Experimental investigation on fracturing process of marble under biaxial compression

Zhaofeng Wang, Xia-Ting Feng, Chengxiang Yang, Yangyi Zhou, Hong Xu,Qiang Han, Yaohui Gao

a State Key Laboratory of Geomechanics and Geotechnical Engineering,Institute of Rock and Soil Mechanics,Chinese Academy of Sciences, Wuhan,430071,China

b University of Chinese Academy of Sciences, Beijing,100049, China

c Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University, Shenyang,110819, China

Keywords:Fracturing mechanism Biaxial compression Acoustic emission (AE)Moment tensor (MT)Hard rock Stepwise crack

ABSTRACT In this study, servo-controlled biaxial compression tests were conducted on marble specimens to investigate their failure characteristics and fracturing process. The complete stress-strain curves were obtained, and the three-dimensional (3D) features of the failure surfaces were acquired by 3D laser scanning. Acoustic emission (AE) monitoring and moment tensor (MT) analysis were used in combination to better understand the fracturing mechanism of marble under biaxial compression. It was noted that a type of 3D stepwise cracking behaviour occurred on the fracturing surfaces of the examined specimens. The stress dropped multiple times, and a repeated fracturing mode corresponding to the repeated stress drops in the post-peak regime was observed. Three substages, i.e. stress stabilisation,stress decrease and stress increase, were identified for a single fracturing mode. Then quantitative and statistical analyses of the fracturing process at each substage were discussed.Based on the testing results,it was found that at the stress stabilisation substage,the proportion of mixed-mode fractures increased.At the stress decrease substage, the proportion of mixed-mode fractures decreased, and the tensile or shear fractures increased. At the stress increase substage, the proportion of mixed-mode or tensile fractures decreased,and the shear fractures increased.Finally,a conceptual model for the stepwise crack formation was proposed.

1. Introduction

In response to excavation of a deep cavern within hard rocks,brittle fractures appear at the excavation boundaries (Feng and Hudson, 2011). As the normal stress on the boundary is zero, the engineering problems occurring near the excavation boundary may be generalised as brittle fracturing process under biaxial compression.The fractures initiate and propagate during the stress redistribution period, and their development eventually leads to brittle failure, e.g. spalling (Liu et al., 2017). In more serious cases,cracks near the excavation boundary can induce energy accumulation, and the free rock slab may violently shoot out after crack coalescence when strain rockbursts occur (Feng, 2017). Hence,advancing our understanding of the fracturing characteristics and process under biaxial compression is of great significance(Feng and Hudson, 2011).

Biaxial compression tests have been widely used to understand the deformation or fracturing characteristics of rocks or concrete.Biaxial proportional loading tests with different stress ratios(σ2／σ1, in which σ1and σ2are the maximum and intermediate principal stresses, respectively) were conducted on 0.2 m × 0.2 m × 0.05 m (length × width × thickness) concrete specimens (Kupfer et al.,1969). Their results indicated that many micro-cracks were formed parallel to the free face of the specimen and that the angle between the major shear crack and the free face was approximately 18°-27°. The biaxial fracturing characteristics of the cubic concrete specimens with side length of 0.0508 m revealed that the splitting macro-cracks, which were formed parallel to the free face, dominated the fracturing process, instead of the shear macro-cracks (Taylor et al., 1972). The influences of the loading boundary and the control method on the fracturing mode in biaxial compression tests carried out on 0.076 m × 0.076 m × 0.025 m (length × width × thickness)Wombeyan marble specimens were studied by Brown (1974).Results indicated that splitting was notable in failed specimens without servo-control, and shear macro-cracks were observed only when solid steel platens were used.

The influence of rock type on the biaxial fracturing behaviour was studied by Papamichos et al.(1994)using biaxial compression tests on rock specimens with dimensions of 0.08 m × 0.09 m × 0.11 m(length × width × thickness). It was found that spalling was observed on the failed Berea sandstone samples, and macro-scale shearing was observed on failed Indiana limestone. The macrofailure surface was observed to be parallel to the σ2direction through biaxial compression tests on rock specimens with dimensions of 0.037 m×0.016 m×0.01 m(Mogi,1967,2006).Threedimensional(3D)crack growth in rocks under biaxial compression was investigated via the biaxial compression test on the cubic rock specimens with side length of 0.1 m(Sahouryeh et al.,2002),and the results indicated that all the specimens failed by splitting parallel to the free surface.Through a novel biaxial loading test on a cubic rock specimen with side length of 0.0508 m (Garg et al., 2018), it was observed that the shear failure plane rotated from the strike plane in the σ2direction to a plane between the σ1and σ2directions with increasing σ2value. The above experimental observations concentrated on the biaxial compressive strength or final failure mode.However, these experimental observations cannot provide insight into the whole fracturing process, especially in the post-peak regime. The complete stress-strain behaviour is difficult to achieve due to experimental limitations. In conventional biaxial proportional loading, sudden brittle failure occurred after the stress reached the peak strength(Hudson et al.,1972).For feedback servocontrolled biaxial loading, the high stiffness, precise control and measurement accuracy were difficult to satisfy considering the rock brittleness in biaxial compression tests (Feng et al., 2016). Some important aspects, such as post-peak progressive fracturing, have been reported in rocks under uniaxial or conventional compression conditions (Martin, 1997). However, the post-peak fracturing behaviour under biaxial compression conditions has rarely been reported.

