Evaluation of rockburst proneness considering specimen shape by storable elastic strain energy

Abstract

To systematically assess the rockburst proneness considering specimen shape, multiple groups of laboratory tests were performed on 5 rock materials in cylindrical and cuboid shapes. The linear energy storage (LES) law of both cylindrical and cuboid rock specimens under uniaxial compressive load was confirmed, and the energy storage coefficient was found to be unrelated to specimen shape. On the basis of LES law, two rockburst proneness indexes, namely the strain energy storage index (W_et) and the potential energy of elastic strain (PES), were modified. Subsequently, the W_et, PES, peak‐strength strain energy storage index ($${W}_{\text{et}}^{{\rm{p}}})$$, and peak‐strength potential energy of elastic strain (PES^p) were used to assess the rockburst proneness of the cylindrical and cuboid specimens. In addition, the fragment ejection course of specimens under test was recorded by a high‐speed camera. Then, the rockburst proneness judgments obtained from the 4 indexes were compared with the qualitative data during rock destruction. The results show that, under similar stress conditions, specimen shape has an ignorable effect on the rockburst proneness as a whole. The judgment accuracy of the two modified indexes, especially that of the PES^p, is favorably improved to evaluate the rockburst proneness of both cylindrical and cuboid specimens. However, misjudgment of $${W}_{\text{et}}^{{\rm{p}}}$$ and PES^p may still occur in the assessment of rockburst proneness as these two indexes only consider the energy state before rock peak strength and the $${W}_{\text{et}}^{{\rm{p}}}$$ is formulated in a ratio form.

INTRODUCTION

Rockburst is an impact failure phenomenon resulting from a sudden release of elastic energy accumulated within highly stressed rocks (He et al., 2012; Li et al., 2017; J. A. Wang & Park, 2001). Understanding rockburst from the perspective of elastic energy is, thus, in line with the essence of this phenomenon (Gong et al., 2022; Xie et al., 2009; Zhang et al., 2012). With mining or tunneling projects developing into greater depths, the incidence of rockburst is more frequent, which brings great challenges to the safe advance of deep rock engineering, as it can cause casualties and damage to construction equipment (Du et al., 2022; Martin & Maybee, 2000; Miao et al., 2016; Sengani, 2018; Si & Gong, 2020). Therefore, the study of rockburst problem is of pressing demand and has attracted much attention.

Rockburst proneness refers to the activity and intensity of a propensity for rockburst to occur under certain stress conditions. In terms of energy, it refers to the property of rock that accumulates strain energy and produces an impact failure (The National Standards Compilation Groups of People's Republic of China, 2010). The estimate of rockburst proneness is crucial for prediction and control of rockburst (Zhang et al., 2017). Accordingly, many indexes have been proposed to evaluate the bursting liability of rock materials by means of the strain energy method. For instance, Neyman et al. (1972) and Szecowka et al. (1973) put forward a method to estimate the bursting liability of rock-like materials. On the basis of this, several years later, the strain energy storage index (W_et) was derived for the evaluation of rockburst proneness (Kidybiński, 1981). Subsequently, the potential energy of elastic strain (PES) was introduced to evaluate the bursting proneness of rocks based on the unloading elastic modulus (Tajdus et al., 2014; Wang and Park, 2001). Notably, for a comprehensive evaluation of coal bursting liability, the Chinese standards (The National Standards Compilation Groups of People's Republic of China, 2010) defined the laboratory testing methods to determine the duration of dynamic fracture (DT), W_et (called elastic strain energy [ESE] index in the standard), bursting energy index (K_E), and uniaxial compressive strength (R_c). However, these 4 indexes still have their respective shortcomings, which inevitably affect the accuracy of coal bursting evaluation even when a comprehensive method is adopted. For instance, K_E would actually overestimate the bursting liability, because in its calculation the pre-peak dissipated energy is included, which should be excluded according to the energy roles in specimen bursting failure; W_et only reflects the energy state at a certain stage before specimen peak strength, and, thus, it cannot reflect the energy state at specimen failure. For these reasons, considering the linear energy storage (LES) law and linear energy dissipation law of uniaxially compressed rocks, Gong et al. (2018, 2019) put forward the residual ESE index (A_EF) and peak-strength strain energy storage index ($${W}_{\text{et}}^{{\rm{p}}})$$ to evaluate the bursting liability of rock materials by laboratory testing. In addition, they also modified the PES index, and proposed the peak-strength potential energy of elastic strain (PES^p) index according to the experimental results from uniaxially compressed rocks in a cylindrical shape (Gong et al., 2020). However, a careful review shows that these existing rockburst proneness indexes based on laboratory testing are all derived from cylindrical rock specimens. In fact, both cylindrical and cuboid specimens are commonly used in laboratory testing (Fairhurst & Hudson, 1999; He et al., 2015; Lunder & Pakalnis, 1997). For example, cylindrical specimens are normally used in the uniaxial and conventional triaxial tests, while cuboid specimens are usually adopted in the biaxial and triaxial tests (Fakhimi et al., 2016; Hauquin et al., 2018; Hosseini et al., 2015; Luo et al., 2022; S. Y. Wang et al., 2006). Thus, it is necessary to comparatively study the bursting proneness of rock materials considering specimen shape. Although some scholars have reported the effect of specimen shape on the mechanical parameters and failure characteristics of rocks (Du et al., 2022, 2020; Liang et al., 2016; Paterson & Wong, 2005; Xu & Cai, 2017; Zhang & Zhao, 2014; Zhao et al., 2015), the investigation into rockburst proneness considering specimen shape is extremely rare. In addition, the estimates of rockburst proneness indexes have been rarely compared with the failure state of rock specimens.

In view of this, herein both cylindrical and cuboid rock specimens were adopted to explore the bursting proneness from the perspective of strain energy storage. More than 40 cuboid and/or cylindrical rock specimens from 5 rock materials were tested for comparative analysis. First, the energy evolution properties and failure characterization of the two-shaped specimens were revealed and compared. Subsequently, according to the experimental results, the bursting proneness of the two-shaped rock specimens was comprehensively analysed in terms of W_et, $${W}_{\text{et}}^{{\rm{p}}}$$, PES, and PES^p. Finally, the judgment results obtained from these 4 indexes were compared according to the failure information of rock specimens.

