1 Introduction

The excavation unloading of underground engineering changes the initial stress equilibrium of rock mass and the stress state of surrounding rock changes from triaxial compression to unidirectional or bidirectional compression. Under high stress, the unloading of surrounding rock may suddenly release the elastic energy accumulated in the rock mass, resulting in a rock burst that is accompanied the by ejection of surrounding rock (He and Zhao 2013; Baud et al. 2014). Dynamic disasters, such as rock bursts, are abrupt, random, and harmful, threatening the lives and safety of construction personnel, and affecting the progress of construction (Zhang et al. 2020; He et al. 2015; Bruno 2023, Chen et al. 2020, 2022, 2023).

The true triaxial unloading test is one of the common methods that are typically used to reveal the mechanism and occurrence conditions of rock bursts (Li et al. 2018; Bai et al. 2019; Zhou et al. 2020, 2022). The results show that crack coalescence types and ultimate failure modes are significantly affected by unloading stress paths. The rock failure mode changes from one involving predominantly shear failure to shear-tension failure as the minimum principal stress decreases (Jiang 2017; Duan et al. 2019), and the rock strength is related to the unloading path (Xu et al. 2019; Zhang et al. 2019, 2021, 2022). Given to the above, many researches have been conducted to investigate the rock mechanical behavior under triaxial unloading conditions. Huang et al. (2001) and Li et al. (2015) conducted a series of conventional triaxial tests on granite specimens under loading and unloading conditions, and the results show that the maximum principal stress would increase, while the minimum principle would decrease. Du et al. (2015) conducted a true triaxial unloading test on granite and sandstone to investigate the impacts of stress path on the rock mechanical behavior. Li et al. (2015) carried out true triaxial unloading tests on granite, sandstone, and cement mortar, and the results show that the rock strength and failure mode are governed by the intermediate principal stress. Chen et al. (2016) and Liu et al. (2017) conducted triaxial compression tests and unloading tests on rock salt and coal samples to study the mechanical behaviors of different types of rocks under unloading conditions. Zhao et al. (2015a, b), Fang et al. (2016), and Qiu et al. (2012) investigated the energy release procedure utilizing the energy theory by true triaxial unloading test with various confining pressures. Zhang et al. (2023), Xu et al. (2023) and Wang et al. (2018) proposed a triaxial creep constitute model by a series of tests on an altered rock.

The above-mentioned researches mainly focus on the mechanical behavior of rock specimen under unloading by experimental tests. The equipment for the true triaxial test in the laboratory is expensive and stress paths for the test are limited, so it is impossible to simulate all possible complex and changeable stress conditions pertaining to a rock burst. The numerical analysis method is also an efficient method to investigate the failure procedure of rock or rock-like materials, in which the non-linear deformation characteristics of rock mass and complex and changeable stress paths can be taken into account. Scholars have simulated the deformation and failure of rock in the true triaxial unloading test by utilizing the finite element method (Tao et al. 2012; Niu et al. 2017), finite difference method (Tu and Cheng 2019; Min et al. 2020), and discrete element method (Hu et al. 2018; Chen et al. 2020). Limited by the element mesh problem, most finite element and finite difference programs cannot simulate the intersection, agglomeration, and coalescence of cracks in the rock. Some scholars have developed programs to simulate the true triaxial failure of rock, but the programs are limited in their universality and are often only applicable to some specific stress path or rock type.

Three-dimensional particle flow code (PFC3D) has been developed based on discrete element theory and its model is composed of particulate matter (Potyondy and Peter 2004). It can accurately simulate micro-mechanical characteristics, crack revolution from microcosmic to macrocosmic, and the kinematic behaviours of rock (Sun et al. 2013). Stress–strain curves, strength, and deformation parameters and macro-failure modes of pre-fissures of rock-like material samples were studied by using PFC3D. Numerical results suggest that shear damage dominates the failure process. The faster the unloading rate, the more likely brittle failure is (Li et al. 2010). The mechanical and progressive failure characteristics of pre-fissured marble samples were investigated under triaxial unloading conditions based on PFC3D (Zhang et al. 2020). Results indicate that the peak strength increases as the initial confining pressures increase or the unloading rate decreases. The mechanical and damage behaviours of anisotropic veined gneiss were explored by PFC3D (Bai and Young 2020). Results show that the unloading of σ2 and σ3 triggers microcracks in the early stages, revealing the role of unloading in rock failure. PFC3D simulation of rock burst implies that the number of cracks during a rock burst rapidly increases and the proportion of shear cracks reduces (Zhao et al. 2015a, b). The debonding of particles is mainly dominated by tensile fracture, supplemented by shear fracture (Zhu et al. 2007). The failure process of rock is more stable and less inclined to exhibit dynamic ejection at a low unloading rate. During unloading under high and low confining pressures, rock is subjected to explosive rock bursts and continuous rock bursts, respectively (Wang et al. 2014).

