The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Veriﬁcation of Butterﬂy-Shaped Plastic Zones

: The research on the formation factors of rock burst is one of the main research directions of rock mechanics in recent years, which is helpful to solve the problem of rock burst accidents. So, in this study, the calculation method of energy released during rock burst is ﬁrst obtained by using different medium models, and then, the formation factors of rock bursts are obtained by comparing the calculation energy with the actual accident energy. The method of energy calculation utilizes the difference between elastoplastic and pure elastic models to innovatively quantify the speciﬁc values of energy released before and after the occurrence of the rock burst. It is considered that the stress and plastic zone state before the occurrence of rock burst have an important inﬂuence on the occurrence of the accident and are one of the formation factors, while the deviatoric stress ﬁeld and butterﬂy-shaped plastic zone create conditions for greater energy release. In addition, the trigger stress constitutes another formation factor. The plastic zone state before rock failure is veriﬁed by the experimental test; the location distribution shape of acoustic emission (AE) events during the later stage of compression failure is approximately the same as theoretical result. The results also preliminarily indicated the fractal characteristics of acoustic emission events distribution before sample failure. The study obtained the formative factors of rock burst accident, which provides a new ideas and references for the research on the formation of rock bursts.


Introduction
The dynamic disasters of rock engineering perplex safe construction in many engineering fields.The so-called dynamic disasters mainly include rock burst occurring in tunnels [1], rock burst occurring in non-coal mines [2], rock burst occurring in coal mines (different from the disasters occurring in non-coal mines, the medium is coal, and the mining impact is greater) [3], coal and gas outbursts occurring in coal mines [4], slope collapses occurring during metro engineering [5], etc.The common point of these dynamic disasters is the instantaneous dynamic failure of coal or rock, which is often related to the medium state and human construction activities [6].In terms of dynamic disasters, coal mine rock burst accidents have occurred more frequently in China in recent years, bringing huge pressure to safety production.The study of the mechanism of geotechnical engineering disasters is a fundamental work that can provide direction for subsequent monitoring [7,8], warning [9], and evaluation [10].
In view of the frequent occurrence of rock burst accidents in coal mines, scholars have also carried out a lot of research, mainly focusing on their occurrence mechanism [11], monitoring and early warning indicators [12] and methods [12,13], and risk treatment measures [11,14].The research on the mechanism of rock burst dynamic disaster is the most important, which has attracted more scholars to study.In recent decades, various mechanisms have been formed, such as early energy theory [15], early strength theory [16], early stiffness theory [17], early rock burst tendency theory [18], intermediate instability theory [19], intermediate three-factor theory [20], intermediate three-criteria theory [21], later dynamic and static load theory [20], later rock burst initiation theory [22], later butterfly rock burst theory [23,24], etc.Some of these mechanisms or theories have put forward some concepts and descriptions, while others have established models and carried out strict mechanical derivation.which may be more scientific.
For the research methods of rock bursts, some studies focus on a specific condition, such as fault influence [25,26], high-strength mining faces [27,28], stress anomaly particularity [29], large vibration interference [30], periodic movement of hard roofs [31], etc.Other studies are conducted only from a certain angle, for example rock strength [32], stress analysis [33], energy analysis [34], special rock properties [20], gas coupling [35].Some use theoretical reasoning and calculation [36,37], some use laboratory research [38,39], some use numerical simulation [40,41], and some only use a conceptual expression.Rigorous mathematical and mechanical reasoning is highly recommended for scientific and engineering problems, and numerical simulation can be used to reflect or verify regular problems.If the simplest calculation method and the simplest model can be established to explain the common problems for geotechnical engineering, the rock burst mechanism will be easier to reveal (like the famous Oakham criterion [42]: if there are two or more different hypotheses about the same phenomenon, the simpler or falsifiable one should be adopted).
As for the stress state when rock burst occurs, it is clear that it must be a situation with certain characteristics, and then the smaller triggering stress leads to the accident [43].How the energy changes before and after the occurrence is directly related to the root cause of the occurrence mechanism.In order to study the mechanism of rock burst more scientifically, the energy conversion before, after, and during its occurrence is calculated from an energy perspective, which is an important research content in the field of rock dynamics.This study first proposes a method to calculate the system energy, and then obtains which mechanical state is more likely to reach the accident energy through this method.The results show that the butterfly-shaped plastic zone state caused by the bias stress field is more dangerous.Finally, the whole process AE monitoring of a loaded sample is carried out from the laboratory scale to verify the butterfly mode before failure.The experiment fully considered the specific situation of the original waveform, and adopted a more appropriate event location method, and finally obtained the butterfly-shaped plastic zone.The energy calculation method, rock burst mechanism, and its verification obtained in this study provide a new idea for the study of the rock burst mechanism.

