Unified description method and system of nonlinear characteristics of surrounding rock considering intermediate principal stress

Abstract

The application provides an elastoplastic unified description method and system considering the nonlinear characteristics of surrounding rock of intermediate principal stress, and belongs to the technical field of computer-aided design. Through the true triaxial mechanical property test of surrounding rock, the full stress-strain curve of surrounding rock is obtained, the characteristic stress points of the stress-strain curve are extracted, and the stress-strain curve is divided into stages; based on the characteristic stress points, the characteristic yield surface in the stress space is constructed; based on the stage characteristics of the stress-strain curve, the plastic deformation capacity representation functions of different stages are respectively constructed, and the plastic deformation capacity representation functions are used for describing the plastic deformation capacity of the stress-strain curve; and the nonlinear hardening-softening equation is constructed to describe the evolution characteristics of the characteristic yield surface. The scheme can more comprehensively and accurately describe the nonlinear characteristics of the stress-strain curve under different minimum principal stresses and intermediate principal stresses.

Description

Technical Field

[0001] This application relates to the field of computer-aided design technology, and in particular to a unified elastoplastic description method and system for the nonlinear characteristics of surrounding rock considering intermediate principal stress. Background Technology

[0002] Laboratory tests are an important means of determining the basic physical properties of surrounding rocks, among which conventional triaxial compression and true triaxial compression tests are the most common. Currently, many conventional triaxial compression and true triaxial compression tests have been conducted, and the nonlinear characteristics of the specimens have been analyzed. However, accurate characterization of the strain hardening and strain softening nonlinearities of surrounding rocks under different stress states has not been achieved, and a unified method for describing the nonlinearity of surrounding rock strength and the nonlinearity of stress-strain curves is lacking.

[0003] Therefore, a unified method is needed to accurately describe these nonlinear mechanical behaviors. Summary of the Invention

[0004] The purpose of this application is to provide a unified elastoplastic description method and system for the nonlinear characteristics of surrounding rock considering the intermediate principal stress, so as to solve or alleviate the problems existing in the prior art.

[0005] To achieve the above objectives, this application provides the following technical solution:

[0006] Firstly, this application provides a unified elastoplastic description method for the nonlinear properties of surrounding rock considering intermediate principal stress, including:

[0007] Mechanical property tests were conducted on the surrounding rock to obtain the full stress-strain curve of the surrounding rock;

[0008] Characteristic stress points are extracted from the stress-strain curve, and the stress-strain curve is divided into stages to obtain multiple stages, including: elastic stage, strain hardening stage, strain softening stage, and residual stage.

[0009] Based on the characteristic stress points, the corresponding characteristic yield surfaces are plotted in the stress space. The characteristic yield surfaces include: the initial yield surface, the peak strength surface, and the residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength.

[0010] Based on the stage-specific characteristics of the stress-strain curve, plastic deformation capacity characterization functions for different stages are constructed. These plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve.

[0011] A nonlinear hardening-softening equation is constructed to describe the evolution characteristics of the characteristic yield surface.

[0012] In conjunction with the first aspect, in one possible implementation, the plastic deformation capacity characterization function is used to characterize the relationship between the plastic shear strain and the minimum principal stress at each characteristic yield surface, or the plastic deformation capacity characterization function is used to characterize the relationship between the plastic shear strain and the intermediate principal stress and the minimum principal stress at each characteristic yield surface.

[0013] In conjunction with the first aspect, in one possible implementation, the evolution of multiple mechanical parameters is used to characterize the evolution of the characteristic yield surface, and the hardening-softening equation is used to describe the evolution relationship of the mechanical parameters with plastic shear strain at each characteristic yield surface.

[0014] In conjunction with the first aspect, the elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the intermediate principal stress further includes:

[0015] The elastic modulus and Poisson's ratio are used to describe the deformation characteristics of the elastic stage of the stress-strain curve; the expansion parameter is used to describe the deformation characteristics of the plastic stage of the stress-strain curve.

[0016] The elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the principal stress provided in this embodiment has the following beneficial effects:

[0017] 1. Traditional descriptive methods usually only focus on the peak strength surface, while the unified elastoplastic descriptive method provided in this application introduces three characteristic yield surfaces: the initial yield surface, the peak strength surface, and the residual strength surface. These three yield surfaces jointly describe the nonlinear strength characteristics of the surrounding rock at different stages under conventional triaxial compression and true triaxial compression tests, thus enabling a more comprehensive description of the strength change process of the surrounding rock from the elastic stage to the plastic stage and then to the residual stage.

[0018] 2. Traditional methods often neglect the variation of plastic deformation capacity with confining pressure or intermediate principal stress. However, the unified elastoplastic description method provided in this application introduces a plastic deformation capacity characterization function, which describes the plastic deformation capacity in the strain hardening and strain softening stages through the plastic shear strain at the characteristic yield surface. This can more accurately characterize the plastic deformation behavior of the surrounding rock under different stress states and realize the brittle-ductile transformation of the stress-strain curve.

[0019] 3. Traditional methods typically use linear or piecewise linear methods to describe stress-strain curves. The elastoplastic unified description method provided in this application introduces nonlinear hardening and softening equations, and describes the evolution characteristics of the yield surface through the change of plastic shear strain, which can more realistically reproduce the nonlinear characteristics of the stress-strain curve of the surrounding rock in the plastic stage.

[0020] In summary, the elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the principal stress provided in this embodiment can achieve the unification of strain hardening-softening, conventional triaxial and true triaxial, and brittle-ductile transformation, thereby more accurately describing the nonlinear characteristics of surrounding rock.

[0021] Secondly, this embodiment provides a constitutive model construction method. The constitutive model is based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering the principal stress provided in any of the above embodiments, including: the constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface;

[0022] The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function.

[0023] The constitutive model adopts The parameters s, a, and s are combined to control the movement of the characteristic yield surface, and hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, thus obtaining a nonlinear hardening-softening equation; where, is the decrease in the material constant of the intact rock, where s and a are both material constants of the rock mass;

[0024] The constitutive model employs a non-associated flow rule to capture the expansion characteristics of the surrounding rock. In conjunction with the second aspect, in one possible implementation, the constitutive model is solved using a numerical algorithm based on the non-convex and non-smooth GZZ criterion, including the following steps:

[0025] Obtain the expression for the GZZ strength criterion in the stress invariant space;

[0026] Based on the expression of the GZZ strength criterion in the stress invariant space, write the equations for each characteristic yield surface;

[0027] At the current stress level and At that point, stress invariant space The nonlinear GZZ criterion uses a linear tangent approximation and employs this linear tangent as the equivalent yield surface equation, where... This represents the second stress deviator invariant. For Lode angle, This represents the average stress.

[0028] In conjunction with the second aspect, in one possible implementation, the expression for the plastic deformation capacity characterization function is as follows:

[0029] ,

[0030] ,

[0031] In the formula, , They are Plastic shear strain at the lower peak strength surface and the residual strength surface; and These are the plastic shear strains at the peak strength surface and the residual strength surface, respectively; and These are the minimum principal stresses. The correlation coefficient of principal stress in the plastic shear strain at peak strength and residual strength when the value is greater than 0; and It is the minimum principal stress The correlation coefficient of principal stress in plastic shear strain at peak strength and residual strength when the stress is equal to 0. It is the middle principal stress. It is the minimum principal stress; Below, the plastic deformation capacity characterization function degenerates into:

[0032] ,

[0033] ,

[0034] In the formula, and They are The correlation coefficient of confining pressure at the lower peak strength and residual strength; and They are Plastic shear strain at the lower peak strength and residual strength. In conjunction with the second aspect, in one possible implementation, the nonlinear hardening-softening equation is specifically:

[0035] During the strain hardening stage, the characteristic yield surface evolves from the initial yield surface to the peak strength surface, as expressed below:

[0036] ,

[0037] ,

[0038] ,

[0039] During the strain softening stage, the characteristic yield surface evolves from the peak strength surface to the residual strength surface, as expressed below:

[0040] ,

[0041] ,

[0042] ,

[0043] In the formula, The subscript represents the decrease in the material constant of the intact rock mass, where s and a are both material constants of the rock mass. Indicates the initial yield surface, subscript Indicates peak intensity surface, subscript Represents the residual strength surface. and These are the normalized plastic shear strains at the pre-peak and post-peak stages under true triaxial conditions, respectively. These are hardening parameters. The parameters are softened. The constitutive model construction method provided in this embodiment considers the influence of the intermediate principal stress on each stage of the stress-strain curve, and can more comprehensively and accurately describe the mechanical behavior of the surrounding rock under different stress states.