MATERIALS AND METHODS

Specimen processing and experimental apparatus

To ensure the universality of the experimental results, 5 rock materials (red sandstone, limestone, marble, green sandstone, and granite, which were collected from Shandong, Hubei, Guangxi, Sichuan, and Hunan provinces of China, respectively) were used for the laboratory tests. These 5 rock materials include 3 main rock types, namely, the stratiform rock, the metamorphic rock, and the igneous rock. Each rock material was prepared into cylindrical and cuboid geometries with a height-to-diameter/width ratio of 2.0 by coring from an identical rock block (the height is about 100 mm, and the diameter or width of specimens is approximately 50 mm). The error tolerance of prepared rock specimens meets the experimental requirements (ASTM 1986; Fairhurst & Hudson, 1999; ISRM, 1979). For comparison, about 16 specimens were prepared for each rock material and divided into the cylindrical group and the cuboid group.

The uniaxial compression experiments were performed on the INSTRON 1346 testing system (Figure 1), whose axial force capacity was 2000 kN (Luo & Gong, 2020a). The deformation measurement of rock specimens was realized by installing an extensometer along the loading direction so that the stress and strain signals of a tested specimen could be caught in real time. To avoid wear on the indenter of the testing machine and accurately measure the axial specimen deformation, rigid spacers were placed at both specimen ends which were lubricated evenly. Moreover, a high-speed (HS) camera system was also adopted to photograph the ejection progress of specimen fragments during the experiment, and a frame could be captured every 0.08 s during rock destruction. Detailed components of the experimental system are shown in Figure 1.

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Test scheme

For each specimen group (the cylindrical or cuboid group) of rock material, the uniaxial compression test was performed in two manners according to the desired stress path. First, one rock specimen was tested in a monotonic uniaxial compression (MUC) experiment at a force-controlled rate of 1.0 MPa/s to obtain its uniaxial compressive strength (σ_c). The single cyclic loading–unloading uniaxial compression (SCLUC) experiment was conducted on the remaining rock specimens after the MUC experiment. On the basis of σ_c, the stress value corresponding to the unloading point (Figure 2b) was preset as σ_k = kσ_c (k = 0.1, 0.3, 0.5, 0.7, 0.9, as shown in Figure 2a). To facilitate the presetting of σ_k, the applied stress of rock specimens was increased at a rate of 1.0 MPa/s until reaching σ_k, and then it was nearly released to zero at the same rate, completing a loading–unloading cycle. Subsequently, the rock specimen was compressed using the displacement loading mode of 0.065 mm/min until global disintegration or residual strength occurred. The purpose of the control model conversion in the SCLUC experiment was to avoid a sudden impact of the test machine which may influence the test results. It should be noted that the same loading method was used in both the cylindrical and cuboid groups to ensure similar loading conditions. For any given specimen group, an additional separate specimen was tested at each stress level in the SCLUC tests. A total of 10 specimen groups were tested in this study. The detailed scheme of carrying out a group of SCLUC tests can refer to Gong et al. (2018).

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Strain energy principle of rock under uniaxial compression

As shown in Figure 2, considering the elastic energy state at each loading stage before rock peak strength, 5 stress levels (or unloading points) were set to carry out the SCLUC experiments. On the basis of stress and strain signals derived in a SCLUC experiment, with reference to the law of energy conservation, the pre-peak ESE and the pre-peak input strain energy (ISE) at a given stress level can be determined by the following equations (Xie et al., 2005): 1$$\left.\begin{array}{c}{U}_{{\rm{e}}}^{k}={\int }_{{\varepsilon }_{0}}^{{\varepsilon }_{1}}\sigma {\rm{d}}\varepsilon \\ {U}_{{\rm{I}}}^{k}={\int }_{0}^{{\varepsilon }_{1}}\sigma {\rm{d}}\varepsilon \\ {U}_{{\rm{d}}}^{k}={U}_{{\rm{I}}}^{k}-{U}_{{\rm{e}}}^{k}\end{array}\right\},$$where $${U}_{{\rm{e}}}^{k}$$, $${U}_{{\rm{I}}}^{k}$$, and $${U}_{{\rm{d}}}^{k}$$ denote the pre-peak ESE, pre-peak ISE, and pre-peak dissipated strain energy (DSE) at a given stress level k, respectively; $${\varepsilon }_{0}$$, $${\varepsilon }_{1}$$, and $${\varepsilon }_{{\rm{p}}}$$ refer to the permanent strain, the strain at an unloading point, and the total strain at a peak stress point, respectively (Figure 2b).

EXPERIMENTAL RESULTS AND ANALYSIS

Stress–strain relations

The stress–strain relations of the 5 tested rock materials differ due to the change in specimen geometry (or shape). As can be seen in Figure 3a, the stress–strain relations of various rock materials obtained from the MUC tests are different. However, the variation trends in the stress and strain of both rock cylinders and cuboids are similar. In other words, these stress–strain curves roughly include 4 segments: the initial nonlinear segment, the elastic linear segment, the nonlinear plastic segment, and the post-peak failure segment. The σ_c of 5 rock materials, whether the cylinder or cuboid, conforms to the following ranking: granite > limestone > red sandstone > green sandstone > marble. From the stress–strain relations of both cylindrical and cuboid specimens, it can be inferred that the marble shows apparent plasticity, as its post-peak curve can be completely measured. However, the other 4 rock materials display the characteristics of brittle rocks, whose post-peak curves drop suddenly. The drop of post-peak curves of cuboid rocks becomes more moderate compared with that of rock cylinders. Figure 3b displays the typical stress–strain relations in the SCLUC experiment. It is also noted that a single loading–unloading procedure in uniaxial compression conditions exerts little effect on the stress–strain curves.

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Failure processes of cylindrical and cuboid rock specimens

Figure 4 shows the representative failure process of the 5 rock materials with cylindrical and cuboid specimen geometries. From these captured images, it can be identified that specimens of different rock materials possess distinctive impact potentials. Among the 5 rock materials, granite seems to exhibit the most severe ejection during specimen fracturing. This can be evidenced by two aspects. (I) Specimen fragments from the granite are ejected quickly accompanied by a loud sound. (II) The duration of fragment ejection is short. As shown in Figure 4e, the fragment ejection of the cylindrical granite only lasts for 2.040 s. In contrast, the spalling duration of the marble specimens is longer. It is found that the fracturing duration of cuboid marble lasts for 92.552 s (Figure 4c), and only a small amount of rock particles are stripped from the marble matrix due to the influence of gravity, showing the obvious characteristics of no impact potential.

