J. Cent. South Univ. (2016) 23: 1474-1485

DOI: 10.1007/s11771-016-3200-3

Experimental and numerical study on loading rate effects of rock-like material specimens containing two unparallel fissures

HUANG Yan-hua(黄彦华), YANG Sheng-qi(杨圣奇), ZENG Wei(曾卫)

State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: A series of laboratory experiments and PFC numerical simulations for rock-like material specimens containing two unparallel fissures were carried out. On the basis of experimental and numerical results, the stress-strain curves, mechanical properties, AE events, cracking behavior and energy characteristics were analyzed to reveal the macro-mechanical behavior and meso-mechanism of pre-fissured specimens under different loading rates. Investigated results show that: 1) When the loading rate is relatively low, the stress-strain curves show a brittle response. When the loading rate is relatively high, the curve shows a more ductile response. Both of the peak strength and elastic mudulus increase with the increase of loading rate, which can be expressed as power functions. 2) Four crack types are identified, i.e., tensile crack, shear crack, far-field crack and surface spalling. Moreover, the tensile crack, far-field crack and surface spalling are under tensile mechanism, while the shear crack is under shear mechanism.3) The drops of the stress-strain curves all correspond to the crack initiation or coalescence, which is also linked to a sudden increasing in the accumulated micro-crack curve. 4) Both of the maximum bond force and energy have the similar trend with the increase of loading rate to peak strength, which indicates that the trend of peak strength can be explained by the meso-mechanics and energy.

Key words: rock mechanics; two pre-existing fissures; strength parameters; crack coalescence; particle flow simulation

1 Introduction

Loading rate has significant effects on rock mechanical properties, and rock materials are fractured differently at varied loading rates. Therefore, it is of great importance to study the effects of the loading rate on the mechanical behavior of rock materials. ZHANG et al [1] carried out fracture toughness tests for Fangshan gabbro and marble with a wide range of loading rates. Their experimental results showed that the static fracture toughness is nearly a constant, while the dynamic fracture toughness increases with the loading rate. LIANG et al [2] tested uniaxial compression for salt rock with different loading strain rates and they found that the strength of salt rock is only slightly affected by loading strain rate. GONG and ZHAO [3] conducted indirect tensile tests on sandstone under quasi-static loading and dynamic loading, using a servo-hydraulic machine and a split Hopkinson pressure bar. The results showed that the indirect tensile strength of the sandstone increases with the loading rate. ZHAO et al [4] simulated the uniaxial tensile strength of sandstone using the distinct lattice spring model (DLSM) and they thought that microstructure causes a strain rate dependency. LIANG et al [5] reported their laboratory experimental results on granite at different strain rates, subjected to uniaxial compression. Their SEM images indicated that the fracture modes change from intergranular to transgranular fracture with increasing loading rate. From the above investigations [1-5], we know that mechanical properties and fracture process of rock are rate- dependent.

However, most of them dealt with rocks as a kind of intact material. In real rock engineering, mechanical properties of rock material are greatly affected by a variety of structural surfaces. In fact, open fissure is one of the most common structural surfaces. To better understand the influence of open fissures on mechanical characteristic and cracking process, a series of experimental studies have been performed on model- rock or rock materials with parallel fissures by SHEN [6], WONG and CHAU [7], BOBET [8], GEHLE and KUTTER [9], FUJII and ISHIJIMA [10], LI et al [11], WONG and EINSTEIN [12], YANG et al [13] and ZHOU et al [14]. But rock masses are generally composed of unparallel fissures. In recent years, LEE and JEON [15], YANG et al [16] and HAERI et al [17-18] carried out uniaxial compression or tensile experiments on rock containing unparallel fissures (a kind of new fissure geometry) in order to investigate fracture mechanism of rock containing intermittent unparallel fissures. To study the cracking processes under different loading rates, ZHANG and WONG [19] made uniaxial compression simulation for one fissure- and two fissures- contained model specimens, which analyzed the effect of loading rate on strength and cracking behavior. But up to now, no laboratory experiments have been carried out on model-rock specimens containing two unparallel fissures with different loading rates.