In previous work,the main faults in brittle rocks were found to follow a rather tortuous path, which was characterized by steplike or en echelon arrangements (Chang and Haimson, 2000;Hoek and Martin,2014).En echelon cracks,which occur as unique sets of subparallel fractures, were commonly found in natural fault damage zones (Aydin,1978). En echelon fractures generally initiate first, weakening a zone in the rock. Then these fractures connect as the rock bridges between them are broken.As a result,faults will form(Cheng et al.,2015).Some early studies suggested that a macroscopic fault plane develops in the central portion of the sample and grows by merging with the existing tensile microcracks in a stepwise manner (Hallbauer et al.,1973). An idealized failure model (Peng and Johnson, 1972) based on observations concerning micro-crack evolution was proposed, which showed that a simple shear fault plane develops due to the bending and ultimate fracturing of the small rock columns or beams that are bounded by the vertical cracks. According to acoustic emission(AE) results, a localised fracturing process started with stepwise growth of many tensile cracks, which eventually merged into a shear macro-fault plane(Reches and Lockner,1994).A conceptual model of a staircase-like crack was suggested to explain the mechanism of crack coalescence: the coalescence of interacting tensile cracks would be accompanied by the weakening and collapse of the bridging material between the cracks (Eberhardt et al., 1999). The above studies were mainly based on observations of section or faulting process among pre-cut flaws in thin rock sheets under uniaxial compression conditions,and these conceptual mechanisms were proposed for a two-dimensional(2D) case. The experimental results of 2D crack growth under uniaxial and biaxial compression conditions cannot be applied to modelling of 3D fractures under biaxial compression(Germanovich et al.,1994;Sahouryeh et al.,2002).In addition,the coalescence of natural cracks is also different from that of pre-cut flaws (Golshani et al., 2006).

The fracturing process in brittle rocks or rock-like materials has been extensively studied in laboratory tests (e.g. Hoek and Bieniawski, 1965; Bieniawski, 1967; Scholz, 1968; Lin et al.,2014; Kong et al., 2018) and numerical simulations (e.g.Gopalaratnam and Ye,1991; Liu et al., 2007; Li et al., 2016). The macro-mechanisms of crack initiation and interaction were reported in the results of particle flow code (PFC) simulated compression tests on Lac du Bonnet granite (Diederichs et al.,2004), and rock compression tests with digital image correlation were used to reveal the occurrences of surface spalling(Kao et al.,2016). Another laboratory method, i.e. AE, is realised by considering the released energy due to cracking as transient elastic waves. AE provides a method to describe the fracturing process without affecting the sample integrity (Mogi, 1962). Consequently, this method has been widely applied in the study of fracturing process (Li et al., 2010; Lin et al., 2018; Rodríguez and Celestino, 2019), crack propagation (Bunger et al., 2015), and failure characteristics(Xu et al.,2012)involved in brittle materials such as rocks. To reveal the fracture mode of the crack, moment tensor(MT)inversion is used to analyse the source mechanisms of AE events. Pioneering work on the MT analysis on rocks was performed in the study of earthquake sources (Gilbert, 1971).Following this work, MT analysis has been proven to be effective in evaluating rock cracking mechanisms (Ohtsu, 1991, 1995; Lei et al., 1992; Chang and Lee, 2004; Charalampidou et al., 2015;Liu et al.,2015). However,quantitative and statistical analyses of the fracturing process of rocks under biaxial compression, performed by combining AE monitoring and MT analysis, are rarely reported.

In the present study,with the 3D failure plane scan method,AE monitoring and MT analysis, the failure characteristics and fracturing process of marble under biaxial compression were studied.Complete stress-strain curves including the post-peak region were obtained by means of a rigid indenter and a servo-controlled loading system (Feng et al., 2016). The failure planes were determined with the 3D scan method to obtain the 3D feature of the surface, while the crack shape was also acquired from the 2D failure image. The temporal evolutions of the AE activity, lateral strain (ε2) and volumetric strain (εV) were studied to understand the characteristics of the whole fracturing process. Based on several AE event locations and MT analysis results, quantitative and statistical results on the fracturing process were obtained.

