1 Introduction

Rockburst, which is often accompanied by loud noise, vibration, and air waves, is generally induced by sudden release of accumulated elastic energy during coal mining (Zhang et al. 2017; Hadi 2021; Dennis and Black 2019; Wei et al. 2022; He et al. 2023). After rockburst, the supporting structure of bolts and cables failed, the surrounding rocks of the roadways collapsed and roof fell, and the internal equipment of the roadways was damaged. In severe cases, workers working inside the roadways may also be injured, resulting in massive property losses and casualties in the coal mine. The methods of preventing and controlling this type of dynamic disaster include regional management, impact support, and local stress relief. Among them, regional management is mainly realized by optimization of mining arrangement, small-coal-pillar or no-coal-pillar mining, liberated layer mining, and regional hydraulic fracturing, while the specific measures of impact support include high-prestress support, anti-impact anchors (cables), U-beam support, and impact support brackets (Song et al. 2014; Chen et al. 2021; Miao et al. 2016; He et al. 2018). In addition, borehole stress relief, characterized with convenient construction, simple operation, and strong applicability, is widely applied to rockburst prevention and control (Kang et al. 2023; Sun et al. 2023; Wang et al. 2022; Liu et al. 2015a, b). Constructing boreholes in coal rock can considerably weaken their ability to store elastic deformation energy, thus raising the critical conditions and reducing disaster occurrence (Zhang et al. 2019; Kang et al. 2022).

However, the effect and measures of stress relief were the focus of most relevant previous researches. In fact, despite its positive role in weakening the strength of the surrounding rock, a wide range of stress relief may affect the stability of the support structure (Piotr and Zbigniew 2020; Pu et al. 2019; Gong et al. 2022). After drilling, the initial failure vertical stress of the rock decreases, and the surrounding rock is more susceptible to damage, indicating that the unloading of the borehole causes damage to the surrounding rock, thus inducing a strength weakening effect (Si et al. 2022; Ma et al. 2022). A large number of stress relief boreholes would destroy the surrounding rock’s self-carrying structure, reduces the support structure’s bearing capacity, and weakens the support system’s impact resistance. Consequently, the roadway cross-section will be strongly contracted or even closed in the presence of even a low-energy impact event. The depths of pressure relief boreholes in a burst-prone roadway are usually 15–25 m, much larger than the depth of anchor support in the surrounding rock (generally 6–8 m). If the pressure relief boreholes within the anchoring range are filled, the integrity and strength of the surrounding rock there can be improved, and the bearing capacity of the support structure can be enhanced accordingly. Meanwhile, the pressure relief space of the deep boreholes in the surrounding rock is still retained. In this way, the anti-impact effect of pressure relief and the support of the burst-prone roadway can be realized synergistically.

In general, scholars paid more attention to the failure mode, fracture expansion and energy dissipation of coal rock containing pores and fractures, and seldom discussed the mechanical properties of hole filling and the energy evolutions of coal rock. Crack development in hole-containing rocks under loading is more complex. Extensive studies have been conducted on the crack initiation conditions, influencing factors, expansion rules of hole-containing rocks. Major factors affecting crack expansion include rock properties, drilling hole shape, stress level, number of drilled holes, etc. Zhang et al.(2019) and Zhu et al. (2022) studied the expansion law of cracks around coal boreholes and believed that the degree of crack development, energy release and pressure relief effect are related to borehole density. Subsequently, domestic and foreign scholars have carried out a large number of studies on the crack initiation conditions, influencing factors and expansion rules of rocks containing holes. The expansion and evolution of cracks are closely related to factors such as rock material properties, hole shape, stress level and number of holes (Zhu et al. 2005; Wong et al. 2006). Liu et al. (2015a, b) used acoustic emission technology to study the spatiotemporal evolution of micro-cracks in coarse-grained granite samples containing circular openings. The AE positions observed on the surface of the granite samples were consistent with macro-cracks; Wu et al. (2020) based on DIC and CT methods, studied the failure forms of rocks containing holes, which are tensile extension, shear and tensile forms; Zuo et al. (2019) systematically studied the macro/mesoscopic failure of deep rocks or coal-rock combinations under different loading conditions. The mechanism reveals the influence of the mechanical properties of coal and rock on the failure form of the combined structure.

