Study on coal wall spalling mechanism of large mining height working face based on folding mutation theory

Abstract

Aiming at the unclear mechanism behind the sudden instability of coal wall spalling, this study takes a fully-mechanized mining face with large mining height as an example. Through theoretical analysis and numerical simulation, the energy release mechanism governing the instability of the coal shell structure is revealed, and a theoretical failure model for spalling prediction is established. Based on the mechanical characteristics of coal shell failure and employing the theory of plastic strain localization in coal mass, a catastrophe model for coal wall instability was developed by analyzing the incremental relationship between coal shell deformation and the work done by deviatoric stress. The research outcomes derived quantitative expressions for energy release and proposed the expression for deviatoric stress strain energy, clarifying that energy exceeding a critical threshold is the fundamental cause of sudden spalling. Furthermore, expressions for the release of volumetric strain energy and gas internal energy were derived, revealing the coupling mechanism between internal and external stresses. The research outcomes can be directly applied to predict the risk of rib spalling by monitoring the degree of energy accumulation and to optimize support parameter design based on the energy release threshold, providing theoretical guidance for safe mining in high-stress coal seams. This study achieves a breakthrough from phenomenological description to mechanistic quantification, holding significant practical value for the prevention and control of rib spalling disasters in fully-mechanized mining faces.

Introduction

In recent years, with the significant improvement of coal mining equipment and safety management, the mining thickness of the working face has also gradually increased. Due to the increase of mining height, the load and exposure area of the coal wall continue to increase, which enhances the difficulty of surrounding rock control. In this case, the strong ground pressure can induce the rockburst and the instability of the coal wall with a large height, and the operation failure of the mining equipment can be caused, which seriously affects the safe production of the coal mine^{1,2,3,4,5,6,7,8,9,10}. The development of China’s large mining height fully mechanized mining technology can be summarized as four technological revolutions: the transition from manual drill-and-blast mining, conventional mechanized mining, and fully mechanized mining to intelligent mining. Post-2000, in response to the demand for thick seam extraction, China developed the “fully mechanized top coal caving mining with large mining height” technology, overcoming the challenge of extracting ultra-thick coal seams (e.g., 14–20 m) in a single pass and achieving a recovery rate exceeding 88%. For instance, the world’s first intelligent fully mechanized face with an ultra-large 10-meter mining height was successfully completed at the Caojiatan Coal Mine of Shaanxi Coal Group, achieving a monthly output of 1.55 million tons. This project utilized intelligent winches and precision penetration technology, reducing manual intervention by over 70%.

Research on the issue of coal wall rib spalling primarily focuses on its characteristics, mechanisms, and control measures¹¹. In terms of influencing factors and mechanism analysis, Tian et al.¹², Li et al.¹³ and Liu et al.¹⁴ have explored the factors affecting coal wall stability and their sensitivity, Chen et al.¹⁵, Yang et al.¹⁶ and Xiong et al.¹⁷ analyzed characteristics such as nonlinear failure and asymmetric damage, Si et al.¹⁸ and Li et al.¹⁹ elucidated the spalling mechanism based on damage fracture mechanics and plate structural mechanics. Regarding the construction of mechanical models, Wang et al.²⁰ and Fu et al.²¹ studies have established a roof-coal interaction model, Wang et al.²² and Li et al.²³ studies a Bishop mechanical model for coal wall sliding instability, Chen et al.²⁴ established a dynamic thin-plate rupture model, Guo et al.²⁵ analyzed the distribution of plastic zones and instability criteria. In the area of control techniques and methods, measures such as adjusting support strength and modifying coal stress states and cutting height. Ma et al.^26,27 proposed to prevent spalling, while the energy release characteristics associated with rockburst induced by coal wall instability in deep roadways have also been investigated. Methodologically, research commonly integrates theoretical analysis, Yang et al.²⁸, Li et al.^29,30, Xue et al.³¹ and Pan et al.³² illustrated the primary modes and locations of spalling from both macro and micro perspectives. These existing achievements provide a critical foundation for the systematic understanding and prevention of coal wall rib spalling. Xue et al.³³, Li et al.³⁴ and Zhao et al.³⁵ investigated the influencing factors of coal wall rib spalling, quantified these factors while performing a sensitivity ranking, and constructed the analysis structure diagram of coal wall spalling characteristics.

Current research primarily exhibits the following limitations, Firstly, most existing studies simplify the coal wall as a continuous medium for analysis, overlooking the critical characteristic that the coal mass forms a load-bearing shell structure prior to failure, which consequently fails to explain the instantaneous and abrupt nature of rib spalling. Secondly, prevailing energy analysis methods largely remain at the macroscopic level of energy conservation, lacking quantitative descriptions of the mesoscopic energy conversion mechanisms, such as the work of deviatoric stress and the increment of strain energy. The innovations of this study are as follows. A coal wall instability folding catastrophe model based on plastic strain localization is constructed. For the first time, the mechanism of sudden spalling driven by multi-source energy coupling is quantified from the perspective of energy threshold, which realizes the theoretical breakthrough from phenomenon description to quantitative early warning, and provides a direct basis for engineering prevention and control. To summarize, traditional studies on coal wall spalling have largely relied on empirical statistics or macroscopic mechanical analysis, which fail to capture the energy release mechanism at the moment of instability. In response, this study employs shell structure theory to conceptualize the spalling process as the instability of a “coal shell.” This framework overcomes the limitations of conventional continuum models and better reflects the actual shell-like failure characteristics observed in situ. While existing research on strain localization has focused predominantly on shear band formation, this work innovatively integrates plastic deformation localization theory with catastrophe theory to describe the complete process of energy accumulation and release within the localized zone. Furthermore, whereas most energy-based models emphasize a single energy form, this study simultaneously derives expressions for the release of both distortional strain energy and volumetric strain energy, clarifying the synergistic role of multiple energy types during instability. Finally, by incorporating the spatial constraints of the coal shell structure, the proposed model resolves the difficulty faced by classical progressive catastrophe models in representing structural effects in geotechnical contexts.

Theory/calculation

Relationship between coal wall displacement, deformation and increment of work and energy

According to the actual measurement of the working face 42,105 of the Buertai coal mine, the fracture points of the rib spalling were mostly concentrated in the middle of coal walls, as shown in Fig. 1. Field observation results indicate that the rib spalling fracture points are concentrated in the middle of the coal wall, which validates the rationality of the “spherical symmetry failure” hypothesis proposed in this paper. It can be found that the coal wall rib spalling is a spherical symmetry problem, and the coal mass before rib spalling is divided into many coal shell structures, as shown in Fig. 2. As can be seen from Fig. 2, the mechanical behavior of this hypothesis is almost consistent with that of the coal wall failure in coal seam, and it can reasonably explain the failure behavior of such coal seams. According to the Saint Venant’s principle, within the coal shell diameter of the rib spalling, the stress distribution in the coal mass, roof stress and floor stress of the rib spalling of the coal wall is approximately spherical symmetry, as shown in Fig. 3. Based on the property of the spherical symmetry, the Lode strain parameters in the elastic stage are the same as those in the plastic stage, meeting the strain strength additivity condition³¹. According to the physical relationship of a single curve of the Nᴫющин³², the physical relationship between the stress intensity σ_i of coal and strain strength ε_i can be expressed as follows:

$$\left. \begin{gathered} {\sigma _i}{\text{=}}\frac{3}{{2\left( {1{\text{+}}\mu } \right)}}E{\varepsilon _i}=BE{\varepsilon _i},\;{\varepsilon _i}<{\left( {\frac{1}{k}} \right)^{\frac{1}{k}}}{\varepsilon _0}={\varepsilon _c} \hfill \\ {\sigma _i}=BE{\varepsilon _i}\exp \left[ { - {{\left( {\frac{{{\varepsilon _i}}}{{{\varepsilon _0}}}} \right)}^k}} \right],\;{\varepsilon _i} \geqslant {\left( {\frac{1}{k}} \right)^{\frac{1}{k}}}{\varepsilon _0}={\varepsilon _c} \hfill \\ B=\frac{3}{{2\left( {1{\text{+}}\mu } \right)}} \hfill \\ \end{gathered} \right\}$$

(1)

where σ₁, σ₂, σ₃ and τ₁, τ₂, τ₃ represent the three-dimensional stress and three-dimensional strain of any stress unit of coal shell; k is the homology index of the curve. When k = 1, then ε₀ = ε_c; the larger the k, the steeper the yield section of the curve, as shown by the solid line in Fig. 4. E is the modulus of elasticity. Equation (1) presents the functional relationship between stress and strain.

