Study on Creep Characteristics of High-Volume Fly Ash-Cement Backfill Considering Initial Damage

Abstract

To reveal the long-term deformation behavior of high-volume fly ash-based backfill under continuous mining and backfilling, a fly ash–cement backfill material with 73.0% fly ash content was developed, and creep characteristic tests considering initial damage were conducted. The results demonstrate that: (1) A calculation method for the initial damage of backfill based on stress–strain hysteresis loop cycles is proposed, with cumulative characteristics of initial damage across mining phases analyzed; (2) Creep behaviors of backfill affected by initial damage are investigated, revealing the weakening effect of initial damage on long-term bearing capacity; (3) An enhanced, nonlinear plastic damage element is developed, enabling the construction of an HKBN constitutive model capable of characterizing the complete creep behavior of backfill materials. The research establishes a theoretical framework for engineering applications of backfill materials with early-age strength below 5 MPa, while significantly enhancing the utilization efficiency of coal-based solid wastes.

1. Introduction

Coal remains the dominant energy source underpinning China’s national economic development. Currently, over 60% of coal production is utilized for thermal power generation, and this coal-centered energy consumption structure is unlikely to be fundamentally altered in the short term [1,2]. However, the fly ash generated from coal combustion poses severe environmental threats, particularly in the ecologically fragile coal-power energy bases of northwestern China [3,4,5,6]. The development of high-volume fly ash-based backfill material technology, when combined with appropriate backfill mining methods, represents the most promising solution for addressing the accumulation challenges of coal-based solid wastes like fly ash.

Fly ash-based backfill materials demonstrate significant advantages in resource utilization, low-carbon environmental protection, and cost reduction [7,8,9]. Fly ash demonstrates remarkable utilization efficiency in backfill applications, achieving incorporation rates of 70%~90% by mass [10]. This high-volume substitution translates to significant carbon mitigation, with each cubic meter of produced backfill reducing CO₂ emissions by 0.3~0.5 metric tons compared to conventional cement-based mixtures [11]. Research findings demonstrate that fly ash-based backfill materials exhibit a 40%–60% reduction in Global Warming Potential compared to conventional cement-based materials, as quantified through comprehensive life cycle assessment studies [12,13]. A mining operation successfully implemented fly ash–slag composite backfill (40% fly ash—60% slag by mass), achieving 30% reduction in backfill material costs and 100% compliance with 28d strength requirements [14].

The main techniques for activating fly ash reactivity include alkaline activation, sulfate activation and ultrafine grinding [15,16,17,18,19,20]. The research shows that the 28d uniaxial compressive strength (UCS) of fly ash polymer can be increased to 25~35 MPa by optimizing the proportion of alkaline activator [21]. Sulfate can promote the formation of C-S-H gel and ettringite, and the pozzolanic reaction efficiency is increased by 40% [22]. Using superfine grinding to make its specific surface area not less than 600 m²/kg can significantly shorten the setting time of filling material and reduce the porosity to less than 15% [23,24].

In addition, further performance optimization can be achieved through co-activation effects in composite systems [25,26,27,28]. CaO in slag can promote the depolymerization of fly ash and form dense C-(A)-S-H gel [29]. The backfill materials of the fly ash cement system can reduce the hydration heat by 40%, and maintain the UCS not less than 10 MPa for 7 days [30]. At the same time, the content of fly ash has a significant impact on the mechanical properties of backfill materials [31,32,33,34]. When the content is 50%, the backfill materials will continue to creep at constant speed for 120 h at a 30% stress level, and the long-term strength decay rate will not exceed 15%. When the content is 60%, the 28d UCS is 8~12 MPa. When the content is more than 70%, the 7d UCS is only 40%~60% of the cement-based materials, causing a 20%~30% prolongation in mining cycle duration.

The continuous mining and backfilling (CMB)method synchronizes mining and backfilling operations by rapidly filling the goaf with backfill materials [35]. It leverages lateral confinement to compensate for the early-stage strength deficiency of the backfill materials, thereby helping to address the limitation of prolonged mining cycles caused by the low early strength of high-volume fly ash-based backfill materials [36,37]. When the UCS of the backfill is not less than 5 MPa, it can effectively restrain the expansion of the surrounding rock’s plastic zone and reduce roadway maintenance costs by 30% [38]. However, under conditions such as repeated disturbances and high-stress environments, the backfill is prone to creep damage [39,40]. Research on backfill creep damage primarily involves laboratory experiments focusing on creep characteristics, constitutive models, and material optimization [41,42]. The research shows that the higher the binder content in cemented backfill, the smaller the instantaneous creep deformation and rate, and the later the onset of the accelerated creep failure stage [43,44]. Under different stress levels, the creep failure mode of backfill transitions from brittle to ductile, with acoustic emission events directly correlated to crack propagation [45,46].

The long-term strength of sulfur-containing tailings backfill decreases due to expansive products generated by reactions between sulfides and cement, which can exacerbate long-term creep damage [47]. A modified Burgers model, considering stress levels and damage accumulation, can effectively characterize the creep behavior under varying binder contents [48]. A nonlinear creep model can be applied to describe the interaction mechanism between surrounding rock deformation and backfill creep under high-stress conditions in deep mining [49]. The addition of glass fibers, polypropylene fibers, or alkaline agents can inhibit sulfide-induced corrosion, enhancing crack resistance and creep durability [50]. By adjusting parameters such as the cement-to-tailings ratio and slurry density, the long-term stability and economic efficiency of backfill can be balanced, reducing the consumption of cementitious materials [51,52]. However, under the CMB method, the backfill experiences cyclic loading and unloading disturbances due to the mining of adjacent mining roadways [53]. Only after all mining roadways are filled does the backfill enter the creep stage. Therefore, investigating the creep characteristics of high-volume fly ash-based backfill materials with consideration of initial damage is critical, which represents the key to improving mining efficiency in CMB operations and enhancing the comprehensive utilization of fly ash.