2. Experimental procedures and methods

2.1. Specimen preparation

The marble block of the Baishan Group(T2b)was sampled from the tunnel of the Jinping Underground Laboratory Phase II(CJPL-II)project,where rockbursts and spalling frequently occurred(Li et al.,2012; Feng et al., 2018). An X-ray diffraction analysis (Fig. 1a)indicated that the dominant minerals(Fig.1b)(Gao and Feng,2019)of the marble are dolomite(88.08%)with some calcite(11.92%).The basic physico-mechanical properties of the rock are shown in Table 1.

Fig.1. Basic mineral composition information of CJPL-II marble:(a)The X-ray diffraction analysis of the marble;and(b)The petrography image of the marble thin section slide(Gao and Feng, 2019).

Table 1Basic physico-mechanical properties of the CJPL-II marble.

During sampling,transportation and processing,the integrity and freshness of the rock block was guaranteed,and mechanical damage to rocks was avoided by minimizing collisions. Each specimen used in the test was fabricated into a cuboid with dimensions of 0.05 m×0.05 m×0.1 m(length×width×thickness)in accordance with International Society for Rock Mechanics and Rock Engineering(ISRM) recommendations (Feng et al., 2016, 2019) and with the requirement of the aspect ratio under true triaxial compression.The geometric error was controlled within ±0.01 mm, and the requirements concerning smoothness and flatness were both satisfied.In addition,two faces of the specimen were confirmed to be parallel to the free face of the tunnel,and during the loading process,the σ3(minimum principal stress) direction was set perpendicular to the free face of the tunnels of the CJPL-II project.

2.2. Biaxial compression testing programme

The tests were conducted with the Lavender 508 system at Northeastern University in China, which is a novel true triaxial apparatus (Fig. 2a) developed to understand the fracturing behaviour of brittle hard rocks under both loading and unloading stress paths (Feng et al., 2016). The apparatus used in this study was designed for true triaxial compression tests,with which reasonable post-peak deformation behaviour was observed. With the use of a high-stiffness frame and accurate servo-controlled minor principal strain loading system, repeatable and precise post-peak failure responses were successfully obtained (Feng et al.,2016).

As shown in Fig. 2b, the biaxial compression test consisted of three loading phases:(a)σ1and σ2were simultaneously increased to the target values at a rate of 0.5 MPa/s according to the ISRM suggestion (Feng et al., 2019); (b)σ2was kept constant during the independent increase in σ1at 0.5 MPa/s;and(c)σ1was increased by controlling ε3at a rate of 0.005‰/s until failure.

2.3. Acoustic emission monitoring

In this study, a PCI-2 AE monitoring system was used. The threshold of AE signals was 40 dB,and the sampling frequency was 1 MHz. During the experimental process, four faces of each specimen were clamped in place by rigid blocks. AE sensors, of which the operating frequency was 200-750 kHz, were placed on the other two free faces.Each sensor was equipped with a preamplifier of 40 dB. Vaseline was smeared between the AE sensors and the interfaces of the specimens to ensure a good contact. To improve the location accuracy,eight sensors were used as shown in Fig.3ac. Additionally, an asymmetrical distribution of the sensors was applied to enhancing the convergence rates.

Fig. 2. Basic information of biaxial loading process: (a) Biaxial loading apparatus (Feng et al., 2016), and (b) Loading procedure.

This study adopted an AE event location algorithm (Xu et al.,2012), based on the least squares algorithm combined with Geiger’s method to establish the locations. Before locating the AE events, the AE signals were filtered to remove noise, and the rock fracturing-related signals were obtained. The Akaike information criterion Capwas adopted to acquire the accurate arrival times of Pwaves,and the amplitude of the first motion was determined as the first local peak after P-wave arrival, as shown in Fig. 3e (Liu et al.,2015). Several pencil break tests were carried out on the surfaces of the specimens,and the location errors of 90%of the events were determined to be less than 10 mm.The results of the homogeneous P-wave velocity model indicated that the location precision met the micro-crack monitoring requirements of this study(Xu et al.,2012).

