Theoretical Research on Rockburst under Dynamic Load and Triaxial Loading and Unloading Conditions

Theoretical Research on Rockburst under Dynamic Load and Triaxial Loading and Unloading Conditions

Jideng Huang

Department of Architecture and Civil Engineering, Huaiyin Institute of Technology, Huai’an

Wei Shen

Department of Architecture and Civil Engineering, Huaiyin Institute of Technology, Huai’an

Abstract – This article focuses on the problem of rockburst in deep rock masses, and studies the mechanical mechanisms of rockburst induced by triaxial loading and unloading as well as dynamic loading. It constructs a relevant theoretical framework and proposes engineering application methods, providing theoretical support for the prevention of rockburst in deep underground engineering.

1. INTRODUCTION

   In As underground construction progresses deeper into the strata, the “three-high” geological environment of high stress, high permeability pressure, and high temperature has become the core challenge faced by engineering construction [1]. Rockburst, as a sudden dynamic failure disaster that occurs in deep rock masses under stress disturbance [2], has become a key geological hazard issue that restricts the safe construction and stable operation of deep underground engineering. The formation and triggering of rockburst are the coupled results of fracture evolution, energy accumulation, and sudden release of the rock mass under the action of a complex stress field [3-4]. The triaxial loading-unloading effect caused by engineering excavation will change the stress distribution of the rock mass, induce the emergence and expansion of fractures, and dynamic disturbances such as blasting construction, mechanical vibration, and seismic activities will impose additional loads on the high- stress surrounding rock, becoming the key triggers for rockburst [5]. Currently, a large number of scholars have conducted extensive research on the rockburst problem, achieving many results in aspects such as rock mass damage evolution, fracture expansion laws, and energy release characteristics [6-8]. They have proposed rockburst analysis methods based on strength criteria,

   energy theory, and catastrophe theory, providing theoretical support for the assessment of rockburst

   focus on the rockburst mechanism under static loading-unloading or simple dynamic loading conditions, and do not adequately consider the coupling effect of triaxial loading-unloading and dynamic disturbances in deep engineering. There is still no systematic theoretical framework for the formation, triggering, and instability of rockburst under combined static and dynamic loads. In view of this, this paper focuses on the core scientific issues of rockburst in deep rock masses and conducts research on the mechanical mechanism of rockburst induced by triaxial loading-unloading and dynamic loading.

2. ECHANICAL THEORY OF FRACTURE EVOLUTION UNDER

   TRIAXIAL LOADING AND UNLOADING

   1. Three-dimensional Stress state and Damage Constitutive Relationship

      The deep rock mass is in a complex triaxial stress state. Its stress tensor [11] can be expressed as:

      Here, 1 represents the maximum principal stress (usually in the vertical direction), 2 is the intermediate principal stress, and 3 is the minimum principal stress (in the radial unloading direction).

      During the excavation unloading process, the minimum principal stress 3 rapidly decreases, forming an unloading stress path, while 1 and 2 may increase or remain relatively stable due to stress adjustment.

      Based on the continuous damage mechanics theory [12-14], the constitutive relationship of rocks under triaxial loading and unloading can be expressed as:

      tendencies [9-10]. However, existing studies mostly

      In the equation, E represents the elastic modulus of the rock, is the Poisson ratio, D is the damage variable (0 D 1), is the Kronecker symbol, and is the plastic strain

      tensor.

      The evolution of the damage variable D is closely related to the fracture density and the propagation state, and can be defined as:

      In the formula, KI represents the I-type stress intensity factor, and K(IC) represents the I-type fracture toughness of the rock. For elliptical cracks, the stress intensity factor can be calculated as:

      Here, n represents the normal stress on the

      crack surface, a is the major axis of the crack, b is the minor axis of the crack, is the orientation angle of the crack, and F is the shape coefficient

      Here, Ad represents the damaged area, A represents the total cross-sectional area, N represents the number of cracks per unit area, l represents the average length of the cracks, and

      · denotes the average value.

      For the true triaxial unloading condition, the damage evolution equation considering the effect of the intermediate principal stress can be established as:

      function.

