A method for establishing a dynamic triaxial statistical damage constitutive model of rock

Abstract

The present invention discloses a method for establishing a dynamic triaxial statistical damage constitutive model of rock, belonging to the basic research technology field of the dynamic constitutive model of rock in deep underground engineering. The method includes introducing the nonlinear Mohr-Coulomb and Hoek-Brown strength criteria, a strength criterion of rock dynamic strength and strain rate dependence based on thermal activation and viscous mechanism, and deriving a nonlinear rock dynamic strength criterion considering the strain rate effect; constructing a method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rock, establishing a dynamic triaxial statistical damage constitutive model of rock; and carrying out a sensitivity analysis of the dynamic triaxial statistical damage constitutive model of rock. The method effectively establishes a dynamic triaxial statistical damage constitutive model of rock applicable to high confining pressure and high strain rate, so as to accurately simulate the nonlinear deformation process of deep high-confining pressure rock under dynamic loads, and can be applied to the basic research technology field of the dynamic constitutive model of rock in deep underground engineering.

Description

Technical Field

[0001] The present invention belongs to the technical field of basic research on the dynamic constitutive model of rocks in deep underground engineering, and particularly relates to a method for establishing a dynamic triaxial statistical damage constitutive model of rocks applicable to high confining pressure and high strain rate. Background Technique

[0002] In projects such as water conservancy and hydropower, highways, railways, mines, and civil engineering, the deformation analysis and calculation of rock structures (such as slope engineering, underground structures and tunnels, foundation engineering, etc.) are essential in the design of various projects and structures, and are also important indicators for the stability control of various projects and structures.

[0003] Deep rocks are usually in a high in-situ stress environment characterized by high confining pressure, and their mechanical properties and deformation laws will change significantly, resulting in strong non-linear characteristics of surrounding rock deformation. The damage behavior of rocks is directly related to the stability and safety of rock engineering. Therefore, it is of great significance to carry out research on the constitutive relationship of rocks in deep high in-situ stress environments. However, at present, the research on the constitutive relationship of rocks mainly focuses on constitutive models applicable to low confining pressure strength criteria, so it is difficult to ensure the calculation accuracy of the constitutive model obtained under high surrounding rock conditions. This may lead to a large deviation between the final analysis result and the actual project, or even misjudgment. At the same time, rock engineering structures often bear dynamic loads such as earthquakes, blasting, and impact, which cause changes in the corresponding loading speed or strain rate, etc. These factors will also have an important impact on the mechanical behavior of rocks. Obviously, the existing constitutive models under static loads are difficult to reflect this dynamic effect.

[0004] Therefore, in order to more accurately describe and predict the dynamic mechanical behavior of rocks, it is particularly important to develop a dynamic constitutive model of rocks that can reflect the influence of actual loading speed and strain rate. Summary of the Invention

[0005] The purpose of the embodiment of the present invention is to provide a method for establishing a dynamic triaxial statistical damage constitutive model of rocks applicable to high confining pressure and high strain rate. To this end, a new non-linear dynamic strength criterion is derived, a method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rocks is constructed, the statistical damage theory is introduced into the dynamic constitutive model of rocks, a method for calculating model parameters is proposed, a dynamic triaxial statistical damage constitutive model of rocks is established, and the influence of different probability density distribution function forms on the constitutive model curve is considered, and a sensitivity analysis of the dynamic triaxial statistical damage constitutive model of rocks is carried out, so as to solve at least one technical problem involved in the background technique.

[0006] In order to solve the above technical problems, the present invention is implemented as follows:

[0007] A method for establishing a dynamic triaxial statistical damage constitutive model of rocks applicable to high confining pressure and high strain rate includes the following steps:

[0008] Step S1: Introduce the non-linear Mohr-Coulomb and Hoek-Brown strength criteria, the strength criteria of rock dynamic strength and strain rate dependence based on thermal activation and viscous mechanisms, and establish a non-linear rock dynamic strength criterion considering the strain rate effect.

[0009] Step S2: Calculate the rock dynamic yield strength parameters, and combine with the test data to verify the applicability of the non-linear dynamic strength criterion.

[0010] Step S3: Construct a method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rocks, and introduce the statistical damage theory into the rock dynamic constitutive model.

[0011] Step S4: Establish a rock dynamic triaxial statistical damage constitutive model for simulating the rock dynamic deformation process, and propose a calculation formula for the constitutive model parameters.

[0012] Step S5: Through the comparative analysis with the rock dynamic triaxial test data, verify the rationality of the rock dynamic triaxial statistical damage constitutive model.

[0013] Step S6: Focus on the existing Weibull distribution function describing the strength of rock micro-elements, and introduce two probability density functions, namely the Harris distribution and the Gaussian distribution. Considering the influence of different probability density distribution function forms on the constitutive model curve, conduct a sensitivity analysis of the rock dynamic triaxial statistical damage constitutive model.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0015] 1. The present invention considers the non-linear influence characteristics of rock dynamic strain rate on rock strength, introduces the non-linear Mohr-Coulomb and Hoek-Brown strength criteria, and the uniaxial dynamic strength criteria of rock based on thermal activation and viscous mechanisms that depend on strain rate, and establishes a non-linear rock dynamic strength criterion applicable to high confining pressure conditions, providing a new idea for establishing the statistical damage constitutive model of deep underground engineering and expanding the application scope of the statistical damage constitutive model.

[0016] 2. The present invention establishes a method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rocks, introduces the statistical damage theory, and establishes a statistical damage constitutive model for simulating the rock dynamic triaxial deformation process and its parameter determination method; through the comparative analysis of the model theoretical curve and the test curve, it shows that both the models based on the non-linear Mohr-Coulomb criterion and the Hoek-Brown criterion of the present invention can simulate the non-linear deformation process, and have good simulation curves, with rationality and superiority.

[0017] 3. Based on the established dynamic statistical damage model, the present invention introduces two distribution functions, namely the Harris distribution and the Gaussian distribution, to establish a dynamic triaxial statistical damage constitutive model for rocks. Through comparative research with the existing Weibull distribution model, the model curves based on the Weibull distribution and the Gaussian distribution can better reflect the softening property of rocks and are more in line with the actual situation. Different probability distribution functions have their advantages and limitations, and the probability distribution function can be selected and adjusted according to specific circumstances.