Previous studies believe that drilling can change the mechanical properties of coal rock and reduce burst propensity and stress conditions through structure alteration, so as to achieve the purpose of preventing and controlling rockburst. However, constructing a large number of holes will induce the formation of broad damage zones in the roadway anchoring structure in engineering practice, thus rendering the anchored surrounding rock unstable. Filling the holes with coal gangue and foamed cement materials greatly improves the strength and water resistance of the filling body, and greatly improves the load-bearing capacity of the structure (Qiu et al. 2022; Zhang et al. 2022). After the fractured rock is filled, its compression and shear mechanical properties are significantly improved, specimens with a filled flaw have larger crack initiation stress levels and smaller crack inclination angles than specimens with an unfilled flaw (Sinha and Singh 2000; Miao et al. 2018). Overall, the strength of the specimen increases by increasing the filling strength, and crack filling has a significant improvement effect on rock strength (Chang et al. 2018). Therefore, investigation into the mechanical properties of burst-prone coal samples with fillings has certain engineering significance for guiding the roadway anchoring.

This study is aimed at revealing the mechanical properties and energy evolutions of burst-prone coal samples with holes and fillings. To this end, first, burst-prone coal samples with varying hole sizes and filling materials were prepared. Subsequently, their mechanical properties were tested with the aid of a uniaxial compression apparatus, and their data of stress–strain curves, acoustic emission (AE) parameters, spatial deformation characteristics were obtained. Based on the obtained data, the influences of hole sizes and filling material properties on the deformation damage and energy evolution of the coal body were analyzed. The research presented in this study boasts certain theoretical and engineering importance for grasping the deformation damage law and EE mechanism of coal bodies filled with varying materials and for further improving theories concerning rockburst roadway pressure relief and anchoring.

2 Experimental scheme

Binchang Mining Area, where the coal seams are prone to burst, is one of the severest rockburst-struck areas in China. In order to control rockburst, dense large-diameter holes need to be drilled on both sides of the mining roadway for pressure relief during excavation and mining. The diameter, row spacing, and depth of the drilled holes are usually 153 mm, 1–3 m, and 15–20 m, respectively. After the collapse of a large-diameter hole, a fractured zone will be formed around, which then develops into a pressure relief zone, transferring the concentrated high stress from two sides of the roadway to the deep. Nevertheless, the fractured zone induced by large-diameter holes in the shallow surrounding rock significantly breaks the anchoring structure’s integrity, reduces the anchor bolts’ (cables) anchoring force, and weakens the anchoring structure’s bearing capacity, thus aggravating the deformation of the roadway. In view of this fact, in-situ burst-prone coal was selected as the research object, and samples with different-sized drilled holes were prepared; next, the holes were filled to simulate on-site grouting for reinforcing the fractured zone within the anchoring range of the roadway after large-diameter hole construction.

2.1 Sample preparation

Experimental samples were selected from the 402101 haulage roadway in Hujiahe Coal Mine, Binchang Mining Area. The coal seam is 20 m thick and buried at a depth of 680 m. Firstly, in accordance with relevant sampling requirements, coal blocks with a size of about 200 mm × 200 mm × 200 mm were collected in the direction perpendicular to the coal bedding surface. After being collected, they were sealed with plastic film and transported to the laboratory for standard sample processing. It was identified by relevant department that the collected coal blocks were highly burst-prone. Afterwards, according to the ISRM test standard, the large coal blocks were processed into 50 mm × 50 mm × 100 mm standard samples whose surfaces were flat without obvious cracks. Then, to ensure that their flatness and error were up to standard, the 6 surfaces of each sample were polished using fine sandpaper. In addition, to avoid large differences in the mechanical properties of the samples, they were subject to ultrasonic wave detection, and those with large differences in the wave velocity were removed. Subsequently, special high-pressure water knife equipment was used to drill holes in the middle of the rectangular samples for one time. The holes, being 4–10 mm in diameter, penetrated the entire samples (Fig. 1). Finally, two schemes (brittle and ductile filling materials) were adopted to prepare hole-filled coal samples for the purpose of simulating the states of drilled holes filled with materials of different properties. The brittle filling material was cement slurry (42.5 grade cement, cement: water = 1: 0.5, curing time 28 d); and the ductile filling material was high-strength polyurethane (compressive strength after solidification > 45 MPa).

