Triaxial Creep Behavior of Gangue–Gypsum Cemented Backfill and Applicability Verification of the Burgers Model

Abstract

Gangue backfilling has become an important technique for promoting environmentally friendly and low-carbon coal mining. The long-term creep behavior of cemented backfill plays a critical role in maintaining stope stability and controlling surface subsidence during long-term service. Although considerable research has been conducted on cemented tailings backfill, systematic investigations on the triaxial creep evolution, long-term strength characteristics, confining pressure effects, and the applicability of the classical Burgers model for gangue–gypsum cemented backfill under engineering-relevant confining pressures remain limited. In this study, the experimental scheme was designed based on field monitoring data from practical backfill mining operations, which indicate that the in situ backfill generally remains stable without significant deformation or instability under normal working conditions. Multi-stage loading triaxial creep tests were conducted on gangue–gypsum cemented backfill under confining pressures of 1, 2, 3, and 4 MPa. The creep deformation characteristics were analyzed using Chen’s superposition method, while the long-term strength was computed via inflection point method of isochronous stress–strain curves. The parameters of the Burgers creep model were identified using the Levenberg–Marquardt optimization algorithm, and numerical verification was performed using FLAC3D. Our findings demonstrate that the creep deformation process of the backfill consists of three typical stages: instantaneous deformation, attenuated creep, and steady-state creep, and no accelerated creep was observed within the applied stress range. The absolute creep strain surges nonlinearly with increasing stress level (S_L), whereas higher confining pressure significantly suppresses the creep response of the material. Within the investigated stress range, the backfill exhibits mainly linear viscoelastic behavior, and its critical long-term strength is not less than 0.9 times the failure deviatoric stress (qf). Although confining pressure enhances the long-term strength, the strengthening effect weakens as the confining pressure increases. Model fitting outcomes imply that Burgers model precisely describes the creep behavior of gangue–gypsum cemented backfill under all test conditions, with correlation coefficients (R²) exceeding 0.97. The identified parameters show systematic variation with S_L, reflecting stiffness degradation and viscous evolution during loading. Numerical simulation results agree well with the experimental data, providing theoretical guidance for mixture proportion optimization, long-term stability evaluation, and stope support parameter design in gangue backfill mining engineering.

1. Introduction

After coal extraction, untreated underground goafs may lead to a series of serious environmental and geological problems in mining areas, such as ground subsidence, degradation of groundwater systems, and other related hazards. These issues pose long-term risks to the ecological environment and threaten the sustainable development of mining regions [1,2]. Gangue backfill mining technology provides an effective solution to these problems by transporting coal gangue—solid waste produced during coal mining—back into underground voids. Through this approach, underground spaces generated by mining activities can be filled while simultaneously reducing surface gangue accumulation and the associated environmental pollution and land occupation. Consequently, gangue backfilling technology is widely regarded as an important technical pathway for achieving green mining and promoting sustainable development in the coal industry [3,4].

In practical mining operations, the cemented backfill interacts with the roof and floor strata to form a coupled bearing structure. Under the continuous load transferred from overlying rock layers, the backfill remains subjected to long-term compressive stress conditions [5,6,7]. Compared with surrounding rock masses, cemented backfill generally exhibits relatively lower stiffness and strength. This mechanical contrast often causes stress redistribution among backfill and surrounding rock mass [8,9]. When long-term in situ stress is combined with mining disturbances, the backfill gradually experiences time-dependent deformation, commonly referred to as creep. This rheological behavior represents a key mechanical factor controlling the long-term stability of underground backfill structures [10,11,12]. Engineering practice and laboratory investigations have demonstrated that excessive creep deformation in underground supporting systems may fail to effectively control strata movement and surface subsidence. Furthermore, the progressive accumulation of internal damage may trigger accelerated creep failure, eventually causing instability of the backfill body and even severe mining-related disasters [13,14]. Therefore, investigating the creep characteristics of cemented coal gangue backfill is essential for evaluating the long-term stability of backfill mining systems.