[0044] Thirdly, this embodiment provides a unified elastoplastic description system for the nonlinear characteristics of surrounding rock considering intermediate principal stress, including:

[0045] The test unit is configured to conduct mechanical property tests on the surrounding rock and obtain the full stress-strain curve of the surrounding rock;

[0046] The feature extraction unit is configured to extract feature stress points from the stress-strain curve and divide the stress-strain curve into multiple stages, including: elastic stage, strain hardening stage, strain softening stage and residual stage.

[0047] The drawing unit is configured to draw the corresponding characteristic yield surface in the stress space based on the characteristic stress point. The characteristic yield surface includes: initial yield surface, peak strength surface, and residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength.

[0048] The first building unit is configured to construct plastic deformation capacity characterization functions for different stages based on the stage characteristics of the stress-strain curve. The plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve. The second building unit is configured to construct a nonlinear hardening-softening equation to describe the evolution characteristics of the characteristic yield surface.

[0049] Fourthly, this embodiment provides a constitutive model construction system, which is based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering the principal stress provided in any of the above embodiments, including:

[0050] The constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface;

[0051] The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function.

[0052] The constitutive model adopts , , The parameter combination is used to control the movement of the characteristic yield surface, and the hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, so as to obtain the nonlinear hardening-softening equation; where, This represents the decrease in the material constant of the intact rock. and All are material constants of the rock mass;

[0053] The constitutive model uses the non-associated flow rule to capture the expansion characteristics of the surrounding rock. Attached Figure Description

[0054] Figure 1 This is a flowchart illustrating a unified elastoplastic description method for considering the nonlinear characteristics of surrounding rock under principal stress, according to some embodiments of this application.

[0055] Figure 2 These are typical stress-strain curves, where (a) is under conventional triaxial conditions and (b) is under true triaxial conditions.

[0056] Figure 3 This is a schematic diagram of the surrounding rock loading process, where (a) is... (a) Stress-strain curves under the given conditions, (b) Schematic diagram of crack development, (c) Effect of confining pressure (under conventional triaxial conditions) or intermediate principal stress (under true triaxial conditions) on strength.

[0057] Figure 4 This is a schematic diagram showing the influence of plastic deformation capacity on the nonlinear stress-strain curve at different stages, where (a) represents the strain hardening stage and (b) represents the strain softening stage.

[0058] Figure 5 This is a schematic diagram of the evolution characteristics of the yield surface under different plastic deformation capacities, where (a) represents the strain hardening stage and (b) represents the strain softening stage.

[0059] Figure 6 This is a schematic diagram illustrating the nonlinear description of the stress-strain curve of the surrounding rock.

[0060] Figure 7 For GZZ standards in Schematic diagram of the shape in space, where (a) is the shape of the GZZ criterion in the stress invariant space, and (b) is the shape of the GZZ criterion in the stress invariant space. middle The linear equivalent tangent criterion at the location.

[0061] Figure 8 Hardening parameters ( ) and softening parameters ( A schematic diagram illustrating the effect of stress-strain curves.

[0062] Figure 9 This is a schematic diagram of the calculation process for the UHS model in FLAC3D.

[0063] Figure 10 This is a schematic diagram of the geometric relationship of the flow law on the π plane. Detailed Implementation

[0064] Example 1:

[0065] This embodiment provides a unified elastoplastic description method for the nonlinear characteristics of surrounding rock considering intermediate principal stress, including:

[0066] Step S101: Conduct mechanical property tests on the surrounding rock to obtain the full stress-strain curve of the surrounding rock.

[0067] The surrounding rock of deeply buried tunnels often exhibits significant strain hardening and strain softening characteristics. The full stress-strain curve of the surrounding rock can be obtained by conducting true triaxial tests on the surrounding rock.

[0068] Under conventional triaxial conditions, stress-strain curves of different rocks under different confining pressures were obtained by conducting conventional triaxial mechanical tests. Specifically, representative surrounding rocks were collected and processed into standard rock specimens according to the recommendations of the International Society for Rock Mechanics. The specimens were polished to ensure that the parallelism and non-uniformity errors were less than 0.02 mm and 0.05 mm, respectively. To ensure their uniformity and homogeneity, specimens with similar size and physical properties were carefully selected. Compression tests under different confining pressures (0 MPa, 10 MPa up to the initial geostress) were conducted using a rock mechanics testing system, such as MTS815. The test process included the following three steps: (1) Preloading, applying an axial pressure of 2 kN to the specimen to stabilize it; (2) Applying the confining pressure at a rate of 0.1 MPa / s to a constant value and maintaining stability; (3) Using an axial displacement control module, loading was performed at a constant loading rate of 0.001 mm / s until the specimen reached the residual failure stage.

[0069] Under true triaxial conditions, stress-strain curves of siltstone under different minimum principal stresses and intermediate principal stresses were obtained by conducting true triaxial mechanical tests. Specifically, representative surrounding rocks were collected and processed into standard specimens according to the recommendations of the International Society for Rock Mechanics. To ensure their uniformity and homogeneity, specimens with similar size and physical properties were carefully selected. The rock specimens were polished, and the dimensional error and perpendicularity error on each side were controlled within 0.01 mm and 0.02 mm, respectively. Compression tests under different minimum principal stresses and intermediate principal stresses were carried out using a rock mechanics testing system, such as a Mogi-type true triaxial rock mechanics testing system. The test process included the following three steps: (1) Hydrostatic pressure loading stage: Hydrostatic pressure was applied at a rate of 0.5 MPa / s. (2) Middle principal stress loading stage: In this stage, the minimum principal stress remains constant, and the middle principal stress and the maximum principal stress increase simultaneously to the middle principal stress level at a loading rate of 0.5 MPa / s; (3) Maximum principal stress loading stage: In this stage, the middle principal stress and the minimum principal stress remain unchanged, and the maximum principal stress continues to increase.

[0070] Step S102: Extract characteristic stress points from the stress-strain curve and divide the stress-strain curve into stages to obtain multiple stages, including: elastic stage, strain hardening stage, strain softening stage and residual stage.

[0071] The characteristic stress is extracted using the strain curve method for illustration, such as... Figure 2 As shown. Figure 2 In the middle (a), the axial stress-axial strain curve, axial stress-radial strain curve, volumetric strain-axial strain curve and crack volumetric strain-axial strain curve are shown. Figure 2 In the middle (b), there are the axial stress-axial strain curves, axial stress-strain curves in the direction of intermediate principal stress, axial stress-strain curves in the direction of minimum principal stress, volumetric strain-axial strain curves, and crack volumetric strain-axial strain curves under true triaxial conditions.

[0072] Specifically, under conventional triaxial conditions, the axial stress-axial strain curves and the axial stress-radial strain curves are obtained directly from extensometer measurements; the volumetric strain is calculated using the following formula:

[0073] (1)

[0074] In the formula, For volumetric strain, This represents the total volume increment. For the total volume, Strain in the direction of maximum principal stress The strain is in the direction of minimum principal stress.

[0075] Volumetric strain of rock during compression deformation It mainly consists of two parts: one part is the elastic volumetric strain of the specimen. The other part consists of volume changes caused by the closure of internal microcracks, the initiation, propagation, and penetration of new cracks. Therefore, the volumetric strain of the crack can be calculated using the following formula:

[0076] (3)

[0077] In the formula, For crack volume strain, For elastic volumetric strain, For elastic modulus, Poisson's ratio, For the maximum principal stress, It represents the minimum principal stress / confining pressure.

[0078] Under true triaxial conditions, the axial stress-axial strain curves, the axial stress-strain curves along the intermediate principal stress direction, and the axial stress-strain curves along the minimum principal stress direction were obtained directly from extensometer measurements. The total volumetric strain of the specimen... It can be calculated using the following formula:

[0079] (3)

[0080] Elastic volumetric strain and crack volumetric strain The calculation formula is as follows:

[0081] (4)

[0082] Characteristic stress points include: crack closure stress point, crack initiation stress point, initial yield stress point, peak stress point, and residual stress point. According to... Figure 2 The characteristics of the curve show that the deviatoric stress-axial strain curve of the specimen can be divided into 6 stages by 5 characteristic stress points:

[0083] Compaction Stage: During this stage, the rock primarily undergoes the closure of primary cracks, and the stress-strain curve exhibits an upward concave growth. This stage typically constitutes a small portion of the stress-strain curve, which is related to the density and shape of the primary cracks in the rock. The characteristic stress corresponding to the endpoint of this stage is the crack closure stress. The crack closure stress on the curve The point where the crack closes is called the crack closure stress point;

[0084] Elastic stage: In this stage, the original fractures in the rock have been completely closed, and the rock can be assumed to be an isotropic homogeneous elastic material. The stress-strain curve is linear, and the initiation stress is... Defined as the characteristic stress at the end of this stage, the crack initiation stress on the curve The point where the stress occurs is called the crack initiation stress point;

[0085] Stable crack growth stage: During this stage, cracks in the rock begin to propagate and some tiny cracks appear parallel to the axial stress. The total volumetric strain increment is less than the elastic volumetric strain increment. The deviatoric stress-axial strain curve still shows approximately linear growth, while the deviatoric stress-radial strain curve deviates slightly from the linear segment.