As shown in Fig. 3, due to the joint extrusion of the coal wall by the roof stress and the floor stress, the range of the weakening area inside the coal mass gradually increases and is forced to protrude to the working face space, resulting in the gradual increase of the horizontal displacement u(r) of the coal wall. According to the stress environment of coal walls, the displacement d of coal walls mainly includes deviator strain displacement u_d(r) and average strain displacement u_v(r), which can be measured by strain strength ε_i and average strain strength ε_m. The u(r) can be expressed as follows:

$$\left. \begin{gathered} u\left( r \right)={u_d}\left( r \right)+{u_v}\left( r \right)=r\left[ {\frac{{{\varepsilon _i}\left( r \right)}}{2}+{\varepsilon _m}\left( r \right)} \right] \hfill \\ {u_d}\left( r \right)=\frac{{r{\varepsilon _i}\left( r \right)}}{2} \hfill \\ {u_v}\left( r \right)=r{\varepsilon _m}\left( r \right) \hfill \\ \end{gathered} \right\}$$

(2)

Fig. 1

On-site photo of rib spalling in the working face.

Fig. 2

Schematic diagram of the rib spalling in the working face.

Fig. 3

Load, displacement and deformation of coal shell before rib spalling.

Fig. 4

Relationship between stress intensity and strain intensity of coal medium.

Before the occurrence of rib spalling, there is a horizontal displacement increment of the coal shell. The energy change caused by the internal strain of the coal wall is assumed to be δP, and The work of the external force on the corresponding displacement of the coal shell is δQ. If the kinetic energy of coal shell in rib spalling is excluded, the incremental balance relationship between work and energy of coal shell can be obtained according to the principle of energy conservation.

$$\delta P - \delta Q=0$$

(3)

According to the integral mean value theorem, δP in Eq. (2) can be expressed as follows:

$$\begin{aligned} \delta P&=\int_{a}^{b} {\left[ {{\sigma _i}\delta {\varepsilon _i}+3{\sigma _m}\delta {\varepsilon _m}} \right]} S\left( r \right)dr \hfill \\ &=\left[ {{\sigma _i}\left( {{\zeta _1}} \right)\delta {\varepsilon _i}\left( {{\zeta _1}} \right)+3{\sigma _m}\left( {{\zeta _1}} \right)\delta {\varepsilon _m}\left( {{\zeta _1}} \right){\text{+}}p\left( {{\zeta _1}} \right)\delta \omega \left( {{\zeta _1}} \right)} \right] \times S\left( {{\zeta _1}} \right)\left( {b - a} \right) \hfill \\ {\text{ }}&=\delta {P_p}+\delta {P_t}+\delta {P_s},{\text{ }}{\xi _1} \in \left( {a,b} \right) \end{aligned}$$

(4)

According to the balance equation of transportation force, the geometric relationship between displacement and strain, and the relationship between stress intensity and average stress, the energy analysis of external force on rib spalling in the coal wall is conducted. The work increment of external force (δQ) is decomposed into external work increment of deviatoric stress of the coal wall (δQ_p), external work increment of volume strain (δQ_t) and work increment of the seepage force (δQ_s):

$$\begin{aligned} \delta Q&={\sigma _r}\left( b \right)S\left( b \right)\delta {u_b} - {\sigma _r}\left( a \right)S\left( a \right)\delta {u_a} \hfill \\ &=\left[ {{\sigma _i}\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right)+3{\sigma _m}\left( \zeta \right)\delta {\varepsilon _m}\left( \zeta \right) - \beta {F_S}\delta u} \right] \times S\left( \zeta \right)\left( {b - a} \right) \hfill \\ &=\delta {Q_p}+\delta {Q_t}+\delta {Q_s},{\text{ }}\zeta \in \left( {a,b} \right) \hfill \\ \end{aligned}$$

(5)

In Eq. (4), S(r) is the total area of coal shell that is about to rib spalling, and it can be expressed as follows:

$$S\left( r \right)=2\pi {\zeta ^2}\left\{ {1 - \cos \left[ {\left( {{\theta _o}+{\psi _o}} \right)/2} \right]} \right\}$$

(6)

where S(r) includes the area S^eβ0 of the residual part (diagonal β₀) at the end of the coal shell surface in an elastic state, and σ_m is the average stress in Eq. (5).

$$\begin{gathered} \delta {Q_p}=S\left( \zeta \right)\left( {b - a} \right){\sigma _i}\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right) \hfill \\ \delta {Q_t}=S\left( \zeta \right)\left( {b - a} \right)3{\sigma _m}\left( \zeta \right)\delta {\varepsilon _m}\left( \zeta \right) \hfill \\ \delta {Q_s}=S\left( \zeta \right)\left( {b - a} \right)\left[ { - \beta {F_S}\left( \zeta \right)\delta u\left( \zeta \right)} \right] \hfill \\ \end{gathered}$$

(7)

Through the calculation of Eqs. (3)–(5), ζ₁ = ζ can be obtained. In the process of deformation and displacement before the rib spalling, the variation of external force work δQ_p, δQ_t and δQ_s are related to the internal energy δP_p, δP_t and δP_s respectively:

$$\begin{gathered} \delta {P_p} - \delta {Q_p}=0 \hfill \\ \delta {P_t} - \delta {Q_t}=0 \hfill \\ \delta {P_s} - \delta {Q_s}=0 \hfill \\ \end{gathered}$$

(8)

In view of the instability of the coal wall, the expansion of the weakening deformation zone in the coal mass and the extrusion of the internal stress in the coal mass are analyzed.

The relation between work and energy increment of the coal shell under plastic deformation and expansion

Based on the localization theory of plastic deformation in coal and rock failure, plastic deformation or fracture initiates and concentrates at the crack tip. When the coal–rock mass reaches its limit state, it releases energy in the form of elastic energy upon unloading. Figure 5 illustrates the average σ_i - ε_i curve considering only the effect of plastic deformation localization. To effectively analyze the instability and fracture mechanism associated with the localized plastic deformation during coal wall rib spalling, the curve was modified in accordance with Eq. (1) and actual conditions. A smoother constitutive relationship for the yield segment, characterized by a curve index n < m, was introduced, as shown in Fig. 5.

$${\sigma _i}=B{E_{O1}}{\varepsilon _i}\exp \left[ { - {{\left( {{\varepsilon _i}/{\varepsilon _{O1}}} \right)}^n}} \right]$$

(9)

In Fig. 5, point c represents the peak strength and is taken as the dividing point of the plastic failure of the coal mass. When some of the bearing capacity of the plastic deformation concentration area of the coal mass decreases along the softening section of the curve, the rest of the coal mass is unloaded along the c-O straight line with the slope of BE. When the stress increment is δσ_i < 0, the strain increment of the former is δε_ip (> 0), and the strain increment of the latter is δε_i.e. (< 0). In this process, δσ_i is the same, but their strains are quite different, indicating that the plastic deformation area absorbs energy and the elastic deformation area releases energy. The post-peak softening behavior at point c aligns with rock plastic softening laws. Figure 5 reveals a binary path: coal plastic zone softens along the curve while elastic zone unloads along c-O line, breaking the conventional continuum model’s unitary assumption. This discovery uncovers that coal wall spalling stems from energy inversion critical effect, when plastic zone’s energy absorption capacity saturates, instability releases energy via catastrophic model.

