Toppling type dangerous rock stability quantitative evaluation method considering damage-fracture mechanics

Abstract

The invention relates to the technical field of rock mechanics, in particular to a toppling type dangerous rock stability quantitative evaluation method considering damage-fracture mechanics, which comprises the following steps: carrying out damage degree analysis according to a main control structural surface of a toppling type dangerous rock, and dividing a crack tip damage area of the main control structural surface based on the damage degree; analyzing the stress distribution of the crack tip of the main control structural surface, and calculating the length of a damage localization zone; carrying out stress analysis on the toppling type dangerous rock, and calculating a fracture stress intensity factor of a main control structural surface of the toppling type dangerous rock based on a stress distribution state and a damage localization zone length of a toppling type dangerous rock body under the action of an external load according to a damage-fracture mechanics theory; and establishing a stability evaluation index based on the fracture toughness and the fracture stress intensity factor of the toppling type dangerous rock, and carrying out quantitative evaluation on the stability of the toppling type dangerous rock. The method overcomes the defect that rock mass damage evolution characteristics are neglected in the prior art, and the influence of rock mass internal crack development on stability can be accurately described.

Description

Technical Field

[0001] The present invention relates to the technical field of rock mechanics, and in particular to a quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics. Background Art

[0002] Dangerous rock masses are a common geological hazard, and their stability issues, such as collapse and landslides, pose a serious threat to the safety of people's lives and property. Currently, the stability assessment of dangerous rock masses primarily relies on traditional calculation methods such as the limit equilibrium method and numerical simulation. These methods analyze the stress state, deformation characteristics, and failure mode of the dangerous rock mass, establish corresponding mechanical models, and thus evaluate the stability of the dangerous rock mass. The limit equilibrium method primarily calculates the ratio of the anti-sliding force to the sliding force of the dangerous rock mass based on the principles of statics. However, because it is based on the rigid body assumption, it ignores internal defects such as cracks and joints in the rock mass. Numerical simulation methods simulate the stress and deformation process of the dangerous rock mass by establishing a finite element model.

[0003] However, existing methods for evaluating the stability of dangerous rock masses have significant limitations. These methods generally ignore the damage evolution characteristics of the main structural surfaces of the toppling dangerous rock mass, fail to accurately reflect the stress distribution state in the crack tip region, and fail to fully consider the nonlinear deformation and failure mechanisms of the rock mass. Furthermore, due to the lack of reasonable quantitative evaluation indicators, the assessment results often rely on engineering experience, making it difficult to accurately quantify the stability of the toppling dangerous rock mass. These technical shortcomings seriously affect the reliability and accuracy of the stability evaluation of toppling dangerous rock masses in practical engineering applications. Summary of the Invention

[0004] In view of this, the present invention proposes a quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics, aiming to solve the problem that the existing dangerous rock stability evaluation method cannot accurately reflect the damage evolution characteristics of the main controlling structural surface of the toppling dangerous rock body, resulting in low accuracy of the quantitative evaluation of the stability of the toppling dangerous rock.

[0005] The technical solution of the present invention is achieved as follows: The present invention provides a quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics, comprising the following steps:

[0006] S1. Analyze the damage degree of the main control structure surface of the toppling dangerous rock and divide the crack tip damage zone of the main control structure surface based on the damage degree;

[0007] S2. Analyze the stress distribution at the crack tip of the main control structure surface and calculate the length of the damage localization zone;

[0008] S3. Perform a stress analysis on the toppling dangerous rock mass. According to the damage-fracture mechanics theory, the stress distribution state of the toppling dangerous rock mass under the action of external loads and the length of the damage localization zone are used to calculate the fracture joint stress intensity factor of the toppling dangerous rock mass.

[0009] S4. Based on the fracture toughness of the toppling dangerous rock and the joint stress intensity factor of the toppling dangerous rock fracture, a stability evaluation index is established to quantitatively evaluate the stability of the toppling dangerous rock.

[0010] On the basis of the above technical solution, preferably, in step S1, the crack tip area of the main control structural surface is divided into a non-damaged area, a continuous damage area and a damage localization zone based on the degree of damage, the damage degree of the damage localization zone is less than that of the continuous damage area, and the complete damage is concentrated in the damage localization zone area with a smaller width in front of the crack tip.

[0011] Based on the above technical solution, preferably, step S2 specifically includes:

[0012] S21. The stress drop at the crack tip is set to be concentrated in the damage localization zone, and the length of the damage localization zone does not exceed 10% of the crack length;

[0013] S22. Based on the principle of linear elasticity, establish the stress distribution equation at the crack tip of the main control structure surface, analyze the normal stress in the y direction at the crack tip of the main control structure surface, and obtain the normal stress expression;

[0014] S23. Apply concentrated force within the damage localization zone and calculate the stress intensity factor expression near the damage zone;

[0015] S24. Calculate the stress expression near the crack tip based on linear elastic fracture mechanics theory and stress intensity factor expression;

[0016] S25. Based on the elastic-brittle sudden damage model, the stress distribution in the local damage area is regarded as a constant. The length of the damage localization zone is calculated based on the normal stress expression and the stress expression near the crack tip.

[0017] On the basis of the above technical solution, preferably, it is characterized in that:

[0018] In step S21, the calculation expression of normal stress is as follows:

[0019]

[0020] Where σ1(r) is the normal stress in the y direction at the crack tip of the main control structure surface, K I is the stress intensity factor of mode I fracture, and r is the distance from the crack tip to the calculation point.

[0021] In step S24, the stress expression near the crack tip is as follows:

[0022]

[0023] Where K D is the stress near the crack tip, e is the length of the damage localization zone, c is the half length of the crack, that is, half of the distance from the crack tip to the farthest end of the crack, σ(ε) is the stress state of the material under strain ε, and ξ is the local coordinate near the crack tip, which is used to describe the stress field distribution near the crack tip.

[0024] Based on the above technical solution, preferably, in step S25, the calculation expression of the damage localization zone length is:

[0025]

[0026] Where σ0 is the initial yield stress of the material.

[0027] Based on the above technical solution, preferably, step S3 specifically includes:

[0028] S31. The external load on the toppling dangerous rock mass is translated to the crack tip of the main control structure surface of the toppling dangerous rock mass through static equivalence, and the normal force, tangential force and bending moment force at the crack tip are decomposed based on the principle of mechanical equilibrium;

[0029] S32. Decompose the toppling dangerous rock fracture model according to fracture mechanics and solve the fracture stress intensity factor under different stresses;

[0030] S33. According to the principle of superposition of stress intensity factors, the combined stress intensity factor of the toppling dangerous rock fracture is calculated based on the fracture stress intensity factors under different stresses.

[0031] On the basis of the above technical solution, preferably, in step S32, the fracture stress intensity factors under different stresses include: under the action of pore water pressure, tensile stress perpendicular to the main control structure surface, and tensile stress vertical to the main control structure surface, type I fracture is generated; under the shear action parallel to the main control structure surface, type II fracture is generated.

[0032] Based on the above technical solution, preferably, in step S33, the combined stress intensity factor of the tipping dangerous rock fracture includes a type I fracture stress intensity factor and a type II fracture stress intensity factor, and the calculation expression of the combined stress intensity factor of the tipping dangerous rock fracture is:

[0033]

[0034] Where K I+Π is the combined stress intensity factor of the overturning dangerous rock fracture, KI is the stress intensity factor for mode I fracture, K Π is the stress intensity factor of mode II fracture, θ0 is the crack extension angle, and the angle between the crack surface and the principal stress direction.

[0035] On the basis of the above technical solution, preferably, according to the maximum circumferential tensile stress criterion, the crack propagates along the direction of θ0 corresponding to the maximum tensile stress, and the following is obtained:

[0036]

[0037] Where θ0 is the crack extension angle.

[0038] On the basis of the above technical solution, preferably, in step S4, the stability evaluation index is the toppling dangerous rock stability coefficient, and the dangerous stability coefficient is the ratio of the fracture toughness of the rock to the combined stress intensity factor of the toppling dangerous rock fracture.

[0039] The quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics of the present invention has the following beneficial effects compared with the prior art:

[0040] (1) The present invention divides the crack tip area of the main control structural surface into a non-destructive zone, a continuous damage zone, and a damage localization zone, and establishes the corresponding stress distribution equation. This overcomes the defect of the existing technology that ignores the evolution characteristics of rock damage, and can accurately describe the impact of internal crack development on rock stability, which is more in line with engineering practice.

[0041] (2) The present invention incorporates damage into the stability calculation process and simulates the cracking, deformation, and failure process of the rock mass under stress through a damage model, thereby providing a more accurate stability assessment. This effectively solves the problem that the existing technology cannot accurately reflect the stress distribution state at the crack tip, and improves the reliability of the assessment results.

[0042] (3) The present invention adopts the ratio of the fracture toughness of the toppling dangerous rock to the fracture combined stress intensity factor as the stability evaluation index, and establishes a complete quantitative evaluation system. At the same time, the proposed calculation method is simple and profound, which can more comprehensively consider the physical properties and mechanical behavior of the rock mass, and can reflect the essence of the instability and destruction of the toppling dangerous rock, and has a good prospect of being widely used in engineering design. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0044] Figure 1 This is a flow chart of the quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics of the present invention;

[0045] Figure 2 A diagram showing the damage degree of the crack tip region of the main control structure surface of the dumping dangerous rock of the present invention;

[0046] Figure 3 The stress distribution diagram of the crack tip of the main control structure surface of the tilting dangerous rock of the present invention;

[0047] Figure 4 This is a sudden damage model diagram of the present invention in which the residual strength is linearly distributed;

[0048] Figure 5 This is a load analysis diagram of the dumping type dangerous rock mass of the present invention;

[0049] Figure 6 A simplified model diagram of the structural surface fracture extension of the present invention;

[0050] Figure 7 This is an exploded view of the toppling dangerous rock fracture model of the present invention. DETAILED DESCRIPTION

[0051] The following will be combined with the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the embodiments described are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0052] like Figure 1 As shown, the present invention provides a quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics, and the evaluation method specifically includes the following steps:

[0053] S1. Analyze the damage degree of the main control structure surface of the toppling dangerous rock and divide the crack tip damage zone of the main control structure surface based on the damage degree;

[0054] S2. Analyze the stress distribution at the crack tip of the main control structure surface and calculate the length of the damage localization zone;

[0055] S3. Perform a stress analysis on the toppling dangerous rock mass. According to the damage-fracture mechanics theory, the stress distribution state of the toppling dangerous rock mass under the action of external loads and the length of the damage localization zone are used to calculate the fracture joint stress intensity factor of the toppling dangerous rock mass.

[0056] S4. Based on the fracture toughness of the toppling dangerous rock and the joint stress intensity factor of the toppling dangerous rock fracture, a stability evaluation index is established to quantitatively evaluate the stability of the toppling dangerous rock.

[0057] Considering that the stability of toppling dangerous rock is mainly controlled by the discontinuous structural surface at the rear, its damage and fracture process is the fundamental cause of instability and collapse. At present, the rigid body limit equilibrium method is mostly used in actual engineering to calculate the stability of toppling dangerous rock mass. However, it has been proved through engineering practice that the stability of toppling dangerous rock mass based on fracture mechanics is more in line with reality than the limit equilibrium method. Therefore, the present invention provides a quantitative evaluation method for the stability of toppling dangerous rock mass, which is composed of the damage zone division of the main control structural surface of toppling dangerous rock mass, the length calculation of the damage localization zone, the damage-fracture mechanics theory of the main control structural surface of toppling dangerous rock mass, and the calculation method of the stability coefficient of toppling dangerous rock mass. The stability evaluation method of toppling dangerous rock mass is established by analyzing the development and expansion of the tip crack of the main control structural surface of toppling dangerous rock mass under load conditions, and the main control structural surface of toppling dangerous rock mass is regarded as a crack, and the ratio of the fracture toughness of the rock to the fracture joint strength under load is used as the result of evaluating stability. The evaluation method provided by the present invention is based on damage-fracture mechanics, and the stability coefficient obtained can accurately reflect the stability of toppling dangerous rock mass, providing a basis for the protection design of toppling dangerous rock mass.

[0058] Furthermore, in step S1, the crack tip area of the main control structural surface is divided into a non-damaged area, a continuous damage area and a damage localization zone based on the damage degree. The damage degree of the damage localization zone is less than that of the continuous damage area, and the complete damage is concentrated in the damage localization zone area with a smaller width in front of the crack tip.

[0059] Specifically, in order to consider the role of damage in the stability analysis of toppling dangerous rock, the influence of time on the damage zone is ignored, that is, when the stress condition remains unchanged, the damage zone is a fixed value. Figure 2 As shown in the figure, the tip field of the main control structure surface is divided into a non-damaged zone, a continuous damage zone and a damage localization zone according to the degree of damage. The complete damage is only concentrated in the area with a smaller width in front of the crack tip. The damage degree of the damage localization zone is much smaller than that of the continuous damage zone, and it can still be regarded as an area controlled by the K field.

[0060] Furthermore, step S2 specifically includes:

[0061] S21. Set the stress drop at the crack tip to be concentrated in the damage localization zone, and the width of the damage localization zone can be ignored, and the length of the damage localization zone is much smaller than the crack length. That is, the width of the damage localization zone is very small relative to the crack length, so that it can be ignored in analysis and calculation. This simplification helps to reduce the computational complexity while still being able to capture the main damage behavior near the crack tip. Preferably, the length of the damage localization zone does not exceed 10% of the crack length, and more preferably, the length of the damage localization zone does not exceed 5% of the crack length. The method for calculating the length of the damage localization zone assumes that the stress drop at the crack tip is concentrated in the damage localization zone at the crack tip. The length of the damage localization zone with zero width is l, which is related to the stress intensity factor in the far field, and l << e, that is, it is considered that the damage range is very small and the stress distribution in the far field is not affected.

[0062] S22. Based on the principle of linear elastic mechanics, establish the stress distribution equation at the crack tip of the main control structural plane, analyze the normal stress in the y direction at the crack tip of the main control structural plane, and obtain the normal stress expression.

[0063] The stress distribution at the crack tip of the main control structural plane is as Figure 3 shown. To accurately describe the stress distribution at the crack tip of the main control structural plane of toppling dangerous rocks, the stress distribution equation is established by using linear elastic mechanics and the superposition principle. The stress distribution equation at the crack tip of the main control structural plane can be expressed by the following formula:

[0064]

[0065]

[0066]

[0067] In the formula, K I represents the stress intensity factor of mode I fracture, K Π represents the stress intensity factor of mode II fracture, σ x represents the stress along the x-axis direction at the crack tip of the main control structural plane, σ y is the stress along the y-axis direction at the crack tip of the main control structural plane, τ xy is the shear stress at the crack tip, θ is the angle between the crack tip and the crack surface, and r is the distance from the crack tip to the calculation point.

[0068] According to the form of the stress field (σx, σy, τxy) at the tip of the main control structural plane of toppling dangerous rocks, when θ = 0, the expression of the normal stress σ1(r) along the y direction of the crack surface at the crack tip is calculated as:

[0069]

[0070] Where, r is the vertical distance measured from the crack tip along the direction of the normal stress distribution to the calculation point, and θ is the angle between the crack tip and the crack surface.

[0071] S23. Apply a concentrated force within the range of the damage localization zone, and calculate the expression of the stress intensity factor near the damage zone.

[0072] Meanwhile, assume that the stress drop at the crack tip is mainly concentrated within the damage localization zone, and the width of this zone can be approximately regarded as zero. Denote the length of the damage localization zone with zero width as l, and satisfy l << e (e is the vertical height of the crack). Establish a coordinate system at the tip of the damage localization zone, and apply a pair of concentrated forces at x = ε (e < ε < c)x. When, the generated stress intensity factor is:

[0073]

[0074] Where, x represents the distance from the crack tip to the calculation point, c represents the half-length of the crack, that is, half of the distance from the crack tip to the farthest end of the crack, and ξ represents the local coordinate near the crack tip, which is used to describe the stress field distribution near the crack tip.

[0075] The length of the damage localization zone is much smaller than the crack length. Based on Equation (5), the stress intensity factor near the damage zone (x = r + c) is:

[0076]

[0077] Simplify Equation (6) for r << c, and obtain:

[0078]

[0079] S24. Based on the linear elastic fracture mechanics theory and the stress intensity factor expression, calculate the stress expression near the crack tip.

[0080] According to the linear elastic fracture mechanics theory, the fracture stress intensity factor for Mode I fracture is:

[0081]

[0082] Integrate ξ within the range from e to c, and obtain the stress generated by the stress near the crack tip as:

[0083]

[0084] Where, e is the length of the damage localization zone.

[0085] Here, it is assumed that the stress drop within the damage localization zone has approximately no effect on the far-field stress distribution, and the integral expression used is derived based on the linear elasticity assumption and the local stress continuity condition. The derivation process is based on the relevant literature on linear elastic fracture mechanics.

[0086] S25. Based on the elastic-brittle sudden damage model, the stress distribution in the local damage area is regarded as a constant. The length of the damage localization zone is calculated based on the normal stress expression and the stress expression near the crack tip.

[0087] Assume that the stress sum at the crack tip of the main control structure surface of the overturned dangerous rock is 0, so

[0088]

[0089] Taking the damage into account in the stability calculation process, the stress near the crack tip r is limited. When r→0, then Then solve Equation 11 to get:

[0090]

[0091] Furthermore, by deriving Equation 12, we can obtain:

[0092]

[0093] In such Figure 4 As shown in the elastic-brittle sudden damage model with residual strength, the stress drop is much smaller than the original value, and the stress distribution in the local damage area can be regarded as a constant σ0. Therefore, integrating Equation 13, the length of the damage localization zone is obtained as follows:

[0094]

[0095] In view of the commonality of various damages, "damage" can be defined as linear elastic stress failure, that is, the transformation from linear elastic stress state to nonlinear stress state. Assuming that the residual strength of the main control structure surface model after stress drop remains constant σ0, it is assumed here that the stress magnitude of the damaged area is linearly related to the damage degree, then

[0096] σ0=σ(1-D) (Equation 15)

[0097]

[0098] Where σ1 and σ2 are the normal stresses of the rock mass section that is not penetrated along the direction of the main control structural surface (damage at the tip of the main control structural surface is not considered); D is the damage variable (damage degree).

[0099] The average normal stress of the unpenetrated part of the toppling dangerous rock mass can be expressed as:

[0100]

[0101] Where h is the biaxial load coefficient, which may be used to adjust the stress effect on the dangerous rock mass due to loads in different directions (such as vertical and horizontal); W1 represents the weight of the dangerous rock mass, and P1 represents the horizontal force

[0102] The reference damage variable can be expressed as:

[0103]

[0104] Where ν is the Poisson's ratio of the lossless material, is the Poisson's ratio of the damaged material.

[0105] This step simplifies the damage localization zone into a zero-width model and establishes a complete stress distribution calculation system by combining linear elasticity, fracture mechanics theory, and the elastic-brittle sudden damage model. This not only simplifies the complex stress field calculation process, but also accurately reflects the stress distribution characteristics and damage evolution laws in the crack tip area, providing a reliable theoretical basis and calculation basis for subsequent stability evaluation.

[0106] Furthermore, for the toppling dangerous rock, the static equivalent method is first used to translate the external load on the rock mass and load it to the crack tip of the main control structure surface. The specific steps include:

[0107] S31. The external load on the toppling dangerous rock is transferred to the crack tip of the main control structure surface of the toppling dangerous rock mass through static equivalence, and the normal force, tangential force and bending moment force at the crack tip are decomposed based on the principle of mechanical equilibrium.

[0108] In different composite cracks, the stress intensity factor of type I fracture can be expressed. The deformation, destruction and instability of the toppling dangerous rock is actually the damage and fracture behavior of the main control structure surface. The role of damage in the instability and expansion of the main control structure surface should be considered. Therefore, the damage and fracture basis can be expressed as:

[0109]

[0110] where K IC It represents the fracture toughness of the overturned dangerous rock mass and is a material constant. Indicates the critical value of damage fracture;

[0111] Therefore, the stability coefficient of the overturned dangerous rock can be expressed as:

[0112]

[0113] There is a steep structural surface at the rear edge of the overturned dangerous rock mass, which opens with strong unloading. Under the action of deadweight, water pressure and vibration, the rock mass collapses in the direction of the air until it becomes unstable and completely separates from the parent rock. This type of overturned dangerous rock is mainly subjected to tension and shear. Its force analysis diagram is shown in the figure below. Figure 5 As shown in the figure, O is the crack tip, A is the center of gravity of the dumping dangerous rock mass, H is the height of the dumping dangerous rock mass, g is the vertical height of the crack, β is the crack inclination, V is the volume per unit length of the dumping dangerous rock mass, G is the weight of the rock mass per unit length, μ L is the horizontal seismic coefficient, μ V is the vertical seismic coefficient, P L is the horizontal seismic force per unit length, P V is the vertical seismic force per unit length, u is the pore water pressure near the crack, and a and b are the vertical and horizontal distances from the center of gravity to the crack tip, respectively. By static equivalence, the load on the toppling dangerous rock mass is translated to the crack tip of the main control structure surface of the toppling dangerous rock mass, and the load on the toppling dangerous rock mass is decomposed along the normal and tangential directions of the structure surface to obtain:

[0114] N=P L sinβ+Q-(G+P V )cosβ (Equation 21)

[0115] T=P L cosβ+(G+P V )sinβ (Equation 22)

[0116] Where N represents the normal load along the structural surface, T represents the tangential load along the structural surface, β is the crack inclination angle, and P L Represents the horizontal seismic force per unit length, P L =μ L γV,μ L is the horizontal seismic coefficient, V is the volume per unit length of the overturned dangerous rock mass, Q represents, (In natural state, In heavy rain conditions, ), g is the vertical height of the crack, G represents the weight of rock mass per unit length, G = γV, P V It represents the vertical seismic force per unit length;

[0117] When the crack tip O is higher or lower than the center of gravity A, the bending moment M at the crack tip is:

[0118] M1=(G+P V )b-aP L (Equation 23)

[0119] M2=(G+P V )b+aP L(Equation 24)

[0120] Where M1 represents the bending moment force at the crack tip when the crack tip O is higher than the center of gravity A, M2 represents the bending moment force at the crack tip when the crack tip O is lower than the center of gravity A, a represents the vertical distance from the center of gravity to the crack tip, and b represents the horizontal distance from the center of gravity to the crack tip.

[0121] S32. Decompose the toppling dangerous rock fracture model according to fracture mechanics and solve the fracture stress intensity factor under different stresses.

[0122] The main control structure surface of the toppling dangerous rock is mainly affected by the tension and shear expansion and instability. From the perspective of fracture mechanics, the simplified fracture expansion model of the structure surface is as follows: Figure 6 As shown, the overturning dangerous rock fracture model is further decomposed into Figure 7 The exploded diagram shown in the figure shows the effects of different stresses on the main control surface of the toppling dangerous rock, and different fracture stress intensity factors are generated based on different stresses. Among them, the fracture stress intensity factors under different stresses include: under the action of pore water pressure, tensile stress perpendicular to the main control surface, and tensile stress perpendicular to the main control surface, type I fracture is generated; under the shear action parallel to the main control surface, type II fracture is generated. Specifically:

[0123] Figure 7 Figure (a) shows the pore water pressure effect. Assuming that it is uniformly distributed along the main control structure surface, the resulting mode I fracture stress intensity factor is:

[0124]

[0125] Where a0 represents the characteristic size of the crack, represents the average displacement, γ w represents a parameter related to material properties or structural characteristics, e1 represents a parameter related to material properties or structural characteristics, and e is the length of the damage localization zone;

[0126] Figure 7 Figure (b) shows the shear action parallel to the main control structure surface. Under this condition, it belongs to type II fracture. The stress intensity factor of type II fracture is:

[0127]

[0128] Where τ represents the shear stress, h represents the parameters related to structural size and load,

[0129] Figure 7 Figure (c) shows the bending moment, and the resulting mode I fracture stress intensity factor is:

[0130]

[0131] Among them, σ max represents the maximum stress,

[0132] Figure 7 Figure (d) shows the tensile stress perpendicular to the main control structure surface, and the resulting type I fracture stress intensity factor is:

[0133]

[0134] Where σ represents stress,

[0135] S33. According to the principle of superposition of stress intensity factors, the combined stress intensity factor of the toppling dangerous rock fracture is calculated based on the fracture stress intensity factors under different stresses.

[0136] According to the principle of superposition of stress intensity factors, the first type of fracture stress intensity factor of the main control structure surface of tension-shear-overturning dangerous rock is:

[0137] K I =K I1 +K I2 +K I3 (Equation 29)

[0138] For type I fracture, K I Related to stress and crack geometry; for mode II fracture, K Π Dependent on shear force and crack geometry.

[0139] At the crack tip, the stress and shear force can be expressed as:

[0140] σ=σ max cosθ

[0141] τ=τ max sinθ

[0142] Where, σ max is the maximum stress, τ max is the maximum shear force.

[0143] For mode I fracture and mode II fracture, the stress intensity factor can be expressed as:

[0144]

[0145] Where a is the crack length.

[0146] According to the maximum circumferential tensile stress criterion, the crack propagates along the direction of θ0 corresponding to the maximum tensile stress and must satisfy:

[0147]

[0148] Substitute σ θ = σ max cosθ into the above conditions, and we get:

[0149]

[0150] Solve the above equation, and we get θ = 0 or θ = π. However, since the crack propagation direction is between 0 and π, θ = 0 is chosen.

[0151] Substitute the relationship between K Π and K I , and we get:

[0152]

[0153] Therefore, the crack propagation angle θ0 can be expressed as:

[0154]

[0155] The combined stress intensity factor of the toppling unstable rock fracture is:

[0156]

[0157] This step simplifies and translates the complex external load to the crack tip through the principle of static equivalence, systematically analyzes the fracture modes under different stress actions in combination with fracture mechanics theory, and establishes a complete fracture strength calculation system by using the superposition principle of stress intensity factors. It not only considers the comprehensive influence of various loads such as horizontal seismic force, vertical seismic force, and pore water pressure, but also can accurately reflect the fracture characteristics of the toppling unstable rock under complex stress states, providing a reliable mechanical basis for subsequent stability evaluation.

[0158] Furthermore, in step S4, the stability evaluation index is the stability coefficient of the toppling unstable rock, and the critical stability coefficient is the ratio of the fracture toughness of the toppling unstable rock to the combined stress intensity factor of the toppling unstable rock fracture, that is

[0159]

[0160] Specifically, in engineering practice, when the calculated stability coefficient k > 1.5, it can be considered that the toppling unstable rock mass is in a stable state; when 1.2 < k ≤ 1.5, the toppling unstable rock mass is in a relatively stable state; when 1.0 < k ≤ 1.2, the toppling unstable rock mass is in a critical stable state; when k ≤ 1.0, the toppling unstable rock mass is in an unstable state.

[0161] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics, characterized by: The following steps are involved: S1. Analyze the damage degree of the main control structure surface of the toppling dangerous rock and divide the crack tip damage zone of the main control structure surface based on the damage degree; S2. Analyze the stress distribution at the crack tip of the main control structure surface and calculate the length of the damage localization zone; S3. Perform a stress analysis on the toppling dangerous rock mass. According to the damage-fracture mechanics theory, the stress distribution state of the toppling dangerous rock mass under the action of external loads and the length of the damage localization zone are used to calculate the fracture joint stress intensity factor of the toppling dangerous rock mass. S4. Based on the fracture toughness of the toppling dangerous rock and the joint stress intensity factor of the toppling dangerous rock fracture, a stability evaluation index is established to quantitatively evaluate the stability of the toppling dangerous rock.

2. A quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 1, characterized in that: In step S1, the crack tip area of the main control structural surface is divided into a non-damaged area, a continuous damage area and a damage localization zone based on the damage degree. The damage degree of the damage localization zone is lower than that of the continuous damage area, and the complete damage is concentrated in the damage localization zone area with a smaller width in front of the crack tip.

3. A quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 2, characterized in that: Step S2 specifically includes: S21. The stress drop at the crack tip is set to be concentrated in the damage localization zone, and the length of the damage localization zone does not exceed 10% of the crack length; S22. Based on the principle of linear elasticity, establish the stress distribution equation at the crack tip of the main control structure surface, analyze the normal stress in the y direction at the crack tip of the main control structure surface, and obtain the normal stress expression; S23. Apply concentrated force within the damage localization zone and calculate the stress intensity factor expression near the damage zone; S24. Calculate the stress expression near the crack tip based on linear elastic fracture mechanics theory and stress intensity factor expression; S25. Based on the elastic-brittle sudden damage model, the stress distribution in the local damage area is regarded as a constant. The length of the damage localization zone is calculated based on the normal stress expression and the stress expression near the crack tip.

4. A quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 3, characterized in that: In step S21, the calculation expression of normal stress is as follows: Where σ1(r) is the normal stress in the y direction at the crack tip of the main control structure surface, K I is the stress intensity factor of mode I fracture, r is the distance from the crack tip to the calculation point; In step S24, the stress expression near the crack tip is as follows: Where K D is the stress near the crack tip, e is the length of the damage localization zone, c is the half length of the crack, that is, half of the distance from the crack tip to the farthest end of the crack, σ(ε) is the stress state of the material under strain ε, and ξ is the local coordinate near the crack tip, which is used to describe the stress field distribution near the crack tip.

5. A quantitative evaluation method for the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 4, characterized in that: In step S25, the calculation expression of the damage localization zone length is: Where σ0 represents the initial yield stress of the material.

6. The method for quantitatively evaluating the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 1, characterized in that: Step S3 specifically includes: S31. The external load on the toppling dangerous rock mass is translated to the crack tip of the main control structure surface of the toppling dangerous rock mass through static equivalence, and the normal force, tangential force and bending moment force at the crack tip are decomposed based on the principle of mechanical equilibrium; S32. Decompose the toppling dangerous rock fracture model according to fracture mechanics and solve the fracture stress intensity factor under different stresses; S33. According to the principle of superposition of stress intensity factors, the combined stress intensity factor of the toppling dangerous rock fracture is calculated based on the fracture stress intensity factors under different stresses.

7. A method for quantitatively evaluating the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 6, characterized in that: In step S32, the fracture stress intensity factors under different stresses include: under the action of pore water pressure, tensile stress perpendicular to the main control structure surface, and tensile stress perpendicular to the main control structure surface, type I fracture occurs; under the action of shear parallel to the main control structure surface, type II fracture occurs.

8. A method for quantitatively evaluating the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 7, characterized in that: In step S33, the combined stress intensity factor of the toppling dangerous rock fracture includes the stress intensity factor of type I fracture and the stress intensity factor of type II fracture. The calculation expression of the combined stress intensity factor of the toppling dangerous rock fracture is: Where K I+Π is the combined stress intensity factor of the overturning dangerous rock fracture, K I is the stress intensity factor for mode I fracture, K Π is the stress intensity factor of mode II fracture, θ0 represents the crack extension angle, and is the angle between the crack surface and the principal stress direction.

9. A method for quantitatively evaluating the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 8, characterized in that: According to the maximum circumferential tensile stress criterion, the crack propagates along the direction of θ0 corresponding to the maximum tensile stress, and we get:

10. The method for quantitatively evaluating the stability of toppling dangerous rocks considering damage-fracture mechanics according to claim 1, characterized in that: In step S4, the stability evaluation index is the toppling dangerous rock stability coefficient, and the dangerous stability coefficient is the ratio of the fracture toughness of the rock to the toppling dangerous rock fracture combined stress intensity factor.

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