Numerical Investigation on the Creep-Induced Microdamage Evolution in Rock

Abstract

Rock creep, a key factor in the long-term stability of deep geotechnical engineering, remains challenging to study due to the complexity of its microscopic damage mechanisms. Laboratory creep tests are limited by long durations and scale effects, while phenomenological models cannot fully capture the underlying processes. This study employs the parallel-bonded stress corrosion (PSC) model in PFC2D to simulate sandy mudstone’s creep behavior, systematically correlating macroscopic creep deformation with microscopic damage evolution and energy conversion. The model reproduces the four stages of the idealized creep curve and quantifies the effects of axial stress level and confining pressure on creep lifetime, rate, and failure mode. Increasing axial stress shortens creep lifetime; every 10% increase raises the creep rate by a factor of 4–14, and high stress enhances nonlinear deformation, producing stair-stepping curves due to unstable microcrack propagation. In contrast, confining pressure prolongs lifetime; at 90% uniaxial compressive strength (UCS), 15 MPa extends it from 2.78 h to ~25 years. Confinement also enhances ductility by suppressing tensile stresses and delaying damage accumulation. This study reveals the coupling mechanism of stress-corrosion-induced subcritical crack propagation and energy dissipation, clarifies the microscopic origin of stepped creep curves, and provides a micromechanical framework for long-term stability evaluation in deep geotechnical engineering.

1. Introduction

The creep behavior of stressed surrounding rock masses is a key factor in assessing the safety and stability of geotechnical structures such as crude oil storage facilities, nuclear waste repositories, and mines and excavations at depth [1,2,3,4,5,6,7,8]. In view of the significance of the time-dependent deformation characteristics of rock masses, many researchers have performed extensive creep experiments and the corresponding theoretical analysis to study the time-dependent deformation responses of many rock types under multiple stress loading conditions, such as uniaxial [5,9,10,11], triaxial [12,13,14], and shear [15,16]. However, creep tests are time-consuming [17,18], especially under low stress levels [19]. Even so, the creep duration in laboratory tests is still far shorter than that in practical engineering applications and problems. Moreover, though many theoretical creep models and analytical solutions (e.g., [20,21]) have been proposed and successfully applied to predict the mechanical behaviors of rocks under creep conditions, most of the existing models are limited to the domain of macroscopic phenomenological description, and they are unable to fully reveal the inherent complicated microscopic mechanisms related to rock creep behavior [22]. The significance of some parameters in these creep models remains physically obscure and imprecise.

Numerical simulation can greatly extend the temporal and spatial scale of creep tests (e.g., from laboratory scale to field and engineering scale), enrich the experimental scenarios by readily setting different complicated loading conditions to numerical models, and help better study the mechanism of the creep deformation process. Classic numerical simulation methods based on continuum mechanics, such as the finite element method (FEM), finite difference method (FDM), and boundary element methods (BEM), have been widely utilized to study the rock creep behavior. For example, Xu et al. [10] proposed a two-dimensional FEM model to reproduce the rheological behavior of heterogeneous brittle rocks, where the heterogeneity was described by introducing a randomly distributed local failure field and local material property deterioration following an exponential softening law. The localizations of deformation and microdamage were well captured by this model. Lu et al. [23] proposed a dual-scale model to simulate the creep-induced deformation behavior and cracking of heterogeneous brittle rock materials. Using a damage constitutive law built at the element scale, the presented model directly linked the time-dependent deteriorated mechanical properties and damage-induced anisotropy with microcracking processes. The applicability of this model was validated by FEM simulations of biaxial creep tests. Yang et al. [24] derived a nonlinear visco-elasto-plastic creep equation formed by connecting a Schiffman body, a Hooke body, and a nonlinear visco-plastic body in series and verified its credibility with FDM simulations of multi-step loading creep test on diabase specimens.