Fig. 1
figure 1

Coal samples with varying hole sizes and filling materials

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2.2 Experimental system and method

The experiment was performed with the aid of the TAW-3000 electro-hydraulic servo testing machine in State Key Laboratory for Efficient Mining and Clean Utilization of Coal Resources. The AE signals from coal deformation were acquired by a rapid digital AE monitoring system (MISTRAS, the United States). The strain field and crack expansion pattern during coal deformation and failure were monitored by a MatchID 2D HR full-field strain measurement system. A total of 21 samples in 7 types were prepared, namely the intact samples, the drilled samples with hole diameters of 10 mm, 8 mm, 6 mm, and 4 mm without filling, and the drilled samples filled with cement slurry (hole diameter 10 mm) and polyurethane (hole diameter 10 mm). To ensure their sudden failure, the samples were stably loaded at a constant stress loading rate of 0.1 kN/s. Before the experiment, the high-speed camera was placed facing the drilled holes and preheated for 30 min to guarantee its thermal stability. Besides, the AE probes were fully coupled with the samples using Vaseline. When the testing machine started to load, the high-speed camera and the AE monitoring system were opened simultaneously. During the experiment, the monitoring software automatically collected and processed data and displayed stress–strain curves, strain field time-history curves, AE time-history curves, and other data in real time. The experimental system is displayed in Fig. 2.

Fig. 2
figure 2

Photo and schematic diagram of the experimental system

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3 Experimental results and analysis

3.1 Stress–strain curves of the tested samples

Coal samples with varying hole sizes and filling materials were subject to uniaxial compression experiments, and then their mechanical parameters were processed. As exhibited in Table 1, the mechanical properties of the drilled and filled coal samples differ notably from those of the intact coal sample. The intact coal sample has an average peak strength of 12.06 MPa, while the value of this index is 11.67 MPa, 10.32 MPa, 9.53 MPa, 8.14 MPa, 8.60 MPa, and 11.75 MPa for the 4-mm-hole sample, the 6-mm-hole sample, the 8-mm-hole sample, the 10-mm-hole sample, the 10-mm-hole sample filled with cement slurry, and the 10-mm-hole sample filled with polyurethane, respectively, a decline of 3.23% 14.43%, 20.98%, 32.5%, 28.69%, and 3.40%, respectively. Besides, the drilled and filled coal samples also present certain changes in other indexes compared with the intact coal sample. To be specific, their average peak strains rise by 17.41%, 39.29%, 64.96%, 87.95%, 104.91%, and 111.16%; the average elastic moduli drop by 14.21%, 27.43%, 46.95%, 63.46%, 39.34%, and 7.99%; the average dynamic failure times increase by 99.04%, 98.08%, 103.19%, 199.20%, 101.44%, and 230.03%; and the average impact energy indexes decrease by − 3.37%, 1.01%, 11.11%, 22.56%, 35.69%, and 29.63%, respectively. Overall, after hole drilling, the indexes related to burst propensity all decline to some extent, and the decline is more significant for those with a larger hole diameter. The larger the hole diameter, the more significant the decrease in these related indexes. Compared with the drilled coal samples without filling, the samples filled with cement slurry and polyurethane have higher elastic modulus, peak strain, and peak strength, and their indexes related to burst propensity also decline to a certain extent. This indicates that hole filling can not only enhance the strength and deformation resistance of the drilled coal samples, but also weaken their burst propensity.

Table 1 Average mechanical parameters under varying hole sizes and filling materials
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The typical stress–strain curves of coal samples with varying hole sizes and filling materials were plotted to analyze their difference (Fig. 3, where the samples with hole diameters of 8 mm and 4 mm were not included considering the similar laws of data). Clearly, all the curves include five stages, i.e., initial compaction, elastic deformation, plastic deformation, yield failure, and post failure.

Fig. 3
figure 3

Stress–strain curves of the tested samples

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3.2 Analysis on the strain field

According to above analysis, the hole size and filling material exert a notable effect on the uniaxial compressive strength of the samples. In addition, judging from the patterns of curves in Fig. 3, incompatible deformation and stress concentration occurred during the loading process, which is a self-adjustment of stress in the heterogeneous coal samples under external loads, mainly resulting from changes in the homogeneity of the samples caused by drilling and filling. In the hope of exploring the strain field evolutions of the samples during uniaxial compression, four typical samples were selected to analyze the overall deformation morphologies of the samples, which are the intact coal sample, the 10-mm-hole sample without filling, the 10-mm-hole sample filled with cement slurry, and the 10-mm-hole coal sample filled with polyurethane (Table 2).

Table 2 Distribution of maximum principal strain field in typical coal samples with varying hole sizes and filling materials
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As can be observed from Table 2a, under a lower stress (10% peak stress), the intact sample is relatively uniform with a small strain overall. As the stress grows, an obvious ascending trend of the strain can be noticed in the lower part and local areas of the sample, showing local strain concentration. The strain values on the bottom and in the lower right part are relatively large, which is attributed to the heterogeneity of the sample and the presence of primary cracks. When the peak stress is reached, a strain concentration zone appears in the lower part, where obvious cracks have been formed along with the ejection of coal bodies that peel off from the lower part of the sample.