Previous research showed that strength and crack initiation, propagation and coalescence behavior of pre-fissured specimens varied with different loading rates [19]. They mainly investigated the macroscopic mechanical behavior based on numerical simulation. But the mesoscopic mechanism of loading rate effects on cracking behavior is inconclusive. Particle flow code (PFC) can be effectively applied to analyzing the meso-mechanism of rock materials [20]. The software has been widely used for the rock strength failure behavior and mechanism analysis [15, 21-27]. Therefore in this research, to investigate the macro-mechanical properties and meso-mechanism of brittle rock under different loading rate, a coupled experimental–numerical study on strength and crack coalescence behavior in rock-like materials specimens containing two unparallel fissures was carried out. First, a rock mechanics servo- controlled testing system was used to conduct uniaxial compression tests on rock-like material specimens containing two unparallel fissures. The fracturing process was observed using photographic monitoring and acoustic emission (AE) was obtained using AE monitoring techniques. And then, PFC simulation was carried out for two unparallel fissures-contained specimens under different loading rates. On the basis of the simulated results, the meso-mechanism of loading rate effects of specimen containing two pre-existing fissures was analyzed.

2 Experimental material and loading procedure

Based on a great amount of trial and error calibration tests, we fabricated the model specimen by compounding C42.5 cement, silica flour and water at a mass ratio of 1.0:0.8:0.35. The microscopic structure of rock-like materials in the present study is shown in Fig. 1 by plane polarization and SEM observation. Most of the quartz sands are of irregular lumps in geometric shape and they are uniformly mixed together.

Table 1 shows the comparison of marco-mechanical properties of rock-like materials in this study and brittle sandstone [28-29]. From Table 1, we can conclude that the mechanical properties of rock-like materials are similar to those of brittle sandstone.

The two open fissures were created by inserting steels with the thickness of 0.8 mm and the width of 12 mm into model specimen before pouring the model materials. Fissure ① was horizontal, and fissure ② was inclined. α was the angle of fissure ②, 2a was fissure length and 2b was ligament length. Angle α was 15°, and 2a and 2b were fixed to 12 mm and 16 mm, respectively. The loading rates were designed to 0.0004, 0.002, 0.01 and 0.05 mm/s. The strain rates were 0.04×10^-4, 0.2×10^-4, 1.0×10^-4 and 5.0×10^-4 s^-1, which belong to the quasi- static loading. All of the unaxial compression tests were carried out by MTS815.02 rock mechanics servo-controlled testing system. Moreover, acoustic emission (AE) events were recorded with an AE21C-06 acoustic emission system. A HD video camera was used to capture the whole deformation process of specimens. A detailed description of the tested specimens containing two unparallel fissures is given in Table 2.

Fig. 1 Microscopic structure of rock-like materials in the present study:

Table 1 Physico-mechanical properties of rock-like materials and brittle sandstone

3 Experimental results analysis

3.1 Axial stress-strain behavior

Figure 2 shows the axial stress-strain curves of rock-like materials specimens containing two unparallel fissures under different loading rates uniaxially. In Fig.2, we can see that stress-strain curves have a general same shape, i.e., all of them undergo initial compaction, linear elastic deformation, nonlinear deformation and post-peak softening. Loading rate effects on stress-strain curves show as follows: 1) More stress-drops which are caused by the crack initiation or coalescence can be observed under relatively low rates, especially those nearby the peak stress; while no stress-drops are found under relatively high rates. 2) When the loading rate is relatively low, the curves show a brittle response, namely, the axial stress drops abruptly after the peak strength; when the loading rate is relatively high, the curve shows a more ductile response. 3) The peak stress and the slope of curves are significantly influenced by loading rate.

3.2 Strength and deformation parameters

In accordance with the axial stress-strain curves of pre-fissured specimens shown in Fig. 2, we can obtain the mechanical parameters (e.g. the peak strength). Figure 3 depicts the influence of loading rate on the peak strength and elastic modulus of specimens containing two unparallel fissures. The mechanical parameter of 0.2×10^-4 s^-1 was the average value of two specimens (e.g. C15-3 and C15-4).

The effect of loading rate on peak strength of rock-like material specimen containing two unparallel fissures is shown in Fig. 3(a). As the loading rate increases, the peak strength increases. When the loading rates are 0.2×10^-4 s^-1, 1.0×10^-4 s^-1 and 5.0×10^-4 s^-1, the increments of strength are about 7.5%, 11.9% and 24.3%, respectively. The effect of loading rate on elastic modulus of rock-like material specimen containing two unparallel fissures is shown in Fig. 3(b). The elastic modulus value also increases with the increase of loading rate. When the loading rates are 0.2×10^-4 s^-1, 1.0×10^-4 s^-1 and 5.0×10^-4 s^-1, the increments of modulus are about 7.5%, 11.9% and 24.3%, respectively. The increment trend of peak strength and elastic modulus indicate that both the sensitivity of strength and deformation parameters increase with increasing loading rates. Moreover, the relationships between peak strength, elastic modulus and loading rate can be presented as power functions, which are shown as follows:

σ_p=28.66x^0.0821, R²=0.9407                      (1)

E_s=11.848x^0.0832, R²=0.9926                     (2)

3.3 AE behavior and crack coalescence process

Figure 4 presents the axial stress-time curves, the output of AE counts and accumulated AE counts of specimens containing two unparallel fissures under different loading rates. From Fig. 4, we can see that the evolution of stress-time curves can be simply divided into the following four phases: the phase of micro-defect closure, the phase of linear elastic deformation, the phase of nonlinear deformation and the phase of post-peak stress softening. A detailed description on accumulated AE counts evolution process can be described as follows. 1) In the stage of micro-defects closure, no new cracks initiate, thus no AE counts emerge. 2) In the stage of linear elastic deformation, when the stress reaches some extent, cracks begin to initiate. Some AE counts can be seen in this phase. 3) In the stage of nonlinear deformation, AE events occur frequently, and accumulated AE counts curve continues to rise. 4) In the stage of post-peak stress softening, AE events are active, and accumulated AE counts curve increases significantly.

The real-time crack coalescence behavior of C15-4 specimen is analyzed during the entire deformation, as shown in Fig. 5. The number in Fig. 5 shows the orders of crack initiation, while the superscript letters on the number means that the cracks from different tips are simultaneously initiated.

As shown in Fig. 5, the first crack 1^a and crack 1^b initiate at a distance away from the tips of the horizontal fissure and wing crack 1^c initiates at the under tip of the inclined fissure at point 1 (σ₁=9.58 MPa) in the stress- time curve. At the time, the corresponding stress drops to 8.48 MPa rapidly and a big AE event is found. When the specimen is reloaded to point 2 (σ₁=14.60 MPa), crack 2 emerges from under tip of the inclined fissure, and propagates downwards to the bottom of specimen. Meanwhile, the axial stress drops narrowly and a big AE event appears abruptly. With the increase of deformation, the axial stress increases linearly. When the axial stress of the specimen is loaded to point 3 (σ₁=18.85 MPa), axial tensile crack 3 occurs near the under tip of the inclined fissure and propagates downwards to the bottom edge of specimen. Afterwards, the axial stress continues to increase with the increase of deformation. When the axial stress reaches point 4 (σ₁=19.29 MPa), crack 4^a initiates from the right tip of the horizontal fissure and two far-field crack 4^b and crack 4^c occur at the top and bottom of specimen, respectively. Due to obvious damage caused by the initiation of cracks 4^a, 4^b and 4^c, the biggest AE event emerges in the AE curve and the corresponding axial stress drops rapidly from 19.29 MPa to 16.13 MPa. When the stress reloads to point 5 (σ₁= 22.42 MPa), i.e. the peak strength, two far-field cracks 5^a and 5^b can be found near axial tensile crack 4^b. After the peak strength, the axial stress shows multiple drops. Finally, when the specimen is loaded to point 6 (σ₁= 9.30 MPa), a surface spalling failure 6 can be observed between crack 4^b and crack 5^a.

Table 2 Tested rock-like materials specimens containing two unparallel fissures in this research

Fig. 2 Stress-strain curves of pre-fissured specimens under different loading rates

Fig. 3 Effects of loading rate on peak strength (a) and elastic modulus (b)

Fig. 4 AE distribution curves of pre-fissured specimen under different loading rates:

Fig. 5 Crack evolution process of pre-fissured specimen obtained by experiment:

3.4 Cracking mode and mechanism

Figure 6 shows the ultimate failure modes of specimens containing two unparallel fissures under uniaxial compression with respect to different loading rates. The cracking modes of specimens containing two unparallel fissures are obviously more complex than those of intact specimen. Moreover, the crack coalescence modes of specimens containing two unparallel fissures were different with different loading rates.

When the loading rates were 0.04×10^-4 s^-1 and 0.2×10^-4 s^-1, two crack coalescences occurred in the specimen between the two left tips and two right tips of fissures. One of the two coalescences was due to the propagation of tensile crack initiated from the left tip of the inclined fissure towards the left tip of the horizontal fissure. The other was induced by the propagation of tensile crack initiated from or near the right tips of the horizontal fissure towards the right tip of the inclined fissure.

When the rate was 1.0×10^-4 s^-1, two crack coalescences were found in the specimen between the two tips of the horizontal fissures and right tip of the inclined fissure. One of the two coalescences resulted from the propagation of tensile crack initiated from the right tip of the inclined fissure towards the left tip of the horizontal fissure. The other was induced by the propagation of tensile crack initiated from the middle of the horizontal fissure towards the right tip of the inclined fissure.

