A method for determining the safety factor of slopes applicable to smoothed particle hydrodynamics

Abstract

The present invention relates to a method for determining the safety factor of a slope applicable to smoothed particle hydrodynamics, belonging to the technical field of slope stability analysis. Based on the geometric parameters of the slope height and slope angle, the present invention establishes a numerical calculation model of the slope; determines the mechanical parameters of the slope rock and soil materials through mechanical tests, and constructs a strength attenuation strain softening model; uses the linked list search method to search and pair the domain particles of the slope numerical calculation model; solves the particle stress and strain according to the strength attenuation strain softening model; performs stress adjustment according to the particle stress and strain state; calculates the particle acceleration according to the adjusted particle stress, and updates the particle field variables; wherein the particle field variables include velocity, density and displacement; selects displacement characteristic points, and judges the slope stability according to the triple criterion. If the criterion is satisfied, the safety factor is output to evaluate the overall stability state of the slope; otherwise, it enters the automatic reduction program and continues to iteratively solve the slope stability until the criterion requirements are met.

Description

Technical Field

[0001] The present invention relates to a method for determining the safety factor of a slope applicable to smooth particle hydrodynamics, belonging to the technical field of slope stability analysis. Background Art

[0002] Slope stability analysis is a key research issue in the field of geotechnical mechanics. Numerical prediction of this issue can provide theoretical guidance for engineering design and practice. In the past few decades, the limit equilibrium method (LEM) has been widely used for slope analysis, such as the methods of Fellenius, Bishop, Janbu, Morgenstern, and Spencer. These methods have a common feature that they can calculate the safety factor of the slope and predict the slip surface without the stress-strain relationship of the soil mass and initial slope conditions. However, this method has some important defects: it ignores the deformation characteristics of the soil mass, and many assumptions need to be made about the interlayer force and the shape and position of the failure surface before slope stability analysis. Therefore, to solve the soil deformation problem, Zinkiewicz et al. proposed the strength reduction finite element method (FEM). This method considers the stress-strain relationship of the soil mass, makes no assumptions about the direction of the sliding force and the shape and position of the failure surface, and the slip surface is automatically confirmed, and the slope naturally fails. However, this method has low calculation efficiency, and the calculation accuracy depends to a large extent on the user's experience. For geotechnical engineering, the finite element method has more advantages than the LEM method. However, in geotechnical engineering, problems involving large deformations and failures such as landslide problems, in this case, even if the modified Lagrangian method is used, the finite element method will have numerical accuracy defects due to the mesh distortion problem.

[0003] For the above reasons, meshless methods or particle methods have developed rapidly and are applied to simulate large deformation problems of continuous media or dispersed media. So far, the most popular method is the discrete element method (DEM). It simulates the contact between particles through a spring and damping system to track the movement of a large number of particles. The advantage of this method is that it can handle large deformation and failure problems and is relatively simple and easy to implement in computer code. However, its accuracy is low, and the contact model is difficult to determine. Smooth particle hydrodynamics (SPH) is a pure Lagrangian meshless method that discretizes the problem domain into a series of particle sets with field variables, such as mass, density, stress tensor, etc. By transforming the partial differential equation of the continuous medium into the particle motion equation, and then solving it with the updated Lagrangian numerical format. Compared with other grid-based numerical methods, smooth particle hydrodynamics (SPH) can handle large deformation and post-failure problems well due to its adaptive characteristics.

[0004] However, the current safety factor calculation method of smooth particle hydrodynamics (SPH) has limitations although it has high calculation accuracy, and has defects such as cumbersome calculation and the need to solve eigenvalues through multiple iterations. Summary of the Invention

[0005] In view of the technical problems existing in the existing smooth particle hydrodynamics (SPH) safety factor calculation method, such as cumbersome calculation and the need to iteratively solve eigenvalues multiple times, the present invention provides a method for determining the slope safety factor applicable to smooth particle hydrodynamics. That is, based on the SPH framework, mechanical tests are used to obtain the strength parameter characteristics of slope rock and soil materials and their dynamic attenuation laws. Based on the ideal elastoplastic constitutive model, a strain-softening model characterizing the attenuation of material strength parameters is constructed. Based on the automatic strength reduction method and the maximum shear strain increment, the position of the critical slip surface under the static state of the slope is determined. Furthermore, in combination with the theory of dynamic progressive failure of the slip surface, the displacement mutation curve of the safety factor characteristic point of the slope is solved in numerical simulation calculations, truly reflecting the stability state of the slope under its own weight, improving the numerical calculation efficiency and ensuring the accuracy and reliability of the results.