      Under triaxial unloading conditions, as the minimum principal stress 3 rapidly decreases, the effective normal stress on the crack surface increases, significantly promoting the initiation and propagation of tensile cracks. Experimental observations show that during the true triaxial unloading process, when the intermediate principal stress 2 is sufficiently large, rock failure is mainly plate fracture, forming plate-like

      cracks parallel to the excavation surface. The mechanism of this transformation in the failure

      In the formula, Y represents the damage energy release rate, S and s are material constants,

      mode can be explained by the crack interaction theory [16-17]:

      and d is the increment of the plastic multiplier.

      The damage energy release rate Y can be expressed as:

      Here,Ktotal represents the total stress intensity factor, Kapplied represents the stress intensity

      Here, represents the equivalent stress,

      factor generated by the far-field stress, and KI

      represents the contribution of adjacent cracks j to

      and the function f

      characterizes the influence

      the stress field of crack i.

      The relationship between the crack

      of the intermediate principal stress on the damage

      evolution. Experimental studies have shown that as the ratio of the intermediate principal stress 2/1 increases, the failure mode of the rock

      propagation velocity vc and the stress intensity factor can be described by the following empirical formula:

      gradually shifts from shear failure to plate fracture failure. The critical condition for this transition can be described by the modified

      Wiebols-Cook criterion:

      Here, vmax represents the maximum crack

      propagation speed (usually 0.3 – 0.5 times the longitudinal wave velocity of the rock), and m is

      a material constant. Under unloading conditions, due to the rapid increase of the stress intensity

      Here, c represents cohesion, represents internal friction angle, and b is the intermediate principal tress influence coefficient (0 b 1).

   2. Fracture initiation and propagation criteria

      During the triaxial unloading process of the rock, the initiation of fractures follows a combined mechanism of the tensile initiation criterion and the shear initiation criterion [15]. According to the maximum circumferential stress theory, the initiation criterion for tensile fractures is:

      KI=KIC (7)

      factor KI, the crack propagation shows an accelerating characteristic, which is an important manifestation of the precursor of rockburst.

   3. Energy evolution and rockburst susceptibility

   The essence of rockburst is a process of intense energy release. Energy analysis is the core method for evaluating the tendency of rockburst. During the triaxial loading and unloading process, the total energy U of the rock system can be decomposed as:

   U=Ue+Up+Ud+Uk (11)

   Here, Ue represents elastic strain energy, Up represents plastic deformation energy, Ud

   represents damage dissipation energy, and Uk represents kinetic energy. Elastic strain energy is the main energy source for rockburst, which can be calculated as:

   89.0%. These stress threshold values correspond to different energy state transition points.

   The rockburst susceptibility evaluation index Wx can be defined from an energy perspective as:

   Here, Ue represents the elastic strain energy at the peak stress point, and Ue

   During the unloading process, as the damage progresses (D increases), the elastic modulus E(1- D) decreases, resulting in the release of the stored elastic strain energy. According to Zhang Ying’s experimental study [18], for granite under triaxial cyclic loading and unloading conditions, the ratio ci/f of the initial cracking stress ci to the peak strength f is within the range of 37.0% to 44.8%, and the ratio cd/f of the damage stress cd to the peak strength is within the range of 81.2% to

   residual

   represents the residual elastic strain energy after failure. Experimental studies have shown that when the confining pressure is less than 20 MPa, the rockburst susceptibility of granite is relatively low; when the confining pressure reaches 30 MPa, the rockburst susceptibility begins to increase rapidly.

   The comparison of the failure modes and energy characteristics of rocks under different stress conditions is detailed in Table 1 below.

   Table 1 Rock failure modes and energy characteristics under different stress states

   Stress state

   The ratio of the intermediate principal stress (/)

   Main destruction Energy release characteristics mode

   Rockburst susceptibility

   Low confining pressure state

   0.1-0.3 Shear failure Progressive energy release, with a

   high proportion of dissipated energy

   Lower

   Medium confining pressure state

   0.3-0.6

   Shear – plate fracture composite failure

   The rate of energy release has increased, and the proportion of elastic strain energy has risen.