However, rock material is a typical kind of porous medium, with intrinsic characteristics of heterogeneity, discontinuity, and anisotropy. The foregoing numerical simulation methods based on continuum mechanics are no longer applicable when the physical processes and micromechanisms inside rocks (e.g., temporal and spatial evolution of microdamage) are required to be well replicated and linked to the macroscopic deformation behavior. Therefore, the discrete element method (DEM) [25] could be more advantageous in representing the realistic microstructure of rock materials and tracking the progressive localization of deformation and damage (e.g., the accumulation and coalescence processes of microcracks in rocks) during the loading process. In this regard, particle flow code (PFC) based on DEM has been extensively adopted to study the mechanical behaviors of rock materials. In PFC, a rock material is usually represented by a dense assembly of non-uniform-sized cylindrical (2D) or spherical (3D) particles cemented at the contacts with a bond, namely the bonded-particle model (BPM) [26]; therefore, the heterogeneous microstructure evolution and micromechanical interaction (e.g., crack propagation) in rocks can be dynamically observed and tracked in real time. Microscale cracking, mesoscale propagation and coalescence of cracks, and macroscale rupture can be well replicated and linked by PFC. Recently, many scholars have used PFC simulations to study the rock creep behavior. For example, Li et al. [27] presented a new combined model to replicate the thermal–mechanical rheological behavior, which blends Burger’s model with the linear parallel bond model in PFC. A comparison between PFC modelling and experimental observation showed that this model can well capture the creep-induced and thermally induced microdamage evolution inside a rock specimen. Potyondy [28] introduced the damage-rate theory to the parallel bond model and established the parallel-bonded stress corrosion (PSC) model, which was found to successfully reproduce the complete creep deformation behavior and time-dependent microcrack evolution inside rocks. Liu and Cai [29] proposed a grain-based stress corrosion (GSC) method according to the PSC model to replicate the creep failure of brittle rock. The results indicated that both the short-term and long-term mechanical properties of Lac du Bonnet granite can be well predicted using this model, which showed better performance than the PSC model under low stress levels. However, there is still limited research on the relationship between macroscopic creep behavior and microdamage evolution and energy conversion, as well as the effect of stress level and confining stress on creep and the underlying microscopic mechanism.

Hence, based on the PSC model, the present study adopts PFC2D simulation to study rock creep behaviors. Both microdamage evolution and energy conversion in the specimen during the creep process are analyzed quantitatively. At the same time, the effect of the axial stress level and the confining stress on the macroscopic creep characteristics and microscopic damage process inside rock are explored.

2. Methodology

2.1. PSC Model and Numerical Implementation

Cohesive frictional geomaterials (e.g., rock and concrete) are modelled as bonded particle systems in PFC [30]. The built-in parallel bond model (PBM) [31] in PFC2D, as shown in Figure 1a, has been extensively used to model the mechanical response of rock and cementitious materials [6,32,33,34]. The PBM can transmit both contact forces and contact moments between two adjacent particles, thereby limiting the relative rotation of the cemented particles. As the stress state in the PBM satisfies the failure criterion, the PBM breaks and degenerates into an unbonded linear contact model.

The progressive subcritical microcrack growth driven by stress corrosion has been generally assumed as the major internal micromechanical mechanism that induces nonlinear brittle creep deformation [29,35,36]. Based on this understanding and the foregoing PBM, Potyondy [28] proposed a parallel-bonded stress corrosion (PSC) model, as presented in Figure 1b. The essence of the PSC model is to introduce a damage rate of the parallel bond diameter ν to the diameter ($\overline{D}$) of PBM, where ν is determined by Equation (1).

$$\nu ={\displaystyle \frac{\mathrm{d}\overline{D}}{\mathrm{d}t}}=\left\{\begin{array}{c}0, \overline{\sigma }<{\overline{\sigma }}_i\\ -{\alpha }_1{\mathrm{e}}^{{\alpha }_2(\overline{\sigma }/{\overline{\sigma }}_c)}, {\overline{\sigma }}_i\le \overline{\sigma }<{\overline{\sigma }}_c\\ -\infty , {\overline{\sigma }}_c\le \overline{\sigma }\end{array}\right.$$

(1)

where $\overline{\sigma }$ is the tensile stress of PBM, ${\overline{\sigma }}_i$ is the critical tensile stress describing the crack initiation, ${\overline{\sigma }}_c$ is the tensile strength of PBM, and ${\alpha }_1$ and ${\alpha }_2$ are two material constants varying with the temperature and chemical environment. From Equation (1) and Figure 1b, it can be seen that time-dependent subcritical microcrack propagation occurs only when ${\overline{\sigma }}_i\le \overline{\sigma }<{\overline{\sigma }}_c$.

