A Viscoelastic-Plastic Creep Model for Initial Damaged Coal Sample Affected by Loading Rate

Abstract

Underground engineering rock masses are significantly affected by stress redistribution induced by mining or adjacent engineering disturbances, leading to initial damage accumulation in coal-rock masses. Under sustained geostress, these masses exhibit pronounced time-dependent creep behavior, posing serious threats to long-term engineering stability. Dynamic loading effects triggered by adjacent mining activities (manifested as medium strain-rate loading) further exacerbate damage evolution and significantly influence creep characteristics. In this study, coal samples with identical initial damage were prepared, and graded loading creep tests were conducted at rates of 0.005 mm·s⁻¹ (50 microstrains·s⁻¹), 0.01 mm·s⁻¹ (100 microstrains·s⁻¹), 0.05 mm·s⁻¹ (500 microstrains·s⁻¹), and 0.1 mm·s⁻¹ (1000 microstrains·s⁻¹) to systematically analyze the coupled effects of loading rate on creep behavior. Experimental results demonstrate that increased loading rates markedly shorten creep duration, with damage rates during the acceleration phase showing nonlinear surges (e.g., abrupt instability at 0.1 mm·s⁻¹ (1000 microstrains·s⁻¹)). Based on experimental data, an integer-order viscoelastic-plastic creep model incorporating stress-dependent viscosity coefficients and damage correlation functions was developed, fully characterizing four behaviors stages: instantaneous deformation, deceleration, steady-state, and accelerated creep. Optimized via the Levenberg–Marquardt algorithm, the model achieved correlation coefficients exceeding 0.96, validating its accuracy. This model clarifies the impact mechanisms of loading rates on the long-term mechanical behavior of initially damaged coal samples, providing theoretical support for stability assessment and hazard prevention in underground engineering.

1. Introduction

The original stress field distribution in coal-rock masses has been significantly altered by large-scale subsurface resource extraction and adjacent engineering disturbances [1]. The heterogeneous distribution of secondary stress fields induces progressive accumulation of initial damage in coal-rock masses, critically compromising the long-term stability of underground engineering structures [2]. When subjected to dynamic loading effects from adjacent mining operations (characterized by medium strain-rate loading), peripheral rock masses undergo complex stress path evolution. This process accelerates initial damage accumulation and induces pronounced time-dependent creep behavior [3]. Such time-dependent deformation not only exacerbates structural degradation but may also trigger cascading engineering hazards. While substantial progress has been made in understanding creep mechanisms of coal-rock combinations, systematic investigations remain inadequate regarding the synergistic coupling effects between medium strain-rate loading and initial damage. This critical knowledge gap hinders precision formulation of countermeasures against engineering disasters.

To accurately characterize the long-term deformation behaviors of coal-rock masses, substantial research efforts have been dedicated to investigating their creep characteristics under specific environmental-mechanical conditions. Wang et al. [4] conducted uniaxial creep tests on coal-rock composites with varying height ratios, revealing the influence of axial static load pressure and coal-rock height ratio on creep characteristics. Hou et al. [5] performed experimental investigations on the creep behavior of coal specimens under combined compression-shear loading, providing a basis for support pillar design and long-term stability analysis in deep mining scenarios. Yang et al. [6] developed a novel creep constitutive model for stratified siltstone formations and implemented it in computational instability analysis of mining operations. This innovative approach demonstrates superior accuracy in simulating creep behavior compared to conventional methodologies. Wang et al. [7] systematically investigated the creep damage, nonlinear deformation, and sudden instability characteristics of gas-bearing coal under varying axial stress, confining pressure, and methane pressure conditions, thereby quantifying the critical influence of methane pressure on the time-dependent mechanical behaviors of coal. Zhao et al. [8] investigated the shear creep deformation behavior of anchored rock masses under hydro-mechanical coupling conditions, revealing the critical influence of moisture content on the time-dependent mechanical properties of rock materials. Li et al. [9] developed a moisture-dependent creep constitutive model for coal-rock composites, with its predictive capability rigorously validated through multiscale numerical simulations under complex hydro-mechanical coupling conditions. Yang et al. [10] systematically investigated the time-dependent deformation characteristics of highly friable and soft coal-rock masses under varying moisture content conditions, revealing the critical occurrence of secondary creep stages under specific hydro-mechanical coupling thresholds. Tarifard et al. [11] conducted a comprehensive evaluation of the long-term effects of creep deformation and groundwater interaction on tunnel lining loads in weak rock masses, revealing the critical coupling between rock mass creep behavior and hydrogeological conditions that dominate structural degradation mechanisms. Liu et al. [12] developed an anisotropic time-dependent deformation and damage constitutive model for rocks under true triaxial compression, thereby providing enhanced mechanistic insights into the long-term stability evolution of engineering structures in complex stress regimes. Yin et al. [13] conducted short-term uniaxial creep experiments on sandstone-coal composite samples, revealing the influence of sandstone on the creep behavior of coal. Zhang and Wang et al. [14] developed a nonlinear damage creep model for rocks under cyclic loading and unloading. This model explains how rock behaves differently in transient, steady-state, and accelerated creep stages, providing theoretical support for predicting rock deformation. Huang et al. [15] experimentally investigated the creep behavior of coal with initial damage and elucidated the influence of strain history on the deformation characteristics of initially damaged coal specimens. King [16] conducted creep tests on shale under different temperatures and axial stress levels, obtaining creep curves of shale at various temperatures and stress levels. Y. Fujii [17] performed triaxial creep tests on granite and sandstone, acquiring axial creep, circumferential creep, and volumetric creep curves. B. Ladanyi [18] considered the rheological effects of rock and indicated through monitoring the stress and deformation of the initial lining in underground chambers that the calculation results of the viscoelastic model were lower than actual values. Ito H [19,20] carried out decade-long bending creep tests on granite and limestone via creep experiments. Pomeroy [21] studied the creep characteristics of coal at room temperature in the laboratory. Danesh [22] investigated the creep characteristics of coal and their impacts on permeability, analyzing the permeability characteristics of two gases under different confining pressures and pore pressures.

