A Mechanical Model for the Progressive Failure of Slabbing Roadway-Side Backfill Bodies

Abstract

Slabbing failure of roadway-side backfill bodies critically threatens gob-side entry retaining stability. This study establishes an elastic thin-plate model with edge cracks, employing an innovative load transformation to reduce the three-dimensional in situ stress state to the combined action of roof–floor uniform load and equivalent axial bending moment. Based on fracture mechanics and elastic-plastic theory, the stress intensity factor $K_1$ and crack initiation load q are derived in closed form. Results show that q is positively correlated with plate thickness t and bending moment M and negatively with crack length a in the dominant range. Applying the nonlinear Hoek–Brown criterion, the failure zone width ${\mathrm{r}}_{\mathrm{p}}$ at the crack tip is shown to exhibit an approximately exponential relationship with $K_1$ for unbolted backfill. Introduction of tensioned bolts via a stress concentration factor $\mathsf{\eta }$ transforms the failure zone growth from exponential to asymptotic saturation, quantitatively confirming the crack-arresting effect. A sensitivity analysis identifies plate thickness as the dominant parameter. The model bridges the gap between initial slabbing and progressive V-shaped notch formation.

1. Introduction

As underground engineering projects progressively advance into deep geological formations, the failure mechanisms of deep surrounding rock have become critical scientific and technological challenges [1]. The unique characteristics of deep rock masses—high in situ stress, high geothermal temperature, high seepage pressure, and excavation disturbance effects—result in mechanical responses fundamentally different from those in shallow caverns. Regular slabbing fracture is a common failure phenomenon in hard brittle surrounding rock of deep caverns [2], characterized by surrounding rock being cut by primarily Mode I tensile cracks into multiple sets of plate-like structures approximately parallel to the roadway wall, with slab thickness generally increasing from the free surface toward the deeper rock mass.

Slabbing is defined herein as a brittle failure mode in which the surrounding rock or backfill body is split by approximately parallel Mode I tensile cracks into plate-like structures oriented sub-parallel to the excavation surface, with plate thickness generally increasing with depth from the free face. Progressive failure is defined as a consecutive failure process wherein pre-existing edge cracks in the already-formed thin plates initiate, propagate, and coalesce under sustained loading, ultimately leading to structural disintegration and V-shaped notch formation, rather than instantaneous collapse.

The phenomenon of slabbing failure has attracted extensive research attention. Fairhurst et al. [3] provided early descriptions linking slabbing to tensile crack propagation and coalescence. Cai [4] proposed that densely distributed onion-skin-like cracks represent a common manifestation of slabbing failure. Ortlepp [5,6] suggested that slabbing failure could be conceptualized as macroscopic cracks, with maximum tangential stress oriented parallel to the slabbing plane. Martin et al. [7], through statistical analysis of 178 pillar failure cases in Canadian hard rock mines, indicated that progressive slabbing and spalling is the dominant failure mode when the width-to-height ratio of hard rock pillars is less than 2.5. Lee et al. [8] observed slabbing phenomena during V-shaped failure formation in biaxial compression tests on Lac du Bonnet granite. Stacey et al. [9] proposed a tensile strain criterion for brittle rock fracture to predict slabbing failure depth. Diederichs [10] provided a mechanistic framework distinguishing spalling from conventional shear failure in highly stressed brittle rock. Kaiser and Cai [11] proposed design guidelines for rock support in burst-prone grounds based on rock failure processes including slabbing and spalling. More recently, Luo et al. [12] investigated slabbing failure characteristics of surrounding rock in deep tunnels through true triaxial experiments, and Nie et al. [13] examined the fragmentation mechanism of deep hard rock via coupled laboratory and numerical approaches.

Gob-side entry retaining is a mining technique that maintains the roadway along the goaf boundary behind the working face, overcoming problems of excavation-mining interference and goaf isolation [14,15]. High-water rapid-setting materials have been developed for roadside backfill, offering high early strength and rapid stiffness increase. However, field investigations in the study mine revealed that the high-water backfill body, after approximately 3–5 days of exposure following working face passage, developed visible vertical tensile cracks on its exposed surface, with crack spacing of 0.2–0.5 m, oriented approximately parallel to the roadway wall. Measured slab thicknesses ranged from 30 to 120 mm, with thinner slabs concentrated near the free surface. The maximum tangential stress at the observation depth (approximately 600 m) was estimated at 15–20 MPa. Zhou et al. [16,17] conducted experimental studies on the slabbing mechanism using α-type high-strength gypsum specimens (150 mm × 150 mm × 150 mm, uniaxial compressive strength 32–36 MPa, tensile strength 4.5–5.2 MPa, and elastic modulus 8.5–10.2 GPa) with pre-existing flaws under uniaxial compression, revealing crack propagation, slab buckling, and fragment ejection phenomena, as shown in Figure 1. Figure 1a and Figure 1b respectively show the slab cracking failure phenomenon of high-strength gypsum under different conditions. Zhang et al. [18] presented a case study on gob-side entry retaining with high-water backfill, providing field-monitored stress–strain data. Chen et al. [19] analyzed the interaction between roadside backfill and surrounding rock in deep gob-side entry retaining. Hu et al. [20] characterized hardened high-water materials as exhibiting high brittleness, rapid setting, and strength regeneration.

Theoretical models for slabbing failure have been developed by various researchers. Liu et al. [21] derived critical stress and displacement for spalling range based on Kirchhoff plate theory. Li [22] studied rock slab buckling instability via energy dissipation analysis, obtaining critical buckling load and crack number. Feng et al. [15] established an orthotropic thin-plate model and derived the critical load for slabbing buckling rockburst under biaxial stress. These studies primarily address the initial slabbing condition of surrounding rock or rely on numerical approaches for backfill stability analysis.