Creep testing is widely recognized as a fundamental method for investigating the creep behavior of geotechnical materials. Common experimental approaches include uniaxial compression creep tests, triaxial creep tests, and multi-stage loading or unloading creep tests [10,15,16]. Previous studies have examined creep properties of distinct rock types and the effects of factors such as moisture content, confining pressure, initial damage, and freeze–thaw cycles on rock creep behavior [17,18,19,20,21]. Existing research indicates that the creep process of most geotechnical materials typically develops through three characteristic stages: instantaneous deformation, steady-state creep, and accelerated creep. In addition, the long-term strength of rocks is generally estimated to be approximately 60%–80% of their short-term compressive strength [17,18]. With increasing water content, higher degrees of initial damage, or repeated freeze–thaw cycles, the rock mechanical characteristics tend to deteriorate gradually. As a result, creep strain and creep rate increase significantly, while the failure time is reduced [19,20,21,22,23]. To accurately characterize the creep response of geotechnical materials, a variety of rheological models have been proposed. These models can generally be categorized into empirical models, theoretical models, and component models [24,25]. Among them, component models—constructed using basic rheological elements such as elastic, viscous, and plastic components—are widely used because of their clear physical meaning, relatively simple parameter systems, and intuitive mechanical interpretation [24,25]. Previous studies have confirmed that creep models established through combinations of these basic components are capable of effectively capturing the deformation characteristics of many geotechnical materials [24]. The classical Burgers model, consisting of a Maxwell model serially arranged with a Kelvin model, represents a typical four-element rheological model. This model can simultaneously describe instantaneous elastic deformation and steady-state creep behavior and has therefore been widely applied in rheological analyses of rock engineering problems. Nevertheless, given the inherent heterogeneity of geotechnical materials, the mechanical properties involved in creep deformation often exhibit nonlinear evolution as damage accumulates over time [26]. For this reason, numerous nonlinear rheological models have been proposed by incorporating nonlinear components, damage mechanics, or aging damage theory to better capture the accelerated creep stage.

Although existing research has significantly improved the understanding of the mechanical properties of cemented backfill materials and the creep behavior of geotechnical materials, several key issues remain insufficiently explored. First, most studies on cemented backfill focus primarily on short-term mechanical properties and failure characteristics under different material proportions or loading conditions [8,9,10,11,12,13,14]. In contrast, analysis for the long-term creep behavior of cemented coal gangue backfill, particularly under triaxial compression conditions that more closely represent the actual in situ stress environment, remains relatively limited. Second, many previous creep studies mainly concentrate on intact rock materials affected by factors such as water content, initial damage, or freeze–thaw cycles [17,18,19,20,21,22,23,26]. The deformation evolution, damage mechanisms, and nonlinear rheological responses of cemented coal gangue backfill subjected to long-term loading have not yet been fully clarified. Third, most existing rheological models were originally developed for natural rock materials, and the validity of the classical Burgers model to describe the complete creep process of cemented coal gangue backfill still requires further experimental verification and theoretical analysis.

To address these issues, this study investigates cemented coal gangue paste backfill as the research object. A series of triaxial multi-stage loading creep tests were carried out with different confining pressure environments to explore the creep behavior and deformation evolution characteristics of the backfill material. Furthermore, the effectiveness of the classical Burgers model for depicting the creep response of cemented coal gangue backfill was examined. The outcomes of this research provide useful experimental evidence and theoretical support for parameter design in cemented backfill mining, along with evaluating the long-term integrity of backfill bodies in underground goafs under complex geological and high in situ stress conditions in deep coal mines.

2. Specimens and Test Methods: Production in Dezhou City, Shandong Province, China

2.1. Test Materials and Equipment

The gangue–plaster cemented backfill used in this study was prepared using crushed coal gangue, fly ash, and cementitious materials. The coal gangue was collected from the Zhaoguan Coal Mine located in Qihe. Its basic physical and mechanical characteristics are summarized in

Table 1. Physical–mechanical parameters of coal gangue.