This study investigates the creep behavior of backfill materials under initial damage conditions through laboratory experiments, using high-volume fly ash–cement backfill materials in the context of CMB. The outcomes provide fundamental data for understanding creep mechanisms in backfill subjected to mining disturbances. Furthermore, this work offers theoretical support for enhancing the recycling utilization rate of coal-based solid wastes and achieving energy conservation and carbon reduction in coal mining operations.

2. Creep Testing Methods for Backfill Materials

2.1. Sample Preparation

To meet the requirements for backfill material preparation and underground transportation, high-volume fly ash–cement backfill material is formulated into two components: Material A and Material B. Material A serves as the main structural component, providing primary load-bearing capacity, while Material B functions as a binder, reinforcing agent, and quick-setting component. For this experiment, fly ash was selected as Material A, and gypsum, quicklime, cement, and additives were chosen as Material B. By mass ratio, Material A consists mainly of 50 parts fly ash and 0.5~1 part additive, while Material B comprises 1~3 parts gypsum, 5~10 parts lime, 2~6 parts cement, and 0.5~1 part additive [54]. Based on previous backfill mixture ratio tests, an optimal mass ratio of fly ash, gypsum, lime, cement, and additives was determined as 27:1:5:3:1 (with a fly ash content of approximately 73.0%) and a water-to-solid ratio of 0.85:1. The backfill material mixing scheme is shown in

Table 1. The mix design of high-volume fly ash–composite backfill materials.

Water/g	Fly Ash/g	Gypsum/g	Lime/g	Cement/g	Additive/g	Total Mass/g
540.54	335.28	12.42	62.09	37.25	12.42	1000

.

The preparation of the backfill samples consists of six steps: ① Materials preparation; ② Mixing with water to form slurry; ③ Pouring the slurry into molds; ④ Forming and demolding; ⑤ Samples curing; ⑥ Dimensional verification. The sample preparation process is illustrated in Figure 1. Steps ④ and ⑤ are critical stages in specimen preparation. In Step ④, the sealed backfill slurry should be left undisturbed for 24~48 h in an environment with a temperature of 20 ± 5 °C, avoiding direct sunlight. The molds should only be removed after the slurry has fully solidified and formed. In Step ⑤, the demolded backfill samples are first inspected for defects by immersing them in a saturated Ca(OH)₂ solution. Qualified samples are then placed in a constant temperature and humidity curing chamber with a relative humidity of 95% for 28 days. The spacing between samples should be maintained at 10~20 mm.

2.2. Testing Program

The cumulative damage characteristics of backfill were investigated using cyclic loading–unloading tests. The creep properties of backfill considering initial damage were studied through stepwise creep loading tests.

2.2.1. Cyclic Loading–Unloading Tests

In the context of CMB engineering, the initial damage to the backfill can be attributed to disturbances caused by adjacent mining and backfilling operations in mining roadways. The number of disturbances experienced by the backfill depends on the mining phases (typically no more than four phases), with the first-phase backfill undergoing the most disturbances, making it the weakest zone. Taking a typical four-phase scenario as an example, the number of loading–unloading cycles for the backfill in each phase is 6, 4, 2, and 0, respectively. This determines the cyclic loading–unloading frequency.

Uniaxial physical and mechanical property tests on backfill samples show that the average compressive strength, tensile strength, and shear strength are 5.23 MPa, 1.05 MPa, and 1.56 MPa, respectively. The test parameters include a loading rate of 0.05 mm/s, a maximum axial stress of 5.00 MPa, and a confining pressure of 1.00 MPa, with the axial stress not being fully unloaded to zero. The loading protocol is as follows: The first-stage axial stress increases from 0 MPa to 2.00 MPa, then unloads to 1.50 MPa. For subsequent stages, the peak axial stress increases by 0.60 MPa per level, with unloading always to 1.50 MPa. If the sample does not fail when the axial stress reaches 5.00 MPa, a displacement-controlled loading at 0.05 mm/s is applied until failure occurs.

2.2.2. Stepwise Creep Loading Tests

The maximum load for the stepwise creep test on backfill was set at 90% of the conventional triaxial deviatoric stress strength (σ_ds, σ_ds = σ₁ − σ₃), while the minimum load was set at 40% of the conventional triaxial deviatoric stress strength. The test parameters included an axial loading rate of 0.05 mm/s and a confining pressure of 1.00 MPa. The loading protocol was as follows: The axial stress started at 40% of the deviatoric stress strength, with each subsequent level increased by 10% until reaching 90%. Each creep stage lasted 2 h. If the specimen did not fail upon reaching 90% of the deviatoric stress strength, the stress increment was reduced to 2.5% per level for further loading.

2.3. Testing Equipment

The cyclic loading–unloading tests and stepwise creep loading tests were conducted using a GDS triaxial testing system, as illustrated in Figure 2. The test mainly involves the axial pressure control system and the confining pressure control system. By controlling the axial servo press, the loading and unloading of the sample under axial compression can be realized. Silicone oil is injected into the closed chamber through the confining pressure controller to exert confining pressure on the test piece.

3. Initial Damage Characteristics of Backfill

Taking the CMB divided into four phases as an example, the initial damage characteristics of the backfill are analyzed. The backfill undergoes a maximum of six loading–unloading cycles, as shown in Figure 3. It can be observed from the figure that dilation occurs during the second loading cycle, corresponding to a deviatoric stress of 1.49 MPa and a volumetric strain of approximately 0.15%. As the number of cycles increases and the deviatoric stress level rises, the area of the hysteresis loop gradually expands. The axial strain increments for each loading–unloading hysteresis loop are 0.02%, 0.02%, 0.04%, 0.11%, 0.16%, and 0.38%, respectively. When the deviatoric stress reaches 4.27 MPa, the specimen fails, corresponding to an axial strain of 1.58%. Its residual strength is about 2.59 MPa, approximately 55% of the peak stress.

