Identification of stiffness turning point and determination of three-directional crack strain of rock under true triaxial stress condition

Abstract

The application provides a true triaxial stress condition rock stiffness turning point identification and three-directional crack strain determination method, and relates to the technical field of rock mechanics test and data processing. The method comprises the following steps: identifying a stable stiffness stage, and determining the initial position of the subsequent stiffness continuous decline as the stiffness turning point. Before the turning point, the crack closure strain of the three principal stress directions is determined according to the crack closure variable and the initial defect additional flexibility. After the turning point, the rock mesoscopic representative volume unit is divided into an elastic load medium and a crack evolution medium, and the three-directional crack expansion strain is determined according to the volume fraction of the crack evolution medium, the strain concentration relationship and the plastic deformation. The crack closure strain and the crack expansion strain are segmented and combined to obtain the three-directional crack strain and the volume crack strain in the whole loading process. The application can distinguish the deformation caused by the initial crack closure and the stress-induced crack expansion, and provides a basis for the rock crack evolution and anisotropic deformation analysis under the true triaxial condition.

Description

Technical Field

[0001] This invention relates to the field of rock mechanics testing and data processing technology, and in particular to a method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions. Background Technology

[0002] Deep rock masses are often in a state of unequal principal stress, where Σ1 > Σ2 > Σ3. In the initial loading stage, pores and initial micro-fractures gradually close, increasing rock stiffness. Subsequently, stress-induced fractures develop, and stiffness decreases from a stable high value. This transition point can serve as a crucial characteristic connecting the initial defect closure and fracture propagation.

[0003] Existing methods often determine feature points by differencing adjacent points or manually selecting linear segments, which is easily affected by noise, non-equidistant sampling, and subjective judgment. When high stiffness manifests as a plateau, even a single maximum stiffness point is insufficient to stably characterize stage transitions. Furthermore, traditional fracture volumetric strain is the algebraic sum of triaxial fracture strain, failing to distinguish the deformation differences between the intermediate principal stress direction and the minimum principal stress direction. Therefore, a unified method is needed to identify stable stiffness stages, determine inflection points, and separate triaxial fracture strain throughout the entire fracture process. Summary of the Invention

[0004] To address the aforementioned shortcomings in the existing technology, this invention provides a method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions. This method aims to reduce the dependence of stiffness inflection point identification on single-point fluctuations and human experience, and to determine the fracture strain in the three principal stress directions during true triaxial loading under the control of the same inflection point.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows: This solution provides a method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions, including the following steps: The triaxial stress and total triaxial strain of the rock sample were obtained during the true triaxial loading process. The obtained data were synchronized, zero-point corrected, and the effective loading segment with the maximum principal strain increasing monotonically was extracted. Based on the effective loading segment, the stable stiffness stage is determined by the local tangent stiffness of the maximum principal stress-maximum principal strain curve, and the starting position of the continuous decrease in stiffness after the stable stiffness stage is determined as the stiffness inflection point. Before the stiffness inflection point, the crack closure strain in the three principal stress directions is calculated based on the normalized loading variables, crack closure variables in each direction, and the initial defect additional compliance. After the stiffness inflection point, the representative volumetric unit of the rock microstructure is divided into elastic bearing medium and fracture evolution medium. Based on the volume fraction of fracture evolution medium, the triaxial strain concentration relationship of the two types of medium and the triaxial cumulative plastic strain of fracture evolution medium, the macroscopic fracture propagation strain in the three principal stress directions is calculated respectively. Among them, the two types of medium are elastic bearing medium and fracture evolution medium. The crack closure strain before the stiffness inflection point and the macroscopic crack propagation strain after the stiffness inflection point are segmented and merged to obtain the triaxial crack strain during the entire loading process, and the volumetric crack strain is determined by the sum of the triaxial crack strains.

[0006] The beneficial effects of the present invention are: (1) Based on the local tangential stiffness and its continuous change characteristics, the stiffness inflection point can be identified, which can reduce the influence of single-point fluctuation, reduce the dependence on human experience, and improve the stability and objectivity of the identification results.

[0007] (2) Divide the crack closure and expansion stages with the same stiffness inflection point, and determine the triaxial crack strain before and after the inflection point respectively, so as to realize the continuous characterization of crack deformation throughout the loading process.

[0008] (3) It can obtain the fracture strain and volumetric fracture strain in the three principal stress directions respectively, reflecting the directional differences of fracture deformation under true triaxial conditions, and providing a basis for rock fracture evolution analysis and failure mode identification.

[0009] Furthermore, the determination of the stiffness inflection point includes: Based on the effective loading segment, the statistical value of the highest stiffness data before the peak is used as the stiffness reference value. The interval in which the local tangential stiffness reaches the preset proportion of the reference value and remains stable in multiple consecutive sampling points is determined as the stable stiffness stage. After the stable stiffness stage, when the local tangential stiffness of multiple consecutive sampling points decreases as a whole and the cumulative decrease reaches a preset threshold, the position that first meets the condition is determined as the stiffness inflection point.

[0010] The beneficial effects of the above-mentioned further scheme are: by jointly determining the stable stiffness stage and the continuous decrease condition, the false identification caused by local stiffness fluctuations and accidental anomalies can be reduced, and the starting position of the stiffness changing from stable to continuous decrease can be accurately determined, thereby improving the stability and repeatability of stiffness inflection point identification.

