Creep failure characteristics and damage creep model of red layer soft rock based on Perzyna viscoplastic theory

Abstract

Rock creep is essentially a process of damage accumulation. According to the damage evolution of rock under creep conditions, the TAW2000 triaxial test system is used to carry out triaxial creep test on red layer soft rock taken from Yibo Tunnel in Leshan, Sichuan Province, China, and analyzed the creep deformation rule under different confining pressures. Meanwhile, based on the Nishihara model and Weibull distribution function and Perzyna viscoplastic theory, an improved viscoelastic-plastic creep model which can describe the whole process of rock creep failure was established. The critical point damage variable is defined by dividing the creep stage, so that the acceleration creep start time can be determined more accurately. The results show that: (1) The model curves in this paper fit well with the test data, indicating that Weibull distribution function is feasible to describe rock creep damage, and the accuracy and rationality of the model in this paper are verified. (2) Based on Perzyna viscoplastic theory, a more accurate viscoplastic strain expression was established to describe accelerated creep. (3) By defining the critical point damage variables of different creep stages, the relationship between rock creep deformation and damage can be better reflected, which makes up the shortcoming that Nishihara model cannot describe accelerated creep, and enriches the creep constitutive theory of rock materials.

Introduction

In underground engineering, surrounding rock often exhibits significant rheological behavior in complex environments, which is typically reflected through the establishment of rheological models to characterize its deformation properties¹. Existing rheological models can be broadly classified into three categories: empirical models, integral models, and component models². Among them, empirical models are developed based on measurement data of construction field and classical rheological theories, integral models are constructed by integrating classical models using rheological theory, and component models are established using series and parallel rules³. Of these three models, component models are the most widely used. This approach has clear model parameters, and its derivation process is simple, making it widely adopted by scholars both domestically and internationally⁴. But, most existing component models rarely address the internal micro-element damage in rocks, making it difficult to determine the relationship between microscopic level damage to microelements and creep deformation.

In recent years, a large number of scholars have conducted extensive research on the rheological constitutive theory of rocks, and have achieved fruitful results^5,6. Jing et al.⁷ addressing the long-term stability of the surrounding rock in deep soft rock tunnel roofs, built a strain-softening model considering the three stages of rock rheology. The model was validated using field measurement data, and they proposed the concept of the elastic creep modulus. Li et al.⁸, through uniaxial creep fatigue tests, studied the interaction mechanism between fatigue and creep within the internal structure of salt rock, analyzed the evolution of salt rock creep fatigue damage over time under different load levels. Liu et al.⁹ through uniaxial cyclic loading and unloading creep tests, studied the creep deformation rules of sandstone under different temperatures, and proposed a fractional-order viscoelastic-plastic creep model considering initial temperature-induced damage and validated the model based on experimental results. Xia et al.¹⁰ using a Discrete Element Method-Finite Element Method (DEM-FEM) coupled modeling approach, developed a multi-phase numerical model of soft rock creep in tunnel excavation, considering macroscopic and mesoscopic creep damage in rock masses, and the model was validated with field experimental data. Wang et al.¹¹ used the FLAC3D numerical simulation method, combined with a roadway surrounding rock creep model, analyzed the stability of exposed filling materials rules of change over time, and also examined the instability mechanism of the filling material under different creep rates and burial depth conditions. Wang et al.¹², focused on the creep issues of shale during unconventional shale gas extraction, through nanoindentation creep tests and uniaxial compression creep tests to study the long-term deformation characteristics of shale, and analyzed the creep damage mechanisms of shale at both macroscopic and microscopic scales and compared the differences in the parameters of the Burgers model at these two scales. Guner et al.¹³ applied four different constant loads to TSL (Thermal-Set Liquid) materials in indoor test, observed the deformation characteristics during long-term curing, and analyzed the time-dependent properties, creep characteristics, and rupture modes of TSL materials, and developed a four- element nonlinear creep model based on the Burgers model. Liu et al.¹⁴ based on the Kinetic energy theory, Perzyna’s viscoplastic theory, and the Nishihara model, established a unified creep constitutive model capable of describing the entire process of attenuation creep, steady-state creep, and accelerating creep. The model was validated through experimental triaxial creep tests. Chen et al.¹⁵ proposed a new microstructure-based creep constitutive model, aimed at comprehensively characterizing the complex mechanical behavior of anisotropic claystone under saturated conditions. The proposed constitutive model integrates elastic–plastic deformation, time-dependent behavior, and induced damage, enabling a more accurate simulation of rock creep behavior under actual working conditions. Ran et al.¹⁶ conducted single-stage and multi-stage uniaxial compression creep tests to study the creep failure mechanisms of cemented gangue filling materials in high-strain zones, and analyzed the creep failure modes of cemented gangue filling and the evolution rules of strain energy. Meng et al.¹⁷ used the particle discrete element method to establish a numerical model for gangue under triaxial compression and proposed a new criterion for rock debris damage, and explored the relationship between the expansion coefficient and time effect of broken rock mass in the caved zone, especially under triaxial compression conditions, in the roof of mined-out areas in the caving zone. Abolfazl et al.¹⁸ analyzed the effectiveness of their model for simulating the accelerated creep stage of limestone by comparing the model’s performance with creep test results from different types of rocks. Ma et al.¹⁹ conducted nanoindentation creep tests to calculate the mechanical parameters of common rock-forming minerals (such as feldspar, quartz, and mica), including Young’s modulus, hardness, fracture toughness, creep strain rate, stress index, activation volume, and maximum creep displacement, and analyzed the creep mechanical properties of granite from a microscopic perspective. Mohammadreza et al.²⁰ studied the mechanical behavior and failure prediction of rocks under constant loads over time. Guo et al.²¹ investigated the creep mechanical behavior of crushed mudstone through uniaxial compression creep tests, analyzed the mechanism of action of original particle composition and axial stress on the compaction characteristics of crushed mudstone. Song et al.²² used acoustic emission technology to perform true triaxial creep tests on typical thin-layer siliceous slate, revealing the anisotropic crack characteristics of the slate as they evolved over time.