In Table 2b, in the drilled coal sample, the strain is obviously concentrated around the hole under a lower stress (10% peak stress), and strain concentration gradually intensifies with the increase of stress. In addition, strain concentration also appears in the middle-left side of the sample, which is mainly induced by the presence of primary cracks, yet its scale is smaller than that around the drilled hole. At the peak stress, multiple strain concentration zones are formed around the drilled hole, and noticeable cracks appear in the coal body around the hole. Finally, the cracks gradually penetrate the sample, resulting in overall failure.

In Table 2c and d, in the filled samples, strain concentration around the drilled holes is insignificant under a lower stress. As the stress increases, strain concentration appears near the drilled holes. Strain concentration zones can be observed in the lower and upper parts of the sample filled with brittle material (cement), while they emerge around the drilled hole in the sample filled with ductile material (polyurethane). The different morphologies of the stress concentration zones primarily arise from the varying properties of the filling materials. Polyurethane has certain deformability and strength which allow it to deform adaptively to reduce the strain concentration around the drilled hole. At the peak stress, the coal sample filled with cement material exhibits strain concentration zones along the drilled hole, which are generated by tensile cracks. Finally, the tensile cracks penetrate the sample, leading to overall failure. The coal sample filled with polyurethane shows strain concentration zones in the lower, left, and upper parts of the hole, which are jointly generated by tensile and shear cracks. Moreover, this sample has a noticeably higher strain value overall compared with the other three types of samples.

For the intact sample, the principal strains on the sample surface are randomly distributed, and there is no obvious change pattern as the stress increases, but the strain value gradually increases. As the stress increases, the maximum principal strain is concentrated toward the shear plane area. Compared with other drilling and filling methods, the maximum principal strain of the intact sample is the smallest and the uneven deformation is not obvious. Due to the large changes in stress distribution around the hole, there are large differences in crack type and starting position before and after filling, so the location of the maximum principal strain is also significantly different. For unfilled boreholes, the fracture sequence is mainly caused by the concentration of tensile stress at the top and bottom of the borehole causing tensile cracks, and then the tangential compressive stress on the side wall causes plate cracks. For filled boreholes, due to the concentration of radial tensile stress, interface drag first occurs at the interface on the side of the filling hole. Due to the concentration of radial and tangential stress, local principal strains are concentrated in the filling material, and then tensile cracks occur under the action of stress. Gradually enter the coal sample matrix. On the whole, in the drilled coal samples, strain concentration is easily formed near the hole, and cracks usually expand and interconnect along the strain concentration zone. Besides, hole filling can effectively ease strain concentration, especially with ductile material which can not only weaken the strain concentration effect near the drilled hole, but also enhance the deformability of the sample.

4 Energy evolutions of coal samples with varying hole sizes and filling materials

4.1 Stress and AE evolutions of coal samples with varying hole sizes and filling materials

The AE monitoring system was employed to acquire AE energy and cumulative count during uniaxial compression. The monitoring results are depicted in Fig. 4.

Fig. 4
figure 4

Evolution law of stress and AE in coal samples with varying hole sizes and filling materials

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In Fig. 4a, for the intact coal sample, the AE energy and cumulative count are both low under low external loads in the compaction stage (0–200 s), and the expansion of internal cracks is insignificant. As the load increases (200–450 s), the AE energy and cumulative count gradually rise. Such a result suggests that the internal primary cracks gradually expand under the external loads, and some new cracks gradually emerge. However, the AE energy is still at a relatively low level (below 4000 × 106 aJ), which means that most of the cracks are still microcracks. When the external load approaches the peak load of the coal sample, the AE energy and cumulative count surge, the AE energy reaching as high as 9000 × 106 aJ. At this time, the cracks develop the most remarkably. They gradually penetrate the sample, resulting in dynamic failure.

