A Fractional-Order Creep-Damage Model for Carbonaceous Shale Describing Coupled Damage Caused by Rainfall and Blasting

Abstract

In order to better understand the shear creep behavior of weak interlayers (carbonaceous shale) under the coupling effect of the rainfall dry–wet cycle and blasting vibration, as well as quantitatively characterize the coupled damage of the rainfall dry–wet cycle and blasting vibration, a series of shear creep tests were carried out. The results show that the combined damage of the rainfall dry–wet cycle and blasting vibration greatly intensifies the creep effect of carbonaceous shale, leading to an increase in deceleration creep time, an increase in steady-state creep rate, and a decrease in long-term strength. The coupling damage of the rainfall dry–wet cycle and blasting vibration in carbonaceous shale was quantitatively characterized. Based on the fractional-order theory, a fractional-order creep-damage constitutive model (DNFVP) was established by introducing the Abel dashpot to describe the coupled damage of the rainfall wet–dry cycle and blasting vibration and the nonlinear creep acceleration characteristics. The three-dimensional creep equation of the model was derived. The effectiveness of the DNFVP model was verified through the inversion of model parameters and fitting of experimental data, providing a basis for in-depth research on the long-term stability of high slopes in mines with weak interlayers.

1. Introduction

The gently inclined weak interlayers (dip angle less than 25°) of carbonaceous shale in the Permian strata in southwestern China exhibit significant creep aging characteristics [1,2,3], posing a serious threat to the long-term stability of open-pit mine slopes. To ensure the safety and economy of mining, it is necessary to study the creep behavior of carbonaceous shale’s weak interlayers.

In the past few decades, many uniaxial or triaxial creep experimental studies on soft rocks have been carried out, focused on mudstone [4,5,6], salt rock [7,8], clay rock [9,10], shale [11,12,13], limestone [14,15,16], sandstone [17,18], etc. There are also some studies on shear creep tests, and the effect of single-factor damage on the shear-creep characteristics of soft rock has been generally considered. Wang et al. [19] conducted cyclic shear creep tests on shear-zone soil and explained the movement pattern of landslides in the Three Gorges Reservoir area. Ye et al. [20] analyzed the development of damage and the propagation of microcracks during the shale shear creep process. Yin et al. [21] pointed out that as the creep time increases, the Permian carbonaceous shale evolves from a dense to a loose structure. The influence of water on the shear-creep characteristics of soft rocks is significant [22,23]. Ma et al. [24] developed a direct shear apparatus for simulating dry–wet cycles, providing reference for shear tests of soft rocks under rainfall dry–wet cycles. Wei et al. [25] conducted dry–wet cyclic shear creep tests on carbonaceous shale and analyzed the influence of dry–wet cycles on the creep characteristics of carbonaceous shale. Long-term blasting vibration is a unique factor that affects the stability of mining slopes, which is different from other areas of engineering [26,27,28]. Yuan [29] and Zhou [30] developed a shear apparatus that simulates dynamic disturbances, which can test the shear slip characteristics of rocks under different dynamic disturbances. Zhu et al. [31] developed a rock-creep impact testing machine, which can conduct uniaxial creep tests under impact disturbance. However, research on the shear-creep characteristics of soft-rock impact disturbance has not been reported yet. Existing research has mostly analyzed the creep behavior of soft rocks under single-factor disturbance. Considering the obvious creep deformation characteristics affected by rainfall and blasting vibration in the monitoring data of mines in southwestern China, the Permian carbonaceous-shale weak interlayers in southwestern China are affected by rainfall and long-term blasting vibration [3,32], and the mechanical parameters continue to deteriorate, leading to shear creep deformation on high mine slopes, where the weak interlayers are used as the bottom sliding surface until landslides occur [33,34,35]. Therefore, it is crucial to carry out shear creep tests on the effects of rainfall and blasting vibration.

To quantitatively characterize the creep behavior of rocks, various creep constitutive models have been established. Lin et al. [36] proposed using the Kachanov creep-damage law to describe the time characteristics of shear strength during the accelerated creep stage of rocks and established a nonlinear viscoplastic element based on time-varying shear strength to reflect the accelerated creep stage of rocks. Zheng et al. [37] established an improved creep model that considers the effects of stress state and time on creep parameters and proposed a new method for determining creep parameters. Li et al. [38] established a seepage-damage shear creep model for shale and analyzed the variation patterns of model parameters. Zhong et al. [39] established a VEPD model considering dry–wet cycle damage. Yao et al. [40] established a non-stationary fractional-order creep model considering water content. Hu et al. [41] established a shear creep model that considers both initial and vibration damage to rock masses. The above constitutive models provide reference for the study of creep mechanics but mainly focus on the characterization of nonlinear characteristics and single-factor disturbance damage, and they are not suitable for the Permian carbonaceous-shale weak interlayers affected by rainfall dry–wet cycles and blasting vibration. It is necessary to establish a constitutive model that can describe the nonlinear characteristics of carbonaceous-shale weak interlayers and quantitatively characterize the factors of rainfall and blasting vibration damage to carbonaceous shale.