[0086] Unstable crack propagation stage: Once the stress reaches the unstable crack propagation stress... This marks the entry of rock material into plasticity; therefore, the unstable crack propagation stress is often defined as the initial yield stress of the rock in elastoplastic mechanics, and the unstable crack propagation stage is also known as the strain hardening stage. During this stage, the stress-strain curve deviates significantly from linearity, and the crack will propagate directionally along a virtual fracture surface. The unstable crack propagation stress on the curve... The point where it is located is called the unstable crack propagation stress point, also known as the initial yield stress point;

[0087] Post-peak stage: Once the stress exceeds the peak stress The stress-strain curve will show a stress drop, and a macroscopic fracture surface will form; this stage is also known as the strain softening stage. The peak stress on the curve... The point where the stress occurs is called the peak stress point;

[0088] Residual stage: As axial strain continues to increase, axial stress will enter a plateau segment, and this plateau stress is called residual stress. .

[0089] In this embodiment, three characteristic stresses (initial yield stress, peak stress, and residual stress) of the plastic stage are extracted from the above five characteristic stresses to prepare data for subsequent drawing of characteristic yield surfaces.

[0090] Step S103: Based on the characteristic stress points, draw the corresponding characteristic yield surfaces in the stress space. The characteristic yield surfaces include: the initial yield surface, the peak strength surface, and the residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength.

[0091] The process of plotting three characteristic yield surfaces can be briefly described as follows: First, key characteristic stress points are extracted from the stress-strain curve, including: initial yield stress, peak stress, and residual stress. The initial yield stress corresponds to the starting point of the unstable crack propagation stage, marking the transition of the rock from the elastic stage to the plastic stage; the peak stress is the maximum value on the stress-strain curve, representing the maximum bearing capacity of the rock; and the residual stress is the stress value of the plateau segment of the stress-strain curve, representing the final stable state of the rock after failure. Second, the characteristic stress points are mapped to stress space, that is, characteristic stress points under different confining pressures or intermediate principal stresses are mapped to stress space (e.g., ...). plane or (Plane), forming a characteristic yield surface. For example, conventional triaxial conditions ( = The abscissa in stress space represents the minimum principal stress. The vertical axis represents the maximum principal stress. For each confining pressure condition (i.e., different...) The stress state points corresponding to the initial yield stress, peak stress, and residual stress are plotted on [the map]. On the plane, connecting the initial yield stress point, peak stress point, and residual stress point under all confining pressure conditions forms three characteristic yield surfaces: the initial yield surface—describing the boundary where the rock begins to enter plasticity; the peak strength surface—describing the boundary where the rock reaches its maximum bearing capacity; and the residual strength surface—describing the final state boundary of the rock after failure. Under true triaxial conditions (… For each confining pressure ( ) and intermediate principal stress ( The combination of initial yield stress, peak stress, and residual stress is plotted on [the map]. On the plane, all stress state points are then connected to form three characteristic yield surfaces.

[0092] Specifically, such as Figure 3 The figure shows the effect of confining compression under conventional triaxial conditions and the middle principal stress under true triaxial conditions on strength. It should be noted that... Figure 3 In (c), the horizontal axis corresponds to the minimum principal stress under conventional triaxial conditions and true triaxial conditions, respectively. and central principal stress In a conventional three-axis configuration, with Figure 3 (a) Taking the stress-strain curve under certain conditions as an example, by combining the initial yield stress point, peak stress point, and residual stress point of the stress-strain curve, we can obtain... The three corresponding stress state points. Similarly, according to and The stress-strain curves under the given conditions are obtained, and the stress state points corresponding to the three characteristic stresses on each curve are obtained. Connecting the stress state points of the initial yield stress, peak stress, and residual stress respectively yields three characteristic yield surfaces, namely the initial yield surface [0], the peak strength surface [1], and the residual strength surface [2]. Under true triaxial conditions, according to a certain Below, different The stress-strain curve can also be obtained under the following conditions. The three characteristic yield surfaces in stress space will not be elaborated upon here.

[0093] in, , The preset minimum principal stress value, , The values ​​can be different.

[0094] In this embodiment, the characteristic yield surface can be used to uniformly describe the nonlinear mechanical behavior of rocks and serve as the basis for an elastoplastic constitutive model. For example, the characteristic yield surface can be used to determine the state of the surrounding rock. That is, during the loading process of the surrounding rock, the characteristic yield surface can be used to determine the stage of the surrounding rock. Specifically: the initial yield surface can be used to determine whether the rock has entered the plastic stage; the peak strength surface can be used to determine whether the rock has entered the post-peak softening stage; and the residual strength surface can be used to describe the final state of the rock after failure.

[0095] As loading progresses, when the surrounding rock is in the pre-peak stage and the stress exceeds the initial yield stress point, the stress state of the surrounding rock lies between the initial yield surface and the peak strength surface, indicating that the surrounding rock has entered the plastic stage and is in the strain hardening stage of the stress-strain curve. When the surrounding rock is in the post-peak stage and the stress has not reached the residual stress point, the stress state of the surrounding rock lies between the peak strength surface and the residual strength surface, indicating that the surrounding rock is in the strain softening stage of the stress-strain curve. When the surrounding rock is in the post-peak stage and the stress is the residual stress, the stress state of the surrounding rock lies on the residual strength surface, indicating that the surrounding rock is in the residual stage of the stress-strain curve.

[0096] In other words, based on the completed characteristic yield surface drawing, the initial yield surface (such as...) Figure 3 (c) The yield surface labeled [0] is used to determine whether the rock has entered the plastic stage; the peak strength surface (such as Figure 3 The yield surface (c) labeled [1] can be used to determine whether the rock has entered the post-peak softening stage (strain softening stage); the residual strength surface (such as Figure 3 The yield surface labeled [2] in (c) represents the final failure state of the rock. These three yield surfaces are used together to describe the nonlinearity of rock strength.

[0097] Step S104: Based on the stage characteristics of the stress-strain curve, construct plastic deformation capacity characterization functions for different stages. The plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve.

[0098] To accurately characterize the plastic deformation capacity of rock during the plastic stage, the plastic shear strain, which varies with confining pressure or intermediate principal stress, is used to describe the plastic deformation capacity of the stress-strain curve. That is, the plastic deformation capacity characterization function is used to characterize the plastic shear strain and minimum principal stress at each characteristic yield surface. The relationship between the plastic shear strain and the principal stress at each characteristic yield surface, or the plastic deformation capacity characterization function, is used to characterize the plastic shear strain and the principal stress at each characteristic yield surface. Minimum principal stress The relationship between them. The latter is the plastic deformation capacity characterization function under true triaxial conditions, while the other is the plastic deformation capacity characterization function under conventional triaxial conditions. The plastic deformation capacity characterization function under true triaxial conditions can be degenerated into the plastic deformation capacity characterization function under conventional triaxial conditions, thereby achieving a unified description of the plastic deformation capacity characterization function.

[0099] Figure 4 The influence of plastic deformation capacity at different stages on the nonlinearity of the stress-strain curve is shown, where (a) is the strain hardening stage and (b) is the strain softening stage. This represents the plastic shear strain at the peak stress. This represents the plastic shear strain at the residual stress point.

[0100] like Figure 4 As shown in (a), it is assumed that the initial yield stress, peak stress, and residual stress remain constant under different stress-strain curves, and only the plastic shear strain at the peak stress is increased. The magnitude of the strain at the peak stress point increases, indicating an increase in the proportion of the strain hardening stage. The stress-strain curve in the strain hardening stage shifts downward accordingly, while the strain at the residual stress point remains unchanged. Therefore, the stress-strain curve in the post-peak strain softening stage is steeper, indicating a decrease in the proportion of the strain softening stage.

[0101] like Figure 4 As shown in (b), only the plastic shear strain at the residual stress is increased. The magnitude of the stress-strain value does not change the stress-strain curve during the strain hardening stage, including the initial yield stress point and the peak stress point. However, the stress-strain curve decreases more slowly during the post-peak strain softening stage, indicating an increase in the proportion of the strain softening stage. Therefore, the plastic deformation capacity has a significant impact on the nonlinearity of the stress-strain curve, not only changing the proportion of each stage of the stress-strain curve but also altering the nonlinearity of the stress-strain curve during both the strain hardening and strain softening stages.