The numerical implementation of the PSC model is not provided in PFC; therefore, we used the built-in FISH language to develop an iterative numerical procedure as follows:

• The bonded particle system is firstly run to reach a state of static equilibrium under a specified constant stress condition, where the equilibrium ratio limit is smaller than $f_r$.
• The maximum tensile stress of each PBM ($\overline{\sigma }$) is obtained by considering the PBM as a beam with a circular cross section, and the stress state is calculated using the beam theory. The elapsed time for the next PBM breakage ($t_n$) is estimated by Equations (2) and (3). By dividing $t_n$ into $n_t$ copies, each stress corrosion time increment is $\Delta t=t_f/n_t$. The diameter of each PBM ($\overline{D}$) is decreased by $v∆t$ to a new diameter ($\overline{D}\prime =\overline{D}-v∆t$) after a time increment of $∆t$, where the decreasing rate $v$ of each bond is computed by Equation (1). The diameter of each PBM, i.e., the initial diameter in each PSC model, is determined by $\overline{D}=\overline{\lambda }\mathrm{m}\mathrm{i}\mathrm{n}(D_1,D_2)$, where $\overline{\lambda }$ is the bond radius multiplier and $D_1$ and $D_2$ are the diameters of the connected particles.
• When one stress corrosion time increment ($∆t$) is completed, the bonded particle system will be relaxed again. If at least one bond breakage occurs, $∆t$ will be redetermined as the procedure of (2). If not, the time increment of $∆t$ will be multiplied by $f_m$ (i.e., $∆t=f_m∆t$) after $n_t$ times calculations in order to automatically modify the value of the time increment and accelerate computation.
• Steps (1)–(3) are continuously repeated until rock sample failure.

$$t_n=\left\{\begin{array}{c}+\infty , \overline{\sigma }<{\overline{\sigma }}_i\\ {\displaystyle \frac{\overline{D}-{\overline{D}\prime }_p}{{\alpha }_1}}{e}^{-{\alpha }_2(\overline{\sigma }/{\overline{\sigma }}_c)},{\overline{\sigma }}_i\le \overline{\sigma }<{\overline{\sigma }}_c\end{array}\right.$$

(2)

$${\overline{D}\prime }_p=\mathrm{m}\mathrm{a}\mathrm{x}(\overline{D}{\overline{\lambda }}_a,{\displaystyle \frac{-{\overline{F}}^n\pm \sqrt{{\left({\overline{F}}^n\right)}^2+24\left|{\overline{M}}^s\right|{\overline{\sigma }}_c}}{2{\overline{\sigma }}_c}},{\displaystyle \frac{\left|{\overline{F}}^s\right|}{{\overline{\tau }}_s}})$$

(3)

where ${\overline{\tau }}_c$ is the tangential bond shear strength; ${\overline{F}}^n$, ${\overline{F}}^s$, and ${\overline{M}}^s$ are the normal force, the tangential force, and the moment acting on the bond, as shown in Figure 1a; and ${\overline{\lambda }}_a$ is the diameter parameter, which prevents the bond diameter ($\overline{D}$) from falling below an allowable value (e.g., a negative value).

Notably, this study employs PFC2D, in which the torsional moment on bonds is zero. Consequently, torsion on bonds is not included in the calculation. However, if PFC3D is used, this moment should be incorporated into Equations (2) and (3) to determine the maximum stress based on beam theory. The reasons for adopting the 2D model PFC2D rather than the 3D model PFC3D to simulate the cylindrical specimen in realistic experiments are as follows: (1) Although a 3D model more closely resembles the real specimen, it is significantly more computationally demanding, particularly for rock creep simulations using the PSC model, which involves complex implementations. A 2D model can greatly reduce computational costs while still capturing the essential mechanical characteristics and mechanisms, such as stress distribution and crack propagation. (2) Microdamage evolution in a 2D model is more intuitive and easier to visualize and analyze. In contrast, 3D models introduce complex spatial distributions of microdamage, making visualization and interpretation more difficult. (3) It is a conventional practice in PFC simulations to use 2D models to represent cylindrical samples in realistic experiments. However, simplifying a 3D model to 2D inevitably introduces limitations. A 2D model cannot fully capture the three-dimensional crack propagation paths, which may cause some deviations in describing the failure mode. In addition, only in-plane forces are transmitted between particles in 2D, and components such as torsional moments are neglected. Despite these limitations, by calibrating the microscopic particle parameters, the 2D model in this study successfully reproduces the macroscopic mechanical response, and its consistency with experimental results confirms its applicability for the core research objectives.

2.2. Determination of Micromechanical Parameters Using Experimental Data

The investigated sandy mudstone samples throughout this study were taken from a one-kilometer-deep tunnel of the Kouzidong coal mine in Fuyang City in China’s Anhui Province. Dry cylindrical specimens with a core diameter of 50 mm and a height of 100 mm were fabricated for uniaxial compression tests and uniaxial creep tests, which provided further experimental data for numerical studies. In the uniaxial compression test, the samples were loaded with a constant displacement rate of 0.01 mm/min. A representative experimental stress–strain behavior is presented in Figure 2. In the uniaxial creep tests, the samples were axially loaded first with a loading velocity of 0.01 mm/min to the stress level of 90% of the uniaxial compressive strength (UCS) and then continuously maintained for 24 h. A typical axial strain–time curve of sandy mudstone specimens is presented in Figure 3. The mechanical tests were conducted using an MTS816 electrohydraulic servo rock mechanics testing system. The axial strain was calculated by dividing the measured axial deformation by the initial specimen height, while the axial stress was determined by dividing the applied load by the cross-sectional area of the rock sample. Based on these measurements, the experimental axial stress–strain curve from the uniaxial compression test and the axial strain–time curve from the uniaxial creep test were obtained, as shown in Figure 2 and Figure 3, respectively.