Therefore, in order to better understand the shear creep behavior of carbonaceous shale, shear creep tests were conducted on carbonaceous shale. The nonlinear deformation and failure evolution law of carbonaceous shale under the influence of rainfall and blasting vibration was analyzed, and the nonlinear shear creep-damage constitutive model was established to describe the coupled damage of the rainfall dry–wet cycle and blasting vibration on carbonaceous shale and provided a theoretical basis for the long-term stability research of high slopes with gently inclined weak interlayers.

2. Specimen Preparation and Experimental Procedures

2.1. Specimen Material and Test Equipment

The carbonaceous-shale specimen (Figure 1) was taken from the Permian strata in southwestern China. The mineral composition is mainly clay minerals and calcite, which are composed of gravel debris, sand (particle) debris, and carbonaceous and muddy cementitious materials. The specimen size was 75 mm × 75 mm × 150 mm, and three cylindrical holes with a diameter of 8 mm × 37.5 mm were drilled at equal intervals on the top of the specimen. The unevenness error of the end face should not exceed 0.05 mm.

The soft-rock shear-creep test system with a simulation of coupled rainfall seepage and blasting vibrations was used for the experiment (Figure 2a). The specimen was placed in a shear box (Figure 2b). During the shearing process, the simulation of the rainfall dry–wet cycle and blasting vibration of the specimen can be achieved, which is more in line with engineering geological conditions of the carbonaceous-shale weak interlayer. The dry–wet cycle and vibration path are shown in Figure 2c.

2.2. Test Design

The normal pressure values for the experiment are 0.2 MPa, 0.6 MPa, 1.0 MPa, and 1.4 MPa. The number of dry–wet cycles (m) is 0, 4, 8, 12, and 16. The number of vibrations (n) is 0, 12, 24, 36, and 48, with a single vibration energy of 1 J. Taking m = 8 + n = 24 as an example, the experimental plan is shown in Figure 3. The graded loading method was adopted, and shear loads were applied at 13%, 27%, 40%, 53%, 67%, and 80% of the shear strength of natural-moisture-content specimens. The data collection frequency during the test was three times/min. During the experiment, the indoor temperature was controlled at 25 ± 0.5 °C.

3. Test Results and Analysis

Taking the creep shear test of carbonaceous shale under the normal stress of 0.6 MPa as an example, some test curves are shown in Figure 4. From Figure 4, it can be seen that under low shear stress, the specimens exhibit both deceleration creep and steady-state creep. Under destructive shear stress, the specimen enters the accelerated creep stage after a period of steady-state creep, with a sharp increase in creep rate, and eventually experiences rapid failure along the shear plane. This indicates that the damage to the specimen has reached its limit, which is consistent with the phenomenon of sudden instability and failure of high slopes with carbonaceous-shale weak interlayers after long-term creep deformation [42,43].

Rainfall dry–wet cycles and blasting vibrations both lead to varying degrees of damage to carbonaceous shale, manifested by different shear-creep characteristics. By comparing Figure 4a–c, it can be seen that as the number of dry–wet cycles increases, the creep deformation of the specimen gradually increases. When m = 0 or 8, the specimen fails under the sixth level of shear stress (Figure 4a,b). When m = 16, the specimen fails under the fifth level of shear stress (Figure 4c), indicating that as the number of dry–wet cycles increases, the shear stress level that the specimen can withstand decreases. Compared with m = 0, the long-term strength of the specimen with m = 8 and 16 decreased by 6.77% and 22.92%, respectively (Figure 5). The more dry–wet cycles there were, the greater the decrease in the long-term strength of the specimen. The number of dry–wet cycles showed an exponential negative correlation with long-term strength. As the number of dry–wet cycles increases, the deceleration creep time of the specimen increases. Taking the fourth level of shear stress as an example, the deceleration creep times of the specimens with m = 0, 8, and 16 are 0.57 h, 1.37 h, and 1.82 h, respectively. This indicates that dry–wet cycles cause mineral dissolution and internal structural damage in the specimens. Under stable shear stress loads, the time required for internal stress adjustment in the specimens increases with the number of dry–wet cycles.

The water pressure seepage in the dry–wet cycle causes the clay minerals to expand and soften, which dissolve and precipitate along the seepage path, resulting in an increase in the internal pore structure of the specimen. Air pressure seepage leads to a decrease in the moisture content of the specimen, rearrangement of mineral particles, and damage to the internal structure of the specimen. Under repeated dry–wet cycles, the internal structure of the specimen continuously damages, and the long-term strength gradually decreases. It is different from the previous dry–wet cycle methods such as vacuum saturation and high-temperature drying before mechanical testing [44,45,46], which avoids damage caused by the seepage pressure and temperature of the specimen before mechanical testing and can more accurately simulate the dry–wet cycle seepage process under stress field conditions in engineering environments.

