Introduction

As metal mines enter deep mining, the geological structure of the deposit and the conditions of the ore body’s existence deteriorate, and the complex high-stress environment is prone to inducing rock bursts, roof-fall, bottom drums, and other disasters and accidents of safety production in the stopes and roadways1,2,3. Rock damage is the process of internal micro-cracks initiation, expansion and coalescence, while the coalescence of microcracks is often viewed as a cumulative damage process in rocks, which determines the rock’s destabilization, destructive process, and sustainable usage capacity. Therefore, accurately characterizing the evolution of rock fracture damage is crucial for stability monitoring in deep underground mining engineering and disaster early warning4,5,6.

From a macroscopic failure perspective, the high stress damage characteristics of rock have been extensively studied7,8,9, primarily focusing on avoiding damage by reducing stress concentrations through decompression10,11,12. However, exploring the mesoscopic damage behavior during macroscopic failure serves as the foundation of the support design of high stress rock mass. In recent years, acoustic emission, industrial CT scanning and nuclear magnetic resonance have emerged as the most commonly applied exploration methods for investigating the micro-mechanical properties13,14,15,16. The propagation of fractures in brittle rocks is significantly influenced by the stress state and heterogeneity of the rock matrix. Additionally, temperature and seepage also contribute to the expansion of fractures17. In the study of permeability enhancement in rock formations, hydraulic fracturing serves as a widely adopted technique for carbonate reservoir modification, which increases the permeability of reservoirs by injecting high-pressure fluids into the formation to create fractures in the formation18. Lai et al.19 conducted a uniaxial compression test on rocks, revealing that an increased loading rate enhances rock fragmentation, shifts the failure mode from tensile to shear, and positively correlates with the fractal dimension of both the microstructure and macrostructure of rock fractures. Liu et al.20 argue that an escalating stress gradient expedites rock deterioration, facilitates swift fracture along dominant main cracks, triggers a shift from tensile to shear failure, and leads to elevated elastic energy storage and release rates during rockbursts.

In the context of comprehending mesoscopic damage behavior leading to macroscopic failure in high-stress rock masses, the Bonded Block Model (BBM) technique emerges as an alternative approach, categorized under Discrete Element Models (DEM)21. Unlike conventional methods emphasizing stress concentrations and decompression strategies, BBM offers a more nuanced and intricate portrayal of the rock mass structure. Zhang et al.22 employed a 3D bonded block model (BBM) to assess the strength and permeability of fractured rock mass, validating its reliability and revealing that factors such as low fracture stiffness, friction angle, and steep fractures contribute to the weakening of rock mass, thereby providing novel insights into fractured rock properties. Sinha et al.23 emphasized that the importance of inelastic blocks in capturing the transition from strain-softening to pseudo-ductile behavior while investigating pillar damage mechanisms and rock-support interaction in rock masses, utilizing BBM approach. As evidenced in prior research, BBM effectively describes rock mechanics, taking into account both macro and micro-sized fracture characteristics. Since the exploration of the fractal dimension of fractures enhances our comprehension of rock failure mechanisms, strength, stability analysis, and the assessment of physical parameters. Further investigation into the fractal dimension of fractures can provide crucial theoretical underpinnings and practical guidance for the design and safety assessment of rock engineering projects24,25. The integration of BBM’s macro–micro perspective with insights gained from fractal dimension exploration fosters a holistic approach to understanding and applying rock mechanics principles.

In line with this approach, uniaxial compression tests were conducted to investigate the failure mode and acoustic emission (AE) signals during the fracture processes in rock masses. The examination of fracture processes involved utilizing a proposed bonded block model within a 3D discrete element method. Subsequently, a fractal damage renormalization group mechanical model was employed to assess the fractal dimension in critical failure states. Expanding the analyses of static fracture process, simulations were conducted to replicate the fracturing of synthetic samples characterized by various block sizes, shapes, and hole defects. This paper elucidates and discusses the mechanisms governing the evolution of fractal dimensions in deep rock fracture processes, along with quantitative methods for evaluating rock mass degradation.

Brief background of high stress induced fracture damage evolution of rock mass

Site overview

Sanshandao gold mine, as the earliest large-scale gold deposit of fracture zone alteration rock type in China, exhibits a remarkably developed tectonic structure, primarily comprising faults. The Sanshandao-Cangshang fault stands out as the largest, while the NW-directed Sanshandao-Sanyuan fault serves as a secondary fault, belonging to the fault structure resulting from re-activation activity post-metallogenetic period26. The secondary faults are predominantly distributed in the central section of the Cangshang fracture zone and the NNE-NEE oriented fractures with the lower plate, serving as the primary ore-controlling structure of the Sanshandao Mine27. The rock mass within the fracture zone undergoes intense weathering and tectonic movements, resulting in highly developed fractures and a weakened surrounding rock strength. Consequently, issues such as collapsing and roofing frequently arise during roadway excavation and service operation28.