Numerical Methodology
Rock burst accidents occur in rock masses, but rock masses are often relatively large and difficult to study via mechanical models and analysis.So, this study narrowed down the scope of the study and reflected big problems through small models.Due to the fact that regular knowledge is only obtained through model establishment and energy calculation, and the stress and situation of small model are less different from the actual environment in which the tunnel is located, the research method of the small model is considered feasible.In addition, considering the feasibility of the calculation, the method proposed in this study is completed by means of FLAC 3D numerical simulation and theoretical calculation, the rock mass (total volume is Ω) involved in the rock burst accident is picked out separately as the research object in the numerical simulation, and a hole is set in the middle of it to respect mining space.The rock mass is subjected to an isotropic force, which can be simplified as a three-dimensional stress of (P 1 , P 2 , P 3 ), where P 1 , P 2 , and P 3 are the main force.The rock mass is composed of many units (f(x, y, z)), and the force of these units can also be simplified as (σ 1i , σ 2i , σ 3i ), where σ 1i , σ 2i , and σ 3i are the main stress.The energy of each element in this mechanical state can be calculated by Equation (1) [32].
where E i is the elastic modulus of the corresponding element and µ i is the Poisson ratio of the corresponding element.
The stress state before the rock burst accident is named as pre-state stress field (PSSF), after being affected by a trigger stress field (named TSF; a high probability arises from roof fracture or coal rock fracture [44]), the stress state after being applied is named as the later state stress field (LSSF).During FLAC3D numerical simulation, in the model, the loaded media can be pure elastic or elastoplastic, and the energy of the two media under the named PSSF and LSSF states are U PSSF , U PSSF , U LSSF , and U LSSF , respectively.When the model is purely elastic, whether it is PSSF or LSSF, all elements are purely elastic, and the formula can be expressed as Equation ( 2), However, when the model is elastoplastic, under the action of PSSF, some elements will become elastoplastic (Ω e ).Under LSSF, a part of elastoplastic elements (∆Ω p ) will be added compared with PSSF, so the energy under U PSSF and U LSSF states can be expressed as Equations ( 3) and (4), respectively.
where V represents all units of the rock mass.V e and V p represent the units of pure elastic and elastoplastic, respectively.Under the action of PSSF and LSSF, the energy difference (D PSSF/LSSF ) between the pure elastic model and elastic-plastic model rock mass is expressed as Equation (5).The difference of energy difference between PSSF and LSSF is the total energy in the process of mechanical state change.However, the elastic wave energy (W) [45] needs to be multiplied by β (elastic wave energy conversion coefficient, the value is 1~10% [27,46]), as in Equation (6).
The calculation chart process of energy release during the mechanical state change process of rock mass from PSSF to LSSF is denoted in Figure 1.Firstly, calculate the energy difference between the pure elastic and elastic-plastic models under the mechanical state of PSSF (D PSSF ), and then calculate the energy difference under the mechanical state of LSSF (D LSSF ).The difference between D PSSF and D LSSF is the released energy from PSSF to LSSF.When calculating the release energy, it is not simply the difference between U PSSF and U LSSF when it is the elastic-plastic model, but the stored energy under the pure elastic model also needs to be considered in order to calculate the accurate release energy value.This study comprehensively considered the differences between pure elastic and elastic-plastic models under two mechanical states, and eliminated the influence of model size.The obtained energy calculation process and results can be compared with the actual energy release of on-site accidents, which can be used to infer the most likely occurrence condition of rock burst, and is of great significance to reveal the mechanism of rock burst.energy release of on-site accidents, which can be used to infer the most likely occurrence condition of rock burst, and is of great significance to reveal the mechanism of rock burst.