By comparing Figure 4a,d,e, it can be seen that as the number of vibrations increases, the creep deformation of the specimen gradually increases. When n = 0 or 24, the specimen failed under the sixth level of shear stress (Figure 4a,d). When n = 48, the specimen failed under the fifth level of shear stress (Figure 4e), indicating that as the vibration increases, the shear stress level that the specimen can withstand decreases. Compared with n = 0, the long-term strength of the specimen with n = 24 and 48 decreased by 4.68% and 17.19%, respectively (Figure 6). The more vibrations there were, the greater the decrease in the long-term strength of the specimen. There was an exponential negative correlation between vibration and long-term strength. As the number of vibrations increases, the deceleration creep time of the specimen increases. Taking the fourth level of shear stress as an example, the deceleration creep times of the specimens with n = 0, 24, and 48 are 0.57 h, 0.94 h, and 1.38 h, respectively. This indicates that vibration causes the shear off of mineral particles and the internal structure damage in the specimen. Under stable shear stress loads, the time required for adjusting the internal stress of the specimen increases with the increase in vibration times.

The damage to the specimen caused by vibration is transient, manifested as sudden changes in strain and strain rate, with the sudden change in strain rate being more intuitive (Figure 4d,e). After vibration, the strain rate of the specimen suddenly changes and then stabilizes rapidly, during which the structure of the specimen has already been damaged. As the number of vibrations increases, the sudden change in strain rate caused by vibration gradually decreases, indicating that the elastic energy inside the specimen is gradually consumed and the viscoplasticity of the specimen is enhanced.

The vibration waveguide generated by vibration causes the local position of the specimen to be affected by stress concentration, causing the surrounding mineral particles to shear off and the initiation of cracks [47]. Under repeated vibration, cracks inside the specimen, especially on the shear plane, gradually initiate, propagate, and penetrate, resulting in a nonlinear accumulation of damage [48], ultimately leading to shear failure.

The coupling of dry–wet cycles and blasting vibrations leads to increased damage to the specimen. Compared with m = 0 + n = 0 (Figure 4a), the long-term strength of the specimen with m = 8 + n = 0 (Figure 4b), m = 0 + n = 24 (Figure 4d), and m = 8 + n = 24 (Figure 4f) decreased by 6.77%, 4.68%, and 17.71%, respectively. This indicates that the combined damage of dry–wet cycles and blasting vibration on the specimen is much greater than the influence of single-factor action.

The dry–wet cycle and blasting vibration affect the clay minerals and calcite in the specimen, respectively. The dry–wet cycle leads to dissolution, mud formation, and the precipitation of clay minerals. Blasting vibration causes stress concentration inside the specimen, resulting in the shearing off of mineral particles. Under the combined effect of two factors, the specimen continuously generates a positively correlated accumulation of damage, and the shear strength parameters sharply decrease.

4. Fractional-Order Creep-Damage Model and Parameter Identification

To more accurately describe the degree of damage caused by dry–wet cycles and vibration to the specimen, it is necessary to establish a creep constitutive model that quantifies the damage caused by dry–wet cycles and vibration.

4.1. Fractional-Order Abel Dashpot

In 1965, Leibniz first mentioned the problem of fractional derivatives [49]. Since the 1970s, with the increasing research and application fields of numerous scholars, fractional calculus has developed rapidly. Among them, let β∈R, and when p − 1 < β < p, p∈N, the Caputo fractional calculus can be defined as

$${\displaystyle \frac{d^{\beta }f(t)}{d{t}^{\beta }}}={\displaystyle \frac{1}{\Gamma (p-\beta )}}{\displaystyle {\int }_a^t\frac{f^{(p)}(\tau )}{{(t-\tau )}^{\beta -p+1}}d\tau =\frac{f^{(p)}(a){(t-a)}^{p-a}}{\Gamma (p-\beta +1)}+\frac{1}{\Gamma (p-a+1)}{\displaystyle {\int }_a^t{(t-\tau )}^{p-\beta }{f}^{(p+1)}(\tau )d\tau }}$$

(1)

where $\Gamma (·)$ is the gamma function, which is defined as Equation (2).

$$\Gamma (z)={\displaystyle {\int }_0^{\infty }{e}^{-t}{t}^{z-1}dt}$$

(2)

The advantages of fractional calculus theory in describing nonlinear data have led many scholars to construct different fractional-order models. Among them, the Abel dashpot model (Figure 7a) is widely used, which can be used to describe the deformation characteristics of materials with the coexistence of elasticity and viscosity. Its constitutive relationship is as follows

$$\tau (t)=\eta {\displaystyle \frac{d^{\beta }\epsilon (t)}{d{t}^{\beta }}},(0\le \beta \le 1)$$

(3)

where η is the generalized viscosity coefficient of Abel dashpot, and β is the derivative order.

According to Caputo fractional calculus, when τ_t = τ₀, i.e., the material is under creep conditions, its creep equation is

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{\eta }}{\displaystyle \frac{t^{\beta }}{\Gamma (1+\beta )}},(0\le \beta \le 1)$$

(4)

The creep strain curves of Abel dashpot with different values of fractional order β are shown in Figure 8. As shown in Figure 8, when β = 0, the strain remains constant, and the Abel dashpot exhibits an ideal elastic body. When β = 1, the strain increases linearly with time, and the Abel dashpot exhibits an ideal viscous body. When 0 < β < 1, the creep curve exhibits a nonlinear gradient process, and the Abel dashpot exhibits both elastic and viscous characteristics, which can be used to describe the viscoelastic deformation characteristics of the material between ideal elasticity and ideal viscosity.