Causes of roadway destabilization

The deep rock mass of Sanshandao gold mine is characterized by “three highs and one disturbance”: high geostress, high geothermal temperature, high water pressure, and intense mining disturbance29. Measurements of geostress field at the − 960 m level reveal a maximum horizontal stress of 45.31 MPa, with projections indicating a further increase in stress as mining depth progresses. With the progression of mining depth, it is anticipated that the maximum horizontal principal stress will exceed 50 MPa30. As a coastal mine with ore bodies extending beneath sea level, the Sanshandao gold mine encounters a complex water-rich environment attributed to intricate fault structures and well-developed joints and fractures in the rock mass31,32. Fault activity leads to a heat-gathering effect, causing deep high-temperature heat flow to ascend, thus exposing underground mining activities to a high-stress corrosive environment. Although pipe slit anchor support has been historically employed in deep mining, intense pressure activities and water surges frequently result in corrosion, dislodgment, slipping, and breakage of the anchors, leading to damage such as falling, slipping, breaking, and perimeter rock sheet gangs, thereby significantly affecting roadway stability (Fig. 1).

Figure 1
figure 1

(a). Crushing of the roof under high stress, (b) expansion of fractures in the surrounding rock due to mining activities.

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Experimental studies of damage mode of rock mass

The granite utilized in the experiment originated from the − 930 m level of the Xinli mining area in Sanshandao. The rock, predominantly silver–white, comprises quartz, plagioclase, and biotite as its main constituents. Complete fragments of rock core were precisely processed into cylindrical specimens with dimensions of Φ50 mm × Φ100 mm, adhering to the guidelines set by the International Society of Rock Mechanics (IRSM) and relevant regulations in China for rock mechanics testing. In our study, we aimed to investigate the effect of characteristics of fracture propagation under uniaxial compression. To accomplish this, a total of 20 samples were selected and randomly divided into 4 groups, with each group containing 5 samples. The items examined in the study encompassed stress–strain curves, specimen failure mode, and characteristics of the AE signals during the destruction phase of the specimen. For the experimental apparatus, the ZTR-276 rock triaxial test system (depicted in Fig. 2) was utilized, which was calibrated in accordance with standard procedures to ensure accuracy and precision. Stringent quality control measures were implemented, ensuring that the sample size error was within 0.03 mm, the end unevenness did not exceed 0.05 mm, and the deviation between the end face and the axis was limited to 0.25°.

Figure 2
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ZTR-276 rock triaxial test system.

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The acoustic emission signal represents the energy released during the process of crack initiation to propagation with rock deformation. This process of partial damage evolution is intricately linked, and the cumulative energy throughout the entire deformation and failure process can be computed by aggregating the energy generated at each deformation stage. From the uniaxial compression test of 20 groups of rock samples, four typical failure modes were selected. As shown in Fig. 3, UC-I and UC-II represent typical tensile-shear failures, UC-III depicts a typical pure tensile failure, and UC-IV illustrates a typical pure shear failure.

Figure 3
figure 3

Four typical uniaxial compression failure specimens, (a) UC-I, (b) UC-II, (c) UC-III, (d) UC-IV.

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Figure 4 illustrates the time-varying curves of stress and acoustic emission (AE) energy signals observed during typical failure processes of diverse rock samples. The AE signal is discernibly segmented into distinct phases throughout the compression process, commencing from the initial compression phase and culminating at the peak load. Specifically, the OA phase corresponds to a period of stable loading, whereas the AB phase signifies the initiation of fracture formation, and the BC phase marks the progression and expansion of fractures. The findings unveil discernible patterns in the acoustic emission signals across different rock samples during the failure process. Based on the distribution of acoustic emission signals, the loading process can be segmented into three stages: stable loading, fracture formation, and fracture expansion. During the stable loading stage, a minor signal emerges initially, gradually diminishing as crack closure ensues. Transitioning to the fracture formation stage, shear stress surpasses the shear strength in weak areas of the rock mass, triggering the formation and propagation of micro-cracks under increasing driving force. This process is accompanied by the emission of small elastic waves and the gradual emergence of minor acoustic emission signals. However, the energy accumulation curve progresses gradually during this stage. In the fracture expansion stage, elevated stress levels cause shear stress in the rock mass to exceed its shear strength, resulting in the connection of numerous primary and incipient cracks to form macroscopic cracks. As these cracks propagate, they ultimately lead to the destabilization and fracturing of the entire sample. Throughout this stage, elastic waves continuously emit energy, resulting in high-value signals, while the energy accumulation curve exhibits a marked increase, concomitant with the progression of macroscopic cracks penetration and expansion.

Figure 4
figure 4

The temporal evolution of stress and acoustic emission across various samples, (a) UC-I, (b) UC-II, (c) UC-III, (d) UC-IV.