Fig. 5

The σ_i - ε_i curve of plastic deformation concentration area.

The influence area of plastic deformation inside the coal mass is in the crack tip area w. There are a large number of fully developed cracks in the coal shell before the occurrence of rib spalling. The area of the influence area of plastic deformation in the coal shell is counted and recorded as S^m(ζ); the area of other intact parts in the coal shell is recorded as S^nφo(ζ), and their relationship to the aforementioned S^nβo is as follows:

$${S^{n{\varphi _0}}}\left( \zeta \right)=S\left( \zeta \right) - {S^m}\left( \zeta \right) - {S^{n{\beta _o}}}$$

(10)

Assuming that S^nφo(ζ) and S^nβo have the same elastic modulus E, then Eq. (11) can be obtained.

$${S^n}\left( \zeta \right)={S^{n{\varphi _0}}}\left( \zeta \right)+{S^{n{\beta _o}}}\left( \zeta \right)$$

(11)

In this way, δP_p in the first formula of Eq. (1) in Eq. (8) can be divided into δP_p1 and δP_p2.

$$\left. \begin{gathered} \delta P_{p}^{n}{\text{=}}{\sigma _i}\delta \varepsilon _{i}^{n}{S^n}\left( \zeta \right)\left( {b - a} \right)={\sigma _i}\delta S_{i}^{n}\left( \zeta \right)\left( {b - a} \right) \hfill \\ \delta P_{p}^{m}{\text{=}}{\sigma _i}\delta \varepsilon _{i}^{m}{S^m}\left( \zeta \right)\left( {b - a} \right)={\sigma _i}\delta S_{i}^{m}\left( \zeta \right)\left( {b - a} \right) \hfill \\ \end{gathered} \right\}$$

(12)

Since the cracks are evenly and randomly distributed in the coal shell, S^nφo(ζ) and S^m(ζ) are the statistical areas, Sⁿ(ζ) and S^m(ζ) can deform freely without constraining each other in Eq. (12).

$$\delta S_{i}^{m}\left( \zeta \right)={S^m}\left( \zeta \right)\delta \varepsilon _{i}^{m},\;\delta S_{i}^{n}\left( \zeta \right)={S^n}\left( \zeta \right)\delta \varepsilon _{i}^{n}$$

(13)

where $\delta \varepsilon _{i}^{m}$ in Eq. (13) and strain increment $\delta {\varepsilon _i}\left( \zeta \right)$ in the average sense can cause coal shell deformation of an equal amount. The extensive development of cracks within the coal shell directly leads to an increase in the area of the plastic deformation affected zone, denoted as S_m(ζ). As indicated in Eq. (12) and Eq. (13), the uniform random distribution of cracks allows the statistical region S_nρo(ζ)and S_m(ζ)to deform freely without mutual constraint, resulting in the equivalence between the strain increments δε_i^m and δε_i(ζ), which consequently induces the overall deformation of the coal shell. As the crack density increases, the plastic zone continues to expand and accumulates strain energy. Once its load-bearing capacity descends along the softening curve, the elastic zone rapidly releases energy, ultimately triggering instability-induced rib spalling of the coal shell along the crack network.

$${S^n}\left( \zeta \right)\delta \varepsilon _{i}^{n}\left( \zeta \right)+{S^m}\left( \zeta \right)\delta \varepsilon _{i}^{m}\left( \zeta \right)=S\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right)$$

(14)

From the first formula in Eq. (1), it can be found that $\delta \varepsilon _{i}^{n}=\delta \left( {{\sigma _i}/BE} \right)<0$, and S^m(ζ) is much less than S (ζ). Therefore, $\delta \varepsilon _{i}^{m}\left( \zeta \right)>\delta {\varepsilon _i}\left( \zeta \right)$, i.e. σ_i(ξ) can change if dσ_i < 0. Therefore,

$$\delta \varepsilon _{i}^{m}\left( \zeta \right)=\chi \delta {\varepsilon _i}\left( \zeta \right), {\text{ and }}\chi >1$$

(15)

where χ is related to the properties of coal mass, and only its average sense is considered here. The ratio $\delta \varepsilon _{i}^{m}/\delta {\varepsilon _i}$ varies at different corresponding points in Figs. 4 and 5. By introducing χ, the relationship between the strain increment during plastic deformation localization and the strain increment in the average sense can be established, and the relationship between stress intensity and strain intensity described in Eq. (1) can still be used to analyze the instability of coal walls.

Establishment of folding catastrophe model of rib spalling

When there is horizontal displacement δu₁ > 0 in the coal mass to the working face, and the radial displacement caused by corresponding eccentric stress δu₂ > 0, the coal wall is subjected to the stress in the coal mass, and the plastic deformation area S^m(ζ) of the coal mass expands and consumes energy δP_p^m; while in the plastic deformation area Sⁿ(ζ) of the coal mass, due to unloading and releasing performance δP_pⁿ, if δP_pⁿ < δP_p^m, the overall structure of the coal wall is stable; if the rib spalling occurs, the external stress of the coal mass needs to do work to provide energy δQ_p, according to the principle of energy conservation. Therefore, the energy δP_p^m consumed by the expansion of the plastic deformation area S^m(ζ), the elastic energy δP_pⁿ released by the unloading in the elastic deformation area Sⁿ(ζ), and the energy δQ_p provided by the work done by the external stress of the coal satisfy the balance of work and energy increment.

$$\delta P_{p}^{m} - \delta P_{p}^{n} - \delta {Q_p}=0$$

(16)

After Eq. (1) is divided by $\delta S_{i}^{m}$ to differentiate, the equilibrium equation of coal shell displacement and deformation before rib spalling in Fig. 3 can be obtained:

$$\frac{{\delta P_{p}^{m}}}{{\delta S_{i}^{m}}} - \frac{{\delta P_{p}^{n}}}{{\delta S_{i}^{m}}} - \frac{{\delta {Q_p}}}{{\delta S_{i}^{m}}}=0$$

(17)

To ensure the radial displacement increment du₂>0, the external input energy is required to produce an increment $\delta S_{i}^{m}$> 0 in the area of the corresponding plastic deformation concentration area ${S^m}\left( \zeta \right)$, let

$$W=\frac{{\delta {Q_p}}}{{\delta S_{i}^{m}\left( \zeta \right)}}$$

(18)

The W is called the energy input rate. From the physical meaning of Eqs. (3), (16)–(18), when W > 0, the quasi-static deformation of coal shell occurs before rib spalling; If $W \to 0$, it indicates that the external force on the coal wall surface is not required to do work, that is, only the energy transfer occurs inside the coal shell (the strain energy generated in the elastic region is transferred to the plastic region). The area expansion of S^m(ζ) indicates that the coal shell structure tends to be in a critical state. Therefore, W = 0 can be used as the critical condition to judge the structural instability or rib spalling of the coal wall. W = 0 signifies that the coal wall system is in a critical state of instability. At this point, the external energy input and internal energy dissipation achieve instantaneous balance, with the energy required for the expansion of the plastic zone being entirely supplied by the strain energy released from the unloading of the elastic zone, eliminating the need for additional external work. The system is at the boundary between stability and instability, where even minor disturbances can trigger the equilibrium point to cross the critical axis (the F-1 axis), leading to sudden instability and failure of the coal wall. This critical condition provides a clear energy threshold criterion for engineering early warning.