[0018] 4. The method provided by the present invention has simple steps, fully considers the actual engineering requirements, has a wide application range and broad application prospects, effectively makes up for the deficiency that the model curve in the prior art cannot well simulate the nonlinear change process of rocks under deep high in-situ stress, and has very important significance for the simulation method of the dynamic deformation process of rocks in deep underground engineering structures. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained according to these drawings, where:

[0020] Figure 1 is a flowchart of the method for establishing a dynamic triaxial statistical damage constitutive model for rocks applicable to high confining pressure and high strain rate provided by the embodiment of the present invention;

[0021] Figure 2 is a relationship diagram of the yield strength and strain rate of rocks under different confining pressures provided by the embodiment of the present invention (Mohr-Coulomb criterion, 0 ≤ σ3 ≤ σ c ); where Figure 2 (a) is a relationship diagram of the yield strength and strain rate of rocks under the confining pressure σ3 = 5 Mpa provided by the embodiment of the present invention; Figure 2 (b) is a relationship diagram of the yield strength and strain rate of rocks under the confining pressure σ3 = 15 Mpa provided by the embodiment of the present invention; Figure 2 (c) is a relationship diagram of the yield strength and strain rate of rocks under the confining pressure σ3 = 25 Mpa provided by the embodiment of the present invention;

[0022] Figure 3 is a relationship diagram of the yield strength and strain rate of rocks under the confining pressure σ3 = 25 Mpa provided by the embodiment of the present invention (Mohr-Coulomb criterion, σ3 ≥ σ c );

[0023] Figure 4Relationship diagram of rock yield strength and strain rate under different confining pressures provided by the embodiments of the present invention (Hoek-Brown criterion); among them, Figure 4 (a) Relationship diagram of rock yield strength and strain rate under the confining pressure σ3 = 5 Mpa provided by the embodiments of the present invention; Figure 4 (b) Relationship diagram of rock yield strength and strain rate under the confining pressure σ3 = 15 Mpa provided by the embodiments of the present invention; Figure 4 (c) Relationship diagram of rock yield strength and strain rate under the confining pressure σ3 = 25 Mpa provided by the embodiments of the present invention;

[0024] Figure 5 (a)-(i) Comparison diagram of theoretical and experimental curves provided by the embodiments of the present invention (Hoek-Brown criterion, Mohr-Coulomb criterion);

[0025] Figure 6 Influence diagram of strain rate on the rock deformation process under different confining pressures provided by the embodiments of the present invention (Mohr-Coulomb criterion, 0 ≤ σ3 ≤ σ c ); among them, Figure 6 (a) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 5 Mpa provided by the embodiments of the present invention; Figure 6 (b) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 15 Mpa provided by the embodiments of the present invention; Figure 6 (c) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 25 Mpa provided by the embodiments of the present invention;

[0026] Figure 7 Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 25 Mpa provided by the embodiments of the present invention (Mohr-Coulomb criterion, σ3 ≥ σ c );

[0027] Figure 8 Influence diagram of strain rate on the rock deformation process under different confining pressures provided by the embodiments of the present invention (Hoek-Brown criterion); among them, Figure 8 (a) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 5 Mpa provided by the embodiments of the present invention; Figure 8 (b) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 15 Mpa provided by the embodiments of the present invention; Figure 8 (c) Influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 25 Mpa provided by the embodiments of the present invention;

[0028] Figure 9The comparison diagram between the theory obeying the Harris distribution and the test curve provided by the embodiments of the present invention (Hoek-Brown criterion, σ3 = 25 Mpa, );

[0029] Figure 10 (a) to (f) are the comparison diagrams between the theory obeying the Gaussian distribution and the test curve provided by the embodiments of the present invention (Hoek-Brown criterion);

[0030] Figure 11 The influence diagram of strain rate on the rock deformation process under different confining pressures provided by the embodiments of the present invention (Hoek-Brown criterion); among them, Figure 11 (a) is the influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 5 Mpa provided by the embodiments of the present invention; Figure 11 (b) is the influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 15 Mpa provided by the embodiments of the present invention; Figure 11 (c) is the influence diagram of strain rate on the rock deformation process under the confining pressure σ3 = 25 Mpa provided by the embodiments of the present invention. Detailed implementation manners

[0031] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0032] Please refer to Figure 1 As shown, the embodiments of the present invention provide a method for establishing a rock dynamic triaxial statistical damage constitutive model, including the following steps:

[0033] Step S1, introduce the nonlinear Mohr-Coulomb and Hoek-Brown strength criteria, and establish a nonlinear rock dynamic strength criterion considering the strain rate effect based on the rock dynamic strength and strain rate-dependent strength criterion of the thermal activation and viscous mechanisms;

[0034] Step S2, calculate the rock dynamic yield strength parameters, and verify the applicability of the nonlinear dynamic strength criterion in combination with the test data;

[0035] Step S3, construct a method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rocks, and introduce the statistical damage theory into the rock dynamic constitutive model;

[0036] Step S4, establish a rock dynamic triaxial statistical damage constitutive model for simulating the rock dynamic deformation process, and propose a calculation formula for the constitutive model parameters;

[0037] Step S5. Verify the rationality of the rock dynamic triaxial statistical damage constitutive model through comparative analysis with the rock dynamic triaxial test data.

[0038] Step S6. Focus on the existing Weibull distribution function describing the strength of rock micro-elements, introduce two probability density functions, namely the Harris distribution and the Gaussian distribution, consider the influence of different forms of probability density distribution functions on the constitutive model curve, and conduct a sensitivity analysis of the rock dynamic triaxial statistical damage constitutive model.

[0039] In Step S1, establish a non-linear rock dynamic strength criterion considering the strain rate effect, specifically including:

[0040] Establish a rock dynamic strength - strain rate relationship criterion based on the thermal activation and viscous mechanisms, which is expressed by the following formula:

[0041]

[0042] Introduce the non-linear Mohr-Coulomb criterion, which is expressed as:

[0043]

[0044]

[0045] In the formula, σ1 and σ3 respectively represent the maximum and minimum principal stresses of the rock; c is the static cohesion of the rock; is the internal friction angle; σ c represents the uniaxial compressive strength of the rock.

[0046] When 0 ≤ σ3 ≤ σ c , let σ3 = 0 and substitute it into Equation (2) to obtain the cohesion under the uniaxial compression condition of the rock, which can be expressed as:

[0047]

[0048] Considering that the change in the dynamic strength of the rock under dynamic loading is determined by the change in cohesion, and the internal friction angle changes very little with the loading rate, it can be ignored. Therefore, use the uniaxial dynamic stress Y in Equation (1) to replace the uniaxial static stress σ1 in Equation (4) to obtain the dynamic cohesion c d , that is:

[0049]

[0050] Let c = c d , and substitute it into Equation (2) to obtain the rock dynamic strength criterion σ 1d under complex stress conditions, that is:

[0051]

[0052] Since Equation (6) must satisfy the rock strength standard under static conditions, the last two terms on the right side of Equation (6) must both be 0 in static problems, that is:

[0053]

[0054] Therefore, by comparing Equation (2) and Equation (7), we can obtain:

[0055]

[0056] Substituting Equation (8) into Equation (6) again, then Equation (6) can also be rewritten as:

[0057]

[0058] In general rock engineering, since the temperature change is very small, the temperature T can be regarded as a constant. Let A1 = C2T (constant), and substituting it into Equation (9), we can obtain the proposed non-linear dynamic strength standard, which can be expressed as:

[0059]

[0060] Where:

[0061]

[0062]

[0063]

[0064] When σ3 ≥ σ c Let σ3 = 0 and substitute it into Equation (3), the cohesion under uniaxial compression conditions of the rock can be obtained, which can be expressed as:

[0065]

[0066] Replacing the uniaxial static stress σ1 in Equation (14) with the uniaxial dynamic stress Y, the dynamic cohesion can be obtained, that is:

[0067]

[0068] Therefore, let c = c d , and substitute it into Equation (3), the rock dynamic strength criterion under complex stress conditions can be obtained, that is:

[0069]

[0070] Considering that Equation (16) must also satisfy the rock strength criterion under static conditions, the last two terms on the right side of Equation (16) must both be 0 in static problems, that is:

[0071] σ1 = σ3 + C1 (17)

[0072] Therefore, by comparing Equation (3) and Equation (17), we can obtain:

[0073]

[0074] Substituting Equation (18) further into Equation (16), Equation (16) is rewritten as:

[0075]

[0076] In general rock engineering, due to the small change in temperature, the temperature T can be regarded as a constant. Therefore, we can let A2 = C2T (constant) and substitute it into Equation (19) to obtain the nonlinear dynamic strength criterion, which is expressed by the following equation:

[0077]

[0078] Where:

[0079]

[0080] Introducing the failure criterion based on Hoek - Brown, then:

[0081]

[0082] In the formula, σ1 and σ3 are the maximum and minimum principal stresses of the rock respectively; m and s are both constants related to the rock characteristics;

[0083] Let σ3 = 0 and substitute it into Equation (2) to obtain s under uniaxial compression conditions of the rock, which can be expressed as:

[0084]

[0085] Replace the uniaxial static stress σ1 in Equation (23) with the uniaxial dynamic stress Y to obtain the dynamic s d , that is:

[0086]

[0087] Let s = s d , and substitute it into Equation (22) to obtain the rock dynamic strength criterion under complex stress states, which can be expressed as:

[0088]

[0089] Considering that Equation (25) must also satisfy the rock strength criterion under static conditions, the last term on the right side of Equation (25) must be 0 in static problems, that is:

[0090]

[0091] Therefore, by comparing Equation (22) and Equation (26), we can obtain:

[0092]

[0093] Substituting Equation (27) into Equation (25) again, Equation (25) can also be rewritten as:

[0094]

[0095] In general rock engineering, due to the small change in temperature, the temperature T can be regarded as a constant. Let A3 = C2T (constant), and substitute it into Equation (28) to obtain the nonlinear dynamic strength criterion, which is expressed by the following formula:

[0096]

[0097] α3 = m (30)

[0098]

[0099] In step S2, calculate the rock dynamic yield strength parameter, and combine the test data to verify the nonlinear dynamic strength criterion, specifically including:

[0100] Introduce the experimental study on the uniaxial dynamic characteristics of rocks under the confining pressure condition of 5 - 25 Mpa, and obtain the static and dynamic stress - strain curves of rocks under different confining pressures. On this basis, measure the yield strength of rocks under different confining pressure conditions, and the specific data are shown in Table 1.

[0101] Table 1 Static yield stress of rocks under different confining pressures

[0102] <![CDATA[σ3 / Mpa]]> Static yield stress / Mpa 5 19.52 15 29.62 25 41.89

[0103] By non - linearly fitting the static stress - strain curve, calculate the rock static yield strength index, where the cohesion c y is 6.04 Mpa, and the internal friction angle is 5.78°, and the correlation coefficient of the fitting is as high as 0.9941. The constants m related to rock characteristics are 0.1307, and s is 0.2911, and the fitting coefficient is as high as 0.9961. Through the study of the rock dynamic stress - strain curves under different confining pressures and loading rates, the rock dynamic yield stress under the above - mentioned conditions is obtained, as shown in Table 2.

[0104] Table 2 Dynamic Yield Stress of Rock under Different Confining Pressures and Strain Rates

[0105]

[0106]

[0107] The dynamic yield strength of rock under different confining pressures is fitted by using Equations (10), (20), and (29) to obtain the dynamic yield strength parameters of rock under different strength criteria, as shown in Tables 3, 4, and 5.

[0108] Table 3 Dynamic Yield Strength Parameters of Rock (Mohr-Coulomb Criterion, 0 ≤ σ3 ≤ σ c )

[0109]

[0110] Table 4 Dynamic Yield Strength Parameters of Rock (Mohr-Coulomb Criterion, σ3 ≥ σ c )

[0111]

[0112] Table 5 Dynamic Yield Strength Parameters of Rock (Hoek-Brown Criterion)

[0113]

[0114] Substitute these static and dynamic yield strength parameters into Equations (10), (20), and (29) to construct the dynamic strength curves of rock under different confining pressures and different strength criteria. These curves are shown in Figure 2 , Figure 3 , Figure 4 . For the convenience of comparison, the experimental data of dynamic strength are also included in the figure to verify the rationality of the nonlinear dynamic strength criterion.

[0115] Step S3 specifically includes:

[0116] On the basis of the Lemaitre strain equivalence hypothesis, further improve the model. On this basis, the rock under dynamic loading is regarded as two parts: damaged and undamaged. Furthermore, it is assumed that the rock no longer has bearing capacity after failure, and all loads are borne by the undamaged rock.

[0117] Construct a dynamic damage model of rock, and the expression of this model can be represented as:

[0118] 0 ≤ σ3 ≤ σ c

[0119] (i = 1, 2 or 3) (32)

[0120] In the formula, σ id is the dynamic macroscopic nominal stress of the rock, and σ id is the dynamic effective stress of the rock (the stress in the rock mass where there is no undamaged part), and D d is the dynamic damage variable or damage factor of the rock (i.e., the ratio of the cross-sectional area of the damaged part of the rock to the total area of the rock). Since σ id is the stress borne by the undamaged part of the rock, assuming that the undamaged part of the rock obeys the generalized Hooke's law, then:

[0121]

[0122] In the formula, E d and μ d are the dynamic elastic modulus and dynamic Poisson's ratio of the rock, respectively; is the microscopic strain of the material of the undamaged part; i, j, k = 1, 2, 3 or 2, 3, 1 or 3, 2, 1. Since the damaged and undamaged materials are closely mixed, their deformation can be regarded as a cooperative relationship, that is:

[0123]

[0124] In the formula, ε i is the dynamic macroscopic strain of the rock. Substituting equations (32) and (34) into equation (33), we can get:

[0125] σ id = E d ε i (1 - D d ) + μ d (σ jd + σ kd )(35)

[0126] First, assume that the strength or failure of the rock microelement obeys the Weibull distribution, then its probability density function can be expressed as:

[0127] P(F) = m / F0(F / F0) m-1 exp[-(F d / F0) m (36)

[0128] In the formula, F is the distribution variable of the Weibull distribution of microelement failure, and m and F0 are the Weibull distribution parameters.