Extensive research has been conducted by scholars both domestically and internationally on creep models of coal-rock masses. Kachanov [23] proposed the concepts of continuous damage factor and effective stress, and studied creep failure in metals; Chan [24] developed a creep model to describe creep damage in shale, arguing that damage induces inelastic flow until the creep failure stage; Ayari [25] established a creep model under cyclic loading/unloading conditions based on continuous damage theory and cyclic loading creep tests; Tomanovic [26] developed a creep model for the time-dependent deformation of soft rock materials based on mudstone creep tests; Maranini [27] conducted creep tests on granite under confining pressures ranging from 0 to 40 MPa, established a non-associated constitutive equation and viscoplastic potential function considering instantaneous creep behavior, and validated the model’s correctness; Mazzotti [28] proposed an isotropic model for creep damage in concrete under uniaxial compression, considering that nonlinear viscous strain changes are related to high stress levels, and studied the time-dependent damage and strength of concrete through numerical simulation; Hellmich [29] established an early-age creep constitutive model for concrete, considering the thermo-mechanical coupling mechanism in early pores, and identified the model parameters. Guo et al. [30] investigated the creep damage characteristics and constitutive modeling of pre-damaged coal. They proposed a novel creep damage variable incorporating both loading stress levels and temporal effects and further developed a fractional-order creep damage constitutive model to capture time-dependent nonlinear behaviors. Jiang et al. [31] investigated the failure mechanical properties and acoustic emission characteristics of soft rock-coal composites under dynamic disturbances, and systematically analyzed the damage evolution mechanisms of such composite systems under transient dynamic loading conditions. Yang et al. [32] investigated the creep rupture behavior and permeability evolution of sandstone containing pre-existing dual defects under high-temperature thermal treatment. Their findings provide critical theoretical support for optimizing underground coal gasification (UCG) and carbon capture and storage (CCS) technologies in deep geological formations. Ding et al. [33] analyzed the fracture response and damage evolution characteristics of creep-damaged coal under dynamic loading conditions, revealing the underlying mechanisms governing disaster initiation and propagation in deep mining environments. Bao et al. [34] investigated the time-dependent creep behavior of weakly layered rocks in slope strata under coupled damage interactions, and established a fractional-order nonlinear creep constitutive model that explicitly accounts for both initial damage and creep-induced progressive damage evolution. These studies on the creep characteristics of coal-rock masses under various conditions have provided critical theoretical and practical guidance for accurately predicting their long-term stability. Numerous creep models have been proposed to characterize these behaviors, with a predominant focus on how alterations in initial damage states influence creep time-dependent deformation stages. Furthermore, standardized specimen preparation methods for initially damaged coal-rock samples have been systematically developed to ensure experimental reproducibility and engineering relevance. However, research on the effects of varying loading rates on the creep characteristics of coal specimens remains notably insufficient. Current studies have largely neglected the critical influence of dynamic loading conditions on the long-term mechanical behavior of damaged coal pillars. However, research on the effects of varying loading rates on the creep characteristics of coal specimens remains notably insufficient. Current studies have largely neglected the critical influence of dynamic loading conditions on the long-term mechanical behavior of damaged coal pillars.

Coal-rock masses, as quintessential heterogeneous geological materials, exhibit pronounced nonlinear creep behavior arising from the intricate interplay between their inherent structural complexity and external multiaxial stress regimes [35]. Studies have shown [36,37,38,39,40,41,42,43,44] that dynamic loading is the key factor accelerating creep deformation in coal and rock masses, and it has a particularly significant impact on the damage mechanisms of coal specimens. Previous investigations into the creep behavior of damaged coal samples have primarily been conducted through scientific experiments, numerical simulations, and theoretical analyses. However, the coupled effects of loading rates and pre-existing damage conditions on creep characteristics remain insufficiently explored. In this study, coal specimens with controlled initial damage were systematically prepared, and a series of multi-stage creep tests were implemented under four distinct loading rates spanning 0.005–0.1 mm·s⁻¹ (50 to 1000 microstrains·s⁻¹). A viscoelastic-plastic constitutive model integrating initial damage mechanisms was formulated, with the nonlinear creep model’s validity being rigorously validated through experimental data. These findings are intended to serve as a theoretical framework for long-term stability evaluations of coal-rock masses and to provide technical references for underground engineering design.