At present, most of the true triaxial rock burst tests focus on analyzing rock deformation and failure characteristics, and the particle ejection of rock burst are mainly reproduced by the PFC simulation. It is unclear whether rock particle ejection completely corresponds to rock burst, and it is impossible to accurately distinguish static or dynamic failure of rock in the true triaxial failure test. Moreover, the transfer mechanisms and conditions of static-dynamic failure of rock are unclear. In the present study, the mechanical behavior and failure characteristic of granite specimens under a true triaxial unloading condition are discussed, and the transfer mechanisms of static-dynamic failure of granite are proposed. The analysis is carried out in two stages. First, a true triaxial loading test on granite specimen is conducted to obtain the stress–strain curve and failure characteristics, micro-parameters calibrations are then performed by comparing the experimental results to the numerical ones to obtain the micro-parameters that can represent the granite in the numerical model. Second, a series of numerical analysis are performed to find out the critical stress combination for the static-dynamic failure of granite specimens.

2 Numerical modelling setup and calibration

2.1 Experimental settings

The granite specimens were sampled from a deep-buried tunnel project. Rockburst occurs frequently during the construction of this tunnel project, which seriously threatened the safety of personnel and engineering. According to the in-situ investigation, the regional stress field is obtained as σh1 > σv > σh2, and the maximum horizontal stress is regarded as the principal stress in this investigation field (Yan et al. 2022). Since the drilling and blasting method was adopted in the construction of this tunnel project, a sudden unloading path in one direction was selected to reproduce the tunnel excavation.

A mechanic servo-controlled loading system with a maximum axial compressive loading capacity of 1000 MPa and a maximum lateral compressive loading capacity of 200 MPa was utilized for the triaxial compressive experiments (Fig. 1). In the triaxial compression loading test, the confining pressures σ2 and σ3 are first increased to their pre-set values by the displacement control loading method. Then, keeping σ2 and σ3 constant, and increasing σ1 with a loading rate of 0.1 mm/min until the failure of the rock specimen. While in the unloading test, the confining pressures σ2 and σ3 are first increased to their pre-set values the same as in the loading test, then, keeping σ2 and σ3 constant, and increasing σ1 to 80% of the peak strength with a loading rate of 0.1 mm/min. At last, unloading σ3 as depicted in Fig. 2 with different loading rates of 0.3, 0.45, and 0.55 mm/min. The loading and unloading paths are shown in Fig. 2. The granite specimens used in the present study were polished by the fresh rocks that were collected on-site with a dimension of 50 mm × 50 mm × 100 mm (see Fig. 3a), according to the Standard of Test Method for Engineering Rock Mass (GBT50266-1999) .

Fig. 1
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Testing system for specimens under triaxial compressive experiment

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Fig. 2
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Loading and unloading paths during the experimental and numerical process

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Fig. 3
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Schematic diagram of experimental and numerical modelling