However, research on the post-slabbing progressive failure mechanism of already-formed thin plates—particularly in backfill bodies—remains insufficient. Existing models focus predominantly on the conditions leading to initial slabbing fracture formation, while the subsequent progressive failure governed by edge crack propagation and coalescence has received less theoretical attention. Furthermore, the quantitative treatment of bolting effects within a fracture mechanics framework for backfill progressive failure has not been previously addressed.

In contrast to existing studies, Ref. [15] established an orthotropic thin-plate model for the critical load of initial slabbing buckling; our work focuses on the post-slabbing stage, modeling pre-existing edge cracks and their progressive propagation. Ref. [21] derived critical stress and displacement for spalling onset using Kirchhoff plate theory; our model extends beyond the onset by introducing fracture mechanics to quantify the stress intensity factor and crack initiation load governing subsequent propagation. Ref. [22] investigated rock slab buckling instability via energy dissipation; our work addresses the further failure mechanism after slab formation, incorporating bending moment effects and the crack-arresting role of tensioned bolts.

This study presents the following three novel contributions: (i) an original load transformation method converts the complex three-dimensional loading state of the backfill into the combined effect of roof–floor strata load and an equivalent axial bending moment, enabling a tractable thin-plate mechanical model; (ii) analytical expressions for the stress intensity factor $K_1$ at the tip of pre-existing edge cracks are derived, allowing quantitative characterization of progressive crack propagation and coalescence leading to V-shaped notch failure; and (iii) the mechanical contribution of tensioned bolts is incorporated through a stress concentration factor η, analogous to a riveted stiffened plate model, providing a novel quantitative framework to evaluate the crack-arresting effect of bolting on the failure zone width ${\mathrm{r}}_{\mathrm{p}}$.

The geometric symmetry of the thin-plate model, the load symmetry decomposition with respect to the neutral plane, and the symmetry inherent in the Mode I crack-tip stress field are fundamental to the model formulation and align with the scope of this journal.

This paper aims to establish a theoretical framework for the progressive failure mechanism of slabbing roadway-side backfill bodies, derive analytical expressions linking macroscopic loads to mesoscopic crack-driving forces, and quantify the crack-arresting effect of tensioned bolts within a fracture mechanics framework.

2. Mechanical Model for Slabbing Roadway-Side Backfill Body

2.1. Load-Bearing Behavior of the Backfill Body and Coal Mass

In gob-side entry retaining, the final state on both sides of the retained roadway is a stable surrounding rock condition. Before the next working face mining commences, the goaf on the left side of the backfill body essentially stabilizes, while the solid coal on the right side remains nearly in its original rock state. Figure 2 presents a simplified loading model.

As illustrated in Figure 3, the load borne by the backfill body and the solid coal originates from the lateral pressure p and the load of the roof and floor strata. The assumption of identical lateral pressure p on both sides is a deliberate simplification for analytical tractability. In reality, the goaf side exerts active pressure with a lower coefficient, while the solid coal side sustains higher reaction pressure. However, the critical parameter for the backfill plate model is the bending moment M resolved on the backfill body alone, which is governed primarily by the lateral pressure acting directly on the backfill. The lateral pressure on the solid coal does not enter the backfill plate formulation. Therefore, if the actual lateral pressure is lower than the assumed uniform p, the resultant bending moment M decreases and $K_1$ is reduced, making the predicted crack initiation load a conservative lower-bound estimate for backfill strength design. The contribution of M to $K_1$ is linear, allowing straightforward adjustment when site-specific lateral pressure coefficients become available.

2.2. Establishment of the Thin-Plate Mechanical Model

Under loading and unloading conditions, the initiation, propagation, interaction, and coalescence of pre-existing defects such as cracks represent a significant manifestation of failure in brittle materials [23]. For deep underground roadways, the stress redistribution following excavation causes a sharp decrease in radial stress on the free surface, while tangential stress increases. The surrounding rock near the roadway surface is approximately in a uniaxial compression state. Under combined roof–floor load and lateral stress, the surrounding rock is split into thin plates.

While existing studies have largely focused on the conditions leading to the initial formation of slabbing fractures, the subsequent progressive failure of already-formed thin slabs—particularly in backfill bodies—has received less theoretical attention. To address this gap, the present mechanical model is specifically established to describe the post-slabbing behavior where pre-existing edge cracks propagate and coalesce under sustained loading.

The following justifications support the application of elastic thin-plate theory to the studied backfill body:

(i)

Material homogeneity and isotropy: High-water rapid-setting backfill materials are grouted in a fluid state and harden into a relatively uniform paste-like body. The ettringite-dominated matrix exhibits substantially greater macroscopic uniformity than naturally fractured rock masses, with the absence of large-scale bedding planes and tectonic joints. Field core sampling [18] reports elastic modulus variation within ±12% across a single backfill panel, which is acceptable for an analytical model.

(ii)

Porosity justification: High-water backfill materials exhibit a water-to-solid ratio of 2.0–3.0:1, yielding a porosity of 30–50% after hardening. Cracks propagate predominantly along ettringite crystal boundaries at micron scale—two to three orders of magnitude below the slab thickness (t ≈ 30–120 mm). Therefore, pores act as distributed micro-defects that reduce effective elastic modulus rather than as macroscopic structural discontinuities, and homogenization into an equivalent elastic continuum is justified at the plate scale.

(iii)

Thin-plate and small-deflection theory: The slabbing fractures naturally partition the backfill body into plates whose thickness (30–120 mm) is considerably smaller than the height (2–5 m) and length dimensions, satisfying the geometric aspect ratio requirement of Kirchhoff theory (t/H ≤ 0.06). Field observations indicate that the lateral deflection prior to visible crack propagation is typically less than the plate thickness, satisfying the prerequisite of small-deflection theory [15].

