Performance of rock crack stress thresholds determination criteria and investigating strength and confining pressure effects

Construction and Building Materials 243 (2020) 118263

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Performance of rock crack stress thresholds determination criteria and investigating strength and confining pressure effects Abbas Taheri a, Yubao Zhang b,⇑, Henry Munoz c a

School of Civil, Environmental and Mining Engineering, The University of Adelaide, Adelaide, SA 5005, Australia College of Energy and Mining Engineering, Shandong University of Science and Technology, Qingdao, Shandong 266590, China c Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo, 153-8505 Japan b

h i g h l i g h t s  Volumetric strain response is the most promising parameter to identify crack stress thresholds.  Ratios of CC, CI and CD to peak strength are found to be 0.21, 0.33 and 0.80.  Crack damage threshold increases with a rock strength increase.  With an increase in confining pressure, CC ratio decreases and CD ratio increases.

a r t i c l e

i n f o

Article history: Received 25 January 2019 Received in revised form 20 January 2020 Accepted 22 January 2020

Keywords: Rock Crack stress thresholds Uniaxial compression Triaxial compression Strength Confining pressure

a b s t r a c t Identifying crack stress thresholds of rocks under different loading conditions, e.g. long-term monotonic or cyclic loads, are crucial in the design of rock excavations and underground openings. The evaluation of the performance of damage thresholds, i.e. crack closure, crack initiation and crack damage stresses, for a variety of rocks, is crucial to assess the most suitable method to capture the stages of rock strength degradation for rock having different strengths subjected to different confining pressures expected to in the field. In this study, several uniaxial and triaxial tests on different rock specimens (Uniaxial compressive strengths from 5 to 278 MPa) were performed to investigate the application and performance of different criteria to determine damage threshold values, using their stress–strain response. It was found that the volumetric strain response of the rock is the most promising parameter to identify crack stress thresholds. For the rocks having different strength values and subjected to different confining pressures, on average, ratios of crack closure, crack initiation and crack damage to the peak strength are found to be 0.21, 0.33 and 0.80 respectively. Effect of rock strength on crack closure and crack initiation stress thresholds was found to be negligible. Crack damage threshold, however, increases with a rock strength increase. This was mainly attributed to the brittle failure behaviour of stronger rocks when compared to weaker rocks. In triaxial compressive testing, crack closure partially occurs during rock consolidation and therefore, crack closure stress decreases with confining pressure increase. Furthermore, a limiting effect on the generation and further growing of lateral cracks in the rocks did lead to an increase in the stress to induced crack damage with an increase in the confining pressure. On the other hand, the effect of confining pressure on the crack initiation stress was observed to be different in different rocks. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction Over the last 40 years, the detailed analysis of the uniaxial compression stress–strain response of rock is shown that uniaxial compressive strength (UCS) is not sufficient to evaluate strength characteristics of intact rock [1–5]. It is therefore essential to have ⇑ Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (H. Munoz). https://doi.org/10.1016/j.conbuildmat.2020.118263 0950-0618/Ó 2020 Elsevier Ltd. All rights reserved.

reliable criteria to estimate the progressive damage of rocks subjected to different loading conditions. Numerous studies have concluded that the pre-peak fracture damage mechanisms for rock are the closure, initiation, and propagation of micro and macro cracks formed by the compression during uniaxial and triaxial tests [6– 13]. The damaging process of rock can be broken down into five stages: (1) Crack closure (CC), (2) Linear elastic deformation, (3) Crack initiation (CI); that is stable crack growth, (4) Crack damage (CD); that is unstable crack growth, and (5) failure and post-peak