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In addition, it is noted that even for an identical material type, the rockburst proneness seems to be slightly different due to the change of specimen geometry. Figure 4b records the ejection process of limestone specimens in uniaxial compression. From the observations of limestone failure, the cylindrical specimens seem to be subject to a stronger impact potential than the cuboid ones. There are two evidence for the failure observation. For one thing, the amount of residual matrix of cylindrical limestone seems to be less than that of the cuboid specimens. For the other, it is also noted that the failure duration of cuboid limestone (1.360 s) is nearly three times longer than that of cylindrical limestone (0.480 s). However, the failure duration of cylindrical red sandstone specimen is very close to that of the cuboid one (Figure 4a). Generally, a shorter failure duration may characterize a stronger bursting intensity. The above analysis indicates that the bursting proneness of rock materials can be estimated according to their ejection performance. Specimen destruction intensity could be directly characterized by the ejection of rock fragments recorded via HS camera images. It is observed from Figure 4 that the ejection process of different rock materials in the two shapes is generally consistent. Overall, specimen shape does exert a certain impact on the destruction process of specimens of the same rock materials. However, the impact on the destruction intensity of rocks can be ignored to some extent.

LES law of cylindrical and cuboid specimens

With the aid of stress-strain relations, the pre-peak ISE and pre-peak ESE at various stress levels are calculated via Equation (1). With an extremely high coefficient of determination R², the mutual relation between the pre-peak ESE and pre-peak ISE at varied stress levels was determined, which was termed the LES law (Figure 5). It is found that for rock materials in both cylindrical and cuboid shapes, with increasing pre-peak ISE, the increasing rate of pre-peak ESE satisfies a high-to-low sequence as follows: limestone > granite > red sandstone > green sandstone > marble. These linear energy correlations of all tested rock materials can be uniformly expressed as 2$${U}_{{\rm{e}}}^{k}=a{U}_{{\rm{I}}}^{k}+b,$$where a represents the uniaxial compression energy storage coefficient (UCESC), and b denotes a fitting parameter caused by experimental data dispersion between separate rock specimens.

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To accurately predict the peak ESE $${U}_{{\rm{e}}}^{{\rm{p}}}$$ and peak DSE $${U}_{{\rm{d}}}^{{\rm{p}}}$$ of rocks is crucial for the evaluation of rockburst proneness. Herein, it is realized by means of the linear correlations between the two pre-peak strain energies. The $${U}_{{\rm{e}}}^{{\rm{p}}}$$ and $${U}_{{\rm{d}}}^{{\rm{p}}}$$ of both cylindrical and cuboid rocks are predicted by using 3$$\left.\begin{array}{c}{U}_{{\rm{e}}}^{{\rm{p}}}=a{U}_{{\rm{I}}}^{{\rm{p}}}+b\\ {U}_{{\rm{d}}}^{{\rm{p}}}={U}_{{\rm{I}}}^{{\rm{p}}}-{U}_{{\rm{e}}}^{{\rm{p}}}\end{array}\right\},$$where $${U}_{{\rm{I}}}^{{\rm{p}}}$$ can be calculated by integrating the entire pre-peak strain–strain relations. Thus, Equation (3) can be converted into 4$$\left.\begin{array}{c}{U}_{{\rm{e}}}^{{\rm{p}}}=a{\int }_{0}^{{\varepsilon }_{{\rm{p}}}}\sigma {\rm{d}}\varepsilon +b\\ {U}_{{\rm{d}}}^{{\rm{p}}}=(1-a){\int }_{0}^{{\varepsilon }_{{\rm{p}}}}\sigma {\rm{d}}\varepsilon -b\end{array}\right\}.$$

In Equation (4), $${\varepsilon }_{{\rm{p}}}$$ refers to the peak strain of rock specimens (Figure 2a). It should be pointed out that in the SCLUC test, envelopes of the entire pre-peak strain–strain curves were used for the determination of $${U}_{{\rm{I}}}^{{\rm{p}}}$$ (Luo & Gong, 2020b).

The related coefficients of determination R² are summarized in Table 1. It is noted that the linear relation between the pre-peak ISE and pre-peak ESE occurs in the specimens of both cylindrical and cuboid shapes. Especially, the UCESC (a) of cylindrical specimens is consistent with that of the cuboid specimens and follows a descending order: limestone > granite > red sandstone > green sandstone > marble. As a result, it is inferred that correspondingly the energy storage ability of the 5 rock materials should obey the following order: limestone > granite > red sandstone > green sandstone > marble, as it has been documented that the energy storage coefficient can reflect the energy storage ability of rock. For an identical rock material, the UCESCs of the cylindrical and cuboid specimens are nearly identical (Table 1), which indicates a very low sensitivity of a to the geometry of rock specimens. Thus, it is reasonable to consider that the LES law can eliminate the shape effect of rock specimens.

Table 1 Linear regression functions of the pre-peak ESE and pre-peak ISE relations

Specimen geometry	Material type	Linear functions	R2
Cylinder	Red sandstone	$${U}_{{\rm{e}}}^{k}=0.7512{U}_{{\rm{I}}}^{k}+2.0487$$	0.9990
Limestone	$${U}_{{\rm{e}}}^{k}=0.8650{U}_{{\rm{I}}}^{k}-0.2236$$	0.9985
Marble	$${U}_{{\rm{e}}}^{k}=0.5657{U}_{{\rm{I}}}^{k}+1.5331$$	0.9698
Green sandstone	$${U}_{{\rm{e}}}^{k}=0.7068{U}_{{\rm{I}}}^{k}+0.8557$$	0.9964
Granite	$${U}_{{\rm{e}}}^{k}=0.8053{U}_{{\rm{I}}}^{k}+0.0686$$	0.9993
Cuboid	Red sandstone	$${U}_{{\rm{e}}}^{k}=0.7742{U}_{{\rm{I}}}^{k}-0.0126$$	0.9993
Limestone	$${U}_{{\rm{e}}}^{k}=0.8502{U}_{{\rm{I}}}^{k}-0.0584$$	0.9993
Marble	$${U}_{{\rm{e}}}^{k}=0.5532{U}_{{\rm{I}}}^{k}+1.1111$$	0.9795
Green sandstone	$${U}_{{\rm{e}}}^{k}=0.6919{U}_{{\rm{I}}}^{k}-0.6793$$	0.9990
Granite	$${U}_{{\rm{e}}}^{k}=0.8106{U}_{{\rm{I}}}^{k}+0.7273$$	0.9993

