CN113806966B - Construction method of nonlinear rock fatigue constitutive model based on rheological model application - Google Patents

Abstract

The invention discloses a method for constructing a nonlinear rock fatigue constitutive model based on the application of a rheological model, which includes the following steps: constructing a rock mass fatigue constitutive model; obtaining viscous element parameters described by fractional order; obtaining rock mass fatigue constitutive model based on fractional order derivatives. The strain expression of the body fatigue constitutive model ε _eve ; obtain the rock mass fatigue constitutive model ε _eve (N) based on the fractional derivative after introducing the Mittag‑Leffler function; construct a nonlinear fatigue constitutive that reflects the mechanical characteristics of the accelerated fatigue stage Model ε _vp ; obtain the damage variable D after coupling the macroscopic initial damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass; obtain the nonlinear fatigue constitutive model ε _vp (N) that considers the mechanical properties of the accelerated fatigue stage of the initial damage; obtain A nonlinear fatigue constitutive model describing the entire process of rock mass fatigue considering initial damage. The present invention establishes a fatigue constitutive model that describes the entire process of rock mass fatigue by combining nonlinear fatigue constitutive models that describe different fatigue stages. The constitutive model has simple parameters and clear physical and mechanical meanings.

Description

Construction method of nonlinear rock fatigue constitutive model based on rheological model application

Technical field

The invention relates to the technical field of rock structural plane constitutive models, and in particular to a method for constructing a nonlinear rock fatigue constitutive model based on the application of rheological models.

Background technique

During the construction of geotechnical engineering and the mining of underground ores, rock masses are often subjected to periodic cyclic fatigue loads. The macro and micro evolution of the displacement field and stress field within the rock mass during this process is an important reason for the ultimate failure of rock mass engineering. Starting from the constitutive model, the mechanical mechanism of rock mass can be effectively revealed, for example, in the document "He, M., et al., Experimental investigation and damage modeling of salt rock subjected tofatigue loading. International Journal of Rock Mechanics and Mining Sciences, 2019.114 :p.17-23." conducted a study on the damage evolution rules of salt rock under different stress amplitudes, loading frequencies and loading rates, and proposed a fatigue life prediction model based on stress, frequency and loading rate; Another example is the document "Li, T., et al., Nonlinear behavior and damage model for fractured rock under cyclic loading based on energy dissipation principle. Engineering Fracture Mechanics, 2019.206: p.330-341." Based on indoor tests, The damage model of fractured rock mass was established based on the principle of energy dissipation; the document "Liu, Y. and F. Dai, Adamage constitutive model for intermittent jointed rocks under cyclic uniaxial compression. International Journal of Rock Mechanics and Mining Sciences, 2018.103: p.289- 301." Based on Lemaitre's strain equivalence principle, a coupled fatigue constitutive model for discontinuous jointed rock mass is derived; the document "Meng, Q., et al., Research onnon-linear characteristics of rock energy evolution under uniaxial cyclic loading and unloading conditions. Environmental Earth Sciences, 2019.78(23)." Based on the influence of loading rate and lithology on rock energy evolution under cyclic loading conditions, a nonlinear energy evolution model is proposed.

The above constitutive model has well studied the mechanical properties of rock mass under cyclic fatigue loading, but there are problems such as unclear physical and mechanical meaning of parameters and insufficient description of important displacement field evolution rules. In the study of constitutive models, model parameters The clarity of the meaning of physical mechanics is an important factor in determining the value of model engineering applications. Therefore, it is necessary to establish a rock fatigue constitutive model with clear physical and mechanical meaning.

Contents of the invention

Based on this, the purpose of the present invention is to provide a method for constructing a nonlinear rock fatigue constitutive model based on the application of a rheological model, and to establish a constitutive model that describes the entire process of rock mass fatigue. The parameters of this constitutive model are simple and have physical and mechanical significance. clear.

In order to solve the above technical problems, the present invention adopts the following technical solutions:

The present invention provides a method for constructing a nonlinear rock fatigue constitutive model based on the application of a rheological model, which includes the following steps:

Construct a rock mass fatigue constitutive model ε _eve corresponding to the mechanical behavior of the attenuation and stable fatigue stages during the cyclic loading process of the rock mass, and a rock mass fatigue constitutive model corresponding to the mechanical behavior of the attenuation and stable fatigue stages during the cyclic loading process of the rock mass. The constitutive equation of ε _eve is ε _eve = ε _e + ε _ve ; where ε _e represents the strain in the instantaneous loading stage, and ε _ve represents the strain corresponding to the decay and stable fatigue stages;

Introducing Riemann-Liouville fractional integration Integrate the points/> Substitute into the constitutive equation ε _eve = ε _e + ε _ve of the rock mass fatigue constitutive model ε _eve to obtain the strain expression of the rock mass fatigue constitutive model ε _eve based on fractional derivatives;

The Mittag-Leffler function is introduced to obtain the rock mass fatigue constitutive model ε _eve (N) based on fractional derivatives after the Mittag-Leffler function is introduced;

Construct a nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage;

Based on the strain equivalent theory, the damage variable D after the coupling of the macro initial damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass is obtained;

Introduce the coupled damage variable D into the nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage, and obtain the nonlinear fatigue constitutive model ε _vp (N) that considers the mechanical characteristics of the accelerated fatigue stage considering the initial damage;

The rock mass fatigue constitutive model ε _eve (N) based on fractional derivatives after introducing the Mittag-Leffler function is connected in series with the nonlinear fatigue constitutive model ε _vp (N) that considers the mechanical properties of the accelerated fatigue stage of initial damage, and is considered The initial damage is a nonlinear fatigue constitutive model that describes the entire process of rock mass fatigue.

In summary, the present invention provides a method for constructing a nonlinear rock fatigue constitutive model based on the application of a rheological model. It will combine nonlinear fatigue constitutive models that describe different fatigue stages and establish a fatigue model that describes the entire process of rock mass fatigue. The constitutive model has simple parameters and clear physical and mechanical meaning.

Description of drawings

Figure 1 shows the similarity of rock mass deformation and damage mechanisms under rheology and cyclic loading and unloading conditions: (a) creep conditions, (b) cyclic loading and unloading conditions; stages I, II, and III: attenuation, stability, and acceleration stages.

Figure 2 shows the nonlinear fatigue constitutive model based on fractional derivatives.

Figure 3 shows the nonlinear fatigue constitutive model in the acceleration stage.