From Eq. (13), we can obtain $\delta S_{i}^{n}\left( \zeta \right)={S^n}\delta \varepsilon _{i}^{n}={{{S^n}\delta {\sigma _i}} \mathord{\left/ {\vphantom {{{S^n}\delta {\sigma _i}} {BE}}} \right. \kern-0pt} {BE}}$. Then

$$f=\frac{{BE}}{{{S^n}}}$$

(19)

where f is the compressive strength of Sⁿ. Combined with the Eqs. (12), (17) and (18), Eq. (20) can be obtained:

$${\sigma _i}\left( \zeta \right)\frac{{\delta {\sigma _i}\left( \zeta \right)}}{{\delta S_{i}^{m}\left( \zeta \right)f}} - {\sigma _i}\left( \zeta \right) - W\frac{1}{{\left( {b - a} \right)}}=0$$

(20)

Equation (21) can be obtained from formula 2 in Eq. (1).

$${\sigma _i}=BE{\varepsilon _i}\exp \left[ { - {{\left( {\frac{{{\varepsilon _c}}}{{{\varepsilon _0}}}} \right)}^k}{{\left( {\frac{{{\varepsilon _i}}}{{{\varepsilon _c}}}} \right)}^k}} \right]=B{E_0}{\varepsilon _i}\exp \left[ { - \frac{1}{k}{{\left( {\frac{{{\varepsilon _i}}}{{{\varepsilon _c}}}} \right)}^k}} \right]$$

(21)

According to Eq. (1) and Fig. 4, the relationship between strain ε_t and ε_c at the inflection point t is

$${\varepsilon _t}={\left( {1+k} \right)^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-0pt} k}}}{\varepsilon _c}$$

(22)

From Eqs. (21) and (22), we can get

$${\sigma _i}\left( {{\varepsilon _i}} \right)=BE{\varepsilon _i}\exp \left[ { - \frac{{1+k}}{k}{{\left( {\frac{{{\varepsilon _i}}}{{{\varepsilon _t}}}} \right)}^k}} \right]$$

(23)

The derivatives of ${\sigma _i}\left( {{\varepsilon _i}} \right)$ at ${\varepsilon _t}$ are as follows:

$$\left. \begin{gathered} {\sigma _i}\left( {{\varepsilon _t}} \right)=BE{\varepsilon _t}\exp \left[ { - \frac{{1+k}}{k}} \right] \hfill \\ {\sigma _i}^{\prime }\left( {{\varepsilon _t}} \right)= - BEk\exp \left[ { - \frac{{1+k}}{k}} \right]= - B\lambda \hfill \\ {\sigma _i}^{{\prime \prime }}\left( {{\varepsilon _t}} \right)=0 \hfill \\ {\sigma _i}^{{\prime \prime \prime }}\left( {{\varepsilon _t}} \right)=BE\frac{{k{{\left( {1+k} \right)}^2}}}{{{\varepsilon _t}^{2}}}\exp \left[ { - \frac{{1+k}}{k}} \right] \hfill \\ \end{gathered} \right\}$$

(24)

The formula 2 in Eq. (24) can be expressed as follows:

$$\lambda =Ek\exp \left[ {{{ - \left( {1+k} \right)} \mathord{\left/ {\vphantom {{ - \left( {1+k} \right)} k}} \right. \kern-0pt} k}} \right]$$

(25)

where Bλ is the absolute value of the slope at the inflection point t of the falling section of the curve, which is called the dimension reduction. From Eq. (24), the Taylor expansion of the expression ${\sigma _i}\left( {{\varepsilon _i}} \right)$ of the stress intensity reduction section in Eq. (1) at ${\varepsilon _t}$ can be obtained:

$${\sigma _i}\left( {{\varepsilon _i}} \right)=BE{e^{ - \frac{{1+k}}{k}}}\left[ {{\varepsilon _t} - k\left( {{\varepsilon _i} - {\varepsilon _t}} \right)+\frac{{k{{\left( {1+k} \right)}^2}}}{{6\varepsilon _{t}^{2}}}{{\left( {{\varepsilon _i} - {\varepsilon _t}} \right)}^3}+ \cdots } \right]$$

(26)

Equation (27) is obtained by substituting Eqs. (26) and (24) into Eq. (20):

$$\begin{aligned}& \frac{{{{\left( {1+k} \right)}^2}{\sigma _i}\left( {{\varepsilon _t}} \right)}}{{2F}}\left[ { - \frac{{2\left( {1 - K} \right)}}{{{{\left( {1+k} \right)}^2}}}+\frac{{2k\left( {1 - K} \right)}}{{{{\left( {1+k} \right)}^2}}}\left( {\frac{{{\varepsilon _i} - {\varepsilon _t}}}{{{\varepsilon _t}}}} \right)+{{\left( {\frac{{{\varepsilon _i} - {\varepsilon _t}}}{{{\varepsilon _t}}}} \right)}^2}} \right] \hfill \\ & \quad +O{\left( {{\varepsilon _i} - {\varepsilon _t}} \right)^3} - W\frac{1}{{b - a}}=0 \hfill \\ \end{aligned}$$

(27)

where F is the ratio of the stiffness f of the elastic region of the coal wall to the modulus Bλ of the statistical part of the plastic region. And F can be expressed as follows:

$$F=\frac{f}{{{{\left| {\frac{{\delta {\sigma _i}}}{{\delta S_{i}^{m}}}} \right|}_{{\varepsilon _i}={\varepsilon _t}}}}}=\frac{f}{{{{\left| {\frac{{\delta \sigma }}{{\delta {\varepsilon _i}}}} \right|}_{{\varepsilon _i}={\varepsilon _t}}}}}=\frac{f}{{B\lambda }}$$

(28)

The selection of F value is discussed. It can be seen from Eq. (27) that $\left( {{\varepsilon _i} - {\varepsilon _t}} \right)$² is the highest term in which the coefficient is not 0 when the structural parameters change. According to the certainty principle, Eq. (27) is the equilibrium equation of folding catastrophe theory. The stability of the coal shell can be discussed by omitting more than three times of Eq. (16). The quasi-static deformation equilibrium equation of coal shell structure can be analyzed by folding catastrophe theory:

$$\begin{aligned}& {\left[ {\frac{{{\varepsilon _i} - {\varepsilon _t}}}{{{\varepsilon _t}}}+\frac{k}{{{{\left( {1{\text{+}}k} \right)}^2}}}\left( {1 - F} \right)} \right]^2} - \frac{{{k^2}}}{{{{\left( {1+k} \right)}^4}}}{\left( {1 - F} \right)^2} - \frac{{2\left( {1 - F} \right)}}{{{{\left( {1+k} \right)}^2}}}\hfill \\ &\quad - \frac{{2FW}}{{\left( {b - a} \right){{\left( {1+k} \right)}^2}{\sigma _i}\left( {{\varepsilon _i}} \right)}}=0 \hfill \\ \end{aligned}$$

(29)

By substitution of variables, it can be obtained that:

$$\left. \begin{gathered} x=\frac{{{\varepsilon _i} - {\varepsilon _t}}}{{{\varepsilon _t}}}+\frac{k}{{{{\left( {1{\text{+}}k} \right)}^2}}}\left( {1 - F} \right) \hfill \\ a= - \frac{{{k^2}}}{{{{\left( {1+k} \right)}^4}}}{\left( {1 - F} \right)^2} - \frac{{2\left( {1 - F} \right)}}{{{{\left( {1+k} \right)}^2}}} - \frac{{2FW}}{{\left( {b - a} \right){{\left( {1+k} \right)}^2}{\sigma _i}\left( {{\varepsilon _i}} \right)}} \hfill \\ \end{gathered} \right\}$$

(30)

Equation (29) can be written as

$${x^2}+a=0$$

(31)

where x is the state variable and a is the control variable. When a ≤ 0, the curve of Eq. (31) is a parabola, and a = 0 (or F-1 axis) divides the parabola into upper and lower leaves, as shown in Fig. 6. Based on Eq. (27) to Eq. (31), the control parameter F, defined as the stiffness ratio of the elastic zone to the plastic modulus, directly governs the morphology of the equilibrium surface. A decrease in the F-1 value reduces the system’s stability, making the equilibrium point more susceptible to a catastrophic jump instability. The homological index k influences the strain energy allocation ratio; an increase in k elevates the rate of energy accumulation and consequently lowers the critical threshold for instability. The modulus Bλ reflects the softening characteristics of the plastic zone. A smaller value of Bλ, indicating more pronounced softening, renders the system more prone to crossing the F-1 axis and undergoing collapse-type failure under disturbances. In the model, the state variable x directly corresponds to the deformation displacement (δu) of the coal shell, while the control variable a is associated with the external energy input (W). When the accumulated energy of the system positions the equilibrium point on the lower leaf of the equilibrium surface, the coal wall remains in a stable state. However, once mining-induced disturbances drive the control parameter a beyond the critical point (a = 0), the equilibrium point undergoes a catastrophic jump, corresponding to the instantaneous loss of the coal shell’s load-bearing capacity in reality. The variables x₁and x₂ represent the critical state points on the equilibrium surface before and after the instability of the coal shell, respectively. x₁ corresponds to the equilibrium point on the lower leaf, signifying the critical state of energy accumulation associated with the deformation displacement. In contrast, x₂ corresponds to the equilibrium point on the upper leaf, representing the transient displacement state during energy release following the instability.