When the rate was 5.0×10^-4 s^-1, one crack coalescence was observed in the specimen between the two fissures. The coalescence was due to the propagation of tensile crack from the right tip of coalescing horizontal fissure towards the right tip of coalescing inclined fissure.

According to the classification of crack and coalescence types in the previous studies [30-33], and the analysis of crack initiation, propagation and coalescence process in the above section, we can summary the crack types observed in this study. Four different crack types were identified based on their geometry and crack propagation mechanism by analyzing the ultimate failure modes (Fig. 6). The four crack types are tensile crack (TC), shear crack (SC), far-field crack (FC) and surface spalling (SS), respectively. And we have marked them in Fig. 6. All crack types can be categorized as follows.

1) Tensile crack: The tensile crack usually initiates from or with a distance to the tips of fissures. They often emerge with an angle to the direction of fissures, and then gradually depart towards the maximum principal stress, and finally develop along the axial direction.

2) Shear crack: The shear crack may initiate from the tips of fissures, and may emerge during the propagation of other cracks. The growth path of the former type of shear crack is parallel to the direction of fissures approximately.

3) Far-field crack: The far-field crack often initiates far away from the fissures. The direction of far-field crack may be vertical or horizontal.

4) Surface spalling: The surface spalling is always induced by overall failure after tensile cracks.

In order to investigate the micro-structure of cracks after failure in the pre-fissured specimen, Fig. 7 presents the local magnification of the four different crack types. From Fig. 7, it can be seen that the fracture face of tensile crack, far-field crack and surface spalling are all very rough and no frictional sliding can be observed, which means that they all show tension mechanism. However, friction traces and powders can be observed on the section of shear crack, which indicates that the shear crack is under shear mechanism.

4 PFC simulation analysis

Based on the laboratory experimental results, the effects of loading rate on stress-strain curve, mechanical parameters, AE characteristic and cracking mode were analyzed. But the mechanism of loading rate is still unclear, resulting from the insufficient experimental data. In order to explore deeply the effects of loading rate on strength and cracking behavior of rock-like material containing two unparallel fissures, in this section we will carry out a systematic numerical simulation on specimens with two unparallel fissures using bonded- particle model (BPM) in particle flow code (PFC). To reveal the meso-mechanism of loading rate, it is of great importance to analyze the meso-stress field, which is almost impossible to be obtained from the viewpoint of experiment [27].

Fig. 6 Failure modes of pre-fissured specimen under different loading rates:

Fig. 7 Micro-structure of different types of maro-crack:

4.1 Meso-parameters calibration

In BPM, the macro-mechanical parameters are determined by the meso-mechanical parameters, such as friction coefficient, normal and shear bond stiffness, normal and shear bond strength. The meso-mechanical parameters are calibrated by comparing with the laboratory experimental results of intact rock-like materials specimens. Table 3 shows the meso-parameters used in the PFC^2D model for rock-like specimens in this research.

Table 3 Meso-parameters used in PFC^2D model for rock-like material

Figure 8 and Table 4 show the comparison of stress-strain curves and mechanical properties obtained by experimental and numerical intact rock-like materials specimen under uniaxial compression. It can be seen that the numerical curve agrees well with the experimental ones, except for the initial stage of the curve. The initial stage can not be reflected in BPM. Moreover, the macro-mechanical parameters of intact specimen obtained by simulation are all approximately equal to those obtained by experiment, except for the peak strain due to the lack of initial phase.

The meso-mechanical parameters used in pre- fissured specimen are the same as those in intact specimen [15, 21-27]. It is no necessary to calibrate themeso-parameters of pre-fissured specimen again. It needs to be noted that the loading rates used in PFC are not exactly the same as the physical rate, but are obtained by numerical calibration [34].

Fig. 8 Comparison between experimental and numerical stress–strain curves of intact rock-like materials specimen

Table 4 Comparison between experimental and numerical mechanical parameters

4.2 Cracks coalescence behavior

Figure 9 shows the micro-crack evolution curves of pre-fissured specimens during the deformation process. In Fig. 9, the superscript letter “s” on the specimen number stands for simulation. In general, the micro-crack characteristics of pre-fissured specimen are very similar to the AE characteristics as shown in Fig. 4. In a word, a number of micro-cracks evolve during the crack initiation, propagation and coalescence process. From Fig. 9, some micro-cracks’ characteristics can be summarized as follows. No micro-cracks were observed at low stress levels. Some micro-cracks were observed when the stress increased to the elastic yielding stress, which can also be concluded from a slow increasing of the accumulated micro-cracks. When the stress was nearly loaded to the peak strength, the micro-cracking was the most active and the accumulated micro-cracks increased fastest. When the specimen entered the stage of post-peak strength, the micro-cracks increased with the axial strain due to the crack propagation and coalescence. Finally, the micro-cracks kept a low speed increment with the increase of axial strain.