Figure 4 is a schematic diagram of the initial damage to the rock mass.

Figure 5 shows the nonlinear fatigue constitutive model reflecting the mechanical behavior of the rock mass in three stages under cyclic loading conditions.

Figure 6 shows the strain-cycle number curve under different E _K conditions.

Figure 7 shows the strain-cycle number curve under different γ conditions.

Figure 8 is the damage variable D-cycle number curve chart under different δ.

Figure 9 shows the rock sampling and cutting instrument.

Figure 10 shows the geometric diagram and indoor actual picture of the splitting test specimen.

Figure 11 shows the geometric diagram and indoor actual picture of the three-point bending test specimen.

Figure 12 shows the cyclic fatigue test equipment.

Figure 13 is a schematic diagram of the loading and unloading of the cyclic fatigue test.

Figure 14 is a conventional splitting test load-strain curve.

Figure 15 is a diagram of the loading process of the three-point bending specimen.

Figure 16 is a conventional three-point bending test load-displacement curve.

Figure 17 is the tensile stress-strain curve of the cyclic loading and unloading test.

Figure 18 is the peak strain-cycle number curve of the splitting fatigue test.

Figure 19 shows the comparison between the calculated values of the model proposed in this article and the splitting fatigue test values.

Figure 20 is the strain-cycle number curve of the three-point bending fatigue test.

Figure 21 shows the comparison between the experimental values of the three-point bending fatigue test and the calculated values of the model proposed in this article.

Figure 22 is a schematic flowchart of a method for constructing a nonlinear rock fatigue constitutive model based on rheological model application provided by an embodiment of the present invention.

Detailed ways

In order to further understand the characteristics, technical means, and specific purposes and functions of the present invention, the present invention will be described in further detail below in conjunction with the accompanying drawings and specific implementation modes.

Please refer to Figure 1. The deformation and damage of rock mass under cyclic loading and rheological tests also increase nonlinearly with an independent variable such as time, and are all caused by crack evolution. They have the same three characteristic stages: decay stage (stage Ⅰ), stable stage (stage Ⅱ) and acceleration stage (stage Ⅲ). These essentially similar mechanical behaviors indicate the failure mechanism of deformation and damage of rock mass under cyclic loading and rheological tests, that is, the evolution of cracks in space and time. There are significant similarities with deformation features.

Based on the similarity between rock mass fatigue and rheology, the combination of element models in rock mass rheology is applied to the study of rock mass fatigue characteristics, thereby facilitating the construction of rock mass fatigue constitutive models.

Figure 22 is a schematic flow chart of a method for constructing a nonlinear rock fatigue constitutive model based on the application of a rheological model provided by an embodiment of the present invention. As shown in Figure 22, the nonlinear rock fatigue constitutive model based on the application of a rheological model The model construction method specifically includes the following steps:

Step S1: Construct a rock mass fatigue constitutive model ε _eve that describes the mechanical behavior of the attenuation and stable fatigue stages during the rock mass cyclic loading process, and describes the rock mass fatigue corresponding to the mechanical behavior of the attenuation and stable fatigue stages during the rock mass cyclic loading process. The constitutive equation of the constitutive model ε _eve is ε _eve = ε _e + ε _ve .

in, ε _e represents the strain in the instantaneous loading stage, ε _ve represents the strain corresponding to the attenuation and stable fatigue stages; σ represents the upper limit stress of loading, and _EM represents the Hooke body, which represents the deformation corresponding to the instantaneous strain generated under cyclic loading. Modulus; E _K is characterized by the deformation modulus corresponding to the attenuation and stable fatigue stage strains generated under cyclic loading; η _K is the viscous element parameter, characterized by the strain rate during cyclic loading; ε _Ek is the deformation modulus E _K corresponding to Strain, ε _ηk is the strain corresponding to the viscous element parameter η _K.

Step S2: Introduce Riemann-Liouville fractional integration Integrate the points/> Substitute into the constitutive equation ε _eve = ε _e + ε _ve of the rock mass fatigue constitutive model ε _eve to obtain the strain expression of the rock mass fatigue constitutive model ε _eve based on fractional derivatives; among them, the Riemann-Liouville fraction is introduced Points/> Used to describe the viscous element parameter η _K .

Among them, step S2 introduces Riemann-Liouville fractional integration integrate fractions Substitute into the constitutive equation ε _eve = ε _e + ε _ve of the rock mass fatigue constitutive model ε _eve to obtain the strain expression of the rock mass fatigue constitutive model ε _eve based on fractional derivatives. The specific operations include:

Step S21: Introduce Riemann-Liouville fractional integration Among them,/> ξ, N are parameters of the integral equation f(N); Γ(γ) is the Gamma function, γ∈(0,1),/>

Step S22: Calculate the γ-order calculus of f(N); where, the γ-order calculus of f(N) satisfies

Step S23: Integrate the Riemann-Liouville fraction Substituting into the constitutive equation ε _eve = ε _e + ε _ve of the rock mass fatigue constitutive model ε _eve , the following formula is obtained:

Step S24: Obtain the expression of ε _ve Among them, the expression of ε _ve is obtained based on the initial condition N=0, ε _ve =0, and fractional differential theory/> Fractional differential theory is recorded in the literature "Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. Amsterdam: Elsevier; 2006.";

Step S25, combine the formula in step S24 and _the constitutive equation ε _eve of _{the rock mass fatigue constitutive model ε eve, and obtain the strain expression of the rock mass fatigue constitutive model ε eve based on} fractional derivatives/>

Step S3: Introduce the Mittag-Leffler function to obtain the rock mass fatigue constitutive model ε _eve (N) based on fractional derivatives after introducing the Mittag-Leffler function. The fatigue constitutive model based on fractional derivatives after introducing the Mittag-Leffler function is obtained. The model ε _eve (N) effectively describes the mechanical behavior of rock mass in the decay and stable fatigue stages.

Among them, the method of step S3, the specific operations include:

Step S31. Introduce Mittag-Leffler function For formula/> Perform the conversion to get:

Since the formula Medium cumulative term/> The calculation amount is heavy. When the number of cycles N is large, it will be very difficult to perform calculations and parameter inversion. After the Mittag-Leffler function is introduced, the calculation amount is effectively reduced and the cost of parameter inversion is reduced.