[0011] Furthermore, the calculation of the crack closure strain in the three principal stress directions includes: Before the stiffness inflection point, a normalized loading variable is constructed based on the total strain in the direction of the maximum principal stress corresponding to the start point of the effective loading segment and the stiffness inflection point. The triaxial fracture closure variables are determined based on the normalized loading variables and the fracture closure parameters in each direction; The initial defect additional compliance in each direction is updated using the triaxial crack closure variable, and the crack closure strain is accumulated point by point to obtain the crack closure strain in the three principal stress directions. No new crack closure strain is added after the stiffness inflection point.

[0012] The beneficial effects of the above-mentioned further scheme are: by uniformly characterizing the loading process before the inflection point through normalized loading variables, and calculating the crack closure strain in three directions respectively, the directional differences of the initial defect closure deformation can be reflected; at the same time, the stiffness inflection point is taken as the position where the closure strain stops accumulating, avoiding the repeated inclusion of the crack propagation stage and the closure stage, and improving the rationality of the crack strain separation results.

[0013] Furthermore, the calculation of macroscopic crack propagation strain in the three principal stress directions includes: After the stiffness inflection point, the representative volumetric units of the rock microstructure are divided into elastic bearing medium and fracture evolution medium. Initialize the two types of medium states at the stiffness inflection point and irreversibly update the volume fraction of the fracture evolution medium during loading; The local strain of the two types of media is calculated based on the mechanical parameters of the two types of media, the ellipsoidal shape of the fracture evolution region, and the triaxial strain condensation matrix containing off-diagonal terms. Based on the triaxial strain concentration relationship of the two types of media, the triaxial cumulative plastic strain of the fracture evolution medium is calculated according to the pressure-related yield condition and non-associated plastic flow, and multiplied with the current volume fraction of the fracture evolution medium to obtain the triaxial macroscopic fracture propagation strain.

[0014] The beneficial effects of the above-mentioned further scheme are as follows: by dividing the rock response after the inflection point into two parts, elastic bearing and fracture evolution, and combining the irreversible increase of the volume fraction of the fracture evolution medium, the continuous development of the fracture propagation process can be characterized; at the same time, by adopting the triaxial strain concentration relation including off-diagonal terms and pressure-related plasticity calculation, the coupling effect between the three principal stress directions can be considered, and the triaxial macroscopic fracture propagation strain can be obtained respectively, thereby improving the directionality and rationality of the fracture propagation strain determination under true triaxial conditions.

[0015] Furthermore, determining the volumetric crack strain includes: The corresponding cumulative crack closure strain is taken before and at the stiffness inflection point; After the stiffness inflection point, the crack strain in each principal stress direction is taken as the sum of the cumulative crack closure strain in the corresponding direction at the stiffness inflection point and the current macroscopic crack propagation strain in the corresponding direction. The cumulative crack closure strain and the sum of strains are combined in segments to obtain the triaxial crack strain during the entire loading process; The volumetric fracture strain is determined by the sum of the triaxial fracture strains.

[0016] The beneficial effects of the above-mentioned further scheme are: by merging the crack closure strain before the inflection point and the crack propagation strain after the inflection point in segments, the continuous triaxial crack strain of the entire loading process can be obtained, avoiding the omission or double counting of deformation in the two stages; and then the volumetric crack strain can be determined by the sum of the triaxial crack strains, realizing a unified characterization of the directional characteristics of crack deformation and the overall volume change. Attached Figure Description

[0017] Figure 1 This is the overall flowchart of the present invention.

[0018] Figure 2 The diagram shows the relationship between the maximum principal stress and the total triaxial strain under the working conditions of Σ2=5MPa and Σ3=0MPa.

[0019] Figure 3 The diagram shows the results of identifying the stable stiffness stage and stiffness inflection point.

[0020] Figure 4 The three-dimensional crack closure variables and crack closure strain diagrams are shown before the inflection point.

[0021] Figure 5 This is a diagram showing the evolution of the medium volume fraction in the fracture after the inflection point.

[0022] Figure 6 The graph shows a comparison between experimental and calculated values ​​of the triaxial stress-strain relationship under the conditions of Σ2=5MPa and Σ3=0MPa.

[0023] Figure 7 The diagram shows the evolution of triaxial and volumetric crack strain under the conditions of Σ2=5 MPa and Σ3=0 MPa. Detailed Implementation