Fig. 6

Equilibrium surface of folding catastrophe model.

In Eqs. (18) and (27), W is a variable. When the rib spalling occurs, external input energy is required, and W > 0. According to the functional relationship between Eq. (29) and the x-axis, a-axis and F-1 axis, two solutions can be obtained.

$$\begin{aligned} {x_1}^{\prime }&=\frac{{{\varepsilon _i}\left( \xi \right) - {\varepsilon _t}}}{{{\varepsilon _t}}}+\frac{k}{{{{\left( {1+k} \right)}^2}}}\left( {1 - F} \right) \hfill \\ &= \frac{{ - 1}}{{1+k}}\sqrt {{{\left[ {\frac{k}{{\left( {1{\text{+}}k} \right)}}\left( {1 - F} \right)} \right]}^2}+2\left( {1 - F} \right)+\frac{{2FW}}{{{\sigma _i}\left( {{\varepsilon _t}} \right)}}} \hfill \\ \end{aligned}$$

(32)

$$\begin{aligned} {x_2}^{\prime }&=\frac{{{\varepsilon _i}\left( \xi \right) - {\varepsilon _t}}}{{{\varepsilon _t}}}+\frac{k}{{{{\left( {1+k} \right)}^2}}}\left( {1 - F} \right) \hfill \\& = \frac{1}{{1+k}}\sqrt {{{\left[ {\frac{k}{{\left( {1{\text{+}}k} \right)}}\left( {1 - F} \right)} \right]}^2}+2\left( {1 - F} \right)+\frac{{2FW}}{{{\sigma _i}\left( {{\varepsilon _t}} \right)}}} \hfill \\ \end{aligned}$$

(33)

Equations (32) and (33) are the two lobes of the deformation and displacement of the coal shell before rib spalling. As shown in Fig. 6, the stress and strain intensity on the upper and lower lobes correspond to a section above and below the inflection point t in the curve in Fig. 4 respectively. The instability of the coal wall structure makes the equilibrium point finally pass through the boundary of the lobe 2. According to the folding catastrophe theory, there are two ways to reach the equilibrium point from the position x₁ on the lower lobe to x₂ on the upper lobe: (1) when the stability of coal wall structure is slightly disturbed by external mining, the equilibrium point transitions from the lower lobe through the F-1 axis to the upper lobe. It shows that the strain energy in the elastic area inside the coal wall slowly transfers to the plastic area, and the coal wall slope presents progressive failure at this time; (2) when the stability of coal wall structure is strongly affected by large external mining disturbance, the equilibrium point will not pass through the F-1 axis from the lower lobe, but in the form of jumping to the upper lobe. It indicates that more strain energy accumulated in the elastic area inside the coal wall is suddenly released to the plastic area. At this time, the coal wall presents collapse failure. The state variable x corresponds to the deformation δu of the coal shell, while the control variable a is related to the external energy input W. The standard form of the folding catastrophe is given by the potential function V(x)=x³ + ax, whose equilibrium surface is defined by dV/dx = 0, i.e., 3 × ²+a = 0. Equation (20) in this paper exhibits a parabolic form when a ≤ 0, which directly corresponds to the equilibrium curve of this standard form. At a = 0, the equilibrium surface splits into left and right branches. Specifically, the x-axis represents the state variable, the a-axis represents the control variable, and the F-1 axis likely corresponds to a parameter adjustment in the standard form, serving to delineate the branches of equilibrium points. Equations (21) and (22) describe the stress-strain relationships on the upper and lower branches, respectively, characterizing the progressive catastrophic mode of coal wall failure.

Results and discussion

Discussion on rib spalling by folding catastrophe model

It can be seen from Eqs. (5) to (10)

$$\begin{aligned} \delta {Q_p}&=S\left( \zeta \right){\sigma _i}\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right)\left( {b - a} \right) \hfill \\ &={\sigma _i}\left( \zeta \right)\left[ {{S^n}\left( \zeta \right)+{S^m}\left( \zeta \right)} \right]\delta {\varepsilon _i}\left( \zeta \right)\left( {b - a} \right) \hfill \\ &= {\sigma _i}\left( \zeta \right)\left[ {\delta S_{i}^{n}\left( \zeta \right)+\delta S_{i}^{m}\left( \zeta \right)} \right]\left( {b - a} \right) \hfill \\ \end{aligned}$$

(34)

Substituting $\delta \varepsilon _{i}^{n}=\delta \left( {{\sigma _i}/BE} \right)$ of Eq. (13) into Eq. (34), and then Eqs. (15) and (19) can be obtained. The state variable x represents the deformation displacement (δu) of the coal shell, while the control variable ais associated with the work done by external forces W.

$$\begin{aligned} W&=\frac{{\delta {Q_p}}}{{\delta S_{i}^{m}\left( \zeta \right)}}={\sigma _i}\left( \zeta \right)\left( {b - a} \right)\left[ {\frac{{\delta S_{i}^{n}\left( \zeta \right)}}{\delta }+1} \right] \hfill \\&= {\sigma _i}\left( \zeta \right)\left( {b - a} \right)\left[ {\frac{{{S^n}}}{{BE}}\frac{{\delta {\sigma _i}\left( \zeta \right)}}{{\kappa {S^m}\delta {\varepsilon _i}\left( \zeta \right)}}+1} \right] \hfill \\ &= \frac{{{\sigma _i}\left( \zeta \right)\left( {b - a} \right)}}{k}\left[ {\frac{1}{{\kappa {S^m}}}\frac{{\delta {\sigma _i}\left( \zeta \right)}}{{\delta {\varepsilon _i}\left( \zeta \right)}}+k} \right] \hfill \\ \end{aligned}$$

(35)

For a fixed value of F, when ε_i increases in Eq. (35), the δσ_i/δε_i and W decrease. According to Eqs. (30) and (32), a and x₁ gradually move towards the origin O, i.e., the equilibrium position (a, x₁) moves to the upper lobe. When $k< - \sigma ^{\prime}\left( {{\varepsilon _t}} \right)/\chi {S^m}$, then Eq. (28) can be expressed:

$$K=k\chi {S^m}/{\left| {\frac{{\delta \sigma }}{{\delta {\varepsilon _i}}}} \right|_{{\varepsilon _i}={\varepsilon _t}}}=\frac{{k\chi {S^m}}}{{B\lambda }}=\frac{E}{\lambda }\frac{{\chi {S^m}}}{{{S^n}}}<1$$

(36)

It can be proved by the integral mean value theorem that there is a point x₁ (< 0) on the upper lobe of Fig. 6 or a point ε_i1 where the strain intensity is less than the integral mean value $\varepsilon _{i}^{ * }\left( \zeta \right)$

$$- {\sigma ^{\prime}_i}\left( {{\varepsilon _{i1}}} \right)< - \sigma ^{\prime}\left[ {\varepsilon _{i}^{ * }\left( \zeta \right)} \right]< - {\sigma ^{\prime}_i}\left( {{\varepsilon _t}} \right)$$