[0129] The damage variable can be obtained as:

[0130]

[0131] The dynamic statistical damage evolution model of rock triaxial can be expressed as:

[0132]

[0133] Adopt the two non-linear rock dynamic yield strength criteria proposed above, namely equations (10), (20), and (29), to establish the following rock dynamic micro-element strength measurement method, which can be expressed as:

[0134] (M-C criterion, 0 ≤ σ3 ≤ σ c ) (39)

[0135] (M-C criterion, σ3 ≥ σ c )(40)

[0136] (H-B criterion) (41)

[0138] In the formula, are the maximum and minimum principal stresses, i.e., the maximum and minimum effective stresses, acting on the undamaged part of the rock, respectively; other parameters or variables have the physical meanings described above. Therefore, through equations (32) and (35), the rock dynamic micro-element strength F d can also be expressed as:

[0139]

[0140] (M-C criterion, 0 ≤ σ3 ≤ σ c )(42)

[0141] (M-C criterion, σ3 ≥ σ c ) (43)

[0143] (H-B criterion)(44).

[0144] Step S4 specifically includes:

[0145] Based on the above rock dynamic damage model and damage evolution model, substituting equation (38) into equation (35), a rock dynamic triaxial statistical damage constitutive model that can describe the rock dynamic deformation can be obtained.

[0146]

[0147] The above-mentioned rock dynamic triaxial statistical damage constitutive model that includes multiple parameters to simulate the rock dynamic deformation process, i.e., equation (45), will be discussed in detail below on how to determine these model parameters.

[0148] First, the constitutive model based on the non-linear Mohr-Coulomb criterion:

[0149] When \(0\leqslant\sigma_3\leqslant\sigma\) c Since the conventional dynamic triaxial test is usually under equal confining pressure (\(\sigma\) 2d =\(\sigma\) 3d =\(\sigma_3\), where \(\sigma_3\) is the static confining pressure), the first equation in Equation (45) can be rewritten as:

[0150] \(\sigma\) 1d = E d \(\varepsilon\) i exp[-(F d / F0) m + 2\(\mu\) d \(\sigma_3\) (46)

[0151] where F d can be obtained from Equation (42) by setting \(\sigma\) 2d =\(\sigma\) 3d =\(\sigma_3\), and can be expressed as:

[0152]

[0153] Therefore, according to the extreme value characteristics of the conventional dynamic triaxial stress-strain curve of rock, from Equation (47), we can obtain:

[0154]

[0155] In the formula, \(\sigma\) 1dc , \(\varepsilon\) 1dc are the dynamic stress and dynamic strain at the peak point of the conventional dynamic triaxial stress-strain curve of rock measured by the test curve, respectively. In addition, the peak point stress \(\sigma\) 1dc and strain \(\varepsilon\) 1dc must also satisfy Equation (46), that is:

[0156] \(\sigma\) 1dc = E d \(\varepsilon\) 1dc exp[-(F dc / F0) m + 2\(\mu\) d \(\sigma_3\) (49)

[0157] where F dc can be obtained by substituting \(\sigma\) 1d =\(\sigma\) 1dc and \(\varepsilon\) i =\(\varepsilon\) 1dc into Equation (47), that is:

[0158]

[0159] Therefore, by combining Equation (48) and Equation (49), we can obtain:

[0160]

[0161]

[0162] When σ3≥σ c At this time, since the conventional dynamic triaxial test is usually under equal confining pressure (σ 2d =σ 3d =σ3, where σ3 is the static confining pressure), the first equation in Equation (45) can be rewritten as:

[0163] σ 1d =E d ε i exp[-(F d / F0) m +2μ d σ3 (46)

[0164] Among them, F d can be obtained from Equation (43) by setting σ 2d =σ 3d =σ3, and can be expressed as:

[0165]

[0166] Therefore, according to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Equation (46), we can obtain:

[0167]

[0168] In the formula, σ 1dc , ε 1dc are respectively the dynamic stress and dynamic strain at the peak point of the conventional rock dynamic triaxial stress-strain curve measured by the test curve. In addition, the peak point stress σ 1dc and strain ε 1dc must also satisfy Equation (46), that is:

[0169] σ 1dc =E d ε 1dc exp[-(F dc / F0) m +2μ d σ3 (49)

[0170] Among them, F dc can be obtained by substituting σ 1d =σ 1dc and ε i =ε 1dc into Equation (53), that is:

[0171]

[0172] Therefore, by combining Equation (48) and (49), we can obtain:

[0173]

[0174]

[0175] Second, the constitutive model based on the non - linear Hoek - Brown criterion:

[0176] Since the conventional dynamic triaxial test is usually under equal confining pressure (σ 2d = σ 3d = σ3, where σ3 is the static confining pressure), the first equation in Equation (45) can be rewritten as:

[0177] σ 1d = E d ε i exp[-(F d / F0) m +2μ d σ3 (46)

[0178] where F d can be obtained from Equation (3 - 12) by setting σ 2d = σ 3d = σ3, and can be expressed as:

[0179]

[0180] Then, according to the extreme value characteristics of the conventional dynamic triaxial stress - strain curve of rock, from Equation (46) we can get:

[0181]

[0182] In the formula, σ 1dc , ε 1dc are the dynamic stress and dynamic strain corresponding to the peak point of the conventional dynamic triaxial stress - strain curve of rock respectively, and can be obtained from the test curve. In addition, the peak point stress σ 1dc and strain ε 1dc must also satisfy Equation (46), that is:

[0183] σ 1dc = E d ε 1dc exp[-(F dc / F0) m +2μ d σ3 (49)

[0184] where F dc can be obtained by substituting σ 1d = σ 1dc and ε i = ε 1dc into Equation (57), that is:

[0185]

[0186] Thus, by combining Equation (48) and Equation (49), we can obtain:

[0187]

[0188]

[0189] where D d σ can be 1d = σ 1dc and ε i = ε 1dc substituted into Equation (35), and by setting σ 2d = σ 3d = σ3, we can obtain: It can be expressed as:

[0190]

[0191] The above Equation (51) and Equation (52), Equation (55) and Equation (56), Equation (59) and Equation (60) are the methods for determining the model parameters m and F0 proposed by the present invention. However, the dynamic stress-strain curve obtained from traditional rock dynamic triaxial tests is usually a deviator stress and deviator strain relationship curve, which cannot be directly obtained from the test data. Therefore, conversion is required. Thus, through conventional rock dynamic triaxial tests, the peak stress of the rock dynamic triaxial dynamic stress-strain curve is obtained strain Then:

[0192]

[0193]

[0194] In the formula, ε c is the initial strain generated when the rock specimen is subjected to the hydrostatic pressure σ3, that is, when the axial stress σ 1d is loaded to the confining pressure σ3; It is directly obtained from the stress and strain at the peak point of the test curve. Generally, it is considered that the rock is not damaged under hydrostatic pressure. Therefore, according to the second formula in Equation (45), from σ 1d = σ 2d = σ 3d = σ3, the method for measuring the initial strain can be obtained, that is:

[0195] ε c = (1 - 2μ d )σ3 / E d (64)

[0196] In step S5, combined with the test data, verify the constructed constitutive model of dynamic triaxial statistical damage considering deep rocks, specifically including:

[0197] Introduce the test data and test curves. According to the test data and the theoretical formula in step S5, calculate the parameter values under different confining pressure conditions and strain rate conditions, and determine them by using the method proposed above in the present invention. The determination results of these parameters are listed in Tables 6, 7, and 8.