The above Abel dashpot creep equation can be used to describe the first two stages of material creep under low stress but cannot describe the accelerated creep stage. When creep enters the acceleration stage, its viscosity coefficient decreases with increasing creep time. Assuming that the viscosity coefficient is a non-stationary coefficient, the relationship between viscosity coefficient and time can be expressed as an exponential function.

$$\eta (\tau ,t)={\eta }_0{e}^{-bt},(\tau \ge {\tau }_s)$$

(5)

where η₀ is the initial value of the generalized viscosity coefficient, b is the coefficient related to the material, and τ_s is the yield stress of the material.

Substituting the unsteady viscosity coefficient η (τ, t) into Equation (4), we obtain the Abel dashpot containing the unsteady viscosity coefficient, which is called the unsteady Abel dashpot. As shown in Figure 7b, its creep equation is

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{{\eta }_0{e}^{-bt}}}{\displaystyle \frac{t^{\beta }}{\Gamma (1+\beta )}}(0\le \beta \le 1,\tau \ge {\tau }_s)$$

(6)

Let β = 0.5 and plot creep curves have different values of coefficient b (Figure 9). From Figure 9, it can be seen that the value of coefficient b determines the deformation growth rate of the creep curve. The larger the value of coefficient b, the faster the deformation growth rate of the curve. From the rate of deformation growth on the curve, it can be seen that the unsteady Abel dashpot can describe the acceleration creep stage.

4.2. Fractional-Order Creep Constitutive Model

The first two stages of the creep model of carbonaceous shale belong to viscosity–viscoelasticity, which can be described by the Burgers model (Figure 10a). The Burgers model [11,36] is composed of the Maxwell model and Kelvin model in series, and its creep equation is

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{E_1}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1}}t+{\displaystyle \frac{{\tau }_0}{E_2}}(1-e^{-{\displaystyle \frac{E_2}{{\eta }_2}}t})$$

(7)

The Burgers model contains viscous elements, and the Kelvin model is also a type of viscoelastic body. The Abel dashpot (0 < β < 1) can also be used to describe the viscoelasticity of materials and has only one element, which is simpler in structure compared to the Burgers model. Therefore, an improved Burgers model was obtained by replacing the viscous element and Kelvin model in the Burgers model with the Abel dashpot (Figure 10b).

According to the series-connection rule, the state equation for improved Burgers model is

$$\left\{\begin{array}{l}{\dot{\epsilon }}_1={\displaystyle \frac{1}{E}}{\dot{\tau }}_1\\ {\tau }_2(t)={\eta }_1{\displaystyle \frac{d^{\beta }{\epsilon }_2(t)}{d{t}^{\beta }}},(0\le \beta \le 1)\\ \begin{array}{c}\epsilon ={\epsilon }_1+{\epsilon }_2\\ \dot{\epsilon }={\dot{\epsilon }}_1+{\dot{\epsilon }}_2\end{array}\\ \tau ={\tau }_1={\tau }_2\end{array}\right.$$

(8)

The constitutive equation for the improved Burgers model can be solved as follows

$${\displaystyle \frac{d^{\beta }\epsilon }{d{t}^{\beta }}}={\displaystyle \frac{1}{E}}{\displaystyle \frac{d^{\beta }\tau }{d{t}^{\beta }}}+{\displaystyle \frac{\tau }{{\eta }_1}}$$

(9)

Furthermore, when τ = τ₀, the creep equation of the improved Burgers model obtained by solving Equation (9) is

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{E}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1}}{\displaystyle \frac{t^{\beta }}{\Gamma (1+\beta )}},(0\le \beta \le 1)$$

(10)

The creep of carbonaceous shale includes three stages. The first two stages can be described by the improved Burgers model, but the improved Burgers model cannot describe the accelerated creep stage. During the accelerated creep stage, rocks exhibit significant nonlinear characteristics. To describe the third stage of creep in carbonaceous shale, based on the improved Burgers model, a non-stationary Abel dashpot that can describe the acceleration creep stage was connected in series. A nonlinear fractional-order viscoelastic–plastic creep constitutive model for carbonaceous shale was established, which is called the NFVP model (Figure 10b). Here, E is the instantaneous elastic modulus, η₁ and β₁ are the viscosity coefficient and fractional-order derivative order of the Abel dashpot, b, η₂ and β₂ are the material coefficient, viscosity coefficient and fractional-order derivative order of the unsteady Abel dashpot, τ_s is the yield strength of the carbonaceous shale, and ε₁, ε₂, and ε₃ are the strains of the elastic element, Abel dashpot, and unsteady Abel dashpot.

When τ < τ_s, in the NFVP model, only the elastic element and the Abel dashpot (i.e., improved Burgers model) play a role. At this time, the creep equation of the carbonaceous shale is the creep equation of the improved Burgers model.