Step S32, according to the formula It can be known/> And then get the formula The expression process of γΓ(γp) is as follows:

Then the constitutive equation of the fatigue constitutive model ε _eve (N) based on fractional derivatives after introducing the Mittag-Leffler function is obtained:

Step S4: Construct a nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage; the constitutive equation of the nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage satisfies the formula

Among them, σ is the upper limit stress of loading; eta _vp is the viscous element coefficient; D is the damage variable;/> It is the first derivative of strain and cycle number N in the nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage, reflecting the growth rate of strain with the number of cycles; σ _S is the threshold stress value for the rock mass entering the accelerated fatigue stage, In this embodiment, the value of σ _S is 0.75 times the peak stress, which is recorded in the literature "Geranmayeh Vaneghi, R., et al., Fatigue damage response of typical crystalline and granular rocksto uniaxial cyclic compression. International Journal of Fatigue, 2020.138 ." and "Ma, L., et al., Mechanical properties of rock salt under combined creep and fatigue. International Journal of Rock Mechanics and Mining Sciences, 2021.141."

The constitutive equation of the fatigue constitutive model based on fractional derivatives after introducing the Mittag-Leffler function from the above It can be seen that the fatigue constitutive model based on fractional derivatives after introducing the Mittag-Leffler function can effectively describe the mechanical behavior in the decay and stable fatigue stages, but it lacks reflection of the accelerated fatigue stage. Therefore, on this basis, the present invention Construct a nonlinear fatigue constitutive model ε _vp that can reflect the mechanical characteristics of the accelerated fatigue stage. As can be seen from Figure 1, the nonlinear characteristics of rock mass are most significant when entering the accelerated fatigue stage. Therefore, when constructing the corresponding nonlinear fatigue constitutive model that reflects the mechanical characteristics of the accelerated fatigue stage, the nonlinear evolution of its mechanical characteristics is a key consideration. .

The nonlinear fatigue constitutive model constructed by the present invention to reflect the mechanical characteristics of the accelerated fatigue stage is shown in Figure 3. The nonlinear fatigue constitutive model that reflects the mechanical characteristics of the accelerated fatigue stage consists of a plastic element representing the threshold stress value σ _S and The viscous elements representing the first derivative of strain and cycle number are connected in parallel; when the loaded stress and cycle number reach the threshold, the nonlinear fatigue constitutive model that reflects the mechanical characteristics of the accelerated fatigue stage is triggered and begins to function.

Step S5: Based on the strain equivalent theory, obtain the damage variable D after coupling the macro initial damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass.

Among them, the method of step S5, the specific operations include:

Step S51. Based on the strain equivalent theory, the expression of the damage variable D after coupling the macroscopic initial damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass satisfies D=D _ma +D _mi -D _ma D _mi ; where, D _ma is the macro initial damage of the rock mass; D _mi is the microscopic damage of the rock mass. It can be seen from the expression D=D _ma +D _mi -D _ma D _mi that when there is only microscopic damage in the rock mass, that is, macroscopic damage D _ma =0, the damage variable D after coupling is equal to microscopic damage D _mi ; when microscopic damage D _mi = 0, that is, when only macroscopic damage occurs to the rock mass, the damage variable D after coupling = D _ma , indicating that the expression of the damage variable D after coupling can be well adapted to the study of mechanical properties under coupling conditions of macroscopic and microscopic damage. .

During the formation process of rock mass, it is affected by various geological processes, as well as external factors such as stress, weathering, and crustal movement after formation, which cause natural macroscopic defects such as joints inside the rock mass. As shown in Figure 4, the existence of these initial damages is serious. Affects the mechanical properties of rock mass. Therefore, when conducting research on the mechanical properties of rock mass, it is necessary to consider both the initial damage and the microscopic damage caused by cyclic loading. Under the assumption of strain equivalent theory, refer to the reference "Liu, HY, et al., A dynamic damage constitutive model for a rock mass with persistent joints. International Journal of Rock Mechanics and Mining Sciences, 2015.75: p.132-139.", The damage variable after coupling the initial damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass can be expressed as: D=D _ma +D _mi -D _ma D _mi .

Step S52, through the formula Obtain the macro initial damage D _ma of the rock mass, where E _C is the elastic modulus of the rock mass containing natural macro defects, and _EO is the elastic modulus of the complete rock mass.

Step S53: Determine the relationship between the microscopic damage D _mi and the number of cycles N through the Kachanov damage law, where the Kachanov damage law equation is:

A and δ are material constants measured by experiments; ω is the applied load; is the first-order derivative of the microscopic damage variable D _mi of the rock mass with respect to the number of cycles N; in this embodiment, as shown in Figure 1, the relationship between the deformation of the rock mass and the number of cycles under cyclic loading conditions and the deformation and time under creep conditions The relationship is similar. Based on the above characteristics, the present invention determines the relationship between the microscopic damage D _mi and the number of cycles N through the Kachanov damage law in creep.

When the number of cycles N=0, the rock mass sample is not subjected to cyclic loading, and _Dmi =0 at this time. Based on the above initial conditions, and After integral processing, the expression of D _mi can be obtained as

When the rock mass is completely destroyed, D _mi =1, and the corresponding number of complete damage cycles N _C satisfies:

N _C =[A(δ+1)ω ^δ ] ^-1 ;

combine And N _C =[A(δ+1)ω ^δ ] ^-1 , it can be obtained that the rock mass damage evolution equation satisfies the conditions under cyclic loading conditions:

Step S54. Since the nonlinear fatigue constitutive model ε _vp of the present invention that reflects the mechanical characteristics of the accelerated fatigue stage describes the mechanical behavior of the rock mass in the accelerated fatigue stage, when damage begins to occur, N = N _S , so the formula The actual number of damage cycles in is/> The number of actual complete damage cycles is:/> Under this condition, change the formula in step S53/> Change to:

Among them, Ns is the number of cycles when the rock mass enters the accelerated fatigue stage; N _F is the number of cycles when the rock mass is completely destroyed, that is, the fatigue life.

When N=Ns, _Dmi =0, and when N= _NF , _Dmi =1.

Step S55: Change the formula and formula/> Substituting into the formula D = D _ma + D _mi - D _ma D _mi in step S51, the expression of the damage variable D after considering the coupling of the initial macroscopic damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass can be obtained to satisfy:

Step S6: Introduce the coupled damage variable D into the nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage, and obtain the nonlinear fatigue constitutive model ε _vp (N) that considers the mechanical characteristics of the accelerated fatigue stage considering the initial damage. ; Among them, the damage variable D after coupling is the damage variable D after the coupling of the initial macroscopic damage D _ma of the rock mass and the microscopic damage D _mi of the rock mass.