[0024] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0025] Example like Figure 1 As shown, this invention provides a method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions, the implementation method of which is as follows: The triaxial stress and total triaxial strain of the rock sample were obtained during the true triaxial loading process. The obtained data were synchronized, zero-point corrected, and the effective loading segment with the maximum principal strain increasing monotonically was extracted. Based on the effective loading segment, the stable stiffness stage is determined using the local tangent stiffness of the maximum principal stress-maximum principal strain curve. The starting position of the continuous decrease in stiffness after the stable stiffness stage is determined as the stiffness inflection point. The implementation method is as follows: Based on the effective loading segment, the statistical value of the highest stiffness data before the peak is used as the stiffness reference value. The interval in which the local tangent stiffness reaches a preset proportion of the reference value and remains stable within multiple consecutive sampling points is determined as the stable stiffness stage. After the stable stiffness stage, when the local tangent stiffness of multiple consecutive sampling points decreases overall and the cumulative decrease reaches a preset threshold, the earliest position that meets the condition is determined as the stiffness inflection point. Before the stiffness inflection point, the crack closure strain in the three principal stress directions is calculated based on the normalized loading variables, crack closure variables in each direction, and the initial imperfection additional compliance. The implementation method is as follows: Before the stiffness inflection point, normalized loading variables are constructed using the total strain in the maximum principal stress direction corresponding to the effective loading segment start point and the stiffness inflection point; the three-dimensional crack closure variables are determined based on the normalized loading variables and the crack closure parameters in each direction; the initial imperfection additional compliance in each direction is updated using the three-dimensional crack closure variables, and the crack closure strain is accumulated point by point to obtain the crack closure strain in the three principal stress directions. After the stiffness inflection point, no new crack closure strain is added. After the stiffness inflection point, the representative volumetric unit of the rock microstructure is divided into an elastic bearing medium and a fracture evolution medium. Based on the volume fraction of the fracture evolution medium, the triaxial strain concentration relationship of the two types of media, and the triaxial cumulative plastic strain of the fracture evolution medium, the macroscopic fracture propagation strain in the three principal stress directions is calculated. The two types of media are the elastic bearing medium and the fracture evolution medium. The implementation method is as follows: After the stiffness inflection point, the representative volumetric unit of the rock microstructure is divided into an elastic bearing medium and a fracture evolution medium; the states of the two types of media are initialized at the stiffness inflection point, and the volume fraction of the fracture evolution medium is irreversibly updated during loading; the local strain of the two types of media is calculated based on the mechanical parameters of the two types of media, the ellipsoidal shape of the fracture evolution region, and the triaxial strain concentration matrix containing off-diagonal terms; based on the triaxial strain concentration relationship of the two types of media, the triaxial cumulative plastic strain of the fracture evolution medium is calculated according to the pressure-related yield condition and non-associated plastic flow, and multiplied with the current volume fraction of the fracture evolution medium to obtain the triaxial macroscopic fracture propagation strain. The fracture closure strain before the stiffness inflection point and the macroscopic fracture propagation strain after the stiffness inflection point are segmented and merged to obtain the triaxial fracture strain during the entire loading process. The volumetric fracture strain is determined by the sum of the triaxial fracture strains. The implementation method is as follows: After the stiffness inflection point, the fracture strain in each principal stress direction is taken as the sum of the cumulative fracture closure strain in the corresponding direction at the stiffness inflection point and the current macroscopic fracture propagation strain in the corresponding direction; the cumulative fracture closure strain and the sum of strains are segmented and merged to obtain the triaxial fracture strain during the entire loading process; the volumetric fracture strain is determined by the sum of the triaxial fracture strains.

[0026] In this embodiment, the true triaxial loading test data of the rock specimen is acquired and preprocessed as follows: The rock specimen is independently loaded along three mutually orthogonal directions. After the intermediate principal stress Σ2 and the minimum principal stress Σ3 reach their target values, they are kept constant or fluctuate within an allowable range, and the loading displacement in the direction of the maximum principal stress is continued to increase until the predetermined termination condition is met. The sampling sequence number or time, maximum principal stress Σ1, intermediate principal stress Σ2, minimum principal stress Σ3, and corresponding total strain are simultaneously collected. 1. 2 and 3. Synchronize and align the triaxial stress-strain data according to the sampling sequence number or time. Using the formal loading start point as a reference, zero out the total strain in all three directions and capture the total strain. The data interval that shows a monotonically increasing overall trend and includes a pre-peak rising segment and a post-peak softening segment is considered the effective loading segment. Stress is positively expressed as compressive stress, and strain in all three directions uses a unified sign convention. 1 represents the total strain in the direction of the maximum principal stress. 2 represents the total strain in the direction of the intermediate principal stress. 3 represents the total strain in the direction of minimum principal stress.

[0027] In this embodiment, the local tangent stiffness is calculated as follows: Taking the first... k Centered on a sampling point, select the sampling point and the points before and after it. m 2 adjacent sampling points, forming a group containing 2 m +1 neighboring data sets. The maximum principal stress Σ1 and the total strain in the direction of the maximum principal stress within this neighboring data set. 1. Perform univariate linear least squares fitting, and use the slope of the fitted line as the first... k Local tangent stiffness at each sampling point: (1.1) in: ; In the formula, E t,k For the first k Local tangent stiffness at each sampling point m For the first k The number of adjacent sampling points selected on one side of each sampling point; a 1, k and Σ a 1, k These represent the average total strain and the average maximum principal stress in the direction of the maximum principal stress within adjacent data sets. For the first in the neighboring data group j Total strain in the direction of maximum principal stress at each sampling point For the first in the neighboring data group j The maximum principal stress at each sampling point j This represents the sampling point number within the adjacent data group, with a value range of [value range missing]. k - m to k + m .

[0028] The number of sampling points in the neighboring data set is pre-set to an odd number, preferably 5 to 11. Multiple neighboring sampling points are used to fit the local slope together to reduce the impact of fluctuations in a single sampling point on the local tangent stiffness calculation results. For sampling points where a complete neighboring data set cannot be obtained at either end of the effective loading segment, their local tangent stiffness is not calculated, nor are they used for stiffness inflection point identification.