The PFC2D sample was 100 mm × 50 mm, the same as the foregoing experimental sample, containing 5659 cylindrical particles. The aim of the calibration was to ensure the numerical mechanical response is in accordance with the experimental mechanical response, and this was achieved by continuously adjusting the micromechanical parameters. As the PBM and PSC models were adopted in our study, three kinds of micromechanical parameters needed to be calibrated, namely particle parameters, PBM parameters, and PSC parameters. As for the particle ones, the particle density ($\rho$) was set to the same value as the experimental sample. The particle size parameters ($R_{min}$ and $R_{max}$) were selected based on the available computing capacity. These values also fell within the range reported in previous PFC2D simulation studies [28]. The particle effective stiffness and bond effect stiffness ($E_c$ and ${\overline{E}}_c$) and the corresponding ratios of normal to shear stiffness ($k_n/k_s$ and ${\overline{k}}_n/{\overline{k}}_s$) were set the same to reduce the free parameter number. The particle friction coefficient ($\mu$) mainly influences the post-peak response. Therefore, based on a comparison between numerical and experimental curves, a reasonable non-zero value was used for $\mu$. The primary objective of this study is to investigate the intrinsic creep behavior of rock induced by microdamage evolution under axial stress, with particular emphasis on the stress corrosion mechanism of parallel bonds. Friction between particles and walls introduces additional lateral constraints and shear forces, which are not the focus of this research. To eliminate the interference of contact friction on stress transmission, the walls are modeled as frictionless. This ensures that the observed creep deformation and microcrack evolution are driven primarily by internal bond degradation rather than by external frictional interactions. In laboratory uniaxial compression and creep tests, it is common practice to polish the samples and apply lubricants between the specimen ends and platens to establish approximately “frictionless” conditions. This reduces the end effects and promotes a uniform axial stress distribution within the specimen. Our adoption of frictionless walls in the simulations mirrors these experimental measures, thereby enhancing the comparability between numerical and laboratory results. For this reason, opposing frictionless walls were used as the loading platens in the model. Then, a set of PBM parameters was determined by the trial-and-error method to replicate the mechanical properties of the experimental sample (Figure 2). The local damping coefficient of particles was set to 0.2 in this study. This coefficient is dimensionless and typically ranges from 0.0 to 0.7, with values of 0.1–0.3 generally suitable for quasi-static simulations, while higher values, such as 0.7, are used for rapid relaxation. Calibration of this parameter was performed using a trial-and-error approach through uniaxial compression tests. If oscillations appeared in the stress–strain curve, the coefficient was increased. Conversely, if the peak strength was significantly lower than the experimental results, it indicated that the damping was too high, and the coefficient was reduced accordingly. Next, with PBM parameters fixed and some parameters ($f_r$, $n_t$, $f_m$, and ${\overline{\lambda }}_a$) determined according to the recommended values from Potyondy, the micromechanical parameters of PSC in the PFC2D sample were also calibrated according to the experimental creep data (see Figure 3) by the trial-and-error process [6]. Through this calibration scheme, the micromechanical parameters were determined and are listed in

Table 1. The micromechanical parameters adopted in the numerical sample.