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behaviour [14]. Therefore, the identifying performance of crack stress thresholds is critical to revealing the rock damage evolution and stability. The crack damage (CD) stress threshold marks the yielding strength of rock and is a reliable estimate for the upper limit of true long-term strength for rock mass [11,15], which is better to quantify the in-situ state of damage surrounding tunnel excavation [16]. Moreover, the crack initiation (CI) stress threshold could determine the lower limit of the long-term strength and is usually used to analyse spalling in tunnels [17,18]. Munoz et al. [19] found that the brittleness index plays a vital role in crack damage stress of rocks. Pepe et al. [20] studied the correlation between crack initiation-crack damage stress levels and the failure strength of different rocks and proposed reliable models to predict crack initiation and damage stress levels. However, the correlations between crack stress thresholds (including CC, CI, and CD) and UCS of intact rocks has not been comprehensively investigated in the previous studies. Different researchers used different methods to define crack stress threshold values (CC, CI, CD) of rock material [11,21–31]. The most common methods based on stress–strain relations are rock volumetric strain, crack volumetric strain, lateral deformation stiffness, and tangent Young’s modulus methods. In addition, other methods involve the use of scanning electron microscopy, photoelastic, laser speckle interferometry, ultrasonic probing, and acoustic emission (AE) techniques [18,23]. It is noted that the AE properties would be of great help to detect the failure evolution of rocks [32–33]. However, there is no mention of the suitable method to define crack stress thresholds in the ‘‘The Complete ISRM Suggested Methods for Rock Characterization, Testing and Monitoring: 1974–2006” of the ISRM. Rock engineering designs and applications must recommend the most reliable methods for determining crack stress thresholds in the laboratory. Moreover, experimental results undertaken by different researchers to study

pre-peak crack stress thresholds under different confining pressures on rocks are different. Gowd and Rummel [34] found that in a porous rock with an increase in confining pressure, the ratio between crack stress thresholds and rock strength increases. Chang and Lee [24] showed that damage thresholds of Hwangdeung granite and Yeosan marble, except the crack closure stress, increased linearly with confining pressure. Bruning et al. [31] found that the crack initiation threshold in the brittle rock are highly dependent on confining pressure. Cai et al. [35], however, interpreted that crack initiation stress is independent of confining pressure. Nicksiar and Martin [36] suggested that the initiation of new cracks is only slightly inhibited by confining stress. Ning et al. [30] found that the variation in confinement has no influence on the crack initiation stress threshold and crack damage stress threshold of coal specimens. Nevertheless, there is no study on the effect of confining pressure on the three crack stress thresholds (i.e. CC, CI and CD) in a wide range of rocks. In this study, a series of uniaxial and triaxial tests were conducted on different rocks. Based on the stress–strain relations in experimental results, the most reliable criteria to predict prepeak crack stress thresholds were identified. Furthermore, the effects of rock strength and confining pressure on crack stress thresholds were investigated.

2. Experiments 2.1. Sample preparation In total 74 tests including 26 uniaxial tests and 48 triaxial tests under different confining pressures, are performed. As shown in Tables 1 and 2, uniaxial and triaxial tests are undertaken on 13 different rocks having a wide range of strength values, including one schist, three sandstones, one bluestone, five limestones and

Table 1 Summary of uniaxial compression tests. Rock name

Rock type

Grain Size

Number of tests

UCS range (MPa)

Brukunga Hawkesbury Kapunda Massangis Savonniere Chassagne Rocheron Tuffeau Granodiorite American Black Radiant Red Alvand A Alvand B

Schist Sandstone Bluestone Limestone Limestone Limestone Limestone Limestone Granite Granite Granite Granite Granite

/ Medium / Fine Fine Fine Fine Fine / / / Medium Medium

3 6 2 3 2 1 1 2 1 1 1 2 1

89–100 31–36 150, 187 81, 84, 95 19, 24 112 207 5, 7 187 278 259 96, 108 157

Table 2 Summary of triaxial compression tests. Rock name

Rock type

Grain Size

Number of tests

Confining pressure values (MPa)

Brukunga Hawkesbury Kapunda Massangis Savonniere Chassagne Rocheron Tuffeau Alvand A Alvand B

Schist Sandstone Bluestone Limestone Limestone Limestone Limestone Limestone Granite Granite