2.4. Moment tensor analysis

According to quantitative seismology theory (Ohtsu, 1995; Liu et al., 2015), the elastic displacement u at location x and time t induced by cracking can be represented as follows:

wvhere Gip，qis the spatial derivative of Green’s function,S(t)is the source time of the crack motion, and mijrepresents the MT components. Considering that all MT components have the same time dependency (Ohtsu,1995; Liu et al., 2015), the amplitude A(x) of the first motion of the AE signal is determined to have occurred at point x and can be simply calculated as follows:

where R represents the distance between the source and the detection point;r is the direction vector;Rdis the direction cosine vector of r; sv is the sensor installing vector, as shown in Fig. 4b;Ref(sv，r) represents the reflection coefficient between r and sv,and the determination of Ref(sv，r) can be found in the literature(Katsuyama,1996;Grosse and Ohtsu,2008);and parameter Csis a sensor sensitivity coefficient, which is determined by pencil break tests before the experiments. When all of the parameters were determined, the six independent components, or mijof the moment tensor MT of each AE event, as shown in Fig. 4a, can be calculated.

Fig.3. The AE monitoring setup and method.(a)The sensor position on the right side and(b)the left side.(c)The sensor setup on the specimen and loading directions.Meanwhile,the actual specimen setup was also provided. (d)A typical rock fracturing hit signal in amplitude-time domain and its amplitude threshold.(e) The calculation method (Liu et al.,2015) for the P-wave arrived time and the amplitude of the first motion, which is based on the local peak of Akaike information criterion Cap.

Fig.4. The moment tensor analysis method(Liu et al.,2015).(a)Moment tensor MT and its components mij in a microelement in xi coordinate system.(b)The spatial position of the crack and the sensor,in which n and l represent the normal and tangential vectors of the crack surface F,respectively;sv is the sensor setting vector;and u is the displacement of the sensor with amplitude A. Besides, R is the distance between sensor and crack, and r is the direction vector. (c) Three elementary modes (tensile, mixed mode and shear) for the cracking.

The eigenvalue was decomposed into a shear component X,hydrostatic component Z, and deviatoric component Y. With the principal eigenvalue of MT, the normalization was presented as follows:where mmax,mintand mminrepresent the maximum,intermediate and minimum principal eigenvalues,respectively.An identification based on the eigenvalues of MT was used to determine the displacement mode of the source (Gilbert, 1971; Ohtsu,1991; Liu et al., 2015). These were classified into three types of AE sources as shown in Fig.4c:X ＞60%as the shear mode;X ＜40%and Y+Z ＞60% as the tensile mode; and 40% ＜X ＜60% as the mixed mode.Moreover, this classification method was also applied to the mechanism analysis of micro-cracks in the laboratory (Liu et al.,2015) and fractures in the field (Xiao et al., 2016), and good results were obtained.

2.5. 3D scan and 2D crack measurement of the failure surface

In this paper,the 2D crack shape was measured on the exposed σ1-σ3plane. After biaxial loading, failed specimens were immediately removed and photographed in a light box so that the images would capture the original crack shape. The location of all the lamps,the camera and the specimen remained fixed,and the image resolution(300 dpi)was not changed.As shown in Fig.5,the final failure image was transformed into the greyscale image and the crack image was obtained. Then the crack images were filtered to remove noise(with a grey level greater than 20%).The filtering ratio(20%)is the approximate value of the grey level threshold between the cracks and noises. Although this filtering ratio may not be precise,the same filtering method was applied to all specimens to ensure that they were comparable. The method was applied to recognising the shape of the cracks (Yan et al., 2002; Liu et al.,2013). Notably, this measurement was acquired from a 2D image,in which the actual 3D crack characteristics could not be reflected.To understand the 3D crack characteristics, the Artec3D Spider scanner was used to obtain the failure surfaces via noncontact area scanning. The accuracy of the resulting data is ±0.05 mm over a 180 mm×140 mm domain within a measuring time of 0.5 ms,and high reliable results can be acquired by this scanner.

2.6. Experimental setup and limitations

Nineteen groups of biaxial compression tests were conducted in this study, among which lateral deformation (ε3) servo-controlled loading was applied in 17 tests, while axial displacement(0.006 mm/min)control mode loading was conducted in two tests.Table 2 provides the experimental setup, of which 2D crack measurement, 3D scanning, AE process monitoring, AE event positioning and MT analysis were conducted on several specimens.The experimental setup contained three components, i.e. biaxial compression loading system, crack measurement system and AE process monitoring system.

(1) Considering that the actual intermediate principal stress of strata is approximately 65 MPa,seven σ2loading levels were then set, i.e. 0 MPa, 25 MPa, 50 MPa, 75 MPa, 90 MPa,100 MPa and 150 MPa.The highest stress level was 150 MPa,which is close to the uniaxial compressive strength(166.45 MPa)of marble,as given in Table 1.In addition,more than one specimen was tested for each σ2loading level except 90 MPa.