The initial damage (D₀) of the backfill sample is defined as the ratio of the cumulative dissipated energy during the cyclic loading–unloading process to the work done by the press on the specimen [55,56,57]. Thus, after the n-step cyclic loading–unloading test, the initial damage of the backfill sample can be expressed as

$$D_0\left(n\right)={\displaystyle \frac{{\displaystyle \sum _{m=1}^n{\displaystyle {\int }_{{{\epsilon }^{\prime }}_m}^{{\epsilon }_m}\left[{\sigma }_m\left(\epsilon \right)-0.5\right]\mathrm{d}\epsilon }-{\displaystyle \sum _{m=1}^n{\displaystyle {\int }_{{{\epsilon }^{\prime }}_m}^{{\epsilon }_m}\left[{{\sigma }^{\prime }}_m\left(\epsilon \right)-0.5\right]\mathrm{d}\epsilon }}}}{{\displaystyle {\int }_{{{\epsilon }^{\prime }}_m}^{{\epsilon }_m}[{{\sigma }^{\prime }}_m\left(\epsilon \right)-0.5]\mathrm{d}\epsilon }+{\displaystyle \sum _{m=1}^n{\displaystyle {\int }_{{{\epsilon }^{\prime }}_m}^{{\epsilon }_m}\left[{\sigma }_m\left(\epsilon \right)-0.5\right]\mathrm{d}\epsilon }-{\displaystyle \sum _{m=1}^n{\displaystyle {\int }_{{{\epsilon }^{\prime }}_m}^{{\epsilon }_m}\left[{{\sigma }^{\prime }}_m\left(\epsilon \right)-0.5\right]\mathrm{d}\epsilon }}}}}$$

(1)

where σ_m(ε) and σ′_m(ε) denote the deviatoric stresses during the m-th cycle’s loading and unloading phases, respectively; ε and ε′_m represent the axial strains at the peak (vertex) and valley (nadir) points of the stress–strain hysteresis loop for the m-th loading–unloading cycle.

The integral in Equation (1) can be computed using the “Analysis → Mathematics → Integrate” function in Origin. This yields the energy parameters and damage degree of the backfill sample during each loading–unloading cycle, as summarized in

Table 2. Energy and damage parameters of the sample after two cycles.

Number of Cycles	Elastic Strain Energy	Dissipative Energy	Work Done on the Sample	Damage Variable
1	0.017 V1	0.004 V1	0.020 V1	0.173
2	1.668 V2	0.892 V2	2.560 V2	0.348

,

Table 3. Energy and damage parameters of the sample after four cycles.

Number of Cycles	Elastic Strain Energy	Dissipative Energy	Work Done on the Sample	Damage Variable
1	0.038 V1	0.008 V1	0.046 V1	0.182
2	0.277 V2	0.083 V2	0.361 V2	0.231
3	0.885 V3	0.362 V3	1.247 V3	0.290
4	1.643 V4	1.118 V4	2.761 V4	0.404

and

Table 4. Energy and damage parameters of the sample after six cycles.

Number of Cycles	Elastic Strain Energy	Dissipative Energy	Work Done on the Sample	Damage Variable
1	0.026 V1	0.006 V1	0.031 V1	0.178
2	0.117 V2	0.033 V2	0.150 V2	0.220
3	0.267 V3	0.099 V3	0.366 V3	0.270
4	0.47 V4	0.223 V4	0.694 V4	0.327
5	0.854 V5	0.534 V5	1.388 V5	0.385
6	1.755 V6	1.429 V6	3.174 V6	0.450

. Here, V_i denotes the axial compression volume of the specimen in the i-th cycle (where i = 1, 2, 3, 4, 5, 6).

The table shows that after 2, 4, and 6 loading–unloading cycles, the cumulative damage of the backfill sample reaches 0.35, 0.40, and 0.45, respectively. Therefore, the initial damage levels D₀ for the creep tests are set as 0, 0.35, 0.40, and 0.45, representing the influence of mining–backfilling disturbances at the 4th, 3rd, 2nd, and 1st phases, respectively. Additionally, due to the inherent variability among specimens, it is challenging to ensure identical damage values across all samples. Thus, the allowable error range for initial damage is controlled within 5%.

4. Creep Characteristics of Backfill Considering Initial Damage

4.1. Axial Deformation Characteristics

During the creep test, the GDS device recorded data every 10 s. To avoid excessive density of data points interfering with the test results, the original data was diluted. Under each stress level, the duration of stress maintenance was 2 h, and one data point was selected every 120 s. The transition duration from a lower stress level to a higher one was 2 min, during which one data point was selected every 20 s. The variation curve of the axial strain (ε₁) of the specimen with time was obtained, as shown in Figure 4. In the figure, σ_c represents the peak maximum deviatoric stress, in MPa.

By comparing Figure 4a–d, it can be observed that intact samples and samples with initial damage exhibit similar creep characteristics. Under low stress levels, they primarily undergo instantaneous deformation, decelerating creep, and steady-state creep stages. At higher stress levels, they mainly experience instantaneous deformation, decelerating creep, steady-state creep, and accelerating creep stages. The detailed findings include:

(1).

For intact samples: When the stress level does not exceed 0.9 σ_c, the sample undergoes a brief decelerating creep stage before the creep curve stabilizes into a horizontal line. When the stress level reaches 0.925 σ_c, the creep process exhibits three distinct stages: decelerating creep, steady-state creep, and accelerating creep. The instantaneous deformation and creep deformation are 1.13% and 0.15%, respectively.

(2).

For samples with D₀ = 0.45: When the stress level does not exceed 0.7 σ_c, the samples experience a short decelerating creep stage before the creep curve stabilizes. When the stress level reaches 0.8 σ_c, the decelerating creep stage becomes nearly indistinguishable, with the creep process transitioning directly from steady-state creep to accelerating creep. The instantaneous deformation and creep deformation are 0.61% and 0.21%, respectively.

(3).

The total creep durations of the samples are 835 min, 644 min, 612 min, and 555 min, respectively. Compared to the intact sample (835 min), the creep durations of the damaged samples are reduced by 22.87%, 26.71%, and 33.53%, respectively.

(4).

The ratio of creep deformation to total deformation for the samples is 10.24%, 12.09%, 14.12%, and 17.40%, respectively. Compared with the intact sample, the creep deformation proportion increases by 18.07%, 37.89%, and 69.92% for damaged samples with varying degrees of initial damage.