/ Medium / Fine Fine Fine Fine Fine Medium Medium

8 14 5 3 4 2 3 4 3 2

4.6, 9.6, 14.7, 20.6, 30.8–35.2 1.3, 2.6, 3.9, 4–4.1, 5, 8.02–8.04, 10.1 5.0, 10.0, 15.0, 20.1, 30.0 3.4, 6.8, 10.0 1.0, 2.0, 3.0, 5.0 9.8, 14.8 8.6, 17.2, 26.0 1.0, 3.0, 5.0, 6.0 11.2, 16.8, 22.4 14.6, 22.0

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five granites. Uniaxial compressive strength of the rocks is ranging between 5 and 278 MPa. All the samples for different rock type are obtained by drilling in the same direction. Therefore, any possible effect of anisotropy due to existing layering is eliminated. 2.2. Experimental procedure A closed-loop servo-controlled testing machine with a loading capacity of 1000 kN and a loading rate capability in the range 0.001–10 mm/s were used to carry out uniaxial and triaxial compression tests. As shown in Fig. 1, the Hoek cell with capacity up to 65 MPa was used to achieve specimen confinement. The loading system is equipped with a linear variable differential transformer (LVDT) to measure axial displacement externally. Two external LVDTs were used as well to measure the axial displacement of the sample. The results of these LVDTs could be more accurate as they do not include deformations of apparatus, unlike the apparatus LVDT. The measurement made by apparatus LVDT and other two LVDTs, however, may include bedding errors [37]. To have a more accurate measurement, axial and lateral strains were measured throughout the test from the beginning of loading until post-peak state using pairs of strain gauges being attached to the sample. By this local measurement arrangement, it was possible to measure axial, lateral and volumetric strains very accurately. The load and strain data are acquired automatically by a data acquisition system. Uniaxial compression tests were performed under the constant axial displacement rate of 0.05 mm/min. To undertake triaxial compression tests, in the beginning, the axial and confining pressures were increased simultaneously at a rate of 0.1 MPa/s until the desired confining pressure was reached. After that, the axial pressure was increased to the first deviatoric stress level at a controlled axial displacement rate of 0.05 mm/min while keeping the confining pressure constant. By this arrangement, the samples in triaxial compression testing were preconsolidated before applying deviator stress.

Fig. 2. Crack stress threshold points (q: deviator stress; e1: axial strain, e3: lateral strain; evol: volumetric strain; Elat: lateral deformation stiffness; Evol: volumetric deformation stiffness; Etan: Tangent Young’s modulus).

Table 3 Eleven critical points for the definition of crack stress thresholds. Stress threshold

Point number

Method

Crack closure

1

Point where the horizontal section of the calculated crack volumetric strain-axial strain curve begins Point where a linear section of axial stiffness starts Point where the flat section in the lateral stiffness curve ends

2.3. Identifying methods for crack stress thresholds Fig. 2 shows eleven critical points that are adopted by different researchers to determine rock damage thresholds. The crack closure stress may be defined by three different criteria (points 1, 5 and 7). Four points represent criteria of crack initiation determination (points 2, 8, 10 and 11), and three different methods are

5 7 Crack initiation

2

8 10 11

Strain gauges

Crack damage

3

9

Reversal point in volumetric strain-axial stress curve Point where volumetric stiffness curve changes from positive to negative Point of interception of volumetric strain curve

4

Point when axial stress starts to drop

6

Lateral extensometer

Membrane Failure

Point where the horizontal section of the calculated crack volumetric strain-axial strain curve ends Point where a linear section of axial stiffness ends Point where volumetric stiffness has the first large deviation from linear behaviour The maximum point of crack volumetric strain

Rock sample

LVDTs Hoek cell Fig. 1. Experimental set-up and deformation measurement.

suggested to define crack damage stress (points 3, 6 and 9). Lastly, point 4 represents rock failure, namely, peak strength point. The critical points introduced in Fig. 2 are defined based on previous studies [7,10,11,23,27,38] and are described in Table 3. The volumetric strain method (points 3 and 9) is practical because of its specific physical meaning. However, the measurement accuracy of axial strain and lateral strain of the rock sample

