1 Introduction

The Guizhou Jinfeng Gold Mine is a proven super-large Carlin-type gold deposit, which is located in the Yunnan-Guizhou-Guangxi area, China, boasting resource reserves of 126.25 t. Despite this, the evaluation of gold resource potential in the area is only 34.16%, indicating significant untapped exploration and development potential (Chen et al. 2011; Liu et al. 2021a). Owing to extensive mining activities, the current depth of the mine has reached 800 m, and the surrounding rock is primarily composed of sandstone. The geological structure of this area is intricate because of the impact of various tectonic movements, resulting in highly developed joints, cracks, and other defects within the rock mass (Wu et al. 2019; Zheng et al. 2021). This geological complexity exacerbates rock mass damage and directly affects the stability of the roadway (Zhu et al. 2015). Moreover, external force disturbances and stress redistribution lead to the proliferation of macro- and micro-flaws in the roadway's surrounding rock, significantly influencing its mechanical properties, contingent upon the spatial distribution and geometric shape of these defects (Su et al. 2020). However, rock failure merely represents the final stage of crack development, providing limited insight into the history and characteristics of its damage evolution. To comprehensively understand rock failure, attention must be directed towards tracing the distribution of the initial cracks, crack growth, and damage evolution process. By examining the dynamic evolution of microcracks and material damage, researchers can identify the underlying mechanisms and instabilities that lead to failure (Bai et al., 2008; He et al. 2022).

Considerable progress has been made by both domestic and internationall researchers in understanding the laws governing crack initiation, propagation, and damage evolution in cracked rocks. Li et al. (2012) utilized a split Hopkinson pressure bar to link dynamic and static stresses, investigating the failure mode of a sandstone sample containing prefabricated holes. Their study identified the primary factors contributing to rock failure. Similarly, Liu et al. (2020) investigated granite failure characteristics through rock mechanics tests by employing fractal theory to analyze granite fracture patterns. Wong et al. (2006) conducted splitting failure tests on rock samples with prefabricated cracks, using physical and numerical tests. Their study investigated the initiation, propagation, and mode of failure of rock fractures, focusing on exploring the influence of geometric parameters such as the length and slope of the preset crack. Yin et al. (2018) conducted comprehensive physical and numerical tests on sandstone specimens with fractures and holes, examining mechanical properties, crack initiation types, and breakthrough modes influenced by factors including fracture angles, lengths, and distances from holes. Zhao et al. (2020) conducted numerical compression tests on brittle rock with cracks using EDEM, exploring both uniaxial and biaxial loading scenarios. Their investigation focused on examining the impact of the crack dip angle and confining pressure on the mechanical properties and failure behavior of the sample. Manouchehrian et al. (2012) employed PFC2D for biaxial compression tests on rocks with single fractures, exploring the impact of various constraint stresses on crack initiation and progression. Furthermore, Liu et al. (2021b) utilized numerical simulations to investigate the influence of sandstone on the macro-mechanical behavior and meso-damage evolution, quantifying the damage evolution process based on the fractal dimension.

Heterogeneity significantly influences the mechanical properties of rocks in multiple directions, emphasizing the need to consider variability in rock mechanical research (Behnam et al. 2022). Although previous studies have examined the initiation, propagation, and coalescence mechanisms of cracks and defects in rocks, few have investigated the macro nonlinear mechanical behavior and meso-damage evolution mechanism considering the real mesostructure and crack dip angle. Therefore, integrating the fracture dip angle and rock heterogeneity into numerical models can enhance the understanding of the damage characteristics and crack mechanisms in brittle materials such as rocks.

In this study, micro-CT scanning, digital image processing (DIP), and the rock failure process analysis system (RFPA3D) were used to construct a three-dimensional (3D) numerical model of jointed sandstone, capturing its real mesostructure. It simulated the meso-damage progression of prefabricated cracked jointed sandstone under uniaxial compression at various dip angles, while investigating the impact of jointed sandstone heterogeneity and prefabricated cracks with varying dip angles on their failure modes and meso-damage mechanical properties. Additionally, this study employed fractal theory to elucidate the evolution of rock meso-damage, offering a novel quantitative approach for studying damage development in brittle materials such as rocks.