2. Experimental Method and Procedure

2.1. Experimental Equipment and Standard Samples Preparation

Experiments were conducted at 25 °C using an MTS815 electrohydraulic system (MTS Systems Corporation, Eden Prairie, MN, USA) capable of applying axial loads and stabilizing confining pressures [15]. The apparatus integrates five subsystems: digital control, hydraulic actuation, axial loading, pore pressure regulation, and confining pressure stabilization. Pre-damaged coal specimens were prepared using the circumferential displacement-controlled mode of the MTS815 mechanical testing system to maintain structural stability during the yielding phase and prevent sudden fragmentation. The circumferential displacement loading rate was configured at 0.002 mm·s⁻¹, with each loading cycle achieving a 1 mm displacement increment. Unloading phases employed force-controlled mode, progressively increasing axial load removal rates to accomplish stress relaxation. The circumferential displacement sensor and coal specimen are shown in Figure 1b.

Undisturbed coal specimens from Xiaojihan Mine (Yulin, China) were machined into 50 mm × 100 mm cylinders following ISRM protocols [44]. Specimens with P-wave velocities of 1550–1750 m·s⁻¹ (identified via ultrasonic testing [45,46]) were selected for creep analysis. Conventional testing yielded an elastic modulus of 0.5 GPa, uniaxial compressive strength of 10.7 MPa, Poisson’s ratio of 0.27, cohesion of 5.3 MPa, and friction angle of 25.2°, providing baseline mechanical parameters for viscoelastic modeling [47].

2.2. Initially Damaged Coal Sample Preparation Method

The damage parameter D quantitatively characterizes the evolution of internal micro-defects, with its mathematical framework enabling the quantification of crack initiation, propagation, and damage accumulation dynamics, while revealing critical precursor signatures preceding structural instability [45,46]. Based on energy conservation principles, the damage variable D is mathematically formulated as

$$D={\displaystyle \sum _{k=1}^i{U}_d^k}/({\displaystyle \sum _{k=1}^i{U}_d^k}+U_e^k)$$

(1)

where $U_d^k$ is the plastic strain energy produced in the kth cycle, kJ, and $U_e^k$ is the elastic strain energy produced in the kth cycle, kJ, which can be calculated by the area enclosed by the stress-strain curve.

Based on the damaged coal specimen preparation methodology and the damage variable calculation formula (Equation (1)), the stress–strain curves of coal specimens with varying initial damage severity levels under cyclic loading–unloading conditions are illustrated in Figure 2 [39].

Experimental observations revealed that with increasing loading cycles, the elastic stored energy, plastic strain energy accumulation, and damage parameter D within specimens exhibited sustained growth, confirming the enhancing effect of cyclic loading on material degradation processes [15]. Based on the damage evolution model (Equation (1)), the D-values of coal specimens reached 0.312 and 0.448 after three and five loading cycles, respectively. To investigate damage rate sensitivity, a pre-damaged specimen system with D = 0.3 was established, and multi-rate triaxial mechanical tests were conducted, including conventional triaxial compression tests (TCT) at loading rates of 0.005, 0.01, 0.05, and 0.1 mm·s⁻¹ (50, 100, 500, and 1000 microstrains·s⁻¹), as well as triaxial compression creep tests (TCCT) under sustained stress conditions. Generally, the study of strain rate laws typically involves using the loading displacement velocity for research. Therefore, in this paper, the term “loading rate” exclusively refers to the loading displacement velocity [48].

2.3. Triaxial Compressive Strength (TCS) of Initially Damaged Coal Specimen

Triaxial compression tests (TCT) were conducted on initially damaged coal specimens with a damage level of approximately 0.3 using the MTS815 electrohydraulic system under a confining pressure of 2 MPa. Loading was applied at rates of 0.005 mm·s⁻¹ (50 microstrains·s⁻¹), 0.01 mm·s⁻¹ (100 microstrains·s⁻¹), 0.05 mm·s⁻¹ (500 microstrains·s⁻¹), and 0.1 mm·s⁻¹ (1000 microstrains·s⁻¹). The Triaxial Compressive Strength (TCS) of the coal specimens under these loading rates were measured, as summarized in

Table 1. Triaxial compressive strength (TCS) of coal specimens under different loading rates.

Loading Rates/mm·s⁻¹ (Microstrains·s−1)	Triaxial Compressive Strength (TCS)/MPa
0.005 (50)	30.50
0.010 (100)	30.00
0.050 (500)	34.25
0.100 (1000)	39.43

.

2.4. Triaxial Graded Creep Tests

The creep load was determined as 90% of the triaxial compressive strength (TCS) of initially damaged coal specimens, subsequently partitioned into six equal increments from 40% to 90% of the triaxial compressive strength (TCS). A stepped loading protocol with 2 h dwell time per stress level was implemented under 2 MPa confining pressure, with the triaxial multistage creep loading path detailed in Figure 3.