EVALUATION OF ROCKBURST PRONENESS CONSIDERING SPECIMEN SHAPE

Modification of rockburst proneness indexes using LES law

Given that the ESE stored in rocks is the driven power for rockburst, the W_et and PES are two indexes commonly used to evaluate the rockburst proneness, as they are simple in expression and convenient for calculation. The W_et and PES are determined by Equations (5) and (6), respectively (Gong et al., 2020; Kidybiński, 1981): 5$$\left.\begin{array}{c}{W}_{\mathrm{et}}=\displaystyle \tfrac{{U}_{{\rm{e}}}^{\mathrm{et}}}{{U}_{{\rm{d}}}^{\mathrm{et}}}\\ \text{None}\,\text{bursting}\,\text{proneness}:\,{W}_{\mathrm{et}}\lt 2.0\\ \text{Low}\,\text{bursting}\,\text{proneness}:\,{W}_{\mathrm{et}}=2.0\mbox{--}4.99\\ \text{High}\,\text{bursting}\,\text{proneness}:\,{W}_{\mathrm{et}}\gt 5.0\end{array}\right\},$$ 6$$\left.\begin{array}{c}\mathrm{PES}=\displaystyle \tfrac{{\sigma }_{{\rm{c}}}^{2}}{2{E}_{{\rm{u}}}}\\ \text{Very}\,\text{low}\,\text{bursting}\,\text{proneness}:\,\text{PES}\le 50\,\mathrm{kJ}/{{\rm{m}}}^{3}\\ \text{Low}\,\text{bursting}\,\text{proneness}:\,\text{PES}=50\mbox{--}100\,\mathrm{kJ}/{{\rm{m}}}^{3}\\ \text{Medium}\,\text{bursting}\,\text{proneness}:\,\text{PES}=100\mbox{--}150\,\mathrm{kJ}/{{\rm{m}}}^{3}\\ \text{High}\,\text{bursting}\,\text{proneness}:\,\text{PES}=150\mbox{--}200\,\mathrm{kJ}/{{\rm{m}}}^{3}\\ \text{Very\unicode{x000A0}high\unicode{x000A0}bursting\unicode{x000A0}proneness}:\,\text{PES}\gt 200\,\mathrm{kJ}/{{\rm{m}}}^{3}\end{array}\right\},$$where $${U}_{{\rm{e}}}^{\text{et}}$$ and $${U}_{{\rm{d}}}^{\text{et}}$$ are the pre-peak ESE and pre-peak DSE at a stress level ranging from 0.7 to 0.9 (Gong et al., 2022); $${\sigma }_{{\rm{c}}}$$ and $${E}_{{\rm{u}}}$$ represent the peak strength and unloading elastic modulus of rock specimens under uniaxial compressive load, respectively. According to the definition of W_et, it is determined as a ratio of ESE to DSE before rock peak strength. In fact, generally, rockburst occurs provided that the stress reaches its peak. Thus, the ESE to DSE ratio at peak strength, namely the ratio of peak ESE to peak DSE, seems more rational for rockburst proneness evaluation (Gong et al., 2019).

In addition, the PES was determined based on an assumption that the unloading curve at peak strength is a straight line (Figure 6). However, the assumed unloading curve is inconsistent with the actual one, because the elastoplasticity of rocks determines that the unloading curve is nonlinear. The above analysis suggests that these two indexes still have flaws in their determinations. Herein, by using the LES law of both rock cylinders and cuboids, the peak ESE and peak DSE can be predicted. On this basis, the W_et and PES are modified into the $${W}_{\text{et}}^{{\rm{p}}}$$ and PES^p respectively, as expressed by the following equations (Gong et al., 2019, 2020): 7$$\left.\begin{array}{c}{W}_{\text{et}}^{{\rm{p}}}=\displaystyle \tfrac{{U}_{{\rm{e}}}^{{\rm{p}}}}{{U}_{{\rm{d}}}^{{\rm{p}}}}\\ \text{None}\,\text{bursting}\,\text{proneness}:\,{W}_{\text{et}}^{{\rm{p}}}\lt 2.0\\ \text{Low}\,\text{bursting}\,\text{proneness}:\,{W}_{\text{et}}^{{\rm{p}}}=2.0\mbox{--}5.0\\ \text{High}\,\text{bursting}\,\text{proneness}:\,{W}_{\text{et}}^{{\rm{p}}}\gt 5.0\end{array}\right\},$$ 8$$\left.\begin{array}{c}{\text{PES}}^{{\rm{p}}}={U}_{{\rm{e}}}^{{\rm{p}}}=a{U}_{{\rm{I}}}^{{\rm{p}}}+b\\ \text{None}\,\text{bursting}\,\text{proneness}:\,{\text{PES}}^{{\rm{p}}}\le 100\,\text{kJ}/{{\rm{m}}}^{3}\\ \text{Low}\,\text{bursting}\,\text{proneness}:\,100\lt {\text{PES}}^{{\rm{p}}}\le 200\,\text{kJ}/{{\rm{m}}}^{3}\\ \text{Medium}\,\text{bursting}\,\text{proneness}:\,200\lt {\text{PES}}^{{\rm{p}}}\le 300\,\text{kJ}/{{\rm{m}}}^{3}\\ \text{High}\,\text{bursting}\,\text{proneness}:\,{\text{PES}}^{{\rm{p}}}\gt 300\,\text{kJ}/{{\rm{m}}}^{3}\end{array}\right\}.\,$$

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Estimate of rockburst proneness of cylindrical and cuboid specimens