   Medium

   High confining 0.6-1.0

   pressure state

   Plate fracture failure

   Sudden release of energy, with significant increase in kinetic energy

   High

   3

    

   True three-axis Quickly reduce unloading

   Dynamic plate cracking and rockburst

   Extremely high energy release rate, with an excessive amount of energy E being generated

   Extremely high

   Excessive energy E is a direct measure of the intensity of rockburst and is defined as:

   increases, and the rockburst becomes more intense.

   storage input

   residual fracture

3. The theory of crack mutation

   INDUCED BY DYNAMIC LOADING

   storage

   fracture

   Here, Ue represents the elastic strain energy stored before the rockburst, Winput represents the energy input by the equipment after the peak, and Ud represents the energy consumed on the surface of forming new cracks. The experiment shows [19] that as the number of unloading surfaces increases, E gradually decreases, but the energy release rate (EERR)

   1. Mechanical model of dynamic disturbance

      The dynamic disturbances in underground engineering mainly originate from blasting construction, mechanical vibration, fracture of adjacent rock masses, and seismic activities, etc. These disturbances propagate in the form of stress waves and impose additional dynamic loads on

      the surrounding rock that is already in a high- stress state [20]. The motion control equation for rock columns considering shear deformation and rotational inertia is:

      parameter combination (, ) falls within the unstable region, the system will undergo parametric resonance, and the displacement amplitude will grow exponentially, leading to the

      instability of the rock mass.

      The critical condition for the initiation of crack propagation induced by dynamic

      Here, represents the density of the rock, A is the cross-sectional area, u is the lateral displacement, G is the shear modulus, I is the moment of inertia of the cross-section, and F(x,t) represents the dynamic disturbance force.

      disturbances can be derived using the principle of energy balance. Consider a crack with a length of

      a. The expansion condition under dynamic loading is:

      Dynamic disturbances can usually be modeled as a harmonic load superimposed on the

      static stress field:

      Here, s represents the static stress, 0 is the amplitude of the dynamic stress, is the angular frequency, and is the phase angle. For the tunnel surrounding rock, the additional stress field caused by dynamic disturbance can be expressed as:

      Here, G(t) represents the energy release rate, Gc is the critical energy release rate, KI(t) and KII(t) are respectively the I-type and II-type stress intensity factors. Under the action of harmonic dynamic loading, the stress intensity factors can be expressed as:

      Here, KS represents the static stress intensity

      factor, and KI represents the amplitude of the

      Here, fij(r, ) represents the spatial distribution function, is the attenuation coefficient, and r is the distance to the disturbance source.

      dynamic stress intensity factor. When the dynamic load frequency approaches the natural frequency of the fracture system, a dynamic amplification effect occurs:

   2. The parametric resonance theory of fracture

      responses under dynamic loading

      One of the key mechanisms for dynamic disturbances to induce rockburst is parametric resonance [21-22]. When the disturbance frequency matches the inherent frequency of the rock mass system in a specific way, even a very small dynamic load amplitude can cause significant vibration. Considering the motion equation of fractured rock masses under dynamic disturbances:

      Here, m represents the equivalent mass, c is the damping coefficient, k(t) is the time-varying stiffness, x is the displacement, and Fd(t) is the dynamic disturbance force. The time-varying stiffness k(t) reflects the opening and closing effect of the fracture under cyclic loading:

      Rearrange the above equation into the standard form of the Mathieu equation:

      Here, = c/(2m), = 4k0/(m2), = 2k1/(m2), and = (t)/2. According to Floquet theory, the stability of the Mathieu equation is determined by the parameters and . When the

      Here, Kd0 represents the stress intensity factor corresponding to the static dynamic load,

      = /n is the frequency ratio, n is the natural frequency of the fracture system, and is the damping ratio.