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2.2 Numerical modelling

The typically used particle flow code in three dimensions (PFC3D) is adopted to establish the numerical model of granite under the true triaxial loading and unloading test. Since the parallel-bond failure will immediately lead to the reduction of macro stiffness in that PFC code that can reflect the brittleness of granite, a parallel-bond model that could transfer force and moment is selected to simulate the true triaxial failure of granite (Potyondy and Peter 2004). A three-dimensional numerical model with the dimensions of 50 mm × 50 mm × 100 mm is generated with randomly placed particles as shown in Fig. 3, which is consistent with the experimental one. Six walls are first generated as the boundary of grain packing, and the loading and unloading tests are reproduced by moving or removing the walls as shown in Fig. 3c. Although a small particle size could produce a better simulated result, the overall number of particles would lead to an unacceptable calculating time. In the present study, the total number of particles utilized in the numerical analysis is 12,761 as shown in Fig. 3b. The effects of particle size could be ignored when the ratio of dmax to dmin is in the range of 1.5–2.0, where d is the particle diameter (Chen et al. 2022). Given to the above, the maximum particle radius dmax is selected as 3.75 mm, and the minimum particle radius dmin is selected as 2.5 mm, generating a particle radius ratio of 1.5.

2.3 Stress path

The stress paths for the granite failure tests under true triaxial loading and unloading are depicted in Fig. 2. The test schemes were supplemented through PFC3D as outlined in Table 1. In the loading procedure, stresses in all those three directions are applied by moving the “walls” towards to the “balls”. σ1, σ2 and σ3 simultaneously increase to a pre-set value to simulate the confining pressure of underground working, then σ1 gradually increases to the ultimate value when the rock specimens are failure with an increment of 0.001 MPa/step. While in the unloading procedure, σ2 and σ3 keep constant after increasing to the pre-set values the same as in the loading procedure, and then, σ1 increases to the target value σu (80% of the peak strength σf at failure under loading). By keeping σ1 and σ2 constant and deleting a wall in the direction of σ3 as shown in Fig. 3c, sudden unloading is simulated. To obtain the critical stress combination of static dynamic failure, loading method design with 66 groups of simulation schemes with PFC3D are listed in Table 2. In the numerical analysis, the confining stress σ3 is first determined as a certain value, and then give σ2 a range of variations. Keeping σ3 constant, and decrease σ2 gradually until the failure mode of the granite specimen occurs. The stress combination corresponding to this certain σ3 could be determined. Reduce the value of σ3 and repeat the previous operation of σ2, and in finial the critical stress combination of the static-dynamic failure of granite specimen could be obtained.

Table 1 True triaxial failure tests and PFC3D simulations
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Table 2 Loading method design with PFC3D
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2.4 Micro-parameter calibration

The macro-strength of rock materials is significantly affects by the micro-parameters for the particles and bonds. In the present study, the typically used “trial-and-error” method is selected to calibrate the micro-parameters of particles and bonds until the stress–strain curve, peak strength, peak strain, and failure mode obtained by PFC3D are in good accordance with the experimental results. The micro-parameters are finally decided by a series of trial-and-error calculations as summarized in Table 3.

Table 3 Micro-parameters of particles and bonds utilized in numerical analysis
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Figure 4 shows a comparison between the experimental stress–strain curve and the numerical one under the true triaxial loading failure test of granite. In the present study, the transfer mechanism and criteria for static-dynamic failure of granite under a true triaxial unloading test were obtained from a microscopic point of view by the particle flow code, and the experimental test was selected to do the micro-parameter calibration that would be utilized in the subsequent calculation. A detailed analysis of the laboratory results is provided by Liu et al. (2021). Due to the limited number of rock specimens, a large number of numerical simulations have been carried out in the present study, and the transfer mechanism and transfer condition for the static-dynamic failure of granite under true triaxial unloading test are given out. The analytical results were mainly focused on the discussion of bond energy, velocity, displacement, etc. Thus, only the peak strength, slope of the stress–strain curve, and the macrocracks at the specimen surface were compared. The peak strength, and slope of the stress–strain curve obtained from the numerical analysis agree well with those obtained from the experimental test. In the experimental test, small gaps or voids exist in the granite specimens, the stress-stain curve increases nonlinearly in the initial loading stage, while for those numerical ones, the stiffness of particles is constant, leading to stress–strain curve increasing linearly. This is why the physical experiment could not be well reproduced in the initial compaction stage. Based on the relative displacement direction and size of particles adjacent to the fracture, the crack modes were classified into three types: tensile failure, shear failure and mixed failure (Chen et al. 2020). A detailed illustration of crack modes in the numerical analysis is provided by Chen et al. (2020), and the schematic diagram of particle displacement fields for different crack modes is shown in Fig. 5. The crack distribution of granite specimen after failure is also shown in Fig. 4. In the experimental test, the shear cracks are denoted by the red line, and the tensile cracks are denoted by the blue line. A primary long shear crack penetrates from the left top to the right bottom of the rock specimen, and another short shear crack intersects with the primary long one, forming a “V” type shear failure crack. A primary tensile is initiated near the right boundary of the specimen, splitting to the top surface. The failure mode in the numerical analysis is consistent with the test one. It can be concluded that the used micro parameters are reliable to reproduce the experimental stress–strain curve, and those parameters (as in Table 3) are utilized in the subsequent numerical analysis.

