MPa) SD (MPa)

CoV(%)

46G02-02 79 0.18 171 74 74 74 71 70 78 74 2.4 3.3 0.4646G02-03 80 0.25 238 118 122 118 115 111 118 117 3.1 2.7 0.5046G02-04 77 0.25 242 122 115 116 117 100 119 115 6.5 5.7 0.4946G05-01 74 0.28 203 115 119 117 97 110 101 107 9.1 8.5 0.5048G02-01 77 0.29 224 108 112 114 96 98 116 108 7.2 6.7 0.5248G02-02 75 0.31 224 107 113 105 117 115 115 112 4.2 3.7 0.5151G01-01 72 0.27 218 109 117 117 102 103 108 109 5.5 5.1 0.5054G01-02 73 0.29 294 157 118 118 150 146 164 142 19.7 13.9 0.5654G02-01 78 0.26 237 118 132 114 116 113 113 117 6.2 5.3 0.4854G06-01 72 0.24 218 76 73 74 71 88 76 76 5.1 6.7 0.35Mean 76 0.26 227 110 110 107 105 105 111 0.49SD 2.9 0.04 31.3 23.5 19.8 17.6 23.6 19.7 24.5 0.05CoV(%) 3.8 13.8 13.8 21.3 18.1 16.5 22.4 18.7 22.1 11.3

Notes: E, Elastic modulus; v, Poisson’s ratio; USC, uniaxial compressive strength; LSR, Lateral strain response; SD, Standard deviation; CoV, Coefficient of variation; CI, Crack initiation stress. a Brace et al. (1966); b Martin and

Chandler (1994); the ratio of crack initiation stress to peak stress.

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axial stress data using the MAX function in Microsoft Excel. The peak stresses determined by the stress–strain data were 253.39, 246.49 and 259.41 MPa. These values correspond to 253.38, 246.47 and 259.41 MPa, respectively, for the axial strain stiffness method. There is a little difference in the peak strength values obtained by the two methods, indicating that the peak strength can be precisely determined by the axial strain stiffness method.

Once the stress reached its peak value, it did not immediately drop to zero; the sample can sustain a certain stress level, which is its residual stress. As seen in Table 3, the residual strengths determined by the axial stress strain curve were 128.21, 155.73 and 159.32 MPa. Corresponding values for the axial strain stiffness method were 128.58, 155.49 and 159.82 MPa. There is a little difference in the residual strength values from the two methods, indicating that the axial strain method is suit for identifying residual.

5 Conclusions

In rock engineering, CI stress is a vital factor for assessing the long-term strength of the brittle rocks. This study proposed a simple and effective method to estimate CI stress based on the characteristics of the axial strain–stiffness curve. The applicability of the proposed method was examined by experimental study of sandstone samples. The following observations and conclusions have been achieved from the study:

(1) The axial strain stiffness method was proposed to identify the crack-initiation stress, which highlights the undetectable changes in the stress–strain curve and improves the user’s ability to identify the CI threshold. The validity of the axial strain stiffness method has been confirmed by

sandstone testing. The ratios of CI to peak strength have a range of between 0.44-0.51, corresponding to the results reported in the previous studies.

(2) The key aspect of this method is to accurately detect the end point of the nearly horizontal zone by observing the nearly horizontal zone and the change in the axial strain–stiffness curve, which also signals crack initiation.

(3) Axial strain–stiffness curves provide further insight into the rock failure features and can be divided into five stages: (a) irregular fluctuation, (b) nearly horizontal regular fluctuation, (c) irregular fluctuation gradually decreasing to zero, (d) extreme fluctuation and (e) near zero, which mainly correspond to the five stages of the stress–strain curve.

Further experimental studies need to be conducted on other rock types with various confining pressures to explore more the potential application of this approach in identifying the CI stress threshold and to improve data acquisition/processing techniques so that the identification of the key stages in the strain–stiffness curves can be performed with more certainty.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (Grants No. 41772329, 41572283 and 41230635), and the funding of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Nos. SKLGP2017Z001 and SKLGP2013Z004) is gratefully acknowledged. This work is also supported by the Funding of Science and Technology Office of Sichuan Province (Grants No. 2015JQ0020 and 2017TD0018).

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