Results
An example model of 200 × 200 × 1 m (the model and force application direction are shown in Figure 2, the diameter of the hole is 5.6 m, and the application direction of P2 is front and rear) was taken to calculate the rock mass energy under different mechanical states (PSSF as (P1 = 20 MPa, P2 = 20 MPa, P3 = 20 MPa), ΔP as 1 MPa, and only added to P1) and different model conditions, according to the proposed method and the final energy release result.The shear strength, cohesion, friction angle, and tensile strength of the used medium are 1.3 GPa, 3 MPa, 25°, and 1.77 MPa, respectively.

Results
An example model of 200 × 200 × 1 m (the model and force application direction are shown in Figure 2, the diameter of the hole is 5.6 m, and the application direction of P 2 is front and rear) was taken to calculate the rock mass energy under different mechanical states (PSSF as (P 1 = 20 MPa, P 2 = 20 MPa, P 3 = 20 MPa), ∆P as 1 MPa, and only added to P 1 ) and different model conditions, according to the proposed method and the final energy release result.The shear strength, cohesion, friction angle, and tensile strength of the used medium are 1.3 GPa, 3 MPa, 25  From the numerical simulation results, it can be seen that under the conditions of the elastoplastic model, the shape of the plastic zone presents a butterfly shape when P1 is bigger than 50 MPa (η bigger than 2.5, similar as the results in [47]); a more pronounced butterfly shape (extending further) has emerged when P1 = 55 MPa (at this time, η = 2.75) The concentration phenomenon of plastic zone and stored energy distribution around the hole is closely related to the existence of hole.At the same time, the energy difference also continues to increase and expand with the increase in drilling P1, as in Figure 3 (DLSSF dis tribution when P1 = 40 MPa, 50 MPa, 55 MPa, 58.6 MPa).Another interesting point is tha some unit bodies do not release energy when subjected to changes in force, but instead absorb energy (it may be due to the fact that certain unit bodies tend to be subjected to more uniform forces, or the occurrence of tensile phenomena leads to a decrease in the calculated energy value).As the vast majority of unit bodies release energy, the final en ergy result of the entire sample is releasing state.From the numerical simulation results, it can be seen that under the conditions of the elastoplastic model, the shape of the plastic zone presents a butterfly shape when P 1 is bigger than 50 MPa (η bigger than 2.5, similar as the results in [47]); a more pronounced butterfly shape (extending further) has emerged when P 1 = 55 MPa (at this time, η = 2.75).The concentration phenomenon of plastic zone and stored energy distribution around the hole is closely related to the existence of hole.At the same time, the energy difference also continues to increase and expand with the increase in drilling P 1 , as in Figure 3 (D LSSF distribution when P 1 = 40 MPa, 50 MPa, 55 MPa, 58.6 MPa).Another interesting point is that some unit bodies do not release energy when subjected to changes in force, but instead absorb energy (it may be due to the fact that certain unit bodies tend to be subjected to more uniform forces, or the occurrence of tensile phenomena leads to a decrease in the calculated energy value).As the vast majority of unit bodies release energy, the final energy result of the entire sample is releasing state.
The change curve of the storage and different-value of energy are denoted in Figure 4; it can be seen intuitively from the figure that both storage energy of pure elastic medium and elastoplastic medium are increasing with the increase in P 1 , but the increasing speed is different (only describing the change in P 1 is because the values of P 2 and P 3 have not changed in all mechanical states).The different-value of energy increases and acceleration are various with the increase in P 1 , and mainly divided into three stages.The obvious critical points of the three stages are P 1 = 50 MPa, P 1 = 55 MPa, and P 1 = 58.6MPa, respectively.Interestingly, P 1 = 50 MPa is the starting point of the "early butterfly", P 1 = 55 MPa is the starting point of the "late butterfly", and P 1 = 58.6MPa is the value of the "final butterfly".The η values corresponding to the three critical points are 2.5, 2.75 and 2.93, respectively.The above fact shows that the shape of plastic zone is closely related to the different-value of energy, the butterfly shape is strengthened with the increase in P 1 and, meanwhile, the different-value of energy is also increased.