There are two points that need to be explained about the experiment: (1) Due to the specimen preparation methodology for initially damaged coal and the computational approach for damage variables (Equation (1)), special emphasis was placed on selecting test specimens whose measured damage values approximated the target parameters within ±5% tolerance. (2) Given the operational challenges in capturing accelerated creep phases, real-time monitoring of stress–strain behavior was initiated when loading reached 80% of the deviatoric stress threshold. The loading protocol was adaptively refined by implementing 5% deviatoric stress increments with progressive escalation until specimen failure, contingent upon observed strain localization patterns. During the experiment, damaged coal samples with an initial damage of approximately 0.3 were used under a confining pressure of 2 MPa. The stress data of graded loading at different loading rates are shown in

Table 2. Graded stress table under different loading rates.

Loading Rates/mm·s−1 (Microstrains·s−1)	Loading Stress (Proportion of TCS)/MPa
0.005 (50)	12.2 (40%)
	15.3 (50%)
	18.4 (60%)
	21.4 (70%)
	24.5 (80%)
	26.1 (85%)
	27.6 (90%)
0.010 (1000)	12.0 (40%)
	15.0 (50%)
	18.0 (60%)
	21.0 (70%)
	24.0 (80%)
	25.5 (85%)
	27.0 (90%)
0.050 (500)	13.7 (40%)
	17.1 (50%)
	20.6 (60%)
	24.0 (70%)
	27.4 (80%)
	29.1 (85%)
	30.8 (90%)
0.100 (1000)	15.8 (40%)
	19.7 (50%)
	23.6 (60%)
	27.6 (70%)
	31.5 (80%)
	35.5 (90%)

.

3. Creep Properties of Coal Sample with Initial Damage

Triaxial compression creep tests (TCCT) were conducted on initially damaged coal samples by applying incremental stresses at 40%, 50%, 60%, 70%, 80%, 85%, and 90% of their corresponding strength under different loading rates. The creep strain curves for each loading rate are presented in Figure 4.

Figure 4 reveals that under different displacement rates, at lower applied stresses, the creep deformation stabilizes after undergoing a decelerated creep stage. As the stress level increases, the creep transitions into a steady-state creep stage. Upon reaching the final load stage, the creep enters an accelerated creep stage. When the loading rate reached V = 0.1 mm·s⁻¹ (1000 microstrains·s⁻¹), the accelerated creep failure stage exhibited instantaneous and abrupt characteristics, with a dramatic increase in deformation rate culminating in catastrophic instability (Figure 4d).

As shown in Figure 5, the loading rate and stress level have a significant impact on the creep behavior of coal samples. Under low stress conditions, as the loading rate increases, the instantaneous strain of coal samples shows a slight downward trend. This is mainly due to the full development and propagation of internal fractures in coal samples at lower loading rates, which allows new microcracks to continuously form during the application of constant load. In the high-stress state, both extreme rates of V = 0.002 mm·s⁻¹ (20 microstrains·s⁻¹) and V = 0.1 mm·s⁻¹ (1000 microstrains·s⁻¹) experienced creep failure under the sixth-level load, leading to a significant increase in strain. Further analysis reveals that under the same loading rate, creep strain exhibits a transition from linear to nonlinear growth as the load increases, indicating the existence of a stress threshold (24.0–31.5 MPa) that controls the creep stage. Notably, this threshold has a nonlinear correlation with the loading rate: when the loading rate increases from 0.002 mm·s⁻¹ to 0.1 mm·s⁻¹ (20 to 1000 microstrains·s⁻¹), the stress threshold first decreases from 24.7 MPa to 24.0 MPa and then rises back to 31.5 MPa, revealing the complex influence mechanism of the loading rate on the long-term stability of damaged coal samples.

4. Integer-Order Nonlinear Creep Model

4.1. Integer-Order Nonlinear Viscoelastic-Plastic Creep Model

Based on the creep curves of coal samples presented in Section 3, four characteristic deformation stages are identified:

(1) Instantaneous deformation. The instantaneous behavior of coal-rock deformation can be described by the Hookean elastic body.

(2) Decelerated creep. Among existing rheological models, both the Kelvin body and Muruyama body are capable of characterizing the decelerated creep behavior of coal. A critical distinction lies in their activation mechanisms: the Kelvin body exhibits creep deformation under all stress conditions, whereas the Muruyama body incorporates a Saint-Venant element to govern a stress threshold, permitting creep initiation only when the applied stress exceeds this critical value. This study adopts the fractional-order Muruyama body to model the decelerated creep phase.

(3) Steady creep. When the stress level exceeds a critical threshold, a steady creep process emerges in the creep curves of coal pillars. This phase can be characterized by the Bingham body.

(4) Accelerated creep. At stress levels exceeding critical stability limits, the creep curves enter a phase of rapidly increasing deformation rates. Conventional viscous damping coefficients fail to capture this nonlinear acceleration, necessitating the introduction of a damage-coupled viscous coefficient to quantify the instability mechanisms.

This study establishes a nonlinear integer-order viscoelastic-plastic creep model, which is composed of four elements connected in series: a Hook body, a Muruyama body, a Bingham body, and a modified Bingham body (Figure 6).

During the triaxial creep process of coal samples, the decelerated creep and steady creep stages can be described using the Hook body, Muruyama body, and Bingham body. The physical significance of these models is well-established and thus not elaborated upon here. In the accelerated creep phase characterized by higher creep rates, the conventional Bingham body fails to adequately characterize accelerated creep behavior. This study implements a nonlinear modification to the viscous coefficient of the Bingham body, specifically

$${\eta }_3(\sigma ,t)={\eta }_c{e}^{-at}$$

(2)

where $\sigma$ denotes stress, $\eta$ is the viscosity coefficient, and a is $a$ material mechanical parameter with $a$ > 0.

