Method for determining tensile strength of transversely isotropic rock considering size effect
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CENT SOUTH UNIV
- Filing Date
- 2026-03-16
- Publication Date
- 2026-06-12
Smart Images
Figure CN122192897A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of rock mechanics and engineering technology, and in particular relates to a method for determining the tensile strength of transversely isotropic rocks considering size effects. Background Technology
[0002] In rock mechanics and engineering practice, accurately obtaining the mechanical parameters of rock masses is fundamental for stability analysis and engineering design. Tensile strength is one of the key mechanical indicators of rock. Numerous engineering instability cases demonstrate that rock mass failure often begins in areas of tensile stress concentration. However, the tensile strength of rock is far lower than its compressive strength and is extremely sensitive to its internal structure.
[0003] Approximately 75% of rocks in nature, such as sedimentary and metamorphic rocks, exhibit significant anisotropy due to their layered deposition or metamorphism. When this anisotropy manifests as rotational symmetry about a normal axis, the material can be simplified to a transversely isotropic model. This means that in planes parallel to bedding or foliation, the material properties are isotropic; however, in directions perpendicular to this plane, the properties differ. Therefore, the tensile strength of transversely isotropic rocks strongly depends on the angle between the direction of the external load and the principal anisotropic direction of the material (usually the bedding or foliation plane).
[0004] Furthermore, the size effect is another inherent characteristic of quasi-brittle materials like rocks. Strength values measured in the laboratory using centimeter or decimeter-sized specimens cannot directly represent the strength of rock masses at the engineering scale (meters or even tens of meters). This variation in strength with scale stems from the heterogeneity of the distribution of internal defects (such as microcracks, pores, and mineral grains) and their interactions during failure. Existing research has observed two phenomena: a "negative size effect" (strength decreases with increasing size) and a "positive size effect" (strength increases with increasing size), with different dominant mechanisms. Traditional methods typically consider isotropy and the size effect separately, or only one, leading to significant biases in predicting the engineering strength of transversely isotropic rock masses. Summary of the Invention
[0005] The purpose of this invention is to address the shortcomings of existing technologies by proposing a method for determining the tensile strength of transversely isotropic rocks that considers size effects.
[0006] To achieve the above objectives, the present invention employs the following technical solution: a method for determining the tensile strength of transversely isotropic rocks considering size effects, comprising the following steps: S1. Determine the elastic constants of transversely isotropic rocks; S2. Based on the tensile strength criterion of transversely isotropic rocks and the size effect law, a formula for calculating the tensile strength of transversely isotropic rocks considering the size effect is derived. S3. Conduct Brazilian splitting tests with different specimen sizes and loading angles, and calculate the tensile strength of the specimens based on the elastic constants obtained in step S1; S4. Based on the experimental data obtained in step S3, the calculation formula derived in step S2 is fitted to obtain a fitting formula for the tensile strength of transversely isotropic rocks considering the size effect. S5. Based on the fitting formula obtained in step S4, calculate the transversely isotropic rock tensile strength values from indoor test dimensions to engineering dimensions under different loading directions.
[0007] In a further preferred embodiment of the present invention, in step S1, determining the elastic constant of the transversely isotropic rock specifically involves determining the elastic modulus parallel to the transversely isotropic plane. Compared to Poisson And the elastic modulus perpendicular to the transversely isotropic plane Poisson's ratio and shear modulus .
[0008] In a further preferred embodiment of the present invention, step S1 uses a combination of Brazilian splitting tests and numerical simulations to determine the elastic constants, specifically including: A. Brazilian splitting tests were conducted on P-type and N-type disk specimens, and strain was measured using a 45° strain rosette attached to the center of the disk. And calculated using Formula 1 and Formula 2 below. and ; Formula 1 is: ; Formula 2 is: ; B. Set the initial stress concentration factor , , and the measured , , Substituting into Formula 3, we can solve for... , , Temporary values; Formula 3 is: ; C. Use the elastic constants obtained in step B for numerical modeling, conduct Brazilian splitting numerical simulation experiments, and then update the stress concentration factor using Equation 4. , , ; Formula 4 is: ; D. Repeat steps B and C until the calculated result is obtained. , , The value converges.