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The crack-opening mechanism in destabilized ruptures of rocks is identified by analyzing acoustic emission waveform parameters, particularly the RA (rise time/amplitude) and AF (average frequency) values. RA represents the ratio of rise time to amplitude in an acoustic emission event, while AF characterizes the ratio of ringer counts to duration, effectively distinguishing rupture types33. Tensile rupture is characterized by robust transient signals, short duration, and rise time, alongside high ringer counts and amplitude, resulting in low RA and high AF values. In contrast, shear rupture exhibits an opposite pattern. Figure 5 depicts a scatter plot of the RA-AF distribution for rock sample damage, with the diagonal line delineating the boundary between tensile and shear cracks. Cracks above the line indicate tensile rupture, while those below signify shear rupture34, as depicted in Fig. 5.

Figure 5
figure 5

Cracks classification evolution of uniaxial compression, (a) UC-I, (b) UC-II, (c) UC-III, (d) UC-IV.

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In Fig. 5, the crack classification results obtained through RA and AF statistics correspond to the macroscopic failure types of rock, thus validating these statistical outcomes. Under uniaxial compression conditions, higher RA distribution values primarily indicate the presence of tensile cracks, whereas elevated AF values are indicative of a predominance of shear cracks. Moreover, the coexistence of high AF and RA values suggests the presence of tensile-shear composite cracks. During the compaction phase, both tensile and shear crack signals coexist, with a greater prevalence of tensile cracks initially. As random crack initiation and propagation occur, there is an increase in the number of tensile crack signals, particularly those with higher acoustic frequency (AF) values. However, this increase in tensile cracks is accompanied by a relative rise in the proportion of shear cracks, as some of the tensile cracks may evolve into shear cracks or be accompanied by shear cracks in the vicinity. Nevertheless, the overall prevalence of tensile cracks remains higher during this stage. Approaching the peak stress phase, high AF value signals indicative of tensile cracks emerge, alongside a simultaneous increase in shear crack signals. Following the peak stress, the prevalence of tensile crack signals with elevated AF values diminishes, while signals associated with shear cracks, characterized by high RA values, intensify. This progressive trend underscores a growing proportion of shear cracks throughout the process, despite the overall predominant of tensile cracks. Indeed, the prevalence of tensile cracks exceeds that of shear cracks throughout all stages, consistent with the macroscopic failure mode of rocks. Specifically, the tensile-shear failure mode exhibits elevated AF and RA values (Fig. 5a,b), while predominant splitting failure demonstrates high AF values (Fig. 5c), and overall shear failure manifests high RA values (Fig. 5d). Figure 5 presents the AF/RA value of the cumulative acoustic emission signal recorded during the pressurization process of the rock sample. To facilitate comprehension, Table 1 has been incorporated to discernibly differentiate the proportions of tensile and shear cracks at each stage, as the individual AF/RA values depicted in Fig. 5 may be challenging to interpret in isolation.

Table 1 The proportion of shear and tensile cracks in each stage of uniaxial compression test.
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Mathematical model of fracture fractal damage-renormalization group

Damage fractal mechanic model

According to the theory of damage mechanics, the damage metric is defined as \(D(\varepsilon ) = \int_{0}^{\varepsilon } {q(\varepsilon )} d\varepsilon\)35, wherein q(ε) signifies the Weibull probability density function pertaining to the constitutive relationship, and ε denotes the strain of the rock. Based on the brittle failure principle of Weibull36, D(ε) also adheres to a Weibull distribution, assuming strain equivalence.

$$ \frac{1}{1 - D(\varepsilon )} = \frac{1}{{1 - \int_{0}^{\varepsilon } {q(\varepsilon )} d\varepsilon }} = \exp \left[ {\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{b} } \right] $$
(1)

In Eq. (1), εp represents the peak rock strain, while b serves as a shape parameter. Given that the self-similarity characteristics exhibited by the rock damage process during fracture expansion, b can be deemed as a rock-specific parameter. The definition of b is linked to the fractal dimension, allowing for its interpreted as the fractal dimension df37 of D(ε). Upon substitution of b = df into Eq. (1), a simplified fractal damage evolution equation for rocks is derived.

$$ D(\varepsilon ) = 1 - \exp \left[ { - \left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{{d_{f} }} } \right] $$
(2)

Using the damage constitutive equation, the damage model expressed in terms of the fractal dimension of the fracture is as follows.

$$ \sigma (\varepsilon ) = E\varepsilon \exp \left[ { - \frac{1}{{d_{f} }}\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{{d_{f} }} } \right] $$
(3)

Fracture damage renormalization group model

When considering a small unit of rock mass, it is commonly assumed that its failure probability follows a quadratic Weibull distribution, which can be expressed as,

$$ P_{0} (\sigma (\varepsilon )) = 1 - \exp \left[ { - \left( {\frac{\sigma (\varepsilon )}{{\sigma_{f} }}} \right)^{2} } \right] $$
(4)

where σf represents the peak stress, while σ(ε) denotes the stress value at the damage state.