(iv)

Creep and aging consideration: Creep strain accumulates over weeks to months, whereas slabbing failure typically initiates within 3–5 days after working face passage. During this short time window, the time-dependent deformation increment is small relative to the immediate elastic deformation induced by mining-induced stress redistribution. The backfill is therefore treated as an elastic solid for the timescale of interest.

The deflection of the rock slab is necessarily not greater than its own thickness, satisfying the small-deflection theory for thin plates. Therefore, selecting a thin-plate mechanical model is appropriate for the crack initiation and early propagation stage. The post-critical large-deformation behavior (buckling, fragment ejection) occurs after crack coalescence and is beyond the scope of this analysis.

The thin plate is taken with its neutral plane as the xy-plane, oriented parallel to the cavern wall. Figure 4 shows the simplified stress diagram, where the thin plate has thickness t, crack length a, height H, and length L, subjected to a bending moment M and a uniformly distributed load σ₁. The lateral pressure and axial force are combined into the resultant bending moment M acting on the thin plate.

The bending moment M is derived from the combined effect of lateral pressure p from the goaf and the axial force along the roadway direction. As the working face advances, the exposed area increases, and the loads progressively increase. Edge cracks are aligned approximately parallel to the tangential stress σ₁. When σ₁ exceeds the critical load, bending of the thin plate creates new free surfaces, intensifying crack propagation. Repetition of this process leads to coalescence between edge cracks, ultimately forming a V-shaped notch.

For analytical convenience, only two collinear cracks at the top and bottom of the thin plate are investigated. Figure 5 shows the mechanical model for edge crack propagation and its decomposed models. The complex combined loading is decomposed into: ① pure bending moment model and ② uniform roof–floor load model. This decomposition is necessary because the stress intensity factor for each loading mode must be solved separately before superposition yields the total $K_1$.

The coordinate system (x, y) defines the neutral plane. Loading components include uniform roof–floor load ${\mathsf{\sigma }}_1$ and resultant bending moment M (originating from combined lateral pressure and axial force). Geometric parameters are as follows: t (thickness), H (height), L (length), and a (crack length). Boundary conditions: simply supported at roof and floor contacts.

Sub-model ①: pure bending moment M; sub-model ②: uniform roof–floor load ${\mathsf{\sigma }}_1$. The total stress intensity factor is obtained by superposition of the two sub-models. This decomposition is necessary because analytical solutions for $K_1$ are available for each loading case separately, enabling tractable derivation of the combined loading solution.

The stress intensity factor under bending moment M is as follows [24,25]:

$$K_M = F{\displaystyle \frac{12\mathsf{\pi }M}{t^3}}\sqrt{\mathsf{\pi }a}$$

(1)

For practical computation, the curve of F can be approximated by the fitted polynomial, as shown in Figure 6. The results as shown in

Table 1. Typical parameter settings and results.

2c/H	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
F	1.09	0.98	0.87	0.81	0.78	0.77	0.78	0.81	0.87	1.00

.

Here, c = (H/2 − a) is half the distance between the tips of the two collinear edge cracks.

Stress Intensity Factor under Uniformly Distributed Load from Roof and Floor Strata.

Figure 5 illustrates the mechanical model of the thin plate under the uniformly distributed load from the roof and floor strata, as shown in diagram ②. Reference [24] lists the calculation formula for the stress intensity factor in a thin plate subjected to four-point concentrated loads. This paper adopts the approach from reference [24], equating the uniformly distributed load to the action of concentrated stresses at two points on both the top and bottom of the thin plate. Combining this with the problem-solving approach using the body force method for stress intensity factors, the calculation formula for the stress intensity factor under this condition is as follows:

$$K_{ql} = 2F_M·K_M$$

(2)

In Equation (2),

$$K_M=F_M{\displaystyle \frac{3{\mathsf{\sigma }}_{1}L}{t{H}^2}}\sqrt{\mathsf{\pi }a}, F_M=1.122-1.121\mathsf{\xi }+3.740{\mathsf{\xi }}^2+3.873{\mathsf{\xi }}^3-19.05{\mathsf{\xi }}^4+22.55{\mathsf{\xi }}^5, \mathsf{\xi }={\displaystyle \frac{\mathrm{a}}{\mathrm{H}}}.$$

By combining the relationship satisfied by the parameter F_M and the value range of its independent variables, curve fitting was performed using Origin software 2024 to intuitively display its variation pattern. The results show that the variation trend of parameter F_M in Figure 7 is transversely S-shaped. That is, as a/H increases, it first decreases, then increases, and finally decreases again. The inflection points of the curve are around 0.3 and 0.8.

In summary, the stress intensity factor at the tip of the edge crack in the mechanical model of the thin plate of the slabbing backfill body under combined loading is the superposition of the stress intensity factors obtained from the two types of loading described above. Therefore, the stress intensity factor driving the propagation of the edge crack in the thin plate is as follows:

$$K_I = K_{Ml} + K_{ql} = (F{\displaystyle \frac{12zM}{t^3}} + {\displaystyle \frac{6q{L}^2}{t{H}^7}}{F}_M^2)\sqrt{\mathsf{\pi }a}$$

(3)

According to a large number of experimental studies [26], the criterion for compression-shear fracture in rock and concrete is as follows:

$$\mathsf{\lambda }\sum K_I +\left|\sum K_{\mathsf{\Pi }}\right| ={ K}_c$$

(4)

where $\mathsf{\lambda }$ is the compression–shear ratio coefficient for crack propagation; K_c is the fracture toughness of the rock. Since the edge crack studied in this paper is a tensile crack, $K_{\mathsf{\Pi }} = 0$, substituting Equation (3) into Equation (4) yields:

$$\mathsf{\lambda }[(F{\displaystyle \frac{12zM}{t^3}} + {\displaystyle \frac{6q{L}^2}{t{H}^2}}{F}_M^2)\sqrt{\mathsf{\pi }a}] = K_c$$