In summary, there have been detailed discussions on rock element models, but there are relatively few studies that consider rock damage at the microscopic level. This paper is based on the Nishihara model, combined with Weibull distribution function and Perzyna viscoplastic theory, to establish a rock creep model considering micro damage, and verifies the model through triaxial creep test data of red layer soft rock in Yibo Tunnel Sichuan Province, China. The proposed model in this paper provides a deeper understanding of the relationship between deformation and damage of rocks during creep, which is of great significance for predicting the long-term stability and safety of rocks in engineering.

Triaxial creep test

Sample preparation and testing equipment

The red layer soft rock used in this experiment was collected from the Yibo Tunnel construction site with a buried depth of about 800 m. The tunnel site is located at the junction of the famous Sichuan-Yunnan meridional structural system and the Qinghai-Tibet Yunnan-Burma-Indonesia structural system. The particle size distribution is uniform, and no obvious joints were observed on the sample surface. In its natural state, the rock is sorrel, with a dry density ranging from 2.11 to 2.32 g/cm³ and a porosity of 0.62 to 0.71%. To minimize experimental errors caused by sample discreteness, all specimens were taken from the same intact rock block. According to international rock mechanics standards, standard cylindrical specimens with a diameter of 50 mm and a height of 100 mm were prepared. The mineral composition of the red layer soft rock was analyzed using a Japan Rigaku TTR-III X-ray diffraction (XRD) instrument. According to the test results, the primary mineral components of the red layer soft rock are quartz (44.11%), albite (19.12%), calcite (16.08%), microcline (9.31%), montmorillonite (6.69%), illite (1.37%), hematite (1.22%), clinochlore (0.95%), kaolinite (0.73%) and other impurities (0.42%). The standard red layer soft rock specimen and the XRD infrared diffraction results are shown in Fig. 1.

Fig. 1.

Construction site of Yibo tunnel and red layer soft rock specimen and XRD test results.

The triaxial creep tests on the red layer soft rock in this paper were conducted using the TAW2000 multifunctional electro-hydraulic servo triaxial testing system, as shown in Fig. 2. This system, produced by Changchun Chaoyang Testing Machine Equipment Co., Ltd., is a fully automated multifunctional electro-hydraulic servo triaxial testing system specifically designed for rock-like materials. It consists of three modules: the loading module, the testing module, and the control module. The system features independent closed-loop servo control for axial pressure, confining pressure, and pore water pressure. The overall stiffness of the equipment is 7.0 × 10⁹ N/m, with a maximum axial pressure of 1600 kN, a maximum confining pressure of 70 MPa, and a maximum pore water pressure of 70 MPa. And it can perform various loading control modes, meeting the experimental requirements of this study.