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{E}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+{\beta }_1)}},(0\le \beta \le 1)$$

(11)

When τ ≥ τ_s, the NFVP model is fully functional. According to the series-connection rule, its state equation is

$$\left\{\begin{array}{l}{\dot{\epsilon }}_1={\displaystyle \frac{1}{E}}{\dot{\tau }}_1\\ {\tau }_2(t)={\eta }_1{\displaystyle \frac{d^{{\beta }_1}{\epsilon }_2(t)}{d{t}^{{\beta }_1}}},(0\le {\beta }_1\le 1)\\ {\tau }_3(t)={\eta }_2{\displaystyle \frac{d^{{\beta }_2}{\epsilon }_3(t)}{d{t}^{{\beta }_2}}},(0\le {\beta }_2\le 1)\\ {\eta }_3(\sigma ,t)={\eta }_0{e}^{-bt},(\tau \ge {\tau }_s)\\ \begin{array}{c}\epsilon ={\epsilon }_1+{\epsilon }_2+{\epsilon }_3\\ \dot{\epsilon }={\dot{\epsilon }}_1+{\dot{\epsilon }}_2+{\dot{\epsilon }}_3\end{array}\\ \tau ={\tau }_1={\tau }_2={\tau }_3\end{array}\right.$$

(12)

The constitutive equation for solving the NFVP model can be obtained as follows

$${\displaystyle \frac{d^{{\beta }_1}}{d{t}^{{\beta }_1}}}{\displaystyle \frac{d^{{\beta }_2}}{d{t}^{{\beta }_2}}}\epsilon =({\displaystyle \frac{1}{E}}{\displaystyle \frac{d^{{\beta }_1}}{d{t}^{{\beta }_1}}}{\displaystyle \frac{d^{{\beta }_2}}{d{t}^{{\beta }_2}}}+{\displaystyle \frac{1}{{\eta }_1}}{\displaystyle \frac{d^{{\beta }_2}}{d{t}^{{\beta }_2}}}+{\displaystyle \frac{1}{{\eta }_2}}{\displaystyle \frac{d^{{\beta }_1}}{d{t}^{{\beta }_1}}})\tau ,(0\le \beta \le 1)$$

(13)

Furthermore, when τ = τ₀, the creep equation of the NFVP model for τ ≥ τ_s is obtained by solving Equation (13), as follows:

$$\epsilon (t)={\displaystyle \frac{{\tau }_0}{E}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+\beta )}}+{\displaystyle \frac{{\tau }_0}{{\eta }_2{e}^{-bt}}}{\displaystyle \frac{t^{{\beta }_2}}{\Gamma (1+\beta )}},(0\le {\beta }_1\le 1,0\le {\beta }_2\le 1)$$

(14)

4.3. Fractional-Order Creep-Damage Constitutive Model

4.3.1. Definition of Damage Variables

The field of damage mechanics mainly studies the propagation of micro defects (dislocation, micropores, microcrack, etc., as shown in Figure 11) and the mechanical properties of materials containing micro defects under the influence of the external environment and loads. The initiation and propagation of these micro defects can lead to a continuous deterioration of materials, resulting in damage [50]. Material damage will lead to a decrease in its physical and mechanical properties, such as elastic modulus, density, bulk density, etc.

Based on damage mechanics, the damage variable of a material can be defined as

$$D={\displaystyle \frac{A(0)-A(P)}{A(0)}}$$

(15)

where D is the damage variable of the material, A(0) is the initial value of the mechanical parameters of the material, and A(P) is the mechanical parameter value of the material after being affected by various factors. According to Equation (15), when D = 0, the material is not damaged. When D = 1, the mechanical strength of the material is lost and completely destroyed. When 0 < D < 1, the material is in the process of damage accumulation. The larger D is, the greater the entropy increase, and the more severe the damage.

The parameters that continuously damage during the shear creep test of carbonaceous shale under rainfall and blasting vibration are mainly the instantaneous elastic modulus and viscosity coefficient.

• Definition of damage rate for instantaneous elastic modulus.

According to the analysis of creep test data, it can be seen that the instantaneous deformation of the carbonaceous shale continuously decreases and the instantaneous elastic modulus continuously damages due to the influence of rainfall dry–wet cycles and blasting vibration. The damage rate affected by the rainfall dry–wet cycle and blasting vibration is defined by instantaneous elastic modulus, as follows:

$$D_{r1}(m)={\displaystyle \frac{E_r(0)-E_r(m)}{E_r(0)}}$$

(16)

where m is the number of dry–wet cycles, D_r1(m) is the instantaneous elastic-modulus damage rate for m dry–wet cycles, and E_r(0) and E_r(m) are the instantaneous elastic modulus for 0 and m dry–wet cycles, respectively.

$$D_{b1}(n)={\displaystyle \frac{E_b(0)-E_b(n)}{E_b(0)}}$$

(17)

where n is the number of vibrations, D_b1(n) is the instantaneous elastic-modulus damage rate of n vibrations, and E_b(0) and E_b(n) are the instantaneous elastic modulus of 0 and n vibrations, respectively.

2.

Definition of damage rate for viscosity coefficient.