In Fig. 4b, the AE energy and cumulative count evolutions of the drilled coal sample notably differ from those of the intact coal sample. When the external load is low (0–170 s), the evolutions are similar to those of the intact one. As the external load increases further (170–350 s), the AE energy and cumulative count rise rapidly, and the AE energy in this stage is far higher than that of the intact one, the maximum value reaching above 7000 × 106 aJ. This indicates that larger cracks have been formed in the coal sample as the result of drilling. When the external load approaches the peak load of the coal sample, the cumulative count soars, whereas the AE energy remains relatively low. The overall dynamic failure is unobvious, and only local dynamic failure occurs. The evolutions of AE energy and cumulative count in Fig. 4c resemble those in Fig. 4b except for certain differences. With a larger diameter of the drilled hole, the AE energy and cumulative count present a remarkable increase under lower external loads. As the load increases, the AE energy and cumulative count are at high levels. The following findings can be obtained through a comparison of the data of the intact, 6-mm-hole, and 10-mm-hole samples. The intact coal sample experiences longer energy storage in the elastic stage and shorter energy release in the unstable failure stage; meanwhile, its energy release mainly occurs near the stress peak. In contrast, the drilled coal samples exhibit significant AE characteristics throughout the entire loading process, and this phenomenon is increasingly evident for the sample with a large hole size. The presence of the drilled holes not only deteriorates the coal strength and accelerates crack expansion and development, but also changes the time and intensity of energy accumulation and release and reduces the intensity of dynamic failure, which further verifies the effectiveness of pressure relief by hole drilling in preventing and controlling rockburst disasters.

In Fig. 4d, the coal sample filled with brittle material (cement) exhibits continuous AE characteristics throughout the loading process, yet the AE energy and cumulative count are smaller than those of the drilled samples. The AE energy is generally lower than 2000 × 106 aJ, and high-energy events only occur in the stress peak zone. In Fig. 4e, in the coal sample filled with ductile material (polyurethane), the AE energy and cumulative count are at lower levels under low external loads (0–100 s), which is similar to the compaction stage of the intact one. The AE energy and cumulative count gradually increase with the growth of the external load, yet they remain at lower levels. High-energy AE events occur merely in the peak load zone. This indicates that the damage and energy release of the drilled coal samples become more moderate after being filled, especially when filled with the ductile material.

Overall, the drilled coal samples are prone to stress concentration near the holes during uniaxial compression due to its own defects, resulting in chain failure reactions including the initiation, development and rapid expansion of cracks. In this process, they quickly release the accumulated elastic strain energy. Besides, as the drilled coal samples have a significantly lower strength than the intact one, their intensity of energy release is also relatively lower. In the filled coal samples, however, although stress concentration around the drilled holes also tends to induce the initiation, development and rapid expansion of cracks during deformation and failure, the fillings play a certain supporting role and raise the peak value of elastic strain energy release. That is, the fillings increase the difficulty in releasing the elastic strain energy while easing the intensity of energy release. In addition, even though both types of the filled samples undergo the deformation and failure process, during which cracks emerge and expand due to stress concentration around the drilled holes, the ductile filling has a certain flexible supporting and buffering effect, reducing the intensity of elastic energy release by slowly raising the strain value of the coal sample.

4.2 Spatial distribution of AE events under varying hole sizes and filling materials

To analyze spatial distribution of AE events generated by coal samples during loading, events with high AE energy during the failure process were selected to construct the spatial distribution maps based on their spatial positioning and corresponding AE energy. As shown in Fig. 5, the central position of a sphere is the coordinate of the AE event; the size represents its energy level; and the color represents the time when the event occurs (red, yellow, green, and blue represent the event time from the earliest to the latest).

Fig. 5
figure 5

Spatial distribution of AE events generated at different times

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From Fig. 5a, numerous AE events in the intact coal sample are located on the boundary of y = 0, and that high-energy events are mainly located on the shear plane and this boundary. The finding suggests that cracks in the intact coal sample are mostly formed near the shear plane and the boundary of y = 0, which fits well with the experimental result that shear and brittle tensile failure occurs in the sample. Besides, high-energy events occur relatively late, which is also basically consistent with the evolution law in Fig. 4a.

In Fig. 5b, c, AE events, especially high-energy ones, in the drilled coal samples are concentrated near the drilled holes. The major reason is that stress concentration tends to occur near the drilled hole, where the most cracks emerge, develop and expand. Besides, samples with larger drilled holes generate more high-energy events, and high-energy events occur throughout the loading process.

In Fig. 5d, e, the high-energy AE events in the filled coal samples tend to disperse towards the shear plane, compared to those in the drilled samples. Moreover, the sample filled with brittle material generates far more high-energy AE events than the one filled with ductile material, and high-energy AE events occur throughout the loading process.

Overall, hole drilling and hole filling can change the spatial distribution of AE events: drilling makes them concentrated, while filling effectively disperse them. The fundamental reason for changes in the spatial distribution is that the drilled or filled structures can change the fracture morphology of the coal samples.