   3. Dynamic crack propagation and energy release

      The dynamic expansion speed of cracks under dynamic loading has a significant impact on the characteristics of rockburst. According to Freund’s dynamic fracture theory [23], the core breakthrough of Freund’s theory lies in pointing out that the expansion speed of the crack itself will react back on its driving force, thus revealing a crucial positive feedback loop: Dynamic loading Crack begins to accelerate Dynamic stress field changes The crack

      driving force (dynamic stress intensity factor Kd) subsequently changes The crack further accelerates or decelerates. It is precisely this “speed – driving force coupling” relationship that provides the fundamental physical mechanism for explaining the “mutation” process of cracks from slow accumulation to instantaneous out-of- control in rockburst.

      The relationship between crack expansion speed vc and dynamic stress intensity factor Kd is:

      Here, vR represents the Rayleigh wave velocity, which is the theoretical upper limit speed for crack propagation. For rock materials, vR 0.92vs, where vs is the shear wave velocity. The speed formula vc(v) in the Freund theory indicates that there is a theoretical limit speed vR for crack propagation. When the dynamic driving force Kd is much greater than the static fracture toughness KIC, the crack will strive to accelerate and approach vR. This formula reveals the “critical condition” for sudden changes induced by dynamic loading. Under static or quasi-static loading, KS increases slowly, and crack propagation is relatively stable. The instantaneous nature of dynamic loading allows Kd to reach several times the static value in a very short period of time, instantaneously satisfying the condition of Kd >> KIC. Once this

      During the mechanical process of rockburst development and triggering, external dynamic load disturbances will impart significant initial kinetic energy to the pre-existing cracks in the rock mass, enabling them to quickly surpass the low-speed expansion threshold and enter the sensitive response range of the crack expansion rate function g(v). In this state, the energy release efficiency of the crack propagation system increases in a step-like manner, and the elastic strain energy stored in the rock mass is rapidly dissipated, thereby triggering an avalanche-like release of energy and the instability expansion of cracks. Eventually, a sudden change in crack propagation behavior is induced, which perfectly explains why a small dynamic load disturbance can trigger a catastrophic rockburst.

      The energy criterion for dynamic load- induced rockburst can be established based on the excess energy theory:

      Here, E represents the excess energy, Pin(t) is the input power, Pout(t) is the dissipated

      dynamic critical point is crossed, crack

      propagation is no longer constrained by static laws but enters an “Inertial Car” state controlled by its own inertia and wave velocity, exhibiting intense dynamic instability characteristics. Mechanisms such as “parametric resonance”

      power, and Ec is the critical excess energy. The input power mainly comes from the work done by dynamic disturbances:

      reduce the threshold for system instability, and the Freund theory quantitatively describes the intense response behavior of the crack after instability.

      The energy release rate during the dynamic crack propagation process can be expressed as:

      V

      The dissipated power includes the energy consumption of processes such as plastic deformation, damage evolution, and crack propagation:

      G (v)=G0g(v) (25)

      Here, G0 represents the static energy release rate, and g(v) is the velocity function, which satisfies:

      When the excess energy E exceeds the critical value Ec, the system becomes unstable and a rockburst occurs. Experimental studies have shown that the excess energy release rate

      (EERR) is an important indicator for measuring the severity of a rockburst [24]:

      Here, cR represents the Rayleigh wave velocity, and cd represents the expansion wave velocity.

      Freund pointed out that the dynamic energy release rate Gd(v) is a function of the crack velocity v. The function g(v) indicates that when the crack expands at a low speed, the energy release rate changes gradually; but when the speed approaches the Rayleigh wave velocity of the material, g(v) will increase sharply, meaning

   4. The coupling effect between dynamic disturbances and static stress fields

      The response of deep rock masses under dynamic disturbances is the result of the coupling effect between the static stress field and the dynamic stress field. The constitutive relationship of rocks considering combined static and dynamic loading can be expressed as:

      that the energy supply efficiency required for crack propagation increases dramatically.

      Here, Cijkl represents the fourth-order stiffness tensor, p is the plastic strain, and d is the

      characteristic parameters of microseismic signals can serve as important indicators for rockburst

      damage strain. The additional damage evolution equation caused by dynamic perturbation is:

      warning. Based on the research of acoustic emission experiments, the identification features of rockburst precursors include:

      The continuous decline of the b value (the slope in the Gutenberg-Richter law)

      Here, Yd represents the dynamic damage energy release rate, Sd and sd are dynamic damage parameters, is the stress rate, 0 is the reference stress, and m is the rate sensitivity index.