[0198] Table 6 Constitutive model parameters under different confining pressures and strain rates (Mohr-Coulomb criterion, 0 ≤ σ3 ≤ σ c )

[0199]

[0200] Table 7 Constitutive model parameters under different confining pressures and strain rates (Mohr-Coulomb criterion, σ3 ≥ σ c )

[0201]

[0202] Table 8 Constitutive model parameters under different confining pressures and strain rates (Hoek-Brown criterion)

[0203]

[0204]

[0205] Substitute the above parameters into the model provided by the present invention for analysis and calculation, and the theoretical curve of the model of the present invention can be obtained. Plot the theoretical curves of the Hoek-Brown criterion model and Mohr-Coulomb criterion model of the present invention and the test curves in the same graph, as shown in Figure 5 (a)~ Figure 5 (i). It can be seen from this that both models of the present invention can reflect the deformation nonlinearity of the rock dynamic deformation process. As the loading rate and confining pressure increase, the dynamic strength of the rock mass also increases, which is in good agreement with the actual situation, thus verifying the rationality of the established mechanical model.

[0206] In addition, the present invention plots the theoretical curves of the models in the same graph under the same confining pressure conditions, as shown in Figure 6 、 Figure 7 、 Figure 8 for intuitive comparison. Through comparison, it can be found that under the same confining pressure, the deformation behavior of the rock shows significant differences due to the change of strain rate. Specifically, as the strain rate increases, the strength of the rock increases accordingly, and this trend is consistent with the actual geological phenomenon. This change trend verifies the rationality of the theoretical curve of the model and provides strong support for the understanding of the rock dynamic deformation process.

[0207] Based on the above, the model provided by the present invention can well simulate the dynamic deformation process of rocks under high confining pressure, can well reflect the non-linearity of rock deformation, and has obvious rationality and superiority.

[0208] In step S6, the Harris distribution and the Gaussian distribution are introduced to further explore the influence of the probability distribution function on the dynamic deformation of rocks, specifically including:

[0209] Assume that the strength of rock micro-elements follows the improved Harris probability density distribution. Then, according to Li Xiaofeng et al., the rock statistical damage softening model can be expressed as:

[0210]

[0211] In the formula, D is the rock damage variable, F is the random distribution variable of micro-element strength, and a and b are model parameters greater than 0. Since both a and b are greater than 0, then p(F)>0, and the first formula of Equation (65) is monotonically increasing; when the random distribution variable F of rock micro-element strength = 0, D = 0, which is equivalent to an undamaged intact rock material at this time; when F→∞, D→1, then all micro-elements are damaged, and the rock loses its bearing capacity. It can be seen that the change law of the rock damage parameter reflected by the first formula in Equation (65) is close to the actual situation.

[0212] Adopt the following rock dynamic micro-element strength measurement method established based on the Hoek-Brown nonlinear rock dynamic yield strength criterion proposed above, that is, Equation (43), which can be expressed as:

[0213] (H-B criterion) (66)

[0215] In the formula, are the maximum and minimum principal stresses, that is, the maximum and minimum effective stresses, on the undamaged part of the rock respectively; the physical meanings of other parameters or variables are as described above. Then, by using Equations (32) and (34), the rock dynamic micro-element strength F d can also be expressed as:

[0216] (H-B criterion) (67)

[0218] Based on the previously established rock dynamic damage model and damage evolution model, substituting Equation (65) into Equation (34), a dynamic-statistical damage constitutive model that can describe the dynamic deformation of rocks can be obtained, that is:

[0219]

[0220] The above-mentioned rock dynamic triaxial statistical damage constitutive model that includes multiple parameters to simulate the dynamic deformation process of rocks, namely Equation (68), in order to more accurately simulate the behavior of rocks during the dynamic deformation process, a reasonable method for determining model parameters must be established. Therefore, how to determine these model parameters will be discussed in detail next.

[0221] Given that the conventional dynamic triaxial test is usually under equal confining pressure static load (σ 2d =σ 3d =σ3, where σ3 is the static confining pressure), the first equation in Equation (68) can be rewritten as:

[0222]

[0223] where F d can be obtained from Equation (67) by setting σ 2d =σ 3d =σ3, and can be expressed as:

[0224]

[0225] Then, according to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Equation (69), we can obtain:

[0226]

[0227] In the formula, σ 1dc , ε 1dc are the dynamic stress and dynamic strain at the peak point of the test conventional rock dynamic triaxial stress-strain curve respectively, and can be obtained from the test curve. In addition, the peak point stress σ 1dc and strain ε 1dc must also satisfy Equation (69), that is:

[0228]

[0229] where F dc can substitute σ 1d =σ 1dc and ε i =ε 1dc into Equation (70) to obtain, that is:

[0230]

[0231] Then, by combining Equation (71) and Equation (72), we can obtain:

[0232]

[0233]

[0234] The theoretical curves of the Harris distribution model and the Weibull distribution model of the present invention are listed in the same graph as the test curves of Fang Qin et al., as shown in Figure 9 It can be seen from this that the strain-softening model curve of the Harris distribution model of the present invention, which cannot reflect the dynamic deformation process of rocks, deviates from the test curve. Therefore, the simulation effect of the Weibull distribution model provided by the present invention is more in line with the test curve.

[0235] Assume that the logarithmic Gaussian distribution method is used to describe the strength of rock micro-elements:

[0236]

[0237] In the formula, F0 and S0 are the Gaussian distribution parameters and damage variables, which can be expressed as:

[0238]

[0239] In the formula, is the standard Gaussian distribution function. Equation (77) is the damage evolution equation established by the present invention.