2 Experimental methods

2.1 Characterization of the real mesoscopic structure of jointed sandstone based on digital images

The exposed strata of the Jinfeng Gold Mine mainly consist of the Middle Triassic Niluo Formation (T2nl), Bianyang Formation (T2by), Xuman Formation (T2xm), and Lower Triassic Luolou Formation (T1ll). The entire mining area exhibits a complete structural combination of folds and thrusts. Terrigenous clastic rocks and shallow-water carbonate rocks prevail, with fine sandstone, siltstone, limestone, and mudstone constituting the primary lithology (Chen et al. 2011). Joined sandstone samples were collected from the F3 fault fracture zone within the Middle Triassic Bianyang Formation (Fig. 1), located at a depth of 130–150 m and exhibiting a density of 2.70 g/cm3, primarily filled with calcite.

Fig. 1
figure 1

Geological map of the Jinfeng gold ore district (adapted from Chen et al. (2011))

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In this study, the meso heterogeneity of jointed sandstone was characterized using a micro-CT scanning experiment combined with digital image processing technology (Li et al. 2017; Liu et al. 2021a, 2022; Yu et al. 2014). Additionally, a 3D model reconstruction was conducted in combination with RFPA3D. A 2D slice of jointed sandstone (Fig. 2d) was obtained by scanning with a single optical magnification system (Fig. 2c) using a high-resolution industrial CT nanoVoxel-3502E (Fig. 2b) at the Tianjin Sanying Company. The slice dimensions were 40 mm × 100 mm, with an image resolution of 100 pixels × 250 pixels. Each pixel had a gray value ranging from 0 to 255. ImageJ software was selected for threshold segmentation based on the subtle hue shift in the HIS color space (H: hue, I: intensity, S: saturation) to delineate the I value range of different media within the material (Liu et al. 2021a, b; Liu et al. 2022). After iterative debugging, a segmentation threshold of 142 was determined for jointed sandstone (Fig. 2e), where I > 142 corresponds to calcite and I < 142 corresponds to the sandstone matrix. Subsequently, Fig. 2f presents the result of digital image processing from Fig. 2d, where blue represents sandstone and green represents the joint (calcite). This characterization image provides enhanced accuracy in identifying the form and spatial distribution characteristics of the joints (calcite) in the sandstone (Fig. 2f).

Fig. 2
figure 2

Digital image representation process. a Sandstone sample. b NanoVoxel-4000. c Optical amplification system. d 2D CT slice. e Division of I values. f Threshold segmentation feature image of sample

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Utilizing digital image processing technology (DIP), the 2D slice image (Fig. 3a) obtained from micro-CT scanning was imported into RFPA3D for 3D model reconstruction (Hao et al. 2022; Zuo et al. 2022). Batch processing of the 2D slice images resulted in the depiction of the reconstructed 3D model (Fig. 3b). The entire model comprised 2,500,000 cube pixel elements owing to its division into a 100 × 100 × 250 grid (Fig. 3d). Each cube pixel was converted into a cube element. The parameters for each cube element were assigned based on the differences in color and brightness using the Monte Carlo approach (Rubinstein and Kroese 2011). Table 1 lists the mechanical characteristics of the sample material (Liu et al. 2021b). Subsequently, a finite element cubic numerical calculation with dimensions of 40 mm × 40 mm × 100 mm was established (Fig. 3c). Figure 3e and f depict the model after cutting the 3D reconstruction, allowing for a clear observation of the spatial distribution characteristics of internal defects.