Fig. 4
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Comparison of stress–strain curves in laboratory testing and numerical simulation

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Fig. 5
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Schematic diagram of particle displacement fields for different crack modes (Chen et al. 2020)

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2.5 Characteristic stress

In the loading process, the volume of granite specimen compression first, and then expansion with the increment of axial stress, leading to the volume strain first increasing, then decreasing. The total volume strain εv can be calculated as follows:

$$\varepsilon_{v} = \varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3}$$
(1)

The total volume strain εv could be divided into two parts according to the cause of volume change: one is the crack volume strain \(\varepsilon_{v}^{{\text{c}}}\) that is caused by the internal crack closure and expansion, and the other is the elastic volume strain \(\varepsilon_{v}^{e}\) that is induced by the elastic deformation of rock specimen. The relationship among the εv, \(\varepsilon_{v}^{{\text{c}}}\) and \(\varepsilon_{v}^{e}\) could be described as follows:

$$\varepsilon_{v}^{{\text{c}}} = \varepsilon_{v} - \varepsilon_{v}^{e} = \varepsilon_{v} {-}\left( {{1} - {2}v} \right)\left( {\sigma_{1} + \sigma_{2} + \sigma_{3} } \right)/E$$
(2)

The correlations among the stress–strain curve, characteristic stress and the volume stain under the triaxial loading test are depicted as shown in Fig. 6. The crack volume strain \(\varepsilon_{v}^{{\text{c}}}\) decreases at the initial loading stage due to the closure of internal small gaps and voids, and then tend to be stable since no new crack generates. The volume change of rock specimen is mainly affected by the elastic volume strain \(\varepsilon_{v}^{e}\), and the total volume strain εv increases linearly. In the crack propagation stage, a new crack initiates, and the crack volume strain \(\varepsilon_{v}^{{\text{c}}}\) increases. The corresponding stress in this stage is defined as the crack initiation stress σci (see Fig. 6). When the maximum εv is reached, the crack enters the unstable development stage, and the corresponding stress is named as crack damage stress σcd. In the post-peak stage, the peak stress σp is exceeded, and the εv keeps growing as shown in Fig. 6.

Fig. 6
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Correlations among the stress–strain curve, characteristic stress and the volume stain under the triaxial loading test

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3 Mechanical behaviour of granite specimen under true triaxial unloading

3.1 Particle displacement field

Figure 7 shows the evolution of the particle displacement field under true triaxial unloading for granite specimens with the case of σ2 = 40 MPa, σ3 = 30 MPa. With moving towards the upper and lower walls, stress is gradually applied to the granite specimen in the vertical direction. When the applied stress is smaller than the unloading point stress, particles move along the loading direction. The maximum displacement of particles occurs near the upper and lower walls, while in the central of the model, the particle displacement is close to zero (see Fig. 7b). When unloading σ3, the particles near the unloading surface move to coincide with the unloading direction. When σ1 decreases to 80% of the peak strength, several particles are ejected at the upper end of the unloading surface (see Fig. 7c). With the decrement in σ1 to 60% of peak strength, few particles are ejected along the unloading direction at the lower right corner. Particle stratification is observed due to the difference in displacement, forming a sloping cross-plane (see Fig. 7d). When σ1 decreases to 40% of peak strength, a macroscopic failure is generated due to the difference in particle displacement as shown in Figs. 7e–f.

