Method for establishing rock fractional order creep model under disturbance effect

Abstract

According to the method for establishing the rock fractional order creep model under the disturbance effect, test results of rock samples under different confining pressures are obtained, a strain-time curve of the rock samples is drawn according to the creep test results, a proper creep model is obtained in combination with a basic element model, and the rock fractional order creep model is established under the disturbance effect. And respectively adding a damage model and a fractional order model into the creep model, and establishing a damage fractional order creep model. Compared with a traditional model, the fitting result of the improved creep model is consistent with the test result, it is indicated that the improved fractional order model can express the creep behavior of tuff under the complex stress condition, and the defect that the traditional model is difficult to describe the accelerated creep stage is overcome; the evolution law of rock mechanical parameters in the whole creep process is displayed through fractional order change.

Description

Technical Field

[0001] The invention belongs to the technical field of rock engineering, and in particular relates to a method for establishing a rock fractional-order creep model under disturbance. Background Art

[0002] Under disturbance, rocks show obvious nonlinear characteristics. On the one hand, the creep rate increases continuously during the disturbance process, and the expansion of internal cracks in the rock leads to a decrease in rock strength. On the other hand, the decrease in strength will trigger the accelerated creep threshold in advance and cause accelerated creep. Therefore, how to describe the nonlinear creep characteristics of rocks under disturbance is the key and difficulty of constructing a constitutive model. Summary of the invention

[0003] In view of this, the main purpose of the present invention is to provide a method for establishing a rock fractional creep model under disturbance.

[0004] The technical solution adopted by the invention is as follows:

[0005] A method for establishing a rock fractional creep model under disturbance, characterized by comprising the following steps:

[0006] S1, obtain the basic mechanical parameters of rock samples under different confining pressures;

[0007] S2, designing a creep perturbation test of the rock sample according to the basic mechanical parameters of the rock sample under different confining pressures obtained in step S1, carrying out a constant load perturbation creep test of the rock sample under different confining pressures, and obtaining creep test results of the rock sample;

[0008] S3, drawing a strain-time curve of the rock sample according to the creep test result of the rock sample obtained in step S2;

[0009] S4, obtaining a suitable creep model according to the strain-time curve of the rock sample at different confining pressures and disturbance amplitudes obtained in step S3 and in combination with the basic element model;

[0010] S5, under the action of disturbance, introducing the damage model into the creep model selected in step S4 to obtain a damage creep model, and fitting the test results of the rock sample in step S3 with the damage creep model;

[0011] S6, adding a fractional order model to the damage creep model obtained in step S5, and fitting the nonlinear creep characteristics of the rock sample under the disturbance to establish a fractional order creep model.

[0012] The beneficial effects of this application include:

[0013] The fractional-order model is combined with the improved creep damage model to establish a fractional-order damage model that can reflect accelerated creep. By segmenting the test results of different creep stages, the creep model based on variable-order fractional derivatives can well describe the nonlinear creep during the disturbance test. The parameters in the model are determined through segmented fitting, and the theoretical curve is highly consistent with the experimental data. Compared with the traditional model, the fitting results of the improved creep model are more consistent with the experimental results. The fractional-order model can express the creep behavior of tuff under complex stress conditions, making up for the deficiency of the traditional model that it is difficult to describe the accelerated creep stage. The evolution law of rock mechanical parameters in the entire creep process is shown through the change of fractional order. BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The following drawings are only used to illustrate and explain the present invention, and are not used to limit the scope of the present invention, wherein:

[0015] Figure 1 A flow chart of a method for establishing a rock fractional creep model under disturbance in this application;

[0016] Figure 2 Schematic diagram of Kelvin model and Bingham model;

[0017] Figure 3 This is a schematic diagram of the Nishihara model;

[0018] Figure 4 Schematic diagram of viscoplastic element considering damage;

[0019] Figure 5 It is a schematic diagram of the nonlinear damage creep model;

[0020] Figure 6 Schematic diagram of fractional viscoelastic body;

[0021] Figure 7 Schematic diagram of fractional-order nonlinear constitutive model;

[0022] Figure 8 This is a schematic diagram of the software operation interface;

[0023] Fig. 9 Figure 2 is the fitting result diagram under different confining pressures. DETAILED DESCRIPTION

[0024] In order to make the purpose, technical solution, design method and advantages of the present invention clearer, the present invention is further described in detail by specific embodiments in conjunction with the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0025] Reference Figure 1The present invention provides a method for establishing a rock fractional creep model under disturbance, comprising the following steps:

[0026] S1, obtain the basic mechanical parameters of rock samples under different confining pressures;

[0027] S2, designing a creep perturbation test of the rock sample according to the basic mechanical parameters of the rock sample under different confining pressures obtained in step S1, carrying out a constant load perturbation creep test of the rock sample under different confining pressures, and obtaining creep test results of the rock sample;

[0028] S3, drawing a strain-time curve of the rock sample according to the creep test result of the rock sample obtained in step S2;

[0029] S4, obtaining a suitable creep model according to the strain-time curve of the rock sample at different confining pressures and disturbance amplitudes obtained in step S3 and in combination with the basic element model;

[0030] S5, under the action of disturbance, introducing the damage model into the creep model selected in step S4 to obtain a damage creep model, and fitting the test results of the rock sample in step S3 with the damage creep model;

[0031] S6, adding a fractional order model to the damage creep model obtained in step S5, fitting the nonlinear creep characteristics of the rock sample under the disturbance, and establishing a fractional order creep model.

[0032] In the above, the basic mechanical parameters include elastic modulus E, Poisson's ratio μ, cohesion c and internal friction angle

[0033] In the above, reference Figure 2-Figure 3 , in step 4, the steps to select a suitable creep model are as follows:

[0034] In the selection of creep model, tuff has a certain threshold value under long-term load. After experiencing instantaneous elastic deformation and attenuated creep at low stress level, the creep rate of tuff gradually decreases, and the final strain gradually tends to a constant value; after experiencing the above stages at high stress level, the rock will enter a steady-state creep state, and then the creep rate will gradually increase and accelerate creep until the sample is destroyed. Therefore, the model should contain plastic elements; therefore, the model composed of Kelvin model and Bingham model in series is selected as the creep model, which describes the creep characteristics of tuff at different stress levels; this creep model combines the advantages of Kelvin model and Bingham, and can not only reflect most of the creep characteristics of rock, but also reflect the difference in creep morphology of rock at low stress level and high stress level, and it more comprehensively reflects the viscoelastic-elastic-plastic characteristics of rock;

[0035] Among them, the creep equation of the Kelvin model is:

[0036]

[0037] Among them, the creep equation of the Bingham model is:

[0038]

[0039] When the deviatoric stress value is lower than the yield stress, the creep model degenerates into a generalized Kelvin model, and its creep equation is expressed as:

[0040]

[0041] When the deviatoric stress value is higher than the yield stress, the viscoplastic element participates in the work, and the creep equation is expressed as:

[0042]

[0043] In the formula, σ and ε are the total stress and strain of the model, E1 and E2 are elastic moduli, η1 and η2 are viscosity coefficients, and σ s is the yield stress.

[0044] In the above, reference Figure 4-Figure 5 The selected creep model can reflect the instantaneous elasticity, decay creep, steady-state creep and plastic characteristics of rock under different stress levels, but cannot reflect the accelerated creep characteristics of rock samples under high stress levels; therefore, it is necessary to improve the components based on the selected creep model so that it can more accurately reflect the creep characteristics of tuff. Therefore, the damage model is introduced into the creep model. The steps are as follows:

[0045] First, the theoretical formula for the time relationship of the elastic modulus in the creep model is derived:

[0046]

[0047] σ=(1-D)E0ε (6)

[0048] Where: P, Q and R are model parameters, E t is the elastic modulus at time t, E0 is the elastic modulus before damage;

[0049] On this basis, the original damage factor is improved. The initial elastic modulus of the rock sample before being disturbed is E0. After being disturbed, the material properties gradually decay until the construction is completed. After the stress state is stable, the elastic modulus of the material also tends to a stable value (long-term modulus E0). ∞ );

[0050] The damage variable is defined as follows: When the test starts at t = 0, D = 0, and the initial damage is zero; when t → ∞, D(t) = E0-E ∞ / E0, when E ∞ →0, then D=1, indicating that the rock sample is completely damaged; with the gradual increase of time t, the rock mechanical parameters decay, and the damage variable gradually increases but is always less than 1;

[0051] Therefore, the following damage equation can be established:

[0052]

[0053] Then the degradation change of the original damage parameter R(t) in any component model over time can be expressed as:

[0054] R(t)=R t =(1-D)R0 (8)

[0055] Where: R0 is the initial value of the parameter.

[0056] Only when the stress is greater than the yield limit of the rock, the rock sample will be damaged by creep. Referring to the above damage model, the model parameters are simplified and the damage evolution model is established as follows:

[0057]

[0058] The change of viscoplastic element parameters of rock samples in the accelerated creep stage with time conforms to the formula:

[0059] η(t)=η0(1-D) (10)

[0060] D(t)=1-ae -bt (11)

[0061] Where: η0 is the elastic modulus of rock before accelerated creep, a is the material parameter, η t is the time-varying damage elastic modulus. As time increases, the elastic modulus tends to 0. When a=1 and b=0, it degenerates into a basic viscous element.

[0062] Among them, the creep equation of the damaged component is:

[0063] ε=0 σ0<σ (12)

[0064]

[0065] The constitutive equation of the damaged viscous element is:

[0066] ε=0 σ<σ s (14)

[0067]

[0068] Where: η0 is the elastic modulus of rock before accelerated creep, a and b are material parameters;

[0069] Under the action of disturbance, the damage model is introduced into the creep model to form a damage creep model. The constitutive equation of the damage creep model is:

[0070] When the deviatoric stress is less than the yield stress, the constitutive equation of the damage creep model is:

[0071]

[0072] When the deviatoric stress is greater than or equal to the yield stress, the constitutive equation of the damage creep model is:

[0073]

[0074] Where: η t is the time-varying viscosity coefficient, are the first and second order derivatives of stress, are the first and second order derivatives of strain respectively;

[0075] The creep equation of the damage creep model is:

[0076] When the initial condition is t=0 and the deviatoric stress is a constant, the creep equation can be derived from the creep constitutive relation as follows:

[0077] When the deviatoric stress is less than the yield stress, the creep equation is:

[0078]

[0079] When the deviatoric stress is greater than or equal to the yield stress, the creep equation is:

[0080]

[0081] Where: E0, E1, E2 are elastic moduli, η1 is the viscosity coefficient, E0 is the elastic parameter of the rock before accelerated creep, a and b are material parameters, σ s is the critical stress;

[0082] The creep data of the rock sample obtained in step 2 is fitted with the creep equation of the damage creep model, and the fitted expression is:

[0083]

[0084] In the above, reference Figure 6-Figure 7 , step 6 comprises the following steps:

[0085] S60, the damage creep model includes a viscoelastic body, and the viscoelastic body in the damage creep model is improved to a fractional-order viscoelastic body, and the creep equation of the fractional-order viscoelastic body is:

[0086]

[0087] Under the action of disturbance, the fractional-order viscoelastic body becomes a fractional-order nonlinear viscoelastic body, and its expression is:

[0088]

[0089] In the formula, β is the fractional order, η β is the fractional viscosity coefficient, ε(t) is the strain value at a given moment;

[0090] S61, using the variable-order fractional calculus calculation formula to perform variable-order fractional nonlinear viscoelastic body, when the stress is a constant σ0, the expression of the variable-order fractional nonlinear viscoelastic body can be obtained as follows:

[0091]

[0092] Where: γ m =γ(t) is the value of γ(t) at a given time, t m-1 ≤t m , η m is the viscosity coefficient at a given moment, γ m and η m The value of changes in different time periods, Γ is the gamma function; refer to Table 1:

[0093] Table 1 Variation of fractional order γm and viscosity coefficient ηm with time

[0094]

[0095] Under the action of disturbance load, the rock creep morphology is more complicated, and the creep rate increases rapidly, resulting in cracks inside the rock and reduced strength. After the disturbance load ends, the rock threshold is reduced under the action of higher amplitude, and the sample will experience accelerated creep. During the constant load creep test, under different stress levels, the tuff creep morphology of the rock in different time periods is quite different. Therefore, different fractional orders should be selected according to different creep rates.

[0096] Combination Figure 7 It can be seen that:

[0097] 1) The instantaneous elastic deformation of the elastic element when the load is applied at time t0;

[0098] 2) During the period from t0 to t1, the tuff enters the decay creep and steady-state creep stage, and the creep deformation rate gradually decreases. This stage is described by using fractional viscoelasticity.

[0099] 3) After the disturbance starts (t 1～ t2 stage) and the subsequent steady-state creep stage (t 2～ The deformation of segment t3 is relatively complex, so the fractional-order viscoplastic body is used to describe the deformation of this stage;

[0100] 4) When the accelerated creep stage (t 2～ t3 segment), at which the creep deformation rate continues to increase;

[0101] When the stress level is less than the yield stress, the creep equation of the fractional-order nonlinear viscoelastic body is:

[0102]

[0103] When the stress level is greater than the yield stress, the relationship between the various parts of the variable-order fractional-order nonlinear creep model is:

[0104]

[0105] Where: Variable order viscosity coefficient, β, γ1, γ2 are fractional orders, E 1,1+γ2 The Mittag-Leffler function is:

[0106]

[0107] The creep equation of the fractional-order nonlinear damage creep model of rock specimens is obtained as follows:

[0108]

[0109] Where: the damage factor satisfies a≥0, the fractional order β and γ1, γ2 satisfy 0<β≤1, 0<γ1≤1, 0<γ2≤1 respectively;

[0110] S62, fitting the creep data of the rock sample obtained in step 3 with the fractional-order nonlinear damage creep model to obtain the fitted creep equation:

[0111]

[0112] Where: σ1, σ2, σ3 are the stresses of parts 1, 2, and 3 respectively, K is the elastic bulk modulus, and G is the elastic shear modulus.

[0113] The established fractional-order nonlinear damage creep model can not only describe the creep instantaneous deformation and stable creep characteristics under low stress levels, but also describe the unstable creep and accelerated creep under high stress levels. In particular, the fractional-order and damage variables can reflect the differences in creep characteristics at each stage under low-frequency weak disturbance, which is consistent with the experimental phenomena of tuff creep under low-frequency weak disturbance.

[0114] refer to Figure 8-Figure 9 , the fractional-order nonlinear damage creep model established above is verified, and the verification steps are as follows:

[0115] The Levenberg-Marquardt (LM) algorithm was used in combination with 1stOpt software to identify the creep model parameters of rock samples. Based on the LM algorithm, the starting time t of the constant creep stage was s A series of creep test data[t m ,ε1(t m )] to fit the formula;

[0116] The uniaxial and triaxial results are selected for comparative analysis. Since an accelerated creep stage occurs during the creep process, the creep equation is:

[0117]

[0118] Where: the damage factor satisfies a≥0, and the fractional order β and γ1, γ2 respectively satisfy 0<β≤1, 0<γ1≤1, 0<γ2≤1. Among them, t0, t1, t2 are known quantities, and β and γ1, γ2 are determined according to the shape of the creep curve; refer to Fig. 9 The comparison between the tuff creep test results and the fitting results under different confining pressures shows that the fitting results of the fractional-order model are highly consistent with the test results under different confining pressures and stress levels, indicating that the fractional-order Nishihara model established in this paper can not only accurately describe the attenuation and steady-state creep characteristics of tuff, but also reflect the nonlinear creep characteristics of rock after disturbance, as well as the accelerated creep phenomenon after damage accumulation; the fitting of each parameter is shown in Table 2:

[0119] Table 2 Accelerated creep parameter fitting table

[0120]

[0121] The fitting results of fractional order parameters are shown in Table 4.4. It can be seen from Table 4.4 that the fractional order of the model is different in different creep stages, which reflects the changes in the physical and mechanical properties of the material. At t0, the rock produces an instantaneous elastic strain at the moment of loading, and the order of the equation is 0 at this time; in the t0-t1 section, the rock exhibits viscoelasticity, and the order of the equation is controlled by β at this time, and the rock exhibits attenuated creep properties;. In the t1-t2 section, it is the disturbance stage. As the number of disturbances increases, the irreversible deformation of the rock gradually increases, the deformation of the specimen increases, and the order of the equation is controlled by γ1; in the t2-t3 section, it is the accelerated creep stage. As the irreversible deformation of the rock specimen gradually increases, the internal defects of the rock continue to accumulate, and the rock enters the accelerated creep stage, and the order of the equation is controlled by γ2;

[0122] Table 4.4 Fractional order parameter table

[0123]

[0124]

[0125] Based on the above analysis, it can be seen that compared with the traditional model, the fitting results of the improved creep model are more consistent with the experimental results, indicating that the improved fractional-order model can express the creep behavior of tuff under complex stress conditions, making up for the deficiency of the traditional model that it is difficult to describe the accelerated creep stage. The evolution law of rock mechanical parameters in the whole creep process is revealed through the change of fractional order, and its applicability is verified based on experimental data.

[0126] Taking into account the number of components, model parameters and reasonable nonlinearization of components, combined with damage theory and fractional-order model theory, a nonlinear creep model was constructed which can accurately describe the creep characteristics of tuff under disturbance.

[0127] By combining fractional-order elements with improved creep damage elements, a fractional-order damage model that can reflect accelerated creep was established. By segmenting the test results of different creep stages, it was found that the creep model based on variable-order fractional derivatives can well describe the nonlinear creep during the disturbance test. The parameters in the model were determined through segmented fitting, and the theoretical curve was highly consistent with the experimental data.

[0128] The improved fractional damage creep model not only agrees well with the experimental results, but also reveals the evolution law of the gradual deterioration of the mechanical properties of rock materials under the influence of time and disturbance loads during the entire creep process through the change of fractional orders, and verifies its applicability.

[0129] The embodiments of the present invention have been described above, and the above description is exemplary, not exhaustive, and is not limited to the disclosed embodiments. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of the described embodiments. The selection of terms used herein is intended to best explain the principles of the embodiments, practical applications, or technical improvements in the market, or to enable other persons of ordinary skill in the art to understand the embodiments disclosed herein.

Claims

1. A method for establishing a rock fractional creep model under disturbance, characterized in that: The following steps are involved: S1, obtain the basic mechanical parameters of rock samples under different confining pressures; S2, designing a creep perturbation test of the rock sample according to the basic mechanical parameters of the rock sample under different confining pressures obtained in step S1, carrying out a constant load perturbation creep test of the rock sample under different confining pressures, and obtaining creep test results of the rock sample; S3, drawing a strain-time curve of the rock sample according to the creep test result of the rock sample obtained in step S2; S4, obtaining a suitable creep model according to the strain-time curve of the rock sample at different confining pressures and disturbance amplitudes obtained in step S3 and in combination with the basic element model; S5, under the action of disturbance, introducing the damage model into the creep model selected in step S4 to obtain a damage creep model, and fitting the test results of the rock sample in step S3 with the damage creep model; S6, adding a fractional order model to the damage creep model obtained in step S5, fitting the nonlinear creep characteristics of the rock sample under the disturbance, and establishing a fractional order creep model.

2. The method for establishing a rock fractional creep model under disturbance according to claim 1 is characterized in that: In step 4, the steps to select a suitable creep model are as follows: S40, determining the basic element model according to the creep characteristics of rock samples at different stress levels in the creep test results of step 2, wherein the basic element model includes a Kelvin model and a Bingham model; S41, connecting the Kelvin model and the Bingham model in step S40 in series to form the creep model; When the deviatoric stress value is lower than the yield stress, the creep model degenerates into the Kelvin model, and its creep equation is expressed as: When the deviatoric stress value is higher than the yield stress, the viscoplastic element participates in the work, and the creep equation is expressed as: Where t is time, σ and ε are the total stress and strain of the model, E1 and E2 are elastic moduli, η1 and η2 are viscosity coefficients, and σ s is the yield stress.

3. The method for establishing a rock fractional creep model under disturbance according to claim 2 is characterized in that: In step 5, a damage model is introduced into the creep model, and the steps are as follows: S50, theoretical formula of the time relationship of the elastic modulus in the creep model: σ=(1-D)E0ε Where: P, Q and R are model parameters, E t is the elastic modulus at time t, E0 is the elastic modulus before damage; Then the creep equation of the damaged component is: ε=0σ0<σ s The constitutive equation of the damaged viscous element is: ε=0σ<σ s Where: η0 is the elastic modulus of rock before accelerated creep, a and b are material parameters, and η0 is the initial viscosity coefficient; Under the action of disturbance, the damage model is introduced into the creep model to form a damage creep model. The constitutive equation of the damage creep model is: When the deviatoric stress is less than the yield stress, the constitutive equation of the damage creep model is: When the deviatoric stress is greater than or equal to the yield stress, the constitutive equation of the damage creep model is: Where: η t is the time-varying viscosity coefficient, are the first and second order derivatives of stress, are the first and second order derivatives of strain respectively; The creep equation of the damage creep model is: When the initial condition is t=0 and the deviatoric stress is a constant, the creep equation can be derived from the creep constitutive relation as follows: When the deviatoric stress is less than the yield stress, the creep equation is: When the deviatoric stress is greater than or equal to the yield stress, the creep equation is: Where: E0, E1, E2 are elastic moduli, η1 is the viscosity coefficient, E0 is the elastic parameter of the rock before accelerated creep, a and b are material parameters, σ s is the critical stress; S51, fitting the creep data of the rock sample obtained in step 2 with the creep equation of the damage creep model, the fitted expression is:

4. The method for establishing a rock fractional creep model under disturbance according to claim 3 is characterized in that: Step 6 includes the following steps: S60, the damage creep model includes a viscoelastic body, and the viscoelastic body in the damage creep model is improved to a fractional-order viscoelastic body, and the creep equation of the fractional-order viscoelastic body is: Under the action of disturbance, the fractional-order viscoelastic body becomes a fractional-order nonlinear viscoelastic body, and its expression is: In the formula, β is the fractional order, η β is the fractional viscosity coefficient, ε(t) is the strain value at a given moment; S61, variable-order fractional calculus is performed on the fractional-order nonlinear viscoelastic body. The expression of the variable-order fractional-order nonlinear viscoelastic body is: Where: γ m =γ(t) is the value of γ(t) at a given time, t m-1 ≤t m , η m is the viscosity coefficient at a given moment, γ m and η m The value of changes in different time periods, Γ is the gamma function; When the stress level is less than the yield stress, the creep equation of the fractional-order nonlinear viscoelastic body is: When the stress level is greater than the yield stress, the relationship between the various parts of the variable-order fractional-order nonlinear creep model is: The creep equation of the fractional-order nonlinear damage creep model of rock specimens is obtained as follows: Where: the damage factor satisfies a≥0, the fractional order β and γ1, γ2 satisfy 0<β≤1, 0<γ1≤1, 0<γ2≤1 respectively; S62, fitting the creep data of the rock sample obtained in step 3 with the fractional-order nonlinear damage creep model to obtain the fitted creep equation: Where: σ1, σ2, σ3 are the stresses of parts 1, 2, and 3 respectively, K is the elastic bulk modulus, and G is the elastic shear modulus.

5. The method for establishing a rock fractional creep model under disturbance according to claim 1 is characterized in that: In step S1, the basic mechanical parameters include elastic modulus E, Poisson's ratio μ, cohesion c and internal friction angle

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