Among them, the method of step S6, the specific operations include:

Step S61: Change the formula Substitute into the constitutive equation of the nonlinear fatigue constitutive model ε _vp that reflects the mechanical characteristics of the accelerated fatigue stage/> , the constitutive equation of the nonlinear fatigue constitutive model considering the mechanical properties of the accelerated fatigue stage of initial damage can be obtained:

By converting the above formula, we can get:

Step S62. Combined with the initial conditions N = N _S and ε _vp = 0, the formula By solving, the expression of the nonlinear fatigue constitutive model that takes into account the mechanical characteristics of the accelerated fatigue stage of initial damage can be obtained to satisfy:

Step S7: Connect the rock mass fatigue constitutive model ε _eve (N) based on fractional derivatives after introducing the Mittag-Leffler function in series with the nonlinear fatigue constitutive model ε _vp (N) that considers the mechanical characteristics of the accelerated fatigue stage of initial damage. , obtain a nonlinear fatigue constitutive model that describes the entire process of rock mass fatigue taking into account initial damage.

Specifically, in the present invention, as shown in Figures 2, 3 and 5, the rock mass fatigue constitutive model based on fractional derivatives after introducing the Mittag-Leffler function in Figure 2 and the rock mass fatigue constitutive model considering the initial damage in Figure 3 are combined The nonlinear fatigue constitutive model of the mechanical characteristics in the accelerated fatigue stage is connected in series, and the nonlinear fatigue constitutive model reflecting the mechanical behavior of the rock mass in three stages (decay stage, stable stage, and acceleration stage) under cyclic loading conditions can be obtained as shown in Figure 5. The constitutive model is a nonlinear fatigue constitutive model that describes the entire process of rock mass fatigue considering the initial damage.

Among them, the method of step S7, the specific operations include:

Step S71: Connect the rock mass fatigue constitutive model based on fractional derivatives after introducing the Mittag-Leffler function and the nonlinear fatigue constitutive model that considers the mechanical properties of the accelerated fatigue stage of the initial damage in series to obtain a description of the rock mass that considers the initial damage. Nonlinear fatigue constitutive model of the entire fatigue process. Among them, the strain equation of the nonlinear fatigue constitutive model describing the entire process of rock mass fatigue considering the initial damage satisfies:

ε=ε _e +ε _ve +ε _vp ;

Step S72, combine formula

and formula Determine the specific strain equation ε(N) of the nonlinear fatigue constitutive model that describes the entire process of rock mass fatigue considering the initial damage:

Through steps S1 to S7, the present invention can obtain the specific expression formulas of D(N) and ε(N). There are parameters in the formulas whose mechanical properties are not clear enough, such as E _K , γ and δ, and their values vary with the properties of the rock mass. The rock mass fatigue characteristics represented by the E _K , γ and δ values change due to the difference. In the present invention, the values of D (N) and ε (N) under different model parameters E _K , γ and δ can be calculated through the control variable method. Obtaining the reflection of E _K , γ and δ values on rock fatigue characteristics includes the following steps: ① Use the control variable method, assuming that the parameters other than E _K , γ and δ in the fatigue constitutive model ε (N) are constants, Calculate the values of D(N) and ε(N) under different E _K , γ and δ conditions respectively; ② Draw the corresponding DN and ε-N curves to obtain the reflection of the E _K , γ and δ values on the fatigue characteristics of the rock mass. The above method of obtaining the reflection of rock fatigue characteristics by E _K , γ and δ values can obtain the fatigue characteristics reflected by E _K , γ and δ values through calculation and analysis, so that the nonlinear fatigue constitutive model of the present invention can simulate the experimental data. After combination, the mechanical properties of the rock mass can be obtained from the inverted E _K , γ and δ values.

In addition, among the parameters of formula (A), the meanings of E _M , eta _K and eta _VP are relatively clear, representing the instantaneous strain during cyclic loading, the basic rate in the stable phase and the initial rate in the acceleration phase respectively, while the values of E _K , γ and δ The characterized mechanical properties are not yet clear enough. The present invention adopts the control variable method commonly used in sensitivity analysis to analyze the sensitivity of these three parameters.

Assume σ<σ _S , substitute E _M =500MPa, η _K =50GPa, and γ =0.5 into formula (A), take E _K as 1, 3, 5, 7, and 9 GPa, and conduct parameter sensitivity analysis. The results are as shown in the figure 6 shown. It can be seen from the figure that E _K has a significant impact on strain. Specifically, the greater E _K , the smaller the strain in the attenuation and stable fatigue stages. E _K is inversely proportional to the strain in the attenuation and stable fatigue stages. On the other hand, the difference in strain nonlinear characteristics in the attenuation and stable fatigue stages under different E _K is not obvious. E _K mainly represents the strain amount in the attenuation and stable fatigue stages. Since during cyclic loading and unloading, the strains in the attenuation and stable fatigue stages To a large extent, it represents the density of the plastic hysteresis loop and the damage represented by the density. Therefore, E _K also represents the area of the plastic hysteresis loop and the total amount of damage in the decay and isokinetic fatigue stages.

Assume σ<σ _S , substitute E _M =500MPa, E _K =5GPa, η _K =50GPa into formula (A), take γ as 0.1, 0.3, 0.5, 0.7, 0.9, and perform parameter sensitivity analysis on parameter γ, The results are shown in Figure 7. It can be seen from the figure that the ultimate strains under different γ are almost the same, and γ has no effect on the strain amount in the decay and stable fatigue stages. However, the nonlinear characteristics of the curves under different γ are obviously different. Specifically, the larger γ is, the higher the strain increases with the number of cycles, and the more obvious the nonlinear characteristics of the curve are. γ mainly represents the nonlinear characteristics of the attenuation and stable fatigue stages. From the reflection of the strain on the density of the plastic hysteresis loop in the parameter sensitivity analysis of E _K , it can be seen that γ also represents the nonlinear characteristics of the plastic hysteresis loop area expansion and damage variable evolution in the attenuation and stable fatigue stages during cyclic loading and unloading.