(2) The influence of the control mode was considered, because two axial displacement-control mode loading tests were conducted (on specimens 7 and 17). The pre-peak regime and peak stress results were similar to those of the lateral deformation servo-controlled tests at the same σ2level;however, the true post-peak regime cannot be acquired in the axial deformation control mode loading tests(which will be discussed in Section 3). The influence of loading speed was not considered due to the limitation of the servocontrolled system; nevertheless, the same loading speed was applied to ensuring comparability.

(3) The 3D laser scanning was conducted on only six specimens,as their main cracks were persistent.However,almost all the σ2loading levels were included for the scanned specimen.

(4) For some specimens, failure led to fragmentation, and the original crack shape was difficult to measure. For other specimens,cracks were not clear along the boundary surface.Therefore,2D crack shape measurements were conducted on only six specimens with clear test-induced crack shapes.

(5) Due to the sizes of each specimen,sensor and block of linear variable differential transformers(LVDTs),only eight sensors were applied for AE monitoring. As the experimental demand for AE event positioning and MT analysis is high, one accurate fracturing signal should be received by at least four sensors for adequate AE event positioning, while the signal should be received by at least six sensors for reasonable MT analysis.In addition,the total quantity of fracturing signals in marble is smaller than those of other types of rocks,such as sandstone or granite.Due to these experimental limitations,only three successful AE event positioning and one MT analysis results were obtained. Nevertheless, eight AE monitoring datasets, covering all σ2levels, were applied to supporting the analysis.

Fig. 5. The extraction process of the crack shape from the final failure image.

Table 2Experimental setup.

3. Complete stress-strain behaviour and macro-cracking characteristics of marble under biaxial compression

3.1. Repeated stress drop behaviour in complete stress-strain curve

Nineteen stress-strain curves under different levels of σ2were successfully acquired from the biaxial compression tests(σ3=0 MPa),and the representative curves are shown in Fig.6.In some cases,the stress drops repeatedly after reaching its peak value until the final instability occurs (specimens 15, 18 and 23); however, in other cases, the specimen abruptly fails after the load surpasses the rock strength.This repeated stress drop behaviour is not noticeable in the uniaxial compression condition, and the stress drops to the residual state for the axial displacement-control mode loading cases (specimens 7 and 17). Therefore, this behaviour is difficult to observe in traditional uniaxial compression tests or uncontrolled failure tests. Only one successful curve was acquired for the σ2levels of 90 MPa and 150 MPa (specimens 22 and 28,respectively).For biaxial compression,except for the two σ2levels of 90 MPa and 150 MPa,the repeated stress drop behaviour is clear(the proportion of curves with repeated stress drops is larger than 50% for each σ2level). In addition, the stress even starts to drop before reaching its peak value in some cases (specimen 23).

3.2. 3D stepwise cracks on failure surface

The representative final failure images for different σ2levels are shown in Fig. 7. All the failure angles between the main failure plane and σ3direction as shown in the figure are between 60°and 90°, and the failure planes exhibit a high degree of tortuosity. For some of the failed samples, the main crack cut through the specimen, and splitting can be observed at the edges of some failed specimens. In addition, severe failure occurs in some of the specimens (specimens 14 and 17) with many cracks. For all the failed specimens under biaxial compression, the failure surfaces are roughly parallel to σ2direction, even for specimens 14 and 17.

Fig. 8 shows the 3D profiles of the fracture surfaces. A 3D stepwise crack can be observed from the failure surface. The dip directions of the steps are approximately the same, and some of them are approximately parallel to σ2direction. Undulation is observed along the coalescence direction.For specimen 13(Fig.8a),the total undulation is small, and the surface is relatively flat.Nevertheless, this stepwise feature can also be observed in the z contours of specimen 13 (Fig. 8f) to some extent.

A comparison between the average crack shape(average z along σ2direction; red line) calculated from the 3D scan of the surface and the exposed 2D crack shape (black line) at the σ1-σ3plane of specimen 26 is shown in Fig. 8g, and they match well due to the parallel relation between the step surface and σ2direction. In the blue dashed rectangle in Fig. 8g, the path shape is quite similar to that in the blue dashed rectangle in Fig. 8e, although a few undulations exist along σ2direction in the dashed area of Fig.8e.The exposed 2D crack shape at σ1-σ3plane can be applied to studying the 3D undulation of the failure surface to some extent considering that the dip directions of steps are approximately the same.