Furthermore, according to the relationship between stress-strain curve and crack-strain curve shown in Fig.9, it is clear that the stress drops in the stress-strain curves all correspon to the crack initiation or coalescence, which is also linked to a sudden increasing in the accumulated micro-cracks curve. Note that the stress drops in the numerical stress-strain curves are minor than those in experimental curves.

Fig. 9 Micro-cracks numbers evolution curves of pre-fissured specimens under different loading rates:

From Fig. 9, we also can find that the peak strength of pre-fissured specimen simulated by PFC has the similar trend with the increase of loading rate to experimental results. However, the simulated peak strength of pre-fissured specimen is higher than that of experiment because the two-dimensional numerical simulation cannot truly reflect the three-dimensional physical phenomenon by laboratory experiment [15].

Figure 10 shows the influence of loading rate on micro-crack numbers in the ultimate failure specimens. In Fig. 10, we present the tensile micro-crack, shear micro-crack and total micro-crack numbers. Note that the tensile and shear micro-cracks are different from the macro-cracks and the definition of tensile and shear micro-crack can be seen in Refs. [35] and [21]. From Fig. 10, we can conclude that shear, tensile and total micro- crack numbers in the pre-fissured specimen increase with increasing loading rate, which indicates that the fracture extent is higher under higher loading rate. Moreover, there are much more tensile micro-cracks than shear micro-cracks, which means that tensile fracture is the major mode in the pre-fissured specimens. This finding is similar to that of experimental micro-structure shown in Fig. 7.

Fig. 10 Crack numbers of pre-fissured specimens at failure under different loading rates

Figrue 11 presents the simulated crack propagation process of C15-4^s specimen. In Fig. 11, the denoted numbers correspond to the order of cracking shown in Fig. 9, and the superscript letters on the number mean that the cracks from different tips are simultaneously initiated. From Fig. 11, it is clear that the failure of pre-fissured specimen is due to the coalescence of cracks from tips of pre-existing fissures. Crack 1 is initiated from the inclined fissure. And then crack 2 is initiated from the horizontal fissure. Notably, cracks 1^a, 1^b, 2^a and 2^b are not initiated from fissure tips, but with a distance from fissure tips, which may be due to a fact that the fissure is horizontal or the angle between fissure and horizontal direction is small [36-37].

Fig. 11 Crack propagation process of pre-fissured specimens obtained by PFC simulation:

With the increase of deformation, the cracks 1^a and 2^a propagate outside the rock ligament religion, while crack 1^b and 2^b grow inside the rock ligament religion. So, cracks 2^a and 1^b (or 2^b) show different propagation paths. For crack 1^a, it gradually propagates along the direction of the major principal stress, whereas cracks 1^b and 2^b gradually evolve towards the tips of fissure. It is safe to conclude that when the crack propagates inside the ligament region, its growth path is mainly affected by the tips of fissures. While when the crack propagates outside the ligament region, the crack growth path is mainly affected by the direction of axial stress. After cracks 1 and 2 initiate in the specimen, other cracks, e.g. crack 3, 4 and 5 emerge in other position with the increase of axial stress. And two cracks coalescence occurs between the two pre-existing fissures which are induced by the growth of cracks 1^b and 2^b. Finally, ultimate brittle failure is then quickly induced, and crack 6 is initiated in the specimen.

Figure 11 further depicts the comparison between numerical and experimental ultimate failure modes of specimen containing two unparallel fissures under uniaxial compression. From Fig. 11, we can see that the failure mode of pre-fissured specimen obtained by numerical simulation is very similar to the experimental one.

4.3 Energy dissipation characteristics

Suppose that a rock unit deforms under external force and there is no heat exchange with the outside world during the physical process. According to the First Law of thermodynamics:

U=U^d+U^e                                   (3)

where U is the total energy; U^d is the dissipation energy; U^e is the strain energy, energy monitoring mechanism in PFC can obtain different types of energy [38]. In PFC, the boundary energy, i.e. total energy U, is the total power induced by walls, and strain energy U^e is the sum of parallel bond energy and contact energy.