Parameter δ is an important parameter to characterize damage in the accelerated fatigue stage. In order to understand the damage characteristics reflected by δ, assuming σ<σ _S , take N _F =100, N _S =90 are substituted into equation (21), and δ = -0.8, -0.5, 0, 0.5, 1 is taken to perform parameter sensitivity analysis. The results are shown in Figure 8. It can be seen that although the damage evolution endpoints under different δ are consistent, the differences in damage evolution trends are very prominent. The specific performance is as follows: when δ<0, the damage evolves with a rate attenuation trend, and the smaller δ is, the greater the rate attenuation trend. Obviously; when δ=0, the damage evolution trend is linear, and the damage-cycle number curve evolves in the form of a straight line; when δ>0, the damage evolves with a trend of increasing rate, and the larger δ is, the greater the trend of rate growth is. obvious. It can be seen that δ is an important parameter characterizing the damage trend in the accelerated fatigue stage. Through the inversion value of δ, the damage evolution characteristics of the rock mass in the accelerated fatigue stage can be obtained.

At this point, the construction method of the nonlinear rock fatigue constitutive model based on the application of the rheological model has been completely determined. The schematic diagram of the constitutive model is shown in Figure 5. Moreover, the parameters of the constitutive model have clear physical and mechanical meanings and are suitable for the whole process of fatigue mechanics. To describe the behavior, the feasibility of the nonlinear rock fatigue constitutive model construction method based on the rheological model application of the present invention is verified through experimental examples, as follows.

The indoor test data is substituted into the nonlinear rock fatigue constitutive model strain equation ε(N), and its threshold stress value σ _S in the accelerated fatigue stage is determined. The validity and rationality of the model are verified through experiments, and the rock is obtained through the inverted parameters. Body fatigue characteristics.

In order to verify the rationality and effectiveness of the fatigue constitutive model proposed in this invention, an indoor cyclic fatigue loading and unloading test was carried out. Red sandstone, which is widely distributed in geology, was used as the research object. The sample was taken from a red sandstone body in a certain area in Shandong, China. It is a fine sandstone with a light brown color and sugar-like internal particles. After sampling using the drilling core method, professionals are entrusted to cut and process the samples according to test needs. The sampling and cutting instruments are shown in Figure 9. Since in engineering practice, the initiation and expansion of tensile cracks are important causes of rock mass damage, this invention aims at the mechanical properties of tensile crack initiation and expansion under cyclic loading conditions, and develops two types of tensile cracking: splitting and three-point bending. The cyclic fatigue loading and unloading test, which is the main cause of damage, is conducted to explore the mechanical properties of rock mass under cyclic fatigue loading. The splitting test uses a disc-shaped specimen with a prefabricated herringbone crack. The disk diameter d is 100mm and the thickness t is 35mm. A diamond slice is used to cut the prefabricated herringbone crack with a diameter R _S of 60mm and a depth of 20mm. The geometry of the specimen The schematic diagram and indoor actual picture are shown in Figure 10. The elastic modulus of the tensile stress-strain curve of the complete sample is 572.64MPa. The three-point bending test uses a semi-disc-shaped specimen with a crack. The specimen diameter d is 100mm, the thickness t is 35mm, the crack length is 20mm, and the width is 2mm, as shown in Figure 11. The elastic modulus of the tensile stress-strain curve of the complete sample is 840.1MPa.

Test preparation

The test was carried out on the WHY-200/100 microcomputer controlled universal testing machine as shown in Figure 12. The instrument consists of a host computer, measurement and control software, and a measurement and control system. It loads the sample through force or displacement control. It has the advantages of strong stability, high precision and large measuring range, and can meet the requirements of Brazilian splitting and cyclic loading and unloading tests of red sandstone samples. When carrying out the cyclic fatigue test, under the conditions of constant temperature and humidity, fatigue load the specimen according to the loading and unloading method as shown in Figure 13.

Split fatigue test

In order to determine the splitting fatigue test plan, an indoor conventional splitting test was first carried out. Three identical disc-shaped specimens with herringbone cracks were loaded using the stress control method, and the loading rate was 0.1KN/s. After the test, the data was processed. The results are shown in Figure 14. It can be seen that as the loading proceeds, the curve enters the elastic deformation stage from an obvious micro-crack compaction stage, and then brittle failure occurs near the peak load point. Taking the average peak load of the three samples and substituting it into the splitting test tensile strength calculation formula shown in Equation (B), the average tensile strength of the sample can be obtained as 1.903MPa. At the same time, the elastic modulus of the sample is calculated based on the test curve as 753.51MPa.

In the formula, σ _t is the tensile strength; P is the peak load; d, t are the diameter and thickness of the sample.

Since the sample failure in the splitting test is mainly tensile, tensile stress is selected as the upper and lower limit stresses of the cyclic loading and unloading test, and three sets of cyclic fatigue loading and unloading tests with upper and lower limit stress amplitudes of _0.6σt are carried out using this as a variable. . The test plan is shown in Table 1. When the load is damaged or the number of cycles reaches 100, the test ends and the corresponding data is recorded.

Table 1 Splitting fatigue test scheme

Three point bending fatigue test

Similarly, the indoor conventional three-point bending test is first carried out, and the test loading process is shown in Figure 15. Loading is carried out through displacement control, loading from the top center row at a loading rate of 0.1mm/min. After the test is completed, the data is processed, and the results are shown in Figure 16. Since the calculation formula for the tensile strength of the three-point bending test is consistent with that of the splitting test, the average peak load of the three specimens, 2.62KN, is substituted into Equation (B) to calculate the tensile strength of the specimen as 0.953MPa, and the three-point bending design is based on this The fatigue test plan is shown in Table 2. The elastic modulus of the sample was calculated from the test curve and was 381.39MPa.