(5)

Solving Equation (5) gives the crack initiation load for the edge crack as follows:

$$q = {\displaystyle \frac{t{H}^2}{6L^2{F}_M^2}}({\displaystyle \frac{K_c}{\mathsf{\lambda }\sqrt{\mathsf{\pi }a}}} - {\displaystyle \frac{12zMF}{t^3}})$$

(6)

The negative sign on the ordinate indicates direction. (a) Crack initiation load versus crack length for different plate thicknesses. In the dominant range, q decreases with increasing crack length. Within the narrow interval a = 0.0101–0.0150 m, a minor non-monotonic variation of approximately 0.3 MPa is observed (see data below), attributable to the non-monotonic character of F_M in this range, which reflects competing geometric effects of crack lengthening and crack–tip interaction as the two collinear edge cracks approach each other. (b) Crack initiation load versus bending moment. The positive correlation between M and q is attributed to the crack-closure effect of the moment-induced compressive stress on the loaded side, which partially suppresses Mode I opening, as shown in

Table 2. Crack initiation load and result characteristic values.

a (m)	0.0101	0.0105	0.0110	0.0115	0.0120	0.0125	0.0130	0.0135	0.0140	0.0145	0.0150
q (MPa)	−3.82	−3.54	−3.49	−3.51	−3.53	−3.53	−3.52	−3.51	−3.50	−3.49	−3.48

.

Computed data for the non-monotonic interval in Figure 8a. The data confirm that the non-monotonic variation, although small (~0.3 MPa), is systematic and reproducible, arising from the non-monotonic character of F_M (Figure 7), which itself reflects competing effects of crack lengthening and crack–tip interaction. This variation is physically meaningful, but its engineering significance is limited relative to the dominant trends governed by plate thickness and bending moment.

The analysis indicates that the crack initiation load is positively correlated with plate thickness t (due to the cubic dependence of bending stiffness on thickness) and bending moment M (attributed to the crack-closure effect of moment-induced compressive stress) and negatively correlated with crack length a in the dominant range.

3. Analysis of the Thin-Plate Mechanical Model

3.1. Stress Analysis

Since the slabbing backfill body studied in this paper is assumed to be an isotropic material, the expressions for the stress components in the plane problem for isotropic materials, based on Muskhelishvili’s complex variable method, are given as follows:

$${\mathsf{\sigma }}_y + {\mathsf{\sigma }}_x = 2[\mathsf{\phi }\prime (z) + \stackrel{-}{\mathsf{\phi }\prime }(\stackrel{-}{z})] = 4R_{\mathsf{\epsilon }}{\mathsf{\phi }}^{\prime }(z)$$

(7a)

$${\mathsf{\sigma }}_y-{\mathsf{\sigma }}_x+2I{\mathsf{\sigma }}_{xy}=2[\stackrel{-}{z}\mathsf{\phi }″(z)+{\mathsf{\psi }}^{\prime }(\stackrel{-}{z})]$$

(7b)

The above two equations satisfy all fundamental equations of the plane problem. According to the solution method for the stress intensity factor in fracture mechanics [26], the boundary conditions specified for a Mode I crack are:

(1)

On the fracture surface, i.e., for $y = 0$, $\left|x\right| < a$, ${\mathsf{\sigma }}_y ={ \mathsf{\tau }}_{xy} = 0$;

(2)

As $y = 0$, $\left|x\right| > a$, ${\mathsf{\sigma }}_y > \mathsf{\sigma }$, and the closer to $\mathsf{\sigma }$, the larger ${\mathsf{\sigma }}_y$ becomes;

(3)

At infinity, i.e., for $\left|z\right|\to \infty$, ${\mathsf{\sigma }}_x = {\mathsf{\sigma }}_y = \mathsf{\sigma }$, ${\mathsf{\tau }}_{xy} = 0$.

Therefore, the complex function for a Mode I crack is selected as ${\mathrm{Z}}_I(z) = {\displaystyle \frac{\mathsf{\sigma }z}{\sqrt{z^{2 }-{ a}^2}}}$. Combining this with the assumption in theoretical derivations using fracture mechanics that the coordinate origin is set at the center of the crack, simplify $I(x,y)\to \mathsf{\xi }(r,\mathsf{\theta })$ to $y = r\sin \mathsf{\theta }$, ultimately yielding the expressions for the stress components of Mode I edge crack as follows:

$${\mathsf{\sigma }}_x(r,\mathsf{\theta }) = {\displaystyle \frac{K_I}{\sqrt{2\mathsf{\pi }r}}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}(1 - \sin {\displaystyle \frac{\mathsf{\theta }}{2}}\sin {\displaystyle \frac{3\mathsf{\theta }}{2}})$$

(8a)

$${\mathsf{\sigma }}_y(r,\mathsf{\theta })={\displaystyle \frac{K_I}{\sqrt{2\mathsf{\pi }r}}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}\left(1+\sin {\displaystyle \frac{\mathsf{\theta }}{2}}\sin {\displaystyle \frac{3\mathsf{\theta }}{2}}\right)$$

(8b)

$${\mathsf{\tau }}_{\mathsf{\gamma }}(r,\mathsf{\theta })={\displaystyle \frac{K_I}{\sqrt{2\mathsf{\pi }r}}}\sin {\displaystyle \frac{\mathsf{\theta }}{2}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}\cos {\displaystyle \frac{3\mathsf{\theta }}{2}}$$

(8c)