Figure 12 shows the relationship among boundary energy, strain energy, dissipation energy and loading rate. From Fig. 12, we can see that the total energy and strain energy increase with the increase of loading rate. Moreover, the sensitivity of increment increases with increasing loading rates. The higher the loading rate, the more the dissipation energy, which indicates that as the loading rate increases, the energy used for crack initiation and coalescence, and friction between particle and particle increase. It is very clear that the energy obtained by PFC has the similar trend as that of the peakstrength.

Fig. 12 Energy distribution at peak strength under different loading rates:

Strain energy ratio (strain energy normalized by boundary energy) and dissipation energy ratio (dissipation energy normalized by boundary energy) are shown in Fig. 12(b). From Fig. 12(b), the strain energy ratio is in the range of 0.80 (=5.0×10^-4 s^-1)-0.87 which indicates that dominant boundary energy is stored in the specimen. However, the dissipation energy ratio is in the range of 0.13 (=0.04×10^-4 s^-1)-0.20 (=5.0×10^-4 s^-1). In other words, the energy used in crack initiation and coalescence, and friction between particle and particle are inferior in the total boundary energy. Moreover, both of the strain energy ratio and dissipation energy ratio are nonlinearly related to the loading rate, and the extent of change is minor, which shows that strain energy ratio and dissipation energy ratio are not sensitive to the loading rate in the designed range of loading rate.

4.4 Meso-mechanism analysis

Bond between the particles and force chain contain a wealth of meso-information. In this section, we monitor the bond and force chain in the specimen at peak strength to analyze the meso-mechanism of loading rate. Table 5 illustrates the plots of cracks distribution, bond breakage and bonded-force chain in the pre-fissured specimens at peak strength. In Table 5, black color segments stand for micro-tensile cracks, while red color segments stand for micro-shear cracks in the micro- cracks figures. Black color segments represent the parallel bond in the bond figures. The parallel bond forces are represented by the discrete straight line segments: red color stands for tensile force, while black color stands for compressive force, and line thickness and orientation correspond to force magnitude and direction, respectively.

Table 5 Distribution of parallel bond and force chains at peak strength

In the micro-cracks figures, we can see the macro- cracks are formed by the coalescence of micro-cracks. From the local magnification, wing tensile cracks propagate along the direction of the major principal stress. Moreover, the micro-cracks numbers increase with the increase of loading rate, which has been analyzed in above section.

In the parallel bond figures, it can be seen that when the stress increases to limited strength, the bond between the particles breaks and then a micro-crack forms in neighbouring particles. Therefore, the status of bond breakage can reflect the distribution of orientation and overall fracture. In accordance with micro-cracks figures, we can see that the bond breakage zone corresponds to the distribution of micro-cracks and no micro-crack emerges in the intact bond zone.

In the force chain, the tips of pre-fissures are crowded by black segments, which indicates that the fissure tip religions are under compression. From the local magnification, the tip of macro-crack is surrounded by red segments, which means that the macro-crack tip religions are under tension. Furthermore, when the tensile force reaches the limited strength, micro-cracks begin to coalesce. With the loading rate increases, the overall bond chain distribution has a small increase. In order to better understand the bond force, we extract the maximum bond force and list them in Table 5. As we can see, the maximum bond force increases with increasing loading rate. As loading rate increased from 0.04×10^-4 s^-1 to 5.0×10^-4 s^-1, the average maximum bond force of pre-fissured specimen increased from 50.78 kN to 61.59 kN. It is clear that the maximum bond force has the similar trend to peak strength with the increase of loading rate, which means that the variation of the macro peak strength is reflected in the meso-mechanics.

5 Conclusions

1) More stress-drops which are caused by the crack initiation or coalescence can be observed under relatively low rates, especially those nearby the peak stress; while no obvious stress-drops are found under relatively high rate. When the loading rate is relatively low, the curve shows a brittle response, when the loading rate is relatively high, the curve shows a more ductile response. Both of the peak strength and elastic mudulus increase with the increase of loading rate, which can be expressed as power functions.

2) The failure of pre-fissured specimen is induced by the coalescence of cracks initiated from tips of pre-existing fissures. Four crack types are classified, i.e. tensile crack, shear crack, far-field crack and surface spalling. When the crack propagates inside the ligament religion, its growth path is mainly affected by the tips of fissures. Nevertheless, when the crack propagates outside the ligament religion, its growth path is mainly affected by the direction of major principal stress.

3) The stress drops in the stress-strain curves all correspond to the crack initiation or coalescence, which is also linked to a sudden increasing in the accumulated micro-crack curve. Shear, tensile and total micro-crack numbers in the pre-fissured specimen increase with increasing loading rate, and tensile fracture is the major mode in pre-fissured specimens.