[0029] In this embodiment, the stable stiffness stage and stiffness inflection point are identified as follows: The high-value portion is selected from the pre-peak local tangent stiffness sequence, and the median of the local tangent stiffness in this portion is used as the reference stiffness. E H Within the pre-peak loading segment, multiple consecutive sampling points are sequentially selected to form the judgment interval. For any judgment interval, the relative fluctuation amplitude of the local tangent stiffness is calculated. : (1.2) In the formula, E t,max , E t,min and E t,mean These represent the maximum, minimum, and average values ​​of the local tangent stiffness within the continuous interval under investigation.

[0030] A continuous interval is defined as a stable stiffness stage when it simultaneously satisfies the following conditions: (1.3) In the formula, r For reference stiffness ratio, R E,lim This represents the upper limit of allowable relative stiffness fluctuations. The stable stiffness phase should include at least a preset number of consecutive sampling points.

[0031] After the stability stiffness stage identification is completed, starting from the first sampling point after the end of this stage, the results are sequentially counted from the first sampling point. k Using the sampling point as the starting point to be judged, calculate the subsequent sampling point. m The cumulative stiffness decrease of each sampling point relative to that point : (1.4) At the same time, examine the first k To thek + m Stiffness variation between adjacent sampling points among sampling points. When the interval simultaneously satisfies: (1.5) And in this interval m In the group of adjacent stiffness variations, at least one q The group satisfies: (1.6) Then it is assumed that the local tangent stiffness starts from the first... k The sampling points began to show a continuous decline, and the earliest sampling point to meet the above conditions was the first one. k The sampling points were determined as stiffness inflection points. t .

[0032] In the formula, D lim To preset the cumulative stiffness reduction threshold, m The number of sampling intervals spanned for the descent determination; q This represents the minimum number of adjacent data sets for which a decrease in stiffness is required. and For respectively the first j +1 sampling point and the first j Local tangent stiffness at each sampling point.

[0033] In this embodiment, the triaxial crack closure strain before the stiffness inflection point is determined, which is specifically defined as: constructing a normalized loading variable from the start point of the effective loading segment to the inflection point: (1.7) In the formula, For the first k Normalized load variables for each sampling point 1,k For the first k The maximum principal strain at each sampling point 1,t This represents the maximum principal strain at the stiffness inflection point. The normalized loading variable at the start of the effective loading segment is 0, and the normalized loading variable at the stiffness inflection point is 1. When the strain at the start of the effective loading segment has already been normalized to zero, i.e. When 0 1 = 0, the above expression simplifies to: (1.8) No. i The crack closure variables and the current defect additional compliance in each principal stress direction are as follows: (1.9) (1.10) In the formula, For the firstk The sampling point i Crack closure variables in the principal stress directions For the first k The sampling point i The current defect's additional compliance in each principal stress direction, Load variables for normalization of a i Power of 1 a i and b i All are closed-loop parameters. Ld 0 i 1 represents the initial defect with added compliance. The triaxial fracture closure strain is obtained by point-by-point accumulation: (1.11) In the formula, To accumulate up to the number k When the sampling point is _____, the _th i Crack closure strain in each principal stress direction For the first j -1 sampling point to the j Midpoint gap closure variable between sampling points For the first j -1 maximum principal stress at sampling points For the first j -1 sampling point to the j The maximum principal stress increment at each sampling point. xc i , t The first point at the stiffness inflection point i The crack closure variable along each principal stress direction is set at the inflection point. xc i , t =1 indicates that the initial crack closure process in this direction has been completed, and no new crack closure strain will be added after the inflection point, maintaining... cl i , k = cl i , t constant, cl i , t The first point at the stiffness inflection point i Cumulative crack closure strain in each principal stress direction.

[0034] In this embodiment, the triaxial crack propagation strain after the stiffness inflection point is determined as follows: 1) Initialize the two types of media For the stiffness inflection point and the subsequent... kSampling points were used to divide the rock into elastic bearing medium and fracture evolution medium, with volume fractions of 1- c k and c k Among them, the elastic bearing medium is used to characterize the material portion that maintains its elastic bearing capacity, while the crack evolution medium is used to characterize the material portion that undergoes crack propagation and plastic deformation. k The macroscopic stress and total macroscopic strain at each sampling point are determined by the volume fraction weighting of the local stress and total local strain in the three principal stress directions for the two types of media: (1.12) In the formula, i Number the principal stress directions. i =1, 2, and 3 represent the directions of the maximum principal stress, intermediate principal stress, and minimum principal stress, respectively. k Numbering the stiffness inflection point and subsequent sampling points, Σ i,k and i,k The first k The sampling point i Macroscopic stress and total macroscopic strain in each principal stress direction, Σ oh,k and oh,k These represent the local stress and total local strain of the elastic bearing medium, respectively. ci,k and ci,k These represent the local stress and total local strain of the fracture evolution medium, respectively. (Superscript) e and c These represent the elastic bearing medium and the fracture evolution medium, respectively. c k For the first k Volume fraction of fracture evolution medium at each sampling point.

[0035] by t This represents the sampling point corresponding to the stiffness inflection point. When... k = t At that point, the macroscopic stress, total macroscopic strain, and volume fraction of the fracture evolution medium are considered. c t As the initial condition for calculations after the inflection point, the cumulative plastic strain of the fracture evolution medium in the three principal stress directions is initialized to zero: (1.13) In the formula, c,pi,t For the fracture evolution medium at the stiffness inflection point, in the first... iThe cumulative plastic strain along each principal stress direction. The elastic bearing medium adopts an elastic constitutive relation; the fracture evolution medium undergoes elastoplastic updates starting from the stiffness inflection point based on the loading state of subsequent sampling points.