3.2. Displacement Analysis of the Thin-Plate Mechanical Model

The solution for displacement is relatively complex. Due to the small deflection of the thin plate, which does not cause stretching or compression of the neutral plane, and the need to neglect the effect of shear stress, the displacement component expressions for isotropic materials in plane problems, based on Muskhelishvili’s complex variable method, are given as follows [27,28]:

$$2\mathsf{\mu }(u_x + \dot{u_y}) =\dot{ u}\mathsf{\phi }(z) - z\stackrel{-}{{\mathsf{\phi }}^{\prime }}(z) - \stackrel{-}{\mathsf{\psi }}(z)$$

(9)

Equation (9) satisfies all fundamental equations of the plane problem. Based on the boundary conditions of the crack surface, the following new complex function is introduced:

$$\mathsf{\Omega }(z) = z\mathsf{\phi }(z) + \mathsf{\psi }(z)$$

(10)

Substituting Equation (10) into Equation (9) and rearranging yields:

$$2\mathsf{\mu }(u_x + I{u}_y) = \dot{u}\mathsf{\phi }(z) - (z -\stackrel{-}{ z})\overline{\mathsf{\phi }}(z) -\overline{ \mathsf{\Omega }}(z)$$

(11)

In Equation (11), for the plane stress state, $\dot{u} = {\displaystyle \frac{3 - \mathsf{\nu }}{1 + \mathsf{\nu }}}$; for the plane strain state, $u = 3 - 4\mathsf{\nu }$. The specific value of u can be determined according to the specific mechanical problem encountered during the actual calculation process.

In summary, the displacement component expressions near the tip of the edge crack can ultimately be obtained as follows:

$${\mathrm{u}}_x = {\displaystyle \frac{{\mathrm{K}}_{\mathrm{f}}}{2\mathsf{\mu }}}\sqrt{{\displaystyle \frac{\mathrm{r}}{2\mathsf{\pi }}}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}(\stackrel{-}{\mathrm{u}} - 1 + 2{\sin }^2{\displaystyle \frac{\mathsf{\theta }}{2}})$$

(12a)

$$u_y={\displaystyle \frac{K_I}{2\mathsf{\mu }}}\sqrt{{\displaystyle \frac{r}{2\mathsf{\pi }}}}sin{\displaystyle \frac{\mathsf{\theta }}{2}}(\dot{u}+1-2{cos}^2{\displaystyle \frac{\mathsf{\theta }}{2}})$$

(12b)

3.3. Calculation and Analysis of the Failure Zone Width

For jointed rock masses, the nonlinear Hoek–Brown failure criterion is more suitable than the linear Mohr–Coulomb criterion as it accounts for tensile stress regions, disturbance factors, and dynamic changes in mechanical parameters [29]. The Hoek–Brown criterion is as follows [30]:

$${\mathsf{\sigma }}_1 ={ \mathsf{\sigma }}_3 + \sqrt{\mathrm{m}{\mathrm{R}}_{\mathrm{c}}{\mathsf{\sigma }}_3 + \mathrm{s}{\mathrm{R}}_{\mathrm{c}}^2}$$

(13)

where ${\mathrm{R}}_{\mathrm{c}}$ is the uniaxial compressive strength of intact rock; m reflects rock hardness (0.0000001 for heavily disturbed rock masses to 25 for intact hard rock); and s reflects the degree of rock mass fracturing (0 for fractured rock masses to 1 for intact rock).

Parameter determination for high-water backfill: The values of ${\mathrm{R}}_{\mathrm{c}}$, m, s, and μ are determined from laboratory testing and published characterization data. ${\mathrm{R}}_{\mathrm{c}}$ (uniaxial compressive strength) was determined from uniaxial compression tests on high-water backfill specimens (water-to-solid ratio 2.5:1, curing age 7 days) reported by Hu and Cui [20], with tested range of 3.2–5.8 MPa. m and s are estimated following Hoek–Brown classification guidelines [30]: m ≈ 4–8 for carbonate-cemented granular materials (the ettringite-dominated matrix is calcium-sulfoaluminate-based), and s is taken in the range 0.001–0.01 to reflect the porous, micro-cracked state of hardened paste, consistent with back-analysis of backfill pillar strength reported by Zhang et al. [18]. μ (Poisson’s ratio), determined from lateral strain measurements during uniaxial compression, ranges from 0.20 to 0.30 [18,20].

The principal stresses in the near-tip region are obtained from classical elasticity [31]:

$${\mathsf{\sigma }}_{1,2} = {\displaystyle \frac{{\mathsf{\sigma }}_x + {\mathsf{\sigma }}_y}{2}} + \sqrt{({\displaystyle \frac{{\mathsf{\sigma }}_x - {\mathsf{\sigma }}_y}{2}}{)}^2 +{ \mathsf{\tau }}_{xy}^2}$$

(14a)

$${\mathsf{\sigma }}_3=\mathsf{\mu }({\mathsf{\sigma }}_{x }+{\mathsf{\sigma }}_y)$$

(14b)

Substituting Equation (8) into Equation (14), after rearrangement and simplification, the principal stresses in the near-tip region of the edge crack on the thin plate of the slabbing backfill body are obtained as follows:

$${\mathsf{\sigma }}_{1,2} = {\displaystyle \frac{K_J}{\sqrt{2\mathsf{\pi }r}}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}(1 \pm \sin {\displaystyle \frac{\mathsf{\theta }}{2}})$$

(15a)

$${\mathsf{\sigma }}_3={\displaystyle \frac{2\mathsf{\mu }{K}_I}{\sqrt{2\mathsf{\pi }r}}}\cos {\displaystyle \frac{\mathsf{\theta }}{2}}$$

(15b)

Substituting Equation (15) into the Hoek–Brown criterion (Equation (13)) yields the boundary equation for the failure zone [23]:

$$r = {\displaystyle \frac{4A^2}{2\mathsf{\pi }(B \pm \sqrt{B^2 + 4AC}{)}^2}}$$

(16)

In Equation (16), $A = (K_i\cos {\displaystyle \frac{\mathsf{\theta }}{2}}{)}^2[(1 + \sin {\displaystyle \frac{\mathsf{\theta }}{2}}) - 2\mathsf{\mu }{]}^2$, $B = 2m\mathsf{\mu }{R}_c{K}_I\cos {\displaystyle \frac{\mathsf{\theta }}{2}}$, $C = s·R_c^2$.