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Fig. 3. Crack stress threshold points using test results: (a) points 3, 4, 6, 7 and 9; (b) points 1, 2, 3 and 11; (c) points 7; (d) points 6 and 10; (e) points 5 and 8.

is critical to this method. Analogously, both axial strain and lateral strain have a great influence on the volumetric stiffness method (points 6 and 10). The crack volumetric strain method (points 1, 2 and 11) is sensitive to the elastic modulus and Poisson’s ratio. And this method is not appropriate to determine rock damage thresholds for the rock sample with a large number of cracks before testing. The axial stiffness method (points 5 and 8) only depends on axial stress–strain relations. However, it is difficult to precisely find out the onset and end points of the linear section of the axial stiffness curve, so this method has apparent subjectivity. The lateral stiffness method (point 7) could

clearly define the crack closure stress and address the disturbance of axial strain. Nevertheless, this method is sensitive to the intense initial cracks in the rock sample. When the stress-lateral strain relation shows no visible linear behaviour, it is challenging to accurately identify the CC by the flat section in the lateral stiffness curve. As a result, the limitations of each method based on stress–strain relations are visible and a certain degree of subjectivity could be introduced by picking the critical crack stress threshold points. Which method can accurately identify the crack stress thresholds needs to be further verified by contrasting experimental data.

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A. Taheri et al. / Construction and Building Materials 243 (2020) 118263 Table 4 Mean and standard deviations of the crack stress threshold points under all tests. Threshold

Point

Mean value of crack stress thresholds (q/qf)

Standard deviation (SD)

UCS test

Triaxial test

All

UCS test

Triaxial test

All

Crack closure

1 5 7

0.217 0.263 0.304

0.198 0.243 0.250

0.205 0.250 0.265

0.091 0.131 0.108

0.125 0.153 0.175

0.115 0.146 0.161

Crack initiation

2 8 10 11

0.401 0.639 0.642 0.320

0.404 0.487 0.626 0.330

0.403 0.533 0.631 0.327

0.095 0.176 0.150 0.086

0.145 0.192 0.246 0.144

0.132 0.200 0.220 0.128

Crack damage

3 6 9

0.738 0.691 0.864

0.826 0.729 0.812

0.800 0.723 0.826

0.165 0.225 0.273

0.174 0.250 0.278

0.176 0.239 0.277

3. Experimental results and discussions

3.2. Effect of rock strength

3.1. Crack stress thresholds determination

Fig. 4 demonstrates the relations between the normalised damage threshold stress values (i.e. qcc/qf, qci/qf and qcd/qf), that are obtained from the UCS tests, versus the rock strength. As shown in this figure, qcc/qf and qci/qf slightly decrease with an increase in rock strength. This trend is almost negligible. However, the variation of qcd/qf shows a clear trend and increases with rock strength. Munoz et al. [19] developed a method to investigate rock brittleness based on pre-peak and post-peak rock behaviour. They demonstrated that rock brittleness increases with an increase in rock strength. In this situation, axial stress almost increase linearly until the rock approaches the failure. Whereas, in soft rocks, after the crack damage stress, the rock exhibits a considerable inelastic behaviour before failure, as shown in Fig. 5. With the increase of rock strength, the brittleness of soft rocks is increasing, resulting in the rock behaviour changing from inelastic to elastic gradually before peak stress. The gap between the crack damage stress and peak stress of the rock is gradually reduced. Therefore, as demonstrated in Fig. 4c the ratio between crack damage stress and peak strength increases with an increase in peak strength.