Table 2 Three-point bending fatigue test scheme

Split fatigue test results and model verification

The tensile stress-strain curve of the splitting fatigue test is shown in Figure 17. It can be seen from the figure that for specimen S-1 with an upper limit tensile stress of _0.75σt , the plastic hysteresis loop of the specimen curve is in a tight state, and has experienced many times No damage occurred after cyclic loading, indicating that under this loading condition, the internal cracks of the specimen were not fully developed and the deformation was mainly elastic deformation. For the S-2 specimen with an upper limit of tensile stress of _0.85σt , the plastic hysteresis loop of the specimen curve begins to loosen, and failure occurs after about 100 cycles, indicating that the crack development environment of the specimen is better under this condition, but it requires A certain number of cycles. For the S-3 specimen with an upper limit of tensile stress of _0.95σt , the curved plastic hysteresis loop is the sparsest, and fatigue failure occurs after the specimen is subjected to 6 cyclic loads, indicating that the deformation of the specimen under this load condition is mainly plastic deformation, and cracks With sufficient development environment, only a few cycles are needed for destruction. It can be seen that red sandstone has obvious nonlinear characteristics in each fatigue stage, and is greatly affected by the number of cycles and stress. Studying the effects of number of cycles and stress on the fatigue characteristics of engineering rock masses has high engineering value. Since the peak strain can most intuitively reflect the fatigue characteristics of the rock mass, in the present invention, formula (A) is used to describe the relationship between the peak strain and the peak stress at different number of cycles. By processing the data in Figure 17, the peak strain-cycle number curve shown in Figure 18 can be obtained.

Formula (A) is used to fit the data shown in Figure 18. The comparison results between the calculated values and experimental values of the model proposed by the present invention are shown in Figure 19. It can be seen from the figure that the fatigue constitutive model proposed by the present invention agrees well with the test curve, the fitting effect is significant, and the rationality and effectiveness of the model have been effectively verified. The parameter inversion results are shown in Table 3. Combined with the model parameter sensitivity analysis results, it can be seen that: (1) Parameter E _M : S-1<S-2<S-3, indicating that the deformation modulus corresponding to the instantaneous strain of red sandstone increases as the upper limit stress increases; It can be seen from the stress-strain curve in Figure 14 that this is the result of a long compaction stage of sandstone micro-cracks and no obvious softening before the peak stress: the greater the upper limit stress, the higher the proportion of the linear stage with the largest slope in the curve, so The corresponding instantaneous deformation modulus is larger. (2) Parameter E _K obeys S-2<S-1<S-3, which means that in the attenuation and stable fatigue stages, the cyclic fatigue load causes the greatest strain and damage to specimen S-2, followed by S-1, and S -3 minimum. This is because the upper limit stress of sample S-1 is the smallest, the fatigue damage caused by fatigue load on the sample is limited, and the corresponding strain is smaller than that of sample S-2; while for the S-3 sample with the upper limit stress reaching _0.95σt , the attenuation and The stable fatigue stage only exists in a few cyclic loadings in the early stage, so the damage caused by the fatigue load is very small and the corresponding strain is minimal. (3) Parameter γ: S-2<S-1<S-3, indicating that in the decay and stable fatigue stages, the plastic hysteresis loop area of sample S-3 has the most obvious increasing trend with the number of cycles, S-1 times In short, S-2 is the smallest. It shows that in the case of accelerated fatigue, the greater the upper limit stress, the more obvious the damage evolution trend will be in the attenuation and stable fatigue stages of the specimen. However, for low-stress conditions where accelerated fatigue does not occur, the damage evolution trend may not always be inferior to that of high-stress conditions. Obviously. (4) Parameter δ: S-3>S-2, indicating that in the accelerated fatigue stage, the expansion of the plastic hysteresis loop of the S-3 sample is more obvious and the damage evolution is more intense, indicating that the damage evolution characteristics and stress in the accelerated fatigue stage are levels are closely related.

Table 3 Parameter inversion results

Three-point bending fatigue test results and model verification

After the three-point bending fatigue test is completed, the test data is processed to obtain the strain-cycle number curve shown in Figure 20. It can be seen from the figure that, similar to the splitting fatigue test, the curves under different stress levels also have three forms: the upper limit stress is _0.7σt , and the strain-cycle number curve only has attenuation and stable fatigue stages, and no accelerated fatigue occurs. , non-destructive T-1 sample test curve. The upper limit stress is _0.8σt exceeding the threshold stress value of the accelerated fatigue stage. The curve includes the T-2 specimen test curves in the three fatigue stages of attenuation, stability and acceleration. The upper limit stress is _0.9σt which is far beyond the threshold stress value of the accelerated fatigue stage. There are few cycles in the decay and stability and accelerated fatigue stages, and the T-3 sample test curve quickly enters the accelerated fatigue stage and causes damage. It is shown that the stress level has an important influence on the curve shape. Formula (A) is used to fit the data shown in Figure 20, and the calculated values of the model proposed in the present invention are compared with the experimental values, as shown in Figure 21. It can be clearly seen from the figure that the fatigue constitutive model proposed by the present invention can accurately describe the test curve, effectively verifying the rationality and effectiveness of the model. The parameter inversion results are shown in Table 4. Combined with the model parameter sensitivity analysis results, it can be seen that: (1) For parameter E _K : T-2<T-1<T-3, it means that the strain corresponding to the attenuation and stable fatigue stage of sample T-2 is the largest, T-1 time In short, T-3 is the smallest. It shows that in the decay and stable fatigue stages, cyclic fatigue loading causes the greatest damage to specimen T-2, followed by T-1, and T-3 the smallest. This is because the upper limit stress of the T-1 sample is the smallest, and the cumulative amount of fatigue damage in the decay and stable fatigue stages is small; while the upper limit stress of the T-3 sample has reached _0.9σt , and the sample is already very high in the initial stage of cyclic loading. It will soon enter the unstable accelerated fatigue stage, and the decay and stable fatigue stages will have very little time, so the total amount of corresponding damage will be the smallest. (2) Parameter γ: T-2<T-1<T-3, which shows that in the decay and stable fatigue stages, the plastic hysteresis loop area of sample T-3 has the most obvious trend of increasing with the number of cycles, T-1 times In short, T-2 is the smallest. The most obvious phenomenon that the area of the plastic hysteresis loop of sample T-3 increases with the number of cycles shows that the damage characteristics of the attenuation and stable fatigue stages have obvious dependence on the stress level; while the plastic hysteresis loop of sample T-1 grows. The trend is more obvious than that of T-2. Combined with the analysis that the total amount of damage of the T-1 sample is the smallest in the inversion of E _K , it can be seen that in the decay and stable fatigue stages, the total amount of damage is positively correlated with the stress level, but accelerated fatigue does not occur. The damage evolution trend under low stress conditions is not necessarily always less obvious than under high stress conditions. (3) Parameter δ: T-3>T-2, indicating that in the accelerated fatigue stage, the expansion of the plastic hysteresis loop of the T-3 sample is more obvious and the damage evolution is more intense, indicating that the damage characteristics in the accelerated fatigue stage also have an impact on stress. Higher dependency. Comparing the analysis results of the parameter inversion of the splitting fatigue test, it can be seen that the change characteristics of each parameter in the three-point bending fatigue test with the stress level are similar to the parameter inversion characteristics of the splitting fatigue test, which not only shows that there are common fatigue characteristics under the two test conditions It also shows that the fatigue constitutive model proposed by the present invention has wide applicability and good accuracy, and can be suitable for fatigue characteristics research under different boundary conditions.

