Method for establishing rock fractional order creep model under disturbance action

Abstract

The application provides a rock fractional order creep model establishment method under disturbance, according to test results of rock samples under different confining pressures, a strain-time curve of the rock sample is drawn according to the creep test results, and a suitable creep model is obtained by combining a basic element model, a damage model and a fractional order model are added in the creep model under disturbance, and a damage fractional order creep model is established. Compared with the traditional model, the improved creep model fitting result is more consistent with the test result, indicating that the improved fractional order model can express the creep behavior of tuff under complex stress conditions, and make up for the deficiency that the traditional model is difficult to describe the accelerated creep stage, and the evolution law of rock mechanics parameters in the whole creep process is shown by the change of the fractional order.

Description

Technical Field

[0001] This invention belongs to the field of rock engineering technology, and in particular relates to a method for establishing a fractional creep model of rock under disturbance. Background Technology

[0002] Under perturbation, rocks exhibit significant nonlinear characteristics. On the one hand, the creep rate continuously increases during perturbation, and the expansion of internal cracks leads to a decrease in rock strength. On the other hand, the decrease in strength will prematurely trigger the accelerated creep threshold, causing accelerated creep. Therefore, describing the nonlinear creep characteristics of rocks under perturbation is the key and challenging aspect of constructing a constitutive model. Summary of the Invention

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

[0004] The technical solution adopted in this invention is as follows:

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

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

[0007] S2. Based on the basic mechanical parameters of the rock samples under different confining pressures obtained in step S1, design the creep disturbance test of the rock samples, carry out the constant load disturbance creep test of the rock samples under different confining pressures, and obtain the creep test results of the rock samples.

[0008] S3. Plot the strain-time curve of the rock sample based on the creep test results of the rock sample obtained in step S2.

[0009] S4. Based on the strain-time curves of the rock sample obtained in step S3 under different confining pressures and disturbance amplitudes, and combined with the basic element model, obtain a suitable creep model.

[0010] S5. Under the disturbance, a damage model is introduced into the creep model selected in step S4 to obtain a damage creep model, and the test results of the rock sample in step S3 are fitted with the damage creep model.

[0011] S6. A fractional-order model is added to the damage creep model obtained in step S5, and the nonlinear creep characteristics of the rock sample under disturbance are fitted to establish a fractional-order creep model.

[0012] The beneficial effects that this application can produce include:

[0013] By combining the fractional-order model with an improved creep damage model, a fractional-order damage model capable of reflecting accelerated creep was established. By segmenting the experimental results of different creep stages, the creep model based on variable-order fractional derivatives can well describe the nonlinear creep during the perturbation test. Through piecewise fitting, the parameters in the model were determined, and the theoretical curves showed a high degree of agreement 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 shortcomings of the traditional model in describing the accelerated creep stage. The evolution of rock mechanical parameters throughout the creep process was shown through the change of fractional-order order. Attached Figure Description

[0014] The following figures are for illustrative purposes only and are not intended to limit the scope of the invention, wherein:

[0015] Figure 1 A flowchart illustrating the method for establishing a fractional creep model of rock under the disturbance conditions of this application;

[0016] Figure 2 Schematic diagrams of the Kelvin and Bingham models;

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

[0018] Figure 4 A schematic diagram of a viscoplastic element considering damage;

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

[0020] Figure 6 This is a schematic diagram of a fractional-order viscoelastic body;

[0021] Figure 7 This is a schematic diagram of a fractional-order nonlinear constitutive model;

[0022] Figure 8 This is a schematic diagram of the software's user interface;

[0023] Figure 9 The figures show the fitting results under different confining pressures. Detailed Implementation

[0024] To make the objectives, technical solutions, design methods, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the invention.

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

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

[0027] S2. Based on the basic mechanical parameters of the rock samples under different confining pressures obtained in step S1, design the creep disturbance test of the rock samples, carry out the constant load disturbance creep test of the rock samples under different confining pressures, and obtain the creep test results of the rock samples.

[0028] S3. Plot the strain-time curve of the rock sample based on the creep test results of the rock sample obtained in step S2.

[0029] S4. Based on the strain-time curves of the rock sample obtained in step S3 under different confining pressures and disturbance amplitudes, and combined with the basic element model, obtain a suitable creep model.

[0030] S5. Under the disturbance, a damage model is introduced into the creep model selected in step S4 to obtain a damage creep model, and the test results of the rock sample in step S3 are fitted with the damage creep model.

[0031] S6. A fractional-order model is added to the damage creep model obtained in step S5 to fit the nonlinear creep characteristics of the rock sample under disturbance and establish 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 φ.

[0033] In the above, refer to Figures 2-3 In step 4, the steps for selecting a suitable creep model are as follows:

[0034] In selecting the creep model, tuff exhibits a certain threshold value under long-term loading. Under low stress levels, after undergoing instantaneous elastic deformation and decaying creep, the creep rate of tuff gradually decreases, eventually leading to a constant strain. Under high stress levels, after experiencing the above stages, the rock enters a steady-state creep state, followed by a gradual increase in the creep rate, resulting in accelerated creep until the sample fails. Therefore, the selected model should include plastic elements. Thus, a creep model composed of the Kelvin and Bingham models in series was chosen as the creep model to describe the creep characteristics of tuff under different stress levels. This creep model combines the advantages of the Kelvin and Bingham models, reflecting not only most creep characteristics of the rock but also the differences in creep morphology between low and high stress levels. It comprehensively reflects the viscoelastic-plastic properties of the rock.

[0035] The creep equation for the Kelvin model is as follows:

[0036] (1);

[0037] The creep equation for the Bingham model is as follows:

[0038] (2);

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

[0040] (3);

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

[0042] (4);

[0043] In the formula, σ and ε These represent the total stress and strain of the model, respectively. E 1. E 2 represents the elastic modulus. η 1 η 2 represents the viscosity coefficient. σ s Yield stress .

[0044] In the above, refer to Figures 4-5 The selected creep model can reflect the instantaneous elasticity, decay creep, steady-state creep, and plasticity characteristics of rock under different stress levels, but it 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, a damage model is introduced into the creep model, and the steps are as follows:

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

[0046] (5);

[0047] (6);

[0048] In the formula: P, Q and R For model parameters, E t for tElastic modulus at time, E 0 represents the elastic modulus before damage;

[0049] Based on this, the original damage factor was improved, and the initial elastic modulus of the rock sample before disturbance was determined to be: E 0. After being disturbed, the material properties gradually decrease until construction is completed and the stress state stabilizes. The elastic modulus of the material also tends to a stable value (long-term modulus). E ∞ );

[0050] Therefore, the damage variable is defined as: when the experiment begins. t When =0, D =0, initial damage is zero; when t As we approach infinity, D ( t )= E 0- E ∞ / E 0, when E ∞ →0, then D =1 indicates that the rock sample is completely damaged; with time... t As the value gradually increases, the rock mechanical parameters decrease, and the damage variable gradually increases but remains less than 1.

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

[0052] (7);

[0053] Then the original damage parameter in the arbitrary element model R ( t The degradation over time can be expressed as:

[0054] (8);

[0055] In the formula: R 0 is the initial value of the parameter.

[0056] Creep damage will only occur in rock samples when the stress exceeds the rock's yield strength. Referring to the damage model above, the model parameters are simplified to establish the following damage evolution model:

[0057] (9);

[0058] The changes in the viscoplastic element parameters of the rock sample over time during the accelerated creep stage conform to the following formula:

[0059] (10);

[0060] (11);

[0061] In the formula: η 0 represents the elastic modulus of the rock before accelerated creep occurs. a For material parameters, η t The elastic modulus of damage varies with time; as time increases, the elastic modulus tends to 0. a =1, b When =0, it degenerates into a basic viscous element;

[0062] The creep equation for the damaged element is as follows:

[0063] ;

[0064] ;

[0065] The constitutive equation for a damaged viscous element is:

[0066] ;

[0067] ;

[0068] In the formula: η 0 represents the elastic modulus of the rock before accelerated creep occurs. a、b Material parameters;

[0069] Under the influence of disturbance, the damage model is introduced into the creep model to form a damage-creep model, the constitutive equation of which is:

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

[0071] (16);

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

[0073] (17);

[0074] In the formula: η t The viscosity coefficient varies with time. , These are the first and second derivatives of stress, respectively. , These are the first and second derivatives of the strain, respectively;

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

[0076] Initial conditions t When the deviatoric stress is constant and the coefficient of friction is zero, 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] (18);

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

[0080] (19);

[0081] In the formula: E 0、 E 1. E 2 represents the elastic modulus. η 1 represents the viscosity coefficient. E 0 represents the elastic parameter of the rock before accelerated creep occurs. a , b For material parameters, σ s The critical stress;

[0082] Specifically, 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] (20);

[0084] In the above, refer to Figures 6-7 Step 6 includes 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, the creep equation of which is:

[0086] (twenty one);

[0087] Under perturbation, the fractional-order viscoelastic becomes a fractional-order nonlinear viscoelastic, and its expression is:

[0088] (twenty two);

[0089] In the formula, β For fractional order, It is the fractional viscosity coefficient. ε ( t () represents the strain value at a given time.

[0090] S61, using variable-order fractional calculus to perform variable-order fractional calculus on fractional nonlinear viscoelastic bodies, when the stress is constant. σ At time 0, the expression for a variable-order fractional-order nonlinear viscoelastic body can be obtained as follows:

[0091] (twenty three);

[0092] In the formula: For a given time γ ( t The value of ) , Let be the viscosity coefficient at a given time. and The value changes over different time periods. This refers to the gamma function; see Table 1 for reference:

[0093] Table 1. Fractional Order and viscosity coefficient Change over time

[0094]

[0095] Under the action of disturbance load, the creep morphology of rock becomes more complex and the creep rate rises rapidly, leading to cracks and reduced strength within the rock. After the disturbance load ends, the rock threshold decreases under a higher amplitude, and the sample will exhibit accelerated creep. During the constant load creep test, the creep morphology of tuff varies greatly under different stress levels and time periods. Therefore, different fractional order should be selected according to different creep rates.

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

[0097] 1) Elastic element pair t Instantaneous elastic deformation at time 0 when the load is applied;

[0098] 2) In t 0 to t During the first time period, the tuff enters the decay creep and steady-state creep stage, and the creep deformation rate gradually decreases. This stage is described by fractional-order viscoelasticity.

[0099] 3) After the disturbance begins ( t1~ t 2 stages) and subsequent steady-state creep stage ( t 2~ t The deformation in the 3rd stage is relatively complex, so a fractional-order viscoplastic body is chosen to describe the deformation in this stage.

[0100] 4) When the accelerated creep stage occurs ( t 2~ t (3 segments), at which point the creep deformation rate continuously increases;

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

[0102] (twenty four);

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

[0104] (25);

[0105] In the formula: 、 Variable viscosity coefficient, β、 、 For fractional order, for The function is:

[0106] (26);

[0107] The creep equation for the fractional-order nonlinear damage creep model of the rock sample can be obtained as follows:

[0108] (27);

[0109] In the formula: the damage factor satisfies fractional order and 、 Each satisfies , , ;

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

[0111] (28);

[0112] In the formula: σ1, σ2, and σ3 represent the stresses in parts 1, 2, and 3, respectively. K Elastic bulk modulus G Elastic shear modulus.

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

[0114] refer to Figures 8-9 The established fractional-order nonlinear damage creep model is validated using the following steps:

[0115] The Levenberg-Marquardt (LM) algorithm, combined with 1stOpt software, was used to identify the creep model parameters of the rock samples. Based on the LM algorithm, the starting time of the isorheological creep stage was determined. t s Previous creep test data [ t m , ε 1( t m Fit the equation;

[0116] Comparative analysis was performed using uniaxial and triaxial results. Because an accelerated creep stage occurred during the creep process, the creep equation is as follows:

[0117] (29);

[0118] In the formula: the damage factor satisfies fractional order and 、 Each satisfies , .in t 0、 t 1. t 2 is a known quantity. and 、 Determined based on the shape of the creep curve; reference Figure 9Comparison of the experimental and fitting results of tuff creep under different confining pressures shows that the fractional-order model fitting results are in high agreement with the experimental results under different confining pressures and stress levels. This indicates 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 the 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 fractional-order parameter fitting results are shown in Table 3. As can be seen from Table 3, the fractional-order of the model differs at different creep stages, reflecting the changes in the physical and mechanical properties of the material. At the instant of loading, the rock experiences an instantaneous elastic strain, at which point the order of the equation is 0; The rock segment exhibits viscoelasticity, where the order of the equation is controlled by β, and the rock displays decaying creep properties. This segment represents the disturbance stage. As the number of disturbances increases, the irreversible deformation of the rock gradually increases, leading to greater deformation of the specimen and a change in the equation order. Control; in This segment represents the accelerated creep stage. As the irreversible deformation of the rock sample gradually increases, internal defects accumulate, and the rock enters the accelerated creep stage. The equation order changes from [previous stage]. control;

[0122] Table 3. Fractional Parameter Table

[0123]

[0124] In summary, the above analysis shows that the improved creep model fits the experimental results more consistently than the traditional model, indicating that the improved fractional-order model can express the creep behavior of tuff under complex stress conditions, making up for the shortcomings of the traditional model in describing the accelerated creep stage. The changes in the fractional-order order reveal the evolution law of rock mechanical parameters in the whole creep process, and the applicability and rationality are verified by experimental data.

[0125] Taking into account the number of components, model parameters, and reasonable nonlinearization of components, and combining damage theory and fractional-order model theory, a nonlinear creep model that can accurately describe the creep characteristics of tuff under disturbance was constructed.

[0126] 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 curves and experimental data showed high agreement.

[0127] The improved fractional-order damage creep model not only shows a high degree of agreement with the experimental results, but also reveals the evolution law of the gradual deterioration of the rock mechanical properties of the material under the influence of time and disturbance loads throughout the creep process through the change of fractional-order order, and verifies its applicability.

[0128] The various embodiments of the present invention have been described above. These descriptions are exemplary and not exhaustive, nor are they limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the described embodiments. The terminology used herein is chosen to best explain the principles, practical application, or technical improvements to the embodiments in the market, or to enable others skilled in the art to understand the embodiments disclosed herein.

Claims

1. A method for establishing a fractional-order creep model of rock under disturbance, characterized in that, Includes the following steps: S1, to obtain the basic mechanical parameters of rock samples under different confining pressures; S2. Based on the basic mechanical parameters of the rock samples under different confining pressures obtained in step S1, design the creep disturbance test of the rock samples, carry out the constant load disturbance creep test of the rock samples under different confining pressures, and obtain the creep test results of the rock samples. S3. Plot the strain-time curve of the rock sample based on the creep test results of the rock sample obtained in step S2. S4. Based on the strain-time curves of the rock sample obtained in step S3 under different confining pressures and disturbance amplitudes, and combined with the basic element model, obtain a suitable creep model. S5. Under the disturbance, a damage model is introduced into the creep model selected in step S4 to obtain a damage creep model, and the test results of the rock sample in step S3 are fitted with the damage creep model. S6. Add a fractional-order model to the damage creep model obtained in step S5, fit the nonlinear creep characteristics of the rock sample under the perturbation, and establish a fractional-order creep model. S6 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, the creep equation of which is: ； Under perturbation, the fractional-order viscoelastic becomes a fractional-order nonlinear viscoelastic, and its expression is: ； In the formula, β For fractional order, It is the fractional viscosity coefficient. ε ( t () represents the strain value at a given time. S61, performing variable-order fractional calculus on a fractional-order nonlinear viscoelastic body, the expression for the variable-order fractional nonlinear viscoelastic body is: ； In the formula: For fractional order, For a given time The value, , Let be the viscosity coefficient at a given time. and The value changes over different time periods. It is a gamma function; The time phases of the variable-order fractional-order nonlinear damage creep model are divided as follows: t At time 0, the application of the load causes instantaneous elastic deformation; t 0 to t The first time period is divided into the decay creep and steady-state creep stages; t 1 to t The second time period is the phase after the disturbance begins; t 2 to t The third time period is the accelerated creep phase; when t At time 0, the rock experiences an instantaneous elastic strain upon loading, at which point the order of the equation is 0; t 0~ t In section 1, the rock exhibits viscoelasticity; in this case, the order of the equation is controlled by β, and the rock exhibits decaying creep properties. t 1~ t The second stage is the disturbance stage. As the number of disturbances increases, the irreversible deformation of the rock gradually increases, the sample deformation increases, and the equation order changes from... γ 1 Control; in t 2~ t The third stage is the accelerated creep stage. As the irreversible deformation of the rock sample gradually increases, internal defects in the rock accumulate, and the rock enters the accelerated creep stage. The equation order changes from... γ 2 control; 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 components of the variable-order fractional-order nonlinear creep model is as follows: ； In the formula, σ and ε These represent the total stress and strain of the model, respectively. , For elastic modulus, 、 The viscosity coefficient, σ s Yield stress ； The creep equation for the fractional-order nonlinear damage creep model of the rock sample can be obtained as follows: ； In the formula: t is time, and the damage factor satisfies fractional order and 、 Each satisfies , , , , These are the variable-order viscosity coefficients; S62, The creep data of the rock sample obtained in step 3 is fitted with the fractional-order nonlinear damage creep model to obtain the fitted creep equation: In the formula: σ1, σ2, and σ3 represent the stresses in parts 1, 2, and 3, respectively. K Elastic bulk modulus G Elastic shear modulus; The LM algorithm was used in conjunction with 1stOpt software to identify the creep model parameters of the rock samples; based on the LM algorithm, the starting time of the isochronous creep stage was determined. t s Previous creep test data [ t m , ε 1( t m Fit the equation.

2. The method for establishing a fractional-order creep model of rock under disturbance according to claim 1, characterized in that, In step 4, the steps for selecting a suitable creep model are as follows: S40, Based on the creep characteristics of rock samples with different stress levels in the creep test results of step 2, determine the basic element model, which includes the Kelvin model and the Bingham model; S41, The Kelvin model and the Bingham model from step S40 are connected 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 expression is: ； When the deviatoric stress is higher than the yield stress, the viscoplastic element participates in the creep process, and the creep equation is expressed as follows: 。 3. The method for establishing a fractional-order creep model of rock under disturbance according to claim 2, characterized in that, In step 5, a damage model is introduced into the creep model, as follows: S50, the theoretical formula for the time relationship of the elastic modulus in the creep model: ； ； In the formula: P, Q and R For model parameters, E t for t Elastic modulus at time, E 0 represents the elastic modulus before damage; The creep equation for the damaged element is: ； The constitutive equation for a damaged viscous element is: ； In the formula: η 0 represents the elastic modulus of the rock before accelerated creep occurs. 、b Material parameters; Under the influence of disturbance, the damage model is introduced into the creep model to form a damage-creep model, the constitutive equation of which 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: ； In the formula: η t The viscosity coefficient varies with time. , These are the first and second derivatives of stress, respectively. , These are the first and second derivatives of the strain, respectively; The creep equation of the damage creep model is: Initial conditions t When the deviatoric stress is constant and the coefficient of friction is zero, 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: ； In the formula: , , For elastic modulus, η 1 represents the viscosity coefficient. E 0 represents the elastic parameter of the rock before accelerated creep occurs. , b For material parameters, σ s The yield stress; S51, The creep data of the rock sample obtained in step 2 is fitted with the creep equation of the damage creep model. The fitted expression is: 。 4. The method for establishing a fractional-order creep model of rock under disturbance according to claim 1, characterized in that, In step S1, the basic mechanical parameters include elastic modulus E, Poisson's ratio μ, cohesion c, and internal friction angle φ.

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