The constitutive equation of the modified Bingham body is

$$\sigma -{\sigma }_s={\eta }_c{e}^{-at}\dot{\epsilon }$$

(3)

where $\epsilon$ denotes strain, when $\sigma ={\sigma }_c$, the creep equation of the modified Bingham body is

$$\epsilon ={\displaystyle \frac{{\sigma }_c-{\sigma }_s}{a{\eta }_c}}{e}^{at}$$

(4)

By substituting the relevant parameters into Equation (4), the influence of parameter a on creep acceleration is obtained as shown in the figure. It can be observed that an increase in parameter a leads to a higher creep rate in the curves. This demonstrates that the accelerated creep stage can be quantitatively characterized by controlling the magnitude of parameter a, as can be observed in Figure 7.

It is evident that the four components of the nonlinear integer-order viscoelastic-plastic creep model exhibit well-defined physical significance and parametric characteristics. The complete rheological constitutive equations and creep equations for this model are formulated as follows:

(1) Rheological constitutive equations.

When $\sigma <{\sigma }_{s1}$, the rheological model contains only the Hook body, and its constitutive equation is

$$\sigma =E_1\epsilon$$

(5)

where $E$ is the elastic modulus, when ${\sigma }_{s1}\le \sigma <{\sigma }_{s2}$, the rheological model is composed of a Hookean body and a Muruyama body, and each element in the rheological model satisfies the following equations:

$$\left\{\begin{array}{l}{\sigma }_1=E_1{\epsilon }_1,{\sigma }_2-{\sigma }_{s1}=E_2{\epsilon }_2+{\eta }_1{\dot{\epsilon }}_2\\ \sigma ={\sigma }_1={\sigma }_2,\epsilon ={\epsilon }_1+{\epsilon }_2\end{array}\right.$$

(6)

The constitutive equation is obtained as

$$(E_1+E_2)\sigma +{\eta }_1\dot{\sigma }-E_1{\sigma }_{s1}=E_1{E}_2\epsilon +E_1{\eta }_1\dot{\epsilon }$$

(7)

When ${\sigma }_{s2}\le \sigma <{\sigma }_{s3}$, the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The constituent equations for each element in the model satisfy the following relationships:

$$\left\{\begin{array}{l}{\sigma }_1=E_1{\epsilon }_1,{\sigma }_2-{\sigma }_{s1}=E_2{\epsilon }_2+{\eta }_1{\dot{\epsilon }}_2,{\sigma }_3-{\sigma }_{s2}={\eta }_2{\dot{\epsilon }}_3\\ \sigma ={\sigma }_1={\sigma }_2={\sigma }_3,\epsilon ={\epsilon }_1+{\epsilon }_2+{\epsilon }_3\end{array}\right.$$

(8)

From the fifth equation in Equation (8), it can be deduced that

$$\left\{\begin{array}{l}\dot{\epsilon }={\dot{\epsilon }}_1+{\dot{\epsilon }}_2+{\dot{\epsilon }}_3\\ \ddot{\epsilon }={\ddot{\epsilon }}_1+{\ddot{\epsilon }}_2+{\ddot{\epsilon }}_3\end{array}\right.$$

(9)

By substituting the first four equations from Equation (8) into Equation (9), the constitutive equation is obtained as

$${\displaystyle \frac{\ddot{\sigma }}{E_1}}+({\displaystyle \frac{1}{{\eta }_1}}+{\displaystyle \frac{1}{{\eta }_2}}+{\displaystyle \frac{E_2}{{\eta }_1{E}_1}})\dot{\sigma }+{\displaystyle \frac{E_2}{{\eta }_1{\eta }_2}}\sigma -{\displaystyle \frac{{\eta }_2{\sigma }_{s1}+(E_2+{\eta }_1){\sigma }_{s2}}{{\eta }_1{\eta }_2}}=\ddot{\epsilon }+{\displaystyle \frac{E_2}{{\eta }_1}}\dot{\epsilon }$$

(10)

When $\sigma \ge {\sigma }_{s3}$, the rheological model is composed of the Hook body, Muruyama body, Bingham body, and modified Bingham body. The constituent equations for each element in the model satisfy the following relationships:

$$\left\{\begin{array}{l}{\sigma }_1=E_1{\epsilon }_1,{\sigma }_2-{\sigma }_{s1}=E_2{\epsilon }_2+{\eta }_1{\dot{\epsilon }}_2,{\sigma }_3-{\sigma }_{s2}={\eta }_2{\dot{\epsilon }}_3,{\sigma }_4-{\sigma }_{s3}={\eta }_c{e}^{-at}{\dot{\epsilon }}_4\\ \sigma ={\sigma }_1={\sigma }_2={\sigma }_3={\sigma }_4,\epsilon ={\epsilon }_1+{\epsilon }_2+{\epsilon }_3+{\epsilon }_4,\dot{\epsilon }={\dot{\epsilon }}_1+{\dot{\epsilon }}_2+{\dot{\epsilon }}_3+{\dot{\epsilon }}_4,\ddot{\epsilon }={\ddot{\epsilon }}_1+{\ddot{\epsilon }}_2+{\ddot{\epsilon }}_3+{\ddot{\epsilon }}_4\end{array}\right.$$