[0036] 2) Update the volume fraction of the fracture evolution medium For the first stiffness inflection point after the first... k Based on the relative positions of the total strain in the direction of maximum principal stress between the stiffness inflection point and the end of the calculation interval, normalized evolution variables are constructed for each sampling point: (1.14) In the formula, x k For the first k Normalized evolution variables for each sampling point 1,k For the first k Total strain in the direction of maximum principal stress at each sampling point 1,t The total strain in the direction of the maximum principal stress at the stiffness inflection point. 1,e To calculate the total strain in the direction of the maximum principal stress at the end of the interval, t and e These represent the sampling points corresponding to the stiffness inflection point and the end point of the calculation interval, respectively. x (,0,1) indicates that the variable x The range is limited to 0 to 1. When the variable... x When < 0, take the value 0; when 0 ≤ 0, take the value 0. x When ≤1, take x When the variable x When the value is greater than 1, take 1.

[0037] Calculate the first based on normalized evolutionary variables k Candidate values ​​for the volume fraction of fracture evolution medium at each sampling point: (1.15) In the formula, c k These are candidate values ​​for the volume fraction of the fracture evolution medium, calculated based on the current total strain. c t This represents the initial volume fraction of the fracture evolution medium at the stiffness inflection point. c e To calculate the volume fraction of the fracture evolution medium at the end of the interval, a γ and b γ Both are positive evolution parameters, and together they control the rate of change and curve shape of the fracture evolution medium volume fraction as a function of the total strain in the direction of maximum principal stress. For the first k Normalized evolution variables at each sampling point x k of b γ Power of 1.

[0038] To ensure that the volume fraction of the fracture evolution medium is irreversible during monotonic loading, it is updated according to the following formula: (1.16) In the formula, For the first k Updated fracture evolution medium volume fraction for each sampling point The volume fraction of the fracture evolution medium at the previous sampling point, max( c k-1 , c k () indicates taking the larger of the two values ​​to ensure c k ≥ c k-1 min[1,·] indicates that the upper limit of the volume fraction is limited to 1, therefore 0 ≤ c k ≤1 and c k ≥ c k-1 . c t , c e , a γ and b γ For predetermined material parameters, calibration can be performed based on the macroscopic stress-strain response under the corresponding true triaxial test conditions.

[0039] 3) Determine the three-dimensional response of the two types of media. The elastic bearing medium adopts a three-dimensional elastic constitutive relation, according to the first... k Calculate the triaxial local strain increment at each sampling point, and then calculate the triaxial local stress increment. (1.17) in, ; In the formula, ΔΣ oh,k For the first k The elastic bearing medium at the sampling point is in the... i The local stress increment Δ along each principal stress direction oh,k and Δ ej,k The elastic bearing medium is respectively in the firsti The direction of the first principal stress and the first principal stress direction j Local strain increments along the principal stress directions, D e is the elastic stiffness coefficient of strain in the same direction with respect to local stress. L e For the strain pairs in the other two principal stress directions i The coupling stiffness coefficient of local stress in each principal stress direction. K e and G e These are the bulk modulus and shear modulus of the elastic bearing medium, respectively. j ≠ i This indicates summation over the other two principal stress directions.

[0040] The elastic bearing medium in the first k The local stress at each sampling point is: (1.18) In the formula, Σ oh,k -1 and Σ oh,k The first k -1 and the first k The elastic bearing medium at the sampling point is in the... i Local stress in the direction of principal stress.

[0041] The local total strain increment in the fracture evolution medium consists of the elastic strain increment and the plastic strain increment: (1.19) In the formula, Δ ci,k Δ represents the local total strain increment in the fracture evolution medium. c,ei,k and Δ c,pi,k These represent the local elastic strain increment and the local plastic strain increment, respectively, with superscripts indicating the local elastic strain increment and the local plastic strain increment. c Indicates the fracture evolution medium, superscript e and p These represent the elastic and plastic parts, respectively.

[0042] First, assuming that the entire loading increment is an elastic strain increment, calculate the triaxial trial local stress in the fracture evolution medium: (1.20) in, ; In the formula, Σ c,tr i,k For the first k The fracture evolution medium at each sampling point in the first... iThe trial local stresses along the principal stress directions, with the superscript tr indicating the elastic trial state, Σ ci,k -1 represents the local stress in the fracture evolution medium at the previous sampling point; D c and L c These are the same-direction stiffness coefficient and the triaxial coupled stiffness coefficient of the fracture evolution medium, respectively. c and G c These are the bulk modulus and shear modulus of the fracture evolution medium, respectively.

[0043] The average stress of the fracture evolution medium was calculated based on the trial local stress. p tr k and equivalent shear stress q tr k : (1.21) (1.22) Using pressure-dependent yield function and non-associated plastic potential function: (1.23) In the formula, F tr k For the first k Trial yield function values ​​for each sampling point G k Let be the plastic potential function. p This represents the average stress corresponding to the local stress in the fracture evolution medium. q Its equivalent shear stress, α To control the friction parameters related to yield surface pressure, β To control the shear dilatation parameters in the direction of plastic strain increment, k c ( k ) represents the fracture evolution medium as a function of internal variables k Varying yield strength, k As an intrinsic variable characterizing the degree of hardening or softening of the fracture evolution medium, k k-1 The intrinsic variable value of the previous sampling point. For fracture evolution media with internal variables Varying yield strength. When using the non-associated flow rule, α and β They are independent of each other.