5 Influence mechanism of drilled and filled structures on sample failure

According to the above analysis, drilled and filled structures have a remarkable effect on the strength, deformation characteristics, and energy evolutions. Based on the above test results, the influence of drilling and filling on coal mechanics and deformation primarily involves three aspects:

  1. (1)

    Hole drilling can decrease the uniaxial compressive strength and improve the deformability. Meanwhile, it changes the failure mode, significantly reducing the severity of sample failure.

  2. (2)

    After hole filling, the uniaxial compressive strength is enhanced to a certain extent, yet it is still lower than that of the intact one. Besides, the strength and deformation characteristics vary to some extent due to the varying properties of filling materials.

  3. (3)

    Drilled and filled structures can change the emergence, development, and expansion morphology of cracks, which influences the energy evolution in coal.

The strength, deformation characteristics and energy evolutions of the coal samples are essentially affected by stress distribution around the drilled holes in different coal samples. Aiming at exploring the stress distribution and failure mechanism around the drilled holes, a plane mechanical model of filled samples was constructed based on the elastic theory (Fig. 6).

Fig. 6
figure 6

Infinite plane mechanical model of coal containing a hole-filled structure

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Two assumptions are made to simplify the stress problem around the filled sample: first, the coal body and the filling are both uniform, isotropic, and linearly elastic; second, the elastic parameters of the sample are independent of stress variation. Under plane stress conditions, the sample model is considered to be an infinite plate whose center bears a circular hole-filled structure. Additionally, the coal body is assumed to be in complete contact with the filling, and the radial and shear stresses on the contact surface are equal.

Figure 6 depicts the infinite plane mechanical model of coal containing a hole-filled structure under biaxial compression. The compressive stress on the lower and upper sides of the sample is \(p\); the compressive stress on the left and right sides is \(q\); the radius of the filling is \(a\); the elastic moduli of the coal body and the filling are \(E_{1}\) and \(E_{2}\), and the Poisson’s ratios of them are \(\mu_{1}\) and \(\mu_{2}\), respectively.

The lateral stress coefficient of the sample can be defined as:

$$k = \frac{q}{p}$$
(1)

The boundary conditions can be obtained according to the elastic theory:

$$\left. \begin{gathered} \sigma_{r} |_{r = \infty } = p\left[ {\frac{1 + k}{2} - \frac{1 - k}{2}\cos 2\theta } \right] \hfill \\ \sigma_{\theta } |_{r = \infty } = p\left[ {\frac{1 + k}{2} + \frac{1 - k}{2}\cos 2\theta } \right] \hfill \\ \tau_{r\theta } |_{r = \infty } = p\frac{1 - k}{2}\sin 2\theta \hfill \\ \end{gathered} \right\}$$
(2)

where \(\sigma_{r}\), \(\sigma_{\theta }\) and \(\tau_{r\theta }\) are the radial, tangential and shear stresses; \(r\) is the polar radius in the polar coordinate system; and \(\theta\) is the polar angle. Considering the assumption that the coal body is in complete contact with the filling, the contact conditions at \(r = a\) are:

$$\left. \begin{gathered} \sigma_{r} = \sigma^{\prime}_{r} |_{r = a} \hfill \\ \tau_{r\theta } = \tau^{\prime}_{r\theta } |_{r = a} \hfill \\ \end{gathered} \right\}$$
(3)

where \(\sigma^{\prime}_{r}\) and \(\tau^{\prime}_{r\theta }\) are the radial and shear stresses of the filling. According to \(r = \infty\), the Airy stress is described as:

$$\varphi = a_{1} \ln r + a_{2} r^{2} + a_{3} r^{2} \ln r + \left( {a_{21} r^{2} + a_{22} r^{4} + \frac{{a_{23} }}{{r^{2} }} + a_{24} } \right)\cos 2\theta$$
(4)

where \(a_{i}\) and \(a_{ij}\) are unknown coefficients deduced from boundary conditions.

Furthermore, corresponding stress components can be obtained according to the above equations, and when \(r = \infty\), the stress is a finite value. Therefore, when \(a_{3} = a_{22} = 0\) and \(r \ge a\), the stress components of the coal body are:

$$\left. \begin{gathered} \sigma_{r} = 2a_{2} + \frac{{a_{1} }}{{r^{2} }} - \left( {2a_{21} + \frac{{6a_{23} }}{{r^{4} }} + \frac{{4a_{24} }}{{r^{2} }}} \right)\cos 2\theta \hfill \\ \sigma_{\theta } = 2a_{2} - \frac{{a_{1} }}{{r^{2} }} + \left( {2a_{21} + \frac{{6a_{23} }}{{r^{4} }}} \right)\cos 2\theta \hfill \\ \tau_{r\theta } = \left( {2a_{21} - \frac{{6a_{23} }}{{r^{4} }} - \frac{{2a_{24} }}{{r^{2} }}} \right)\sin 2\theta \hfill \\ \end{gathered} \right\}$$
(5)