[0009] In a further preferred embodiment of the present invention, in step S2, the tensile strength criterion for the transversely isotropic rock is the Nova-Zaninetti criterion, the expression of which is Equation 5: ; In the formula and These represent the tensile strength of the rock matrix and the structural plane, respectively. The angle between the loading direction and the transversely isotropic plane.
[0010] In a further preferred embodiment of the present invention, in step S2, the size effect law can simultaneously describe both negative and positive size effect phenomena, as defined by formula 6: ; and formula 7: ; express.
[0011] In a further preferred embodiment of the present invention, in step S2, the formula for calculating the tensile strength of transversely isotropic rock considering size effects is obtained by substituting formula 5 into formulas 6 and 7, i.e., formula 8: ; In a further preferred embodiment of the present invention, step S3, the calculation of the tensile strength value of the specimen based on the elastic constant, specifically involves substituting the peak load P measured in the experiment and the elastic constant obtained in step S1 into formula 9 to calculate the tensile strength value of the specimen. ; Formula 9 is: ; In a further preferred embodiment of the present invention, in step S4, the data fitting specifically refers to: based on the measured tensile strength values corresponding to different specimen diameters d under different loading angles β obtained in step S3. Nonlinear regression analysis was performed on Formula 8 to fit and determine the material parameters in Formula 8, thereby obtaining the fitting formula for tensile strength with determined parameters.
[0012] In a further preferred embodiment of the present invention, in step S4, the material parameters include or including .
[0013] In a further preferred embodiment of the present invention, in step S5, the calculation of the transverse isotropic rock tensile strength value of the engineering dimensions specifically involves substituting the equivalent characteristic dimension D of the target engineering rock mass and the angle B between the engineering stress direction and the isotropic plane of the rock mass into the tensile strength fitting formula determined by the parameters obtained in step S4, and calculating the corresponding predicted tensile strength value.
[0014] The present invention has the following beneficial effects: This method solves a long-standing problem in the engineering strength assessment of transversely isotropic rocks. Traditional methods typically consider material anisotropy (isotropy) and size effects separately, or only one, leading to predictions that do not match engineering realities. This invention creatively couples the Nova-Zaninetti criterion, which describes strength anisotropy, with a theoretical model capable of characterizing both positive and negative size effects, establishing for the first time a unified quantitative relationship model of "tensile strength—specimen size—loading angle." This fills a gap in existing technology and provides a completely new theoretical framework for accurately predicting the tensile strength of rock masses under different stress directions at engineering scales.
[0015] This method combines high theoretical rigor with engineering practicality, forming a complete and reliable technical system. In its implementation, it employs an innovative step of "Brazilian splitting test combined with numerical simulation iterative inversion" to accurately determine five elastic constants of transversely isotropic rocks, laying an accurate foundation for subsequent analysis. Subsequently, strength data is obtained through systematic multivariate Brazilian tests, and calculations are performed using precise transversely isotropic strength formulas, ensuring the reliability of the basic data. Finally, nonlinear fitting is used to determine the material parameters in the theoretical model, transforming the model from a theoretical formula into a quantitative prediction tool for specific rocks. The entire process is clear and highly operable.
[0016] Ultimately, the direct value of this approach lies in providing crucial parameters for the safety and economic design of major engineering projects. By scientifically extrapolating the strength measured from small indoor specimens to actual engineering structural scales (such as slopes, tunnels, and mine pillars), and simultaneously considering the angle between the rock bedding direction and the direction of the engineering stress, a much more reliable tensile strength design value can be obtained than with traditional methods. This significantly improves the accuracy of stability analysis, disaster early warning, and optimization design for underground civil, hydraulic, mining, and energy engineering projects involving layered rock masses (such as shale, slate, and sedimentary rocks), and has significant engineering application value. Attached Figure Description
[0017] Figure 1 This is a flowchart illustrating a method for determining the elastic constants of transversely isotropic rocks, as provided by the present invention.
[0018] Figure 2A schematic diagram of the loading of the Brazilian splitting test on the P-type disk specimen provided by the method of the present invention.
[0019] Figure 3 A schematic diagram of the loading of the Brazilian splitting test on the N-type disk specimen provided by the method of the present invention.