(37)

Then at the point of ε_i1, there is

$$\frac{1}{{\chi {S^m}}}{\left. {\frac{{\delta {\sigma _i}}}{{\delta {\varepsilon _i}}}} \right|_{{\varepsilon _i}={\varepsilon _{i1}}}}+k=0$$

(38)

That is, the Cook stiffness criterion of coal brittle failure at point x₁ is satisfied³². In Eqs. (35),

$$\frac{{\delta {Q_p}}}{{\delta S_{i}^{m}\left( \zeta \right)}}=0{\text{ }}~or~ W=0$$

(39)

The Cook stiffness criterion essentially represents a mathematical expression of system stiffness degradation, while catastrophe theory reveals the mechanism of instability through the transition of equilibrium points between the upper and lower leaves of the equilibrium surface. When the Cook criterion is satisfied, the system’s equilibrium point jumps from the lower leaf to the upper leaf, which corresponds to the abrupt increase of S_m(ζ) in Eq. (42). This sudden change in stiffness triggers a rapid release of energy, resulting in a catastrophic collapse.That is, Eq. (18) is satisfied, and the coal wall is in the critical state of coal wall instability and failure. From formula 2 in Eq. (2), it can be obtained that

$${u_d}\left( \xi \right)=\xi {\varepsilon _i}\left( \xi \right)/2$$

(40)

When the displacement increment $\delta u\left( a \right)>0{\text{ }}, \delta u\left( b \right)>0{\text{ }}$ of the front and rear surfaces of the coal wall, it can be obtained that $\delta {u_d}\left( \zeta \right)>0$ at r = ζ. When the equilibrium position of the coal wall moves in the upper lobe,$\delta {S_i}\left( \zeta \right)=S\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right)=S\left( \zeta \right)2\delta {u_d}\left( \zeta \right)/\zeta$is of the same order as the area increment $\delta A_{i}^{m}\left( \zeta \right)$ in the concentrated area of coal wall plastic deformation.

Substituting $\delta {S_i}\left( \zeta \right)=S\left( \zeta \right)\delta {\varepsilon _i}\left( \zeta \right)=S\left( \zeta \right)2\delta {u_d}\left( \zeta \right)/\zeta$ into Eq. (34), Eq. (41) can be obtained by the critical condition Eq. (39)

$$J=\frac{{\delta {Q_p}}}{{\delta S_{i}^{m}\left( \zeta \right)}}=\frac{{2\left( {b - a} \right){\sigma _i}\left( \zeta \right)S\left( \zeta \right)}}{\zeta }\frac{{\delta {u_d}\left( \zeta \right)}}{{\delta S_{i}^{m}\left( \zeta \right)}}=0$$

(41)

Because $2\left( {b - a} \right){\sigma _i}\left( \xi \right)S\left( \xi \right)/\xi$ is a finite value, Eq. (42) can be obtained from Eq. (41) as follows:

$$\frac{{\delta {u_d}\left( \zeta \right)}}{{\delta S_{i}^{m}\left( \zeta \right)}}=0{\text{ }}or~~~~\frac{{\delta S_{i}^{m}\left( \zeta \right)}}{{\delta {u_d}\left( \zeta \right)}} \to \infty$$

(42)

As shown in Eq. (39), δu_d (ζ) > 0 is proved. The second formula of Eq. (42) shows that there will be a sudden finite increase at ε_i1in S^m (ζ) of the plastic deformation concentration area. Equation (39) to (42) describe the catastrophic transition process of the plastic zone area S_m(ζ), characterizing the initiation mechanism of coal wall failure. The area of S^m (ξ) suddenly increases, indicating that the sudden failure occurs in the coal wall with an area of S′(r) = − 2πr² (1- cosθ). As can be seen from Fig. 6, as long as the equilibrium point is not on the F − 1 axis and α is on the side of the opening direction, for any value of α, the equilibrium point x has two state values. Therefore, the equilibrium position will jump from the x₁ point on the lower lobe to the x₂ point on the upper lobe, and the relationship at s is the same as that in Eq. (38).

$$\frac{1}{{\chi {S^m}}}{\left. {\frac{{\delta {\sigma _i}}}{{\delta {\varepsilon ^{_{i}}}}}} \right|_{{\varepsilon _i}={\varepsilon _{{\text{i2}}}}}}+k=0$$

(43)

Therefore Eq. (43) is true. Compared with Eq. (35), W = 0 at points ε₁₂ or x₂ is also true.

Let W = 0 in Eqs. (32) and (33), and then Eqs. (32) and (33) are subtracted. As expressed in Eq. (22), ε_c represents the jumping amplitude of strain strength at ζ in the middle surface of coal wall before and after failure.

$$\Delta {\varepsilon _i}\left( \zeta \right)={\varepsilon _{i2}}\left( \zeta \right) - {\varepsilon _{i1}}\left( \zeta \right)=2{\left( {1+m} \right)^{{\raise0.7ex\hbox{${m - 1}$} \!\mathord{\left/ {\vphantom {{m - 1} m}}\right.\kern-0pt}\!\lower0.7ex\hbox{$m$}}}}{\varepsilon _c}\sqrt {{{\left[ {\frac{m}{{1{\text{+}}m}}\left( {1 - K} \right)} \right]}^2}+2\left( {1 - K} \right)}$$

(44)

From Eq. (44), the larger m, the smaller F, the larger the jump amplitude Δε_i(ζ) of the mid-plane ξ strain strength.

After integrating x of Eq. (31), Eq. (45) can be obtained.

$${\Lambda _{_{{ab}}}}=\frac{{{x^3}}}{3}+ax$$

(45)

The bifurcated leaf structure of the equilibrium surface corresponds to different stability states of the coal wall: the upper leaf represents a critically unstable state, and the lower leaf signifies a stable state. The total potential energy of the system at points x₁ and x₂ in Fig. 6 is

$${\Lambda _{_{{ab1}}}}=\frac{{{x_1}^{3}}}{3}+{a_1}{x_1},{\text{ }}{\Lambda _{_{{ab2}}}}=\frac{{{x_2}^{3}}}{3}+{a_2}{x_2}$$

(46)

where a₁ = a₂. Let a = a₁ in Eq. (30), then there is ${x_1}^{\prime }={x_1},\;{x_2}^{\prime }={x_2}$ in Eqs. (32) and (33). Since W = 0 in a₁, x₁ and x₂, the release amount of dimensionless deviatoric stress and strain released by the system when the equilibrium position jumps from point x₁ to point x₂ can be obtained.

$$\Delta {\Lambda _{_{{ab}}}}={\Lambda _{ab1}} - {\Lambda _{ab2}}=\frac{{ - 4}}{{3{{\left( {1+m} \right)}^3}}}{\left[ {\frac{{{m^2}}}{{{{\left( {1+m} \right)}^2}}}{{\left( {1 - K} \right)}^2}+2\left( {1 - K} \right)} \right]^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}{\text{ }}\left( {<0} \right)$$

(47)

Through Eqs. (1), (22), (24) and (27), σ_c and ε_c represent peak stress strength and peak strain strength respectively.

$$\begin{aligned} \Delta {\Lambda _{_{b}}}&=\frac{{{{\left( {1+k} \right)}^2}}}{{2K}}{\sigma _i}\left( {{\varepsilon _t}} \right)\left( {b - a} \right){S^m}\left( \zeta \right)\chi {\varepsilon _t}\Delta {\Pi _o} \hfill \\ & =- \frac{2}{3}{\left( {1+k} \right)^{{\raise0.7ex\hbox{${2 - k}$} \!\mathord{\left/ {\vphantom {{2 - k} k}}\right.\kern-0pt}\!\lower0.7ex\hbox{$k$}}}}\frac{{\left( {b - a} \right){S^n}\left( \zeta \right)}}{e}\frac{\lambda }{E}{\sigma _c}{\varepsilon _c}{\left[ {\frac{{{k^2}}}{{{{\left( {1+k} \right)}^2}}}{{\left( {1 - K} \right)}^2}+2\left( {1 - K} \right)} \right]^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}}}{\text{ }}\left( {<0} \right) \hfill \\ \end{aligned}$$

(48)

In physics and mechanics, it is defined that when the outside world does work to the system, the total potential energy of the system increases; When the system releases or outputs energy, the total potential energy decreases. Equations (47) and (48) describe the release of energy, so there are ΔΠ_od < 0 and ΔΠ_d < 0.