(11)

From Equation (11), the constitutive equation is obtained as

$$\begin{array}{l}\ddot{\epsilon }+{\displaystyle \frac{E_2}{{\eta }_1}}\dot{\epsilon }={\displaystyle \frac{\ddot{\sigma }}{E_1}}+{\displaystyle \frac{\dot{\sigma }-{\sigma }_{s1}}{{\eta }_1}}+{\displaystyle \frac{\dot{\sigma }-{\sigma }_{s2}}{{\eta }_2}}+{\displaystyle \frac{\dot{\sigma }-{\sigma }_{s3}}{{\eta }_c}}{e}^{at}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{E_2\dot{\sigma }}{{\eta }_1{E}_1}}+{\displaystyle \frac{E_2}{{\eta }_1{\eta }_2}}(\sigma -{\sigma }_{s2})+{\displaystyle \frac{\sigma -{\sigma }_{s3}}{{\eta }_c}}a{e}^{at}+{\displaystyle \frac{E_2{e}^{at}}{{\eta }_1{\eta }_c}}(\sigma -{\sigma }_{s4})\end{array}$$

(12)

(2) Creep equations

When $\sigma <{\sigma }_{s1}$, $\sigma ={\sigma }_c$, the rheological model contains only the Hook body, and the creep equation is

$$\epsilon ={\displaystyle \frac{{\sigma }_c}{E_1}}$$

(13)

By differentiating both sides of Equation (13), the following result is obtained:

$$\dot{\epsilon }=0$$

(14)

From Equation (14), it can be deduced that when $\sigma <{\sigma }_{s1}$, the creep rate remains zero, indicating that the creep model exhibits only instantaneous deformation.

When ${\sigma }_{s1}\le \sigma <{\sigma }_{s2}$, $\sigma ={\sigma }_c$, the rheological model is composed of the Hook body and Muruyama body. The creep equation is

$$\epsilon ={\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c}{E_1}}$$

(15)

By differentiating both sides of Equation (15), the following result is obtained:

$$\dot{\epsilon }={\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{{\eta }_1}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})$$

(16)

When ${\sigma }_{s2}\le \sigma <{\sigma }_{s3}$, $\sigma ={\sigma }_c$, the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The creep equation of the rheological model is

$$\epsilon ={\displaystyle \frac{{\sigma }_c}{E_1}}+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}t$$

(17)

By differentiating both sides of Equation (17), the following result is obtained:

$$\dot{\epsilon }={\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{{\eta }_1}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}$$

(18)

From Equation (18), it can be deduced that when a = 9, the creep rate decreases with increasing time and eventually approaches a constant value. This demonstrates that the model is capable of describing three deformation stages: instantaneous deformation, decelerated creep, and steady creep.

When $\sigma \ge {\sigma }_{s3}$$\sigma ={\sigma }_c$, the rheological model is composed of the Hook body, Muruyama body, and Bingham body. The creep equation of the rheological model is

$$\epsilon ={\displaystyle \frac{{\sigma }_c}{E_1}}+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}t+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s3}}{a{\eta }_c}}{e}^{at}$$

(19)

By differentiating both sides of Equation (19), the following result is obtained:

$$\dot{\epsilon }={\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{{\eta }_1}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s3}}{a^2{\eta }_c}}{e}^{at}+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}$$

(20)

From Equation (20), it can be deduced that when a = 4, the creep rate increases with time, and the growth rate of creep rate escalates progressively as time advances, resulting in significantly amplified acceleration effects. This demonstrates that the model is capable of characterizing four distinct deformation stages: instantaneous deformation, decelerated creep, steady creep, and accelerated creep.

In summary, the one-dimensional integer-order nonlinear viscoelastic-plastic creep model for the free-face section coal pillar is formulated as follows:

$$\epsilon =\left\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_c}{E_1}}& \hfill \sigma <{\sigma }_{s1}\\ {\displaystyle \frac{{\sigma }_c}{E_1}} + {\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})& \hfill {\sigma }_{s1}\le \sigma <{\sigma }_{s2}\\ {\displaystyle \frac{{\sigma }_c}{E_1}}+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}t& \hfill {\sigma }_{s2}\le \sigma <{\sigma }_{s3}\\ {\displaystyle \frac{{\sigma }_c}{E_1}}+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s1}}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_1}}t})+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s2}}{{\eta }_2}}t+{\displaystyle \frac{{\sigma }_c-{\sigma }_{s3}}{a{\eta }_c}}{e}^{at}& \hfill \sigma \ge {\sigma }_{s3}\end{array}\right.$$

(21)

4.2. Three-Dimensional Extension of the Integer-Order Nonlinear Viscoelastic-Plastic Creep Model

Under three-dimensional stress states, the spherical stress tensor typically induces volumetric deformation in rock elements without altering their shape, while the deviatoric stress tensor causes shape changes without affecting volume. The mathematical expressions for stress and strain states are formulated as follows:

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

(22)

where ${\sigma }_m$ denotes the spherical stress tensor, ${\sigma }_m={\sigma }_{ii}/3$; $S_{ij}$ represents the deviatoric stress tensor, ${\epsilon }_m$ is the volumetric strain tensor, and $e_{ij}$ corresponds to the deviatoric strain tensor.