The criteria presented in Fig. 2 and Table 3 were employed to calculate crack stress thresholds using the tests data. An example of damage thresholds which determined in one of the tests is shown in Fig. 3. By undertaking a statistical analysis mean values of damage threshold stresses (i.e. qcc, qci, qcd) being normalised by the peak deviator stress (qf = r1-r3, r3 = 0 in UCS tests) for all the uniaxial and triaxial compressive tests and their respective standard deviations were determined and presented in Table 4. For crack closure stress threshold, the mean values for point 1, point 5 and point 7 of all the tests are determined to be equal to 0.205, 0.250 and 0.265, respectively. These results are consistent with the previous studies undertaken by Xue et al. [27] and Eberhardt et al. [14]. In addition, the corresponding standard deviations of those three points are slightly different, which are 0.115, 0.146 and 0.161, respectively. The lowest standard deviation was obtained for point 1 at 0.115, showing less scatter in the results. Consequently, point 1 may be adopted as the most suitable method to define crack closure stress. In this case, the mean value of the ratio for crack closure stress threshold (qcc) for the uniaxial and triaxial compression tests, respectively, are equal to 0.217 and 0.198. For crack initiation stress threshold of all the tests, as shown in Table 4, the mean values for point 2, point 8, point 10 and point 11 are 0.403, 0.533, 0.631 and 0.327 respectively. Furthermore, the lowest standard deviations are found for point 11. Therefore, point 11 is considered to be the most reliable method to calculate crack initiation stress (qci). The mean value of qci for the uniaxial and triaxial compression tests, respectively, are almost identical, equal to 0.33. For the crack damage stress threshold of all the compressive tests, as shown in Table 4, the mean values for point 3, point 6, and point 9 are 0.800, 0.723, and 0.826 respectively. The results show that in many of the tests, the sample fails before the volumetric strain becomes negative. These tests, therefore, were excluded when calculating mean and standard deviations values for point 9. As a result, point 9 does not seem to be a suitable method to define crack damage stress. Looking at the standard deviation values, the lowest standard deviation was obtained for point 3. Therefore, point 3 is regarded the best method which can result in the most accurate values for crack damage stress (qcd). As it can be seen in Table 4, larger variations for the mean qcd values is observed for the uniaxial and triaxial compression tests. In this case, the mean value of qcd for the uniaxial and triaxial compressive loading condition is found to be equal to 0.738 and 0.826 respectively. The effect of rock strength and confining pressure, on damage threshold stresses is discussed in the following sections.

3.3. Effect of confining pressure Effects of confining pressure on ten different rocks are examined in this study. As demonstrated in Table 2, confining pressures range from 0 to 10 MPa for Hawkesbury sandstone, 0 to 35 MPa for Brukunga schist, 0 to 30 MPa for Kapunda bluestone, 0 to 22.4 MPa for the Alvand A granite, 0 to 22 MPa for the Alvand B granite, and 0 to 26 MPa for the limestones. Fig. 6 shows the relations of the ratio of the damage stress threshold against confining pressure for ten different rocks. In Fig. 6(a) which shows the results for Hawkesbury sandstone, qcc/qf and qci/qf both display a decreasing trend with the increase of confining pressure. Conversely, qcd/qf increases by a linear fitting as confining pressure increases. In the case of Kapunda bluestone (see Fig. 6(b)), qcd/qf increases with an increase of confining pressure. However, both qcc/qf and qci/qf decrease with confining pressure increase. For Brukunga schist, as shown in Fig. 6 (c), qcd/qf decreases with confining pressure increase while the qcc/ qf and qci/qf show an increasing trend with confining pressure. In the case of Alvand A granite and Alvand B granite (see Fig. 6(d)(e)), qci/qf and qcd/qf increase, while qcc/qf decreases with an increase in confining pressure. Fig. 6(f)-(j), show a similar trend for qcc/qf and qcd/qf values for five limestones with an increase in confining pressure: i.e. an increase in qcd/qf and a decrease in qcc/qf. However, similar to the results obtained for other rocks, for five different limestones qci/ qf show different trends, when confining pressure increases. Table 5 shows the changing trend of crack stress thresholds in different rocks under confined condition. Except for Brukunga schist, we

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Fig. 5. Stress–strain curve of soft rocks in triaxial compression test.

static pressure will result in the partial closure of the existing cracks. As a result of this, closure of existing cracks happens earlier in triaxial tests and, therefore, crack closure stress decreases with an increase in confining pressure. It has been generally recognised that the strength of rock increases with the confining pressure increasing due to the significant contribution of the frictional component in the conventional triaxial test. In triaxial compression testing, due to the effect of confining pressure, the crack opening in the lateral direction is suppressed. Therefore, the shear stress required for unstable cracks propagation increases. Although the mobilisation of friction improves the strength of the rock in the presence of confinement, we observed a different trend for crack initiation stress in different rocks. This needs to be further investigated in future works.