Bulk Density (g/cm3)	Real Density (g/cm3)	Moisture Content (%)	Loss on Ignition (%)	Natural Repose Angle (°)	Porosity (%)	Water Absorption Rate (%)
1.78	2.4885	3.11	3.47	44	28.47	4.2

. Prior to specimen preparation, the gangue was processed through a two-stage crushing system in the mine, and the resulting material had a maximum particle size smaller than 15 mm, which meets the requirement for use as aggregate in the preparation of gangue–plaster cemented backfill. The particle gradation of the crushed coal gangue is outlined in

Table 2. Test results of coal gangue screening experiment.

Particle Size Range (mm)	Trial 1 (1000 g)	Trial 2 (1000 g)	Trial 3 (1000 g)	Average Mass (1000 g)	Average Screen Residue (%)	Average Cumulative Screen Residue (%)
5.0~15.0	214	223	236	224	22.4
3.0~5.0	180	194	152	175	17.5	39.9
2.0~3.0	124	122	118	121	12.1	52.0
1.0~2.0	148	150	152	150	15.0	67.0
0.5~1.0	126	119	129	125	12.5	79.5
0.15~0.5	134	123	138	132	13.2	92.7
0.075~0.15	48	50	55	51	5.1	97.8
≤0.075	26	19	20	22	2.2	100

. Specifically, particles larger than 5 mm account for 22.4%, particles ranging from 5 mm to 0.15 mm constitute 70.3%, and particles smaller than 0.075 mm represent 2.2% of the total mass. Fly ash used in the experiment was obtained from a nearby thermal power plant, and its corresponding physical properties are detailed in

Table 3. Physical and mechanical parameters of fly ash.

Bulk Density (g/cm3)	Real Density (g/cm3)	Moisture Content (%)	Loss on Ignition (%)	Natural Repose Angle (°)	Porosity (%)	Residue on 0.045 mm Sieve (%)
0.65	2.2961	2.46	16.19	37	71.69	72.82

. Ordinary Portland cement (P·O 42.5) was applied as the cementitious binder in the preparation of the backfill specimens.

The triaxial creep experiments were carried out using a TAW-2000 rock triaxial testing system, Changchun Chaoyang Testing Machine Co., Ltd., Changchun, China (Figure 1). The device is computer-controlled and allows automatic data acquisition and processing during the testing process. It is equipped with independent servo-control units for axial loading, confining pressure, and seepage pressure, which ensures high loading precision and sensitivity. By applying different confining pressures, the apparatus can accurately determine the fundamental mechanical properties of the cemented backfill specimens. The testing results obtained from this system provide reliable experimental support for evaluating the mechanical stability of gangue–plaster cemented backfill under various confining pressure conditions.

2.2. Triaxial Creep Test Program

Based on the proportioning design of the gangue–plaster cemented backfill, the mixture ratio adopted in this study is summarized in

Table 4. Mixture ratio of filling material.

Gangue (%)	Fly Ash (%)	Cement (%)	Total (%)
57	14	8	79

. Cylindrical specimens with dimensions of φ50 × 100 mm were prepared for the laboratory tests. After molding, all specimens were cured under standard curing conditions, with temperature maintained at 20 ± 2 °C and relative humidity > 95% over 28 days. Figure 2 illustrates the prepared gangue–plaster cemented backfill specimens.

Before conducting the formal triaxial creep experiments, conventional triaxial compression tests were first performed on the gangue–gypsum cemented backfill under predetermined testing conditions, following the procedures based on “Standard for Geotechnical Testing Method” (GB 50123-2019) [27]. During these preliminary tests, four confining pressure levels were applied. For each confining pressure condition, three parallel specimens were prepared. After the tests were completed, abnormal or invalid data were removed through data screening and discreteness analysis. Given the stress–strain results of the valid specimens under same confining pressure, representative stress–strain curves were obtained using a point-by-point arithmetic averaging approach. Using these averaged curves, the deviatoric stress–axial strain relationships of the cemented backfill under each confining pressure were determined. Subsequently, the corresponding failure deviatoric stress (qf) of the specimens was identified (Figure 3).