      The strength characteristics of rocks under combined dynamic and static loading can be described by the dynamic enhancement factor:

      Here, d represents the dynamic strength, s

      A specific combination pattern of the average frequency and the rising angle

      The accelerated growth of the acoustic emission count rate

      The sudden change point of cumulative energy

      For engineering scenarios affected by dynamic disturbances, such as blasting excavation of tunnels, a dynamic stability criterion can be established based on the parameter resonance theory [26]:

      represents the static strength, is the rate- sensitive coefficient, is the strain rate, and 0 is the reference strain rate.

      For deep-buried caverns, the influence of dynamic disturbances on the failure zone depends on the side pressure coefficient =h/v (the ratio of horizontal stress to vertical stress). Numerical simulation [25] indicates:

      When < 0.5, dynamic disturbances mainly induce spalling failure, and the energy release cycle is longer.

      When 0.5 < < 2.0, dynamic disturbances are prone to cause strain-type rockburst, with a sudden and significant release of energy.

      When > 2.0,the location of rockburst and the failure zone are less affected by dynamic disturbances.

   5. A comprehensive discussion on theory and engineering applications

      The research on rockburst theory ultimately serves engineering practice, providing scientific basis for the stability assessment, disaster warning and prevention of deep underground engineering. Based on the aforementioned mechanical theories, a multi-index comprehensive evaluation system for rockburst susceptibility can be established:

      Here, S(, ) represents the stability function, which is related to the frequency ratio and the damping ratio . By controlling the blasting parameters (such as the amount of explosives and the timing of initiation), the dynamic load frequency can be avoided from the sensitive frequency band of the rock mass, effectively reducing the risk of rockburst.

4. CONCLUSIONS AND OUTLOOK

This thesis conducts a systematic theoretical study on the problem of rockburst under dynamic loading and triaxial loading/unloading. From two core dimensions – the evolution of fractures under triaxial loading/unloading and the sudden change of fractures induced by dynamic loading – it analyzes the mechanical mechanisms of rockburst’s emergence, triggering and instability. Combined with theoretical derivation and experimental conclusions, a theoretical method system for rockburst susceptibility evaluation and engineering prevention and control is formed.

1. 1. In the research on the triaxial loading and unloading effects, based on the continuous damage mechanics, the rock damage constitutive

      and evolution equations with the intermediate principal stress effect were constructed. The

      critical condition for the transformation of the rock failure mode from shear to plate fracture, dominated by the ratio of /, was clarified.

      Here, RBPI represents the rockburst susceptibility index, i is the weight coefficient, Ut is the total input energy, and c is the uniaxial compressive strength. Each weight coefficient can be determined through engineering case analysis or machine learning methods.

      In actual engineering monitoring, the

      Combined with the tensile-compression composite initiation criterion, the relevant formulas for crack initiation and propagation were derived. It was confirmed that the acceleration of crack propagation under unloading is an important precursor of rockburst. From the perspective of energy evolution, the total energy composition of the rock system was

      decomposed, and the evaluation index for rockburst tendency and the calculation method for excess energy were defined. The corresponding laws between the rock failure mode, energy release characteristics and rockburst tendency under low, medium and high confining pressures and true triaxial unloading were revealed. It was clearly stated that the sudden energy release under high confining pressure and true triaxial unloading is the key reason for the high occurrence of rockburst.