The four basic units in Fig. 6 combined to constitute a cell, exhibiting five distinct configurations and comprising a total of 16 states. Each configuration is associated with a unique state. The symbol of ⊗  in Fig. 6 indicates failure units, whereas the symbol of ⊕  represents un-failure units.

Figure 6
figure 6

Five failure/un-failure states of primary cell, (a) un-failure, (b) one unit failure, (c) two units failure, (d) three units failure, (e) all units failure.

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Based on Eq. (4), the probability of the complete destruction of all four cells is \(P_{0}^{4}\), while the likelihood of three cells being fractured and one remaining intact is represented by \(P_{0}^{3} (1 - P_{0} )\). Considering the frequency of occurrence of each scenario, a correlation emerges between the destruction probability of the first-level protocell P1 and the destruction probability of the first-level cell P0.

$$ P_{1} = P_{0}^{4} + 4P_{0}^{3} (1 - P_{0} ) + 2P_{0}^{2} (1 - P_{0} )^{2} { = }2P_{0}^{2} - P_{0}^{4} $$
(5)

During the renormalization process, four first-order primitive cells are transformed into four second-order units. Following renormalization, the same statistical analysis is applied to determine the failure probability P2 of the second-order primitive cell. By iterating this process, the failure probability of the n-order primitive cell can be derived.

$$ P_{n + 1} { = }2P_{n}^{2} - P_{n}^{4} $$
(6)

Equation (6) can be rewritten as,

$$ f(x) = 2x^{2} - x^{4} $$
(7)

The function f(x) = x can be represented as x = 2x2 − x4. Within the range of 0 < x < 1, three stationary points are identified at x = 0, 0.618, and 1. The corresponding values of df/dx​ evaluated at these points are 0, 1.528, and 0, respectively. 0 and 1 are stable stationary points due to \(\left| {df/dx} \right|\)< 1, while 0.618 is an unstable stationary point. f = P∗ = 0.618 represents the critical probability for the rock to produce a fracture, resulting in failure. As observed in Fig. 7, when P < P∗, no damage occurs after a certain number of iterations. Conversely, for P > P∗ and a sufficient number of iterations, damage always occurs.

Figure 7
figure 7

Relationship between the n + 1th probability of protocell fragmentation Pn+1 and the probability of the nth level protocell fragmentation Pn

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Combining Eqs. (3), (4) and critical failure probability P* = 0.618, the failure criterion of fractured rock mass under damage condition can be obtained.

$$ P_{0} (\sigma (\varepsilon )) = 1 - \exp \left[ { - \left( {\frac{{E\varepsilon \exp \left[ { - \frac{1}{{d_{f} }}\left( {\frac{\varepsilon }{{\varepsilon_{p} }}} \right)^{{d_{f} }} } \right]}}{{\sigma_{f} }}} \right)^{2} } \right]{ = }\left\{ \begin{gathered} < 0.618, \, stable \hfill \\ { = }0.618, \, critical \hfill \\ > 0.618, \, failure \hfill \\ \end{gathered} \right. $$
(8)

The rock sample subjected to a peak stress of 110 MPa is analyzed. The failure probability of the fractured rock mass under a uniaxial stress state is determined concerning the uniaxial stress state is determined using Eq. (8), as illustrated in Fig. 8. Various fracture fractal dimensions are represented by different curves. In Fig. 8, the failure probability of the fractured rock mass below the red dotted line indicates stability, whereas above the line, it suggests instability. Figure 7 clearly demonstrates a negative correlation between the fractal dimension of rock mass fractures and the loading limit, indicating that a smaller fractal dimension corresponds to a higher ultimate bearing capacity during compression, while a larger fractal dimension diminishes it. Theoretical calculations reveal that rock mass with a fracture fractal dimension of 1.1, achieves an ultimate bearing capacity of 108 MPa in a stable state, whereas a dimension of 1.5, results in a reduced capacity of 88 MPa. In the context of deep mining excavation, greater rock mass fracturing is associated with a weaker bearing capacity, consistently demonstrating a worsening trend.

Figure 8
figure 8

The failure probability of fractured rock mass varies with stress state axial stress of fractured rock mass with different fractal dimensions.