In laboratory investigations of rock creep behavior, two common loading strategies are typically adopted: single-stage loading and multi-stage loading. In the single-stage loading approach, specimens of the same rock type are tested separately under identical experimental conditions but at different stress levels. This method produces a series of strain–time curves corresponding to different stress states. From a theoretical perspective, such a testing scheme can reliably represent the creep characteristics of the material. However, conducting separate tests for each stress level requires a large number of specimens and significantly increases the testing duration, which limits its practical applicability. In contrast, the multi-stage loading method involves applying increasing stress levels sequentially to the same specimen. After the specimen reaches the steady-state creep stage or after maintaining the load for a specified duration at a given stress level, the applied stress is surged to the next level. This procedure continues until noticeable deformation or failure occurs. In principle, creep data obtained from multi-stage loading tests are commonly processed using the linear superposition principle to derive creep curves corresponding to individual stress levels. Nevertheless, due to the inherent nonlinear rheological behavior of rocks, the experimental creep curves do not always strictly satisfy the assumptions required for linear superposition. Despite this limitation, the multi-stage loading method offers clear practical advantages. By using a single specimen to obtain creep responses under several stress levels, the amount of experimental information can be significantly increased while the overall testing period can be effectively reduced. Consequently, this approach has become the most widely adopted method in laboratory rheological experiments. For this reason, the multi-stage loading scheme was selected for the proposed creep tests.

In the present experiment, the multi-stage loading creep test was designed with five loading stages. The stress level (S_L) is introduced in the following analysis to represent the loading intensity, and its definition is provided in Equation (1)

$$S_L={\displaystyle \frac{q_i}{q_f}}$$

(1)

where q_i indicates the deviatoric stress at a certain loading stage (i = 1~5, q_i = S_L (σ₁ − σ₃)).

In this study, the selected S_L were 0.5, 0.6, 0.7, 0.8, and 0.9. The corresponding q_i was calculated according to Equation (1). For each loading level, the stress was maintained for 2 h to ensure sufficient development of creep deformation [28].

Paste backfill mining techniques were extensively implemented in coal mines located in central and Eastern China. Field monitoring data collected from these mining operations indicate that gangue-cemented paste backfill (GCPB) placed in underground goafs generally remains stable during service, with no evidence of large-scale deformation or structural instability. In addition, in situ strain monitoring results reveal that only limited deformation occurs within the backfill body under normal operating conditions. As an example, the field monitoring data obtained from the No. 2351 paste backfill working face of Daizhuang Coal Mine in Shandong Province are depicted in Figure 4.

Accordingly, instability failure of the cemented backfill was not considered within the scope of this work. Instead, the study focuses on the creep behavior of the material during the stable pre-failure stage, which is essential for evaluating its long-term deformation characteristics. Therefore, when S_L = 0.9, the creep test was terminated after maintaining constant loading for 2 h. The loading procedure is presented in Figure 5, and the corresponding experimental scheme is outlined in

Table 5. Triaxial creep test scheme of gangue paste filling body.

Specimen	σ3 (MPa)	σ1 (MPa)	σ1 − σ3 (MPa)
R-01	1.0	11.0	10.0
		13.0	12.0
		15.0	14.0
		17.0	16.0
		19.0	18.0
R-02	2.0	14.0	12.0
		16.4	14.4
		18.8	16.8
		21.2	19.2
		23.6	21.6
R-03	3.0	17.0	14.0
		19.8	16.8
		22.6	19.6
		25.4	22.4
		28.2	25.2
R-04	4.0	19.5	15.5
		22.6	18.6
		25.7	21.7
		28.8	24.8
		31.9	27.9

.