Divide the energy of the unit body by the volume of the unit body to obtain the energy release density map of the rock mass, and the release energy density map of butterfly related mechanical states are shown in Figure 5.The results show that the distribution of energy release density and plastic zone has a strong management, and has experienced a process from small butterfly to large butterfly.From the value of energy distribution, the greater the force, the greater the density value, and the more concentrated the distribution.The change curve of the storage and different-value of energy are denoted in Figure 4; it can be seen intuitively from the figure that both storage energy of pure elastic medium and elastoplastic medium are increasing with the increase in P1, but the increasing speed is different (only describing the change in P1 is because the values of P2 and P3 have not changed in all mechanical states).The different-value of energy increases and acceleration are various with the increase in P1, and mainly divided into three stages.The obvious critical points of the three stages are P1 = 50 MPa, P1 = 55 MPa, and P1 = 58.6MPa, respectively.Interestingly, P1 = 50 MPa is the starting point of the "early butterfly", P1 = 55 MPa is the starting point of the "late butterfly", and P1 = 58.6MPa is the value of the "final butterfly".The η values corresponding to the three critical points are 2.5, 2.75 and 2.93, respectively.The above fact shows that the shape of plastic zone is closely related to the different-value of energy, the butterfly shape is strengthened with the increase in P1 and, meanwhile, the different-value of energy is also increased.Divide the energy of the unit body by the volume of the unit body to obtain the energy release density map of the rock mass, and the release energy density map of butterfly related mechanical states are shown in Figure 5.The results show that the distribution of energy release density and plastic zone has a strong management, and has experienced a process from small butterfly to large butterfly.From the value of energy distribution, the   The calculation results of the released energy of rock mass are shown in Figure 6, from which it can be seen that the phased characteristics are obvious.The entire process shown in the figure can be divided into three stages: the pregnant period, growth period, and upheaval period, representing the phenomenon of releasing energy from small to large, respectively, and then rapidly increasing.Results show that even with the same TSF, different PSSFs have a great impact on the results of the energy release.When the shape of the plastic zone caused by the PSSF (when there is deviatoric loading [48], and P1/P3 is mostly >2.5) of rock mass is butterfly, the change in the released amount forms an inflection point, and the corresponding η values are 2.5 MPa, 2.75 MPa, and 2.93, respectively, the released energy is increased by 12.5 times form η = 2.5 to η = 2.93.The calculation results of the released energy of rock mass are shown in Figure 6, from which it can be seen that the phased characteristics are obvious.The entire process shown in the figure can be divided into three stages: the pregnant period, growth period, and upheaval period, representing the phenomenon of releasing energy from small to large, respectively, and then rapidly increasing.Results show that even with the same TSF, different PSSFs have a great impact on the results of the energy release.When the shape of the plastic zone caused by the PSSF (when there is deviatoric loading [48], and P 1 /P 3 is mostly >2.5) of rock mass is butterfly, the change in the released amount forms an inflection point, and the corresponding η values are 2.5 MPa, 2.75 MPa, and 2.93, respectively, the released energy is increased by 12.5 times form η = 2.5 to η = 2.93.
The calculated energy of the rock burst can be compared with the actual energy.It can be seen that the released energy of the accident is matched with the calculated energy from the above energy change results, and the formation factors of rock burst are the butterfly plastic zone caused by PSSF and appropriate TSF in addition to the basic conditions (mining space and force relation are easily satisfied).The PSSFs are related to the protolith stress, mining stress, roadway layout, surrounding rock, coal properties, etc.The TSFs are related to the roof breakage event, coal fracture event, blasting vibration, etc. Further, these factors are finally reflected in PSSF and TSF, otherwise it is difficult to meet the energy conditions.
When the energy release reaches the critical value, the energy conditions for the accident can be met, as in Figure 7.The calculated energy of the rock burst can be compared with the actual energy.It can be seen that the released energy of the accident is matched with the calculated energy from the above energy change results, and the formation factors of rock burst are the butterfly plastic zone caused by PSSF and appropriate TSF in addition to the basic conditions (mining space and force relation are easily satisfied).The PSSFs are related to the protolith stress, mining stress, roadway layout, surrounding rock, coal properties, etc.The TSFs are related to the roof breakage event, coal fracture event, blasting vibration, etc. Further, these factors are finally reflected in PSSF and TSF, otherwise it is difficult to meet the energy conditions.When the energy release reaches the critical value, the energy conditions for the accident can be met, as in Figure 7.