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Numerical simulation of deformation process in fractured rock mass

Effect of block size on deformation and failure of fractured rock mass

The Bonded Block Model (BBM) exhibits the advantage of requiring a limited number of input parameters and facilitating ease of calibration. In this study, a manual back-analysis was undertaken to replicate the stress–strain curves presented by Sinha and Walton38 using FLAC3D. The microscopic parameters pertaining to the contact surface in BBM are primarily derived through a rigorous process of parameter calibration. The macroscopic mechanical behavior of rock materials is fundamentally governed by the mechanical characteristics of the interfaces between blocks on the mesoscopic scale. Notably, these mechanical parameters of the contact surface cannot be directly ascertained through laboratory experimentation. Instead, they are inferred from macroscopic physical properties, such as peak strength and elastic modulus, obtained from uniaxial compression tests of rock samples, as detailed in Table 2. The determination of microscopic parameters of the contact surface relies heavily on the application of control variables and an iterative approach involving trial and error methods. The calibration procedure for the mesoscopic parameters of the contact surface comprises the following steps. To determine the mesoscopic mechanical behavior of BBM, firstly, a standard rock sample with dimensions of 50 mm in diameter and 100 mm in length was employed, revealing an average block size of 6 mm. Subsequently, the blocks within BBM are modeled as linear elastomers, and their mechanical properties are outlined in Table 1. For the contact surface, the Mohr–Coulomb constitutive model is adopted, with the primary calibration parameters comprising normal stiffness (kn), tangential stiffness (ks), cohesion (Jc), tensile strength (JT), and internal friction angle (φ). Following Gao’s methodology39, initial estimates for these parameters were set as kn = 104 GPa, kn/ks = 2, Jc and JT both equal to 10 MPa, and the internal friction angle ranging from 10° to 30°. Finally, a uniaxial compression test was conducted on BBM, and the resulting stress–strain curve from numerical simulations was compared to experimental data from the laboratory. In this calibration process, the peak strength and elastic modulus served as key indicators. Through iterative adjustments using the control variable method and repeated trials, the calibration parameters were refined until the relative error between the calculated and experimental peak stress and elastic modulus was maintained within 5%. These optimized parameters were then designated as the final mesoscopic calibration parameters for the contact surface. Table 3 outlines the calibrated micromechanical parameters of the contact interface, with the corresponding stress–strain curves depicted in Fig. 13.

Table 2 Rock macro-mechanical parameters.
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Table 3 Meso-mechanical parameters of block contact interfaces.
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Previous studies have largely overlooked the impact of variations in block size on rock deformation. However, this study has provided valuable insights into the effects of altering block size. Figure 9, 10, and 11 illustrate the distribution patterns of planar fractures in rock specimens under different slicing profiles at peak stress levels. The findings indicate that larger block sizes contribute enhance integrity of the rock mass during failure. For instance, when the block size is 6 mm, fracture propagation area is primarily confined to the lower end of the sample, demonstrating minimal overall penetration. Conversely, as the block size diminishes to 4 mm, there is a discernible transition towards the formation of randomly dispersed discrete cracks, which become the primary mode of penetrating. A further decrease in size to 2 mm reveals more pronounced macroscopic shear fractures, as evident in Fig. 10e, f.

Figure 9
figure 9

Distribution of fractures in different profiles of 6 mm synthetic rock samples, (a) 0°profile, (b) 30°profile, (c) 60°profile, (d) 90°profile, (e) 120°profile, (f) 150°profile.

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Figure 10
figure 10

Distribution of fractures in different profiles of 4 mm synthetic rock samples, (a) 0°profile, (b) 30°profile, (c) 60°profile, (d) 90°profile, (e) 120°profile, (f) 150°profile.

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Figure 11
figure 11

Distribution of fractures in different profiles of 2 mm synthetic rock samples, (a) 0°profile, (b) 30°profile, (c) 60°profile, (d) 90°profile, (e) 120°profile, (f) 150°profile.

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The images presented in Figures 9, 10, and 11 were subjected to binarization using ImageJ, enabling the computation of fracture fractal dimensions via the box counting method. The radar charts in Fig. 12 illustrate the variation in fractal dimensions of fractures across diverse profiles and block sizes. Due to the cylindrical sample geometry, the radar charts exhibit axial symmetry. As depicted in Fig. 12a, a block size of 6 mm yielded a range of fractal dimensions for fractures, spanning from 1.124 to 1.202, with a mean of 1.158, a standard deviation of 0.034, and a coefficient of variation of 2.95%, traversing from 0°to 150° profile. In Fig. 12b, a block size of 4 mm yielded a range of fractal dimension, varying from 1.1438 to 1.2699, with a mean of 1.2011, a standard deviation of 0.046, and a coefficient of variation of 3.87%. Similarly, in Fig. 12c, a reduction in block size to 2 mm resulted in a higher range of fractal dimensions, spanning from 1.2015 to 1.3031, with a mean of 1.2591, a standard deviation of 0.041, and a coefficient of variation of 3.29%. Moreover, the fractal dimension of fractures demonstrated an inverse relationship with block size, as smaller block sizes correlated with larger fractal dimensions. Additionally, the increase in both the standard deviation and coefficient of variation of the fractal dimensions across different profiles within the rock sample suggests an enhancement in macroscopic anisotropy during rock failure.

Figure 12
figure 12

Radar map of the fractal dimension of synthetic rock fractures, (a) block size of 2 mm, (b) block size of 4 mm, (c) block size of 6 mm.