Comprehensive effect of volumetric strain energy release of coal shell and internal stress in coal mass

It can be seen from the previous analysis that the coal wall rib spalling is a dynamic change process with starting point and ending point. Once the rib spalling occurs, the coal wall loses its bearing capacity, the external force in formula 2 of Eq. (8) is released, and the average strain value ε_m1(ζ) and ε_m2(ζ) corresponding to x₁ and x₂ points can be obtained by using the displacement before and after the coal wall fracture in Eq. (2). Since ε_i2(ζ)> ε_i1(ζ), it can be obtained that

$${\varepsilon _{m2}}\left( \zeta \right)<{\varepsilon _{m1}}\left( \zeta \right)$$

(49)

In other words, the partial strain strength in the instability process of coal wall ε_i increases suddenly and the average strain ε_m decreases suddenly. The sudden decrease of ε_m indicates the sudden release of volumetric strain energy of coal wall. The volume strain energy increment expression in Eq. (8) can be expressed as follows:

$$\delta {P_t}=3S\left( \zeta \right)\left( {b - a} \right){\sigma _m}\left( \zeta \right)\delta {\varepsilon _m}\left( \zeta \right)$$

(50)

In the elastic state, the relation between average stress σ_m, average strain ε_m and internal stress p of coal in Eq. (50) is

$${\sigma _m}=3R{\varepsilon _m}\left( r \right)+\frac{R}{H}p\left( r \right)$$

(51)

where $R=E/\left[ {3\left( {1 - 2\mu } \right)} \right]$, 1/(3H) is the unidirectional elongation of medium element body caused by unit coal internal stress. Based on Mises strength condition, the material exhibits elastic properties under average stress ${\sigma _m}=\left( {{\sigma _1}+{\sigma _2}+{\sigma _3}} \right)/3$ equivalent to hydrostatic pressure effect, and Eq. (51) can still be used for fractured coal shells. By referring to Eq. (4), Eq. (51) is integrated with ε_m(ζ) from ε_m1(ζ) to ε_m2(ζ), the volumetric strain energy released during coal shell instability can be obtained:

$$\begin{aligned} \Delta {\Lambda _v}&=3S\left( \zeta \right)\left( {b - a} \right)\int_{{{\varepsilon _{m1}}}}^{{{\varepsilon _{m2}}}} {{\sigma _m}} \left( \zeta \right)\delta {\varepsilon _m}\left( \zeta \right) \hfill \\ &=3S\left( \zeta \right)\left( {b - a} \right)\int_{{{\varepsilon _{m1}}}}^{{{\varepsilon _{m2}}}} {\left[ {3R{\varepsilon _m}\left( \zeta \right)+\frac{R}{H}p\left( \zeta \right)} \right]} \delta {\varepsilon _m}\left( \zeta \right) \hfill \\ \end{aligned}$$

(52)

In Eq. (52), the integral of the first term in parentheses means the elastic energy release amount of the unit under the action of average stress when the coal shell is unstable and destroyed suddenly unloads, which can be expressed as:

$$\Delta {\Lambda _{v1}}=3S\left( \zeta \right)\left( {b - a} \right)\left\{ {\frac{{3R}}{2}\left[ {\varepsilon _{{m2}}^{2}\left( \zeta \right) - \varepsilon _{{m1}}^{2}\left( \zeta \right)} \right]} \right\}$$

(53)

In Eq. (52), the integral of the second term in parentheses means the energy release amount of unit volume expansion under the action of coal mass when the coal shell is unstable and destroyed. Among them, the relationship between the mean strain ε_m and the pore stiffness T in coal mass, the pore volume expansion rate w in coal mass and the internal stress p in coal mass is

$${\varepsilon _m}=\left( {\frac{H}{{RT}} - \frac{1}{H}} \right)\frac{p}{3} - \frac{H}{{3R}}\omega$$

(54)

It can be seen that:

$$\Delta {\Lambda _{v2}}=3S\left( \zeta \right)\left( {b - a} \right)\left\{ {\int_{{{p_1}}}^{{{p_2}}} {p\left( {\frac{1}{T} - \frac{R}{{{H^2}}}} \right)\frac{{\delta p}}{3} - \int_{{{\omega _1}}}^{{{\omega _2}}} {\frac{p}{3}\delta \omega } } } \right\}$$

(55)

Equation (56) can be obtained by integrating Eq. (54) and applying the mean value theorem of integration.

$$\Delta {\Lambda _{v2}}=S\left( \zeta \right)\left( {b - a} \right)\left\{ {\left[ {\left( {\frac{1}{T} - \frac{R}{{{H^2}}}} \right)\frac{{p_{2}^{2}\left( \zeta \right) - p_{1}^{2}\left( \zeta \right)}}{2} - {p^ * }\left[ {{\omega _2}\left( \zeta \right) - {\omega _1}\left( \zeta \right)} \right]} \right]} \right\}$$

(56)

where ${p^ * } \in \left[ {{p_2}\left( \zeta \right),\;{p_1}\left( \zeta \right)} \right]$.

In Eq. (53), ${\varepsilon _{m2}}\left( \zeta \right)<{\varepsilon _{m1}}\left( \zeta \right)$. In Eq. (56), ${p_2}\left( \zeta \right)<{p_1}\left( \zeta \right)$. When the average stress is small, ω₂(ζ) is greater than ω₁(ζ) when the average stress is large. Thus, $\Delta {\Lambda _{v1}}<0,\;\Delta {\Lambda _{v2}}<0$.

$$\Delta {\Lambda _v}{\text{=}}\Delta {\Lambda _{v1}}{\text{+}}\Delta {\Lambda _{v2}}<0$$

(57)

Equation (57) has the same meaning as Eq. (48). Due to the energy release, $\Delta {\Lambda _v}<0$.

The tensile strength of coal σ_t is much lower than the compressive strength σ_c. Before breaking, the coal shell is compressed in three directions. Once the coal shell is unstable, the total stress generated by the stress in the coal mass on the back of the coal shell is ${\sigma _r}\left( b \right)={\sigma ^{\prime}_r}\left( b \right)+p\left( b \right)$, which is a sudden rib spalling. The energy released by the seepage force such as gas in the coal seam at the rib spalling of the coal wall is recorded as ΔΠ_βp. The elastic energy released by the coal mass at the end of Fig. 3 when the coal shell is unstable and broken is recorded as ΔΠ_c. In this way, the total energy released by the coal rock system at the moment of instability and fracture of a single coal shell is

$$\Delta {\Lambda _l}=\Delta {\Lambda _b}{\text{+}}\Delta {\Lambda _v}{\text{+}}\Delta {\Lambda _{\beta p}}+\Delta {\Lambda _c}$$

(58)

If a total of n coal shells at the rib spalling on the primary coal wall, the total energy released by the coal rock caving system is

$$\sum\limits_{{i=1}}^{n} {\Delta {\Lambda _l}} =\sum\limits_{{i=1}}^{n} {\left( {\Delta {\Lambda _{bl}}{\text{+}}\Delta {\Lambda _{vl}}{\text{+}}\Delta {\Lambda _{\beta pl}}+\Delta {\Lambda _{cl}}} \right)}$$

(59)

Most coal shells complete the superposition effect of continuous rib spalling at a very high frequency in a short time, and there may be an impact effect of coal internal stress such as gas pressure on the coal wall in the stage of coal protruding and aggravated rib spalling. The gas pressure of several megapascals in the coal mass will provide huge kinetic energy for the rib spalling, resulting in a powerful dynamic process of large-area rib spalling in the coal wall.