From the boundary Equation (16), when θ = 0, the relationship satisfied by the width of the failure zone in the vertical direction of the edge crack on the thin plate of the slabbing backfill body is obtained:

$$r_p = {\displaystyle \frac{A{\left[K_I(1 - 2\mathsf{\mu })\right]}^4}{2\mathsf{\pi }\left[2m\mathsf{\mu }{R}_c{K}_I \pm \sqrt{(2m\mathsf{\mu }{R}_c{K}_I{)}^2 + 4s{R}_c^2{K}_I^2(1 - 2\mathsf{\mu }{)}^2}\right]}}$$

(17)

Equation (17) indicates a direct and necessary relationship between the size of the failure zone at the tip of the edge crack and the stress intensity factor of the edge crack. To visually demonstrate the specific relationship between the two, this paper assumes different values for the parameters m, μ, $R_c$, and s to generate curves illustrating the relationship between the failure zone width at the tip of the thin-plate edge crack and the stress intensity factor. The expression for the failure zone width at the tip of the thin-plate edge crack (17) contains a ‘±’ sign. Observation reveals that when the ‘±’ in Equation (17) is determined to be ‘+’, the variation in the failure zone width $r_p$ at the tip of the edge crack is more significantly influenced. To more intuitively display the relationship between the failure zone width at the tip of the thin-plate edge crack and the stress intensity factor, this paper ultimately sets the sign in Equation (17) to ‘+’ and uses Origin software to plot the curves for these two variables, as shown below.

The parameter sets for the four curves represent progressive degradation of the backfill body, as shown in

Table 3. Setting values for rock characteristic parameters.

Curve	m	s	R~c~ (MPa)	μ	Physical State Represented
1	8	0.01	5.0	0.20	Intact, early-age backfill
2	7	0.006	4.0	0.23	Slightly micro-cracked
3	5	0.003	3.2	0.26	Moderately degraded (exposed surface)
4	4	0.001	2.5	0.30	Severely weathered/fractured

.

The relationship between $r_p$ and ${\mathrm{K}}_1$ approximates an exponential function. As the backfill body progressively degrades (decreasing m, s, and ${\mathrm{R}}_{\mathrm{c}}$; increasing μ), the failure zone width increases at any given ${\mathrm{K}}_1$. This exponential-like relationship is physically linked to the nonlinear Hoek–Brown criterion as follows: as ${\mathrm{K}}_1$ increases, the crack-tip stress field extends further into the rock mass, and the parabolic strength envelope permits disproportionately larger failure zones at higher stress levels.

In Figure 9, both (a) and (b) illustrate the curve relationships between the width of the crack-tip failure zone and the stress intensity factor. An overview of the two figures reveals that the relationship curve between ${\mathrm{r}}_{\mathrm{p}}$ and K approximates an exponential function-type curve. That is, for given values of the parameters m, μ, ${\mathrm{R}}_{\mathrm{c}}$, and s, the width of the edge crack-tip failure zone exhibits a gradually increasing trend as the stress intensity factor of the edge crack increases, and the curvature of this increase also shows a gradually growing trend. Although one curve in Figure 9b displays local fluctuations in its shape during the growth process, the overall trend still manifests as an exponential increase.

From the variations in each curve and the corresponding parameter values in Figure 9, it can be inferred that as the overall integrity of the backfill body decreases, the stress intensity factor at the tip of the thin-plate edge crack gradually increases, consequently leading to an increase in the width of the edge crack-tip failure zone. Since the slabbing roadway-side backfill body is not formed instantaneously but rather under the sustained action of the load from the roof and floor strata and the lateral bending moment, the following two concurrent processes occur: on the one hand, the slabbing phenomenon progressively extends from shallower to deeper parts; on the other hand, the edge cracks on the slabbing thin plate gradually propagate. Consequently, the values of parameters m, μ, ${\mathrm{R}}_{\mathrm{c}}$, and s differ significantly across different stages or conditions. Specifically, the selected values for parameter m (reflecting the hardness/softness degree of the rock), parameter s (reflecting the fragmentation degree of the rock mass), and parameter ${\mathrm{R}}_{\mathrm{c}}$ (reflecting the uniaxial compressive strength of the rock) all decrease sequentially. In contrast, the selected value for Poisson’s ratio μ, which reflects the actual strength, weathering degree, and development of joints and fissures in the rock, increases sequentially.

4. Analysis of Progressive Failure in Slabbing Roadway-Side Backfill Body with Tensioned Bolts

As coal mining progressively transitions to deep extraction, the manifestation of strata behavior intensifies, roadway convergence becomes more pronounced, and issues such as large deformation and load-bearing failure of the roadway-side backfill body in gob-side entry retaining are increasingly prevalent. This is particularly severe in backfill bodies that are not fully capped (in contact with the roof). Therefore, the effectiveness of roof contact and the support provided to the roadway-side backfill body are critical in gob-side entry retaining. To prevent the dehydration and weathering of ettringite, the primary hydration product in the backfill material, which would reduce the compressive strength of the backfill body, relevant scholars have researched and invented a bagged backfill structure similar to that shown in Figure 10. This aims to mitigate the weathering effect on the roadway-side backfill body and enhance its support strength. The procedure involves, during the preparation stage for roadside backfilling, pre-installing special backfill bags and erecting tensioned bolts of a certain strength within the bags, followed by roadside grouting through reserved openings. According to actual field cases, the effect of roadside backfilling is significant, greatly reducing the slabbing failure phenomenon of the roadway-side backfill body and inhibiting the propagation and failure of edge cracks on the thin plate of the slabbing roadway-side backfill body.