Fig. 2.

TAW2000 triaxial test system.

Conventional triaxial compression test

First, a triaxial compression test was conducted on the red layer soft rock to determine the load levels for the triaxial creep test. Based on the actual burial depth of the tunnel, confining pressures of 5 MPa, 10 MPa, 15 MPa, and 20 MPa were applied in the triaxial compression tests. Prior to loading, a small amount of petroleum jelly was applied to both ends of the sample to reduce the impact of uneven end surfaces on the test results. The specific testing procedure is as follows:

1. Apply the confining pressure to the sample to the preset value and maintain it constant. The loading mode is force-controlled, with a loading rate of 500 N/s.

2. Apply axial pressure to the sample until instability and failure occur. The loading mode and rate are the same as those for the confining pressure.

3. Derive the test data and plot the stress–strain curve.

To minimize errors caused by sample discreteness, three parallel tests were conducted for each confining pressure, and the most representative test results were selected. Figure 3 shows the complete stress–strain curve and failure mode of the red layer soft rock at a confining pressure of 5 MPa and 20 MPa. The mechanical parameters of the red layer soft rock at different confining pressures are presented in Table 1.

Fig. 3.

Whole process stress–strain curve and failure mode of red layer soft rock: (a) Confining pressure 5 MPa; (b) Confining pressure 20 MPa.

Table 1.

Mechanical parameters of red layer soft rock.

Confining pressure σ3, (MPa)	Peak strength σpk, (MPa)	Axial peak strain εpk	Elastic modulus E, (MPa)	Poisson’s ratio μ	Cohesion c, (MPa)	Internal friction Angle φ, (°)
5	52.51	0.0074	6.84	0.316	4.27	32.05
10	65.73	0.0080	7.70	0.308
15	80.19	0.0081	9.01	0.302
20	96.31	0.0088	11.21	0.298

As shown in the figure, the triaxial loading failure process of the red layer soft rock follows a pattern similar to the classic stress–strain curve of rocks, which can be divided into four stages: microcrack compression stage, elastic stage, plastic yielding stage and post-peak stage. Since the red layer soft rock samples were collected from the Yibo Tunnel construction site, with a burial depth of approximately 600 m, the integrity of the samples is relatively good. As a result, the samples exhibited a brittle shear failure mode under different confining pressures and generally conformed to the Mohr–Coulomb strength criterion. As shown in the table, as the confining pressure increased from 5 to 20 MPa, there were significant changes in the mechanical parameters of the red layer soft rock. Among them, the peak strength increased by 45.48%, the peak strain increased by 15.91%, the elastic modulus increased by 38.98%, and the Poisson’s ratio decreased by 5.69%.

Triaxial creep test

To analyze the long-term deformation characteristics of the red layer soft rock, a triaxial compression creep test was conducted, and the specific methods are as follows:

1. Based on the peak strength from the indoor triaxial compression tests, the load levels for the creep tests under different confining pressures were determined. The initial load level was set at 50% of the peak strength, and subsequent load levels were set at 60%, 70%, 80%, and 90% of the peak strength. This ensures that the sample experiences instability and failure within five load levels.

2. Apply the confining pressure to the preset value and maintain it constant. The loading mode is force-controlled, with a loading rate of 500 N/s.

3. Apply the axial load to the sample until the preset value is reached and maintain it constant. The loading mode and rate are the same as those for the confining pressure.

4. After the loading time reaches 48 h, apply the next load level. Repeat the operation until the sample experiences instability and failure.

5. Derive the test data and process it using the Chen loading method. Finally, obtain the axial creep curve and axial creep rate curve of the red layer soft rock under different confining pressures, as shown in Fig. 4.

Fig. 4.

Axial creep curves of red layer soft rock under different load level: (a) Confining pressure 5 MPa; (b) Confining pressure 10 MPa; (c) Confining pressure 15 MPa; (d) Confining pressure 20 MPa.