3. Test and Analysis Outcomes

3.1. Triaxial Compression Creep Properties

The pseudo-triaxial creep data from the multi-stage loading tests were processed using Chen’s superposition method [29]. Based on this approach, the creep curves of the gangue–gypsum cemented backfill subjected to confining pressures of 1, 2, 3, and 4 MPa were reconstructed (Figure 6). Despite the differences in confining pressure, the creep deformation behavior of the specimens exhibits similar evolutionary characteristics. The entire creep process can be categorized into three typical stages: instantaneous deformation, decelerating creep, and steady-state creep. Immediately after each S_L is applied, the specimen experiences a rapid elastic response, resulting in instantaneous deformation. The magnitude of this instantaneous strain increases progressively with increasing S_L. During the subsequent decelerating creep stage, axial strain continues to accumulate while the creep rate declines. This behavior is associated with internal stress redistribution within the cemented skeleton and the gradual compaction of micro-defects and pores. As the creep process develops further, the specimen enters the steady-state creep stage. At this stage, the internal stress state of the cemented structure becomes relatively stable, and the creep rate approaches a nearly constant and low value. With increasing loading time, the axial deformation gradually stabilizes, and no signs of accelerated creep or structural instability are observed within the tested stress range.

To quantitatively evaluate the long-term creep deformation of the cemented backfill, the method from Sun et al. [30] was applied. The absolute creep strain is obtained by subtracting the instantaneous strain from the final strain, generated at the beginning of each loading stage. The calculated absolute creep strains corresponding to different S_L are illustrated in Figure 7. At S_L = 0.5, 0.6, 0.7, 0.8, and 0.9, the absolute creep strains are 0.33, 0.44, 0.65, 1.00, and 1.60%, respectively. These results indicate a clear increasing pattern of absolute creep strain with the surge of S_L, suggesting that the time-dependent deformation of the cemented backfill becomes more pronounced under higher loading conditions. Figure 7 also depicts the correlation between absolute creep strain and S_L at distinct confining pressures.

3.2. Isochronous Stress–Strain Curves

To further analyze the creep behavior and determine the long-term strength of the gangue–gypsum cemented backfill [31], stress–strain data recorded at various time intervals (0, 15, 30, 45, 60, 75, 90, 105, and 120 min) were extracted to construct isochronous stress–strain curves under various confining pressures (Figure 8).

The obtained curves display similar evolutionary patterns under different confining pressure conditions. At relatively low S_L, the curves corresponding to different holding times are closely distributed, indicating that the axial strain varies only slightly with time. As the S_L increases step by step, the entire group of curves gradually shifts toward the axial strain direction. Under same deviatoric stress, the axial strain becomes larger with increasing holding time, revealing a clear time-dependent deformation characteristic of the material. Throughout the investigated stress range, the isochronous curves at different time intervals remain approximately linear and nearly parallel, and the spacing between adjacent curves does not expand sharply as the S_L increases. This behavior suggests that the axial strain develops almost linearly with deviatoric stress. Consequently, the deformation response of the specimen within the test range is dominated by linear elastic and linear viscoelastic behavior, and no obvious nonlinear deformation associated with viscoplastic yielding is observed.

Based on the above observations, it can be inferred that within the applied stress range and loading duration, irreversible viscoplastic flow does not occur in the gangue–gypsum cemented backfill. The critical long-term strength is therefore estimated to be not less than 0.9 times the qf. In addition, as the confining pressure surges, the slope of the isochronous curves becomes steeper while the spacing between curves corresponding to different times becomes smaller. This phenomenon indicates that confining pressure effectively restrains the time-dependent deformation of the backfill and contributes to the improvement in its long-term mechanical stability.

The long-term strength of the backfill, defined as the deviatoric stress at which the stress–strain relationship changes from linear to nonlinear, was determined via the isochronous curve inflection-point method [32]. The calculated long-term strengths under confining pressures of 1, 2, 3, and 4 MPa are 17.0, 21.2, 25.4, and 28.8 MPa, respectively. Compared with the previous confining pressure level, the increments are 24.7, 19.8, and 13.4%, indicating that although confining pressure enhances the long-term strength of the backfill, the magnitude of this strengthening effect gradually diminishes with increasing confining pressure.