According to Eqs. (2) and (5), the assumption of the boundary conditions, etc., it can be derived by calculation (Zhang et al. 2019) that:

$$\left. \begin{gathered} a_{2} = p\frac{1 + k}{4} \hfill \\ a_{21} = p\frac{1 - k}{4} \hfill \\ a_{1} = M_{1} a^{2} p\frac{1 + k}{2} \hfill \\ a_{23} = - M_{2} a^{4} p\frac{1 - k}{4} \hfill \\ a_{24} = M_{2} a^{2} p\frac{1 - k}{2} \hfill \\ \end{gathered} \right\}$$
(6)
$$\left. \begin{gathered} M_{1} = \frac{{(1 - \mu_{1} )\eta - (1 - \mu_{2} )}}{{(1 + \mu_{1} )\eta + (1 - \mu_{2} )}} \hfill \\ M_{2} = \frac{{(1 + \mu_{1} )\eta - (1 + \mu_{2} )}}{{(3 - \mu_{1} )\eta + (1 + \mu_{2} )}} \hfill \\ \eta = \frac{{E_{2} }}{{E_{1} }} \hfill \\ \end{gathered} \right\}$$
(7)

Thus, the stress of the coal model is:

$$\left. \begin{gathered} \sigma_{r} = \frac{(1 + k)p}{2}\left[ {1 + M_{1} \left( \frac{a}{r} \right)^{2} } \right] - \frac{(1 - k)p}{2}\left[ {1 - 3M_{2} \left( \frac{a}{r} \right)^{4} + 4M_{2} \left( \frac{a}{r} \right)^{2} } \right]\cos 2\theta \hfill \\ \sigma_{\theta } = \frac{(1 + k)p}{2}\left[ {1 - M_{1} \left( \frac{a}{r} \right)^{2} } \right] + \frac{(1 - k)p}{2}\left[ {1 - 3M_{2} \left( \frac{a}{r} \right)^{4} } \right]\cos 2\theta \hfill \\ \tau_{r\theta } = \frac{(1 + k)p}{2}\left[ {1 + 3M_{2} \left( \frac{a}{r} \right)^{4} - 2M_{2} \left( \frac{a}{r} \right)^{2} } \right]\sin 2\theta \hfill \\ \end{gathered} \right\}$$
(8)

According to the experimental scheme, the sample model can be divided into three patterns, which are:

(1) When a circular drilled hole exists in the coal sample and the sample is only subjected to upper and lower compressive loads, \(k = q = 0\), \(E_{2} = 0\), \(\mu_{2} = 0\), \(M_{1} = M_{2} = - 1\). Then, stress components surrounding the drilled hole (\(r = a\)) can be obtained by:

$$\left. \begin{gathered} \sigma_{r} = 0 \hfill \\ \sigma_{\theta } = p + 2p\cos 2\theta \hfill \\ \tau_{r\theta } = 0 \hfill \\ \end{gathered} \right\}$$
(9)

(2) When the circular hole is filled with polyurethane whose Poisson’s ratio and elastic modulus are close to those of the coal body, assuming they are approximately the same, \(k = q = 0\), \(E_{2} = E_{1}\), \(\mu_{2} = \mu_{1}\), \(M_{1} = M_{2} = 0\). Then, stress components surrounding the drilled hole (\(r = a\)) are given by:

$$\left. \begin{gathered} \sigma_{r} = \frac{p}{2} - \frac{p}{2}\cos 2\theta \hfill \\ \sigma_{\theta } = \frac{p}{2} + \frac{p}{2}\cos 2\theta \hfill \\ \tau_{r\theta } = \frac{p}{2}\sin 2\theta \hfill \\ \end{gathered} \right\}$$
(10)

(3) When the hole is filled with cement, it is assumed that \(E_{1}\) = 3500 MPa, \(E_{2}\) = 35 GPa, \(\mu_{1} =\) 0.40, \(\mu_{2}\) = 0.20, and \(k = q = 0\), so that the difference of the Poisson’s ratios and elastic moduli between the sample and the coal body is large enough for comparison. Then, stress components surrounding the hole (\(r = a\)) can be expressed as:

$$\left. \begin{gathered} \sigma_{r} = 0.642p - 0.642p\cos 2\theta \hfill \\ \sigma_{\theta } = 0.358p - 0.207p\cos 2\theta \hfill \\ \tau_{r\theta } = 0.736p\sin 2\theta \hfill \\ \end{gathered} \right\}$$
(11)

For the purpose of revealing the failure mechanism of drilled and filled samples, the stress distribution surrounding the drilled holes is calculated based on Eqs. (9)–(11) (Fig. 7).