Fig. 3
figure 3

Schematic diagram of 3D model reconstruction and finite element conversion of jointed sandstone. a 2D CT slice. b 3D model. c 3D model after assignment. d Cube element. e Axial cutting diagram. f Cutting section diagram

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Table 1 Material mechanical parameters of samples (from Liu et al. 2021b)
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To systematically investigate the meso-heterogeneity of jointed sandstone and the effects of prefabricated cracks with various dip angles on the meso-damage evolution and macro-mechanical properties, five sets of fractures were prefabricated in this experiment at different dip angles (α = 0°, 30°, 45°, 60°, and 90°) under constant imaging conditions. The prefabricated cracks, measuring 14 mm in length and 1 mm in width, were centered within the sample (Fig. 4), where α represents the included angle between the horizontal direction and the prefabricated crack. Figure 5 is a simplified force diagram of the numerical model when α = 45°, the loading procedure was simulated using strain analysis, employing a constant displacement loading control in the axial direction with a loading increment of 0.002 mm per step, starting from an initial displacement of 0.001 mm, and loading ceased upon sample damage.

Fig. 4
figure 4

Numerical model of prefabricated cracked jointed sandstone with various dip angles

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Fig. 5
figure 5

Model load diagram

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2.2 The constitutive relationship of damage on the mesoscale

Considering the heterogeneity of materials, it was assumed that the elastic moduli and strengths of sandstone and calcite follow a Weibull random distribution (Weibull 1951; Tang et al. 2007; Zuo et al. 2015). Given that the compressive strength of sandstone significantly surpasses its tensile strength, the Mohr–Coulomb strength criterion with the tensile criterion is utilized as the criterion for element failure. As the size of the basic element decreases, containing fewer defects, its strength increases, and elastic brittleness becomes more pronounced. The relationship between the linear elastic stress and strain under uniaxial compression is expressed as follows (Tang et al. 1997):

$$\left\{ {\begin{array}{*{20}c} {\sigma_{x} } \\ {\sigma_{y} } \\ {\sigma_{z} } \\ {\tau_{xy} } \\ {\tau_{yz} } \\ {\tau_{zx} } \\ \end{array} } \right\} = \frac{E}{{\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)}}\left[ {\begin{array}{*{20}c} {1 - \nu } & \nu & \nu & {} & {} & {} \\ \nu & {1 - \nu } & \nu & {} & {} & {} \\ \nu & \nu & {1 - \nu } & {} & {} & {} \\ {} & {} & {} & {\frac{1 - 2\nu }{2}} & {} & {} \\ {} & {} & {} & {} & {\frac{1 - 2\nu }{2}} & {} \\ {} & {} & {} & {} & {} & {\frac{1 - 2\nu }{2}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{x} } \\ {\varepsilon_{y} } \\ {\varepsilon_{z} } \\ {\gamma_{xy} } \\ {\gamma_{yz} } \\ {\gamma_{zx} } \\ \end{array} } \right\}$$
(1)

When the elastic modulus decreases to a low value, the strength also diminishes to the residual strength level. The damage evolution equation for an elastic brittle model with residual strength under uniaxial tensile stress is (Zhu et al. 2003):

$$\omega = \left\{ {\begin{array}{*{20}c} {0,\varepsilon t0 \le \varepsilon < 0} \\ {1 - \frac{\lambda \varepsilon t0}{\varepsilon },\varepsilon tl \le \varepsilon < \varepsilon t0} \\ {1,\varepsilon \le \varepsilon tl} \\ \end{array} } \right.$$
(2)

where \(\lambda\) represents the residual intensity coefficient of the element, and \(\varepsilon tl\) represents the ultimate tensile strain of the element. The element approaches the state of tensile fracture when its uniaxial tensile strain reaches the ultimate tensile strain.\(\eta\), often known as \(\varepsilon tl = \eta \varepsilon t0\), denotes the ultimate strain coefficient.\(\varepsilon t0\) is the tensile strain associated with the elastic limit, also referred to as the tensile failure strain threshold. Similarly, the damage evolution equation for the shear failure is (Tang et al. 1997):

$$\omega = \left\{ {\begin{array}{*{20}c} {0,\varepsilon < \varepsilon c0} \\ {1 - \frac{\lambda \varepsilon c0}{\varepsilon },\varepsilon \ge \varepsilon c0} \\ \end{array} } \right.$$
(3)