According to the analysis of creep test data, it can be seen that due to the influence of rainfall dry–wet cycles and blasting vibration, the creep deformation of carbonaceous shale continues to increase, and the viscosity coefficient continuously damages. Referring to the definition of damage rate for the instantaneous elastic modulus, the viscosity-coefficient damage rate affected by rainfall dry–wet cycles and vibration is defined as follows

$$D_{r2}(m)={\displaystyle \frac{{\eta }_r(0)-{\eta }_r(m)}{{\eta }_r(0)}}$$

(18)

where m is the number of rainfall dry–wet cycles, D_r2(m) is the viscosity-coefficient damage rate for m dry–wet cycles, and η_r(0) and η_r(m) are the viscosity coefficients for 0 and m dry–wet cycles.

$$D_{b2}(n)={\displaystyle \frac{{\eta }_b(0)-{\eta }_b(n)}{{\eta }_b(0)}}$$

(19)

where n is the number of vibrations, D_b2(n) is the viscosity coefficient-damage rate of n vibrations, and η_b(0) and η_b(n) are the viscosity coefficients of 0 and n vibrations.

3.

Definition of damage rate under the combined action of dry–wet cycle and blasting vibration.

To indirectly determine material damage, Professor Lemaitre proposed the strain equivalence principle [51], which states that the strain caused by the total stress σ acting on the damaged material is equivalent to the strain caused by the effective stress σ′ acting on the non-destructive material, and its expression is

$$\epsilon ={\displaystyle \frac{\sigma }{E^{\prime }}}={\displaystyle \frac{{\sigma }^{\prime }}{E}}$$

(20)

where E and E′ are the elastic modulus of the non-destructive material and the damaged material, respectively.

Most rocks have initial damage, and under the action of force F, the rock damage will expand. Therefore, Zhang et al. defined the initial damage state of rocks as the benchmark damage state and proposed the generalized strain equivalence principle [52]. For any two damage states of materials, the effective stress in the first damage state acting on the strain caused by the second damage state is equivalent to the effective stress in the second damage state acting on the strain caused by the first damage state, and its expression is

$${\sigma }^1{A}^1={\sigma }^2{A}^2$$

(21)

where σ¹ and A¹ are the effective stress and effective bearing area in the first damage state, and σ² and A² are the effective stress and effective bearing area in the second damage state. The strain of two states is

$$\epsilon ={\displaystyle \frac{{\sigma }^1}{E^2}}={\displaystyle \frac{{\sigma }^2}{E^1}}$$

(22)

where E¹ and E² are the elastic modulus of the first and second damage states, respectively. Equation (22) is the generalized strain equivalence principle.

According to the generalized strain equivalence principle, it is proposed that the effective stress in the dry–wet cycle damage state acting on the strain caused by the blasting-vibration damage state is equivalent to the effective stress in the blasting vibration damage state acting on the strain caused by the dry–wet cycle damage state. The damage rate under the combined action of rainfall dry–wet cycles and blasting vibration is defined as

$$D_1(m,n)=D_{r1}(m)+D_{b1}(n)-D_{r1}(m)·D_{b1}(n)$$

(23)

$$D_2(m,n)=D_{r2}(m)+D_{b2}(n)-D_{r2}(m)·D_{b2}(n)$$

(24)

where D₁(m,n) and D₂(m,n) are the damage rate of the instantaneous elastic modulus and viscosity coefficient under the combined action of dry–wet cycle and blasting vibration, respectively.

4.3.2. Damage Rate of Rainfall Dry–Wet Cycle and Blasting Vibration

• Damage rate caused by the rainfall dry–wet cycle.

Based on the creep test data, the creep parameters of carbonaceous shale under different dry–wet cycles were obtained, as shown in

Table 1. Creep parameters of carbonaceous shale under different dry–wet cycles.

Number of Dry–Wet Cycles m	E/GPa	η1/GPa·hβ
0	2.35	1.56
4	1.65	1.09
8	1.43	0.75
12	1.29	0.52
16	1.25	0.39

. The relationship between the parameter damage rates of carbonaceous shale and dry–wet cycles is shown in Figure 12.

The relationship between the creep parameter damage rate of carbonaceous shale and the number of dry–wet cycles is expressed as follows

$$D_{r1}(m)=-0.48\times \exp (-{\displaystyle \frac{m}{4.26}})+0.48$$

(25)

$$D_{r2}(m)=-0.96\times \exp (-{\displaystyle \frac{m}{10.24}})+0.96$$

(26)

2.

Damage rate caused by blasting vibration.

Based on the creep test data, the creep parameters of carbonaceous shale under different vibrations were obtained, as shown in

Table 2. Creep parameters of carbonaceous shale under different vibrations.

Number of Vibration n	E/GPa	η1/GPa·hβ
0	2.35	1.56
12	1.55	1.18
24	1.36	0.94
36	1.07	0.81
48	1.01	0.76

. The relationship between the parameter damage rates of carbonaceous shale and the vibration is shown in Figure 13.

The relationship between the creep parameters damage rate of carbonaceous shale and the number of vibrations is expressed as follows

$$D_{b1}(n)=-0.59\times \exp (-{\displaystyle \frac{n}{16.23}})+0.59$$

(27)

$$D_{b2}(n)=-0.58\times \exp (-{\displaystyle \frac{n}{20.98}})+0.58$$

(28)

3.

Damage rate caused by the rainfall dry–wet cycle and blasting vibration.