Fig. 9 provides the exposed 2D crack shape at σ1-σ3plane for several specimens. Each main crack is roughly separated into inclined, vertical and overlapping cracks based on the angle and the crack shape. When σ2is relatively low (i.e. 0 MPa, 25 MPa and 50 MPa;Fig. 9a-c), the macro-crack shape is irregular, alternating between vertical and inclined. When σ2is relatively high (i.e.75 MPa and 150 MPa;Fig.9d and e),an overlapping pattern appears between two vertical cracks instead of two inclined cracks. Basically,the vertical cracks seem to exhibit a larger width compared to that of the inclined cracks,indicating that vertical cracks may tend to open. Generally, this irregular shape (vertical-inclined/overlapping-vertical; wide-narrow-wide) is evident and is particularly pronounced at moderate σ2levels (Fig. 9c).

4. Fracturing process of marble under biaxial compression

Combined with the AE results, the fracturing process of the marble specimens under biaxial compression is shown in Fig.10.In this figure, the AE hit is the measurement of an AE signal on a channel shown in Fig.3c and d(ASTM E1316-18a,2018;Wong and Guo,2019),and the cumulative energy is the total energy released(which can be calculated from the integral area in the amplitudetime map in all events, as shown in Fig. 3d and e (ASTM E1316-18a, 2018)). The results in Fig. 10c-e correspond to the same σ2level (50 MPa); Fig. 10e is the AE monitoring result with axial displacement-control mode loading.

This paper mainly concentrates on the post-peak fracturing process.From Fig.10a-d and f-h,a repeated and evident AE evolution mode(Fig.10i)can be observed in most of the cases of the postpeak regime.At the first stage of the evolution mode,the stress remains approximately stable,with localised fluctuations,and the AE hit remains high and the cumulative energy increases slowly.At the second stage,the stress drops suddenly,the AE hit generally remains high, and the cumulative energy increases. At the third stage, the stress increases slowly,the AE hit remains low,and the cumulative energy roughly remains constant in most cases. For Fig. 10f, the mode of several stress drops may be not evident due to the periodic noises.In this context,these three substages are defined as the stress stabilisation,stress decrease,and stress increase.

To further understand this kind of evolution mode, Fig.11a, b and d provides the temporal evolution of the AE hit and released energy with the lateral strain(ε2)and the volumetric strain(εV)of specimen 15. Based on the AE evolution mode and corresponding stress variation,four stress drops were identified in the post-peak regime.Each stress drop is divided into three substages,as shown in Fig.11d: stress stabilisation, stress decrease and stress increase substages, in sequence. As shown in Fig.11d, except for the stress variation and AE evolution shown above,at the stress stabilisation substage, the lateral strain ε2generally decreases. At the stress decrease substage, ε2generally increases. At the stress increase stage, ε2generally decreases. Table 3 compares the results obtained from these four stress drops. Some significant similarities were observed among the substages of different stress drops.Notably,some misfits appear in stress drop 4.As shown in Fig.11c,the energy release of stress drop 4 is small, which indicates a low level of fracturing activities. This lesser fracture development discriminates drop 4 from the other stages and might cause differences among them. Generally, based on the similar strain and stress variations and AE evolution characteristics, a repeated fracturing mode exists in the post-peak regime of marble under biaxial compression, corresponding to repeated stress drop behaviour.

Fig. 7. Representative final failure images under different σ2 levels: (a) σ2 = 0 MPa, (b) σ2 = 0 MPa (axial displacement control mode), (c) σ2 = 25 MPa, (d) σ2 = 50 MPa, (e)σ2=50 MPa,(f)σ2=50 MPa(axial displacement-control mode),(g)σ2=75 MPa,(h)σ2=90 MPa,(i)σ2=100 MPa,and(j)σ2=150 MPa.Meanwhile,the fractures have been traced by blue dashed lines.

Fig.8. 3D scanning result and related analysis:(a)-(e)3D scanning results of the failure surfaces of different specimens;(f)Contour figures of z corresponding to(a)for specimen 13,and the black line was the approximate boundary for z=1.5 mm(yellow);and(g)2D crack shape(crack image and black line)and average crack shape(red line)calculated from(e) for specimen 26. Besides, the blue dashed area in (g) was corresponding to the blue dashed area in (e).

Fig. 9. The exposed 2D crack shapes on the σ1-σ3 plane of several specimens at different σ2 levels: (a) σ2 = 0 MPa, (b) σ2 = 25 MPa, (c) σ2 = 50 MPa, (d) σ2 = 75 MPa, and (e)σ2 =150 MPa ‘V’, ‘I’ and ‘O’ represents the vertical crack, inclined crack and overlapping part, respectively.

Fig. 10. AE monitoring results for different specimens under different σ2 levels: (a)-(h) AE hit and Cumulative energy evolutions for different specimens; and (i) A schematic diagram for the repeated AE evolution mode. Meanwhile, (e) is the AE monitoring result of the axial displacement control mode loading test.