[0102] Figure 5 This diagram illustrates the evolution of the yield surface under different plastic deformation capacities, where (a) represents the strain hardening stage, (b) represents the strain softening stage, and Δγ represents the plastic shear strain increment. For plastic shear strain, , These represent the normalized plastic shear strains before and after the peak, respectively. , Let be the plastic shear strains at the peak strength and residual strength, respectively. The plastic deformation capacity of the stress-strain curve is described by the plastic shear strain varying with the confining pressure or intermediate principal stress, such as... Figure 5 As shown, in stress space, and The two coordinate systems correspond to the conventional triaxial and true triaxial cases, respectively. Taking the conventional triaxial system as an example, the dashed lines in the figure represent the plastic shear strain isosurface, the solid lines represent the characteristic yield surface, the red solid lines represent the peak strength surface (labeled [1]), and the green solid lines represent the initial yield surface (labeled [0]). Observation Figure 5 As shown in (a) and (b), during the strain hardening stage, the plastic shear strain at the peak strength surface gradually increases with increasing confining pressure; while during the strain softening stage, the difference in plastic shear strain between the residual strength surface and the peak strength surface gradually decreases with increasing confining pressure. Therefore, for both the strain hardening and strain softening stages, it is necessary to construct plastic deformation capacity characterization functions to describe the plastic shear strain at the characteristic yield surface under conventional triaxial and true triaxial conditions.

[0103] Specifically, the plastic deformation capacity characterization function can be fitted based on experimental data to obtain an expression for the plastic deformation capacity characterization function.

[0104] For example, under conventional triaxial conditions, the plastic shear strain at the peak strength surface and the plastic shear strain at the residual strength surface are established in relation to the minimum principal stress. The functional relationship between them, and the expression for the function characterizing plastic deformation capacity, are as follows:

[0105] , (5)

[0106] Under true triaxial conditions, the expression for the plastic deformation capacity characterization function is as follows:

[0107] , (6)

[0108] In the formula, This represents the plastic shear strain at the i-th characteristic yield surface, where i represents the index of the characteristic yield surface. A value of i = 1 indicates the plastic shear strain at the peak strength surface, and a value of i = 2 indicates the plastic shear strain at the residual strength surface. Indicates confining pressure. Indicates the principal stress.

[0109] In practice, as long as the chosen function form can describe the plastic shear strain at the peak strength surface, the plastic shear strain at the residual strength surface, and the minimum principal stress reflected in the experimental data, it is acceptable. The relationship between the two yield surfaces can be determined using this functional form, which expresses the relationship between the plastic shear strain and the minimum principal stress at the two yield surfaces. The functional relationship between them is described. Similarly, as long as the chosen functional form can describe the plastic shear strain at the peak strength surface, the plastic shear strain at the residual strength surface, and the minimum principal stress reflected in the experimental data, it is acceptable. Middle principal stress If the functional relationship between them is known, other functional forms can be used to describe the experimental data.

[0110] Step S105: Construct nonlinear hardening-softening equations to describe the evolution characteristics of the characteristic yield surface.

[0111] The stress-strain curves during rock loading are simplified, and the stress-axial strain curves of the compaction stage, elastic stage, and stable crack propagation stage are simplified to linear elasticity. In this embodiment, the hardening-softening equation is used to describe the nonlinear characteristics of the stress-strain curves.

[0112] In this embodiment, the evolution of multiple mechanical parameters is used to characterize the evolution of the characteristic yield surface. The hardening-softening equation is used to describe the evolution relationship of the mechanical parameters with plastic shear strain at each characteristic yield surface. For both strain hardening and strain softening stages, it is necessary to construct equations describing the evolution of mechanical parameters with plastic shear strain under conventional triaxial and true triaxial conditions, i.e., to construct hardening-softening equations, the expressions of which are:

[0113] (7)

[0114] In the formula, M is a mechanical parameter. This represents plastic shear strain.

[0115] Taking a conventional three-axis model as an example, Figure 6This is a schematic diagram of the nonlinear description of the stress-strain curve of the surrounding rock. The left and right figures are the characteristic yield surfaces in the stress space. The red figure is the peak strength surface [1], the green figure is the initial yield surface [0], and the purple figure is the residual strength surface [2]. The labels of each characteristic yield surface are [1], [0], and [2]. The three characteristic yield surfaces divide the stress space into different regions, and thus divide the stress-strain curve into four stages: the linear elastic stage, the strain hardening stage, the strain softening stage, and the residual stage. In the above four stages, the linear elastic stage does not accumulate plastic shear strain, and the stress state point is always located below the initial yield surface. Therefore, the linear elastic theory can be used to describe the axial stress-axial strain. After entering the plastic stage, in addition to the plastic deformation capacity, the form of the stress-strain curve is also affected by the hardening-softening equation. The hardening-softening equation is used to describe the evolution characteristics of yield. Figure 6 As shown, as the characteristic yield surface evolves from the initial yield surface to the peak strength surface, the stress-strain curve changes from the stress point of crack instability propagation. Evolution to peak stress point As the yield surface evolves from the peak strength surface to the residual strength surface, the stress-strain curve evolves from the crack peak stress point to the residual stress point. The rock mechanical parameters remain unchanged in the residual stage.

[0116] In some embodiments, the method further includes: using Poisson's ratio to describe the deformation characteristics of the elastic stage for the circumferential strain curve under conventional triaxial conditions or the strain curves under the intermediate principal stress direction and minimum principal stress direction under true triaxial conditions; and using expansion parameters to describe the deformation characteristics of the plastic stage.

[0117] Specifically, for the circumferential strain curve under conventional triaxial conditions or the strain curves under true triaxial conditions along the directions of intermediate principal stress and minimum principal stress, the deformation characteristics can be described by Poisson's ratio in the elastic stage, and by expansion parameters, such as the expansion angle, in the plastic stage.

[0118] In summary, the nonlinear characteristics of rock strength can be described by introducing a characteristic yield surface, and the stress-strain curve can be divided into stages. The nonlinear characteristics of the stress-strain curve can be reproduced by combining the plastic shear strain, hardening / softening equation, expansion angle and elastic parameters generated in the strain hardening and strain softening stages.

[0119] This embodiment provides a unified elastoplastic description method for the nonlinear characteristics of surrounding rock considering principal stress, comprising three elements: "characteristic yield surface," "plastic deformation capacity characterization function," and "hardening-softening equation." First, characteristic stress points are extracted from the stress-strain curve and extended to a characteristic yield surface in stress space, thus describing the nonlinear characteristics of strength. Next, a function characterizing plastic deformation capacity (i.e., the plastic deformation capacity characterization function) is introduced to describe the proportion of strain hardening and strain softening stages in the stress-strain curve. Finally, a nonlinear hardening-softening equation is used to reproduce the stress-strain curve form of the strain hardening and strain softening stages, achieving an accurate and unified characterization of the nonlinearity of strain hardening and strain softening in the surrounding rock. This not only more comprehensively describes the strength change process of the surrounding rock from the elastic stage to the plastic stage and then to the residual stage, but also more accurately characterizes the plastic deformation behavior of the surrounding rock under different stress states, more realistically reproducing the nonlinear characteristics of the stress-strain curve in the plastic stage.

[0120] Based on the same inventive concept, this embodiment provides a unified elastoplastic description system for the nonlinear characteristics of surrounding rock considering intermediate principal stress, the system comprising:

[0121] The test unit is configured to conduct mechanical property tests on the surrounding rock and obtain the full stress-strain curve of the surrounding rock;

[0122] The feature extraction unit is configured to extract feature stress points from the stress-strain curve and divide the stress-strain curve into multiple stages, including: elastic stage, strain hardening stage, strain softening stage and residual stage.

[0123] The drawing unit is configured to draw the corresponding characteristic yield surface in the stress space based on the characteristic stress point. The characteristic yield surface includes: initial yield surface, peak strength surface, and residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength.

[0124] The first building unit is configured to construct plastic deformation capacity characterization functions for different stages based on the stage characteristics of the stress-strain curve. The plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve.

[0125] The second building block is configured to construct a nonlinear hardening-softening equation to describe the evolution characteristics of the characteristic yield surface.

[0126] The elastoplastic unified description system for the nonlinear characteristics of surrounding rock considering the principal stress provided in this embodiment can realize the steps and processes of the elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the principal stress provided in any of the above embodiments, and achieve the same technical effect, which will not be repeated here.

[0127] Example 2:

[0128] This embodiment provides a constitutive model construction method, which is based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering the principal stress provided in any of the foregoing embodiments, including:

[0129] The constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface;

[0130] The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function.