Fig. 7
figure 7

Evolution of the particle displacement field under true triaxial unloading (σ2 = 40 MPa, σ3 = 30 MPa; Unit: m/s). Remark: σci is the crack initiation stress; σu is the unloading point stress; and σp is the peak stress, relatively

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3.2 Particle velocity field

Figure 8 shows the evolution of the particle velocity field under true triaxial unloading for granite specimen with the case of σ2 = 40 MPa, σ3 = 30 MPa. When σ1 is less than the unloading point stress σu, the particle velocity is almost the same, and the maximum value of particle velocity increases from 0.26 to 0.36 m/s when σ1 increases from σci to σu (see Figs. 8a, b). When unloading σ3, and σ1 decreases to 80% of peak strength, the direction of particle velocity near the unloading surface changes dramatically and vertically to the direction of σ1. The maximum ejection velocity of particles is about 37.80 m/s (see Fig. 8c). When σ1 decrease to 60% of peak strength, the velocity of particles on the left upper side is small, while for those on the right lower side is great, leading to an inclined staggered plane is generated along the diagonal of rock specimen as shown in Fig. 8d. The particle velocity decreases when σ1 changing from 40% σp to 10% σp as depicted in Figs. 8e, f, and the maximum particle ejection velocity is almost immune to the variation of σ1.

Fig. 8
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Evolution of the particle velocity field under true triaxial unloading (σ2 = 40 MPa, σ3 = 30 MPa; Unit: m/s)

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3.3 Evolution of microscopic energy

A series of energies can be recorded in PFC3D, taking into account the type of particles, the contact relationship,and the bonding relationship. The main energies in the present study are summarized as follows (Potyondy and Peter 2004): the Eboundary defines as the total input energy at the wall boundary, and Estrain denotes the elastic energy stored in contact. The parallel bonding strain energy is the main energy loss caused by a bond failure in the process of rock specimen failure and denotes Epbstrain. The energy dissipated by slip (Eslip) and local damping denotes (Edamping, a body-based damping scheme termed local damping is available to remove additional kinetic energy) are the main components of dissipated energy. Ekinetic denotes the total kinetic energy. Those mechanical significances are summarized in Table 4.

Table 4 Description of microscopic energy
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Figure 9 shows the microscopic energy evolution in true triaxial unloading numerical failure test granite specimen with σ2 = 40 MPa, σ3 = 30 MPa. In the loading phase, the microscopic energy increases nonlinearly with the increment in ε1, and a larger ε1 leads to a great growth rate of microscopic energy. The Eboundary is the greatest energy, which is much larger than the others. Epbstrain is greater than the Estrain, illustrating that the dissipation energy is more than the elastic energy stored in contact during a test. Compared with Eboundary, Eslip, Edamping, and Ekinetic are approximately zero, indicating that in the loading stage, energy dissipation caused by a slip, and local damping could be ignored. In the unloading phase, Eboundary continuously increases. Epbstrain slowly increases after a sudden decrease, then gradually decreases. Estrain remains unchanged at first, then begins to decline. Eslip and Edamping tend to increase and the growth rate becomes large. When Ekinetic reaches its peak, the variation of Eslip and Edamping tend to be stable, manifesting that some of the energy dissipation inducing by slip and local damping is transformed into kinetic energy of particles. Compared with the other energies, the variation of Ekinetic is rarely small. While the Ekinetic follows a similar trend to Eboundary, Epbstrain, and Estrain.

Fig. 9
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Microscopic energy evolution in true triaxial unloading numerical failure test (σ2 = 40 MPa, σ3 = 30 MPa)

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4 Judgment criteria for static–dynamic failure of granite

4.1 Judgment criteria for static–dynamic failure

Figures 10 and 11 illustrates the numerical particle displacement and velocity as tabulated in Table 1. The failure modes of granite specimens under true triaxial unloading are classified into three categories, and three typical numerical analysis cases as displayed in Fig. 12 are selected to explain the main failure characteristics. The failure modes of the granite specimen under the triaxial unloading test were divided into three types as shown in Fig. 12. (1) A few particles are ejected near the unloading-free surface and no obvious coalesced crack generates in the specimen (see Fig. 12a); (2) Several particles are ejected at the upper end of unloading surface and a main coalescence shear crack forms in the specimen (see Fig. 12b). (3) A great number of particles are ejected at the unloading surface and an obvious V-shaped pit is formed on the sample surface. A main shear crack and several secondary cracks are formed by fracture coalescence as shown in Fig. 12c.