It can be seen that under the same confining pressure, as the axial load increases step by step, the specimen undergoes instantaneous deformation at the moment of loading, which gradually increases with the axial load. Before loading, the internal pores of the specimen are fully open, and the pore closure during loading provides some deformation for the instantaneous strain. As the loading continues, the original fissures gradually close, and new fissures begin to form. The proportion of instantaneous deformation in the total deformation decreases gradually, while the proportion of creep strain increases, indicating that new fissures are progressively forming inside the specimen, and the damage level is gradually increasing. But, based on the ratio of instantaneous strain to creep strain in the total strain, the proportion of instantaneous strain is much larger than that of creep strain, indicating that the damage caused by instantaneous deformation to the specimen is more severe. Taking the confining pressure of 5 MPa as an example, at the first three load levels, the specimen exhibits only decaying and stable creep. At the fourth load level, after a period of decaying and stable creep, the specimen enters the acceleration stage, and the creep time shortens, indicating that the increase in load level makes the specimen more prone to instability and failure.

In addition, at the first few load levels before creep destruction, the axial strain of red layer soft rock exhibits the transient creep stage and the steady-state creep stage. The transient creep stage lasts for a short time, with a rapid decrease in the creep rate, while the steady-state creep stage lasts relatively longer, with the creep rate remaining nearly constant. When the load level reaches the final stage, the axial strain of red layer soft rock under different confining pressures first undergoes the transient creep and steady-state creep stages, and then enters the accelerated creep stage, where the creep rate increases sharply, ultimately leading to specimen failure. According to Fig. 4, by comparing the differences in creep rate changes of red layer soft rock under different confining pressures, and calculating the relationship between the steady-state creep rate and load level under different confining pressures, as shown in Fig. 5, it is observed that, under the same confining pressure, as the load level increases, the steady-state creep rate of red layer soft rock follows an exponential growth trend. When the load level reaches the failure load level, the creep rate increases by an order of magnitude. Under the same creep rate, the higher the confining pressure, the higher the corresponding load level.

Fig. 5.

Creep rate distribution curve of red layer soft rock: (a) Creep strain rate under different confining pressure; (b) Relationship between steady-state creep rate and load level.

Model establishment

Consideration of statistical damage viscoplastic strain rate

In classical rheological element models, the Nishihara model can well describe the creep process of rock-like materials. In this model, E₀ is the instantaneous elastic modulus, E₁ is the viscoelastic modulus, η₁ is the viscoelastic viscosity coefficient, η₁ is the viscoplastic viscosity coefficient, σ is the stress, and σ_s is the long-term strength. According to existing research, the Nishihara model effectively describes the decaying creep stage and the stable creep stage of rock-like materials. However, for the accelerated creep stage, the Nishihara model has a large error²³. Therefore, a new model needs to be built to make up for the shortcomings of the Nishihara model. Since the Perzyna viscoplastic theory has a better presentation the viscoplastic characteristics of rocks and also accurately describe the deformation behavior during the creep phase of the rock accelerates²⁴. Therefore, this paper adopts the Perzyna viscoplastic model to replace the viscoplastic portion of the Nishihara model. According to the Perzyna viscoplastic theory, the viscoplastic strain rate can be expressed as:

1

where έ_vp is Perzyna viscoplastic strain rate, %; η is the viscosity coefficient, GPa h; F is the yield function; {m} is the direction of viscoplastic flow; φ(F) is the an arbitrary function of the yield function F, typically represented as:

2

where β is the material parameter; σ_s0 is the initial yield strength, MPa.

Generally, the initiation of internal defects in rock-like materials is the main cause of their deformation, and statistical damage theory describes the damage degree of materials based on the ratio of the number of micro-damage units N_f to the total number of micro units N. Therefore, statistical damage theory can be used to describe the evolution law and creep behavior of the yield function in the accelerated creep stage of rock-like materials. Figure 6 is a schematic of the typical rock creep curve divided into segments. When the rock creep transitions from the attenuation stage to the stable stage, the corresponding time is t₁, and the number of damaged micro-units is N₁, with a lower limit of N₁^*; when the rock creep transitions from the stable stage to the accelerated stage, the corresponding time is t₂, and the number of damaged micro-units is N₂, with a lower limit of N₂^*.

Fig. 6.