[0044] when F tr k When the stress is ≤0, the calculated stress does not exceed the yield surface, and the fracture evolution medium is treated as an elastic state: Δ c,p i,k =0,Σ ci,k =Σ c,tr i,k ,when F tr k When the stress is greater than 0, the calculated stress exceeds the yield surface, and the fracture evolution medium enters the plastic state. The triaxial plastic strain increment is determined based on the plastic potential function. ， In the formula, Σ ci,k To deduct the increment of plastic strain, the first k The fracture evolution medium at each sampling point in the first... i Corrected local stress in the direction of each principal stress; Δ l k For the first k The plastic multiplier increment at each sampling point is used to control the development range of plastic strain within this loading increment; G k / Σ c i,k For the plastic potential function with respect to the first i The partial derivatives of the local stresses in each principal stress direction are used to determine the distribution ratio of the plastic strain increment in the three principal stress directions.

[0045] After deducting the plastic strain increment, the triaxial local stress is corrected according to the elastic relationship: (1.24) In the formula, , is the local stress correction factor caused by the increase in plastic strain in the same direction of the fracture evolution medium. , which is the local stress coupling correction factor caused by the plastic strain increment in the other principal stress directions. K c and G c These are the bulk modulus and shear modulus of the fracture evolution medium, respectively.

[0046] Plastic multiplier increment Δ l k Determined based on the modified local stress satisfying the yield uniformity condition: (1.25) In the formula, For the first k The yield function values ​​corresponding to the local stress state after correction of the fracture evolution medium at each sampling point. To control the friction parameters related to yield surface pressure, p k and q kThese represent the mean stress and equivalent shear stress corresponding to the corrected triaxial local stress, respectively. For the first k The updated intrinsic variables after each sampling point ( k c ) represents the internal variables of the fracture evolution medium. The current yield strength under action. From this, the [number]th yield strength is obtained. k The triaxial local stress, triaxial elastic strain increment, and triaxial plastic strain increment of the fracture evolution medium at each sampling point.

[0047] 4) Calculate the three-dimensional strain condenser tensor The fracture evolution region is equivalent to an ellipsoidal region, and the 3×3 normal component matrix S is determined based on its shape, orientation, the stiffness of the two types of media, and the current plastic state. ell k And calculate: (1.26) (1.27) In the formula, I is a 3×3 identity matrix, and S ell k For the first k The normal component matrix of the ellipsoidal fracture evolution region at each sampling point is used to reflect the influence of its shape and orientation on triaxial deformation, C e C is the normal stiffness matrix of the elastic bearing medium. c ep,k For the first k Normal elastic-plastic stiffness matrix of the fracture evolution medium at each sampling point Let Ad be the volume fraction of the fracture evolution medium, Adk be the strain concentration matrix of the fracture evolution region relative to the elastic bearing medium, and Aek and Ack be the strain concentration matrices of the elastic bearing medium and the fracture evolution medium, respectively.

[0048] The triaxial local strain increments for the two types of media are as follows: (1.28) In the formula, Δ i,k For the first k The sampling point i The total macroscopic strain increment in each principal stress direction , oh, k and Δ ci,k These represent the local total strain increments in this direction for the elastic bearing medium and the fracture evolution medium, respectively.

[0049] Starting from the first sampling point after the stiffness inflection point, using the stress, strain, and plastic state of the previous sampling point as the initial state, the triaxial local strain increment, local stress, and plastic strain increment of the fracture evolution medium are calculated sequentially for both types of media; the updated C2 is used. c ep,k Recalculate the strain set matrix above, and use the updated state for the next sampling point, C c ep,k For the first k The normal elastoplastic stiffness matrix of the fracture evolution medium at each sampling point.

[0050] In this embodiment, the triaxial fracture strain is combined, which is achieved as follows: the triaxial plastic strain of the fracture evolution medium is accumulated and converted into macroscopic fracture propagation strain according to its volume fraction: (1.29) In the formula, To accumulate up to the number k -1 sampling point, fracture evolution medium at the first i Plastic strain Δ in each principal stress direction c,pi,k and c,pi,k The first k The fracture evolution medium at each sampling point in the first... i The increment of plastic strain and the cumulative plastic strain in each principal stress direction. gr i,k For the first k The sampling point at the th sampling point i Macroscopic crack propagation strain in each principal stress direction.

[0051] The triaxial crack strain and volumetric crack strain during the full loading process are as follows: (1.30) (1.31) In the formula, cl i,k The first one determined before the stiffness inflection point i Crack closure strain in each principal stress direction cr i,k For the first part of the full loading process i Crack strain in the principal stress direction cr v,k For volumetric crack strain, t This represents the sampling point number corresponding to the stiffness inflection point. , and The first k Crack strain along the directions of maximum principal stress, intermediate principal stress, and minimum principal stress at each sampling point; superscript c, p , gr , cl and cr These represent the fracture evolution medium, plastic strain, fracture propagation strain, fracture closure strain, and fracture strain, respectively.

[0052] The following is in conjunction with the appendix Figure 1-Figure 7 The embodiments of the present invention will be further described below.