According to the creep test data, the creep parameters of carbonaceous shale with m = 0 + n = 0, m = 8 + n = 0, m = 0 + n = 24, m = 8 + n = 24 were obtained, as shown in

Table 3. Creep parameters of carbonaceous shale under the coupling effect of rainfall dry–wet cycle and blasting vibration.

Number of Dry–Wet Cycles m	Number of Vibration n	E/GPa	η1/GPa·hβ
0	0	2.35	1.56
8	0	1.43	0.75
0	24	1.36	0.94
8	24	0.84	0.45

.

According to the definition of damage rate under the combined action of the rainfall dry–wet cycle and blasting vibration, the instantaneous elastic-modulus damage rate and viscosity-coefficient damage rate of the specimen with m = 8 + n = 24 are 65% and 71%, respectively. The corresponding instantaneous elastic modulus and viscosity coefficient are 0.83 and 0.45, which are basically the same as the creep parameters obtained from the experiment. This indicates that the definition of damage rate under the combined action of the rainfall dry–wet cycle and blasting vibration is reasonable and reliable. The expression for the relationship between the creep parameter damage rate and the number of dry–wet cycles and vibrations obtained from Equations (25)–(28) is

$$D_1(m,n)=-0.28\times \exp (-{\displaystyle \frac{m}{4.26}}-{\displaystyle \frac{n}{16.23}})-0.2\times \exp (-{\displaystyle \frac{m}{4.26}})-0.31\times \exp (-{\displaystyle \frac{n}{16.23}})+0.79$$

(29)

$$D_2(m,n)=-0.56\times \exp (-{\displaystyle \frac{m}{10.24}}-{\displaystyle \frac{n}{20.98}})-0.4\times \exp (-{\displaystyle \frac{m}{10.24}})-0.02\times \exp (-{\displaystyle \frac{n}{20.98}})+0.98$$

(30)

4.3.3. Creep-Damage Constitutive Model

By substituting D₁(m,n) and D₂(m,n) into the constitutive equation of the NFVP model, a nonlinear viscoelastic–plastic creep-damage constitutive model for carbonaceous shale considering rainfall dry–wet cycles and blasting vibration damage was established, called the DNFVP model (Figure 10c). The creep constitutive equation is

$$\epsilon (t)=\left\{\begin{array}{l}{\displaystyle \frac{{\tau }_0}{E(1-D_1(m,n))}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1(1-D_2(m,n))}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+{\beta }_1)}},(0\le {\beta }_1\le 1,\tau <{\tau }_s)\\ \\ {\displaystyle \frac{{\tau }_0}{E(1-D_1(m,n))}}+{\displaystyle \frac{{\tau }_0}{{\eta }_1(1-D_2(m,n))}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+{\beta }_1)}}+{\displaystyle \frac{{\tau }_0}{{\eta }_2{e}^{-bt}}}{\displaystyle \frac{t^{{\beta }_2}}{\Gamma (1+{\beta }_2)}},\\ (0\le {\beta }_1\le 1,0\le {\beta }_2\le 1,\tau \ge {\tau }_s)\end{array}\right.$$

(31)

According to the theory of elastic–plastic mechanics, the three-dimensional creep equation of the DNFVP model is

$${\epsilon }_{ij}(t)=\left\{\begin{array}{l}{\displaystyle \frac{{(S_{ij})}_0}{2G(1-D_1(m,n))}}+{\displaystyle \frac{{(S_{ij})}_0}{{\eta }_1(1-D_2(m,n))}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+{\beta }_1)}}+{\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}}\\ (0\le {\beta }_1\le 1,{(S_{ij})}_0<{\tau }_s)\\ {\displaystyle \frac{{(S_{ij})}_0}{2G(1-D_1(m,n))}}+{\displaystyle \frac{{(S_{ij})}_0}{{\eta }_1(1-D_2(m,n))}}{\displaystyle \frac{t^{{\beta }_1}}{\Gamma (1+{\beta }_1)}}+{\displaystyle \frac{{(S_{ij})}_0}{{\eta }_2{e}^{-bt}}}{\displaystyle \frac{t^{{\beta }_2}}{\Gamma (1+{\beta }_2)}}+{\displaystyle \frac{{\sigma }_m{\delta }_{ij}}{3K}}\\ (0\le {\beta }_1\le 1,0\le {\beta }_2\le 1,{(S_{ij})}_0\ge {\tau }_s)\end{array}\right.$$

(32)

4.4. Parameter Identification

To verify the DNFVP model, the shear-creep test data of carbonaceous shale with m = 8 + n = 24 were fitted (Figure 14), and the creep parameters and fitting coefficients of the DNFVP model were listed (

Table 4. Parameter identification of DNFVP model for m = 8 + n = 24.