5. Quantitative and statistical analyses of fracturing process

In this section,the total fracturing process is quantitatively and statistically analysed, especially the repeated fracturing mode in the post-peak regime.Due to the experimental limitation outlined in Section 2.6, only three representative AE event positioning results (specimens 9,15 and 23) are analysed, and one MT analysis result (specimen 15) is studied. Nevertheless, the repeated fracturing mode can be observed in the biaxial compression results for almost all the σ2levels tested,as discussed in Section 4.Fortunately,this mode is fairly evident in specimen 9 and especially sound in specimens 15 and 23 (Figs.10 and 11).

The AE event positioning (Fig.12a, c and d) or the MT analysis(Fig.12b)results of three specimens are shown in Fig.12.The results indicate a good agreement between the AE event clusters and final macro-crack (purple dashed curves in Fig.12) observations. Based on the MT analysis,the temporal evolution of the source type proportions of specimen 15 is shown in Fig.13a. The purple, blue and white parts represent the proportions of tensile, mixed-mode and shear event sources, respectively. Fig.13 shows that tensile event(approximately 45%)is the main type of AE source.The proportion of mixed-mode or shear event type is approximately 20%-30%.The peak of the proportion of tensile event type occurs at stress drop 2.At the same time,the energy release of stress drop 2 is the largest,as shown in Fig.11c,with the most serious brittle failure that occurs at stress drop 2. The temporal evolution of the energy release is also shown in Fig. 14a. The peak of energy release occurs at the peak stress.Before the peak stress,the energy release is low.In addition,the energy release events are active during stress drop 2.

The evolutions of the source type proportion and the release energy at the pre-peak stage are analysed in Figs.13b and 14b.This stage is separated into two parts, i.e. A-B and B-C, which are defined in Fig.11b.At the A-B stage,the proportions of tensile and mixed-mode events decrease while the proportion of shear events increases.Meanwhile,there are a few AE events with low released energy.At the B-C stage,the proportion of tensile events increases while the proportion of mixed-mode events decreases. The AE activity is dormant at the B-C stage, while an event with high released energy occurs as the stress approaches its peak.The results indicate that events, which first accumulate energy and then suddenly release energy, occur at the B-C stage.

Stress drop 2 is analysed in Figs.13c and 14c. At the stress stabilisation stage,the proportions of tensile and mixed-mode events increase while the proportion of shear events decreases. Meanwhile, events with moderate released energy occur. At the stress decrease stage, the proportions of the tensile and mixed-mode events decrease while the proportion of the shear events increases. In addition, several events with a moderate energy release occur during the stress drop.At the stress increase stage,the proportions of the tensile and mixed-mode events decrease while the proportion of the shear event increases. AE events are barely observed,and a few events with a low energy release appear at the end of the stage.

Fig. 11. The AE monitoring results and deformation evolution during the loading for specimen 15. (a) The stress plotted against the volumetric strain. (b) The AE hit and the cumulative energy monitoring results.(c)The energy release of different deformation stages.(d)The lateral strain ε2 plotted against time.Besides,the loading sequence(Phase I,II and III)is also provided in(b).In addition,several characteristic stress points are marked in(b).Points O,A,B and C are the biaxial loading stress,crack initiation stress based on AE characteristics, crack damage stress based on volumetric strain evolution and peak stress, respectively. Points D, E, F and G are the local stress peaks in the post peak regime.Moreover, three substages have been divided by red dotted line in (d).

Table 3Comparison of four stress drop stages in specimen 15.

The changes in the source type proportion of all the stress drops and the average process are shown in Fig.15. The temporal evolutions of the substages at different stress drops are similar to the source type proportion evolution in stress drop 2.The mixed-mode proportion increases at the stress stabilisation stage; the mixedmode proportion decreases while the shear or tensile proportion increases at the stress decrease stage; the tensile or mixed-mode proportion decreases while shear proportion generally increases at the stress increase stage.In addition,the source type proportion change at the stress increase stage is similar to that at the A-B stage,as shown in Fig.13b.

Fig.12. Comparison between the location results and final failure image,including(a)the amplitude and(b)the mechanism of AE events for specimen 15,and the amplitude of AE events for (c) specimen 9 and (d) specimen 23. The scatter size is proportional to the magnitude of the amplitude in (a), (c) and (d).

Fig.13. Temporal evolution of fracturing mechanisms in specimen 15,in which(a)the evolution of the source type proportion during loading,(b)the source type proportion for the pre-peak stage and (c) for the stress drop 2 stage.