From an engineering perspective, these results suggest that in deep stopes subjected to high confining pressure, the potential improvement in long-term strength achieved solely through confining pressure is relatively limited. Therefore, enhancing the intrinsic load-bearing capacity of the backfill material—such as by optimizing the cement content or aggregate gradation—becomes increasingly important. In contrast, in shallow mining areas or regions with relatively weak lateral confinement, the strengthening effect of confining pressure is more pronounced. This characteristic can be utilized to appropriately control the amount of cementitious materials while still satisfying the requirements for long-term stability.

4. Determination of Creep Parameters for Gangue–Paste Cemented Backfill

4.1. Burgers Constitutive Model

The Burgers rheological model is composed of a Maxwell element serially connected with a Kelvin element (Figure 9). The corresponding 1D creep constitutive equation of Burgers model is given below [6]:

$$\epsilon ={\displaystyle \frac{{\sigma }_0}{E_{\mathrm{M}}}}+{\displaystyle \frac{{\sigma }_0}{{\eta }_{\mathrm{M}}}}t+{\displaystyle \frac{{\sigma }_0}{E_{\mathrm{K}}}}(1-{\mathrm{e}}^{-{\displaystyle \frac{E_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}t})$$

(2)

where $E_{\mathrm{M}}$ indicates instantaneous elastic modulus and $E_{\mathrm{K}}$ indicates the viscoelastic modulus; while ${\eta }_{\mathrm{M}}$ and ${\eta }_{\mathrm{K}}$ indicate the viscosity coefficients.

4.2. 3D Creep Equation

Under 3D stress conditions, the stress tensor ${\sigma }_{ij}$ of rock mass can be separated into two components: the deviatoric stress tensor $S_{ij}$ and the spherical (hydrostatic) stress tensor ${\sigma }_m$. In a similar manner, the strain tensor ${\epsilon }_{ij}$ can also be expressed as the combination of two parts, namely the deviatoric strain tensor $e_{i}j$ and the spherical strain tensor ${\epsilon }_m$. The corresponding decomposition relationship can therefore be written as follows:

$$\left\{\begin{array}{l}{\sigma }_{ij}=S_{ij}+{\delta }_{ij}{\sigma }_m\hfill \\ {\epsilon }_{ij}=e_{ij}+{\delta }_{ij}{\epsilon }_m\hfill \end{array}\right.$$

(3)

where ${\delta }_{ij}$ indicates the Kronecker delta. The ${\sigma }_m$ and ${\epsilon }_m$ are given by

$$\left\{\begin{array}{l}{\sigma }_m={\displaystyle \frac{1}{3}}({\sigma }_1+{\sigma }_2+{\sigma }_3)={\displaystyle \frac{1}{3}}{\sigma }_{ii}\hfill \\ {\epsilon }_m={\displaystyle \frac{1}{3}}({\epsilon }_1+{\epsilon }_2+{\epsilon }_3)={\displaystyle \frac{1}{3}}{\epsilon }_{ii}\hfill \end{array}\right.$$

(4)

Under 3D stress conditions, the elastic component follows the generalized Hooke’s law as follows:

$$\left\{\begin{array}{l}{S}_{ij}=2G{e}_{ij}\hfill \\ {\sigma }_m=3K{\epsilon }_m\hfill \end{array}\right.$$

(5)

where $G$ indicates the shear modulus and $K$ indicates the bulk modulus.