These testing findings confirm that initial damage significantly compromises the mechanical integrity of backfill material, primarily through three manifestations: reduction in failure stress thresholds, contraction of total creep lifespan, and elevation in creep deformation contribution.

4.2. Volume Deformation Characteristics

The volumetric strain (ε_V, ε_V = ε₁ + 2 ε₃) time-history curves of the specimens are shown in Figure 5. Comparative analysis of (a) to (d) reveals that both intact specimens and those with initial damage exhibit similar evolutionary trends in volumetric strain, progressing sequentially through compression, stabilization, dilatancy, and failure stages. When the deviatoric stress remains below the yield strength, axial deformation (ε₁) dominates, resulting in overall volume compression. As the deviatoric stress approaches the yield strength, circumferential deformation (ε₃) begins to increase significantly, with its growth rate becoming comparable to or even exceeding that of axial deformation. This leads to stabilization of the volumetric strain curve, followed by gradual expansion. Once the deviatoric stress exceeds the yield strength, the circumferential deformation rate (v₃) surpasses the axial deformation rate (v₁), and the disparity between them progressively widens, making volume dilatancy increasingly pronounced. With continued creep progression or further increase in deviatoric stress, the specimen enters the accelerated creep stage, exhibiting distinct creep dilatancy before ultimately undergoing creep failure. The detailed findings include:

(1)

The volumetric dilatancy and creep-induced expansion of specimens typically initiate during the stress adjustment phase, a critical period characterized by the nucleation, propagation, and subsequent coalescence of microcracks, which progressively deteriorate the material’s load-bearing capacity.

(2)

At stress levels ≤ 0.5 σ_c, the specimen exhibits predominantly axial compressive deformation, resulting in net volumetric contraction. In the stress range of 0.6 σ_c to 0.7 σ_c, the specimen’s circumferential strain rate approaches or surpasses its axial strain rate, resulting in stabilization and subsequent dilatation of the volumetric strain curve. When the stress level reaches or exceeds 0.8 σ_c, the specimen exhibits a circumferential strain rate that consistently surpasses the axial strain rate, marking the onset of significant creep effects, which is defined as the SCE index. Progressive creep development or additional stress elevation beyond this threshold drives the specimen into the accelerated creep phase, ultimately leading to structural failure.

(3)

The SCE serves as a critical indicator marking the transition from stable deformation to imminent failure. Quantitative analysis reveals SCE initiation times of 602 ± 2 min, 486 ± 2 min, 488 ± 2 min, and 490 ± 2 min for respective samples, corresponding to 72.10%, 75.47%, 79.74%, and 88.29% of their total creep durations.

4.3. Characteristics of Volumetric Strain Rate

The volumetric strain rate (v_v) evolution curves of the specimens are presented in Figure 6, where positive values indicate volumetric compression and negative values represent volumetric expansion. Comparative analysis of subplots Figure 5a–d reveals that both intact specimens and those with initial damage exhibit similar evolutionary patterns in volumetric strain rate, which can be distinctly categorized into volumetric compression and expansion phases. The detailed findings include:

(1)

Volumetric compression phase:

The peak volumetric strain rate (v_vmax) decreases with increasing stress levels, indicating that higher stresses induce greater circumferential strains, which partially offset axial compression through lateral expansion. At identical stress levels, intact samples demonstrate higher v_vmax than pre-damaged samples. This suggests that initial damage compromises the material’s elastic capacity, with axial strain dominating volumetric response at low stress levels (≤0.5 σ_c).

(2)

Volumetric expansion phase:

v_vmax increases proportionally with stress levels, demonstrating the circumferential strain’s dominant contribution to volumetric deformation. This confirms that initial damage facilitates circumferential strain development, further evidencing its weakening effect on backfill materials.

5. Construction of HKBN Constitutive Model for Backfill

5.1. Proposing an Accelerated Creep Element

To address the limitations of classical rheological models in accurately characterizing accelerated creep behavior while accounting for the influence of initial damage on backfill materials, an enhanced nonlinear plastic damage model (NPDM) was developed to describe the accelerated creep stage. The plastic damage element with cross-sectional area A in the NPDM framework comprises three distinct components: the undamaged portion (A_r), initial damage (A₀), and creep-induced damage (A_c). The research findings demonstrate that damage initiation occurs exclusively when the applied stress reaches the yield stress threshold (σ_s) [58,59,60]. The undamaged portion of the NPDM is assumed to comply with Hooke’s law of elasticity. Under uniaxial loading conditions, the creep constitutive equation for the NPDM incorporating initial damage effects is expressed as:

$${\epsilon }_N\left(t\right)=\{\begin{array}{ll}0& , \sigma <{\sigma }_s\\ {\displaystyle \frac{{\sigma }_{ds}-{\sigma }_s}{E_N}}{\left[1-{\displaystyle \frac{t}{t_z\left(D_0\right)}}\right]}^{{\displaystyle \frac{1}{r+1}}}& , \sigma \ge {\sigma }_s\end{array}$$

(2)

In the equation: t_z: total creep duration, s; r: material-specific constant of backfill; E_N: elastic modulus of the NPDM model, GPa; σ_ds: deviatoric stress, defined as σ₁–σ₃, MPa.

Based on previous research findings demonstrating the correlation between total creep duration and initial damage in backfill materials, we performed nonlinear regression analysis on the experimental data from Figure 5. The resulting empirical relationship between t_z and D₀ is expressed as:

$$t_z\left(D_0\right)=-139.97e^{{\displaystyle \frac{D_0}{0.41}}}+973.92$$

(3)

By combining the damage evolution law from Equation (3) with the fundamental constitutive relation in Equation (2), we obtain the complete NPDM creep equation accounting for initial damage:

$${\epsilon }_N\left(t\right)=\{\begin{array}{ll}0& , \sigma <{\sigma }_s\\ {\displaystyle \frac{{\sigma }_{ds}-{\sigma }_s}{E_N}}{\left[1-{\displaystyle \frac{t}{-139.97e^{\frac{D_0}{0.41}}+973.92}}\right]}^{{\displaystyle \frac{1}{r+1}}}& , \sigma \ge {\sigma }_s\end{array}$$

(4)

5.2. The Three-Dimensional HKBN Constitutive Model

Based on the above analysis, the connection method of the rheological mechanical components in the HKBN model is shown in Figure 7. Assuming that σ_sB and σ_sN are the initiation stresses for steady-state creep and accelerated creep, respectively, where σ_sB < σ_sN. When σ_ds < σ_sB, only the Hooke body and Kelvin body contribute to deformation, and the HKBN model exhibits instantaneous deformation and decelerating creep. When σ_sB ≤ σ_ds < σ_sN, the Hooke body, Kelvin body, and Bingham body contribute to deformation, and the HKBN model exhibits instantaneous deformation, decelerating creep, and steady-state creep. When σ_ds ≥ σ_sN, all constitutive elements within the HKBN model participate in the deformation process, exhibiting comprehensive three-stage creep behavior.

Thus, the creep equation of the one-dimensional HKBN model can be expressed as:

$$\epsilon \left(t\right)=\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_{ds}}{E_H}}+{\displaystyle \frac{{\sigma }_{ds}}{E_K}}\left[1-\exp \left(-{\displaystyle \frac{E_K}{{\eta }_K}}t\right)\right]& , {\sigma }_{ds}<{\sigma }_{sB}\\ \\ {\displaystyle \frac{{\sigma }_{ds}}{E_H}}+{\displaystyle \frac{{\sigma }_{ds}}{E_K}}\left[1-\exp \left(-{\displaystyle \frac{E_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sB}}{{\eta }_B}}t& , {\sigma }_{sB}\le {\sigma }_{ds}<{\sigma }_{sN}\\ & \\ {\displaystyle \frac{{\sigma }_{ds}}{E_H}}+{\displaystyle \frac{{\sigma }_{ds}}{E_K}}\left[1-\exp \left(-{\displaystyle \frac{E_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sB}}{{\eta }_B}}t\\ +{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sN}}{E_N}}{\left[1-{\displaystyle \frac{t}{-139.97e^{\frac{D_0}{0.41}}+973.92}}\right]}^{{\displaystyle \frac{1}{r+1}}}& , {\sigma }_{ds}\ge {\sigma }_{sN}\end{array}$$

(5)

The stress tensor σ_ij and strain tensor ε_ij of the backfill can be expressed respectively as [61,62]:

$${\sigma }_{ij}={\delta }_{ij}{\sigma }_m+S_{ij}$$

(6)

$${\epsilon }_{ij}={\delta }_{ij}{\epsilon }_m+e_{ij}$$

(7)

In the equation: σ_m: the spherical (or volumetric) stress tensor; S_ij: the deviatoric stress tensor; δ_ij: Kronecker delta; ε_m: the spherical (or volumetric) strain tensor; e_ij: the deviatoric strain tensor.

Therefore, Equation (5) can be rewritten as:

$${\epsilon }_{ij}\left(t\right)=\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_m}{3K_H}}+{\displaystyle \frac{S_{ij}}{2G_H}}+{\displaystyle \frac{S_{ij}}{2G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]& , S_{ij}<{\sigma }_{sB}\\ {\displaystyle \frac{{\sigma }_m}{3K_H}}+{\displaystyle \frac{S_{ij}}{2G_H}}+{\displaystyle \frac{S_{ij}}{2G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{S_{ij}-{\sigma }_{sB}}{{\eta }_B}}t& , {\sigma }_{sB}\le S_{ij}<{\sigma }_{sN}\\ {\displaystyle \frac{{\sigma }_m}{3K_H}}+{\displaystyle \frac{S_{ij}}{2G_H}}+{\displaystyle \frac{S_{ij}}{2G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{S_{ij}-{\sigma }_{sB}}{{\eta }_B}}t& \\ +\left({\displaystyle \frac{{\sigma }_m}{3K_N}}+{\displaystyle \frac{S_{ij}-{\sigma }_{sN}}{2G_N}}\right)\cdot {\left[1-{\displaystyle \frac{t}{-139.97e^{\frac{D_0}{0.41}}+973.92}}\right]}^{{\displaystyle \frac{1}{r+1}}}& , S_{ij}\ge {\sigma }_{sN}\end{array}$$

(8)

where, K_H, G_H, and G_K are the bulk modulus of the Hooke body (GPa), the shear modulus of the Hooke body (GPa), and the shear modulus of the Kelvin body (GPa), respectively.

Since the samples were subjected to hydrostatic pressure loading, Equations (6) and (7) reduce to:

$${\sigma }_m={\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{3}}$$

(9)

$$S_{ij}={\displaystyle \frac{2\left({\sigma }_1-{\sigma }_3\right)}{3}}$$

(10)

By substituting Equations (9) and (10) into Equation (8), the three-dimensional HKBN creep equation under hydrostatic pressure loading is derived as:

$${\epsilon }_{ij}\left(t\right)=\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_{ds}+3{\sigma }_3}{9K_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]& , {\sigma }_{ds}<{\sigma }_{sB}\\ {\displaystyle \frac{{\sigma }_{ds}+3{\sigma }_3}{9K_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sB}}{3{\eta }_B}}t& , {\sigma }_{sB}\le {\sigma }_{ds}<{\sigma }_{sN}\\ {\displaystyle \frac{{\sigma }_{ds}+3{\sigma }_3}{9K_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_H}}+{\displaystyle \frac{{\sigma }_{ds}}{3G_K}}\left[1-\exp \left(-{\displaystyle \frac{G_K}{{\eta }_K}}t\right)\right]+{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sB}}{3{\eta }_B}}t& \\ +\left({\displaystyle \frac{{\sigma }_{ds}+3{\sigma }_3}{9K_N}}+{\displaystyle \frac{{\sigma }_{ds}-{\sigma }_{sN}}{3G_N}}\right)\cdot {\left[1-{\displaystyle \frac{t}{-139.97e^{\frac{D_0}{0.41}}+973.92}}\right]}^{{\displaystyle \frac{1}{r+1}}}& , {\sigma }_{ds}\ge {\sigma }_{sN}\end{array}$$

(11)

5.3. Parameters Fitting for the HKBN Constitutive Model

5.3.1. Parameters Classification and Fitting

The parameters of the HKBN creep model are classified into three categories for fitting: Threshold stresses for initiating different creep stages (σ_sB and σ_sN); Instantaneous deformation parameters (K_H and G_H) that can be directly determined; Creep deformation parameters (G_K, η_K, η_B, K_N, G_N, and r) requiring iterative fitting calculations [63,64,65,66].