Diagram of strain criticality.

According to the research results of Zhang et al.^25,26, the Weibull distribution can accurately describe the micro-unit strength of rocks, and its probability density function is given by:

3

where m and F₀ are the Weibull parameters.

When rock-like materials are subjected to external loads, overall damage and degradation occur. When the external load reaches the yield limit, the number of damaged micro-units N_f can be expressed as:

4

where N is the total number of micro-units inside the rock; N_f is the number of damaged micro-units.

By solving Eqs. (2) and (4) simultaneously, the number of damaged micro-units N₁ at time t₁ in the rock can be expressed as:

5

where F(σ) is the stress state function during the stable creep stage of the rock.

By combining Eq. (5), similarly, the lower limit value N₁^* of the number of damaged micro-units at the critical point N₁ can be expressed as:

6

where σ_s is the long-term strength, MPa; N₁^* is the lower limit value of the number of damaged micro-units at the critical point N₁, taken as the maximum number of damaged micro-units during the attenuation creep stage.

During the rock creep process, the number of damaged micro-units inside the specimen at any given time can be described by Eq. (5)²⁷. Therefore, the viscoplastic strain rate can be expressed as:

7

According to the related flow rule, the flow direction is consistent with the plastic flow direction, so {m} = 1.

Establishment of a rock statistical damage creep model based on Perzyna viscoplastic theory

Through the above model, the viscoplastic strain rate of the material can be accurately determined, but the number of internal microelement damage units N_f and the total number of units N have not yet been determined. In order to accurately classify the various creep stages, the Weibull distribution statistical damage theory is used to establish a reasonable relationship between the number of damaged microelements N_f and the total number of microelements N, by defining the ratio of the number of damaged microelements to the total number of microelements as the damage variable D:

8

Substituting Eqs. (3), (4), and (8) into Eq. (7), and simplifying:

9

where D₁^* is the lower limit value of the damage variable corresponding to the number of damaged micro-elements N₁^* at the critical point; σ^* is the effective stress, MPa.

Based on Fig. 6 and Eq. (9), the creep stages are reclassified using the damage variable, as shown in Fig. 6. When the creep transitions from the attenuation stage to the stable stage (at creep time t₁), the corresponding damage variable is D₁, with a lower limit of D₁^*; when the creep transitions from the stable stage to the accelerated stage (at creep time t₂), the corresponding damage variable is D₂, with a lower limit of D₂^*.According to the Lemaitre strain equivalence assumption principle²⁸, the effective stress of rock-like materials satisfies the following relationship with the stress:

10

Based on Eq. (10), the microelement yield function of rock-like materials can be expressed as:

11

Substituting Eq. (11) into Eq. (9), the viscoplastic strain rate of rock-like materials based on the Weibull distribution is expressed as:

12

Typically, the strain of rock-like materials is a function of stress σ and time t. Under creep conditions, the stress is constant (σ = σ₀), meaning that the creep strain is solely a function of time t. Therefore, based on the Lemaitre strain equivalence hypothesis, the viscoplastic strain of rock-like materials considering damage can be expressed as:

13

where ε_vp is the viscoplastic strain, %; ε(t) is the strain function dependent on time.

According to the distribution law of viscoplastic strain in classical rock-like materials, and it is infinitely approximated using polynomials this can be expressed as:

14

where h(t) is polynomial function dependent on time; i is the polynomial exponent; λ_i is the polynomial constant coefficient.

When the creep time lies between t₁ and t₂, the material is in the stable stage of creep. At this point, the creep strain and time generally follow a linear relationship:

15

Substituting Eqs. (13), (14), and (15) into Eq. (12):

16

When the material’s creep is at the critical point between the attenuation and stable stages, the lower limit value of the damage variable D₁^* can be expressed as:

17

where is the viscoplastic strain at time t₁, %.

Before the critical point at time t₁, the viscoplastic strain ε_vp generated in the rock-like material is less than . Therefore, during the attenuation stage, D < D₁^*. Since the creep deformation of rock-like materials satisfies a certain functional relationship with time, the damage generated in the attenuation stage has the same distribution parameters as the damage generated at the critical point t₁.

By expanding the right-hand side of Eq. (16) using a Taylor series:

18

where o() is the higher-order infinitesimal of ε_vp.