Discussion
The formation factors of rock burst have been obtained in the Section 3. In fact, the  The calculated energy of the rock burst can be compared with the actual energy.It can be seen that the released energy of the accident is matched with the calculated energy from the above energy change results, and the formation factors of rock burst are the butterfly plastic zone caused by PSSF and appropriate TSF in addition to the basic conditions (mining space and force relation are easily satisfied).The PSSFs are related to the protolith stress, mining stress, roadway layout, surrounding rock, coal properties, etc.The TSFs are related to the roof breakage event, coal fracture event, blasting vibration, etc. Further, these factors are finally reflected in PSSF and TSF, otherwise it is difficult to meet the energy conditions.When the energy release reaches the critical value, the energy conditions for the accident can be met, as in Figure 7.

Discussion
The formation factors of rock burst have been obtained in the Section 3. In fact, the conclusive key factor of large ratio PSSF is based on less requirements of TSF under "late

Discussion
The formation factors of rock burst have been obtained in the Section 3. In fact, the conclusive key factor of large ratio PSSF is based on less requirements of TSF under "late butterfly shape" or "final butterfly shape" of the plastic zone, so the required minimum TSF to reach the critical energy value under different stage based on a total energy of 10 7 J is indicated in Figure 8.The results show that in the non-butterfly stage (P 1 less than 50 MPa), the minimum TSF required is 1~14 MPa, while in the butterfly stage (P 1 more than 50 MPa and less than 55 MPa), the value is reduced to 0.2 MPa, and in the late butterfly stage (P 1 more than 55 MPa), the value is even reduced to 0.1 MPa, which is easy to achieve Fractal Fract.2023, 7, 829 9 of 16 in realistic data.The above facts confirm the key factor role of a large ratio PSSF for a rock burst, which is similar to the result of rock bursts caused by larger difference existing between horizontal and vertical stresses in [49].
butterfly shape" or "final butterfly shape" of the plastic zone, so the required minimum TSF to reach the critical energy value under different stage based on a total energy of 10 7 J is indicated in Figure 8.The results show that in the non-butterfly stage (P1 less than 50 MPa), the minimum TSF required is 1~14 MPa, while in the butterfly stage (P1 more than 50 MPa and less than 55 MPa), the value is reduced to 0.2 MPa, and in the late butterfly stage (P1 more than 55 MPa), the value is even reduced to 0.1 MPa, which is easy to achieve in realistic data.The above facts confirm the key factor role of a large ratio PSSF for a rock burst, which is similar to the result of rock bursts caused by larger difference existing between horizontal and vertical stresses in [49].In fact, the implicit equation of the plastic zone boundary under the condition of the uniform medium model is obtained in a previous study [24].The plastic zone distributions of typical mechanical states obtained by Equation (7).The results show that when P1 = 40 MPa, the shape of plastic zone is not butterfly, but when P1 = 50 MPa, the butterfly state is more obvious.When P1 = 55 MPa, the maximum radius of the plastic zone Rmax extends to 15 m, and when P1 = 58.6MPa, the Rmax extends to 90 m.The plastic zone results of previous theoretical calculations are the same as those in this study, while this study focuses more on the variation law of energy corresponding to plastic zone.
Another index of the plastic zone is the area of plastic zone, which not only represents the depth of the plastic zone, but also comprehensively reflects the volume of the plastic zone.Therefore, the variation curves of plastic zone Rmax, area S, and released energy with the increase in P1 are compared in Figure 9. From the comparison results, the three indicators all formed a certain inflection point when η = 2.5, 2.75, and 2.93, but the difference is the rate and degree of change.The most drastic change is the released energy index, followed by Rmax index, and finally S.Although the change degree of the three indexes is In fact, the implicit equation of the plastic zone boundary under the condition of the uniform medium model is obtained in a previous study [24].The plastic zone distributions of typical mechanical states obtained by Equation (7).The results show that when P 1 = 40 MPa, the shape of plastic zone is not butterfly, but when P 1 = 50 MPa, the butterfly state is more obvious.When P 1 = 55 MPa, the maximum radius of the plastic zone R max extends to 15 m, and when P 1 = 58.6MPa, the R max extends to 90 m.The plastic zone results of previous theoretical calculations are the same as those in this study, while this study focuses more on the variation law of energy corresponding to plastic zone.