	Micromechanical Parameters	Values
Particle	The minimum radius of the particle, $R_{min}$ (mm)	0.4
	The maximum radius of the particle, $R_{max}$ (mm)	0.6
	Particle density, $\rho$ (kg/m3)	2619.3
	Local damping coefficient of particles, $f$	0.2
	Effective stiffness of particles, $E_c$ (GPa)	11.5
	Ratio of normal to tangential stiffness of the particle, $k_n/k_s$	0.95
	Particle-particle friction coefficient, $\mu$	0.3
	Stiffness ratios of walls to particles	1.5
	Wall-particle friction coefficient	0
PBM	Effective stiffness of parallel bonds, ${\overline{E}}_c$ (GPa)	11.5
	Ratio of normal to tangential stiffness of the parallel bond, ${\overline{k}}_n/{\overline{k}}_s$	0.95
	Parallel bond normal tensile strength, ${\overline{\sigma }}_c$ (MPa)	17
	Standard deviation of normal mean strength, $△{\overline{\sigma }}_c$ (MPa)	3.5
	Parallel bond tangential shear strength, ${\overline{\tau }}_c$ (MPa)	23
	Standard deviation of tangential mean strength, $△{\overline{\tau }}_c$ (MPa)	4.7
	Bond radius multiplier, $\overline{\lambda }$	1.0
PSC	Material parameter, ${\alpha }_1(\times 10^{-15})$	25
	Material parameter, ${\alpha }_2(\times 10^{-15})$	30
	Critical tensile stress, ${\overline{\sigma }}_i$ (MPa)	1.0
	Implosion multiplier, ${\overline{\lambda }}_a$	0.01
	Equilibrium ratio limit,$f_r$	1.0 × 10−4
	Stress corrosion time increment subdivision factor, $n_t$	4
	Stress corrosion time increment scaling factor, $f_m$	2.0

. In all the following PFC simulations, the micromechanical parameters adopted were consistent. Note that the aim of this work is to employ the PSC model to study microdamage evolution in conjunction with macroscopic creep deformation and to investigate the effects of axial stress level and confining stress. To this end, we first calibrated the PSC parameters against experimental data from the uniaxial compression test and the uniaxial creep test to validate the model’s accuracy. We then created additional scenarios with varying axial stress levels and confining stresses for further analysis. Though extensive validation against experimental data would enhance the reliability of the model, such comprehensive validation is not the primary focus of this study, and we plan to address it in future work.

3. Results and Discussion

Comparisons were made between experimental and numerical stress–strain behaviors under the uniaxial compression test, as shown in Figure 2. Using the micromechanical parameters in Table 1, under the uniaxial compression test, the mechanical properties of the rock sample given by the PFC simulation show good agreement with the experimental curve. The axial stress–strain curves given by the PFC simulation are generally located at the upper part of the laboratory test curve, and the strain at peak strength obtained by the simulation is smaller than that obtained by the laboratory test. This is because there are a lot of defects in the realistic rock materials, such as pores and cracks. The initial closure and compaction of these pre-existing pores and cracks causes larger deformations in laboratory tests [32,37,38].

Figure 3 presents the time-dependent deformation behavior of the rock sample under the axial stress level of 90% uniaxial compressive strength (UCS) when the PSC micromechanical parameters in Table 1 are adopted. The numerical curve agrees quite well with the experimental curve and successfully reproduces the four phases of the idealized creep response [39,40], namely the instantaneous elastic strain, the decelerating creep, the stable creep, and the accelerating creep. This validates the applicability of the PSC model. However, the simulated creep curve has a larger strain rate and amount of deformation in the accelerating creep stage. Note that the numerical curve exhibits an obvious jumpy or stair-stepping pattern, which is absent in the experimental curve. This arises because the parallel bond model (PBM) in PFC2D represents the rock material as an assembly of discrete particles connected by parallel bonds. Macroscopic deformation is governed by the initiation, propagation, and coalescence of microcracks, corresponding to the breakage of these bonds. When the local stress exceeds the bond strength, multiple bonds may break simultaneously, releasing stored strain energy abruptly. This sudden release leads to a sharp drop in load-bearing capacity or a rapid increase in strain, producing the “jumps” observed in the stress–strain or strain–time curves. Due to computational limitations, the numerical model in this study contains only several thousand particles. At this scale, the macroscopic response is strongly affected by localized microcracking events, resulting in the observed discontinuous pattern. If the particle number is increased by several orders of magnitude, approaching that of a real rock specimen, these patterns would be significantly smoothened or even eliminated.

3.1. Creep-Induced Microdamage Evolution and Energy Conversion

Microcrack Development. Figure 2 quantitatively presents the microcrack evolution inside the sample during the loading process of the uniaxial compression test. Figure 3 shows the creep-induced temporal and spatial variation of microcracks inside the sample as the axial stress level is 90%, where $t_f$ is the (creep) lifetime of the sample under a given constant loading and $t$ is the creep time (elapsed time since constant loading was applied). For quantitative comparison purposes, the creep time ($t$) can be normalized by creep failure time ($t_f$) as $t/t_f$. The creep failure time is defined as the moment when an abrupt increase in axial strain occurs, corresponding to the stage at which numerous microcracks propagate and coalesce to form a macroscopic fracture in the sample, as shown in Figure 3 at approximately 10,000 s. It can be seen that the variation of microcrack number is in accordance with that of the axial creep strain, indicating that the growth, coalescence, and penetration of microcracks inside the rock sample are key micromechanical mechanisms controlling the creep deformation behavior of the sample. In the decelerating creep phase (e.g., ${\displaystyle \frac{t}{t_f}}=0.019$), a certain amount of microcracks is generated, and there is a scattered spatial distribution in the sample. In the stable creep phase (e.g., ${\displaystyle \frac{t}{t_f}}=0.803$), the microcrack number grows slowly and stably, indicating the steady propagation of microcracks in certain stress-concentrated locations of the sample. In the accelerating creep stage (e.g., ${\displaystyle \frac{t}{t_f}}=1$), the microcrack number increases significantly and sharply, indicating the unstable propagations of microcracks and the penetrations of multiple macroscopic fracture surfaces.