[0131] The constitutive model adopts The parameters s, a, and s are combined to control the movement of the characteristic yield surface, and hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, thus obtaining a nonlinear hardening-softening equation; where, is the decrease in the material constant of the intact rock, where s and a are both material constants of the rock mass;

[0132] The constitutive model uses the non-associated flow rule to capture the expansion characteristics of the surrounding rock.

[0133] The constitutive model construction method provided in this embodiment will be illustrated below using constitutive models constructed under true triaxial stress and conventional triaxial stress.

[0134] Example 1: Construction of constitutive model under true triaxial stress state

[0135] In the study of the nonlinear mechanical behavior of surrounding rock in deep-buried tunnels, considering the influence of the intermediate principal stress is crucial. Specifically, firstly, the increase of intermediate principal stress affects the proportion of the pre-peak and post-peak stages in the stress-strain curve. As the intermediate principal stress increases, the proportion of the pre-peak strain hardening stage of the surrounding rock may increase, while the proportion of the post-peak strain softening stage may decrease. Secondly, intermediate principal stress has a significant impact on the peak strength of the surrounding rock. Generally, as the intermediate principal stress increases, the peak strength of the surrounding rock first increases and then decreases. Thirdly, intermediate principal stress affects the failure mode of the surrounding rock, and it also has a significant impact on the plastic deformation capacity and expansion characteristics of the surrounding rock. Therefore, this embodiment provides a unified strain hardening and strain softening elastoplastic constitutive model of surrounding rock that considers the influence of intermediate principal stress, aiming to more comprehensively and accurately describe the mechanical behavior of surrounding rock under different stress states, and provide a more reliable theoretical basis for the analysis of the disturbance characteristics of deep-buried tunnel excavation. Since the GZZ strength criterion has a non-convex and non-smooth shape in the principal stress space, this embodiment uses GZZ to measure the initial yield strength, peak strength, and residual strength of the surrounding rock, considering the three-dimensional stress condition of the intermediate principal stress. Normalized plastic shear strain is used as the plastic internal variable. The influence of the intermediate principal stress on the plastic shear strain at the characteristic yield surface under different minimum principal stresses is considered. The relationship between the intermediate principal stress and the plastic shear strain at the characteristic yield surface is fitted to accurately describe the evolution characteristics of mechanical parameters with plastic shear strain. The parameters are used... The parameters k1, k2, and k3 are used to control the movement from the initial yield surface to the peak strength surface during the strain hardening stage and from the peak strength surface to the residual strength surface during the strain softening stage. The strain hardening and strain softening stages of the stress-strain curve are accurately described by the hardening parameter k1 and the softening parameter k2. The expansion characteristics of the surrounding rock are captured by the non-associated flow law, thereby forming a constitutive model considering the middle principal stress. This model can better simulate the strain hardening and strain softening characteristics of the surrounding rock under three-dimensional stress state, providing a basis for subsequent excavation disturbance analysis and support design of deep-buried tunnels.

[0136] In this embodiment, the GZZ strength criterion is used as the characteristic yield surface equation to describe the three characteristic yield surfaces under three-dimensional stress: the initial yield surface, the peak strength surface, and the residual strength surface.

[0137] Considering that the shape of the GZZ criterion is non-convex and non-smooth on the π plane, which limits its use in numerical software, this embodiment also provides a numerical implementation algorithm for the non-convex and non-smooth GZZ criterion to solve the proposed constitutive model.

[0138] The GZZ strength criterion in the stress invariant space ( The expression in () is as follows:

[0139] (8)

[0140] In the formula, Represents the average stress. It is the uniaxial compressive strength of intact rock. ; This represents the second stress deviator invariant. ; For Lode angle, .

[0141] Figure 7 For GZZ standards in A schematic diagram of the shape in space, where (a) is the shape of the GZZ criterion in the stress invariant space; (b) is the shape of the GZZ criterion in the stress invariant space. middle The linear equivalent tangent criterion at the location.

[0142] According to the GZZ strength criterion in the stress invariant space ( Based on the expression in (), the equations for each characteristic yield surface are written as follows:

[0143] (9)

[0144] (10)

[0145] (11)

[0146] (12)

[0147] In the formula,

[0148] = (13)

[0149] (14)

[0150] (15)

[0151] In the formula, , , These are the initial yield surface, peak strength surface, and residual strength surface, respectively. The subscripts [0], [1], and [2] represent the initial yield surface, peak strength surface, and residual strength surface, respectively. , , , , , These are parameters a and b, respectively. The values ​​of the initial yield surface, peak strength surface, and residual strength surface, It is the uniaxial compressive strength of intact rock.

[0152] Considering the equation of the initial yield surface in stress space The nonlinearity of the stress in the model leads to iterative calculations during stress updates for each element, especially in FLAC3D software. Since each element requires iterative calculations in every calculation step, this significantly reduces the model's computational efficiency. Therefore, at the current stress level... and At the stress space The nonlinear GZZ criterion is approximated by a linear tangent, and this linear tangent is used as the equivalent yield surface equation, where, This represents the second stress deviator invariant. For Lode angle, This represents the average stress. Therefore, the linear equivalent yield surface equation at the current stress level... for:

[0153] (16)

[0154] (17)

[0155] (18)

[0156] In the formula, , All are simplified parameters. For Lode angle, , , .

[0157] The above calculation process overcomes the shortcomings of the GZZ criterion, which is non-convex and non-smooth in shape on the π plane.

[0158] In some embodiments, under true triaxial conditions, it is necessary to establish the plastic shear strain and minimum principal stress at the peak strength surface and the residual strength surface, respectively. Middle principal stress The functional relationship between them.

[0159] Considering the influence of the intermediate principal stress on the plastic shear strain at the characteristic yield surface under different minimum principal stresses, the following relationship is used to fit the relationship between the intermediate principal stress and the plastic shear strain at the characteristic yield surface. That is, the expression of the plastic deformation capacity characterization function under true triaxial conditions is as follows:

[0160] (19)

[0161] (20)

[0162] In the formula, , They are Plastic shear strain at the lower peak strength surface and the residual strength surface; and They are The correlation coefficient of confining pressure at the lower peak strength and residual strength; and They are Plastic shear strain at the lower peak strength and residual strength; and These are the plastic shear strains at the peak strength surface and the residual strength surface under true triaxial conditions, respectively. and These are the minimum principal stresses. The correlation coefficient of principal stress in the plastic shear strain at peak strength and residual strength when the value is greater than 0; and It is the minimum principal stress The correlation coefficient of principal stress in plastic shear strain at peak strength and residual strength when the stress is equal to 0. It is the principal stress.

[0163] Under these conditions, the true triaxial condition degenerates into the conventional triaxial condition. Based on the conventional triaxial test data, a linear relationship can be used to analyze the plastic shear strain at the peak strength surface and the plastic shear strain at the residual strength surface with respect to the minimum principal stress. By fitting the relationship between them, we obtain the expression for the corresponding plastic deformation capacity characterization function:

[0164] (twenty one)

[0165] (twenty two)

[0166] It should be understood that the above-described form of the plastic deformation capacity characterization function is merely an exemplary description and should not be construed as a limitation on this embodiment.

[0167] The aforementioned plastic deformation capacity characterization function, considering the influence of the intermediate principal stress and the minimum principal stress, uniformly describes the plastic shear strain at peak strength and residual strength. It can effectively capture the plastic deformation capacity varying with the intermediate and minimum principal stresses; that is, with increasing minimum principal stress, the ductility of the stress-strain curve increases; with increasing intermediate principal stress, the brittleness of the stress-strain curve increases. Simultaneously, in Under these conditions, the plastic deformation capacity characterization function can be degenerated into an expression under conventional triaxial conditions, achieving a unified description of plastic shear strain.

[0168] In some embodiments, during the strain hardening stage, the characteristic yield surface evolves from the initial yield surface to the peak strength surface, as expressed below:

[0169] (twenty three)

[0170] (twenty four)

[0171] (25)

[0172] During the strain softening stage, the characteristic yield surface evolves from the peak strength surface to the residual strength surface, as expressed below:

[0173] (26)

[0174] (27)

[0175] (28)

[0176] In the formula, The subscript represents the decrease in the material constant of the intact rock mass, where s and a are both material constants of the rock mass. Indicates the initial yield surface, subscript Indicates peak intensity surface, subscript Represents the residual strength surface. and These are the normalized plastic shear strains at the pre-peak and post-peak stages under true triaxial conditions, respectively. These are hardening parameters. This is a softening parameter.