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Upon comparing the full stress–strain curves of synthetic rock samples subjected to uniaxial compression (as shown in Fig. 13), it is evident that the block size exerts a significantly influence on the mechanical properties of these samples. Specifically, for a block size of 6 mm, the synthetic rock samples attain a peak stress of 108.1 MPa, corresponding to a peak strain of 0.34%. As the block size is reduced to 4 mm, a decrease in peak stress to 107.1 MPa is observed, while the peak strain increases to 0.41%. Furthermore, upon further reduction of the block size to 2 mm, the peak stress diminishes to 92.7 MPa, accompanied by an augmentation in peak strain to 0.54%. Under identical mechanical parameters for blocks and contact surfaces, an increase in block size results in a reduction of the ultimate bearing capacity and Young’s modulus of the synthetic rock samples. Additionally, a correlation between peak stress and the fractal dimension at failure of synthetic rock samples reveals that 108.1 MPa corresponds to a fractal dimension of 1.158, 107.1 MPa to 1.2011, and 92.7 MPa to 1.2591. These findings are in close accordance with the critical instability condition for fractured rock masses, as derived from Eq. (8). The outcomes of the numerical simulation validate the applicability of the critical instability criterion for fractured rock masses.

Figure 13
figure 13

The full stress–strain curves of synthetic rock samples subjected to uniaxial compression.

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Effect of block shape on deformation and failure of fractured rock mass

The BBM model incorporates two distinct methodologies for constructing rock samples, the tetrahedral block assembly and the voronoi block assembly. In Section "Effect of block size on deformation and failure of fractured rock mass", a thorough analysis of fracture evolution characteristics in rock samples is conducted, specifically focusing on block size under the tetrahedral block assembly mode. Although the two-dimensional voronoi block assembly has been extensively employed in studies investigating crack propagation and energy evolution in brittle rock samples40,41,42,43, the adoption of the three-dimensional voronoi block assembly in the BBM is relatively scarce43. To assess the comparative influence of these two block assembly techniques on crack propagation in a three-dimensional setting, a synthetic rock sample was created utilizing three-dimensional voronoi block assembly, with a size of 6 mm. The macroscopic and detailed observational parameters for the block and contact surfaces adhered to the specifications outlined in Tables 1 and 2. Similarly, fracture evolution characteristics were observed across six profiles spanning 0° to 150°, with the central axis of the circular column rotated by 180°. Figure 14 illustrates the fracture evolution characteristics of these six profiles under peak stress conditions. Upon comparison with Fig. 9, it is evident that the tetrahedral BBM assembly consistently results in shear fractures, exhibiting typical shear failure patterns in terms of macro-mechanical properties. Conversely, BBM formed through voronoi assembly is more inclined to manifest tensile cleavage failures at the macroscopic level, highlighting notable differences in macroscopic damage between the two assembly methods in a three-dimensional context.

Figure 14
figure 14

Distribution of fractures in different profiles of 6 mm voronoi synthetic rock samples, (a) 0°profile, (b) 30°profile, (c) 60°profile, (d) 90°profile, (e) 120°profile, (f) 150°profile.

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Figure 15 illustrates the impact of block shape on fracture fractal dimension. Analysis of the six profiles reveals that the fractal dimension of both rock samples varies within a narrow range of1.10 to 1.20, exhibiting a consistent pattern of evolution, with the exception for a notable deviation observed in the 120° profile. Turning to Fig. 16, it is evident that the peak stress and elastic modulus of synthetic rock samples constructed with the voronoi block model demonstrate significant elevation compared to those utilizing the tetrahedral block model. Specifically, the peak stress registers approximately 11.84% higher, whereas the Young’s modulus displays an increase of about 18.83%.

Figure 15
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The influence of block shape on fractal dimension.

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Figure 16
figure 16

Stress–strain curves of tetrahedral block and voronoi block.