Shear Stress/MPa	E/GPa	η1/GPa·hβ	β1	η2/GPa·hβ	b	β2	R2
0.34 MPa	0.36	0.611	0.222				0.936
0.72 MPa	0.59	0.442	0.023				0.990
1.06 MPa	0.94	0.264	0.015				0.986
1.40 MPa	1.12	0.193	0.012				0.984
1.78 MPa	0.84	0.452	0.020	5.32E6	7.13	0.5	0.993

). Meanwhile, compared with the Burgers model, it can be concluded that the Burgers model can only describe the creep behavior of carbonaceous shale before failure-level shear stress, while the DNFVP model can describe the three-stage nonlinear creep behavior of carbonaceous shale, and the fitting coefficients of the DNFVP model are all greater than 90%, indicating a high degree of fitting. Overall, the DNFVP model has fewer elements and parameters, and it can fit the creep test data of carbonaceous shale well, with high rationality and reliability.

5. Discussion

The long-term stability of open-pit mine slopes is the foundation for ensuring safe mining operations. It is crucial to conduct research on the sliding mechanism and disaster prevention and control for a large number of landslide geological hazards along the weak interlayers of Permian carbonaceous shale [33,34,35]. Therefore, shear creep tests were conducted on the weak interlayers of carbonaceous shale commonly found in the Permian strata in southwestern China, and a fractional-order creep constitutive model for carbonaceous shale was established to describe rainfall dry–wet cycles and blasting vibration damage. The dry–wet cycle of rainfall and blasting vibration have a significant impact on the shear creep behavior of carbonaceous shale. The damage to carbonaceous shale caused by the dry–wet cycle of rainfall is long term. Water pressure seepage (wet) causes the dissolution precipitation of clay minerals in carbonaceous shale, while air pressure seepage (dry) causes the minerals in carbonaceous shale to lose water and redistribute, resulting in structural damage. The damage to carbonaceous shale caused by blasting vibration is instantaneous, and the single damage is small. However, long-term blasting vibration leads to the initiation and propagation of internal cracks in carbonaceous shale, and the damage accumulates continuously. Under the coupling effect of two factors, the degree of damage is exacerbated by the superposition. The macroscopic manifestation of the shear-creep test results is that as the number of dry–wet cycles and vibration increases, the creep displacement increases, the deceleration creep time increases, the steady-state creep rate increases, and the long-term strength decreases. Based on the improved strain equivalence principle, a creep parameter damage-rate index was proposed to characterize the damage caused by rainfall dry–wet cycles and blasting vibrations. This index was introduced into a fractional-order creep constitutive model to quantify the damage caused by rainfall dry–wet cycles and blasting vibrations using the DNFVP model. The DNFVP model provides theoretical guidelines for in-depth research on the creep disaster mechanism of high slopes containing gently inclined weak interlayers in mines using secondary development tools.

In order to further explore the shear-creep characteristics of carbonaceous shale under the effects of rainfall dry–wet cycles and blasting vibrations, and apply them to engineering, we will conduct research on the following aspects in the future: firstly, we mainly analyzed the influence of the number of dry–wet cycles and the number of blasting vibrations on the creep behavior of carbonaceous shale. In the future, we can further refine the parameters of rainfall seepage and blasting vibrations (such as seepage water pressure, seepage time, vibration frequency, vibration amplitude, etc.), and carry out corresponding shear creep tests to clarify the degree of influence of each parameter on the creep behavior of carbonaceous shale. Secondly, we can conduct research on the identification method of DNFVP model parameters to determine the most suitable constitutive model parameters for on-site engineering conditions. Thirdly, the DNFVP model can be further developed using secondary development tools. Based on the DNFVP model for secondary development and numerical simulation software, creep calculations can be carried out on high slopes with gently inclined weak interlayers. Combined with on-site monitoring data and engineering geological characteristics, the disaster mechanism of high slopes controlled by the creep characteristics of weak interlayers can be revealed, providing theoretical guidance for the long-term stability analysis and the prediction of high slopes in mines with gently inclined weak interlayers.

6. Conclusions

Through shear creep tests on the weak interlayers of carbonaceous shale, the creep-damage law of carbonaceous shale under rainfall dry–wet cycles and blasting vibration was studied. A fractional-order creep-damage constitutive model was established, and the following conclusions were obtained.

• Carbonaceous shale has obvious creep characteristics, and especially under the influence of dry–wet cycles and vibration, the nonlinear characteristics of its creep curve are more obvious. As the number of dry–wet cycles and vibrations increases, the deceleration creep time increases, the steady-state creep rate increases, the long-term strength decreases, and the degree of damage intensifies. The damage of the specimen under the coupling effect of dry–wet cycle and vibration is much greater than that caused by a single factor.
• Based on the generalized strain equivalence principle, the coupled damage rate of the rainfall dry–wet cycle and blasting vibration for carbonaceous shale was defined. Based on the creep test results, the expression for the coupled damage of the rainfall dry–wet cycle and blasting vibration is determined.
• Based on the fractional calculus theory and the introduction of the Abel dashpot, a nonlinear fractional shear-creep constitutive model (DNFVP model) was established to characterize the coupled damage caused by rainfall dry–wet cycles and blasting vibration. The three-dimensional creep equation of the model was derived to describe the three-stage nonlinear creep characteristics of carbonaceous shale under the influence of rainfall dry–wet cycles and blasting vibration.
• The DNFVP model was used to fit the creep curve of carbonaceous shale, and the creep parameters of the creep model were determined. The fitting effect of the model was good, and the effectiveness of the model was verified.