(1)

Threshold stresses

As can be seen from Figure 4, the deviatoric stress level at which the backfill sample undergoes volumetric expansion ranges between 0.5~0.6 σ_c, while the stress level for volumetric dilatancy occurs between 0.8~0.9 σ_c. Taking the lower bounds of these stress intervals, we obtain σ_sB = 0.5 σ_c and σ_sN = 0.8 σ_c. For the case of D₀ = 0, the maximum deviatoric stress is 4.59 MPa, yielding σ_sB = 2.30 MPa and σ_sN = 3.67 MPa. For the case of D₀ = 0.45, the maximum deviatoric stress is 4.27 MPa, resulting in σ_sB = 2.14 MPa and σ_sN = 3.42 MPa.

(2)

Instantaneous deformation parameters

The instantaneous deformation parameters K_H and G_H can be determined by solving the system of stress–strain equations for the backfill under different stress levels. It should be noted that when the stress level is below σ_sB or above σ_sN, only a single dataset is available, making it impossible to establish a system of equations. In such cases, this study employs linear fitting to estimate the instantaneous deformation parameters. The fitted results for the instantaneous deformation parameters in the HKBN creep model are presented in

Table 5. Fitting results of instantaneous deformation parameters in the HKBN model (D₀ = 0).

	Stress	0.4 σc	0.5 σc	0.6 σc	0.7 σc	0.8 σc	0.9 σc	0.925 σc
Parameters
σds/MPa		1.84	2.30	2.75	3.21	3.67	4.13	4.25
KH/GPa		7.84	5.63	5.60	9.70	6.81	7.11	2.55
GH/GPa		3.69	2.70	2.37	1.86	1.85	1.72	1.75

,

Table 6. Fitting results of instantaneous deformation parameters in the HKBN model (D₀ = 0.35).

	Stress	0.4 σc	0.5 σc	0.6 σc	0.7 σc	0.8 σc	0.9 σc
Parameters
σds/MPa		1.80	2.26	2.71	3.16	3.61	4.06
KH/GPa		7.11	8.08	6.05	6.39	5.26	7.06
GH/GPa		3.59	3.00	2.86	2.55	2.49	1.95

,

Table 7. Fitting results of instantaneous deformation parameters in the HKBN model (D₀ = 0.40).

	Stress	0.4 σc	0.5 σc	0.6 σc	0.7 σc	0.8 σc	0.9 σc
Parameters
σds/MPa		1.72	2.16	2.59	3.02	3.45	/
KH/GPa		4.96	1.72	0.71	0.89	1.47	/
GH/GPa		0.42	0.47	0.64	0.55	0.47	/

and

Table 8. Fitting results of instantaneous deformation parameters in the HKBN model (D₀ = 0.45).

	Stress	0.4 σc	0.5 σc	0.6 σc	0.7 σc	0.8 σc	0.9 σc
Parameters
σds/MPa		1.71	2.14	2.56	2.99	3.42	/
KH/GPa		4.12	1.03	0.99	1.19	9.02	/
GH/GPa		0.43	0.56	0.55	0.52	2.13	/

.

(3)

Creep deformation parameters

The Levenberg-Marquardt algorithm (LM method) was employed to determine the partitioned creep deformation parameters. In this study, interval calculations were first performed for different stress levels, followed by linear fitting of the creep deformation parameters at remaining stress levels based on the computational results. For instance, the LM method was initially used to fit creep deformation parameters at deviatoric stress levels of 0.4 σ_c, 0.6 σ_c, 0.8 σ_c, 0.9 σ_c, and 0.925 σ_c, respectively. Subsequently, these results were utilized to derive the creep deformation parameters at intermediate stress levels of 0.5 σ_c and 0.7 σ_c through additional fitting procedures. This approach not only ensured nonlinear fitting of creep deformation parameters across all stages (decelerated creep, steady-state creep, and accelerated creep) but also validated the model’s rationality.

The fitting results of creep deformation parameters for the HKBN model under deviatoric stress levels of 0.4 σ_c, 0.6 σ_c, 0.8 σ_c, 0.9 σ_c, and 0.925 σ_c with initial damage levels of 0, 0.35, 0.40, and 0.45 are presented in

Table 9. Fitting results of creep deformation parameters in the HKBN model (D₀ = 0).

	Stress	0.4 σc	0.6 σc	0.8 σc	0.9 σc	0.925 σc
Parameters
GK/GPa		7.42	8.82	16.54	13.25	14.06
ηK/GPa·h		1.45	0.94	1.077	0.67	0.73
ηB/GPa·h		/	8.45	18.04	12.15	9.03
KN/GPa		/	/	18.07	20.60	14.79
GN/GPa		/	/	/	11.33	5.77
r		/	/	0.85	0.76	0.61

,

Table 10. Fitting results of creep deformation parameters in the HKBN model (D₀ = 0.35).

	Stress	0.4 σc	0.6 σc	0.8 σc	0.9 σc
Parameters
GK/GPa		6.12	8.47	15.04	13.50
ηK/GPa·h		1.137	0.93	0.90	0.71
ηB/GPa·h		/	7.69	17.65	11.39
KN/GPa		/	/	16.98	17.99
GN/GPa		/	/	/	7.38
r		/	/	0.265	0.62

,

Table 11. Fitting results of creep deformation parameters in the HKBN model (D₀ = 0.40).