[0240] Adopt the Hoek-Brown nonlinear rock dynamic yield strength criterion proposed above in the present invention, that is, Equation (29), to establish the following rock dynamic micro-element strength measurement method, which can be expressed as:

[0241] (H-B criterion) (66)

[0243] In the formula, are the maximum and minimum principal stresses, that is, the maximum and minimum effective stresses, acting on the undamaged part of the rock respectively; the physical meanings of other parameters or variables are as described above. Thus, by using Equations (32) and (35), the rock dynamic micro-element strength F d can also be expressed as:

[0244] (H-B criterion) (67)

[0246] The rock dynamic damage model and the damage evolution model have been established above. Substituting Equation (77) into Equation (35), a rock dynamic triaxial statistical damage constitutive model that can reflect the dynamic deformation process of rocks can be obtained, which can be expressed as:

[0247]

[0248] In view of the fact that the conventional dynamic triaxial test is usually an equal-confining-pressure static load (σ 2d = σ 3d= σ3, where σ3 is the static confining pressure), then Equation (78) can be rewritten as:

[0249]

[0250] Among them, F d can be obtained from Equation (67) by setting σ 2d = σ 3d = σ3, and can be expressed as:

[0251]

[0252] Then, according to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Equation (79), we can obtain:

[0253]

[0254]

[0255] In the formula, σ 1dc , ε 1dc are the dynamic stress and dynamic strain at the peak point of the test conventional rock dynamic triaxial stress-strain curve respectively, and can be obtained from the test curve.

[0256] From Equations (80) and (81), we can obtain:

[0257]

[0258] In the formula

[0259] Take the logarithm of both sides of Equation (82) and organize:

[0260]

[0261] From Equation (79), we get:

[0262]

[0263] In the formula,

[0264] For the peak point of the test curve under a certain confining pressure, a Φ value can be obtained from Equation (85), and the corresponding X value can be obtained from the Gaussian distribution function table.

[0265] From Equations (84) and (85), the parameters F0 and S0 can be obtained:

[0266]

[0267] F0 = XS0 + lnF dc (88)

[0268] Among them, Fdc σ can be 1d = σ 1dc and ε i = ε 1dc substituted into Equation (70) to obtain, that is:

[0269]

[0270] Determined by the method proposed above in the present invention, the determination results of these parameters are listed in Table 9.

[0271] Table 9 Constitutive model parameters under different confining pressures and strain rates (Hoek - Brown criterion)

[0272]

[0273] Substituting the above parameters into the model provided by the present invention for analysis and calculation, the theoretical curve of the model of the present invention can be obtained. The theoretical curves of the Gaussian distribution model and the Weibull distribution model of the present invention and the test curves are listed in the same graph, as shown in Figure 10 (a) - Figure 10 (f). It can be seen therefrom that both models of the present invention can reflect the deformation non - linearity of the rock dynamic deformation process.

[0274] In addition, the present invention plots the theoretical curves of the models under the same confining pressure condition in the same graph, as shown in Figure 11 (a) - Figure 11 (c) for intuitive comparison. It can be found through comparison that under the action of the same confining pressure, the deformation behavior of the rock will show significant differences due to the change of the strain rate. Specifically, with the increase of the strain rate, the strength of the rock increases correspondingly, and this trend is consistent with the actual geological phenomenon. This change trend verifies the rationality of the theoretical curve of the model and provides strong support for the understanding of the rock dynamic deformation process.

[0275] To sum up, the Gaussian distribution model and the Weibull distribution model provided by the present invention can better simulate the dynamic deformation process of rocks under high confining pressure, well reflect the non - linearity of rock deformation, fully reflect the strain - softening characteristics of the rock dynamic deformation process, and have obvious rationality and superiority.

[0276] The method for establishing a dynamic triaxial statistical damage constitutive model of rocks applicable to high confining pressure and high strain rate provided by the present invention aims at the non-linear deformation mechanical characteristics of deep rocks under the conditions of high confining pressure and high strain rate. Considering the non-linear influence characteristics of the dynamic strain rate of rocks on the rock strength based on the actual engineering background, the non-linear Mohr-Coulomb and Hoek-Brown strength criteria are introduced, and the strength criterion of uniaxial dynamic strength of rocks depending on strain rate based on thermal activation and viscous mechanism is used to derive the non-linear rock dynamic strength criterion considering the strain rate effect, providing a theoretical basis for constructing the dynamic statistical damage constitutive relationship of rocks. A method for measuring the strength of rock micro-elements that can reflect the dynamic deformation characteristics of rocks is established, and the statistical damage theory is introduced into the rock dynamic constitutive model, and a method for calculating model parameters is proposed. Based on this, a dynamic triaxial statistical damage constitutive model of rocks is established. On this basis, through the comparative analysis with the corresponding rock dynamic triaxial test data, the theoretical curve calculated by the model can better simulate the non-linear deformation process of rocks under dynamic loads, indicating the rationality of this method. It is analyzed that under the action of the same confining pressure, the deformation behavior of rocks will show significant differences due to the change of strain rate. Specifically, with the increase of strain rate, the strength of rocks increases correspondingly, and this trend is consistent with the actual geological phenomena. This change trend verifies the rationality of the theoretical curve of the model and provides strong support for understanding the dynamic deformation process of rocks. Finally, based on the established dynamic triaxial statistical damage constitutive model, two distribution functions, namely Harris distribution and Gaussian distribution, are introduced to carry out the sensitivity analysis of the dynamic triaxial statistical damage constitutive model of rocks. Through the comparative study with the existing Weibull distribution model, the influence of different probability distribution forms of micro-element strength on the model curve is analyzed.

[0277] In the present invention, specific examples are used to elaborate on the principle and implementation manner of the present invention. The description of the above embodiments is only used to help understand the method and its core idea of the present invention. At the same time, for those of ordinary skill in the art, there will be changes in the specific implementation manner and application scope according to the idea of the present invention. To sum up, the content of this specification should not be construed as a limitation of the present invention.