Based on the preceding introduction and theoretical analysis of the bagged backfill structure, it is indicated that the mechanism of the backfill body reinforced with tensioned bolts is analogous to the mechanical mechanism of riveted stiffened plates in fracture mechanics. Therefore, this paper draws on this concept to study the thin plate of the slabbing roadway-side backfill body after the application of tensioned bolts [32,33]. Figure 11 illustrates the edge crack propagation in the thin plate after the installation of tensioned bolts.

According to the Stress Intensity Factors Handbook [24] in fracture mechanics, and by applying analytical solution methods, the stress intensity factor for this Mode I crack can be expressed as follows:

$$K_R = \mathsf{\eta }·K_0$$

(18)

In Equation (18), $\mathsf{\eta }$ is termed the stress concentration factor, which can represent the strengthening effect of the bolts on the roadway-side backfill body. $\mathsf{\eta }$ varies with parameters α and β, as shown by the curves in Figure 12.

In Figure 12, $\mathsf{\beta } = (2A_5{E}_5)/(b\mathrm{h}E)$ represents the relative tensile stiffness of the bolt, b is the bolt spacing (row spacing and line spacing), and E and Es are the elastic moduli of the thin plate of the slabbing roadway-side backfill body and the bolt, respectively.

In summary, the stress intensity factor for the edge crack in the thin plate after the application of tensioned bolts is as follows:

$${\mathrm{K}}_{\mathrm{R}} = \mathsf{\eta }·\mathsf{\sigma }\sqrt{\mathsf{\pi }\mathrm{a}}$$

(19)

Based on the boundary Equation (17) derived earlier, the boundary equation for the failure zone width at the tip of the edge crack in the thin plate of the backfill body after the application of tensioned bolts is as follows:

$${\mathrm{r}}_{\mathrm{p}} = {\displaystyle \frac{4{\mathrm{K}}_{\mathrm{R}}^4(1 - 2\mathsf{\mu }{)}^4}{2\mathsf{\pi }\left[2\mathrm{m}\mathsf{\mu }{\mathrm{R}}_{\mathrm{c}}{\mathrm{K}}_{\mathrm{R}} \pm \sqrt{(2\mathrm{m}\mathsf{\mu }{\mathrm{R}}_{\mathrm{c}}{\mathrm{K}}_{\mathrm{R}}{)}^2 + 4\mathrm{s}{\mathrm{R}}_{\mathrm{c}}^2{\mathrm{K}}_{\mathrm{R}}^2(1 - 2\mathsf{\mu }{)}^2}\right]}}$$

(20)

From the previous analysis, it is known that the width of the edge crack-tip failure zone is approximately exponentially related to the crack stress intensity factor. From Equation (19), it can be seen that the stress intensity factor for the edge crack in the reinforced roadway-side backfill body thin plate is primarily influenced by $\mathsf{\eta }$, with a directly proportional relationship between the two. From Figure 12, it can be seen that when $\mathsf{\beta }$ is in the range of (0,1), $\mathsf{\eta }$ gradually increases as 2a/b increases, but as $\mathsf{\beta }$ approaches 1, its growth rate significantly decreases; when $\mathsf{\beta }$ is in the range of (1,∞), $\mathsf{\eta }$ gradually decreases as 2a/b increases, but as β approaches ∞, its rate of decrease significantly increases. Therefore, for the reinforced gob-side filling body, the width of the damage zone at the edge crack-tip varies with the stress intensity factor and the stress concentration factor $\mathsf{\eta }$, with a relatively large initial growth rate that gradually decreases until it approaches infinitesimal. Meanwhile, the failure width of the edge crack in the reinforced filling body shows a trend of initially increasing sharply, then increasing slowly until it approaches a certain constant value.

Sensitivity Analysis

A sensitivity analysis was conducted to quantify the relative influence of key parameters on the crack initiation load q (Equation (6)). Each parameter was varied by ±20% around baseline values, while holding others constant. The baseline parameters are: t = 60 mm, H = 3 m, L = 15 m, a = 0.02 m, M = 2 kN·m/m, and Kc = 0.15 MPa·m^(1/2).

In

Table 4. Load variation values and the resulting influence values of basic parameters.

Parameter	−20% Change in q	+20% Change in q	Sensitivity Ranking
t (thickness)	−48.8%	+72.8%	Highest (cubic dependence)
H (height)	+44.4%	−30.6%	High (quadratic inverse)
L (length)	+56.3%	−30.6%	High (quadratic inverse)
a (crack length)	+11.8%	−10.6%	Moderate (square-root inverse)
M (bending moment)	+3.8%	−3.8%	Low (linear)
K~c~ (fracture toughness)	−20.0%	+20.0%	Moderate (linear)

, the results indicate that plate thickness t exerts the dominant influence due to its cubic contribution to bending stiffness, followed by geometric parameters H and L. The bending moment M has the smallest effect, confirming that the backfill body’s inherent geometry and material toughness primarily govern stability.

5. Discussion

This paper primarily employs theoretical analysis to investigate the progressive failure mechanism of slabbing roadway-side backfill bodies. The key contributions and limitations are discussed below.

Physical interpretation of V-shaped notch formation: The model provides a mechanistic explanation for V-shaped notch formation: the differential crack propagation rate between the free surface (where confinement is zero) and deeper regions, combined with crack coalescence near the mid-height of the plate, naturally explains the characteristic V-shaped geometry observed in field cases. This transition from an initially symmetric two-crack configuration to asymmetric macroscopic failure represents a symmetry-breaking outcome of the governing equations.