Compared with the failure pattern of the rock sample experiencing the uniaxial compression test (Figure 2), the failure pattern of the rock sample under the uniaxial creep test (Figure 3) is more scattered and more fractured. Only 1–2 macroscopic fracture planes are observed in Figure 2, and the total microcrack number developed in the loading process is ~200 when the maximum stress is reached. However, multiple extra orthogonal sub-fracture surfaces can be found in Figure 3, with a total microcrack number being ~1500, which indicates that the microdamage in the sample is greatly deepened under the creep loading condition. In addition, the ratio of the tensile microcrack number to the shear microcrack number is ~7 in Figure 3 but only ~3.5 in Figure 2, indicating that the creep loading condition leads to more tensile failure in the microstructure of the sample.

Energy Conversion. Throughout the loading processes of the uniaxial compression test and the uniaxial creep test, various energies in the specimen were monitored, and they satisfy the following relationship:

$$E_{Boundary}=E_{strain}+E_{pbstrain}+E_{slip}+E_{dashpot}+E_{damp}+E_{kinetic}$$

(4)

where $E_{Boundary}$ is the work done by the loading end on the bonded particle system, namely the total input energy; $E_{strain}$ is the strain energy of the deformed particles; $E_{pbstrain}$ is the strain energy stored in the bonds; $E_{slip}$ is the slip energy dissipated by frictional slip between particles after bond breakage; $E_{dashpot}$ is the energy dissipated by the dashpot forces at the contacts; $E_{damp}$ is the energy dissipated by local damping forces on the particles; and $E_{kinetic}$ is the kinetic energy of particles.

The variation curves of energies in the sample under the uniaxial compression test are shown in Figure 4a. At the beginning of the loading process, the boundary energy is largely transferred into the particle strain energy and bond strain energy, namely the elastic strain of the bonded particle system. Parabolic shapes are reasonably obtained for the curves of $E_{Boundary}$, $E_{strain}$, and $E_{pbstrain}$. When the microcracks in the sample begin to appear at an axial strain of around 0.05–0.10%, the strain energy of the grain framework starts to be dissipated and converted into $E_{slip}$, $E_{dashpot}$, $E_{damp}$, and $E_{kinetic}$, associated with the micromechanical interactions in the realistic rocks such as the initiation, nucleation, and penetration of microcracks; microcrack compaction and closure; relative dislocation along fracture surfaces; grain boundary cracking, opening, and sliding; lattice distortion; and dislocation slip of mineral grains. After the peak of axial stress, a large amount of strain energy is rapidly released through these microscopic processes as the total input energy has exceeded the value that the bonded particle system can store and bear.

Accordingly, the evolution curves of different energies in the sample under the uniaxial creep test are presented in Figure 4b. The variations of energies are also found to be consistent with that of the axial creep strain. For example, the stable energy dissipation on the curves corresponds with the steady propagation of microcracks in the stable creep stage, and the rapid energy dissipation and release on the curves matches the abrupt increase in microcrack number in the accelerating creep phase. The continuous accumulation of strain energy on particles and bonds and the continuous energy dissipation through the microcracking process can be observed throughout the creep process. Due to the subcritical microcrack propagation induced by the stress corrosion mechanism, the mechanical properties of the bonded particle system constantly deteriorate and the deformation of the sample constantly increases, which causes the loading end to continuously do positive work (input energy) on the bonded particle system, leading to further energy accumulation and dissipation in the sample.