Claims

1. A method for establishing a dynamic triaxial statistical damage constitutive model of rock, characterized in that It includes the following steps: Step S1, introduce the non-linear Mohr-Coulomb and Hoek-Brown strength criteria, the strength criteria of rock dynamic strength and strain rate dependence based on thermal activation and viscous mechanisms, and establish a non-linear rock dynamic strength criterion considering the strain rate effect, specifically including: Establish a rock dynamic strength-strain rate relationship criterion based on thermal activation and viscous mechanisms, which is expressed by the following formula: Y = + Tln + bln / + 1] (1) Where Y is the uniaxial dynamic strength of the rock; is the uniaxial dynamic strain rate of the rock; is the uniaxial ultimate strain rate of the rock; b is the maximum amplitude of the contribution of macroscopic viscosity to strength growth; is the strain rate corresponding to the inflection point of the strength–strain rate semi-logarithmic curve of the rock; T is the initial temperature of the rock material; , , n is a material constant; Introduce the non-linear Mohr-Coulomb criterion, which is expressed by the following formula: - = + ,0 （2） - = , （3） In the formula, , respectively represent the maximum and minimum principal stresses of the rock; c is the static cohesive force of the rock; is the internal friction angle; represents the uniaxial compressive strength of the rock; When 0 , let = 0, and the cohesion under uniaxial compression of the rock is obtained and expressed by the following formula: c= （4） Replace the in Equation (4) with the uniaxial dynamic stress Y in Equation (1) to obtain the dynamic cohesive force , which is expressed by the following formula: = + Tln + (5)​ Let \(c = \), substitute it into Equation (2) to obtain the dynamic strength criterion of rock under complex stress conditions , which is expressed by the following formula: = + + Tln + + (6) Equation (6) needs to meet the rock strength standard under static conditions, and we get: = + + （7） Compare Equation (2) and Equation (7), and we get: = （8） Then substitute Equation (8) into Equation (6), and we get: = + + Tln + + (9) Regarding the temperature T as a constant, let = T, substitute it into Equation (9), and the non-linear dynamic strength criterion is obtained, which is expressed by the following formula: = + + ln + + （10） Where: = （11） = （12） p= （13） When = 0 is substituted into Equation (3), the cohesion under uniaxial compression of the rock is obtained and expressed by the following formula: c= （14） Replace the uniaxial static stress in Equation (14) with the uniaxial dynamic stress Y , and the dynamic cohesion is obtained, that is: = + Tln + (15)​ Let \(c = \), and substitute it into Equation (3) to obtain the dynamic strength criterion of rock under complex stress conditions, that is: = + + Tln + (16) Equation (16) needs to meet the rock strength standard under static conditions, that is: = + （17） Compare Equation (3) and Equation (17), and we get: = （18） Substitute Equation (18) into Equation (16), and we get: = + + Tln + (19) Regarding the temperature T as a constant, let = T, substitute it into Equation (19) to obtain the non-linear dynamic strength criterion, which is expressed as follows: = + + ln + （20） Where: = （21） Introduce the failure criterion based on Hoek-Brown, then: （22） In the formula, , are the maximum and minimum principal stresses of the rock respectively; both m and s are constants related to the characteristics of the rock; Let = 0, substitute it into Equation (2), and the s under uniaxial compression condition of rock is obtained, expressed as: s= （23） Replace the uniaxial static stress in Equation (23) with the uniaxial dynamic stress Y , and obtain the dynamic , that is: = （24） Let \(s = \), and substitute it into Equation (22), then the dynamic strength criterion of rock under complex stress state is obtained, which is expressed as: = + （25） Equation (25) must meet the rock strength standard under static conditions, that is: = + （26） Compare Equation (22) and Equation (26), and we get: = （27） Then substitute Equation (27) into Equation (25), and we get: = + （28） Regarding the temperature T as a constant, let = T, substitute it into Equation (28) to obtain the non-linear dynamic strength criterion, which is expressed as follows: = + （29） =m （30） = （31）； Step S2, calculate the rock dynamic yield strength parameters, and combine the test data to verify the rationality of the non-linear dynamic strength criterion; Step S3, construct a rock micro-element strength measurement method that can reflect the rock dynamic deformation characteristics, and introduce the statistical damage theory into the rock dynamic constitutive model; Step S4, establish a rock dynamic triaxial statistical damage constitutive model to simulate the rock dynamic deformation process, and propose a calculation formula for the constitutive model parameters; Step S5, verify the rationality of the rock dynamic triaxial statistical damage constitutive model through comparative analysis with the rock dynamic triaxial test data; Step S6, around the existing Weibull distribution function describing the rock micro-element strength, and introduce two probability density functions of Harris distribution and Gaussian distribution, consider the influence of different probability density distribution function forms on the rock dynamic triaxial statistical damage constitutive model curve, and carry out the sensitivity analysis of the rock dynamic triaxial statistical damage constitutive model.

2. The method according to claim 1, wherein In Step S2, calculate the rock dynamic yield strength parameters, and combine the test data to verify the non-linear dynamic strength criterion, specifically including: Combine the test data, measure the rock yield strength under different confining pressure conditions, calculate the rock static yield strength index by non-linearly fitting the static stress-strain curve, obtain the rock dynamic yield stress through the study of the rock dynamic stress-strain curve under different confining pressures and loading rates, use Equation (10), Equation (20), Equation (29) to fit the rock dynamic yield strength under different confining pressures, obtain the rock dynamic yield strength parameters under different strength standards, substitute the static and dynamic yield strength parameters into Equation (10), Equation (20), Equation (29), construct the dynamic strength curves of rocks under different confining pressures and different strength criteria, and verify the rationality of the non-linear dynamic strength criterion.