Model validation: Direct experimental or field validation of the derived analytical expressions is presently limited. However, the predicted failure sequence—vertical tensile cracks evolving into slab detachment and V-shaped notch formation—qualitatively matches the gypsum model test results of Zhou et al. [16], who observed identical crack patterns under uniaxial compression. Key components of the derivation, including the edge crack stress intensity factor under bending [24] and the body force method for distributed loads, are drawn from validated formulations. The model currently serves as a theoretical framework to reveal dominant mechanisms and trends; dedicated experimental or field validation constitutes a necessary next step.

Model limitations and assumptions:

(i)

Material assumptions: The isotropic homogeneous assumption does not account for possible micro-heterogeneity, residual porosity, or property degradation due to weathering. The elastic thin-plate model is applicable when: slab thickness exceeds the characteristic pore spacing by at least one order of magnitude; material heterogeneity is limited (elastic modulus variation < 15%); and the failure timescale is short relative to characteristic creep time.

(ii)

Boundary conditions: The model assumes ideal contact at roof–backfill and floor–backfill interfaces, without considering potential gap closure, local crushing, or interfacial slip. The predicted crack initiation load may be overestimated under poor contact conditions.

(iii)

Loading path and dynamic effects: The load is applied quasi-statically, and abrupt stress adjustments during periodic weighting events are not included. Static fracture mechanics is appropriate for determining threshold conditions for the pre-dynamic stage; dynamic analysis would be required for post-initiation fragment ejection.

(iv)

Applicability: The model is intended to reveal dominant mechanical mechanisms and trends, rather than to provide highly accurate quantitative predictions for all field conditions. Its predictions should be interpreted in conjunction with numerical simulations and in situ monitoring data.

Connection between initial slabbing and progressive failure: Initial slabbing partitions the backfill body into thin plates via Mode I tensile cracks, creating free surfaces and releasing radial confinement. This transforms the stress state from triaxial compression to approximate plane-stress condition. Under sustained roof–floor load and eccentricity-induced bending moment, tensile stress concentrates at the tips of newly formed edge cracks. When $K_1$ reaches $K_c$, stable crack propagation initiates, marking the transition to progressive failure. Equation (3) directly captures this transition by linking the post-slabbing stress field to the crack driving force.

Mode I assumption justification: Field observations and published experiments consistently show that slabbing fractures in backfill bodies are tensile cracks oriented sub-parallel to the maximum principal stress direction (Mode I). For a plate under combined compression and bending, the dominant crack-tip loading for an edge crack aligned with the compression direction is Mode I opening induced by plate bending. The Mode II contribution for this crack orientation is small. For conditions where crack orientation deviates from the loading direction, mixed-mode (I + II) fracture criteria would be required.

Practical applications and engineering significance: The derived expression for q enables estimation of the critical roof–floor load a slabbing backfill body can sustain, providing a theoretical reference for determining required backfill strength in gob-side entry retaining. The contrasting failure zone growth patterns—exponential for unbolted versus asymptotic saturation for bolted—are explicitly connected to support design as follows: tensioned bolts fundamentally alter the failure propagation mode, suggesting bolt parameters should be designed to maximize η.

Future research directions include: (i) incorporating orthotropic or layered material models to represent stratified backfill structures; (ii) introducing nonlinear foundation models to capture more realistic roof–backfill–floor interaction; (iii) developing time-dependent loading schemes to simulate mining-induced stress redistribution; (iv) extending to mixed-mode (I + II) fracture criteria for deviated crack orientations; and (v) conducting laboratory tests on high-water backfill specimens with pre-fabricated edge cracks to quantitatively validate the derived stress intensity factor expressions.

6. Conclusions

This study presents an analytical investigation into the progressive failure mechanism of slabbing roadway-side backfill bodies in gob-side entry retaining. The main conclusions are:

(1)

A thin-plate mechanical model with edge cracks was established for the slabbing roadway-side backfill body. Through an original load transformation method, the three-dimensional in situ stress state is reduced to the combined action of a uniform roof–floor load ${\mathsf{\sigma }}_1$ and an equivalent bending moment M. The derived stress intensity factor $K_1$ (Equation (3)) and crack initiation load q (Equation (6)) provide closed-form analytical expressions linking macroscopic loading to mesoscopic crack-driving forces. q increases positively with plate thickness t (cubic dependence) and bending moment M and decreases with crack length a in the dominant range.

(2)

Based on the nonlinear Hoek–Brown criterion, the failure zone width ${\mathrm{r}}_{\mathrm{p}}$ at the edge crack tip (Equation (17)) was derived and shown to exhibit an approximately exponential relationship with ${\mathrm{K}}_1$ under unbolted conditions. The growth rate of ${\mathrm{r}}_{\mathrm{p}}$ increases as the backfill body progressively degrades (decreasing m, s, and ${\mathrm{R}}_{\mathrm{c}}$; increasing μ).

(3)

Introduction of tensioned bolts via a stress concentration factor η transforms the failure zone growth from exponential (unbolted) to asymptotic saturation (bolted), where ${\mathrm{r}}_{\mathrm{p}}$ increases sharply initially then approaches a constant. This quantitatively confirms the crack-arresting mechanism of bolting and provides a theoretical basis for optimizing bolt parameters to limit progressive failure. The derived expression for q enables estimation of the critical roof–floor load for backfill strength design.

(4)

The model reveals that V-shaped notch formation in roadway-side backfill bodies arises from differential crack propagation rates between the free surface (zero confinement) and interior regions, governed by the spatial variation in confinement and non-uniform distribution of ${\mathrm{K}}_1$.

Future research directions include: (a) extending the model to orthotropic plates to account for layered backfill construction; (b) incorporating time-dependent material degradation to capture weathering effects; and (c) conducting laboratory tests on high-water backfill specimens with pre-fabricated edge cracks to quantitatively validate the derived expressions.