In general, deviatoric stress is responsible for generating creep deformation, whereas the spherical (hydrostatic) stress component mainly produces elastic volumetric deformation. By assuming that the rock behaves as an isotropic material, the 3D creep constitutive equation of Burgers model is shown below:

$${\epsilon }_{ij}\left(t\right)={\displaystyle \frac{{\sigma }_m}{3K}}+{\displaystyle \frac{S_{ij}}{2G_{\mathrm{M}}}}+{\displaystyle \frac{S_{ij}}{2{\eta }_{\mathrm{M}}}}t+{\displaystyle \frac{S_{ij}}{2G_{\mathrm{K}}}}(1-{\mathrm{e}}^{-{\displaystyle \frac{G_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}t})$$

(6)

For a conventional triaxial compression creep test, the stress condition ${\sigma }_2={\sigma }_3$ is satisfied. By substituting this condition into Equation (6), the axial and radial creep constitutive equations of the Burgers model under constant confining pressure can be derived as follows:

$${\epsilon }_1(t)={\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_{\mathrm{M}}}}+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3{\eta }_{\mathrm{M}}}}t+{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_{\mathrm{K}}}}\left(1-{\mathrm{e}}^{-{\displaystyle \frac{G_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}t}\right)$$

(7)

$${\epsilon }_3\left(t\right)={\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}}-{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{6G_{\mathrm{M}}}}-{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{6{\eta }_{\mathrm{M}}}}t-{\displaystyle \frac{{\sigma }_1-{\sigma }_3}{6G_{\mathrm{K}}}}\left(1-{\mathrm{e}}^{-{\displaystyle \frac{G_{\mathrm{K}}}{{\eta }_{\mathrm{K}}}}t}\right)$$

(8)

4.3. Model Parameter Identification Through Experimental Outcomes

To examine the feasibility of Burgers creep model, the experimental creep curves obtained under different test conditions were fitted using MATLAB vR2025b. During the parameter identification process, the model parameters were obtained via inverse analysis through the least squares method combined with Levenberg–Marquardt optimization algorithm. The resulting fitting parameters and the corresponding comparison curves are presented in

Table 6. Parameter identification on creep model.

Test Conditions	SL	K (GPa)	GM (GPa)	${\mathit{\eta }}_{\mathbf{M}}$ (GPa·h)	GK (GPa)	${\mathit{\eta }}_{\mathbf{K}}$ (GPa·h)	Correlation Coefficient R2
σ3 = 1 MPa	0.5	0.572	1.903	33.333	1.515	0.291	0.979
	0.6	0.748	1.706	40.000	8.000	0.824	0.989
	0.7	0.912	1.488	23.333	7.778	1.054	0.997
	0.8	1.063	1.292	13.333	2.963	0.793	0.999
	0.9	1.243	0.980	20.000	1.132	0.445	0.999
σ3 = 2 MPa	0.5	0.688	1.968	13.333	1.143	0.344	0.986
	0.6	0.832	1.489	16.000	6.857	0.871	0.994
	0.7	0.976	1.298	11.200	4.667	1.129	0.998
	0.8	1.151	1.015	10.667	2.560	1.036	0.999
	0.9	1.306	0.758	12.000	1.358	0.623	0.999
σ3 = 3 MPa	0.5	0.734	1.933	23.333	1.197	0.334	0.981
	0.6	0.896	1.472	9.333	4.000	1.433	0.987
	0.7	1.042	1.189	8.167	13.067	2.332	0.988
	0.8	1.239	0.901	6.222	8.296	2.056	0.986
	0.9	1.432	0.708	3.652	2.471	1.546	0.974
σ3 = 4 MPa	0.5	0.893	1.477	17.222	0.728	0.328	0.991
	0.6	1.027	1.151	15.500	4.133	0.943	0.998
	0.7	1.194	0.968	12.056	3.288	0.905	0.999
	0.8	1.358	0.783	10.333	2.851	1.117	0.998
	0.9	1.600	0.630	23.250	1.550	0.748	0.999

and Figure 10, respectively. The fitting results indicate that the Burgers model provides a satisfactory representation of the creep behavior of gangue–paste cemented backfill.

4.4. Numerical Verification of Model Based on FLAC

To further examine the applicability of Burgers creep model in describing the creep behavior of gangue–gypsum cemented backfill, and to evaluate the physical reliability of the parameters identified in the previous section, a triaxial creep numerical simulation was performed using the FLAC3D 6.0 software.