3 Results and discussion

3.1 Mechanical properties of jointed sandstone under uniaxial compression

For the five sample groups featuring cracks at various dip angles, Table 2 presents the simulation results for the compressive strength and elastic modulus. Significantly divergent peak strength and elastic modulus values were observed across the samples with varying crack angles. As shown in Fig. 6, both the compressive strength and elastic modulus exhibited a linear increase with increasing crack angle. At α = 90°, the sample reached its maximum compressive strength of 43.28 MPa, whereas at α = 0°, the minimum was 22.54 MPa, indicating a discrepancy of 20.74 MPa between the smallest and largest values. Similarly, the elastic modulus peaked at α = 90° with a value of 93.87 GPa, whereas at α = 0°, the minimum was 83.74 GPa, showcasing a difference of 10.13 GPa between the minimum and maximum values. Figure 6 illustrates the linear growth pattern of the compressive strength and elastic modulus with increasing crack angle, reflecting a pronounced anisotropic feature likely attributable to meso heterogeneity resulting from the spatial distribution of different prefabricated cracks. In addition, the presence of joints introduced varying degrees of stress concentration, leading to damage to the joint surfaces and shear slip failure. Notably, a significant disparity existed between the obtained compressive strength and elastic modulus values. These findings align with those of Li et al. (2019), who confirmed a linear relationship between the compressive strength, elastic modulus, and angle of prefabricated cracks, underscoring the high reliability of the numerical simulation results in this study.

Table 2 Simulation results of compressive strength and elastic modulus of sandstone with various dip angles
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Fig. 6
figure 6

Elastic modulus and compressive strength of sandstone at various dip angles

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3.2 Analysis of crack propagation and failure evolution of jointed sandstone

The temporal and spatial distribution characteristics of rock meso-element parameter deterioration and the accumulation of acoustic emission failures accurately replicate the incubation and evolution process of rock cracks (Jia et al. 2021; Liu et al. 2021a). Figure 7 illustrates the fracture process and acoustic emission evolution of the sample, with each ball representing an event of acoustic emission. The acoustic emission energy decreased with decreasing ball diameter and increased with increasing diameter. The blue balls denote shear failure, and the red balls represent tensile failure.

Fig.7
figure 7

Fracture process and acoustic emission evolution diagram of prefabricated cracked jointed sandstone samples with various dip angles

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At α = 0°, shear failure initially occurred along the joint surface and near the prefabricated cracks during the initial loading phase. As the stress increased, a significant degree of tensile failure, accompanied by minor shear failure, arose from the distributed fine point source fractures in the upper and lower joints. Finally, shear slip failure occurred along the joint surface owing to the numerous tensile and shear crack elements, primarily resulting from the presence of joints. At α = 30°, shear failure occurred around the prefabricated crack during the early stages of stress, with wing cracks forming around the crack tip as stress intensified. Increasing the stress led to the expansion of wing cracks as a vast number of shear fracture elements assembled, accompanied by significant tensile failure in the upper and lower joints. Finally, wing cracks penetrated the entire sample because of the presence of numerous tensile and shear failure elements, with prefabricated cracks being the primary contributing factor to the eventual failure of the sample. At α = 45°, wing cracks steadily developed from the tip of the prefabricated crack, accumulating numerous shear crack elements owing to rising stress. Eventually, the formation of a macro shear band, resulting from the aggregation of tensile and shear crack elements, led to macro shear failure, with joints having minimal influence on the final failure. At α = 60°, shear failure initially occurred along the joint surface at the lower end and around the prefabricated crack during the initial loading stage. As stress rose, numerous tensile failures occurred in the upper and lower joints. The wing crack continued to expand due to the aggregation of shear crack elements. The eventual spread of wing cracks throughout the entire sample, due to the accumulation of composite failure elements, was primarily attributed to the prefabricated crack. At α = 90°, tensile failure first occurred at the upper and lower joint surfaces with no element failure at the prefabricated crack tip during the initial loading stage. Therefore, no obvious mechanical response was observed. As the stress increased, the stored energy in the joint was rapidly released, leading to significant tensile failure and eventual macroscopic failure of the test piece, the joint became the primary determinant of the final failure.