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In 3DEC-BBM, the monitoring of the acoustic emission process employs the following methods. The numerical model’s sub-contacts are dynamically monitored in real-time utilizing a self-developed Fish function. Upon obtaining the microscopic calibration parameters of the contact surface, the BBM technology repeats the uniaxial compression process of the synthetic rock sample, dividing the compression stages prior to peak load into ten distinct intervals. Upon reaching the predetermined compression load, the stress state of the sub-contact surface is saved and recorded. The criterion for identifying an acoustic emission signal involves the generation of a micro-crack when the normal and tangential stresses of the sub-contact surface surpass the ultimate strength, subsequently recorded as an AE event. Subsequently, the next loading stage is initiated, and the mentioned steps are repeated until the load attains peak strength, thereby simulating the entire acoustic emission signal process. The spatial distribution of three-dimensional acoustic emission events in synthetic rock samples is depicted in Figs. 17 and 18. In Fig. 17, at low initial stress levels (0–0.2σc), the specimen constructed with tetrahedral block assembly displays minimal AE events, which are attributed to the presence of initial micro-cracks, the closure of damage, and micro-structural surface dislocation. As stress gradually increases during the initial stage of elastic deformation (approximately 0.2–0.5σc), following the closure of initial micro-cracks, a linear relationship emerges between the stress–strain curve. The insufficient stress within the specimen prevents the formation of new cracks, with only partial closure occurring. The limited AE events observed are primarily attributed to dislocation between micro-crack surfaces, as well as deformation and slippage between grains. However, during the later stage of elastic deformation (around 0.5–0.7σc), the increased stress levels induce the emergence and expansion of new micro-cracks within the specimen, leading to a gradual increase in AE events, which corresponds to with the growth rate of cumulative AE signals observed in laboratory tests. During the subsequent plastic stage (approximately 0.7–1.0σc), micro-cracks proliferate rapidly, expanding and penetrating further, ultimately resulting in rock failure. The spatial distribution of acoustic emission events corresponds to the failure shape of the rock sample, with a preponderance of events occurring around the macroscopic failure surface. In Fig. 18, for the synthetic rock sample employing voronoi block assembly, the spatial distribution of AE signals remains relatively uniform throughout the loading phase, up until reaching peak stress is reached, accompanied by a moderate growth rate of events. Following the attainment of peak stress, no conspicuous shear fracture surface generation is observed, and the distribution of AE shear signals appears dispersed. It is evident that utilization of tetrahedral block assembly for simulating AE event characteristics during the evolution of rock uniaxial compression damage maintains a high degree of consistency with indoor test results. This reinforces the notion that the generation and propagation of cracks are fundamental contributors to the deterioration of rock mechanical properties.

Figure 17
figure 17

Tetrahedral block synthesized rock sample simulated acoustic emission signals.

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Figure 18
figure 18

Voronoi block synthesized rock sample simulated acoustic emission signals.

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Effect of hole defects on fracture fractal dimension

From a macroscopic perspective, the stability of underground chambers poses a significant concern within geotechnical engineering. At the microscale, voids are prevalent defects within rock masses, intricately governing their mechanical properties. The initiation, propagation, and penetration of cracks surrounding these voids are notably influenced by factors such as size, shape, and prevailing stress conditions.

A synthetic rock sample with a block size of 6 mm underwent a uniaxial compression test, in which a hole with a radius ranging from 3 to 6 mm was drilled at its center. Figure 19 depicts the crack propagation patterns under peak load for various hole radius. The red cracks denote shear cracks, while the blue cracks represent tensile cracks. The crack growth pattern is consistent with AE characteristics, where tensile cracking predominantly drives the uniaxial compression failure process, ultimately leading to overall shear failure at a macroscopic level. Figure 20 compares the constitutive relationship of the test sample with hole defects to that of an intact rock sample. The stress–strain curve reveals that hole defects markedly alter the mechanical damage evolution traits of the rock. For synthetic rock samples with hole radius ranging from 3 to 6 mm, the ultimate bearing capacity decreases by 39.1% to 48.2%. Larger hole radius correlate with lower compressive strength, accompanied by a corresponding decrease in peak strain from 0.34% in intact samples to 0.217% in samples with a 6 mm block size. Moreover, the slope preceding the peak of the curve indicates that hole defects also diminish the Young’s modulus of the sample, thereby degrading the overall mechanical properties of the synthetic rock sample.

Figure 19
figure 19

Distribution of fractures in different profiles of synthetic rock samples with hole defects, (a1–a3) perpendicular to the axis of the hole, (b1–b3) 72° to the axis of the hole, (c1–c3) 54° to the axis of the hole, (d1–d3) 36° to the axis of the hole, (e1–e3) 18° to the axis of the hole, (f1–f3) parallel to the axis of the hole.

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Figure 20
figure 20

Constitutive relationship of synthetic rock samples with various hole defects.

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The computation of the fractal dimension of cracks within distinct profiles of flaw samples, containing holes and subjected to peak load, is performed. In comparison, the average fractal dimension of fractures across the various profiles of the sample exhibits a decrease when compared to that of intact rock. Figure 21 illustrates that, for a fixed hole radius, the profile perpendicular to the hole axis exhibits the highest fractal dimension of fractures. Conversely, a gradual decrease in the angle of the profile’s interception direction and the hole axis direction results in a continuous decline in the fractal dimension of profile fractures, ultimately reaching its minimum value. By substituting the peak stress of rock samples containing holes and the fractal dimensions of various profiles obtained from Fig. 20 into Eq. (8), the variation trend of failure probability of rock mass across profiles is derived, which closely correlates with the variation trend of the fractal dimension. All failure probability values exceed the critical threshold of 0.618. In the profile aligned with the vertical hole axis, the expanded fracture attains its maximum fractal dimension, gradually diminishing as the angle between the profile and the hole axis decreases. This observation implies that, in samples with hole defects, the degradation of rock mass is most pronounced in the vertical hole axial direction, whereas it remains relatively minimal in the direction parallel to the hole axis. Consequently, cavernous defects exert a limited impact on the fractal dimension of samples, primarily affecting the bearing capacity of rock mass.