Fig. 10
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Displacement distribution of granite specimen under different stress combinations (Unit: m)

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Fig. 11
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Velocity distribution of granite specimen under different stress combinations (Unit: m/s)

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Fig. 12
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Failure modes of granite in true triaxial unloading test simulated by PFC3D (Unit: m/s)

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The changes of Eboundary, Epbstrain, Estrain, Eslip, Edamping, and Ekinetic corresponding to the three failure modes in Fig. 12 match those of microscopic energy in Figs. 10 and 11. The static and dynamic failure cannot be distinguished according to these differences in microscopic energy, so another way must be found. When a rock burst occurs, the energy stored in rock masses could be converted into two parts: energy dissipation by particle breakage, and kinetic energy for particle ejection. In the numerical analysis, the bond between particles breaks, and energy is released instantly under unloading, leading to the particle ejection. In the present study, instantaneous kinetic energy is proposed, taking into account the kinetic energy for particle ejection. The instantaneous kinetic energy is defined as the variation of kinetic energy in a small-time interval from particle bonding breakage to particle ejection, and the total kinetic energy is the accumulation of instantaneous kinetic energy.

The failure of granite specimens for the numerical cases as listed in Table 1 from the perspective of instantaneous kinetic energy could be classified into two types: static failure and dynamic failure. Figure 13 shows the significant differences in instantaneous kinetic energy corresponding to the static failure and dynamic failure of the granite specimen. For the static failure, the instantaneous kinetic energy is almost 0 when the axial strain ε1 is less than 0.039. After a sharp increase at the unloading point, the instantaneous kinetic energy enters a quiet period, varying around 0.05 J m−3. Subsequently, the instantaneous kinetic energy fluctuates in an increment tendency to its peak value of about 0.35 J m−3 when the axial strain ε1 is about 0.0465 and then undergoes a quick drop as shown in Figs. 13a and 14a. While for those dynamic failures, the instantaneous kinetic energy varies around 0 except when the axial strain ε1 is about 0.029 with a sharp increase as depicted in Fig. 13b.

Fig. 13
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Relationship between instantaneous kinetic energy and axial strain in typical numerical cases for different failure mode

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Fig. 14
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Relationship between instantaneous kinetic energy and axial strain of samples under different stress states

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Changes in the instantaneous kinetic energy in each PFC3D simulation scheme in Table 1 are displayed in Fig. 14. Changes in the instantaneous kinetic energy are classified into two categories.

  1. 1.

    For failure mode ① in Fig. 14a, the instantaneous kinetic energy at the unloading point suddenly increases to the maximum and then rapidly decreases and stabilizes at a low value. Thereafter, the instantaneous kinetic energy only suddenly increases by a small amount, as shown in Figs. 13a and 14a. It is thus considered that static failure occurs.

  2. 2.

    In terms of failure modes ② and ③ in Figs. 14b, c, the instantaneous kinetic energy at the unloading point suddenly increases to the maximum and then rapidly reduces to a low value. After stabilizing for a certain time, the instantaneous kinetic energy gradually increases to form a dense area with high instantaneous kinetic energy, as displayed in Figs. 13b, 14b, c. Dynamic failure is thus considered to occur.

Based on changes in the instantaneous kinetic energy, the judgment criteria for static and dynamic failure were proposed. The instantaneous kinetic energy upon static failure is characterized by a sharp increase and sporadic sharp increases, while that upon dynamic failure is characterized by a sharp increase, a short quiet period, and a substantial increase.

4.2 Critical stress for static–dynamic failure

According to the static–dynamic failure characteristics in Sect. 4.1, the results used for determining static or dynamic failure of granite specimens are listed in Table 5. When σ3 ≥ 20 MPa, dynamic failure occurs; when σ3 = 10 MPa, static failure occurs. It can be concluded that the critical stress combination for static–dynamic failure of granite specimen is in the range of 10 MPa < σ3 < 20 MPa. To find out the critical stress combination, the stresses of σ2 and σ3 are firstly decreased from σ2 = σ3 = 20 MPa at an interval of 0.1 MPa, and dynamic failure occurs when σ2 = σ3 ≥ 17.3 MPa.