[0177] The non-associated flow principle is used to capture the swelling characteristics of the surrounding rock. The plastic potential function adopts a form similar to the yield surface equation, introducing a dilatation coefficient to describe the rock's swelling behavior. Specifically, the plastic potential function is as follows:

[0178] (29)

[0179] in, The value ranges from 0 to 1. When the flow rules are in a certain state, they are correlated; otherwise, they are not. In particular, when... At that time, the volumetric strain increment is 0.

[0180] The increment of plastic strain can be calculated using the following formula:

[0181] (30)

[0182] (31)

[0183] In the formula, As a plastic multiplier, Let be the plastic potential function.

[0184] As an example, the numerical implementation process and parameter calibration of this constitutive model are described in detail below.

[0185] According to elastoplastic theory, the total strain increment It can be divided into elastic strain increments and plastic strain increment The expression is as follows:

[0186] (32)

[0187] The stress vector of the current step can be calculated using the following formula:

[0188] (33)

[0189] In the formula, the superscripts N and P represent the stress information of the current step and the previous step, respectively. It is the stress increment:

[0190] (34)

[0191] In the formula, and These are the elastic stress increment and the plastic correction stress, respectively. It is an elasticity matrix.

[0192] Therefore, the calculation of stress increment will be divided into two steps: (1) Elastic conjecture stress (2) Calculation of the stress vector in the current step (Equation 36):

[0193] (35)

[0194] (36)

[0195] exist In space, equation (33) can be rewritten as:

[0196] (37)

[0197] (38)

[0198] In the formula, K and G are the bulk modulus and shear modulus, respectively. (Vector) The representative size is Direction is The shear stress. Vector It represents the increment of elastic shear strain with both magnitude and direction.

[0199] According to equation (34), in In space, the plastic correction stress can be calculated using the following formula:

[0200] (39)

[0201] (40)

[0202] (41)

[0203] In the formula, and These are the plastic correction stress increments in the directions of hydrostatic pressure and plastic stress deviation, respectively. Increment of plastic pure shear stress and Lode increment composition. and These are the increments of plastic volumetric strain and plastic shear strain, respectively. Generated by the increase in hydrostatic pressure, It is generated by the increase in shear stress. and yes The two components, such as Figure 10 As shown. The occurrence is due to The existence of, generally speaking, for geotechnical materials Very small. In this embodiment, an approximate algorithm is used. That is, the current step. It is assumed that:

[0204] (42)

[0205] In the formula, These are weighting coefficients. The value range is 0 to 1. , and These are respectively the previous step and the elastic conjecture stress point. and the current step As can be seen from equation (42), in each constitutive calculation, All of these values ​​are definite, indicating that the stress space in which the constitutive calculation takes place is perpendicular to the stress space. A plane of the axis. Specifically, to simplify calculations, We can assume it to be 0. In this case, = ,once Confirmed The calculation can be simplified to a calculation in two-dimensional stress space (i.e.) Figure 7 In space).

[0206] Based on the foregoing explanation, according to the stress state information from the previous step... and the strain increment information of the current step Based on the obtained material parameters, the elastic conjectured stress is calculated. Then, all the elastic conjectured stresses are converted to It is expressed in space and substituted into the linear equivalent yield surface equation. (Equation (9)). When This indicates that the stress state is elastic, and the calculated elastic conjecture stress is the new stress state. ;if Then, it is necessary to calculate the plastic correction stress. That is, by substituting formulas (29) to (31) into formulas (39) and (40), the plastic correction stress can be calculated by the following formula:

[0207] (43)

[0208] (44)

[0209] because Very small, so the weighting coefficients in formula (42) can be assumed to be... In other words, in each calculation, a circular plastic potential function will be used to approximate the non-circular plastic potential function at the current stress level, thus obtaining a new stress state:

[0210] (45)

[0211] (46)

[0212] (47)

[0213] In the formula, The average stress of the current step, The average stress from the previous step, For the current step , For the previous step .

[0214] Since the yield surface equation used is the linear equivalent yield surface equation of the GZZ equation, the plasticity multiplier... It can be calculated using the following formula:

[0215] (48)

[0216] Substituting formulas (29) and (48) into formula (31) yields the plastic shear strain increment.

[0217] Meanwhile, the plastic shear strain can be calculated using the following formula:

[0218] (49)

[0219] in, This is the accumulated plastic shear strain from the previous step. Based on the calculated plastic shear strain, the plastic intrinsic variables can be updated, thereby updating the material parameters. Finally, the new stress will be transformed into a stress state in Cartesian coordinates. The specific stress transformation method can be performed using existing techniques, and will not be elaborated here.

[0220] The parameter calibration process is described below.

[0221] The constitutive model proposed in this embodiment includes a total of 23 parameters, including: 10 strength-related parameters, 5 strain hardening parameters, 5 strain softening parameters, 1 expansion parameter, and 2 elastic parameters.

[0222] Strength-related parameters are , , , , , , , , , .in, This is the uniaxial compressive strength of intact rock, which can be obtained from uniaxial compression tests on intact rock. The crack instability propagation stress of the rock is obtained by conducting at least three sets of true triaxial compression tests. The results can be obtained by fitting the data using formula (9). , , Similarly, the peak strength and residual strength can be obtained by fitting the data through at least three sets of true triaxial tests. , , , , , Strain hardening parameters control the nonlinearity of the pre-peak strain hardening curve and the pre-peak plastic deformation capacity of the surrounding rock, including: , , , , By conducting a series of triaxial compression tests, the plastic shear strain at the peak strength under triaxial compression conditions was obtained. Then, by fitting the results using formulas (19) to (22), the plastic shear strain at the peak strength was obtained. , , . and It describes the difference in plastic shear strain at peak strength caused by the difference between the intermediate principal stress and the minimum principal stress. Characterization The influence of the middle principal stress on the plastic shear strain at the peak strength when the lower principal stress is not equal to the minimum principal stress. This was investigated by conducting two sets of... The true triaxial compression test was conducted to obtain the plastic shear strain at the peak strength, and then the result can be obtained through fitting. . Characterized The effect of the difference between the intermediate principal stress and the minimum principal stress on the plastic shear strain at the peak strength is investigated. At least two sets of studies on a single constant stress are required. The true triaxial test was conducted to obtain this. The influence coefficient of the lower principal stress on the plastic shear strain at the peak strength is then compared with... The lower peak strength can be obtained by fitting the plastic shear strain at every point. .

[0223] Strain softening parameters affect the nonlinearity of the post-peak strain softening curve and the post-peak plastic deformation capacity of the surrounding rock, including: , , , , . , , The calibration method can be referred to , , The steps involved are not detailed here. , This describes the difference in plastic shear strain at the residual strength caused by the difference between the intermediate principal stress and the minimum principal stress. The method for obtaining this information can be found in [reference needed]. , The execution process will not be elaborated here.

[0224] Unlike existing constitutive models, in this embodiment, only one dilation parameter is used as the initial stage for constructing the proposed constitutive model. The expansion behavior of the surrounding rock during the plastic stage can be directly reflected by the volumetric strain curve, and this parameter can be obtained by fitting the stress-strain curve.

[0225] Therefore, when using the constitutive model proposed in this embodiment to reproduce the stress-strain curve of the surrounding rock, a total of 23 parameters need to be calibrated, requiring at least six sets of tests, including one set of uniaxial tests, one set of conventional triaxial tests, and two sets of... True triaxial test and two sets of constant values True triaxial test under the following conditions.

[0226] Example 2: Constitutive model construction under conventional triaxial stress state

[0227] In deeply buried tunnels, specimens often exhibit brittle failure under low confining pressure, with a significant stress drop occurring after the peak stress-axial strain increment. As confining pressure increases, the strain softening characteristics of the post-peak stress-axial strain curve become more pronounced. With further increases in confining pressure, obvious macroscopic cracks may not form on the specimen surface, and the stress-axial strain curve may not exhibit the stress drop characteristic, leading to ductile failure. With increasing confining pressure, the circumferential deformation of the surrounding rock gradually decreases, while the residual strain gradually increases. Therefore, this example describes a conventional triaxial strain-hardening-softening elastoplastic constitutive model for surrounding rock, abbreviated as the UHS model.

[0228] In this example, the UHS model uses the Hoek-Brown strength criterion as the characteristic yield surface equation to describe the initial yield surface, peak strength surface, and residual strength surface under conventional triaxial stress.

[0229] (50)

[0230] (51)

[0231] (52)

[0232] In the formula, the subscripts [0], [1] and [2] represent the initial yield surface, peak strength surface and residual strength surface, respectively.