      1. For the research on dynamic instability caused by dynamic loading, a dynamic disturbance mechanical model combining harmonic loading and static stress field was established. Based on the theory of parametric resonance, the motion equation of rock mass with fractures was derived, the conditions for system parametric resonance instability were clarified, and it was found that the dynamic amplification effect when the disturbance frequency is close to the rock’s natural frequency would significantly promote the expansion of fractures. Combined with Freund’s dynamic fracture theory, the coupling relationship between crack propagation speed and dynamic stress intensity factor was clarified, the sudden transformation mechanism of cracks from slow accumulation to instantaneous instability under dynamic loading was revealed, the formula for energy release rate of dynamic crack propagation was derived, and it was pointed out that the energy release efficiency would sharply increase when the crack approaches the Rayleigh wave velocity. Based on the excess energy theory, the energy criterion for dynamic rockburst was established, and the excess energy release rate was used as the core indicator to measure the severity of rockburst.
      2. In addition, this article also analyzed the coupling effect of dynamic and static stress fields, constructed the constitutive and dynamic damage evolution equations under combined dynamic and static loading, clarified the influence of rate- sensitive coefficients on the dynamic strength of rock masses, and determined the transformation rules of different fracture patterns in deep-buried caverns under dynamic disturbances dominated by the side pressure coefficient. Finally, by integrating the aforementioned theoretical results, a multi-index comprehensive index system for evaluating the tendency of rockburst was constructed. The identification features of rockburst precursors in microseismic and acoustic emission signals were determined. A dynamic stability criterion based on parameter resonance theory was proposed. It was clearly stated that by controlling the blasting parameters to avoid the sensitive frequency band of the rock mass, the risk of rockburst can be effectively reduced. This provides comprehensive theoretical basis and method support for the evaluation of rockburst stability, disaster warning and

prevention in deep underground engineering, and also lays a foundation for the engineering application and optimization of rockburst theory in the future.

REFERANCE

1. Li H, He M, Cheng T, et al. “Effects of intermedi ate principal stress on strainburst in granite: Insight s from true-triaxial unloading experiments and PFC 3D-GBM simulations, “International Journal of Min ing Science and Technology, vol. 36, no. 2, pp. 29 5-311, 2026.
2. Zhang Y Y, Xu W, Zhao G, et al. “Mechanical be haviors and energy evolution of sandstone under tri axial cyclic loading,” J. Environmental Earth Scien ces, vol. 84, no. 20, pp. 591-591, 2025.
3. Hao J, Fang S, Zhao W, et al. “Mechanical respon ses and time-dependent multifractal behaviors of bri ttle rock under true triaxial stress,” J.Journal of Ro ck Mechanics and Geotechnical Engineering, vol. 1 7, no. 10, pp. 6413-6427, 2025.
4. Wang G, Zhu S, Liu J, et al. “Failure Process of Soft and Hard Coal Composite Structure Subjected to Dynamic Disturbance: Insights from True Triaxia l Experiment,” J. Rock Mechanics and Rock Engin eering, vol. 59, no. 2, pp. 1-23, 2025.
5. Qiao Z, Li K, Li M, et al. “Experimental study of Rockburst in structurally controlled deep-buried ro adways under staticdynamic coupling via true tri axial tests,” J. Theoretical and Applied Fracture M echanics, vol. 141, no. PA, pp. 105244-105244, 20

   26.

6. Wang Y, Liu D, Karakus M, et al. “The Influence of Cyclic Loading Amplitude on Delayed Rockburs t: Insights from True Triaxial Disturbance Experime nts,” J . Rock Mechanics and Rock Engineering, v ol. 59, no. 2, pp. 1-26, 2025.
7. Zhang Y, Xia Y, Huang J, et al. “Experimental inv estigation on the fracture process and infrared radia tion characteristics of structure rockburst under gra dient loading,” J. Infrared Physics and Technology,

   pp. 105565-105565, 2024.