[0020] Figure 4 This is a schematic diagram illustrating the relationship between the tensile strength of transversely isotropic rock, specimen size, and loading angle, provided for an embodiment of the present invention. Detailed Implementation
[0021] To facilitate understanding of the present invention, a more complete description will be given below with reference to the accompanying drawings. Preferred embodiments of the invention are shown in the drawings. However, the invention can be implemented in many different forms and is not limited to the embodiments described herein. Rather, these embodiments are provided to provide a thorough and complete understanding of the disclosure of the invention.
[0022] Please see Figure 1-4 In this invention, a technical solution is provided: A method for determining the tensile strength of transversely isotropic rocks considering size effects, comprising the following steps: Step 1: Determine the elastic constants of transversely isotropic rocks Transversely isotropic materials are fully defined by five independent elastic constants: the elastic modulus parallel to the isotropic plane. Compared to Poisson The elastic modulus perpendicular to the isotropic plane Compared to Poisson and shear modulus perpendicular to the isotropic plane. Accurately obtaining these five constants is fundamental to all subsequent analyses. This method employs an iterative inversion approach combining the Brazilian splitting experiment with numerical simulation, as follows: Figure 1 As shown.
[0023] Sample preparation and Brazilian splitting test Rock sample selection and processing: Select intact rock blocks with significant bedding or foliation structures from the engineering site or rock cores. Process them into two types of disc specimens: P-type specimen: The disk axis is parallel to the transversely isotropic plane (i.e., the bedding plane), such as Figure 2 As shown. The loading direction forms a certain angle with the bedding plane. .
[0024] N-type specimen: The disk axis is perpendicular to the transversely isotropic plane, such as... Figure 3 As shown.
[0025] Multiple specimens should be prepared for each type, and the diameter (d) and thickness (t) must be accurately measured and recorded. A thickness-to-diameter ratio (t / d) of approximately 0.5 is recommended, conforming to standard Brazilian splitting test recommendations.
[0026] Experimental setup and measurement: The disc specimen was subjected to diametrical compression loading using a material testing machine until failure. The peak load P was recorded.
[0027] A 45° strain rosette (three strain gauges at 45° angles to each other) was attached to the center of the specimen to measure the strain at that center. The strain rosette was measured in three directions: horizontal (H), 45° (45°), and vertical (V), and the corresponding strain readings were...
[0028] elastic constant and Preliminary calculations: Based on the stress-strain relationship at the center point of the Brazilian disk in elasticity, the strain rosette readings are first converted to normal strain and shear strain in the xy coordinate system (x-axis is the loading direction): (1) in, and for and The positive strain in the direction, For shear strain.
[0029] For P-type specimens, in the special case where the loading direction is parallel (β=0°) or perpendicular (β=90°) to the isotropic plane, the stress state at the center has a theoretical solution. Using this particular solution, the stress state can be directly calculated using the following formula. and : (2) The elastic modulus and Poisson's ratio calculated here are parallel to the isotropic plane.
[0030] elastic constant , , Iterative inversion Due to the complexity of the constitutive model of transversely isotropic materials, the other three elastic constants... , , It cannot be solved directly from a single experiment; an iterative method is required.
[0031] Compliance matrix of transversely viewed isotropic materials Relating stress to strain: (3) Flexibility coefficient It is the elastic constant and loading angle The function, specifically, is expressed as:
[0032]
[0033]
[0034]
[0035]
[0036]
[0037] The angle between the loading direction and the transversely isotropic plane.
[0038] Establish the inversion equation: Stress at the center point of the Brazilian disk The relationship with the load P can be expressed as: (4) in , , The stress concentration factor is a dimensionless coefficient that varies with the material's elastic constants and the loading angle. related.
[0039] Iterative solution process: Step A (Initial Setup and Solution): Set the initial guess value for the stress concentration factor, which can usually be taken as... (Corresponding to the theoretical value for isotropic materials). The experimentally measured... , , and known , and loading angle Substituting into formula ③, formula ③ now contains three equations. The unknown is... , , The system of equations (functions of the form ) is solved using numerical methods (such as least squares method and Newton's iteration method) to obtain the following results. , , A set of provisional estimates.
[0040] Step B (Numerical Simulation Update): The five elastic constants obtained in the previous step ( The material parameters are input into commercial software (such as FLAC3D and Abaqus) to establish a transversely isotropic Brazilian splitting numerical model that is completely consistent with the experiment (with the same dimensions and loading angle). Numerical simulation was performed to extract the simulated stress value at the center point of the disk. The load obtained from numerical simulation (Should be consistent with the test load P) and model dimensions, update the stress concentration factor using the following formula:
[0041] Step C (Iterative Convergence): Use the updated... , , Replace the initial guess, repeat step B, and solve for the next round of results. Values. Compare the values obtained from two consecutive iterations. If the relative errors are all less than the set convergence tolerance (e.g., 0.1%), the iteration terminates, and the last calculated value is the final determined elastic constant. Otherwise, continue numerical simulation with the new elastic constant (step B), update the stress concentration factor, and invert again until convergence.
[0042] Step 2: Establish a theoretical formula for the tensile strength of transversely isotropic rocks that considers size effects. This step aims to construct a theoretical framework for tensile strength that can simultaneously reflect both orientation anisotropy and size effects.
[0043] Cross-sectional tensile strength criterion for isotropic rocks: The Nova-Zaninetti criterion is used to describe the variation of tensile strength with loading angle β. This criterion is concise in form and has a clear physical meaning; its expression is: (5) in, Indicates the loading angle is The tensile strength at that time. The tensile strength representing the rock matrix (material), that is, the tensile strength perpendicular to the isotropic plane ( =0°) Strength under load. The tensile strength representing the bedding plane, i.e., parallel to the isotropic plane ( The strength under load at 90° (=90°). This formula can well describe the strength at... and The relationship between them changes continuously with angle.
[0044] Size effect law: Size effect law describes strength With feature size (e.g., changes in specimen diameter). To simultaneously cover both negative and positive size effects, this method employs two forms of laws: Negative size effect law (bounded type): Based on the theory of fracture energy release, the strength gradually decreases from the higher values of the small size and tends to a stable value.
[0045] (6) Positive size effect law (fractal type): Based on fractal fracture theory, the strength may increase with the size, which is common in some highly heterogeneous rocks.
[0046] (7) in, , Indicates diameter is The tensile strength of the specimen; , Indicates when the specimen size Theoretical tensile strength at →0 (intrinsic strength of material). This represents the macroscopic ultimate tensile strength as the specimen size d→∞; A dimensionless constant representing the heterogeneity of a material; The largest characteristic size inside the material (such as the largest mineral grain size); Fractal dimension representing crack distribution .
[0047] Coupling formula: Substituting the direction-dependent strength criterion (Equation 5) into the size effect law (Equations 6 and 7), we obtain the result that simultaneously considers size (Equations 6 and 7). ) and direction ( The formula for calculating the tensile strength of ) is: (8) in, , Indicates corresponding to of, =0° and Intrinsic tensile strength of the material in the 90° direction; , Corresponding to of, =0° and =90° macroscopic ultimate tensile strength; , Corresponding to of, =0° and Intrinsic tensile strength of the material in the 90° direction; , , These are material-related size effect parameters.
[0048] Step 3: Conduct Brazilian splitting tests to obtain strength data To calibrate the unknown parameters in formula (8) A systematic experiment is required.
[0049] Experimental Design: Multiple sets of Brazilian disc specimens were processed from the same rock block.
[0050] Size variables: Select at least 3-4 different diameters (e.g., 50mm, 75mm, 100mm, 150mm), keeping the thickness-to-diameter ratio t / d approximately constant.
[0051] Angular variables: For each specimen size, select at least 4-5 different loading angles β (e.g., 0°, 15°, 30°, 45°, 60°, 75°, 90°) covering the entire range from perpendicular to parallel bedding. Each size-angle combination should have 3-5 parallel specimens to ensure repeatability.
[0052] Testing and strength calculation: Brazilian splitting tests were performed on each specimen, and the peak load at failure was recorded. .
[0053] Using the five elastic constants of this type of rock determined in step one The tensile strength was calculated using the Brazilian splitting strength formula applicable to transversely isotropic materials. This formula takes into account the effect of anisotropy on stress distribution. (9) in, That is, the specimen is in terms of size and loading angle The measured tensile strength under the given load. Using this formula, we can obtain the measured load... This is converted into a value that reflects the true tensile strength of the material after eliminating the effects of anisotropic stress concentration. value.
[0054] Step 4: Data fitting to determine the quantitative relationship between strength, size, and angle Data processing: Compile the strength data of all specimens obtained in step three. According to the corresponding size d and angle Organize it.
[0055] Model selection and fitting: Observe the trend of strength with angle under different dimensions, and the trend of strength with size under the same angle. If the strength decreases and tends to stabilize as the size increases, then choose the formula (8). Model (negative size effect). If the strength initially increases and then decreases or continues to increase (within the test size range) with increasing size, then a model may be selected. The model (positive size effect) may require determining the dominant mechanism based on broader size data. Sometimes it may be necessary to fit two models with two sets of data separately for later use.
[0056] Using nonlinear regression analysis (such as curve fitting tools in software like MATLAB, Python SciPy, or Origin), the prepared... - - Experimental data for the selected formula (8) or Fit the data.
[0057] Fitting parameters: Through the fitting process, the material parameters in the formula are determined, i.e. (for (model) or (for Model).
[0058] Relationship established: After fitting, an empirical-theoretical model with defined parameters is obtained, which can quantitatively describe the relationship between "tensile strength - specimen size - loading angle", such as... Figure 4 The surface relationship shown. The model (i.e., the formula (8) after substituting the fitting parameters) is "the fitting formula for the tensile strength of transversely isotropic rocks considering the size effect".
[0059] Step 5: Calculate the tensile strength under the engineering dimensions Once the fitted formula is obtained, it can be applied to engineering prediction.
[0060] Determine engineering parameters: Engineering Dimension (D): The characteristic dimension is determined based on the scope of engineering concern. For example, the thickness of the loosened zone in tunnel excavation, the thickness of the potential sliding body on a slope, or the size of a pillar can be equivalently represented by the characteristic dimension d and substituted into the formula. This may require equivalence based on the specific failure mode.
[0061] Engineering loading direction (β): Analyze the angle between the principal stress direction of key parts in the engineering rock mass and the bedding / foliation direction of the rock strata, and substitute it as β into the formula.
[0062] Strength calculation: Substitute the engineering dimension D and the engineering loading angle B into the fitted formula (equation (8)) obtained in step four, whose parameters are already determined. or In the calculation, the predicted value of the tensile strength T(D,B) of the transversely isotropic rock is directly calculated.
[0063] Application of Results: The calculated T(D,B) represents the predicted tensile strength of rock mass at the engineering scale in the stress direction, taking into account material anisotropy and size effects. This value can be input into numerical analysis models (such as FLAC3D, UDEC, and finite element software) as a material strength parameter, or used in analytical calculations to assess engineering stability. Its reliability is far higher than that of directly using the strength values of small laboratory specimens.
[0064] 3. Key points and precautions for implementation Sample representativeness: The rock sample taken should be able to represent the main lithology, bedding / foliation development characteristics and integrity of the engineering rock mass.
[0065] Test accuracy: The loading conditions in the Brazilian splitting test must be precise, and the strain measurement system must be sensitive and reliable. The reading of the peak load should be accurate.
[0066] Numerical simulation reliability: In the iterative inversion of step one, the numerical model (such as mesh density, boundary conditions, and contact settings) should be verified to ensure that it can accurately reproduce the stress and strain field of the experiment.
[0067] Parameter fitting: The nonlinear fitting in step four may be sensitive to the initial parameter values. Multiple sets of initial values should be tried to ensure that the global optimum is obtained, and the goodness of fit should be evaluated (e.g., ).
[0068] Model applicability: The relationships established by this method are applicable to transversely isotropic rocks. For materials with stronger anisotropy, more complex constitutive and strength criteria may be required. The two size effect laws in formula (8) need to be selected based on the experimental data.
[0069] This invention details a complete technical path from experimentation to theory and then to engineering application. Through rigorous elastic constant inversion, systematic multi-size and multi-angle strength testing, and data fitting guided by theory, a tensile strength prediction model that can simultaneously consider anisotropy and size effects is finally established. This method overcomes the limitations of traditional methods that treat these two aspects separately, providing a more scientific and reliable solution for determining strength parameters in transversely isotropic rock engineering (such as layered slopes, underground caverns in sedimentary rock strata, shale gas extraction, etc.). In practical applications, engineers can refer to the implementation steps in this paper to obtain targeted material parameters and prediction formulas based on the specific conditions of the target rock mass, thereby significantly improving the safety and economy of engineering design.
[0070] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.
Claims
1. A method for determining the tensile strength of transversely isotropic rocks considering size effects, characterized in that, Includes the following steps: S1. Determine the elastic constants of transversely isotropic rocks; S2. Based on the tensile strength criterion of transversely isotropic rocks and the size effect law, a formula for calculating the tensile strength of transversely isotropic rocks considering the size effect is derived. S3. Conduct Brazilian splitting tests with different specimen sizes and loading angles, and calculate the tensile strength of the specimens based on the elastic constants obtained in step S1; S4. Based on the experimental data obtained in step S3, the calculation formula derived in step S2 is fitted to obtain a fitting formula for the tensile strength of transversely isotropic rocks considering the size effect. S5. Based on the fitting formula obtained in step S4, calculate the transversely isotropic rock tensile strength values from indoor test dimensions to engineering dimensions under different loading directions.
2. The method according to claim 1, characterized in that, In step S1, determining the elastic constants of the transversely isotropic rock specifically involves determining the elastic modulus parallel to the transversely isotropic plane. Compared to Poisson And the elastic modulus perpendicular to the transversely isotropic plane Poisson's ratio and shear modulus .
3. The method according to claim 2, characterized in that, Step S1 uses a combination of Brazilian splitting tests and numerical simulations to determine the elastic constants, specifically including: A. Brazilian splitting tests were conducted on P-type and N-type disk specimens, and strain was measured using a 45° strain rosette attached to the center of the disk. And calculated using Formula 1 and Formula 2 below. and ; Formula 1 is: ； Formula 2 is: ； B. Set the initial stress concentration factor , , and the measured , , Substituting into Formula 3, we can solve for... , , Temporary values; Formula 3 is: ； C. Use the elastic constants obtained in step B for numerical modeling, conduct Brazilian splitting numerical simulation experiments, and then update the stress concentration factor using Equation 4. , , ; Formula 4 is: ; D. Repeat steps B and C until the calculated result is obtained. , , The value converges.
4. The method according to claim 1, characterized in that, In step S2, the tensile strength criterion for transversely isotropic rocks is the Nova-Zaninetti criterion, and its expression is Equation 5: ； In the formula and These represent the tensile strength of the rock matrix and the structural plane, respectively. The angle between the loading direction and the transversely isotropic plane.
5. The method according to claim 4, characterized in that, In step S2, the size effect The sizing law can describe both negative and positive size effects simultaneously, as given by Equation 6: ； and formula 7: ； express.
6. The method according to claim 5, characterized in that, In step S2, the formula for calculating the tensile strength of transversely isotropic rock considering size effects is obtained by substituting formula 5 into formulas 6 and 7, i.e., formula 8: 。 7. The method according to claim 1, 2, 3, 4, 5 or 6, characterized in that, In step S3, the calculation of the tensile strength value of the specimen based on the elastic constant specifically involves substituting the peak load P measured in the experiment and the elastic constant obtained in step S1 into formula 9 to calculate the tensile strength value of the specimen. ; Formula 9 is: 。 8. The method according to claims 6 and 7, characterized in that, In step S4, the data fitting specifically involves: based on the measured tensile strength values corresponding to different specimen diameters d under different loading angles β obtained in step S3. Nonlinear regression analysis was performed on Formula 8 to fit and determine the material parameters in Formula 8, thereby obtaining the fitting formula for tensile strength with determined parameters.
9. The method according to claim 8, characterized in that, In step S4, the material parameters include or including .
10. The method according to claim 1, characterized in that, In step S5, the calculation of the transverse isotropic rock tensile strength value of the engineering dimensions specifically involves substituting the equivalent characteristic dimension D of the target engineering rock mass and the angle B between the engineering stress direction and the isotropic plane of the rock mass into the tensile strength fitting formula determined by the parameters obtained in step S4, and calculating the corresponding predicted tensile strength value.