Another index of the plastic zone is the area of plastic zone, which not only represents the depth of the plastic zone, but also comprehensively reflects the volume of the plastic zone.Therefore, the variation curves of plastic zone R max , area S, and released energy with the increase in P 1 are compared in Figure 9. From the comparison results, the three indicators all formed a certain inflection point when η = 2.5, 2.75, and 2.93, but the difference is the rate and degree of change.The most drastic change is the released energy index, followed by R max index, and finally S.Although the change degree of the three indexes is different, they all reflect the mutation phenomenon, which shows that the rock burst mechanism relying on large ratio PSSF and butterfly shape plastic zone is reasonable.different, they all reflect the mutation phenomenon, which shows that the rock burst mechanism relying on large ratio PSSF and butterfly shape plastic zone is reasonable.The above analysis obtained the important influence of butterfly plastic zone based on theoretical calculation and numerical simulation.An experimental verification is more reliable; therefore, the whole process of AE monitoring of sample loading process is implemented.The equipment is denoted in Figure 10a, which mainly includes a universal tester and its control system, an AE acquisition system, and a rock sample.The sample is a kind of sandstone with a rock burst tendency (shown in Figure 10b), and the size is 200 cubic millimeters (a 20 mm diameter central hole is set in the axial direction to represent the roadway, and the direction is assumed to be in the Y direction).The uniaxial loading method is adopted for the experiment, until the sample is completely crushed.The sampling frequency of AE system is 6 MHz, and a total of eight sensors are used for signal acquisition, as in Figure 10c.The study focuses more on the positioning of AE events within the few seconds before rock failure, so as to verify the shape of plastic zone area before failure.The above analysis obtained the important influence of butterfly plastic zone based on theoretical calculation and numerical simulation.An experimental verification is more reliable; therefore, the whole process of AE monitoring of sample loading process is implemented.The equipment is denoted in Figure 10a, which mainly includes a universal tester and its control system, an AE acquisition system, and a rock sample.The sample is a kind of sandstone with a rock burst tendency (shown in Figure 10b), and the size is 200 cubic millimeters (a 20 mm diameter central hole is set in the axial direction to represent the roadway, and the direction is assumed to be in the Y direction).The uniaxial loading method is adopted for the experiment, until the sample is completely crushed.The sampling frequency of AE system is 6 MHz, and a total of eight sensors are used for signal acquisition, as in Figure 10c.The study focuses more on the positioning of AE events within the few seconds before rock failure, so as to verify the shape of plastic zone area before failure.
The study should focus on the failure distribution form in the XZ plane (perpendicular to the Y direction) during 1990~1995s, and the influence of boundary conditions should be removed as far as possible.Therefore, the statistical AE event scatter plot and heat map during 1990~1992s and 1993~1995s are stated in Figure 11.The number of AE events during 1990~1992s is less than that during 1993~1995s.The distribution pattern of event points is not obvious, but the heat map results can basically show the butterfly shape.The distribution of each butterfly leaf is not very regular in the heat map results, which may be caused by the heterogeneity of the sample or the influence of force transmission.The AE event scatter plot and heat map during the whole period of 1990~1995s are shown in Figure 12, which shows a more obvious butterfly shape, and is basically consistent with the results of theoretical calculation and numerical simulation.The formation of butterfly-shaped plastic zones not only reflects the morphology before failure, but also indicates that events are concentrated in certain specific areas (fractal characteristics) and may cause large-scale damage after reaching a certain level.