	Stress	0.4 σc	0.6 σc	0.8 σc	0.9 σc
Parameters
GK/GPa		4.69	9.40	15.71	13.22
ηK/GPa·h		0.80	1.50	0.91	0.72
ηB/GPa·h		/	7.18	16.54	9.14
KN/GPa		/	/	15.66	16.98
GN/GPa		/	/	/	6.45
r		/	/	0.73	0.16

and

Table 12. Fitting results of creep deformation parameters in the HKBN model (D₀ = 0.45).

	Stress	0.4 σc	0.6 σc	0.8 σc	0.9 σc
Parameters
GK/GPa		4.95	7.38	11.01	/
ηK/GPa·h		0.85	1.33	1.18	/
ηB/GPa·h		/	9.03	9.07	/
KN/GPa		/	/	4.90	/
GN/GPa		/	/	/	/
r		/	/	0.35	/

, with corresponding fitting curves illustrated in Figure 8. The coefficient of determination (R²) between fitted data and testing data exceeded 0.98 in all cases, demonstrating excellent agreement.

5.3.2. Parameter Validation

The validation results of model parameters at deviatoric stress levels of 0.5 σ_c and 0.7 σ_c were obtained through linear fitting methodology, as summarized in

Table 13. Validation results of creep deformation parameters in the HKBN model.

	Stress	D0 = 0		D0 = 0.35		D0 = 0.40		D0 = 0.45
Parameters		0.5 σc	0.7 σc	0.5 σc	0.7 σc	0.5 σc	0.7 σc	0.5 σc	0.7 σc
GK/GPa		5.55	9.60	8.79	9.52	5.36	8.51	4.60	6.16
ηK/GPa·h		1.03	0.60	3.27	0.87	1.13	0.58	0.81	0.63
ηB/GPa·h		/	10.89	/	11.54	/	9.32	/	9.36

. These determined parameters were subsequently incorporated into the HKBN constitutive model, with the corresponding validation results graphically presented in Figure 9.

From a correlation perspective, the squared coefficient of determination (R²) between the validated and experimental values was no less than 0.80. Comparative analysis revealed that during the initial creep stage, the validated values were systematically lower than the experimental measurements. This discrepancy may be attributed to two primary factors: (1) the HKBN constitutive model characterizes the complete creep deformation behavior of backfill materials without accounting for stress fluctuation effects on deformation, and (2) the GDS testing system requires approximately 120 s for stress adjustment before entering the stress-holding phase, during which the strain rate is significantly higher than during steady-state creep. Consequently, the validated values exhibit a temporal lag relative to experimental data.

From a correlation standpoint, the coefficient of determination (R²) between the validated and experimental values was high. Notably, after 20~40 min of creep, the two datasets exhibited strong convergence. These findings substantiate the scientific value of the HKBN model for investigating critical engineering problems pertaining to creep deformation mechanisms and long-term bearing behavior in high-volume fly ash–cement backfill materials.

5.3.3. Parameters Sensitivity Analysis

To validate the accuracy and efficacy of the established three-dimensional HKBN model, creep parameter inversion was performed on the accelerated creep curves corresponding to three distinct stress levels (0.925 σ_c, 0.9 σ_c, and 0.8 σ_c) as presented in Figure 8. The inversion results, depicted in Figure 10 and tabulated in

Table 14. Parameter inversion results in the HKBN model.

	Parameters	GK/GPa	ηK/GPa·h	ηB/GPa·h	KN/GPa	GN/GPa	r
Stress
0.8 σc		13.65	0.98	9.12	14.80	6.59	0.51
0.9 σc		13.79	0.75	9.22	14.99	6.32	0.42
0.925 σc		13.87	0.73	9.06	15.12	6.02	0.62

, demonstrate that the three-dimensional HKBN model exhibits robust applicability to backfill samples with varying initial damage states and subjected to different loading stress levels.

As indicated by Equation (11), the coefficient and exponent of the time variable t are the primary factors influencing the model’s effectiveness. For clarity in presentation, two parameters—α and β—are introduced, where α represents the ratio of G_K to η_K, and β is defined as the reciprocal of (r + 1). Since the α value significantly affects the deceleration creep stage of the sample, a sensitivity analysis was conducted. Based on the data presented in Table 9, Table 10, Table 11 and Table 12, the α value ranges from 5 to 20. Accordingly, the creep curves were generated for α values of 5, 10, 15, and 20, as illustrated in Figure 11.

The results demonstrate that as α increases, the sample transitions into the deceleration creep stage more rapidly. However, this parameter does not influence the instantaneous deformation or the subsequent steady-state creep behavior. These findings confirm that the HKBN model effectively captures both the deceleration and steady-state creep stages of the sample.

Equation (11) reveals that the β parameter governs the accelerated creep stage of the specimen. As documented in Table 9, Table 10, Table 11 and Table 12, the parameter R exhibits a range of 0.15 to 0.65. Consequently, corresponding β values were calculated as 0.91, 0.77, 0.67, and 0.59 for r values of 0.1, 0.3, 0.5, and 0.7, respectively. The characteristic creep curves during the accelerated stage are presented in Figure 12.

The results demonstrate higher β values correspond to shorter accelerated creep transition periods. This finding confirms that the HKBN model effectively characterizes accelerated creep behavior across varying stress levels through appropriate β value modulation.

The initial damage parameter (D₀) significantly influences the creep behavior of specimens. Figure 13 presents the accelerated creep stage characteristics for D₀ values of 0, 0.35, 0.40, and 0.45. The comparative analysis reveals that damaged samples exhibit a different creep pattern compared to intact samples. The characteristic steady-state creep phase is virtually absent, with samples progressing directly to accelerated creep and subsequent failure.

Furthermore, when compared with the results shown in Figure 12, the initial damage D₀ demonstrates a more pronounced effect on creep characteristics than the material property parameter r. This finding underscores the critical role of pre-existing damage in determining the creep response and failure mechanism of the materials.