4. Conclusions

Fig. 4. Relation of the normalised damage threshold stresses against the UCS for different rocks: (a) crack closure, (b) crack initiation, (c) crack damage.

observed consistent results for qcc/qf and qcd/qf ratios when confining pressure increases. An increase of confining pressure results in an increase in qcd/qf and a decrease in qcc/qf values. However, the trend of qci/qf values with confining pressure values are different in different rocks and do not show a particular trend. To undertake triaxial compressive testing, axial and confining pressure are increased simultaneously until the prescribed confining pressure is reached. This consolidation procedure under hydro-

To investigate the application of different criteria to determine damage threshold values, using stress–strain relations, uniaxial and triaxial compression tests on 13 different rocks with UCS values ranging from 5 to 278 MPa were conducted. The statistical analysis showed that the point where the horizontal section of the calculated crack volumetric strain-axial strain curve begins is the best method to define crack closure stress. Moreover, the maximum point of crack volumetric strain is the most suitable criteria to recognise crack initiation stress. To define crack damage stress, the reversal point in the volumetric strain curve when plotted against axial stress is recommended. It was found that for the rocks having different strengths and subjected to different confining pressures, the mean values of normalised crack closure, crack initiation and crack damage thresholds are 0.21, 0.33 and 0.80, respectively. For the UCS tests, it was found that rock strength has an almost negligible influence on the crack closure, and crack initiation stresses. The crack damage stress, however, increases with an increase in rock strength. This is because, stronger rocks demonstrate more brittle behaviour, and therefore, a crack damage stress closer to the peak point. In the triaxial compression tests, closure of existing cracks happens earlier during the consolidation stage and, therefore, crack closure stress decreases with an increase in confining pressure. Moreover, in triaxial compression test confinement restricts the crack opening in the lateral direction. Therefore, the axial stress required to create unstable cracks (i.e. crack damage stress) increases. Crack initial stress showed a different trend for different rocks when confining pressure increases.

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Fig.6. Relations of crack stress thresholds against confining pressure in different rocks: (a) Hawkesbury sandstone, (b) Kapunda bluestone, (c) Brukunga schist, (d) Alvand A granite, (e) Alvand B granite, (f) Tuffeau limestone, (g) Massangis limestone, (h) Rocheron limestone, (i) Savonniere limestone, (j) Chassagne limestone.

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Fig. 6 (continued)

Table 5 Change trend of crack stress thresholds in different rocks under the confined condition. Rock name

Rock type

Crack Closure

Crack Initiation

Crack Damage

Brukunga Hawkesbury Kapunda Alvand A Alvand B Tuffeau Massangis Rocheron Savonniere Chassagne

Schist Sandstone Bluestone Granite Granite Limestone Limestone Limestone Limestone Limestone

Increasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing Decreasing

Increasing Decreasing Decreasing Increasing Increasing Decreasing Decreasing Increasing Increasing Decreasing

Decreasing Increasing Increasing Increasing Increasing Increasing Increasing Increasing Increasing Increasing

CRediT authorship contribution statement Abbas Taheri: Conceptualization, Methodology, Writing review & editing. Yubao Zhang: Formal analysis, Validation, Writing - original draft. Henry Munoz: Investigation, Formal analysis. Acknowledgment The second author would like to acknowledge the financial support of the China Scholarship Council (No. 201708370105). References [1] J.A. Hudson, E.T. Brown, C. Fairhurst, Shape of the complete stress-strain curve for rock, in: Proceedings of the 13th US Symposium on Rock Mechanics, Stability of Rock Slopes, ASCE, 1972, pp. 773–795.

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