This study reveals the shape of butterfly-shaped plastic zone of the sample in a specific state, which is caused by a large ratio of the stress field (like the state described in [50,51], the influence of some large ratio of foundation ground stress or geological structure often results in deviatoric stress field, which increases the probability of rock bursts), and resulting in a large amount of concentrated energy release.Based on the initial stress state of the rock, the study considers that a small TSF may lead to a huge release under specific stress conditions, and gives the minimum TSF required to form disasters under different states.It considers that the probability of the occurrence of rock burst is the butterfly-shaped plastic zone shape caused by deviatoric stress field.This can also be demonstrated from the actual size of TSF; in fact, TSF is unlikely to become the dominant factor in most cases, which has been already demonstrated in [52].In addition, this study also preliminarily discovered the fractal characteristics of acoustic emission events distribution before sample failure.The study should focus on the failure distribution form in the XZ plane (perpendicular to the Y direction) during 1990~1995s, and the influence of boundary conditions should be removed as far as possible.Therefore, the statistical AE event scatter plot and heat map during 1990~1992s and 1993~1995s are stated in Figure 11.The number of AE events during 1990~1992s is less than that during 1993~1995s.The distribution pattern of event points is not obvious, but the heat map results can basically show the butterfly shape.The distribution of each butterfly leaf is not very regular in the heat map results, which may be caused by the heterogeneity of the sample or the influence of force transmission.The AE event scatter plot and heat map during the whole period of 1990~1995s The results of this study are based on a calculation method of energy to obtain the energy value before and after the change in the rock mass mechanical state, and the specific expression of dangerous stress state is obtained; that is, when the ratio of maximum principal stress to minimum principal stress reaches the 2.5/2.75/2.93critical point.In addition, an experimental test was used to preliminarily verify the shape of the butterfly-shaped plastic zone on the eve of sample failure.Although the form of uniaxial compression was used for the convenience of the test, this form also showed butterfly shape in numerical simulation and theoretical calculation, which proved to be basically effective, but the butterfly shape under triaxial loading should be obtained more in later experiments.In addition, it is necessary to try measuring vibration signals in more directions, such as three axes [53].The experimental verification has been preliminarily completed, and the on-site verification should be carried out later, and a warning method using the three-dimensional stress field or butterfly plastic zone state can also be studied in future.Furthermore, more detailed three-dimensional stress states [54,55] and anisotropy [56], or particle breakage [57] should also be considered, and the effects of deformation and cracks [58,59] can also be comprehensively considered.This study obtained the energy values before and after the occurrence of rock burst through the innovative energy calculation method proposed.By comparing them with the actual accident energy, the energy cloud maps of different mechanical states and the relationship between released energy were obtained.From an energy perspective, the formation factors of rock bursts were clarified, and the importance of deviatoric stress in PSSF was emphasized.The minimum triggering stress value required to reach the critical value under different PSSFs was obtained, and the butterfly-shaped plastic zone formed by a deviatoric stress field was preliminarily verified at the laboratory scale.This study provides a new approach for studying the mechanism of rock dynamic disasters and lays a certain foundation for monitoring and warning of butterfly-shaped plastic zones.