In the elastic state, assuming the volumetric strain tensor and deviatoric strain tensor are respectively related to the spherical stress tensor and deviatoric stress tensor, the three-dimensional form of Hooke’s law is formulated as

$$\left\{\begin{array}{l}{\epsilon }_m={\displaystyle \frac{{\sigma }_m}{3K}}\\ e_{ij}={\displaystyle \frac{S_{ij}}{2G}}\end{array}\right.$$

(23)

where K is the bulk modulus and G denotes the shear modulus, with their mathematical expressions defined as

$$\left\{\begin{array}{l}K={\displaystyle \frac{E}{3(1-2\mu )}}\\ G={\displaystyle \frac{E}{2(1+\mu )}}\end{array}\right.$$

(24)

From Equation (24), the strain of the elastic material can be expressed as

$${\epsilon }_{ij}^e={\displaystyle \frac{{\sigma }_m}{3K}}{\delta }_{ij}+{\displaystyle \frac{S_{ij}}{2G}}$$

(25)

Assuming that volumetric changes are elastic and that rheological properties are predominantly manifested in shear deformation, the constitutive relation for the viscoelastic three-dimensional body is formulated as

$${\epsilon }_{ij}^{ve}={\displaystyle \frac{1}{2G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}})S_{ij}$$

(26)

The three-dimensional constitutive relation for the viscoplastic body in the rheological model is formulated as

$${\epsilon }_{ij}^{vp}={\displaystyle \frac{1}{{\eta }_2}}〈\varphi ({\displaystyle \frac{F}{F_0}})〉{\displaystyle \frac{\partial Q}{\partial {\sigma }_{ij}}}t$$

(27)

where F denotes the yield function; F₀ represents the initial value of the rock yield function; and Q corresponds to the plastic potential function.

This demonstrates that the three-dimensional integer-order nonlinear viscoelastic-plastic creep model can be obtained by substituting Equation (24), Equation (25), and Equation (26) into the creep Equation (27):

$${\epsilon }_{ij}=\left\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}} + {\displaystyle \frac{S_{ij}}{2G_0}}& \hfill S_{11}<{\sigma }_{s1}\\ {\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}} + {\displaystyle \frac{S_{ij}}{2G_0}}+{\displaystyle \frac{S_{ij}-{\sigma }_{s1}}{2G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t})& \hfill {\sigma }_{s1}\le S_{11}<{\sigma }_{s2}\\ {\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}} + {\displaystyle \frac{S_{ij}}{2G_0}}+{\displaystyle \frac{S_{ij}-{\sigma }_{s1}}{2G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t}) +{\displaystyle \frac{S_{ij}-{\sigma }_{s2}}{{\eta }_2}}t& \hfill {\sigma }_{s2}\le S_{11}<{\sigma }_{s3}\\ {\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}} + {\displaystyle \frac{S_{ij}}{2G_0}}+{\displaystyle \frac{S_{ij}-{\sigma }_{s1}}{2G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t}) +{\displaystyle \frac{S_{ij}-{\sigma }_{s2}}{{\eta }_2}}t+{\displaystyle \frac{S_{ij}-{\sigma }_{s3}}{a{\eta }_c}}{e}^{at}& \hfill S_{11}\ge {\sigma }_{s3}\end{array}\right.$$

(28)

In the triaxial creep test of coal samples with equal confining pressures (${\sigma }_2={\sigma }_3$), substituting into Equation (28) yields the following:

$$\left\{\begin{array}{l}{\sigma }_m={\displaystyle \frac{1}{3}}({\sigma }_1+2{\sigma }_3)\\ S_{ij}={\displaystyle \frac{2}{3}}({\sigma }_1-{\sigma }_3)\end{array}\right.$$

(29)

By substituting Equation (29) into Equation (28), the integer-order nonlinear viscoelastic-plastic creep model under equal confining pressure is obtained as follows:

$$\epsilon (t)=\left\{\begin{array}{ll}{\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}} + {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_0}}& \hfill {\sigma }_1-{\sigma }_3<{\sigma }_{s1}\\ {\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}} + {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_0}}+{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s1}}{3G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t})& \hfill {\sigma }_{s1}\le {\sigma }_1-{\sigma }_3<{\sigma }_{s2}\\ {\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}} + {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_0}}+{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s1}}{3G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t}) +{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s2}}{3{\eta }_2}}t& \hfill {\sigma }_{s2}\le {\sigma }_1-{\sigma }_3<{\sigma }_{s3}\\ {\displaystyle \frac{{\sigma }_1+2{\sigma }_3}{9K}} + {\displaystyle \frac{{\sigma }_1-{\sigma }_3}{3G_0}}+{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s1}}{3G_1}}(1-e^{-{\displaystyle \frac{G_1}{{\eta }_1}}t}) +{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s2}}{3{\eta }_2}}t\\ +{\displaystyle \frac{({\sigma }_1-{\sigma }_3)-{\sigma }_{s3}}{3a{\eta }_c}}{e}^{at}& \hfill {\sigma }_1-{\sigma }_3\ge {\sigma }_{s3}\end{array}\right.$$

(30)

5. Parameter Recognition and Verification of the Creep Damage Model

Parameter identification for the established creep model was conducted using 1stOpt 5.0 software, in which the global optimization Levenberg–Marquardt (LM) [51] algorithm was employed to perform nonlinear fitting on the creep data of coal samples. The fitting curves of coal sample creep data under different loading rates are shown in Figure 8.