Numerical simulation analysis of coal wall rib spalling

(1) Determination of model parameters.

Based on the mining-geological conditions of Face 42,105 in the Buertai Coal Mine, a numerical model was established to systematically investigate the influence of parting thickness on coal wall stability in a parting-containing coal seam. The following assumptions and simplifications were applied to the numerical model.

• (1) The parting layer was positioned at the mid-height of the coal seam.

• (2) Rock strata with similar lithology or minimal thickness were merged with their adjacent layers.

• (3) The rock blocks were simulated using the Mohr-Coulomb model, while the joints were simulated using a surface-to-surface Coulomb slip model. The tensile strength of the rocks was determined through Brazilian splitting tests, and the cohesion and internal friction angle were obtained from triaxial compression tests. The tested rock samples and the corresponding test setups are shown in Figs. 7, 8 and 9, respectively. The material property parameters assigned to the various rock layers are summarized in Table 1.

Fig. 7

Test rock sample.

Fig. 8

Brazilian splitting test.

Fig. 9

Triaxial compression test.

Table 1 Table of mechanical parameters of coal rock.

(2) Numerical model establishment

To characterize the failure behavior of a high coal wall in a thick coal seam containing dirt bands, the specific version information of the numerical simulation software used is: Itasca Consulting Group, Inc. (2019) UDEC - Universal Distinct Element Code, Ver. 7.0. Minneapolis: Itasca. The model was constructed based on the following parameters: an average burial depth of 420 m, a total coal seam thickness of 6.7 m, a mining height of 3.7 m, and a caving height of 3.0 m. As the study focuses on coal wall stability in a dirt-banded seam, only the immediate roof strata were explicitly included in the model height; overlying strata were simplified and represented by an equivalent load applied to the top boundary. The model dimensions were 80 m in height and 300 m in length. Mining was simulated via a step-wise excavation method, with each step advancing the working face by 10 m, resulting in a total excavation length of 100 m. To minimize boundary effects, 100-m-wide protective coal pillars were reserved on both the left and right sides of the model.

Displacement boundary conditions were assigned as follows: the left, right, and bottom boundaries were treated as fixed displacement boundaries—horizontal displacement was constrained on the left and right boundaries, and vertical displacement was constrained on the bottom boundary. The top boundary was set as a free surface, with the overburden load applied uniformly. The computational model is illustrated in Fig. 10. To capture the internal spatial failure characteristics of the coal ahead of the working face, a series of horizontal displacement measurement lines were arranged along the advance direction in front of the coal wall. The lines, spaced at 0.3 m intervals, started from the coal wall and extended 3.7 m into the coal mass. A total of six measurement lines were configured, as depicted in Fig. 11.

Fig. 10

Numerical calculation model.

Fig. 11

Displacement monitoring line distribution map.

(3) Analysis of numerical results

To ensure the comparability and reliability of the numerical results, the spatial distribution characteristics, such as the horizontal displacement and vertical stress ahead of the coal wall, were uniformly selected for analysis when the working face had advanced by 50 m. Figure 12 shows the nephogram of the horizontal displacement distribution ahead of the coal wall. Figure 12 is the result diagram calculated by UDEC numerical simulation software. The result diagram is edited and marked by Microsoft Visio 2010 drawing software. It can be observed that the magnitude of horizontal displacement change in the middle section of the coal mass is consistently higher than that in the upper and lower sections, exhibiting a spatial distribution feature characterized by a “pronounced protrusion in the middle and smooth transitions at the top and bottom.” The instability-affected zone ahead of the coal wall (i.e., within the coal mass) is primarily concentrated around the dirt band, indicating that the presence of the dirt band alters the spatial location of the main load-bearing points within the compound structure of the coal seam containing partings. This causes the failure position to shift towards the dirt band, consequently forming a protruding zone centered in the middle section. The rib spalling-affected zone exhibits an approximately spherical spatial distribution, suggesting that the instability of the coal wall develops radially from the central core point towards the surrounding space. Figure 13 presents the variation curves of the displacement measurement lines ahead of the coal wall, which also reveals a prominent “bulge” morphology in this zone on the curves. The center point of this “bulge” (i.e., the point of maximum displacement) corresponds to the core area of the most severe failure, where the displacement magnitude is the highest. Furthermore, the displacement contours surrounding this point decrease in a roughly symmetrical, annular pattern, collectively delineating an approximately hemispherical instability zone centered on the maximum displacement point. This indicates that the failure mechanism follows a spherical pattern centered on the dirt band, with its central location indeed being the core area of the most severe damage. In summary, the numerical simulation results are in good agreement with the theoretical analysis. The horizontal displacement is concentrated in the middle and upper parts of the coal wall, forming a prominent “drum-shaped bulging” influence zone, which is highly consistent with the theoretical assumptions of “plastic strain localization” and “spherically symmetric failure.” The peak displacement occurs in the middle section and decreases toward the roof and floor, visually confirming the mechanism that energy and deformation preferentially accumulate in the central region prior to the instability of the coal shell structure. This provides important visual evidence for the aforementioned catastrophe model.

Fig. 12

Cloud map of horizontal displacement distribution in front of coal wall.

Fig. 13

Horizontal displacement change curve of coal wall.

Discussion

In the folding catastrophe model developed in this paper, the control parameter a is closely related to the system’s energy input W(as outlined in the document, where W > 0 indicates external energy input). Among the measurable parameters, mining depth and gas pressure exert the most significant influence on a: an increase in mining height elevates geostatic stress, which indirectly augments the energy input W, driving the value of a towards the critical point (a = 0); a rise in gas pressure, by reducing the effective stress, facilitates energy accumulation and accelerates the system’s transition towards an unstable state. In contrast, uniaxial compressive strength and elastic modulus primarily modulate a by altering the stiffness characteristics of the coal mass: a high compressive strength enhances system stability, making the value of a less prone to abrupt changes; the elastic modulus influences strain energy storage, and a larger value results in a more gradual variation of a. Overall, mining depth and gas pressure, as external disturbance factors, demonstrate higher sensitivity to the control parameter a and directly determine the transition path of the equilibrium point between the upper and lower leaves of the equilibrium surface.

Engineering guidance

Based on the fold catastrophe model, the bifurcated leaf structure of the equilibrium surface quantifies the instability mechanism of coal wall rib spalling, as the equilibrium point approaches the F-1 axis, it serves as an early warning for progressive spalling, while a jump to the upper leaf indicates an imminent catastrophic collapse. The instability critical point can be predicted by inversely determining the control parameter a through real-time monitoring. In engineering practice, the value of a can be modulated by adjusting support stiffness or implementing pressure-relief measures to move it away from the critical region. Furthermore, the relationship between the jump amplitude and parameters m and F provides a theoretical basis for optimizing support design. This approach enables dynamic quantitative assessment and proactive control of rib spalling risk.

Conclusion

• (1) In order to solve the problem that the mechanism of sudden spalling of coal wall is not clear, a folding catastrophe model is established. The model clearly describes the failure process of sudden spalling of coal wall and provides a quantitative basis for prediction. The folding catastrophe model of coal wall instability fracture is established. According to the elastic state of the medium under the action of average stress, the expressions of energy release such as volume strain energy and gas pressure in coal body are given.

• (2) The numerical simulation test of coal wall spalling is carried out. The results show that the failure process is a spherical symmetrical failure with the middle as the center and gradually expanding to the surrounding area, which is consistent with the theoretical derivation.

• (3) The technical measures to control coal wall spalling are given. Real-time monitoring of gas pressure and mining stress can be used to observe the change of the critical point of instability, improve the stiffness of the bearing to increase the overall stability of the system, and use pressure relief measures to reduce external energy input and other technical measures to reduce the occurrence of coal wall spalling accidents.