The acoustic emission diagram revealed that most internal failure elements of the sample were tensile failures (red balls), and the aggregation and connection of internal tensile-shear composite failure elements was the internal cause of the formation of macro shear bands. Considering both the micro-heterogeneity of jointed sandstone and the inclination of prefabricated cracks, the analysis of the simulation results (Fig. 7) indicated that the contribution of joints to the final failure of sandstone varied significantly with different crack inclinations. The joints were the main cause of failure at α = 0° and α = 90°. In contrast, the prefabricated cracks predominantly led to failure at α = 30°, α = 45°, and α = 60°. Therefore, the influence mechanism of joints and cracks on rock fractures is inherently distinct, emphasizing the importance of considering fissure distribution in determining their contribution to rock fractures.

4 Fractal characteristics of meso-damage evolution of jointed sandstone

4.1 3D fractal analysis of images based on box dimension

Fractal theory extends the conventional notion of dimension and offers a novel approach to quantify complex forms and phenomena in nature. It is currently widely used in various fields, including rock mechanics, nonlinear science, and geology (Xie et al. 1993; Li et al. 2009; Liu et al. 2021a). The box dimension algorithm, renowned for its mathematical simplicity and experimental measurability, provides a more intuitive understanding of the calculation process. In this study, it is defined as:

$$Ds = \mathop {\lim }\limits_{k \to \infty } \frac{\lg Nrk(A)}{{\lg \frac{1}{rk}}}$$
(4)

where rk denotes the generated shrinking sequence with the element's square box size, Ds denotes the self-similar fractal dimension of the failure zone, and the size rk square box covers the target set A with the fewest possible grids or Nrk(A).

To illustrate the fractal characteristics of meso-damage evolution in jointed sandstone, the first step is to extract the damaged elements from the numerical simulation results. Subsequently, 3D images of these damaged elements need to be reconstructed. In this study, the damaged elements identified in the numerical simulation results were extracted and 3D reconstructed using self-developed MATLAB code. The resulting reconstructed image of the 3D fracture is shown in Fig. 8. To better compare the distribution of internal element failure, combined with stress–strain curves, a total of 5 test points of pre-peak stress σ = 30%σmax, σ = 70%σmax, σ = 90%σmax, σ = 100%σmax, σ = 150%σmax were selected in sequence for fractal characteristic analysis. Among them, σ = 150%σmax refers to the point at which the stress peak drops to 50%σmax, the crack expansion is basically completed or even penetrated at the point.

Fig. 8
figure 8

3D fracture distribution at different stress levels

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The corresponding fractal dimension is calculated using the 3D box dimension algorithm independently developed by our team (Liu et al. 2023). The main flow of the algorithm is as follows: (1) Divide the 3D image into cubes using various scales and save the relevant data; (2) Cover the damaged region of the 3D image with a cube box of side length rk, count the number of cube boxes Nrk(A) within the damaged area, and record pertinent information, continuously adjust the scale of rk to cover the same damaged area, resulting in a dataset of Nrk(A) and rk; (3) Perform regression analysis of Nrk(A) and rk using double logarithmic coordinates, calculate the Ds value from the slope of the fitting line, as illustrated in Fig. 9.

Fig. 9
figure 9

Calculation results of fractal dimension of damage area when azimuth angle is 0°(stress level of 150%)

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4.2 Analysis of internal fracture characteristics in specimens

To make a better quantitative analysis on the change rule of crack expansion volume in the sample, ImageJ software was used to conduct statistical analyses of the crack volumes under varying stress levels in the numerical simulation results. The statistical findings are presented in Fig. 10. During the σ = 30%σmax to σ = 30%σmax stage, characterized as the elastic phase, small cracks gradually emerged along the crack propagation direction around the sample. The crack volume remained relatively stable, indicating an energy-accumulation stage. As the stress increased from σ = 70%σmax to σ = 100%σmax, representing the plastic phase, the release of elastic energy led to a notable increase in the total crack volume, accelerating the growth rate. The subsequent σ = 100%σmax to σ = 150%σmax stage marks the post-failure phase, characterized by the rapid expansion of cracks due to macro damage resulting from numerous microcracks within the sample. Brittleness becomes evident during this stage as sandstone progresses from an initial meso-crack to a macro-fracture. The distribution of joints in sandstone affected the stress concentration at special angles α = 0° and α = 90°. During these instances, the primary factor leading to sample failure was the presence of joints. Failure initiated from the joint surface without wing crack formation at the prefabricated crack tip. As the stress increased, The final joint surface penetrates and causes shear slip failure, resulting in the failure of the specimen, due to the absence of wing-crack initiation and expansion at the tip of the prefabricated crack, the crack volume remained relatively small.

Fig. 10
figure 10

Relationship between different stress levels and crack volume

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4.3 Meso-damage evolution characteristics of jointed sandstone

The internal damage of rock progresses from small point sources to large fracture surfaces, with significant acoustic emission (AE) occurring near the joint surface and the damaged region. Because the entire rock fracture process may be characterized by the progression of AE, monitoring the internal failure of rock through AE can effectively elucidate its meso-damage mechanism and inverse its failure process (Li et al. 2017; Yu et al. 2014).

Table 3 presents the total AE counts and 3D fractal dimensions of the samples at various stress levels, as depicted in Fig. 11, when the stress levels were below 30%, indicating the elastic stage, the fractal dimension remained at 0, suggesting minimal damage within the samples, with AE counts approaching 0. With stress levels below 90%, there was a gradual increase in the AE counts, with a relatively small fractal dimension and consistent changing trend. However, as the stress level surpassed 90%, both the AE counts and fractal dimensions increased rapidly. At the 150% stress level, the AE count reached its peak, and the fractal dimension reached its maximum, indicating complete sample fracturing. For the sample with α = 0°, the cumulative AE was 46.641, and the fractal dimension was 2.252, the lowest among all samples. Conversely, the α = 90° sample exhibited the highest cumulative AE of 228.792 and a fractal dimension value of 2.588, indicating the most severe internal damage and the greatest energy release among all the samples.

Table 3 Calculation results of AE energy and fractal dimension of samples under different stress levels
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Fig. 11
figure 11

a Relationship between fractal dimension and distinct stress levels; b Relationship between cumulative AE and distinct stress levels

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Therefore, the meso-heterogeneity of jointed sandstone and the dip angle of prefabricated cracks significantly influenced the meso-damage evolution, corroborating the findings of Deng et al. (2019). In particular, the mesostructure (joint distribution) in the samples with α = 0° and α = 90° exerted a substantial impact on damage progression. The presence of the joints led to a concentrated stress distribution, prompting the initiation and propagation of wing cracks along the joint surface. Conversely, for samples with α = 30°, α = 45°, and α = 60°, the angle of the prefabricated crack emerged as the decisive factor leading to the final fracture. With an increase in the prefabricated crack angle, both the cumulative number of AEs and 3D fractal dimension of the sample exhibited linear growth. During this phase, the micro-heterogeneity of the jointed sandstone minimally affected the macrofracture. In summary, when α = 0° and α = 90°, the presence of joint cracks in sandstone led to a significant number of AE counts, release of abundant elastic properties, and ultimately, a large fractal dimension.

The relationship between the load, AE count, fractal dimension, and loading steps for the sandstone samples with joints and prefabricated cracks is depicted in Fig. 12. Initially, during the initial stages of loading, the AE count stabilized for each sample, with slow crack expansion and steady changes in the fractal dimension. As the loading stress increased, cracks grew in quantity and size, leading to a gradual increase in both the AE count and fractal dimension, indicating escalating internal damage. With additional loading steps, the AE count experienced rapid growth, indicating accelerated sample deterioration under the load, accompanied by a rapid increase in the fractal dimension. When the AE count peaked, so did the fractal dimension, synchronously. This correlation stems from the aggregation and diffusion of internal failure elements within the test piece, leading to microcrack formation, propagation, and eventual breach, reflecting the transition from meso-element damage to macro-fracture in the development of the test piece. The macro failure of the samples with α = 0° and α = 90° was influenced by the meso heterogeneity of the jointed sandstone. This resulted in a concentrated distribution of AE energy near the peak load, resulting in a maximum AE count and fractal dimension, indicative of good brittleness characteristics. Conversely, the macroscopic failure of samples with α = 30°, α = 45°, and α = 90° was determined by the inclination angle of the prefabricated crack, leading to a concentrated AE energy distribution after the peak load. Hence, the AE count and fractal dimension exhibited a certain hysteresis, characteristic of typical plastic behavior. According to the results in Table 3, the fractal dimensions of the samples with α = 0° and α = 90° were notably large, whereas those of the samples with α = 30°, α = 45°, and α = 60° fell between α = 0° and α = 90°, and this variation is attributed to the different main factors controlling the final fracture of the samples. Hence, the contribution of joints and cracks to the final fracture of rocks should be evaluated based on their distribution, the evolution mechanism of joints and fissures in meso-crack formation in rocks differs significantly.

Fig. 12
figure 12

Relationship diagram of load, AE, and fractal dimension. a α = 0°; b α = 30°; c α = 45°; d α = 60°; e α = 90°

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5 Conclusion

  1. (1)

    The final failure modes of sandstone can be classified into three categories: composite shear failure, shear failure along the joint surface, and tensile failure, attributed to joint and prefabricated cracks. With an increase in the prefabricated fracture inclination, the elastic modulus and compressive strength of the jointed sandstone exhibited linear variations, accompanied by significant anisotropy. The presence of joints can induce and regulate internal crack initiation and propagation in sandstones.

  2. (2)

    The contribution of the joints to the final failure of sandstone varied significantly among the samples with various prefabricated crack angles. For samples with α = 0° and α = 90°, the presence of joints predominantly determined the ultimate failure of the sample, whereas in samples with α = 30°, α = 45°, and α = 60°, the prefabricated crack played a dominant role in the eventual collapse of the sample. Consequently, the contribution of joints and cracks to rock fractures should be assessed based on their distribution, as their influence mechanisms on rock failure differ fundamentally.

  3. (3)

    The mesostructure (joint distribution) of sandstone in samples with α = 0° and α = 90° significantly affected the damage. The existence of the joints led to a concentrated stress distribution, initiating and propagating wing cracks along the joint surface. The failure and coalescence of the joints led to a high number of AEs, released considerable elastic properties, and caused shear slip failure, resulting in a large fractal dimension. Conversely, in samples with α = 30°, α = 45°, and α = 60°, the cumulative number of AEs and the 3D fractal dimension linearly increased with the prefabricated crack angle.

  4. (4)

    The meso-damage progression of sandstone was quantified in terms of the fractal dimension, demonstrating that the damage severity increased with a larger fractal dimension. The samples with α = 0° and α = 90° exhibited large fractal dimensions, whereas those with α = 30°, α = 45°, and α = 60° fell between α = 0° and α = 90° because of the different main factors controlling the final fracture. This study quantitatively characterizes the meso-damage evolution in jointed sandstone, offering a novel approach for quantitatively analyzing the damage evolution characteristics of rocks.