According to Equations (5)–(8), the W_et, $${W}_{\text{et}}^{{\rm{p}}}$$, PES, and PES^p of rock specimens were determined and summarized in Table 2. Additionally, the fragment ejection speed, the fractured sound of rock specimens, and the amount of ejected fragments can directly reflect the failure intensity of rock specimens. During an experiment, the louder the fractured sound, the faster the ejection speed of the fragment, and the greater amount of ejected fragments, the more violent failure the specimen undergoes, and the higher the rockburst proneness is. Therefore, for a check of the applicability of the four rockburst proneness indexes, the data mentioned above was also collected as a unified standard for actual rockburst proneness, as listed in Table 2. The bursting proneness of rock materials is classified into four grades, whose detailed characteristics are as follows:

Table 2 Failure information of cylindrical and cuboid specimens and calculations of bursting proneness indexes

No.	Ejection speed	Fracture sound	Fragment features	Rockburst proneness	$${W}_{\text{et}}$$	$${W}_{\text{et}}^{{\rm{p}}}$$	PES (kJ/m3)	PESp (kJ/m3)
C-RS-1	Fast	Loud	Many, majority on the loading plate	Medium	–	3.13	204.85	242.25
C-RS-2	Fast	Loud	Many, majority on the loading plate	Medium	–	3.12	210.98	252.57
C-RS-3	Fast	Loud	Many, majority on the platform	High	–	3.12	205.64	265.81
C-RS-4	Fast	Loud	Many, majority on the platform	High	–	3.13	187.93	234.18
C-RS-5	Fast	Loud	Many, majority on the platform	High	3.65	3.11	208.92	268.57
C-RS-6	Fast	Loud	Many, majority on the platform	High	2.90	3.12	197.03	244.91
C-RS-7	Fast	Loud	Many, majority on the platform	High	3.49	3.12	217.11	265.57
C-RS-8	Fast	Loud	Many, majority on the platform	High	–	3.12	222.42	254.29
C-RS-9	Fast	Loud	Many, majority on the loading plate	Medium	–	3.12	224.97	265.76
B-RS-1	Fast	Loud	Many, majority on the platform	High	–	3.43	265.36	292.64
B-RS-2	Fast	Loud	Many, majority on the platform	High	–	3.43	273.11	318.89
B-RS-3	Fast	Loud	Many, majority on the platform	High	–	3.43	261.74	305.34
B-RS-4	Fast	Loud	Many, majority on the loading plate	Medium	–	3.43	239.81	270.10
B-RS-5	Fast	Loud	Many, majority on the platform	High	3.84	3.43	258.30	289.09
B-RS-6	Fast	Loud	Many, majority on the loading plate	Medium	3.32	3.43	239.55	268.81
B-RS-7	Fast	Loud	Many, majority on the loading plate	Medium	–	3.43	253.69	279.95
B-RS-8	Fast	Loud	Many, majority on the platform	High	3.74	3.43	250.40	287.98
C-L-1	Fast	Loud	Many, majority on the loading plate	Medium	–	6.35	181.73	189.29
C-L-2	Fast	Loud	Many, majority on the platform	High	–	6.36	216.16	233.06
C-L-3	Fast	Loud	Many, majority on the platform	High	6.26	6.36	238.90	242.67
C-L-4	Fast	Loud	Many, majority on the loading plate	Medium	–	6.35	139.86	172.17
C-L-5	Fast	Loud	Many, majority on the loading plate	Medium	–	6.35	175.78	187.68
C-L-6	Fast	Loud	Many, majority on the platform	High	–	6.38	387.79	392.57
C-L-7	Slow	Slight	A few, majority on the loading plate	Low	–	6.33	117.22	139.59
C-L-8	Fast	Loud	Many, majority on the loading plate	Medium	–	6.35	188.34	192.09
C-L-9	Fast	Loud	Many, majority on the platform	High	6.94	6.37	270.04	289.30
B-L-1	Fast	Loud	Many, majority on the platform	High	–	5.67	190.37	254.56
B-L-2	Slow	Slight	A few, majority on the loading plate	Low	–	5.66	132.30	161.23
B-L-3	Slow	Slight	A few, majority on the loading plate	Low	–	5.67	194.47	220.87
B-L-4	Fast	Loud	Many, majority on the platform	High	–	5.67	231.44	238.16
B-L-5	Slow	Slight	A few, majority on the loading plate	Low	4.93	5.66	118.07	155.77
B-L-6	Fast	Loud	Many, majority on the loading plate	Medium	5.31	5.67	187.37	229.67
B-L-7	Fast	Loud	Many, majority on the loading plate	Medium	6.24	5.67	212.22	226.74
C-M-1	No	No	No	No	–	1.30	40.51	68.16
C-M-2	No	No	No	No	–	1.30	49.63	50.49
C-M-3	No	No	No	No	–	1.30	37.85	58.99
C-M-4	No	No	No	No	–	1.30	36.70	68.90
C-M-5	No	No	No	No	–	1.30	41.43	61.22
C-M-6	No	No	No	No	1.71	1.30	37.70	61.57
C-M-7	No	No	No	No	–	1.30	37.70	70.83
C-M-8	No	No	No	No	2.29	1.30	38.06	56.82
C-M-9	No	No	No	No	1.42	1.30	37.13	58.95
B-M-1	No	No	No	No	–	1.24	30.59	52.97
B-M-2	No	No	No	No	–	1.24	38.37	53.55
B-M-3	No	No	No	No	–	1.24	41.21	48.49
B-M-4	No	No	No	No	–	1.24	38.43	45.94
B-M-5	No	No	No	No	2.36	1.24	33.83	41.81
B-M-6	No	No	No	No	–	1.24	33.90	46.38
B-M-7	No	No	No	No	1.60	1.24	41.77	46.48
B-M-8	No	No	No	No	1.23	1.24	35.64	46.48
C-GS-1	Slow	Slight	A few, majority on the loading plate	Low	–	2.48	134.03	154.82
C-GS-2	Slow	Slight	A few, majority on the loading plate	Low	–	2.48	126.44	152.57
C-GS-3	Slow	Slight	A few, majority on the loading plate	Low	–	2.48	139.71	155.82
C-GS-4	Slow	Slight	A few, majority on the loading plate	Low	–	2.48	137.30	157.91
C-GS-5	Slow	Slight	A few, majority on the loading plate	Low	2.86	2.48	132.28	151.08
C-GS-6	Slow	Slight	A few, majority on the loading plate	Low	2.73	2.48	129.04	144.34
C-GS-7	Slow	Slight	A few, majority on the loading plate	Low	–	2.48	129.22	165.14
C-GS-8	Slow	Slight	A few, majority on the loading plate	Low	2.79	2.48	133.23	148.88
C-GS-9	Fast	Loud	Many, majority on the loading plate	Medium	–	2.48	132.65	165.83
B-GS-1	Slow	Slight	A few, majority on the loading plate	Low	–	2.25	144.90	173.37
B-GS-2	Slow	Slight	A few, majority on the loading plate	Low	–	2.25	138.52	172.44
B-GS-3	Slow	Slight	A few, majority on the loading plate	Low	–	2.25	136.23	173.51
B-GS-4	Slow	Slight	A few, majority on the loading plate	Low	–	2.25	143.41	177.43
B-GS-5	Slow	Slight	A few, majority on the loading plate	Low	2.39	2.25	145.24	172.46
B-GS-6	Slow	Slight	A few, majority on the loading plate	Low	2.23	2.25	143.77	180.56
B-GS-7	Slow	Slight	A few, majority on the loading plate	Low	2.38	2.25	150.95	178.23
B-GS-8	Slow	Slight	A few, majority on the loading plate	Low	–	2.25	138.02	181.90
C-G-1	Fast	Loud	Many, the majority on the platform	High	–	4.14	300.32	356.66
C-G-2	Fast	Loud	Many, the majority on the platform	High	–	4.14	293.69	341.31
C-G-3	Fast	Loud	Many, majority on the platform	High	–	4.14	288.84	333.23
C-G-4	Fast	Loud	Many, majority on the platform	High	–	4.14	288.78	332.70
C-G-5	Fast	Loud	Many, majority on the platform	High	4.40	4.14	313.82	356.46
C-G-6	Fast	Loud	Many, majority on the platform	High	3.77	4.14	293.05	348.47
C-G-7	Fast	Loud	Many, majority on the platform	High	4.52	4.14	291.72	339.69
C-G-8	Fast	Loud	Many, majority on the platform	High	–	4.14	313.34	379.79
B-G-1	Fast	Loud	Many, majority on the platform	High	–	4.30	298.72	357.37
B-G-2	Fast	Loud	Many, majority on the platform	High	–	4.30	294.42	353.33
B-G-3	Fast	Loud	Many, majority on the platform	High	–	4.30	285.36	330.99
B-G-4	Fast	Loud	Many, majority on the platform	High	–	4.30	231.60	283.11
B-G-5	Fast	Loud	Many, majority on the platform	High	–	4.30	299.63	335.42
B-G-6	Fast	Loud	Many, majority on the platform	High	4.50	4.30	301.49	379.32
B-G-7	Fast	Loud	Many, the majority on the platform	High	3.99	4.30	307.85	349.73
B-G-8	Fast	Loud	Many, the majority on the platform	High	–	4.30	295.51	318.92