The numerical model was established to replicate the laboratory testing conditions as closely as possible. A cylindrical specimen with dimensions of φ50 × 100 mm was constructed (Figure 11). The model consisted of 3976 nodes and 1272 elements. During the simulation, the bottom boundary was fixed in the z-direction, while a uniform axial load was adopted to the top surface of the specimen. In addition, uniform radial pressures were applied to the lateral surfaces to simulate confining pressure conditions. The confining pressures were set to 1, 2, 3, and 4 MPa, which matched the loading conditions in the laboratory triaxial compression creep tests.

The built-in Burgers creep constitutive model provided in FLAC3D was assigned directly to the model elements. The material parameters used in the simulation were obtained from the parameter identification results in Table 6. After completing the numerical calculation, the axial displacement of the specimen was extracted and converted into axial strain, which was subsequently relative to the triaxial creep test outcomes. The simulated displacement contour is illustrated in Figure 12, while the comparison between the numerical simulation outputs and the experimental curves is depicted in Figure 13.

The simulated curves agree well with the experimental results (Figure 13), implying that the fitted model reproduces the creep behavior of the specimens with satisfactory accuracy. This comparison further validates the reliability of the identified model parameters. The results also demonstrate that the Burgers creep model can effectively characterize the time-dependent deformation of gangue–gypsum cemented backfill, providing a reliable basis for its application in large-scale numerical simulations of gangue backfill mining engineering.

5. Conclusions

Multi-stage loading triaxial creep tests were adopted on gangue–paste cemented backfill at distinct confining pressures (1, 2, 3, and 4 MPa) to investigate its time-dependent deformation characteristics. Our findings showed that the Burgers creep model was applied for parameter identification and was further verified through numerical simulation. The primary findings can be summarized below:

(1)

By analyzing the step-loading creep test data using Chen’s superposition method, the creep evolution characteristics of gangue–gypsum cemented backfill under different confining pressures were revealed. Within the investigated stress range, the creep process can be divided into three typical stages: instantaneous deformation, decelerating creep, and steady-state creep, while no accelerated creep or instability failure occurred. Moreover, the absolute creep deformation increases progressively with surging S_L, indicating that the time-dependent deformation behavior becomes more pronounced under higher stress conditions.

(2)

The isochronous stress–strain curves were used to further analyze the time-dependent deformation behavior of the backfill. The outcomes imply that within the experimental stress range the material mainly exhibits linear elastic and linear viscoelastic responses, and no viscoplastic yielding was observed. The critical long-term strength of the backfill is therefore estimated to be not less than 0.9 times the qf. The long-term strength values under confining pressures of 1, 2, 3, and 4 MPa were determined using the inflection point method of isochronous curves. Although confining pressure enhances the long-term strength of the backfill, the magnitude of this strengthening effect gradually decreases as the confining pressure increases. These findings provide theoretical guidance for mixture proportion optimization and long-term stability control of gangue backfill mining under different mining depth conditions.

(3)

The absolute creep strain exhibits a nonlinear increasing trend with S_L, while confining pressure significantly suppresses the creep deformation. Under the same S_L, specimens subjected to higher confining pressure show noticeably smaller creep strain than those under lower confining pressure. This phenomenon indicates that lateral confinement effectively inhibits the initiation and propagation of microcracks within the cemented skeleton, thereby delaying internal damage evolution.

(4)

The Burgers creep model provides an effective description of the creep behavior of gangue–paste cemented backfill. The R² of the fitted curves under different test conditions are all greater than 0.97, demonstrating a high fitting accuracy. The identified model parameters exhibit systematic variation with S_L, reflecting the stiffness degradation and viscous evolution of the cemented material during loading. It should be noted that the present study adopts segmental fitting for graded creep curves. Future studies may consider using global parameters to perform unified fitting for the entire multi-stage creep process, which would further improve the applicability and predictive capability of the model.