Table 5 Static-dynamic failure results based on numerical analysis
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By selecting the stress combination with σ3 < 17.3 MPa for simulation based on Table 5, five critical stress combinations of static–dynamic failure of granite specimen are displayed in Table 6. The critical stress Eq. (3) for the dynamic failure of granite specimen is obtained by fitting analysis as shown in Fig. 15. The critical value of σ2 for dynamic failure with various stress combination is bold in Table 6. It can be concluded that dynamic failure of granite specimen occurs when σ3 is identical and σ2 is greater than or equal to the critical value. While if σ2 is less than the critical value, static failure of the granite specimen would occur.

$$\sigma_{1} - \sigma_{2} = 109.41 + 6.84\sigma_{3} - 0.146\sigma_{3}^{2}$$
(3)
Table 6 The critical stress combinations giving rise to a static and dynamic failure of granite specimen
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Fig. 15
figure 15

The fitted relationship between the critical stress combination for the static and dynamic failure of granite

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4.3 Verification of criteria for static–dynamic failure

To verify the reasonability of critical stress combinations of static–dynamic failure of granite specimen, the true triaxial failure test results in previous research (Su et al. 2020; Si and Gong 2020; Zhu et al. 2007) are substituted into Eq. (3) and the results are listed in Table 7. The predicted results are consistent with the actual results, indicating that the proposed critical stress condition of static–dynamic failure can be used to distinguish between static and dynamic failure.

Table 7 Comparison of failure modes for the prediction results and test results
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Mogi established a rock strength criterion considering the effects of intermediate principal stress based on a large number of triaxial compression tests as follows:

$$\tau_{\rm oct} = {{f}}(\sigma_{{\rm m},2} )$$
(4)

and,

$$\tau_{\rm oct} = \frac{1}{3}\sqrt {\left( {\sigma_{1} - \sigma_{2} } \right)^{2} + \left( {\sigma_{2} - \sigma_{3} } \right)^{2} + \left( {\sigma_{3} - \sigma_{1} } \right)^{2} }$$
(5)
$$\sigma_{{\rm m},2} = \left( {\sigma_{1} + \sigma_{3} } \right)/2$$
(6)

where τoct is the shear stress of the octahedron, σm, 2 is the effective intermediate stress.

According to the Mogi-coulomb strength criterion, a fitting curve as shown in Fig. 16 is obtained based on the numerical results of critical stress in the present study, and the square of correlation coefficient R2 is about 0.9764. When the point is above the fitting curve, static failure of granite occurs, while, when the point is below the fitting curve, dynamic failure of granite would occur.

Fig. 16
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Fitting curve of critical stress based on the Mogi-coulomb criterion

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Figure 17 shows the application of the critical stress criterion to the granite true triaxial unloading test results in the literature (Su et al. 2016; Jiang 2017; He et al. 2015; Zhang 2018). The unloading test results are mainly distributed adjacent to the fitting curve obtained above, and the square of correlation coefficient R2 is about 0.9678. It is indicated that the experimental data and the theoretical prediction value determined from the Mogi-Coulomb criterion agree well with each other.

Fig. 17
figure 17

Application of the critical stress criterion to the granite true triaxial unloading test results in the literature (Su et al. 2016; Jiang 2017; He et al. 2015; Zhang 2018)

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5 Discussions: transfer mechanism of static–dynamic failure

Experimental research on rock bursts and slabbing processes of granite implied that shear-type signals increase sharply, as well as their proportion in the rock burst test. However, slabbing continues to develop, and the proportion of the shear-type signals increases gradually in the spalling test. Figure 18 shows the ratios of tensile crack number and shear crack number to total crack number in granite specimens for static and dynamic failure. The ratio of a shear crack number to total crack number gradually increases in static failure, while it suddenly increases in dynamic failure after unloading, which is consistent with the results by Su et al. (2020), Si and Gong (2020), and Zhu et al. (2007).