[0233] Under conventional triaxial conditions, the plastic shear strain at both the peak strength surface and the residual strength surface changes with increasing confining pressure. A linear formula can be used to fit the confining pressure and the plastic shear strain at these two characteristic yield surfaces. Although some fitted formulas have low correlation coefficients, this may be related to the high dispersion of the experimental data or the non-linear relationship between the plastic shear strain at the characteristic yield surface and the confining pressure. However, due to the lack of experimental data, it is very difficult to propose a unified and suitable formula to describe the relationship between the plastic shear strain at the characteristic yield surface and the confining pressure. Therefore, as a first-stage attempt, this embodiment uses a linear formula to approximate the relationship between these two factors:

[0234] (53)

[0235] (54)

[0236] In the formula, and These are the plastic shear strains at the peak strength surface and the residual strength surface, respectively. and These are the confining pressure correlation coefficients at peak strength and residual strength, respectively; and These are the plastic shear strains at the peak strength and residual strength under uniaxial compression, respectively. If the confining pressure has little effect on the plastic shear strain at the characteristic surface, and It will be a very small value.

[0237] The normalized plastic shear strain is used as the internal plastic variable and calculated according to the following definition:

[0238] (55)

[0239] (56)

[0240] In the formula: and These are the normalized plastic shear strains at the pre-peak and post-peak stages, respectively. At the initial yield surface during the pre-peak stage... The value is 0, at the peak intensity surface. =1; As the yield surface increases from 0 to 1, it evolves from the initial yield surface to the peak strength surface. Similarly, in the post-peak stage, the peak strength surface... =0, residual strength surface =1; As the yield surface increases from 0 to 1, it evolves from the peak strength surface to the residual strength surface. Therefore, the proposed model can well describe the three characteristic yield surfaces (such as the initial yield surface, peak strength surface, and residual strength surface) obtained from experiments.

[0241] Considering that the macroscopic mechanical parameters of the surrounding rock exhibit nonlinear evolution characteristics with plastic shear strain during both the strain hardening and strain softening stages, the model will employ parameter... The movement of the yield surface is controlled by s. The strain hardening stage progresses from the initial yield surface [0] to the peak strength surface [1]. The evolution of s is as follows:

[0242] (57)

[0243] (58)

[0244] In the formula These are hardening parameters, and they are inherent properties of the rock. The value should be greater than 0. When the value approaches 0, the stress-strain curve approximates elastoplastic characteristics. As... As the stress increases, the nonlinearity of the strain hardening stage in the stress-strain curve gradually decreases, such as... Figure 8 As shown.

[0245] During the strain softening stage, from the peak strength surface [1] to the residual strength surface [2], The evolution of s is as follows:

[0246] (59)

[0247] In equation (60) This is a softening parameter, also an inherent property of the rock. The value should be greater than 0. As the stress approaches zero, the stress-strain curve decreases approximately according to the brittle-plastic relationship. As the stress-strain curve increases, the nonlinearity of the strain softening stage gradually decreases.

[0248] Flow rules are commonly used to describe the relationship between stress and strain rate during the plastic deformation of rock materials. A flow rule specifies the direction of the plastic deformation rate of a rock under a given stress state. That is, flow rules can be used to determine the ratio of plastic deformation in different directions during incremental loading. Flow rules include associated flow rules and unassociated flow rules. In associated flow rules, the direction of the plastic strain rate is correlated with the direction of the stress increment. Unassociated flow rules allow the direction of the plastic strain rate to be inconsistent with the direction of the stress increment. When describing the plastic strain increment of rock, using associated flow rules often leads to an overestimation of the rock's plastic deformation. Therefore, unassociated flow development is often used to describe the deformation characteristics of rocks. In the UHS model, the plastic potential function used is as follows:

[0249] (61)

[0250] (62)

[0251] In the formula, It is the equivalent cohesion under the current stress state. It is the shear dilatation angle under the current stress state, calculated by the following formula:

[0252] In equation (63), , and These represent the volumetric, maximum, and minimum plastic strain increments, respectively. For simplicity, two constant dilatation angles are used in the UHS model. A constant pre-peak dilatation angle is used before the peak intensity. Following the peak intensity, another constant post-peak dilatation angle was employed. Although the dilatation angle varies with confining pressure and plastic shear strain, using a constant dilatation angle can provide an acceptable approximation of experimental results.

[0253] Therefore, principal plastic strain It can be calculated using the following formula:

[0254] , i=1,2,3(64)

[0255] In the formula, It is a plastic multiplier.

[0256] As an example, the numerical implementation process of this constitutive model is described in detail below.

[0257] The proposed constitutive model is embedded into FLAC3D using the C++ language. The algorithm includes the following steps:

[0258] Step 1: Obtain material parameters.

[0259] Step 2, calculate the new stress vector In the calculation process of the UHS model, a new stress vector is generated. It is the stress vector from the previous loading step. This is obtained by calculating the strain increment vector of the current step. During the calculation process, the elastic conjectured stress is calculated first. :

[0260] (65)

[0261] (66) (67)

[0262] (68)

[0263] In the formula, is the elasticity matrix of the element, K is the bulk modulus, and G is the shear modulus.

[0264] Step 3: Convert the calculated elastic conjecture stress into the form of three principal stresses, and substitute these three principal stresses into the initial yield surface equation. The Chinese character (50) is abbreviated as f.

[0265] Step 4: Determine if f is greater than or equal to 0 (f≥0)

[0266] Step 5, if If this stress state is elastic, then the calculated elastic conjectured stress is the final new stress vector. At this point, the unbalanced force is calculated and convergence is determined. If convergence is achieved, the calculation ends; otherwise, the stress vector from the previous loading step is obtained. Calculate the new stress vector using the strain increment vector of the current step. Then proceed to the next iteration.

[0267] Step 6, if Calculate plastic multipliers The specific steps are as follows:

[0268] Principal stress increment It will be recalculated using the following formula:

[0269] , i=1,2,3(69)

[0270] In the formula, , and These are the elastic principal strain increment, the total principal strain increment, and the plastic principal strain increment, respectively. In FLAC3D, to improve efficiency, the nonlinear yield surface can be linearized. That is, in the nonlinear yield surface, under the current minimum principal stress, the linear Mohr-Coulomb yield surface is continuously used to approximate the nonlinear yield surface. Therefore, the stress state should satisfy the following condition:

[0271] (70)

[0272] (71)

[0273] In the formula, and These are the maximum and minimum principal stresses of the previous step, respectively; and These are the maximum and minimum principal stresses of the current step, respectively. It is the yield surface equation. It is a linear term of the yield surface, and C is a constant of the linear yield surface.

[0274] Considering that the stress state in the current step is obtained by adding the stress increment to the stress state in the previous step, the calculation is as follows:

[0275] , i=1, 2, 3 (72)

[0276] In terms of principal stresses, equation (65) can be written as follows:

[0277] , i=1, 2, 3 (73)

[0278] Substituting equations (64) to (69), (72), and (73) into equations (70) and (71), the plastic multiplier can be calculated using the following formula:

[0279] (74)

[0280] In the formula,

[0281] (75)

[0282] (76)

[0283] Step 7, update the principal stress state.

[0284] New principal stress It can be calculated using the following formula:

[0285] , i=1,2,3(77)

[0286] Step 8, Calculate plastic shear strain .

[0287] Substituting equations (61) and (74) to (76) into equation (64), the plastic strain increment can be calculated. Simultaneously, the plastic shear strain can be calculated using the following equation:

[0288] (78)

[0289] (79)

[0290] (80)

[0291] (81)

[0292] In the formula, It is the accumulated plastic shear strain from the previous step. It is the increment of plastic shear strain in the current step. It is a built-in plastic shear strain calculation method in FLAC3D.

[0293] Step 9: Update the yield surface parameters. The plastic intrinsic variables can be updated based on the calculated plastic shear strain, thereby updating the material parameters.

[0294] Step 10: The new principal stresses will be transformed into a stress state in Cartesian coordinates. Calculate the unbalanced forces and determine if the updated principal stress state has converged. If yes, end the iteration; otherwise, obtain the stress vector from the previous loading step. Calculate the new stress vector using the strain increment vector of the current step. Then proceed to the next iteration.

[0295] In FLAC3D, the calculation process for the UHS model is as follows: Figure 9 As shown.

[0296] The UHS model has 15 parameters, and its parameter calibration process is similar to that in Example 1 above, so it will not be described in detail here.

[0297] It is worth noting that in Example 1, in = In this case, it can be degenerated into Example 2, that is, Example 1 can achieve a unified description of the stress-strain relationship of conventional triaxial and true triaxial surrounding rock.