4) PFC numerical trend of peak strength of pre- fissured specimen agrees well with the experimental result. Both of the maximum bond force and energy have the similar trend to the peak strength with the increase of loading rate, which indicates that trend of peak strength can be explained from the meso-mechanics and energy.

References

[1] ZHANG Z X, KOU S Q, YU J, YU Y, JIANG L G, LINDQVIST P A. Effects of loading rate on rock fracture [J]. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(5): 597-611.

[2] LIANG W G, ZHAO Y S, XU S G, DUSSEAULT M B. Effect of strain rate on the mechanical properties of salt rock [J]. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(1): 161-167.

[3] GONG F Q, ZHAO G F. Dynamic indirect tensile strength of sandstone under different loading rates [J]. Rock Mechanics and Rock Engineering, 2014, 47(6): 2271-2278.

[4] ZHAO G F, RUSSELL A R, ZHAO X, KHALILI N. Strain rate dependency of uniaxial tensile strength in Gosford sandstone by the distinct lattice spring model with X-ray micro CT[J]. International Journal of Solids and Structures, 2014, 51(7): 1587-1600.

[5] LIANG C, WU S, LI X, XIN P. Effects of strain rate on fracture characteristics and mesoscopic failure mechanisms of granite [J]. International Journal of Rock Mechanics and Mining Sciences, 2015, 76: 146-154.

[6] SHEN B. The mechanism of fracture coalescence in compression— Experimental study and numerical simulation [J]. Engineering Fracture Mechanics, 1995, 51(1): 73-85.

[7] WONG R H C, CHAU K T. Crack coalescence in a rock-like material containing two cracks [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(2): 147-164.

[8] BOBET A. The initiation of secondary cracks in compression [J]. Engineering Fracture Mechanics, 2000, 66(2): 187-219.

[9] GEHLE C, KUTTER H K. Breakage and shear behaviour of intermittent rock joints [J]. International Journal of Rock Mechanics and Mining Sciences, 2003, 40(5): 687-700.

[10] FUJII Y, ISHIJIMA Y. Consideration of fracture growth from an inclined slit and inclined initial fracture at the surface of rock and mortar in compression [J]. International Journal of Rock Mechanics and Mining Sciences, 2004, 41(6): 1035-1041.

[11] LI Y P, CHEN L Z, WANG Y H. Experimental research on pre-cracked marble under compression [J]. International Journal of Solids and Structures, 2005, 42(9): 2505-2516.

[12] WONG L N Y, EINSTEIN H H. Systematic evaluation of cracking behavior in specimens containing single flaws under uniaxial compression[J]. International Journal of Rock Mechanics and Mining Sciences, 2009, 46(2): 239-249.

[13] YANG S Q, DAI Y H, HAN L J, JIN Z Q. Experimental study on mechanical behavior of brittle marble samples containing different flaws under uniaxial compression [J]. Engineering Fracture Mechanics, 2009, 76(12): 1833-1845.

[14] ZHOU X P, CHENG H, FENG Y F. An experimental study of crack coalescence behaviour in rock-like materials containing multiple flaws under uniaxial compression [J]. Rock Mechanics and Rock Engineering, 2014, 47(6): 1961-1986.

[15] LEE H, JEON S. An experimental and numerical study of fracture coalescence in pre-cracked specimens under uniaxial compression [J]. International Journal of Solids and Structures, 2011, 48(6): 979-999.

[16] YANG S Q, LIU X R, JING H W. Experimental investigation on fracture coalescence behavior of red sandstone containing two unparallel fissures under uniaxial compression [J]. International Journal of Rock Mechanics and Mining Sciences, 2013, 63: 82-92.

[17] HAERI H, SHAHRIAR K, MARJI M F, MOAREFVAND P. Experimental and numerical study of crack propagation and coalescence in pre-cracked rock-like disks [J]. International Journal of Rock Mechanics and Mining Sciences, 2014, 67: 20-28.

[18] HAERI H, SHAHRIAR K, MARJI M F, MOAREFVAND P. A coupled numerical– experimental study of the breakage process of brittle substances [J]. Arabian Journal of Geosciences, 2015, 8: 809-825.

[19] ZHANG X P, WONG L N Y. Loading rate effects on cracking behavior of flaw-contained specimens under uniaxial compression [J]. International Journal of Fracture, 2013, 180(1): 93-110.

[20] CUNDALL P A, STRACK O D L. A discrete numerical model for granular assemblies [J]. Geotechnique, 1979, 29(1): 47-65.