Fig. 18
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Ratios of tensile crack number and shear crack number to total crack number in granite specimen for static and dynamic failure

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In static failure (see Fig. 18a), tensile cracks and shear cracks separately appear at 12.6 × 104 steps and 17.1 × 104 steps. In the range of 17.1 × 104 to 23.1 × 104 steps, the ratio of the shear crack number to the total crack number (ratio_s) first increases suddenly, then decreases slowly within the range from 0.018 to 0.05. From 23.1 × 104 to 24.6 × 104 steps (corresponding to the unloading phase), the ratio of the shear crack number to the total crack number increases from 0.04 to 0.21 at an average rate of 0.113 × 104/step. The ratio of the tensile crack number to the total crack number (ratio_t) decreases from 0.96 to 0.79, indicating that the shear crack number increases faster than the tensile crack number during static failure. In dynamic failure (see Fig. 18b), the variation of crack number in the loading phase is like those in static failure. In the unloading phase (from 26.2 × 104 to 26.5 × 104 steps), the ratio_s first slightly fluctuates, and then suddenly increases from 0.03 to 0.27 within about 0.3 × 104 steps. The average increment rate of the ratio_t is about 0.801 × 104/step. While the ratio_t suddenly decreases from 0.97 to 0.73 within about 0.3 × 104 steps, suggesting that the number of shear cracks in dynamic failure suddenly increases, but tensile cracks remain dominant.

The crack distribution in static and dynamic failure of granite specimens for typical cases is shown in Fig. 19. In the static failure, 6933 tensile cracks and 1842 shear cracks are generated, and the ratio of a tensile crack number to that of shear crack number is about 3.76. In dynamic failure, the tensile crack number is 10,550, while the shear crack number is 3790, leading to the ratio of the tensile crack number to the shear crack number is about 2.78. Although the total crack number increases in the dynamic failure, the ratio of a tensile crack number to shear crack number decreases, compared with the static failure. The shear crack number increases by 1.06 times while that of tensile cracks is halved, but the shear crack number is still much smaller than that of tensile cracks.

Fig. 19
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Crack distribution during static and dynamic failure for typical cases

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Table 8 lists the ratio of a tensile crack number to that of shear crack number (ratio_ts) for granite specimens under different stress combinations. In static failure, the ratio_ts varies in the range of 3.32–3.76. While in dynamic failure, the ratio_ts varies from 2.51 to 2.91. In the transformation from static failure to dynamic failure, the numbers of tensile and shear cracks both increases, and the number of shear crack increases at a much faster rate than those of tensile crack, resulting in a decrement in ratio_ts. The proportion of shear fracture in particle bonding increases while the tendency to tensile failure weakens; however, tensile failure remains dominant in granite, accompanied by shear failure.

Table 8 Ratio of the tensile crack number to that of shear crack number for granite specimen
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6 Conclusions

  1. 1.

    The failure modes of granite under true triaxial unloading are classified into three categories: ① Few particles are ejected and no obvious crack coalescence is observed in the granite specimen. ② Some particles are ejected at the upper end near the unloading surface and crack coalescence in the sample forms a main shear crack. ③ A great number of particles are ejected near the unloading surface and a V-shaped pit is formed.

  2. 2.

    The instantaneous kinetic energy upon static failure is characterized by a sharp increase and sporadic sharp increases, while that upon dynamic failure is characterized by a sharp increase, a short quiet period, and a substantial increase. The critical stress combination for the transformation of static–dynamic failure of granite specimen is proposed, which is consistent with the actual experimental results.

  3. 3.

    The number of shear cracks gradually increases in static failure, while it suddenly increases in dynamic failure. In the transformation from static failure to dynamic failure, the ratio of a tensile crack number to that of shear crack number decreases. The proportion of shear crack in particle bonding increases while the tendency to tensile failure weakens. However, tensile failure remains dominant in granite, accompanied by shear failure.

  4. 4.

    A judgment criterion is proposed for distinguishing static or dynamic failure of granite specimens under a true triaxial unloading test. Whether the conclusions in the present study are applicable to other rock types or not is still uncertain, and further studies need to be carried out to verify the reliability of the proposed criterion on other rock types.