[0053] The following describes this method using a true triaxial compression test of dolomite under the condition of intermediate principal stress Σ2=5MPa and minimum principal stress Σ3=0MPa. The specimen size is 50×50×100mm, and the compressive stress is positive.

[0054] Data Acquisition and Preprocessing: Maintaining the intermediate principal stress Σ2 and minimum principal stress Σ3 constant, the maximum principal stress Σ1 is monotonically increased until the post-peak softening stage. The maximum principal stress Σ1, intermediate principal stress Σ2, minimum principal stress Σ3, and triaxial total strains Ē1, Ē2, and Ē3 are simultaneously acquired. Using the formal loading start point as a reference, the triaxial total strain is zeroed, and the effective loading segment containing both pre-peak and post-peak data with a monotonically increasing overall strain of Ē1 is extracted. The triaxial stress-strain curve for this condition is shown below. Figure 2 .

[0055] Stiffness inflection point identification: Local tangential stiffness is calculated according to equation (1.1). E t,k In this embodiment, for each calculation point, three adjacent sampling points are selected on each side, resulting in a local slope fitted using seven sampling points. The median of the highest 10% of the positive stiffness before the peak is taken as the reference stiffness. E H At least 5 consecutive sampling points meet the requirements. E t,k ≥0.90 E H And relative fluctuation range R E When the stiffness is no greater than 5%, it is considered to be in the stable stiffness stage. E t,k For the first k Local tangent stiffness at each sampling point.

[0056] After the stable stiffness phase ends, a decrease is determined point by point using six consecutive sampling points; when at least four out of five adjacent stiffness changes satisfy the decrease condition, and the cumulative stiffness decrease is D... k When the stiffness is not less than 5%, the earliest point that meets the condition is determined as the stiffness inflection point. t . Figure 3 The location of the stable stiffness stage and the inflection point is shown. This embodiment yields... 1,t =0.589303%, Σ 1,t=104.12 MPa, where, 1,t Σ represents the maximum principal strain at the stiffness inflection point. 1,t This represents the maximum principal stress at the stiffness inflection point.

[0057] Determining the crack closure strain before the inflection point: From the start of the effective loading segment to the inflection point, calculate the normalized loading variable, crack closure variables in each direction, and triaxial crack closure strain according to equations (1.7) to (1.11); at the inflection point, let... x ci,t =1, x ci,t The first point at the stiffness inflection point i The crack closure variables are defined for each principal stress direction, and no further crack closure strains are added thereafter. The transition states and closure parameters used are shown in Table 1. Figure 4 The evolution of the three-dimensional crack closure variables and the corresponding crack closure strain before the inflection point with the maximum principal strain is given. The three-dimensional crack closure variables all reach 1 at the inflection point. Based on this, the crack closure strain in the three principal stress directions before the inflection point is determined, as shown in Table 1. Table 1 is a table of stiffness transition state and crack closure parameters.

[0058] Table 1

[0059] The strain for crack propagation after the inflection point is determined as follows: k = t The rock is divided into an elastic bearing medium and a fracture evolution medium, and the sampling points after the turning point are calculated point by point according to equations (1.14) to (1.28). The local strain is determined based on the current volume fraction of the fracture evolution medium and the strain concentration relationship between the two types of media. When the fracture evolution medium meets the yield condition, the plastic strain in its three principal stress directions is updated by non-associated plastic flow, and the three-dimensional fracture propagation strain is obtained from this.

[0060] Volume fraction of fracture evolution medium c k The calculation adopts a pre-peak to post-peak segmented evolution method: from the stiffness inflection point to the peak point, the pre-peak parameters are used. a γ,pre and b γ,pre And make the volume fraction at the peak point reach c p =0.445, c p The volume fraction of the fracture evolution medium at the peak point is used; after the peak point, the post-peak parameter is used. a γ,post and b γ,post ,by ce =0.720 was used as the volume fraction at the end of the post-peak calculation. Post-peak evolution was based on the peak value. c p The initial value is used, and in the point-by-point calculation, the larger value between the current candidate value and the volume fraction of the previous sampling point is taken according to formula (1.16) to ensure the volume fraction of the fracture evolution medium. c k Continuous and irreversible. The yield strength of the fracture evolution medium is determined by... k c ( k k )=max[ k c,res , k c0 +H c k k Update, in the formula, k c0 , k c,res and H c These are the initial yield strength, residual yield strength, and strength evolution parameters, respectively. k k For the first k The updated intrinsic variables after each sampling point k c ( k k ) represents the current yield strength of the fracture evolution medium under the action of internal variables. The main calculation parameters are shown in Table 2. Table 2 is the calculation parameter table after the inflection point. Figure 5 The segmented evolution results of the fracture evolution medium volume fraction after the inflection point are presented. It can be seen that after the inflection point, the fracture evolution medium volume fraction continuously increases with loading. After the inflection point, at each sampling point, the strain concentration relationship between the two types of media is updated according to the current volume fraction, and the triaxial fracture propagation strain at that sampling point is obtained by updating the yield judgment and plastic strain of the fracture evolution medium. Then, the calculation proceeds to the next sampling point until the loading endpoint.

[0061] Table 2

[0062] The triaxial fracture strain merging and result explanation are as follows: According to equation (1.29), the cumulative plastic strain in the three principal stress directions of the fracture evolution medium after the inflection point is multiplied by the current volume fraction to obtain the macroscopic fracture propagation strain in the three directions; then, according to equation (1.30), the triaxial fracture closure strain determined by the initial defect closure before the inflection point is segmented and merged to obtain the triaxial fracture strain of the entire loading process. The volumetric fracture strain is obtained by summing the fracture strains in the three directions.