[21] ZHANG X P, WONG L N Y. Cracking processes in rock-like material containing a single flaw under uniaxial compression: A numerical study based on parallel bonded-particle model approach [J]. Rock Mechanics and Rock Engineering, 2012, 45(5): 711-737.

[22] GHAZVINIAN A, SARFARAZI V, SCHUBERT W, BLUMEL M. A study of the failure mechanism of planar non-persistent open joints using PFC2D [J]. Rock Mechanics and Rock Engineering, 2012, 45(5): 677-693.

[23] GHAZVINIAN A, NEJATI H R, SARFARAZI V, HADEI M R. Mixed mode crack propagation in low brittle rock-like materials [J]. Arabian Journal of Geosciences, 2013, 6(11): 4435-4444.

[24] MANOUCHEHRIAN A, MARJI M F. Numerical analysis of confinement effect on crack propagation mechanism from a flaw in a pre-cracked rock under compression [J]. Acta Mechanica Sinica, 2012, 28(5): 1389-1397.

[25] MANOUCHEHRIAN A, SHARIFZADEH M, MARJI M F, GHOLAMNEJAD J. A bonded particle model for analysis of the flaw orientation effect on crack propagation mechanism in brittle materials under compression [J]. Archives of Civil and Mechanical Engineering, 2014, 14(1): 40-52.

[26] SARFARAZI V, GHAZVINIAN A, SCHUBERT W, BLUMFL M, NEJATI H R. Numerical simulation of the process of fracture of echelon rock joints [J]. Rock Mechanics and Rock engineering, 2014, 47(4): 1355-1371.

[27] YANG S Q, HUANG Y H, JING H W, LIU X R. Discrete element modeling on fracture coalescence behavior of red sandstone containing two unparallel fissures under uniaxial compression [J]. Engineering Geology, 2014, 178: 28-48.

[28] LI Shu-chen, WANG Lei, LI Shu-cai, HAN Jian-xin. Post-peak deformation and failure experimental study of rock-like specimens with different inclination angles persistent joints [J]. Chinese Journal of Rock Mechanics and Engineering,2013, 32(S2): 3391-3395. (in Chinese)

[29] ZHUANG X, CHUN J, ZHU H. A comparative study on unfilled and filled crack propagation for rock-like brittle material [J]. Theoretical and Applied Fracture Mechanics, 2014, 72: 110-120.

[30] WONG R H C, CHAU K T. Crack coalescence in a rock-like material containing two cracks [J]. International Journal of Rock Mechanics and Mining Sciences, 1998, 35(2): 147-164.

[31] SAGONG M, BOBET A. Coalescence of multiple flaws in a rock-model material in uniaxial compression [J]. International Journal of Rock Mechanics and Mining Sciences, 2002, 39(2): 229-241.

[32] WONG L N Y, EINSTEIN H H. Crack coalescence in molded gypsum and Carrara marble: Part 1. Macroscopic observations and interpretation [J]. Rock Mechanics and Rock Engineering, 2009, 42(3): 475-511.

[33] YANG S Q, JING H W. Strength failure and crack coalescence behavior of brittle sandstone samples containing a single fissure under uniaxial compression [J]. International Journal of Fracture, 2011, 168(2): 227-250.

[34] HUANG Da, CEN Duo-feng, HUANG Run-qiu. Influence of medium strain rate on sandstone with a single pre-crack under uniaxial compression using PFC simulation [J]. Rock and Soil Mechanicsm, 2013, 34(2): 535-545. (in Chinese)

[35] DIEDERICHS M S. Instability of hard rockmasses: The role of tensile damage and relaxation [M]. University of Waterloo, 2001.

[36] AL-SHAYEA N A. Crack propagation trajectories for rocks under mixed mode I–II fracture [J]. Engineering Geology, 2005, 81(1): 84-97.

[37] ZHANG Zhi-qiang, LI Ning, CHEN Fang-fang, ZHANG Ping. Review and status of research on failure mode of nonpenetrative fractured rock mass [J]. Rock and Soil Mechanics, 2009, 30(s2): 142-148. (in Chinese)

[38] Itasca Consulting Group Inc. Particle flow code [R]. Sudbury: Itasca Consulting Group Inc, 2004.

(Edited by YANG Hua)

Foundation item: Project (BK20150005) supported by the Natural Science Foundation of Jiangsu Province for Distinguished Young Scholars, China; Project (2014YC10) supported by the Fundamental Research Funds for the Central Universities, China

Received date: 2015-06-01; Accepted date: 2015-11-15

Corresponding author: YANG Sheng-qi, Professor, PhD; Tel: +86-516-83995856; E-mail: yangsqi@cumt.edu.cn