3. The method according to claim 2, wherein Step S3 specifically includes: Based on the Lemaitre strain equivalence hypothesis, the rock under dynamic loading is regarded as two parts: damaged and undamaged. Furthermore, it is assumed that the rock no longer has bearing capacity after failure, and all loads are borne by the undamaged rock. A dynamic damage model of the rock is constructed and expressed by the following formula: 0 = (1 ),where i = 1, 2 or 3 (32) In the formula, is the dynamic macroscopic nominal stress of the rock, is the dynamic effective stress of the rock, is the dynamic damage variable or damage factor of the rock; Since is the stress on the undamaged part of the rock, assuming that the undamaged part of the rock obeys the generalized Hooke's law, then: = + ( ) (33) wherein and are the dynamic elastic modulus and dynamic Poisson's ratio of the rock, respectively; is the microscopic strain of the undamaged part of the material; i, j, k = 1, 2, 3 or 2, 3, 1 or 3, 2, 1; Due to the close mixing of damaged and undamaged materials, their deformation is regarded as a cooperative relationship, that is: = （34） wherein, is the dynamic macroscopic strain of the rock. Substituting Eqs. (32) and (34) into Eq. (33), we get: = (1- )+ ( + ) (35) Assume that the strength of rock micro-elements or the failure of rock micro-elements follows the Weibull distribution, then its probability density function is expressed as: P(F)=m0 / (36) where F is the distribution variable of the Weibull distribution of the elemental failure, m and are Weibull distribution parameters; The damage variable is obtained as: = ] （37） The dynamic statistical damage evolution model of the rock is expressed as: = (38) Based on Eqs. (10), (20), and (29), a method for measuring the dynamic micro-element strength of the rock is established and expressed as: = ln （39） = ln （40） = （41） In the formula, , are respectively the maximum and minimum principal stresses borne by the undamaged part of the rock; the dynamic micro-element strength of the rock is expressed as: = ln p (42) = ln （43） = （44）。 4. The method according to claim 3, characterized in that, Step S4 specifically includes: Based on the rock dynamic damage model and the damage evolution model, substituting Eq. (38) into Eq. (35), a rock dynamic triaxial statistical damage constitutive model describing the dynamic deformation of the rock is obtained and expressed by the following formula: = （45） The rock dynamic triaxial statistical damage constitutive model contains multiple parameters simulating the dynamic deformation process of the rock. The parameters are determined by the following method: Constitutive model based on the nonlinear Mohr-Coulomb criterion: When 0 At this time, since the conventional dynamic triaxial test is usually under equal confining pressure, that is = = , is the static confining pressure, the first formula in Equation (45) is rewritten as: = （46） Among them, From Equation (42) and by setting = = it is obtained and expressed by the following formula: = ln p (47) According to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Eq. (47), we get: =0 （48） In the formula, , are the dynamic stress and dynamic strain at the peak point of the conventional rock dynamic triaxial stress-strain curve measured by the test curve, respectively. In addition, the peak point stress and strain satisfy Equation (46), that is: （49） Among them, Substitute = and = into Equation (47) to obtain, that is: = ln p (50) Combining Eqs. (48) and (49), we get: = （51） = （52） When Since the conventional dynamic triaxial test is usually under equal confining pressure, that is = = , is the static confining pressure, the first formula in Equation (45) is rewritten as: = （46） Among them, From Equation (43) and by setting = = it is obtained and expressed by the following formula: = ln （53） According to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Eq. (46), we get: =0 （48） In the formula, , are the dynamic stress and dynamic strain at the peak point of the conventional rock dynamic triaxial stress-strain curve measured by the test curve, respectively; In addition, the peak point stress and strain satisfy Equation (46), that is: （49） Among them, Substitute = and = into Equation (53), we get: = ln （54） Combining Eqs. (48) and (49), we get: = （55） = （56） Constitutive model based on the nonlinear Hoek-Brown criterion: Since the conventional dynamic triaxial test is usually under equal confining pressure, that is = = , is the static confining pressure, the first formula in Equation (45) is rewritten as: = （46） Among them, From formula (43) and by setting = = it is obtained and expressed by the following formula: = （57） According to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Eq. (46), we get: =0 （48） In the formula, , are respectively the dynamic stress and dynamic strain corresponding to the peak point of the conventional rock dynamic triaxial stress-strain curve, which are obtained from the test curve; In addition, the peak point stress and strain satisfy Equation (46), that is: （49） Among them, Substitute = and = into Equation (57), we get, that is: = （58） Combining Eqs. (48) and (49), we get: = （59） = （60） Among them, Substitute = and = into Equation (35) and let = = to obtain, which is expressed by the following formula: =1 （61） Equations (51) and (52), (55) and (56), (59) and (60) are the methods for determining the parameters and ; The peak stress of the dynamic stress-strain curve of rock under dynamic triaxial test is obtained through conventional dynamic triaxial test of rock , strain , then: = + （62） = + （63） In the formula, is the initial strain generated when the rock specimen is subjected to hydrostatic pressure , that is, the axial stress is loaded to the confining pressure ; , is directly obtained from the stress and strain at the peak point of the test curve; assuming that the rock is not damaged under hydrostatic pressure, according to the second formula in Equation (45), from = = the method for measuring the initial strain is obtained, that is: =(1 ) / (64)。 5. The method according to claim 4, characterized in that In step S5, to verify the rationality of the rock dynamic triaxial statistical damage constitutive model, it specifically includes: Combined with the test data and test curves, according to the test data and theoretical formulas, calculate the parameter values under different confining pressure conditions and strain rate conditions. Obtain the theoretical curve of the rock dynamic triaxial statistical damage constitutive model based on the parameter values, compare and analyze it with the test curve, verify the rationality of the rock dynamic triaxial statistical damage constitutive model, and analyze the calculated parameter values.

6. The method according to claim 5, characterized in that, Step S6 specifically includes: Assume that the strength of rock micro-elements follows the improved Harris probability density distribution, then the rock statistical damage softening model is obtained and expressed by the following formula: = （65） wherein is the rock damage variable, F is the random distribution variable of the micro-element strength, , are model parameters greater than 0; Adopt the Hoek-Brown nonlinear rock dynamic yield strength criterion, Eq. (43) to establish the following method for measuring the dynamic micro-element strength of the rock, which is expressed by the following formula: = （66） In the formula, , are respectively the maximum and minimum principal stresses on the undamaged part of the rock; Using equations (32) and (34), the dynamic micro-element strength of the rock is expressed as: = （67） Substitute Eq. (65) into Eq. (34) to obtain a rock dynamic triaxial statistical damage constitutive model that can describe the dynamic deformation of the rock, that is: = (68)。 7. The method according to claim 6, characterized in that, The parameters in the rock dynamic triaxial statistical damage constitutive model are determined by the following method: In view of the fact that the conventional dynamic triaxial test is usually an equal-confining-pressure static load, that is = = , being the static confining pressure, the first formula in Equation (68) is rewritten as: = （69） Among them, From Equation (67) and by setting = = we obtain, which is expressed by the following formula: = （70） According to the extreme value characteristics of the conventional rock dynamic triaxial stress-strain curve, from Eq. (69), we get: =0 （71） In the formula, , are the dynamic stress and dynamic strain at the peak point of the conventional rock dynamic triaxial stress-strain curve in the test, respectively, obtained from the test curve; in addition, the peak point stress and strain satisfy Equation (69), that is: （72） Among them, Substitute = and = into Equation (70) to obtain, that is: = （73） Combining Eqs. (71) and (72), we get: = （74） = (75) Assume that the strength of rock micro-elements is described by the logarithmic Gaussian distribution method: P(F)= exp (76) In the formula, is the damage variable of Gaussian distribution parameter, which is expressed by the following formula: D= dx= (77) In the formula, is the standard Gaussian distribution function, which is the established damage evolution equation; Adopt the Hoek-Brown nonlinear rock dynamic yield strength criterion, that is, Eq. (29) to establish the following method for measuring the dynamic micro-element strength of the rock, which is expressed by the following formula: = （66） Express the dynamic micro-element strength of the rock using Equation (32) and Equation (35). as follows: = （67） Substitute Equation (77) into Equation (35) to obtain a dynamic triaxial statistical damage constitutive model of rock that can reflect the dynamic deformation process of rock, which is expressed by the following formula: = （78） In view of the fact that the conventional dynamic triaxial test is usually under equal confining pressure static load, that is = = , is the static confining pressure, then rewrite Equation (78) as: = （79） Among them, From Equation (67) and by setting = = we obtain, which is expressed by the following formula: = （70） According to the extreme value characteristics of the conventional dynamic triaxial stress-strain curve of rock, from Equation (79), we have: =0 （80） = + { } （81） From Equations (80) and (81), we have: ]= ( ) （82） where A = (83) Take the logarithm of both sides of Equation (82) and rearrange to get: =-2 ( ) ] （84） From Equation (79), we have: = = )= （85） where X = (86) For the peak point of the test curve under a certain confining pressure, a Φ value is obtained from Equation (85), and the corresponding X value is obtained from the Gaussian distribution function table; The parameters are obtained from Equation (84) and Equation (85). : =exp ) ln (87) =X +ln (88) Among them, Substitute = and = into Equation (70), and we get: = （73）。

Images