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

^{1}

^{2}

^{3}

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^{5}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Methodology

^{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].

_{i}is the elastic modulus of the corresponding element and ${\mathsf{\mu}}_{\mathrm{i}}$ is the Poisson ratio of the corresponding element.

_{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 (ΔΩ

_{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.

_{e}and V

_{p}represent the units of pure elastic and elastoplastic, respectively.

_{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).

_{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.

## 3. Results

_{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), ΔP 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°, and 1.77 MPa, respectively.

_{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.

_{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}= 58.6 MPa, 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}= 58.6 MPa 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.

_{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.

## 4. Discussion

^{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 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].

_{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}= 58.6 MPa, 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.

_{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.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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

**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).

**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 9.**Variation curves of plastic zone R

_{max}, area S, and released energy with the increase in P

_{1}.

**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 12.**Scatter plot and heat map of AE events occurred during the whole period of 1990~1995s. (

**a**) Scatter plot of AE events occurred during the whole period of 1990~1995s. (

**b**) Heat map of AE events occurred during the whole period of 1990~1995s.

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**MDPI and ACS Style**

Zhang, W.; Feng, J.; Ren, J.; Ma, J.; Shi, J.; Zhang, J.
The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Verification of Butterfly-Shaped Plastic Zones. *Fractal Fract.* **2023**, *7*, 829.
https://doi.org/10.3390/fractalfract7110829

**AMA Style**

Zhang W, Feng J, Ren J, Ma J, Shi J, Zhang J.
The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Verification of Butterfly-Shaped Plastic Zones. *Fractal and Fractional*. 2023; 7(11):829.
https://doi.org/10.3390/fractalfract7110829

**Chicago/Turabian Style**

Zhang, Wenlong, Jicheng Feng, Jianju Ren, Ji Ma, Jianjun Shi, and Junfeng Zhang.
2023. "The Formative Factors of a Rock Burst Based on Energy Calculations and the Experimental Verification of Butterfly-Shaped Plastic Zones" *Fractal and Fractional* 7, no. 11: 829.
https://doi.org/10.3390/fractalfract7110829