[0063] To verify the rationality of the parameters of the two types of media and the triaxial strain distribution relationship after the inflection point, the calculated macroscopic stress-strain response was compared with the experimental results. Figure 6 A comparison of experimental and calculated curves for the three principal stress directions under Σ2=5MPa and Σ3=0MPa conditions is presented. The results show that the same set of parameters can jointly describe the macroscopic stress-strain response in the three principal stress directions, providing a computational basis for further extraction of triaxial fracture strain.

[0064] Figure 7 The evolution of the three principal stress directions and volumetric fracture strain under this working condition is presented. Before the stiffness inflection point, the fracture strain mainly reflects the deformation caused by the gradual closure of initial defects inside the rock. After the stiffness inflection point, with the increase of the volume fraction of the fracture evolution medium and the accumulation of plastic strain, the degree of fracture propagation in the three principal stress directions gradually differs, with the third-direction fracture propagation without minimum principal stress constraint being more significant. This shows that the present invention can not only obtain the traditional volumetric fracture strain, but also determine the fracture strain in the three principal stress directions separately, thereby characterizing the directional differences in fracture deformation under true triaxial stress conditions.

Claims

1. A method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions, characterized in that, Includes the following steps: The triaxial stress and total triaxial strain of the rock sample were obtained during the true triaxial loading process. The obtained data were synchronized, zero-point corrected, and the effective loading segment with the maximum principal strain increasing monotonically was extracted. Based on the effective loading segment, the stable stiffness stage is determined by the local tangent stiffness of the maximum principal stress-maximum principal strain curve, and the starting position of the continuous decrease in stiffness after the stable stiffness stage is determined as the stiffness inflection point. Before the stiffness inflection point, the crack closure strain in the three principal stress directions is calculated based on the normalized loading variables, crack closure variables in each direction, and the initial defect additional compliance. After the stiffness inflection point, the representative volumetric unit of the rock microstructure is divided into elastic bearing medium and fracture evolution medium. Based on the volume fraction of fracture evolution medium, the triaxial strain concentration relationship of the two types of medium and the triaxial cumulative plastic strain of fracture evolution medium, the macroscopic fracture propagation strain in the three principal stress directions is calculated respectively. Among them, the two types of medium are elastic bearing medium and fracture evolution medium. The crack closure strain before the stiffness inflection point and the macroscopic crack propagation strain after the stiffness inflection point are segmented and merged to obtain the triaxial crack strain during the entire loading process, and the volumetric crack strain is determined by the sum of the triaxial crack strains.

2. The method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions according to claim 1, characterized in that, The determination of the stiffness inflection point includes: Based on the effective loading segment, the statistical value of the highest stiffness data before the peak is used as the stiffness reference value. The interval in which the local tangential stiffness reaches the preset proportion of the reference value and remains stable in multiple consecutive sampling points is determined as the stable stiffness stage. After the stable stiffness stage, when the local tangential stiffness of multiple consecutive sampling points decreases as a whole and the cumulative decrease reaches a preset threshold, the position that first meets the condition is determined as the stiffness inflection point.

3. The method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions according to claim 1, characterized in that, The calculation of the crack closure strain in the three principal stress directions includes: Before the stiffness inflection point, a normalized loading variable is constructed based on the total strain in the direction of the maximum principal stress corresponding to the start point of the effective loading segment and the stiffness inflection point. The triaxial fracture closure variables are determined based on the normalized loading variables and the fracture closure parameters in each direction; The initial defect additional compliance in each direction is updated using the triaxial crack closure variable, and the crack closure strain is accumulated point by point to obtain the crack closure strain in the three principal stress directions. No new crack closure strain is added after the stiffness inflection point.

4. The method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions according to claim 1, characterized in that, The calculation of macroscopic crack propagation strain in the three principal stress directions includes: After the stiffness inflection point, the representative volumetric units of the rock microstructure are divided into elastic bearing medium and fracture evolution medium. Initialize the two types of medium states at the stiffness inflection point and irreversibly update the volume fraction of the fracture evolution medium during loading; The local strain of the two types of media is calculated based on the mechanical parameters of the two types of media, the ellipsoidal shape of the fracture evolution region, and the triaxial strain condensation matrix containing off-diagonal terms. Based on the triaxial strain concentration relationship of the two types of media, the triaxial cumulative plastic strain of the fracture evolution medium is calculated according to the pressure-related yield condition and non-associated plastic flow, and multiplied with the current volume fraction of the fracture evolution medium to obtain the triaxial macroscopic fracture propagation strain.

5. The method for identifying rock stiffness inflection points and determining triaxial fracture strain under true triaxial stress conditions according to claim 1, characterized in that, The determination of volumetric crack strain includes: The corresponding cumulative crack closure strain is taken before and at the stiffness inflection point; After the stiffness inflection point, the crack strain in each principal stress direction is taken as the sum of the cumulative crack closure strain in the corresponding direction at the stiffness inflection point and the current macroscopic crack propagation strain in the corresponding direction. The cumulative crack closure strain and the sum of strains are combined in segments to obtain the triaxial crack strain during the entire loading process; The volumetric fracture strain is determined by the sum of the triaxial fracture strains.

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