No bursting proneness: In this case, the rock specimen after testing nearly remains intact on the loading plate (Figure 7e,f). No ejection phenomenon of rock fragments or particles and fracture sound can be observed during the failure process of rock specimens (Figure 4c). Rock particles only fall down from the specimen matrix under the action of gravity.

[IMAGE OMITTED. SEE PDF]

Low bursting proneness: Gentle failure of the rock specimens is observed. There is a light cracking sound during rock failure and several pieces of fragments are produced. The specimen surface is cracked and peeled off, and the ejection occurs sporadically at a low velocity. For example, during the failure of cylindrical and cuboid green sandstone specimens, the specimen fragments were ejected at a low velocity (Figure 4d). After the test, the remaining specimen matrix was relatively intact and most of the specimen fragments were distributed on the loading plate of the test machine (Figure 7g,h).

Medium bursting proneness: During the destruction progress, rock specimens are fractured with a loud sound. As can be observed in Figure 4a, many fragments of rock specimens were ejected rapidly. However, after the test, most of the rock fragments were distributed in the loading plate of the test machine, as displayed in Figure 7a.

High bursting proneness: A large number of rock fragments were ejected at an HS with a loud fracture sound during specimen failure, which can be observed in HS camera images (Figure 4e). After the test, the majority of the rock fragments were distributed on the platform of the test machine, as shown in Figure 7b,i,j.

Correlation between actual rockburst proneness and estimates from rockburst indexes

The correlation between the estimates obtained from the four rockburst proneness indexes and the actual rockburst proneness is demonstrated in Figure 8. The green, cyan, orange, and red areas in Figure 8 represent the cross areas between the actual bursting proneness and the grading standards of rockburst proneness indexes, which are corresponding to no, low, medium, and high rockburst proneness, respectively. From Figure 8a,b, it can be seen that for the red sandstone, limestone, green sandstone, and granite, estimates of W_et and $${W}_{\text{et}}^{{\rm{p}}}$$ are very similar. For the marble under test, the failure characteristics apparently indicate that the marble in both cylindrical and cuboid shapes is prone to no rockburst. However, misjudgment of W_et on marble specimens sometimes occurs. The above results show that the $${W}_{\text{et}}^{{\rm{p}}}$$ is more accurate than the W_et. Furthermore, it is noticed that the judgment of $${W}_{\text{et}}^{{\rm{p}}}$$ in some cases is still inconsistent with the actual specimen failure phenomenon. This is because the $${W}_{\text{et}}^{{\rm{p}}}$$ is formulated in a ratio form, and only considers the energy state during pre-peak loading of rock specimens. A ratio actually cannot directly reflect the absolute ESE required to trigger a rockburst.

[IMAGE OMITTED. SEE PDF]

From Figure 8c,d, it is observed that the judgment accuracy of PES is low, and most of the data points are distributed out of the cross areas. In contrast, based on an accurate calculation of the peak ESE by using the LES law, judgment accuracy has been significantly improved. As can be seen in Figure 8d, most of the data points are distributed in the cross areas. However, an inaccurate judgment of PES^p still occurs in some cases. This is because the PES^p index only considers the ESE stored at the pre-peak loading stage of rock specimens, although it accurately characterizes the absolute ESE at peak specimen strength.