Figure 21
figure 21

Fracture fractal dimension and failure probability in various sections of hole-defected samples.

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Discussion

Based on the insights from Eq. (8), a negative correlation between the ultimate bearing capacity of intact rock masses and their fractal dimension has been validated in "Effect of block size on deformation and failure of fractured rock mass" section. However, the presence of defects in rock masses frequently leads to deterioration in their ultimate bearing capacity. In the context of deep mining operations in metal mines, various monitoring methodologies are regularly employed to provide early warnings for underground disasters. To investigate the dynamic evolution of fracture zones during deep mining activities, boreholes were drilled in a horizontal haulage roadway extending from − 960 to − 1000 m depth in a gold mine in China. The GD3Q-GA panoramic digital borehole camera system was utilized to conduct field measurements of the fracture zone. Figure 22 illustrates the fracture fitting curve and plane extension diagram. Fracture distribution within the rock mass at the − 960 m level is depicted in Fig. 23, with a fractal dimension of 1.360, whereas the drill core obtained at the − 1000 m level is shown in Fig. 24, with a fractal dimension of 1.422.

Figure 22
figure 22

Crack fitting surface and plane extension.

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Figure 23
figure 23

Borehole fracture spatial distribution and fracture binarization (− 960m level).

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Figure 24
figure 24

Borehole fracture spatial distribution and fracture binarization (− 1000m level).

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As the fractal dimension augments, a marked decrease in the rock mass’s ultimate bearing capacity is observed, accompanied by a pronounced intensification of deterioration. As illustrated in Fig. 8, the uniaxial compressive strength of the rock mass at the − 960 m level attains a value of approximately 98.3 MPa, whereas at the − 1000 m level, it diminishes to about 97.5 MPa. The excavation of roadways or stopes within regions characterized by damaged and fractured rock mass unavoidably exacerbates the deterioration of the rock mass. When the uniaxial ultimate bearing capacity of the rock mass diminishes by approximately 40% to 50%, it underscores that during the excavation of deep rock masses in mines, the influence of several tens of megapascals of geo-stress results in damage and fragmentation of the rock mass. Fractured rock masses pose a significant risk of disasters such as substantial deformation, roof collapse, and surrounding rock collapse, as depicted in Fig. 1. Consequently, it is imperative to intensify monitoring of deep fractured rock masses in mines, ascertain the ultimate strength of fractured rock masses, and estimate the failure probability, thereby ensuring safe production. Corresponding safety protection measures need to be proposed accordingly. The fractal damage-renormalization mechanical model of fractured rock mass proposed in this study provides a quantitative assessment methodology for disaster protection in deep mining operations of metal mines.

Conclusion

  1. (1)

    A mechanical model for fractured rock masses is developed based on the results of laboratory uniaxial compression tests, incorporating principles from damage fractal theory and renormalization group theory. This model serves to elucidate the intricate behavior of fractured rock masses. By determining critical failure probabilities for such rock masses, distinct peak limit loads corresponding to various fracture fractal dimensions are delineated.

  2. (2)

    The block size within the tetrahedron block model of the BBM positively correlates with the ultimate strength and Young’s modulus of synthetic rock samples during uniaxial compression testing. Conversely, it negatively correlates with the fracture fractal dimension of these samples. Notably, the synthetic rock sample with a block size of 6 mm exhibits close alignment with the results observed in laboratory-based uniaxial tests.

  3. (3)

    Although the results of the damage fractal-renormalization mechanical model align with stress–strain curves and fracture fractal dimensions at peak loading conditions for synthetic rock samples composed of 3D voronoi blocks, the simulated AE patterns and rock failure modes throughout the loading process are deemed unsatisfactory. Notably, rock samples synthesized using Voronoi blocks exhibit a greater deviation from laboratory test results than those synthesized with 3D tetrahedron blocks.

  4. (4)

    In synthetic rock samples containing hole defects, the presence of these holes significantly enhances the degree of deterioration, with the effect being more pronounced in larger holes. Furthermore, in synthetic rock samples featuring defects, the area experiencing the highest degree of deterioration is typically oriented vertically to the axis of the hole defects. Consequently, in excavation zones of fractured rock masses, supportive structures such as anchor rods and cables are routinely installed around the perimeter to mitigate rock mass damage and deterioration. The fractal damage-renormalization mechanical model presented in this study provides a quantitative approach for assessing disaster prevention measures in deep metal mining operations to a considerable extent.

  5. (5)

    Considering both experimental and theoretical findings, the BBM simulation technique emerges as a valuable instrument for providing quantitative and anticipatory insights into the failure mode of rock masses. BBM’s capability to track apparent crack propagation and simulate spatial and temporal distributions of AE signals, facilitates the establishment of a link between micro-mechanical behaviors and macro-mechanical responses.