8. Zhang R, Liu Y, Zhu L, et al. “Failure characterist ics and energy evolution process of delayed and in stantaneous basalt rockburst under true triaxial cond itions,” J. International Journal of Rock Mechanics and Mining Sciences, 83105909-105909, 2024.
9. Gu L, Zhou Y, Xiao Y, et al. “Effect of Initial Str ess Difference on Hard Rock Fracture Mechanism Under True Triaxial Excavation Unloading Stress P aths,” J. Rock Mechanics and Rock Engineering, v ol. 57, no. 12, pp. 2024.
10. Feiyue S, Jiaqi G, Junqi F, et al. “Experimental st udy on rockburst fragment characteristic of granite under different loading rates in true triaxial conditi on,” J. Frontiers in Earth Science, 2022.
11. Jin Xiaochuan, Zhou Zonghong, Zhang Yaqi, et al. “Method for Determining the Depth of Rockburst a nd Its Application,” J. Mining Research and Devel opment. vol. 33, no. 01, pp. 69-72, 2013.
12. Wang Susheng, Yang Shengqi, Zhang Qiang, et al. “A Plastic-Piezoelectric Localized Damage Constitut ive Model for Raw Coal,” J. Journal of Engineerin g Science, vol. 47, no. 04, pp. 620-627, 2025.
13. Xu Mengfei, Jiang Annan, Shi Hongtao, et al. “Re search on Rock Mass Elastoplastic Damage Model Based on Hoek-Brown Criterion and Its Stress Refl ection Algorithm,” J. Engineering Mechanics, vol. 37, no. 01, pp. 195-206, 2020.
14. Liu Hongtao, Liu Qinyu, Chen Zihan, et al. “Rese arch on Dynamic Mechanical Response Characterist ics and Dynamic Damage Constitutive Relationship

    of Coal Samples under Confined Pressure,” J/OL. Geomechanics, pp. 1-13, 2026.

15. Zhang Xiangdong, Wang Hao, Jing Pengfei. “Explo ring Macroscopic Fracture Laws Based on “Equival ent Damage” of Rock,” J. Journal of Geohazard an d Disaster Prevention in China, vol. 31, no. 03, pp. 117-125, 2020.
16. Luo Yong, Gong Fengqiang. “Research Progress an d Prospects on Plate Fracture Failure of Surroundin g Rock in Deep Hard Rock Roadways,” J. Coal S cience and Technology, vol. 50, no. 06, pp. 46-60,

    2022.

17. Zhao Penggeng, Deng Hui, Yu Junqi. “Research on Microscopic Damage Constitutive Model Based on Initial Damage and Residual Strength,” J/OL. Geo logical Disasters and Environmental Protection, pp. 1-11, 2026.
18. Zhang Ying, Miao Shengjun, Guo Qifeng, et al. ” Microscopic Energy Evolution of Stress Threshold Values in Granite under Cyclic Loading and Rockb urst Propensity,” J. Journal of Engineering Science, vol. 41, no. 07, pp. 864-873, 2019.
19. Yu J L, Qiao D L, Chao M H, et al. “Excess ene rgy characteristics of true triaxial multi-faceted rapi d unloading rockburst,” J. Journal of Central South University, vol. 31, no. 5, pp. 1671-1686, 2024.
20. Duan Minke, Jiang Changbao, Guo Xianwei, et al. “Experimental Study on Coal Rock Mechanics and Damage Characteristics under True Triaxial Cyclic

    Loading,” J. Chinese Journal of Rock Mechanics a nd Engineering, vol. 40, no. 06, pp. 1110-1118, 20

    21.

21. Deng, Gu . “Buckling mechanism of pillar rockbur sts in underground hard rock mining,” J. Geomecha nics and Geoengineering, vol. 13, no. 3, pp. 168-1

    83, 2018.

22. Qin Hao. “Research on the Mechanism of Tunnel Surrounding Rock Instability and the Mechanism of Impact Mine Pressure,” D. China University of M ining and Technology, 2008.
23. FREUND L B. “Dynamic fracture mechanics,”M. C ambridge: Cambridge University Press, 1990.
24. Wang Hao, Yao Yi, He Liang, et al. “Research on Accurate Three-Dimensional Stress Field Inversion and Spatiotemporal Evolution Mechanism under De ep Non-uniform Rock Mass Mining Conditions,” J. Chinese Mining Engineering, vol. 54, no. 06, pp. 5

    3-60, 2025.

25. Li X, Weng L. “Numerical investigation on fracturi ng behaviors of deep-buried opening under dynami c disturbance,” J. Tunnelling and Underground Spa ce Technology incorporating Trenchless Technology Research, pp. 61-72, 2016.
26. Deng J, Gong F, Zhu H. “Tunnel rockbursts induce d by dynamic disturbances: mechanism and mitigati on,” J. IOP Conference Series: Earth and Environm ental Science, vol. 1331, no. 1, 2024.