Figure 9 shows the mean values of the four rockburst proneness indexes for different types of rocks in both cylindrical and cuboid specimen shapes. For the rocks with poor heterogeneity which displays the obvious difference between the mechanical characteristics of rock specimens (e.g., the tested limestone), it will be more accurate to evaluate the rockburst tendency with PES^p. According to the judgment results of the 4 indexes, the bursting proneness of cylindrical red sandstone is slightly lower than that of the cuboid red sandstone; for limestone and marble, the bursting proneness of cylindrical specimens is somewhat higher than that of the cuboid ones. In summary, the specimen shape can have an influence on the rockburst proneness of rock materials. However, this influence can be ignored, as the judgment results for both rock cylinders and cuboids are at the same bursting proneness level in most cases.

[IMAGE OMITTED. SEE PDF]

DISCUSSION

In the present study, the LES law of both cylindrical and cuboid specimens under uniaxial compression was confirmed. Interestingly, the specimen shape is found to almost have no effect on the LES law. As is proved, the UCESCs of the cylindrical and cuboid specimens of the same rock material are highly similar. Accordingly, it can be concluded that the LES law can eliminate the shape effect of rock specimens. By using the LES law, the peak ESE of both cylindrical and cuboid specimens could be predicted precisely. On this basis, two commonly involved rockburst proneness indexes, namely the W_et and PES, were carefully modified into the $${W}_{\text{et}}^{{\rm{p}}}$$ and PES^p, respectively. In addition, the actual rockburst proneness of the two shaped rock specimens was estimated according to the failure information during testing. The estimates of the four rockburst proneness indexes and the actual bursting proneness were mutually checked. The results indicate that the judgment accuracy of the two modified indexes, especially that of the PES^p, has been significantly improved. However, misjudgments still occurred in the estimates of the $${W}_{\text{et}}^{{\rm{p}}}$$ and PES^p. The reasons for the misjudgments are as follows: (I) These two indexes only consider the pre-peak compressive stage of a rock sample but fail to cover the whole loading process of the rock specimen. The destruction of uniaxially compressed rocks releases ESE and is closely relevant to both the pre-peak loading stage and the post-peak failure stage of rock specimens. Comparatively, the post-peak behaviors are equally and even more important to reflect the bursting liability. Thus, simply examining the energy state before or at the peak strength point cannot fully reflect the whole process of energy evolution and the degree of energy release after impact failure of rock specimens. (II) In contrast to the PES^p, the $${W}_{\text{et}}^{{\rm{p}}}$$ is in the form of a dimensionless ratio, which cannot quantitatively describe the energy released during a bursting of rock specimens. Similarly, the duration of the DT index only focuses on the post-peak failure behaviors of the sample, whereas the bursting energy index considers both the pre-peak and post-peak failure of the sample according to their definitions (The National Standards Compilation Groups of People's Republic of China, 2010). However, the burst energy index is formulated in a ratio form and also includes the DSE at the sample peak point, thus leading to the overestimated results (Gong et al., 2018). Therefore, according to the analysis above, the related rockburst proneness indexes are expected to be further improved by means of overcoming these shortcomings.

The uniformity characteristics of rock materials are crucial in that they can influence the measurement accuracy of rockburst proneness. In this study, both rocks with favorable (e.g., red sandstone) and poor (limestone) uniformity were tested, respectively. As is found, the performance of the rockburst proneness index is different when the uniformity characteristics of different rock types are considered. The $${W}_{\text{et}}^{{\rm{p}}}$$ is stable and the PES^p can vary greatly for rock specimens even from the same rock type. Therefore, $${W}_{\text{et}}^{{\rm{p}}}$$ is suggested for the relatively homogeneous rock when a ratio-formed rockburst proneness index is used and PES^p is suggested for the determination of bursting proneness of rock materials with poor homogeneity.

CONCLUSION

In this study, the uniaxial compression test was conducted on five-rock materials in two specimen shapes. Similar LES law of cylindrical and cuboid rock specimens in uniaxial compression was confirmed. On the basis of an accurate prediction of peak ESE by using the LES law of both cylindrical and cuboid specimens, the W_et and PES indexes were modified into the $${W}_{\text{et}}^{{\rm{p}}}$$ and PES^p indexes, respectively. The judgment accuracy of the two modified indexes was favorably improved for the evaluation of bursting proneness of both cylindrical and cuboid specimens. $${W}_{\text{et}}^{{\rm{p}}}$$ is relatively stable, but it cannot sufficiently reflect the difference in rockburst proneness between individual rock specimens. Among the four rockburst proneness indexes, the PES^p produced the most consistent judgments with the actual failure observations of rock specimens. However, misjudgment still occurs when the PES^p is used to assess the rockburst proneness. The reason is that it only considers the energy state at the pre-peak stage of rock specimens, but fails to cover the post-peak energy state.

ACKNOWLEDGMENTS

This study was supported by the National Natural Science Foundation of China (Grant No. 41877272) and the Fundamental Research Funds for the Central Universities (Grant No. 2242022k30054).

CONFLICT OF INTEREST

The authors declare no conflict of interest.

DATA AVAILABILITY STATEMENT

All data generated or analyzed during this study are included in this published article.

ETHICS STATEMENT

Not applicable.

References

American Society of Testing and Materials (ASTM). Standard test method for unconfined compressive strength of intact rock core specimens. ASTM‐D2938; 1986:390‐391.

Du K, Li XF, Su R, et al. Shape ratio effects on the mechanical characteristics of rectangular prism rocks and isolated pillars under uniaxial compression. Int J Min Sci Technol. 2022;32:347‐362.

Du K, Li XF, Yang CZ, Zhou J, Chen SJ, Manoj K. Experimental investigations on mechanical performance of rocks under fatigue loads and biaxial confinements. J Cent South Univ. 2020;27:2985‐2998.

Fairhurst CE, Hudson JA. Draft ISRM suggested method for the complete stress–strain curve for intact rock in uniaxial compression. Int J Rock Mech Min Sci. 1999;36(3):281‐289.

Fakhimi A, Hosseini O, Theodore R. Physical and numerical study of strain burst of mine pillars. Comput Geotech. 2016;74:36‐44.

Gong FQ, Luo S, Jiang Q, Xu L. Theoretical verification of the rationality of strain energy storage index as rockburst criterion based on linear energy storage law. J Rock Mech Geotech Eng. 2022. [DOI: https://dx.doi.org/10.1016/j.jrmge.2021.12.015]

Gong FQ, Wang YL, Luo S. Rockburst proneness criteria for rock materials: review and new insights. J Cent South Univ. 2020;27:2793‐2821.