Substituting Eq. (18) into Eq. (16):

19

When t → 0, then ε_vp = 0, integrating Eq. (19):

20

Thus, the improved Nishihara model creep equation can be expressed as:

21a

21b

During the creep process of rock-like materials, there is a significant difference between the stable stage and the accelerated stage. In the stable stage, the creep rate is essentially constant and relatively small, while in the accelerated stage, the creep rate increases non-linearly until failure occurs. The viscoplastic strain curve during the accelerated stage is also a function of time t, and the failure mode under the same stress state is generally consistent. According to Fig. 7, when ε_vp > , the creep strain and time satisfy the following nonlinear relationship:

22

Fig. 7.

Model validation curve: (a) Confining pressure 5 MPa; (b) Confining pressure 10 MPa; (c) Confining pressure 15 MPa; (d) Confining pressure 20 MPa.

Substituting Eqs. (13) and (22) into Eq. (12), the viscoplastic strain rate in the accelerated stage can be expressed as:

23

where F₀’,m’ are the Weibull parameters for viscoplastic strain ε_vp during the accelerated stage.

Expanding the right-hand side of Eq. (23) using a Taylor series:

24

Substituting Eq. (24) into Eq. (23) and integrating:

25

where ; A is the integration constant.

In summary, the rock statistical damage creep model based on Perzyna’s viscoplastic theory can be expressed as:

26a

26b

26c

Model verification

Critical parameters

In the process of model derivation in this paper, there are two critical points: the critical point from the attenuation stage to the stable stage (, t₁)and the critical point from the stable stage to the accelerated stage (,t₁). Both sets of critical parameters can be determined from the creep test curve. Table 2 shows the results of the two sets of critical points under different confining pressures.

Table 2.

Critical point parameters.

σ3, (MPa)	Axial stress, (MPa)	t1, (h)	, (%)	t2, (h)	, (%)	σ3, (MPa)	Axial stress, (MPa)	t1, (h)	, (%)	t2, (h)	, (%)
5	26.26	5.08	0.114	–	–	10	32.87	5.10	0.121	–	–
	31.57	4.42	0.132	–	–		40.52	4.86	0.133	–	–
	36.76	4.04	0.168	–	–		47.27	4.44	0.175	–	–
	42.01	3.66	0.216	–	–		54.02	3.82	0.236	–	–
	47.26	2.05	0.228	10.04	0.249		60.78	2.26	0.261	13.32	0.277
15	40.09	5.20	0.146	–	–	20	48.16	5.33	0.148	–	–
	48.11	5.02	0.177	–	–		57.79	5.09	0.181	–	–
	56.13	4.76	0.218	–	–		67.42	4.76	0.225	–	–
	64.15	4.01	0.276	–	–		77.05	4.14	0.281	–	–
	72.17	3.15	0.319	19.08	0.328		86.68	3.28	0.324	19.72	330

Model validation

To verify the accuracy of the model proposed in this paper, parameter inversion was performed on experimental data at different confining pressures using Origin software. Based on the variation patterns of the creep curve of red layer soft rock, it was found that, for example, at a confining pressure of 20 MPa, when the load levels were 48.16 MPa, 57.79 MPa, and 67.42 MPa, the creep exhibited attenuation creep, and Eq. (26a) was used for fitting. When the load level was 77.05 MPa, the creep exhibited both attenuation and stable creep, and Eq. (26b) was used for fitting. When the load level was 86.68 MPa, the creep exhibited attenuation, stable, and accelerated creep, and Eq. (26c) was used for fitting. The parameter inversion results are shown in Table 3. Figure 7 shows the comparison between the experimental and theoretical curves under different confining pressures. It can be seen that under different confining pressures, the creep model based on Perzyna viscoplastic theory and statistical damage theory proposed in this paper fits the experimental curves well, with the fitting correlation coefficients R² all above 0.95. This indicates that the model can accurately describe the three-stage creep deformation characteristics of red layer soft rock, thus validating the accuracy and rationality of the model.

Table 3.

Model parameter inversion results.