Results show that the new integer-order creep model can reflect the decay creep, steady-state creep, and accelerated creep characteristics of coal samples under stress and initial damage. The fitting curves exhibit a high correlation with the creep strain curves of Figure 4 under different loading rates, verifying the accuracy and applicability of the model. The parameter values in nonlinear integer-order creep models under different loading rates are systematically tabulated in

Table 3. Parameter values in nonlinear integer-order creep models with different loading rates.

V/mm·s−1	σ1–σ3	K/GPa	G0/GPa	G1/GPa	η1/GPa·h	η2/GPa·h	ηc/GPa·h	α
0.005	12.2	5.226	1.0014	7.8705	3.2220	-	-	-
	15.3	5.2464	1.0140	11.5109	8.2670	-	-	-
	18.4	0.9280	1.8270	10.5516	4.9341	-	-	-
	21.4	0.4296	1.6345	9.4184	5.1057	1.9520	-	-
	24.5	0.4370	1.8047	10.0072	2.1598	7.1888	-	-
	26.1	0.5314	3.5944	7.3451	5.5411	6.5304	3.2011	76.8
	27.6	0.3781	1.6321	8.4501	5.8510	0.5313	3.2145	45.4
0.01	12.0	6.8729	1.0953	13.1982	2.3849	-	-	-
	15.0	0.9642	2.0594	10.4216	3.5060	-	-	-
	18.0	0.5760	5.9033	10.2757	4.4911	-	-	-
	21.0	0.6740	3.1974	10.8972	8.3033	-	-	-
	24.0	12.830	1.1541	8.4180	3.8084	0.3305	-	-
	25.5	7.9332	1.0650	7.7545	3.7684	3.5917	2.0055	56.3
	27.0	8.6911	1.0220	8.8841	3.2509	3.9112	2.7123	71.0
0.05	13.7	0.7721	3.4427	9.8614	1.4964	-	-	-
	17.1	0.6101	5.7079	7.9588	2.9583	-	-	-
	20.6	1.3977	1.7510	8.8411	2.7169	-	-	-
	24.0	0.5584	6.4483	7.1948	3.4242	1.6842	-	-
	27.4	4.8800	1.2038	7.8248	4.2479	2.1687	-	-
	29.1	0.4242	3.2182	12.3183	5.4993	7.4177	3.0738	46.0
	30.8	0.4579	3.3866	4.8155	1.3154	1.9944	3.2081	8.2
0.1	15.8	0.6634	11.2076	21.1316	3.4952	-	-	-
	19.7	7.4510	1.3789	17.5394	6.2641	-	-	-
	23.6	0.6675	6.8392	21.8978	7.2299	-	-	-
	27.6	1.2806	2.2003	14.7146	12.3827	0.4193	-	-
	31.5	3.0127	1.3061	12.4878	4.2972	4.8699	-	-
	35.5	4.1327	1.4932	12.944	1.5625	1.2069	8.4631	100

.

6. Conclusions

The creep characteristics of damaged coal samples were investigated through stepwise loading creep tests, and a new integer-order viscoelastic-plastic creep model was established. The conclusions are as follows:

(1) Coal samples with identical initial damage were prepared using cyclic displacement-controlled loading–unloading protocols via an MTS servo-controlled testing system. As the number of cycles increases, the cumulative plastic deformation increases and the degree of damage escalates, ultimately yielding coal specimens with an initial damage variable value of 0.3 for this test.

(2) The damaged coal samples exhibit distinct rheological behavior, progressing through three characteristic creep stages: decelerated creep, steady-state creep, and accelerated creep. Compared with intact specimens, coal samples subjected to higher loading rates demonstrate significantly shorter duration and higher deformation rates in the accelerated creep phase, highlighting the critical influence of loading rate on the creep behavior of coal.

(3) Under the condition of maintaining a constant initial damage variable value, when the loading rate is progressively increased from 0.005 mm·s⁻¹ to 0.1 mm·s⁻¹ (50 to 100 microstrains·s⁻¹), the experimental results demonstrate that such rate enhancement significantly reduces the creep duration of coal samples. Notably, the damage rate during the accelerated creep stage exhibits a sharp upward trend, which provides crucial insights for investigating the rate effects in creep deformation mechanisms.

(4) An integer-order viscoelastic-plastic creep model incorporating the initial damage effects of coal samples has been developed, with parameters successfully identified for specimens under varying loading rates. The fitting results demonstrate that the proposed creep model exhibits high correlation coefficients (>0.96) and accurately characterizes the strain–time relationships of damaged coal samples across different loading rate conditions, thereby validating the model’s reliability and applicability.