For further analysis and comparison, the energy evolution during the uniaxial compression test in Figure 4a shows that, after reaching peak stress, the rock sample fails and the accumulated elastic energy is dissipated as slip energy and damp energy. Consequently, the particle strain energy and bond strain energy—representing the energy stored in the grain framework—decrease, while the slip and damp energies increase. In the creep tests shown in Figure 4b, the boundary energy continues to rise due to progressive deformation under constant axial stress. Throughout this process, the particle strain energy and bond strain energy accumulate, storing deformation energy, while microcracks gradually form within the sample, leading to continuous dissipation of energy as slip and damp energies. At approximately 10,000 s, a sudden macroscopic fracture occurs, and the energy stored in the grain framework is abruptly released, dissipating as particle kinetic energy, dashpot energy, slip energy, and damp energy. This abrupt release results in simultaneous decreases in particle strain energy and bond strain energy. These energy evolution patterns highlight the brittle mechanical behavior of the studied rock materials.

3.2. Effect of Axial Stress Level

Uniaxial creep tests under multiple axial stress levels (60, 70, 80, and 90% of UCS) were performed on numerical models. The obtained axial creep strain curves in Figure 5 indicate that the specimens will remain stable for a long time at the axial stress levels of 60 and 70%. The creep failure time ($t_f$) is several days and several months, respectively. However, when the axial stress level increases to 80 and 90%, creep failure of samples will occur in a short time (e.g., 2.78 h under the axial stress level of 90%). Therefore, with the axial stress level increasing, the creep failure time is significantly decreased.

With regard to the axial creep rate, Figure 6 presents the results of linear regression of the steady-state (stable) creep stage under different axial stress levels. It can be seen that the axial creep strain rate in the constant-strain-rate creep phase increases with the axial stress level increasing, appearing to be stabilized at about 1.15 × 10⁻¹¹ s⁻¹, 4.81 × 10⁻¹¹ s⁻¹, 6.57 × 10⁻¹⁰ s⁻¹, and 6.97 × 10⁻⁹ s⁻¹ when the axial stress level is 60%, 70%, 80%, and 90%, respectively. With an increment of 10% in the axial stress level, there is around a 4–14 times increase in the creep rate. This finding indicates that the stress level has a considerable influence on the long-term stability of the rock, which is consistent with the findings on other rocks from previous studies [41,42]. Figure 6 also shows that, with the axial stress level increasing, the axial creep deformation curves are inclined to have stair-stepping patterns, which is in accordance with the results of the linear regression that the higher the axial stress level, the smaller the determination (regression) coefficient ($R^2$), indicating that the nonlinear characteristics of creep deformation of the samples under higher stress levels are strengthened. As a matter of fact, under a high axial stress level, a specimen enters the stage of unstable crack propagation. The unstable growth and penetration of cracks will lead to intensive and rapid strain energy release and dissipation. This can explain why a stair-stepping pattern is observed for the creep deformation curve under high axial stress levels.

Different axial stress levels lead to different failure modes of samples. The spatial distributions of microcracks in the failed samples under various axial stress levels are presented in Figure 7, which shows that the cracking patterns under high axial stress levels exhibit more serrations and branches (wing cracks) in the specimens, leading to a more scattered failure pattern formed by multiple intersecting fractures. This is because higher axial stress levels will result in subcritical crack propagation induced by the stress corrosion mechanism occurring in more places of the sample, leading to more dispersed failure patterns of the samples. However, under lower axial stress levels, the local stress concentration usually occurs at few locations where cracks have been generated during the instantaneous elastic deformation stage. In these locations, the subcritical cracking induced by the stress corrosion mechanism develops slowly and stably with time, which is manifested as relatively concentrated damage zones and fewer fracture surfaces, as presented in Figure 7. It is also found that the fractures in the samples tend to be parallel to the loading direction under lower axial stress levels, and the overall failure mode is closer to the axial splitting than that under higher axial stress levels.

For quantitative comparison purposes, the creep time ($t$) is normalized by creep failure time ($t_f$) as $t/t_f$. Figure 8 shows the temporal evolution curves of total microcracks, tensile microcracks, and shear microcrack numbers in the samples under different axial stress levels, which yields the following conclusions:

• Many microcracks are formed inside the samples after the initial instantaneous elastic deformation stage, with a higher axial stress level resulting in more microcracks generating inside the sample. In the creep failure process of all studied axial stress levels, there are always more tensile microcracks than shear microcracks.
• The creep failure process of the sample under a higher axial stress level is more sudden and more abrupt. A large amount of microcracks occur at the beginning of the accelerating creep phase, resulting in the rapid microstructural and micromechanical deterioration of the samples within a short time. A higher axial stress level causes a sudden rock failure, embodied by the sudden increases in microcrack number curves at the stage of accelerating creep. A lower axial stress level leads to a relatively smooth and gradual rock failure, embodied by the relatively steady increase in microcrack number curves at the stage of accelerating creep, and no obvious points of sudden increase can be observed. This is because the higher axial stress level rapidly applied on the sample causes a large amount of strain energy to accumulate inside the sample, and the energy is quickly released in the subsequent subcritical damage evolution induced by the stress corrosion mechanism, further resulting in the rapid and sudden rock failure.