6. Discussion

(1) To characterize the creep behavior of backfill materials under CMB disturbance conditions, cyclic loading–unloading paths with explicit physical significance were designed for backfill samples. An initial damage variable was introduced to quantitatively represent the disturbance effects induced by adjacent mining roadways. Through a comprehensive analysis of the axial deformation characteristics, volumetric strain evolution, and volumetric strain rate patterns of the backfill materials, an advanced HKBN constitutive model was developed specifically for high-volume fly ash–cement backfills. Systematic parameter identification and validation were subsequently conducted.

In combination with Figure 4, it can be observed that specimens with different initial damage levels exhibit distinct differences in the curve patterns of accelerated creep. For the sample with D₀ = 0, the three stages of creep only become apparent when the stress level reaches 0.925 σ_c. The durations of these stages are approximately 1200 s, 4600 s, and 900 s, respectively, indicating that the sample undergoes a relatively prolonged period of steady-state creep before failure. As the initial damage increases, a stress level of 0.9 σ_c is sufficient to induce sample failure. Comparing Figure 4b,c, it is found that when D₀ = 0.40, the sample already enters the accelerated creep phase during the stress adjustment stage at 0.9 σ_c. Therefore, when acquiring accelerated creep curves, the magnitude of stress adjustment requires a more refined design. A 10% stress increment may lead to missing the optimal curve morphology.

As can be seen from Figure 5, the volumetric change of the samples consistently follows the sequence of “compression → expansion → dilatancy → failure.” Comparing Figure 5a–d, it is evident that the volumetric compression phase always occurs at stress levels no higher than 0.5 σ_c. This suggests that the initial damage has an insignificant influence at relatively low stress levels (maybe not exceeding 0.5 σ_c). The underlying mechanism of this effect requires further experimental validation.

(2) The CMB demonstrates distinct advantages through its adaptable coal extraction techniques and constrained backfill space utilization, effectively compensating for the insufficient early-age strength of high-volume fly ash-based backfills (fly ash content > 70%). In practical applications, the mining area is frequently partitioned into multiple extraction phases (typically 2~4 or more phases), with staggered development of mining roadways. While such operational configurations may alter the stress distribution within backfill materials and consequently modify their creep behavior (resulting in deviations from the conclusions presented in this study), the fundamental research methodology and analytical framework established herein remain fully applicable.

(3) The parameter identification for the creep model was conducted through a categorized analytical approach. The threshold stresses and instantaneous deformation parameters were determined as priority parameters, followed by the determination of creep deformation parameters through optimized fitting procedures. This hierarchical identification strategy significantly reduces computational complexity while maintaining high accuracy (R² > 0.98). Furthermore, an interval-fitting methodology was implemented for different stress levels, whereby model parameters at intermediate stress levels were extrapolated from fitted results and subsequently validated against experimental data. This approach provides support for the reliability of fitting parameters.

(4) This study employs an integer-order creep model. During the transition from instantaneous deformation to creep deformation, certain deviations are observed. However, these deviations progressively diminish with increasing creep duration and eventually approach zero. Comparative studies demonstrate that fractional-order models exhibit superior performance when higher precision is required. Nevertheless, the integer-order model offers distinct practical advantages, including fewer model parameters that facilitate both computation and application, while still maintaining satisfactory correlation (R² > 0.80). Consequently, the integer-order model presents greater engineering value for investigating the creep behavior of backfill materials.

(5) The material category is also one of the important factors influencing the creep characteristics of backfill. The research object of this paper is a fly ash–cement backfill material with a blending ratio of 73%, which belongs to brittle materials. For backfill materials such as loose, paste, and high-water-content types, their load-bearing mechanisms differ, and their creep characteristics may also vary. For example, in loose backfill materials using gangue as aggregate, the bonding force between particles has minimal impact on deformation. The creep behavior primarily reflects the macroscopic mechanical response of mesoscopic movements such as squeezing, interlocking, and crushing among gangue particles. Therefore, this paper has limitations in addressing the creep characteristics of loose backfill materials.

7. Conclusions

Based on the engineering context and mechanical conditions of CMB, this study conducted creep tests on high-volume coal-based solid waste backfill materials using fly ash as a representative raw material. The investigation focused on the creep behavior of backfill materials under mining–backfilling cyclic disturbances, and a corresponding creep constitutive model was established. The principal research findings are as follows:

(1) A high-volume fly ash–cement backfill material compatible with CMB requirements was prepared, with a fly ash content of 73.0%. The cyclic loading–unloading mechanical effects exerted by CMB on backfill were simulated through experimental procedures involving constant confining pressure with multiple axial loading–unloading cycles. Based on the relationship between the dissipated energy and work done by the press machine, a computational method for determining the initial damage of backfill materials was proposed. Using four phases as examples, the study investigated the initial damage characteristics of backfill materials during different mining phases.

(2) Creep tests were conducted on high-volume fly ash–cement backfill samples with initial damage levels of 0, 0.35, 0.40, and 0.45, respectively. The weakening effects of initial damage on the long-term bearing capacity of backfill were systematically investigated through axial strain, volumetric deformation, and volumetric strain rate analyses. The key findings include: (a) The failure stress level decreased from 0.925 σ_c to 0.8 σ_c; (b) The total creep duration was reduced from 835 min to 555 min; (c) The proportion of creep deformation to total deformation increased from 10.24% to 17.40%.

(3) The HKBN constitutive model, capable of describing the entire creep process of backfill, was established by improving a nonlinear plastic damage component. The model parameters were categorized into the threshold stress for initiation, instantaneous deformation parameters, and creep deformation parameters. These parameters were solved using the experimental analysis method, the simultaneous equations method, and the Levenberg-Marquardt method, respectively. Linear fitting was then employed to validate the parameters. The fitting results showed an R² no less than 0.98, while the validation results exhibited an R² no less than 0.80.

The results provide deformation-based criteria for roof management and overlying strata movement control in CMB, offering scientific value for studies on overburden damage-reduction mining, surface deformation control, and shallow water resource protection. Furthermore, this study demonstrates that appropriate coal mining techniques can compensate for the early-stage strength deficiency of high-volume fly ash backfill materials, thereby establishing a theoretical foundation for enhancing the utilization efficiency of coal-based solid waste resources.