Gong FQ, Yan JY, Li XB. A new criterion of rock burst proneness based on the linear energy storage law and the residual elastic energy index. Chin J Rock Mech Eng. 2018;37(9):1993‐2014.

Gong FQ, Yan JY, Li XB, Luo S. A peak‐strength strain energy storage index for rock burst proneness of rock materials. Int J Rock Mech Min Sci. 2019;117:76‐89.

Hauquin T, Gunzburger Y, Deck O. Predicting pillar burst by an explicit modelling of kinetic energy. Int J Rock Mech Min Sci. 2018;107:159‐171.

He MC, Xia HM, Jia XN, Gong WL, Zhao F, Liang KY. Studies on classification, criteria, and control of rockbursts. J Rock Mech Geotech Eng. 2012;4(2):97‐114.

He MC, Zhao F, Cai M, Du S. A novel experimental technique to simulate pillar burst in laboratory. Rock Mech Rock Eng. 2015;48(5):1833‐1848.

Hosseini O, Theodore R, Razavi M, Fakhimi A. Rock burst of underground pillars: Proceedings of the 49th US Rock 836 Mechanics/Geomechanics Symposium, ARMA 15‐0650. San Francisco, CA; 2015.

ISRM. Suggested methods for determining the uniaxial compressive strength and deformability of rock materials. Int J Rock Mech Min Sci Geomech Abstr. 1979;16(2):138‐140.

Kidybiński A. Bursting liability indices of coal. Int J Rock Mech Min Sci Geomech Abstr. 1981;18(4):295‐304.

Li XB, Gong FQ, Tao M, et al. Failure mechanism and coupled static‐dynamic loading theory in deep hard rock mining: a review. J Rock Mech Geotech Eng. 2017;9(4):767‐782.

Liang CY, Zhang QB, Li X, Xin P. The effect of specimen shape and strain rate on uniaxial compressive behavior of rock material. Bull Eng Geol Environ. 2016;75:1669‐1681.

Lunder PJ, Pakalnis RC. Determination of the strength of hard‐rock mine pillars. Can Min Metallurgical Bull. 1997;90(10):51‐55.

Luo S, Gong FQ. Linear energy storage and dissipation laws of rocks under preset angle shear conditions. Rock Mech Rock Eng. 2020a;53(10):3303‐3323.

Luo S, Gong FQ. Linear energy storage and dissipation laws during rock fracture under three‐point flexural loading. Eng Fracture Mech. 2020b;234: [eLocator: 107102].

Luo S, Gong FQ, Li LL, Peng K. Linear energy storage and dissipation laws and damage evolution characteristics of rock under triaxial cyclic compression with different confining pressure. Trans Nonferr Metal Soc China. 2022.

Martin CD, Maybee WG. The strength of hard‐rock pillars. Int J Rock Mech Min Sci. 2000;37:1239‐1246.

Miao SJ, Cai MF, Guo QF, Huang ZJ. Rock burst prediction based on in‐situ stress and energy accumulation theory. Int J Rock Mech Min Sci. 2016;83:86‐94.

Neyman B, Szecowka Z, Zuberek W. Effective methods for fighting rockbursts in Polish collieries: Proceedings of the 5th International Strata Control Conference; 1972:23‐31.

Paterson MS, Wong TF. Experimental Rock Deformation: The Brittle Field. Springer; 2005.

Sengani F. The performance of bolt‐reinforced and shotcreted in‐stope pillar in a rockburst prone areas. Int J Min Geo‐Eng. 2018;52(2):105‐117.

Si XF, Gong FQ. Strength‐weakening effect and shear‐tension failure mode transformation mechanism of rockburst for fine‐grained granite under triaxial unloading compression. Int J Rock Mech Min Sci. 2020;131: [eLocator: 104347].

Szecowka Z, Domzal J, Ozana P. Energy index of natural bursting ability of coal. Transactions of the Central Mining Institute, No. 594; 1973 (in Polish).

Tajdus A, Cieslik J, Tajdus K. Rockburst hazard assessment in bedded rock mass: laboratory tests of rock samples and numerical calculations. Arch Min Sci. 2014;59(3):591‐608.

The National Standards Compilation Groups of People's Republic of China. Methods for test, monitoring and prevention of rock burst‐part 2: classification and laboratory test method on bursting liability of coal. GB/T 25217.2‐2010; 2010.

Wang JA, Park HD. Comprehensive prediction of rockburst based on analysis of strain energy in rocks. Tunn Undergr Space Technol. 2001;16(1):49‐57.

Wang SY, Lam KC, Au SK, Tang CA, Zhu WC, Yang TH. Analytical and numerical study on the pillar rockbursts mechanism. Rock Mech Rock Eng. 2006;39(5):445‐467.

Xie HP, Li LY, Peng RD, Ju Y. Energy analysis and criteria for structural failure of rocks. J Rock Mech Geotech Eng. 2009;1(1):11‐20.

Xie HP, Peng RD, Ju Y, Zhou HW. On energy analysis of rock failure. Chin J Rock Mech Eng. 2005;24(15):2603‐2608 (in Chinese).

Xu YH, Cai M. Numerical study on the influence of cross‐sectional shape on strength and deformation behaviors of rocks under uniaxial compression. Pure Appl Geophys. 2017;84:129‐137.

Zhang CQ, Feng XT, Zhou H, Qiu S, Wu W. Case histories of four extremely intense rockbursts in deep tunnels. Rock Mech Rock Eng. 2012;45:275‐288.

Zhang CQ, Lu JJ, Chen J, Zhou H, Yang FJ. Discussion on rock burst proneness indexes and their relation. Rock Soil Mech. 2017;38(5):1397‐1404 (in Chinese).

Zhang QB, Zhao J. A review of dynamic experimental techniques and mechanical behaviour of rock materials. Rock Mech Rock Eng. 2014;47(4):1411‐1478.

Zhao XG, Cai M, Wang J, Li PF. Strength comparison between cylindrical and prism specimens of Beishan granite under uniaxial compression. Int J of Rock Mech Min Sci. 2015;76:10‐17.