[0298] In summary, the constitutive model construction method provided in this embodiment describes the initial yield surface, peak strength surface, and residual strength surface under three-dimensional stress using the GZZ criterion, and... This paper expresses the stress-strain curve in three-dimensional space, considering the influence of the intermediate principal stress on the plastic shear strain at the characteristic yield surface. The traditional formula describing the plastic shear strain at the characteristic yield surface is extended, defining a three-dimensional formula for describing the plastic shear strain at the characteristic yield surface. A hardening / softening rule (hardening-softening equation) is introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages of the stress-strain curve. Furthermore, the non-associated flow rule is used to capture the expansion characteristics of the surrounding rock, constructing a unified strain hardening and strain softening elastoplastic constitutive model of the surrounding rock considering the intermediate principal stress. The numerical calculation implementation process in FLAC3D is illustrated. This model can accurately describe the influence of the intermediate principal stress on each stage of the stress-strain curve, providing a basis for disturbance analysis and support design in deep-buried tunnel excavation.

[0299] Based on the same inventive concept, this embodiment provides a constitutive model construction system. This system is implemented based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering principal stresses provided in any of the above embodiments. The system includes:

[0300] The constitutive model uses the GZZ strength criterion as the characteristic yield surface equation to describe the initial yield surface, peak strength surface, and residual strength surface under three-dimensional stress.

[0301] The constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface;

[0302] The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function.

[0303] The constitutive model adopts , , The parameter combination is used to control the movement of the characteristic yield surface, and the hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, so as to obtain the nonlinear hardening-softening equation; where, This represents the decrease in the material constant of the intact rock. and All are material constants of the rock mass;

[0304] The constitutive model uses the non-associated flow rule to capture the expansion characteristics of the surrounding rock.

[0305] The constitutive model construction system provided in this embodiment can implement the steps and processes of the constitutive model construction method provided in any of the above embodiments and achieve the same technical effect, which will not be described in detail here.

[0306] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

Claims

1. A unified elastoplastic description method for the nonlinear characteristics of surrounding rock considering intermediate principal stress, characterized in that, include: True triaxial mechanical property tests were conducted on the surrounding rock to obtain the full stress-strain curves of the surrounding rock; Characteristic stress points are extracted from the stress-strain curve, and the stress-strain curve is divided into stages to obtain multiple stages, including: elastic stage, strain hardening stage, strain softening stage, and residual stage. Based on the characteristic stress points, the corresponding characteristic yield surfaces are plotted in the stress space. The characteristic yield surfaces include: the initial yield surface, the peak strength surface, and the residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength. Based on the stage-specific characteristics of the stress-strain curve, plastic deformation capacity characterization functions are constructed for different stages considering the influence of medium stress. These plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve. A nonlinear hardening-softening equation is constructed to describe the evolution characteristics of the characteristic yield surface; The expression for the plastic deformation capacity characterization function is as follows: ， ， In the formula, , They are Plastic shear strain at the lower peak strength surface and the residual strength surface; and These are the plastic shear strains at the peak strength surface and the residual strength surface under true triaxial conditions, respectively. and These are the minimum principal stresses. The correlation coefficient of principal stress in the plastic shear strain at peak strength and residual strength when the value is greater than 0; and It is the minimum principal stress The correlation coefficient of principal stress in plastic shear strain at peak strength and residual strength when the stress is equal to 0. It is the middle principal stress. It is the minimum principal stress.

2. The elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the intermediate principal stress as described in claim 1, characterized in that, The plastic deformation capacity characterization function is used to characterize the relationship between the plastic shear strain and the intermediate principal stress and minimum principal stress at each characteristic yield surface.

3. The elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the intermediate principal stress as described in claim 1, characterized in that, The evolution of mechanical parameters is used to characterize the evolution of the yield surface. The hardening-softening equation is used to describe the evolution of the mechanical parameters with plastic strain at each characteristic yield surface.

4. The elastoplastic unified description method for the nonlinear characteristics of surrounding rock considering the intermediate principal stress as described in claim 1, characterized in that, Also includes: The elastic modulus and Poisson's ratio are used to describe the deformation characteristics of the elastic stage of the stress-strain curve; Expansion parameters are used to describe the deformation characteristics of the plastic stage of the stress-strain curve.

5. A constitutive model construction method, characterized in that, The constitutive model is implemented based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering the principal stress as described in any one of claims 1 to 4, including: The constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface; The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function. The constitutive model adopts The parameters s, a, and s are combined to control the movement of the characteristic yield surface, and hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, thus obtaining a nonlinear hardening-softening equation; where, is the decrease in the material constant of the intact rock, where s and a are both material constants of the rock mass; The constitutive model uses the non-associated flow rule to capture the expansion characteristics of the surrounding rock.

6. The constitutive model construction method according to claim 5, characterized in that, The constitutive model is solved using a numerical implementation algorithm based on the non-convex and non-smooth GZZ criterion, comprising the following steps: Obtain the expression for the GZZ strength criterion in the stress invariant space; Based on the expression of the GZZ strength criterion in the stress invariant space, write the equations for each characteristic yield surface; At the current stress level and At that point, stress invariant space The nonlinear GZZ criterion uses a linear tangent approximation and employs this linear tangent as the equivalent yield surface equation, where... This represents the second stress deviator invariant. For Lode angle, This represents the average stress.

7. The constitutive model construction method according to claim 5, characterized in that, Below, the plastic deformation capacity characterization function degenerates into: ， ， In the formula, and They are The correlation coefficient of confining pressure at the lower peak strength and residual strength; and They are Plastic shear strain at the lower peak strength and residual strength.

8. The constitutive model construction method according to claim 6, characterized in that, The nonlinear hardening-softening equation is as follows: During the strain hardening stage, the characteristic yield surface evolves from the initial yield surface to the peak strength surface, as expressed below: ， ， ， During the strain softening stage, the characteristic yield surface evolves from the peak strength surface to the residual strength surface, as expressed below: ， ， ， In the formula, The subscript represents the decrease in the material constant of the intact rock mass, where s and a are both material constants of the rock mass. Indicates the initial yield surface, subscript Indicates peak intensity surface, subscript Represents the residual strength surface. and These are the normalized plastic shear strains at the pre-peak and post-peak stages under true triaxial conditions, respectively. These are hardening parameters. This is a softening parameter.

9. A unified elastoplastic description system for the nonlinear characteristics of surrounding rock considering intermediate principal stress, characterized in that, include: The test unit is configured to conduct true triaxial mechanical property tests on the surrounding rock and obtain the full stress-strain curve of the surrounding rock; The feature extraction unit is configured to extract feature stress points from the stress-strain curve and divide the stress-strain curve into multiple stages, including: elastic stage, strain hardening stage, strain softening stage and residual stage. The drawing unit is configured to draw the corresponding characteristic yield surface in the stress space based on the characteristic stress point. The characteristic yield surface includes: initial yield surface, peak strength surface, and residual strength surface, which are used to jointly describe the nonlinear characteristics of rock strength. The first building unit is configured to construct plastic deformation capacity characterization functions for different stages based on the stage characteristics of the stress-strain curve. The plastic deformation capacity characterization functions are used to describe the plastic deformation capacity of the stress-strain curve. The second building block is configured to construct a nonlinear hardening-softening equation to describe the evolution characteristics of the characteristic yield surface; The expression for the plastic deformation capacity characterization function is as follows: ， ， In the formula, , They are Plastic shear strain at the lower peak strength surface and the residual strength surface; and These are the plastic shear strains at the peak strength surface and the residual strength surface under true triaxial conditions, respectively. and These are the minimum principal stresses. The correlation coefficient of principal stress in the plastic shear strain at peak strength and residual strength when the value is greater than 0; and It is the minimum principal stress The correlation coefficient of principal stress in plastic shear strain at peak strength and residual strength when the stress is equal to 0. It is the middle principal stress. It is the minimum principal stress.

10. A constitutive model construction system, characterized in that, The constitutive model construction system is implemented based on the elastoplastic unified description method of the surrounding rock nonlinear characteristics considering the principal stress as described in any one of claims 1 to 4, including: The constitutive model uses the GZZ strength criterion to describe the initial yield surface, peak strength surface, and residual strength surface; The constitutive model uses normalized plastic shear strain as the plastic internal variable, and fits the plastic shear strain at the characteristic yield surface based on the experimental results to obtain the expression of the plastic deformation capacity characterization function. The constitutive model adopts , , The parameter combination is used to control the movement of the characteristic yield surface, and the hardening parameter k1 and softening parameter k2 are introduced to describe the nonlinear characteristics of the strain hardening and strain softening stages, so as to obtain the nonlinear hardening-softening equation; where, This represents the decrease in the material constant of the intact rock. and All are material constants of the rock mass; The constitutive model uses the non-associated flow rule to capture the expansion characteristics of the surrounding rock.

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