Table 4 Three-point bending fatigue test model parameter inversion results

Compared with existing constitutive model construction methods, the present invention has the following beneficial effects:

1. Aiming at the mechanical behavior of rock mass in the attenuation and fatigue stages, a nonlinear fatigue constitutive model based on fractional derivatives is proposed. This model can not only describe the relationship between fatigue deformation of rock mass and the number of cycles, but also reflect the fatigue process. Nonlinear characteristics; the model parameter E _K can reflect the plastic hysteresis loop area and total damage amount in the attenuation and stable fatigue stages; γ can represent the nonlinear characteristics of the plastic hysteresis loop area expansion trend and damage variable evolution in the attenuation and stable fatigue stages.

2. For the accelerated fatigue stage with severe damage evolution, Kachanov damage law is used to describe the microscopic damage of the rock mass, and the damage expression considering the initial damage is obtained by combining the coupling expression of macroscopic damage and microscopic damage; the damage expression is applied to the viscous element To describe the damage evolution, establish a nonlinear fatigue constitutive model that reflects the mechanical characteristics of the accelerated fatigue stage. The model parameter δ can reflect the damage evolution characteristics of the rock mass in the accelerated fatigue stage. The larger δ is, the more severe the rock mass damage will be. Different fatigue stages will be described. A combination of nonlinear fatigue constitutive models is used to establish a fatigue constitutive model that describes the entire process of rock mass fatigue. The parameters of this constitutive model are simple and the physical and mechanical meaning is clear.

3. Comparing the calculated values of the fatigue constitutive model proposed by the present invention with the indoor splitting and three-point bending fatigue test values, it is found that the nonlinear fatigue mechanical behavior of the rock mass can be accurately reflected by the fatigue constitutive model proposed by the present invention, which verifies that the fatigue constitutive model proposed by the present invention can be accurately reflected. The invention proposes the rationality and reliability of the model. According to the model parameter inversion results, it can be seen that stress and cycle number are important factors affecting the evolution of fatigue damage.

(1) Decay and stable fatigue stages: Under different stress conditions, damage accumulates as the number of cycles increases. However, under low stress conditions, the specimen does not undergo fatigue damage, and the damage accumulates slowly with the increase of the number of cycles, and converges to a certain characteristic value; as the stress level increases, the accumulation of damage in the attenuation and stable fatigue stages begins to increase, and In the case of accelerated fatigue, the greater the upper limit stress, the higher the damage evolution rate.

(2) Accelerated fatigue stage: The higher the stress level, the more intense the damage evolution, and the fewer cycles required for fatigue damage to occur, indicating that the damage evolution characteristics of the accelerated fatigue stage are closely related to the stress level.

The above-described embodiments only express several implementation modes of the present invention, and their descriptions are relatively specific and detailed, but should not be construed as limiting the scope of the present invention. It should be noted that, for those of ordinary skill in the art, several modifications and improvements can be made without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the appended claims.

Claims (10)

1. The construction method of the nonlinear rock fatigue constitutive model based on rheological model application is characterized by comprising the following steps,

constructing a rock fatigue constitutive model epsilon corresponding to mechanical behaviors describing attenuation and stable fatigue stages in cyclic loading process of rock mass _eve Describing rock fatigue constitutive model epsilon corresponding to mechanical behaviors of attenuation and stable fatigue stage in cyclic loading process of rock mass _eve The constitutive equation of (2) is epsilon _eve ＝ε _e +ε _ve The method comprises the steps of carrying out a first treatment on the surface of the Wherein ε _e Characterized by strain, ε, of the transient loading phase _ve Characterized by strain corresponding to the decay and stabilization fatigue phase;

introduction of Riemann-Liouville fractional integrationScore integration +.>Substituted into rock fatigue constitutive model epsilon _eve Constitutive equation ε of (2) _eve ＝ε _e +ε _ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained _eve Is a strain expression of (2);

introducing a Mittag-Leffler function to obtain a Mittag-Leffler functionRock fatigue constitutive model epsilon based on fractional derivative _eve (N)；

Construction of nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage _vp ；

Based on strain equivalence theory, obtaining macroscopic initial damage D of rock mass _ma And rock mass microscopic damage D _mi A damage variable D after coupling;

introducing the coupled damage variable D into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage _vp In the method, a nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained _vp (N)；

Rock fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function is introduced _eve (N) nonlinear fatigue constitutive model epsilon with mechanical properties of accelerated fatigue phase considering initial damage _vp (N) connecting the two components in series to obtain a nonlinear fatigue constitutive model which is used for describing the whole fatigue process of the rock mass and considers initial damage; the strain equation of the nonlinear fatigue constitutive model satisfies the following conditions: epsilon=epsilon _e +ε _ve +ε _vp 。

2. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1, wherein the rock fatigue constitutive model epsilon corresponding to mechanical behaviors of damping and stabilizing fatigue stages in the cyclic loading process of the descriptive rock mass _eve The constitutive equation of (2) is epsilon _eve ＝ε _e +ε _ve In (a) Wherein σ is characterized by the upper stress limit of the load, E _M The deformation modulus is characterized by the deformation modulus corresponding to the instantaneous strain generated under cyclic loading; e (E) _K The deformation modulus is characterized by attenuation generated under cyclic loading and corresponding to strain in a stable fatigue stage; η (eta) _K Is a viscous element parameter characterized by a strain rate at cyclic loading; epsilon _Ek For modulus of deformation E _K Corresponding strain, ε _ηk Is the parameter eta of the viscous element _K Corresponding strain.

3. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1 or 2, wherein the steps introduce a Riemann-liooville fractional integralScore integration +.>Substituted into rock fatigue constitutive model epsilon _eve Constitutive equation ε of (2) _eve ＝ε _e +ε _ve In the method, a rock fatigue constitutive model epsilon based on fractional derivative is obtained _eve The specific operations of the method of strain expression of (a) include,

introduction of Riemann-Liouville fractional integration Wherein (1)>ζ and N are parameters of an integral equation f (N); gamma (Gamma) is Gamma function, gamma E (0, 1), and->

Calculating gamma calculus of f (N); wherein the gamma-order calculus of f (N) satisfies

The Riemann-Liouville fraction was integratedSubstituted into rock fatigue constitutive model epsilon _eve Constitutive equation ε of (2) _eve ＝ε _e +ε _ve The following formula is obtained:

acquisition of epsilon _ve Expression of (2)

Combination formulaConstitutive equation epsilon of rock fatigue constitutive model _eve ＝ε _e +ε _ve Obtaining rock fatigue constitutive model epsilon based on fractional derivative _eve Strain expression +.>

4. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 3, wherein the step of obtaining epsilon _ve Expression of (2)The method comprises the following specific operations:

epsilon based on the initial condition n=0 _ve =0 and fractional differential theory to obtain ε _ve Expression of (2)

5. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 3, wherein the steps introduce a Mittag-Leffler function to obtain a rock fatigue constitutive model epsilon based on fractional derivative after introducing the Mittag-Leffler function _eve The method of (N), the specific operations include,

step S31, introducing Mittag-Leffer function For formula-> Conversion is carried out to obtain:

step S32, according to the formulaIt can be seen that->Thereby obtaining the formulaThe process of representation of gamma Γ (γp) is as follows:

thereby obtaining the fatigue constitutive model epsilon based on the fractional derivative after introducing Mittag-Leffler function _eve Constitutive equation of (N):

6. the method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 5, wherein the method comprises the following steps: the steps construct a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage _vp Nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage _vp The constitutive equation of (2) satisfies the formulaWherein sigma is the upper stress of the load; η (eta) _vp Is a viscous element coefficient; d is a damage variable; />Nonlinear fatigue constitutive model epsilon for reflecting mechanical characteristics of accelerated fatigue stage _vp First derivative of medium strain and cycle number N; sigma (sigma) _S The threshold stress value for the rock mass entering the accelerated fatigue phase.

7. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 6, wherein the method comprises the following steps: the sigma _S The value was 0.75 times the peak stress.

8. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 1, wherein the method comprises the following steps: the step is based on strain equivalence theory to obtain macroscopic initial damage D of rock mass _ma And rock mass microscopic damage D _mi The method of damaging the variable D after coupling, the specific operations include,

step S51, based on strain equivalence theory, rock mass macroscopic initial damage D _ma And rock mass microscopic damage D _mi The expression of the impairment variable D after coupling satisfies d=d _ma +D _mi -D _ma D _mi The method comprises the steps of carrying out a first treatment on the surface of the Wherein D is _ma Macroscopic initial damage to the rock mass; d (D) _mi Is microscopic damage to the rock mass;

step S52, through the formulaObtaining macroscopic initial damage D of rock mass _ma Wherein E is _C Modulus of elasticity, E, of rock mass containing natural macroscopic defects _O The modulus of elasticity of the complete rock mass;

step S53, determining microscopic damage D through Kachanov damage law _mi Relationship with cycle number N, wherein the Kachanov's law of damage equation is:

A. delta is the material constant determined by the test; omega is the applied load; />Is a rock microscopic damage variable D _mi A first derivative with respect to the number of cycles N; wherein, pair-> Integrating to obtain D _mi Expression->D when the rock mass is completely destroyed _mi =1, corresponding complete damage cycle number N _C Satisfy N _C ＝[A(δ+1)ω ^δ ] ^-1 ；

Bonding ofN _C ＝[A(δ+1)ω ^δ ] ^-1 Obtaining the rock mass damage performance under the cyclic loading conditionThe equation of the chemical formula satisfies the condition->

Step S54, the formula in step S53The method is changed into that:

wherein Ns is the number of cycles of the rock mass into the accelerated fatigue phase; n (N) _F For the number of cycles at the moment of complete destruction of the rock mass, the actual number of cycles of damage +.>True complete injury cycle +.>When n=ns, D _mi ＝0，N＝N _F At time D _mi ＝1；

Step S55, the formula is calculatedAnd formula->The formula d=d substituted into step S51 _ma +D _mi -D _ma D _mi In the method, the initial macroscopic damage D of the rock mass is considered _ma And rock mass microscopic damage D _mi The expression of the impairment variable D after coupling satisfies:

9. according to claimThe method for constructing the nonlinear rock fatigue constitutive model based on rheological model application is characterized by comprising the following steps of: the step introduces the damage variable D after coupling into a nonlinear fatigue constitutive model epsilon reflecting the mechanical characteristics of the accelerated fatigue stage _vp In the method, a nonlinear fatigue constitutive model epsilon considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained _vp The method of (N), the specific operations include,

step S61, formulaSubstituted into nonlinear fatigue constitutive model epsilon reflecting mechanical characteristics of accelerated fatigue stage _vp Constitutive equation of->In the method, constitutive equation of a nonlinear fatigue constitutive model considering mechanical characteristics of initial damage in an accelerated fatigue stage is obtained:

the conversion is carried out on the above materials to obtain the following components:

step S62, combining the initial conditions n=n _S ，ε _vp =0, for the aboveSolving to obtain a nonlinear fatigue constitutive model expression considering the mechanical characteristics of the initial damage in the accelerated fatigue stage, wherein the nonlinear fatigue constitutive model expression satisfies the following conditions:

10. The method for constructing a nonlinear rock fatigue constitutive model based on rheological model application according to claim 9, wherein the method comprises the following steps: the steps are to introduce a rock fatigue constitutive model epsilon based on fractional derivative after Mittag-Leffler function _eve (N) nonlinear fatigue constitutive model epsilon with mechanical properties of accelerated fatigue phase considering initial damage _vp (N) series connection, a method for obtaining a nonlinear fatigue constitutive model describing the whole process of rock mass fatigue by considering initial damage, the specific operation comprises,

step S71, connecting a nonlinear fatigue constitutive model based on fractional derivatives and a nonlinear fatigue constitutive model considering initial damage in series to obtain a nonlinear fatigue constitutive model describing the whole process of rock body fatigue considering initial damage, wherein a strain equation of the nonlinear fatigue constitutive model meets the following conditions:

ε＝ε _e +ε _ve +ε _vp ；

step S72, combining formulas

Sum formulaDetermining a specific strain equation epsilon (N) of a nonlinear fatigue constitutive model describing the whole process of rock mass fatigue taking initial damage into consideration: