A Method for Constructing a Modified Constitutive Model of Rock Considering the Pore Compaction Stage

Abstract

This invention discloses a method for constructing a modified constitutive model of rock considering the pore compaction stage. First, the rock is divided into several micro-elements containing different defects, and it is assumed that these micro-elements conform to the generalized Hooke's law, that the strength of each micro-element follows the Weibull statistical distribution, and that the failure criterion for the micro-elements is the Drucker-Prager criterion. Next, a uniaxial compression constitutive model considering the pore compaction stage of the rock is established based on the Drucker-Prager failure criterion and the Weibull distribution. Then, the established model is compared and verified with experimental data. Finally, a modified uniaxial compression constitutive model of rock considering the pore compaction stage is established, and the modified model of this invention is verified using the uniaxial compression test results of coarse-grained marble after high-temperature damage. The method of this invention has clear physical meanings for the parameters, is well-organized, logically sound, has a wide range of applications, is simple and easy to implement, and can well reflect the uniaxial compression mechanical behavior of rock considering the pore compaction stage.

Description

Technical Field

[0001] This invention belongs to the field of rock mechanics and geotechnical engineering technology, specifically relating to a method for constructing a modified constitutive model of rock that considers the pore compaction stage, which can be widely used in deep mineral resource mining, deep-buried tunnel construction and slope engineering treatment. Background Technology

[0002] Rocks are heterogeneous materials with microcracks and micropores, formed by various mineral crystals and cementing materials under long-term geological tectonic activity. Due to initial damage, rock materials exhibit complex mechanical properties under compression, with a typical deformation pattern of plastic-elastic-plastic. Micropores are the main carriers of the nonlinear mechanical properties of rock materials, such as the nonlinear closure effect of micropore compression and plastic dissipation caused by frictional slip in closed pores. In related engineering design and construction, accurately characterizing the stress and deformation characteristics of rocks is crucial, including the numerical simulation analysis methods commonly used to solve engineering problems, the core of which is still the establishment of constitutive models.

[0003] Numerous scholars have conducted extensive research on constitutive models for rock materials. Examples include Yuan Xiaoping's elastoplastic damage constitutive model based on the Drucker-Prager criterion; Zhu Qizhi's quasi-brittle rock friction damage constitutive model based on micromechanics; and Fang Jingnian's coupled elastoplastic damage constitutive model. Most of these studies focus on the elastoplastic stage of rock material deformation, neglecting the nonlinear closure process of microcracks under compressive stress. For rock materials with significant microcrack closure effects, the aforementioned model-building methods often lead to significant deviations between the model fitting results and actual engineering conditions. Summary of the Invention

[0004] The purpose of this invention is to address the problem that the model fitting results of existing technologies deviate significantly from actual engineering conditions. It provides a simple, intuitive, and reasonable method for constructing a modified constitutive model of rocks that considers the pore compaction stage. This method is used to characterize the deformation and failure characteristics of rocks under uniaxial compression, thereby accurately describing the entire process of rock deformation and failure under uniaxial compression. This has important theoretical significance and application value for the safe construction, operation, and maintenance of engineering projects.

[0005] To achieve the above-mentioned objectives of this invention, a method for constructing a modified constitutive model of rock considering the pore compaction stage is provided. Based on the Drucker-Prager failure criterion and Weibull distribution, a uniaxial compression constitutive model considering the pore compaction stage of rock is established. After establishing the model, it is modified based on the Sigmoid function to establish a modified uniaxial compression constitutive model of rock considering the pore compaction stage. Specifically, the following steps are used:

[0006] Step 1: First, the rock needs to be divided into several micro-elements containing different defects, and the following assumptions are made:

[0007] ① The infinitesimal element conforms to the generalized Hooke's law;

[0008] ② The intensity of each infinitesimal element follows the Weibull statistical distribution law;

[0009] ③The strength failure criterion for the infinitesimal element is the Drucker-Prager criterion.

[0010] Step 2: The probability density function formula for the Weibull statistical distribution is:

[0011]

[0012] In the formula, m and S0 are the Weibull distribution scale and morphological parameter, respectively, and S is the infinitesimal random intensity distribution variable;

[0013] Step 3: The Drucker-Prager failure criterion formula is as follows:

[0014]

[0015] In the formula, I1 is the first invariant of the stress tensor, with units of MPa; J2 is the second invariant of the deviatoric stress tensor, with units of MPa. It is a constant related to the internal friction angle Φ of the rock;

[0016] Step 4: The damage and failure of the rock can be represented by the damage variable D, and the formula is:

[0017]

[0018] In the formula, σ * σ represents the effective stress, in MPa.

[0019] Step 5: Substituting the Weibull statistical distribution function from equation (1) into equation (3) and integrating, the rock damage variable D can be simplified to the following formula:

[0020]

[0021] Step 6: Substituting equation (2) of the Drucker-Prager failure criterion into equation (4), we can obtain the expression for the assumed rock damage variable D, which is:

[0022]

[0023] Step 7: According to the damage mechanics hypothesis, the damaged rock micro-element cannot withstand stress. Therefore, the effective stress that the rock can withstand is defined as σ.* The effective stress of rock and the stress borne by a intact rock micro-element are related by the following formula:

[0024]

[0025] Step 8: As assumed in Step 1, the generalized Hooke's Law formula is:

[0026]

[0027] In the formula, σ1, σ2, and σ3 represent the effective stresses, respectively, in MPa; E is the elastic modulus, in GPa; ν is Poisson's ratio.

[0028] Step 9: Under uniaxial compression test conditions, σ2=σ3=0, then from the above equation (7), we can further obtain the effective stress. The relationship between the elastic modulus E and the strain ε1 is given by the following formula:

[0029]

[0030] Step 10: According to the definitions of I1 and J2 in the Drucker-Prager failure criterion, the following relationship can be obtained:

[0031]

[0032] Step 11: Combine the rock damage variable D expression (5) and effective stress. Substituting equations (8) and (9) into equation (6), we obtain the relationship for the stress curve of a rock micro-element, which is:

[0033]

[0034] Step 12: Further, simplify the above equation to facilitate subsequent parameter solving for the curve. The simplification process is as follows:

[0035]

[0036] In the formula, a1 is the curve parameter;

[0037] Step 13: Substitute equation (11) into equation (10) to obtain the rock yield hardening constitutive model, which is:

[0038] σ1=Eε1 exp[-(a1ε1) m (12)

[0039] Step 14: Verify equation (12), and find a deviation from the experimental data. Correct it by defining the proportion of rock involved in deformation as f(x). The corrected expression for the uniaxial compressive stress-strain curve of the rock is:

[0040] σ1=Eε1 exp[-(a1ε1) m ]·f(x) (13)

[0041] Step 15: The correction function f(x) can be obtained by compressing the Sigmoid function, as shown in the formula:

[0042]

[0043] In the formula, b and c are the parameters of the correction function, which can be obtained by fitting; x is the independent variable, which is the strain ε1 in this constitutive model, without units, expressed as a percentage.

[0044] Step 16: Substitute equation (14) into equation (13) to obtain the modified uniaxial compression constitutive model of the rock, the formula is:

[0045]

[0046] Preferably, in step 4, the rock damage variable D is defined as the rock failure probability, which occurs when the rock micro-element reaches the failure condition.

[0047] Preferably, in step 14, the stress-strain curve of the rock under uniaxial compression is obtained using the curve fitting plugin of Matlab based on formula (12). Since the strain ε1 is extremely small at the beginning of loading, a1ε1 approaches 0, exp[-(a1ε1)] m The slope of the curve approaches 1, so the initial slope of the curve is close to a straight line, resulting in poor fitting of formula (12) in the pore compaction stage. According to the above inference, the rock is almost undamaged in the early stage of loading. However, due to the presence of pores and the influence of weathering, natural rock masses have natural damage in the initial state. Therefore, formula (12) cannot represent the pore compaction stage and needs to be corrected.

[0048] Preferably, in step 15, a large number of uniaxial compressive stress-strain curves of rocks are summarized, and the correction function f(x) grows in an S-shape before the elastic stage and approaches 1 after reaching the elastic stage. The Sigmoid function can meet the above requirements.

[0049] Compared with existing technologies, the rock modified constitutive model construction method considering the pore compaction stage of the present invention has the following features and advantages:

[0050] First, the damage model constructed in this invention is logically sound and intuitively reasonable. Based on the Drucker-Prager failure criterion and Weibull distribution, a uniaxial compression modified constitutive model considering the rock crack closure stage is established, demonstrating strong logical coherence. Comparison and verification with experimental data reveals that the model is insufficient to characterize the highly nonlinear initial crack closure stage. Therefore, a Sigmoid function is further introduced for compressive displacement and used as a correction function for the aforementioned model. The resulting constitutive model considering the pore compaction stage better reflects the mechanical behavior of rock under uniaxial compression.

[0051] Second, this method determines the parameters of the rock damage model clearly, has a wide range of applications, and is simple and easy to implement. The parameters in the model are not constants; they vary for different types and sizes of rocks with different damage levels. These parameters can be obtained by fitting relevant experimental data, making it simple, easy to implement, and highly adaptable. Attached Figure Description

[0052] Figure 1 This is a schematic diagram of a typical uniaxial compressive stress-strain curve for rock.

[0053] Figure 2 This is a schematic diagram of the rock yield hardening constitutive model result before modification, based on the rock modified constitutive model construction method considering the pore compaction stage of the present invention.

[0054] Figure 3 This is a schematic diagram of the Sigmoid function curve;

[0055] Figure 4 This is a schematic diagram comparing the uniaxial compressive stress-strain curve of coarse-grained marble after heat damage treatment at 200℃ in the example with the fitting results of the constitutive model of the present invention. Detailed Implementation

[0056] To describe the present invention, the following detailed description, in conjunction with the accompanying drawings and embodiments, provides a method for constructing a rock constitutive model that considers the pore compaction stage.

[0057] Figure 1 This is a schematic diagram of a typical uniaxial compressive stress-strain curve for rock. Figure 2 This is a schematic diagram of the pre-correction rock yield hardening constitutive model result of a method for constructing a modified constitutive model of rock considering the pore compaction stage according to the present invention. Figure 2 It can be seen that the constitutive model before the modification exhibits linear growth from the initial stage, which deviates from the highly nonlinear pore closure in the actual situation. Further modification of the model is required. Figure 3 The diagram shows the curve of the Sigmoid function, which satisfies the requirement that the correction function f(x) grows in an S-shape before the elastic phase and approaches 1 after the elastic phase is reached. Figure 4This is a schematic diagram comparing the uniaxial compressive stress-strain curves of coarse-grained marble after heat damage treatment at 200℃ in the examples with the fitting results of the constitutive model of the present invention. Figure 4 It can be seen that the modified rock constitutive model can fit the uniaxial mechanical behavior of rock well. This invention provides a method for constructing a modified rock constitutive model considering the pore compaction stage, specifically employing the following steps:

[0058] S1: First, the rock needs to be divided into several micro-elements containing different defects, and the following assumptions are made:

[0059] ① The infinitesimal element conforms to the generalized Hooke's law;

[0060] ② The intensity of each infinitesimal element follows the Weibull statistical distribution law;

[0061] ③The strength failure criterion for the infinitesimal element is the Drucker-Prager criterion.

[0062] S2: The probability density function formula for the Weibull statistical distribution is:

[0063]

[0064] In the formula, m and S0 are the Weibull distribution scale and morphological parameter, respectively, and S is the infinitesimal random intensity distribution variable;

[0065] S3: The formula for the Drucker-Prager failure criterion is:

[0066]

[0067] In the formula, I1 is the first invariant of the stress tensor, with units of MPa; J2 is the second invariant of the deviatoric stress tensor, with units of MPa. It is a constant related to the internal friction angle Φ of the rock;

[0068] S4: The rock damage variable D is defined as the rock failure probability. When a rock element meets the failure condition, the rock element fails. Rock damage can be represented by the damage variable, and the formula is:

[0069]

[0070] In the formula, σ * σ represents the effective stress, in MPa.

[0071] S5: Substituting the Weibull statistical distribution function from equation (1) into equation (3) and integrating, the rock damage variable D can be simplified further, and the formula is:

[0072]

[0073] S6: Substituting the Drucker-Prager failure criterion formula (2) into formula (4), we can obtain the expression for the assumed rock damage variable D, which is:

[0074]

[0075] S7: According to the assumptions of damage mechanics, a damaged rock micro-element cannot withstand stress. Therefore, the effective stress that the rock can withstand is defined as σ. * The effective stress of rock and the stress borne by a intact rock micro-element are related by the following formula:

[0076]

[0077] S8: As assumed in step 1, the generalized Hooke's law formula is:

[0078]

[0079] In the formula, σ1, σ2, and σ3 represent the effective stresses, respectively, in MPa; E is the elastic modulus, in GPa; ν is Poisson's ratio.

[0080] S9: Under uniaxial compression test conditions, σ2=σ3=0, then from the above equation (7), we can further obtain the effective stress. The relationship between the elastic modulus E and the strain ε1 is given by the following formula:

[0081]

[0082] S10: According to the definitions of I1 and J2 in the Drucker-Prager failure criterion, the following relationship can be obtained, and the formula is:

[0083]

[0084] S11: Combined expression for rock damage variable D (5), effective stress Substituting equations (8) and (9) into equation (6), we obtain the relationship for the stress curve of a rock micro-element, which is:

[0085]

[0086] S12: Further, simplifying the above equation facilitates subsequent parameter solving for the curve. The simplification process is as follows:

[0087]

[0088] In the formula, a1 is the curve parameter;

[0089] S13: Substituting equation (11) into equation (10), we obtain the constitutive model for rock yield hardening, which is:

[0090] σ1=Eε1 exp[-(a1ε1) m (12)

[0091] S14: Using the curve fitting plugin of Matlab, the stress-strain curve of the rock under uniaxial compression was verified based on formula (12). It was found that there was a deviation from the experimental data. Since the strain ε1 is extremely small in the early stage of loading, a1ε1 approaches 0, exp[-(a1ε1)] m The initial slope of the curve approaches 1, so the initial slope of the curve is close to a straight line, resulting in poor fitting of formula (12) in the pore compaction stage. According to the above inference, the rock is almost undamaged in the early stage of loading. However, due to the presence of pores and the influence of weathering, natural rock masses have natural damage in the initial state, so formula (12) cannot represent the pore compaction stage and needs to be corrected. Define the proportion of rock involved in deformation as f(x), and the corrected expression of the uniaxial compressive stress-strain curve of the rock is:

[0092] σ1=Eε1 exp[-(a1ε1) m ]·f(x) (13)

[0093] S15: Summarizing the characteristics of numerous uniaxial compressive stress-strain curves in rocks, the correction function f(x) exhibits an S-shaped growth before the elastic stage, and approaches 1 after reaching the elastic stage. The Sigmoid function can satisfy these requirements. The correction function f(x) can be obtained by applying the compressive displacement to the Sigmoid function, as shown in the formula:

[0094]

[0095] In the formula, b and c are the parameters of the correction function, which can be obtained by fitting; x is the independent variable, which is the strain ε1 in this constitutive model, without units, expressed as a percentage.

[0096] S16: Substituting equation (14) into equation (13), we obtain the modified uniaxial compression constitutive model of the rock, which is:

[0097]

[0098] Example

[0099] This embodiment uses the rock modified constitutive model considering the pore compaction stage proposed in this invention to fit the uniaxial compression experimental data of coarse-grained marble under heat treatment at 200℃. The specific steps are as follows:

[0100] Step 1: Conduct a uniaxial compression test on the coarse-grained marble after heat damage treatment at 200℃, and plot the stress-strain curves based on the experimental data. (See the graph for details.) Figure 4 ;

[0101] Step 2: Using experimental data and curves of coarse-grained marble under 200℃ heat treatment, the original rock yield hardening constitutive model and correction function were fitted to obtain the corresponding morphological parameter m, curve parameter a1, and fitted values ​​of correction function parameters b and c. The fitting parameter results are shown in Table 1 below;

[0102] Step 3: Substitute each parameter into equation (15) to obtain the fitting curve of the uniaxial compression constitutive model of coarse-grained marble under heat treatment at 200℃.

[0103] Step 4: Compare the fitting curve of the uniaxial compression correction constitutive model of coarse-grained marble with the experimental data of coarse-grained marble. The comparison results are shown in [the table below]. Figure 4 Therefore, the model proposed in this invention can better describe the mechanical behavior of coarse-grained marble and can characterize the initial pore compaction stage of the rock stress-strain curve.

[0104] Table 1. Model Fitting Parameters in the Examples

[0105]

Claims

1. A method for constructing a modified constitutive model of rock considering the pore compaction stage, characterized in that, Includes the following steps: S1: First, the rock needs to be divided into several micro-elements containing different defects, and the following assumptions are made: ① The infinitesimal element conforms to the generalized Hooke's law; ② The intensity of each infinitesimal element follows the Weibull statistical distribution law; ③The strength failure criterion for the infinitesimal element is the Drucker-Prager criterion; S2: The probability density function formula for the Weibull statistical distribution is: （1） In the formula, m and S0 are the Weibull distribution scale and morphological parameter, respectively, and S is the infinitesimal random intensity distribution variable; S3: The Drucker-Prager failure criterion formula is: （2） In the formula, I1 is the first invariant of the stress tensor, in MPa; J2 is the second invariant of the deviatoric stress tensor, in MPa; ∂ is a constant related to the internal friction angle Φ of the rock; S4: The damage and failure of the rock is represented by the damage variable D, and the formula is: （3） In the formula, σ * is the effective stress of stress σ, with the unit of MPa; S5: Substituting the Weibull statistical distribution function from equation (1) into equation (3) and integrating, the rock damage variable D is further simplified, and the formula is: （4） S6: Substituting the Drucker-Prager failure criterion formula (2) into formula (4), we obtain the expression for the assumed rock damage variable D, which is: （5） S7: Define the effective stress borne by the rock as σ * The effective stress of rock and the stress borne by a intact rock micro-element are related by the following formula: （6） S8: As assumed in step 1, the generalized Hooke's law formula is: （7） In the formula, σ*1, σ*2, and σ*3 are the effective stresses corresponding to σ1, σ2, and σ3, respectively, with units of MPa; E is the elastic modulus, with units of GPa; and ν is Poisson's ratio. S9: Under uniaxial compression test conditions, σ2=σ3=0, then from the above equation (7), we can further obtain the relationship between the effective stress σ*1 and the elastic modulus E and the strain ε1, which is: （8） S10: According to the definitions of I1 and J2 in the Drucker-Prager failure criterion, the following relationship is obtained: （9） S11: Combine the rock damage variable expression (5), the effective stress σ*1 relationship (8) and (9), and substitute them into equation (6) to obtain the stress curve relationship of the rock micro-element. The formula is: （10） S12: Further, simplifying the above equation facilitates subsequent parameter solving for the curve. The simplification process is as follows: （11） In the formula, a1 is the curve parameter; S13: Substituting equation (11) into equation (10), we obtain the constitutive model for rock yield hardening, which is: （12） S14: Verification of equation (12) revealed a deviation from the experimental data. This was corrected by defining the proportion of rock involved in the deformation as f(x). The corrected expression for the uniaxial compressive stress-strain curve of the rock is: （13） S15: The correction function f(x) is obtained by compressing the Sigmoid function, and the formula is: In the formula, b and c are the parameters of the correction function, obtained by fitting; x is the independent variable, which is the strain ε1 in this constitutive model, without units, expressed as a percentage. S16: Substituting equation (14) into equation (13), we obtain the modified uniaxial compression constitutive model of the rock, which is: 。 2. The method for constructing a modified constitutive model of rock considering the pore compaction stage as described in claim 1, characterized in that: In step S4, the rock damage variable D is defined as the rock failure probability. When the rock micro-element reaches the failure condition, the rock micro-element will fail.

3. A method for constructing a modified constitutive model of rock considering the pore compaction stage as described in claim 1 or 2, characterized in that: In step S15, a large number of uniaxial compressive stress-strain curves of rocks are summarized. The correction function f(x) grows in an S-shape before the elastic stage and approaches 1 after reaching the elastic stage. The Sigmoid function satisfies the above requirements.

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