Fig. 7
figure 7

Maps of stress distribution surrounding three types of structural holes in coal samples (\(r = a\))

Full size image

In Fig. 7a, the radial stress surrounding the drilled hole in the unfilled sample is 0, while the minimum concentrated stresses appear at 0° (360°) and the maximum concentrated stresses exist in the 90° direction in the polyurethane-filled and cement-filled samples, with values of \(p\) and 1.284\(p\), respectively. In Fig. 7b, for the unfilled sample, the lower and upper positions of the drilled hole witness circumferential tensile stress concentration, the maximum value—\(p\) appearing at 90°, which agrees with the experimental result that the strain concentration arises at the bottom and top of the drilled hole (Table 2). The left and right sides of the drilled hole witness circumferential compressive stress concentration, the maximum value 3\(p\) appearing at 0°. After hole filling, the circumferential stress around the drilled hole becomes distributed in a notably different patten. For the polyurethane-filled sample, the minimum circumferential stress mostly occurs at the bottom and top of the drilled hole, with a value of 0, whereas the maximum stress is concentrated on the left and right sides, with a value of \(p\). For the cement-filled sample, the minimum circumferential stress appears on the left and right sides, with a value of 0.151\(p\), yet the maximum stress is concentrated at the top and bottom, with a value of 0.565\(p\). In Fig. 7c, the shear stress surrounding the drilled hole in the unfilled sample is 0. In the polyurethane-filled and cement-filled samples, however, the minimum concentrated stresses around the hole appear at 135° (315°), which are − 0.5\(p\) and − 0.736\(p\) respectively, while the maximum concentrated stresses appear in the 45° direction, which are 0.5\(p\) and 0.736\(p\), respectively.

Overall, hole drilling reduces coal strength and change coal deformability, while hole filling enhances coal mechanical properties. Besides, the degree of enhancement is closely correlated with the mechanical properties of the filling. Compared with cement filling, polyurethane filling can improve coal mechanical properties more effectively. Among the tested samples, the mechanical properties of the intact one are the finest, followed by the 6-mm-hole, polyurethane-filled, and 10-mm-hole ones in turn. As can be concluded through the experimental results and theoretical derivation, hole filling has an obvious reinforcement effect on coal, while the degree of reinforcement varies with filling materials. The strength and deformation characteristics of coal are mainly affected by the emergence, development, and expansion of cracks, yet cracking is determined by stress. Therefore, the mechanical behavior of the drilled coal samples is closely correlated with the distribution of stress surrounding the drilled holes. In the different drilled or filled samples, smaller radial, circumferential, and shear stresses and a lower concentration degree of compressive stress around the drilled holes correspond to better mechanical properties.

6 Conclusions

  1. (1)

    After hole drilling, the indexes related to burst propensity of the coal samples all decline to a certain extent. The larger the hole diameter, the more significant the decline in these indexes. Hole filling can not only enhance the strength and deformation resistance of coal, but also reduce its burst propensity.

  2. (2)

    Strain concentration tends to occur around holes, and cracks usually expand and interconnect along the strain concentration zone. Hole filling can effectively reduce strain concentration, especially with ductile materials which significantly lower the strain concentration degree near holes.

  3. (3)

    During uniaxial compression, the coal samples with drilled holes can rapidly release the accumulated elastic strain energy, but the intensity is less compared to that of the intact coal sample. After hole filling, the intensity of energy release is reduced, and the decrease is more notable in the sample filled with the ductile material.

  4. (4)

    Analytical solutions of stress distribution around the circular holes in the coal samples with different elastic properties were derived. Based on the experimental results and theoretical derivation, hole filling can weaken the stress concentration degree around holes, thus enhancing coal strength and deformability. Moreover, the fillings can achieve a better filling effect if their Poisson’s ratio and elastic modulus are closer to those of the coal body.

  5. (5)

    For impact-prone coal seams, not only borehole pressure relief must be considered, but also the effects of borehole pressure relief and filling on the stability of the surrounding rock in the roadway can be achieved, so as to achieve comprehensive control of impact-prone coal seams pressure relief and surrounding rock stability.