σ3, (MPa)	Axial stress , (MPa)	E0, (GPa	E1, (GPa)	η1, (GPa h)	η2, (GPa h)	F0	m	λ0	λ1	η3, (GPa h)	F0′	m′	λ0′	λ1′	λ2	A	R2
5	26.26	45.51	159.63	1130.13	992.53	–	–	–	–	–	–	–	–	–	–	–	0.97
	31.57	62.37	367.21	2586.02	1531.18	–	–	–	–	–	–	–	–	–	–	–	0.97
	36.76	85.23	662.85	4166.69	1706.51	–	–	–	–	–	–	–	–	–	–	–	0.96
	42.01	101.89	948.16	5136.39	2617.59	15.37	5.21	6.39	2.85	–	–	–	–	–	–	–	0.95
	47.26	127.54	1568.32	5295.19	3490.23	6.95	6.58	2.03	7.96	1837.52	3.57	1.14	2.58	-1.85	-0.48	9.11	0.95
20	48.16	105.34	241.98	1412.66	1359.63	–	–	–	–	–	–	–	–	–	–	–	0.98
	57.79	114.29	435.97	2938.66	1988.54	–	–	–	–	–	–	–	–	–	–	–	0.98
	67.42	124.58	985.64	4629.65	2031.55	–	–	–	–	–	–	–	–	–	–	–	0.97
	77.05	137.16	1148.37	5406.73	2845.11	5.02	2.86	3.05	7.54	–	–	–	–	–	–	–	0.96
	86.68	187.49	2013.85	5403.25	3921.57	4.62	4.31	1.14	3.88	2041.69	1.96	2.03	1.13	-0.37	-5.01	2.88	0.95

To further verify the capability of the model proposed in this paper to describe the accelerated creep stage of red layer soft rock, fifth-stage creep test data under different confining pressures were extracted and fitted using both the proposed model and the Nishihara model. The fitting results are shown in Fig. 8. During the attenuation and stable creep stages, the model curve proposed in this paper closely matches the Nishihara model curve and both can effectively describe the creep characteristics of red layer soft rock during these two stages. However, as time progresses and creep enters the accelerated stage, the error between the Nishihara model and the experimental data gradually increases. In contrast, the model proposed in this paper can better describe the accelerated creep characteristics of red layer soft rock. This indicates that the proposed model has a significant advantage over the Nishihara model in describing the attenuation, steady-state, and accelerated creep behavior of rocks.

Fig. 8.

Comparison between creep model of this paper and Nishihara model: (a) Confining pressure 5 MPa; (b) Confining pressure 10 MPa; (c) Confining pressure 15 MPa; (d) Confining pressure 20 MPa.

Conclusion

1. Based on the Weibull distribution function, the creep damage variable of rock was defined to explain rock damage from a microscopic perspective. By defining a critical damage variable, the different deformation characteristics of rock in the attenuation, steady-state, and accelerated creep stages were effectively distinguished, laying a foundation for the subsequent establishment of creep models.

2. Based on Perzyna’s viscoplasticity theory, an expression for the viscoplastic strain rate of rock was established. By incorporating this expression into the damage variable defined by the Weibull distribution function, a viscoplastic strain expression capable of describing accelerated creep was obtained, providing a more precise description of the relationship between damage-induced deformation and time during the creep process.

3. Using the Nishihara model as the foundation, and combining it with the Weibull distribution function and Perzyna’s viscoplasticity theory, a new nonlinear creep model was established. This model can effectively describe the entire creep failure process of red-bed soft rock and more accurately reflect the relationship between rock creep deformation and damage, and it makes up for the shortcomings that the Nishihara model cannot describe accelerated creep.

Author contributions

Conceptualization, L.C.; Data curation, L.C. and J.H.; Formal analysis, S.Z.; Investigation, J.J. and J.T.; Methodology, L.C.; Writing-original draf, L.C. and J.H.; Writing-review and editing, L.C. and J.H. All authors have read and agreed to the published version of the manuscript.

Funding

University-local government scientific and technical cooperation cultivation project of Ordos Institute-LNTU (YJY-XD-2024-A-007). The Youth Fund of Liaoning Provincial Department of Education (JYTQN2023198). Ordos Iconic Innovation Team (TD20240005). Ordos industrial innovation and entrepreneurship talent team.

Data availability

All data, models, and code generated or used during the study appear in the submitted article.

Declarations

Competing interests

The authors declare no competing interests.