Conclusions
In this study, the released energy of rock burst accident is obtained by means of energy calculation, the formation factors of rock bursts are obtained by means of the calculation method, and the formation factors of butterfly-shaped plastic zone are obtained and verified by means of laboratory tests.It is easier to achieve quantification and accuracy from the perspective of energy, and the obtained factors of rock burst formations are clearer, which has certain scientific value.The main conclusions are as follows: (1) The energy conditions of rock burst accidents are obtained, which lays the foundation for determining the formation factors of rock bursts.(2) The formation factors of rock burst are the butterfly plastic zone caused by PSSF and appropriate TSF, and PSSF plays a leading role, which determines the possibility of rock burst accidents.The PSSF that leads to the butterflyshaped plastic zone represents a dangerous state of deviatoric stress field, in which a small triggering stress can lead to large-scale energy release and rock failure.(3) The butterfly failure mode has been preliminarily verified at the laboratory scale through the location of AE events, and the rock sample shows a butterfly-shaped plastic zone before uniaxial loading failure.(4) This study also preliminarily discovered the fractal characteristics of acoustic emission events distribution before the sample failure.

Funding:
The authors gratefully acknowledge the financial support by Qingdao Huanghai University Doctoral Research Fund Project (2022boshi02) and National Natural Science Foundation of China (52174111).

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Figure 1 .
Figure 1.Calculation chart process of energy release during the mechanical state change process of rock mass from PSSF to LSSF.

Figure 1 .
Figure 1.Calculation chart process of energy release during the mechanical state change process of rock mass from PSSF to LSSF.

Figure 2 .
Figure 2. Example model and its external force state.

Figure 2 .
Figure 2. Example model and its external force state.

Figure 3 .
Figure 3. DLSSF distribution when P1 = 40 MPa, 50 MPa, 55 MPa, 58.6 MPa (the horizontal and vertical coordinates represent the width and height of the model, corresponding to Figure 1).

Figure 3 .
Figure 3. D LSSF distribution when P 1 = 40 MPa, 50 MPa, 55 MPa, 58.6 MPa (the horizontal and vertical coordinates represent the width and height of the model, corresponding to Figure 1).

FractalFigure 4 .
Figure 4. Change curve of storage and different-value of energy.

Figure 4 .
Figure 4. Change curve of storage and different-value of energy.

FractalFigure 5 .
Figure 5. Release energy density map of butterfly related mechanical states (the horizontal and vertical coordinates represent the width and height of the model, corresponding to Figure 1).

Figure 5 .
Figure 5. Release energy density map of butterfly related mechanical states (the horizontal and vertical coordinates represent the width and height of the model, corresponding to Figure 1).

Figure 7 .
Figure 7.The formation factors of rock burst based on energy calculation.

Figure 7 .
Figure 7.The formation factors of rock burst based on energy calculation.

Figure 8 .
Figure 8. Required minimum TSF to reach the critical energy value under different stage.

Figure 8 .
Figure 8. Required minimum TSF to reach the critical energy value under different stage.

Figure 9 .
Figure 9. Variation curves of plastic zone Rmax, area S, and released energy with the increase in P1.

Figure 9 .
Figure 9. Variation curves of plastic zone R max , area S, and released energy with the increase in P 1 .

Fractal
Fract.2023, 7, x FOR PEER REVIEW 12 of 17 (a) Equipment layout (b) Test rock sample and the AE sensor (c) AE sensors layout

Figure 10 .
Figure 10.Equipment used for experimental verification.

Figure 10 .
Figure 10.Equipment used for experimental verification.

FractalFigure 11 .Figure 11 .
Figure 11.Scatter plot and heat map of AE events occurred during the period of 1990~1992s and 1993~1995s.(a) Scatter plot of AE events occurred during the period of 1990~1992s.(b) Heat map of AE events occurred during the period of 1990~1992s.(c) Scatter plot of AE events occurred during the period of 1993~1995s.(d) Heat map of AE events occurred during the period of 1993~1995s.

Figure 11 .Figure 12 .Figure 12 .
Figure 11.Scatter plot and heat map of AE events occurred during the period of 1990~1992s and 1993~1995s.(a) Scatter plot of AE events occurred during the period of 1990~1992s.(b) Heat map of AE events occurred during the period of 1990~1992s.(c) Scatter plot of AE events occurred during the period of 1993~1995s.(d) Heat map of AE events occurred during the period of 1993~1995s.