6. Conceptual model for formation of three-dimensional stepwise cracks

Combining the quantitative and statistical analyses of the fracturing process(Sections 4 and 5),the similar characteristics of these substages in the post-peak regime are shown in Table 4. Based on these characteristics,conceptual models for each substage are also shown in Table 4.

As shown in Table 4, the possible fracturing process of these substages is as follows: (1) The stress stabilisation stage is characterised by the opening and extension of many wing cracks(Nemat-Nasser and Hori,1987)throughout the specimen.This process will continue until a few local fractures are formed(Rudnicki and Rice,1975). (2) The stress decrease stage is characterised by the coalescence and sudden unstable opening of the wing cracks.In a study of rock-like material(Tung and Tue,2015),the growth of a structurally localised fracture zone was observed with a decrease in stress in the post-peak regime. In addition, in the work of Wawersik and Brace(1971), it was found that in the post-peak regime of granite, spalling occurs, and steeply inclined shear fractures involve into an open fault during stress drop stages V and VII, respectively. (3) At the stress increase stage, shearing of the pre-existing flaw surface around the tip of the main crack is believed to occur.In addition,the shearing-induced plasticity was observed to be self-sustained with energy dissipation, and stress increases as the strain hardening behaviour increases (Khan and Huang,1995). The aforementioned three processes repeat until the main stepwise crack cuts through the entire specimen. This conceptual model is based on observations and relevant previous studies,and it matches with the testing results well. However, it is only possibly one explanation for the deformation, energy and fracturing characteristics shown in Table 4.

Fig.14. Temporal evolution of the energy release in specimen 15,in which(a)the evolution of the energy release during loading,(b)the energy release evolution for the pre-peak stage, and (c) for the stress drop 2 stage.

Fig.15. Statistical results for the change of the source type proportion at different deformation substages of specimen 15.

Table 4Characteristics and possible fracturing process of three substages in the post peak regime.

7. Conclusions

In this study, servo-controlled biaxial compression tests on marble specimens were conducted to examine the failure characteristics and fracturing process.Complete stress-strain curves and 3D failure surface characteristics were acquired.In accordance with the AE event positioning and MT analysis results from the tests,the characteristics and formation of the cracks were studied. Furthermore,a conceptual model describing crack growth in the post-peak regime was proposed.Therefore,based on the experimental results in this study, the following conclusions were drawn:

(1) In the majority of marble specimens under biaxial compression, the stress repeatedly drops until instability finally occurs. The stress drops multiple times. In addition,the stress drop even starts before reaching its peak value in some cases.

(2) A 3D stepwise crack can be observed on the failure surface.The dip directions of the steps are approximately the same,and some of them are approximately parallel to σ2direction.Undulation is observed along the coalescence direction. For the exposed 2D crack shape at the σ1-σ3plane, an irregular shape (vertical-inclined/overlapping-vertical; wide-narrow-wide) is evident, and it is particularly pronounced at moderate σ2levels.

(3) A repeated fracturing mode corresponding to the repeated stress drop in majority of the cases of the post-peak regime is observed. At the stress stabilisation substage, the stress remains approximately constant, with many localised fluctuations,and the lateral strain ε2generally decreases;the AE hit generally remains high level,and the stored energy tends to release.At the stress decrease substage,the stress drops,and ε2generally rises;the AE hit generally remains high level,and the energy release increases.At the stress increase substage,the stress increases slowly,and ε2generally decreases;the AE hit remains low level,and the energy release is low.

(4) Based on the quantitative analyses of the source types of AE events,it is found that during the stress stabilisation substage,the proportion of mixed-mode events increases. During the stress decrease substage, the proportion of mixed-mode events decreases, and the proportion of tensile or shear events increases. During the stress increase substage, the proportion of mixed-mode or tensile events decreases, and the proportion of shear events generally increases.The source type evolutions of the stress increase substage are similar to those of the crack initiation stage in the pre-peak regime.

Declaration of competing interest

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Acknowledgments

The authors sincerely acknowledge the financial support received from the Key Research Program of Frontier Sciences, Chinese Academy of Sciences (CAS) (Grant No. QYZDJ-SSW-DQC016).We would like to thank Key Laboratory of Ministry of Education on Safe Mining of Deep Metal Mines, Northeastern University for giving us the opportunities to conduct the experimental test.Appreciation is extended to all the teachers and staffs here, especially Prof. Jianpo Liu for his selfelss guidance. Besides, we are grateful to Mr. Yan Zhang, Mr. Yong Han and other members of Mechanical Response of Deep Hard Rock(MRDHR) group for their generous assistance with the experimental operation.