3.3. Effect of Confining Stress

Confining stress is another important factor that influences rock creep behaviors and the corresponding time-dependent microdamage evolution processes. Two extra lateral loading platens were used to apply a specified value of confining stress (5, 10, and 15 MPa). Figure 9 presents the axial creep deformation curves of the rock sample under different confining stresses as well as the corresponding temporal evolution of microcrack numbers inside the rock sample. The following conclusion can be drawn from this figure:

• The confining stress significantly increases the creep failure time of the rock sample. Under an axial stress level of 90% UCS, the creep failure time for a rock sample without lateral confinements is about 2.78 h, which increases to ~17 days, ~1 year, and ~25 years when the lateral confining stress is 5, 10, and 15 MPa, respectively. This reveals that imposing lateral confinements can greatly improve the long-term stability of rock structures, effectively preventing and delaying the occurrence of creep failure. The micromechanism is that the higher confining stress compacts the bonded particle system and restricts the tensile stresses between cemented particles, which reduces the subcritical cracking caused by the stress corrosion mechanism inside the sample and lengthens the creep failure time.
• With the lateral confining stress increasing, the value of axial instantaneous elastic strain decreases, indicating that confining stress increases the instantaneous deformation stiffness of the specimen by restricting the lateral deformation of the specimen.
• The higher confining stress leads to a smoother axial creep curve, as presented in Figure 9d, indicating weakened brittleness and strengthened ductility of the rock sample. It can be seen that the higher the confining stress, the less likely for the rock sample to eventually exhibit an instantaneous brittle fracture. Under a lower lateral confining stress, the duration of the accelerating creep phase in the axial creep curve is short, while under a higher lateral confining stress, the accelerating creep phase lasts for a longer period. This is because lateral confinement strengthens the bearing capacity of the rock sample. With the confining stress increasing, the number of microcracks generated in the sample increases, indicating that more damage is required to cause creep failure of the sample under confining stresses.

Figure 10 further presents the failure pattern of the rock sample under different lateral confining stresses. It can be seen that when the confining stress is applied, the ultimate failure pattern of the sample is basically similar, with the macroscopic fracture surfaces aligned at ~45° to the axial loading direction. As the confining stress increases, this trend becomes more apparent. In addition, a higher confining stress leads to a more dispersed, more extensive, and more uniform microcrack distribution in the rock sample.

4. Conclusions

This study aimed to better understand the creep-induced microdamage evolution and energy conversion inside rock as well as determine the effect of axial stress level and confining stress on the macroscopic mechanical response and microstructural interactions. Using the PSC model, a series of PFC simulations were performed, and the main findings are summarized below:

• The PSC model accurately reproduces the four-stage nonlinear creep deformation of rocks—instantaneous elastic strain, decelerating creep, steady-state creep, and accelerating creep—with numerical curves in close agreement with uniaxial creep test results. Calibration of particle, PBM, and PSC parameters captures the correlation between macroscopic stress–strain behavior and microscopic crack evolution, verifying the model’s effectiveness in revealing creep mechanisms.
• Creep-induced microdamage evolution and energy conversion correspond closely to the axial creep deformation stages: dispersed cracks in the decelerating stage, slow growth in the steady stage, rapid increase in the accelerating stage, and coalescence fracture surface upon failure. Compared with uniaxial compression, creep produces 7.5 times more cracks, with a higher proportion of tensile cracks, indicating that stress corrosion more readily induces tensile microdamage.
• Axial stress level strongly influences creep lifetime: specimens remain stable at 60–70% uniaxial compressive strength (UCS) but fail rapidly at 80–90% UCS. Each 10% stress increase raises the creep rate by 4–14 times. At high stress, creep curves exhibit stepped, nonlinear patterns caused by rapid strain energy release from sudden microcrack expansion, and failure modes feature more scattered, serrated, and branched cracks, forming multiple fracture surfaces.
• Confining pressure markedly extends creep life and alters failure characteristics. At 90% UCS, 15 MPa confinement extends life from 2.78 h to ~25 years by reducing tensile stress between particles and slowing subcritical crack growth. It also increases stiffness, enhances ductility, and shifts failure to shear at ~45° to the axis, with more uniformly distributed microcracks. This demonstrates that lateral restraint improves long-term rock stability by suppressing tensile damage and dispersing stress concentrations.