[0006] A method for determining the slope safety factor applicable to smooth particle hydrodynamics is as follows:

[0007] (1) Based on the geometric parameters of the slope height and slope angle, establish a numerical calculation model of the slope;

[0008] (2) Determine the mechanical parameters of the slope rock and soil materials through mechanical tests, where the mechanical parameters of the slope rock and soil materials include density, elastic modulus, Poisson's ratio, cohesion, internal friction angle, and dilation angle; construct a strength attenuation strain-softening model;

[0009] (3) Use the linked list search method to search and pair neighboring particles in the numerical calculation model of the slope;

[0010] (4) According to the strength attenuation strain-softening model, solve the particle stress and strain;

[0011] (5) Perform stress adjustment according to the particle stress and strain state;

[0012] (6) Calculate the particle acceleration according to the adjusted particle stress and update the particle field variables; where the particle field variables include velocity, density, and displacement;

[0013] (7) Select displacement characteristic points and judge the slope stability according to triple criteria. If the criteria are met, output the safety factor and evaluate the overall stability state of the slope; otherwise, enter the automatic reduction program and continue to iteratively solve the slope stability until the criteria requirements are met.

[0014] The construction method of the strength attenuation strain-softening model in step (2) is

[0015] Use the nonlinear softening theory to approximately characterize the attenuation law of the strength parameters of rock and soil materials, and establish a model with the cumulative plastic deviator strain e p as the independent variable and the internal friction angle The expression of the strain-softening model relationship with cohesion χ as the dependent variable is

[0016]

[0017] Where: χ c is the residual cohesion of the slope; χ k is the peak cohesion of the slope; Similarly, represent the residual internal friction angle and the peak internal friction angle respectively; λ is the plastic scalar factor; h and ζ represent the yield function and the plastic potential function respectively; ε is the strain; dk is the softening control coefficient; c e is the elastic stiffness matrix; sig is the element stress tensor.

[0018] The particle stress and strain calculation method in step (4) is

[0019]

[0020] Where: sig is the element stress tensor; str is the element rotation tensor; G and K represent the shear modulus and the bulk modulus respectively; a ψ is the element plastic potential function coefficient; J2 is the second deviatoric stress invariant; s is the element deviatoric stress tensor; δ is the Dirac function; λ represents the plastic scalar factor.

[0021] The stress adjustment method in step (5) is

[0022]

[0023]

[0024] Where represents the components of the artificial stress tensor in the xx, xy, and yy directions respectively; θ i is the projection angle; Rs i 'xx 、Rs i 'yy represent the stress components in the xx and yy directions on the original coordinate axes; rho is the particle density; represents the components of the stress tensor in the xx, yy, and xy directions respectively.

[0025] The particle field variable update method in step (6) is

[0026]

[0027] u n+1 / 2 =u n +Δtv n+1 / 2

[0028] where: rho is the density; Δt is the time step; v is the velocity; u is the displacement; n, n+1 / 2, and n-1 / 2 represent the current iteration time step, the predicted half iteration time step, and the previous half iteration time step, respectively.

[0029] The method for judging the slope stability by the triple criterion in step (7) is

[0030] 1) Introduce the displacement increment tolerance value ΔR;

[0031] 2) Select the displacement characteristic points, and judge the relationship between the displacement increment of the characteristic points and ΔR. If the displacement increment of the characteristic points is greater than the tolerance value, that is, ΔR≥u n+1 -u n , then enter the next level of judgment, that is, enter step 3), otherwise exit the conditional judgment and enter the automatic strength reduction program iteration;

[0032] 3) Judge the acceleration of the characteristic points. If the output acceleration of this point is positive, that is then it is considered that the displacement of this point increases as a concave function, and enter the next level of judgment, that is, enter step 4), otherwise exit the conditional judgment and enter the automatic strength reduction program iteration;

[0033] 4) Judge the speed growth of the characteristic points in the preset iteration steps. If the speed is always increasing, that is then judge the displacement mutation of this point and output the safety factor, otherwise exit the conditional judgment and enter the automatic strength reduction program iteration.

[0034] The preset iteration steps in step 4) are 2000 to 3500.

[0035] The algorithm of the automatic strength reduction program is

[0036] χ t+1 =χ t / (ln itimestep / 1000+1 +0.5)

[0037]

[0038] where χ t 、χ t+1 represent the particle cohesion at time t and t+1, respectively; similarly represent the particle internal friction angle at time t and t+1, respectively; itimestep represents the number of iteration steps.

[0039] The method for evaluating the overall stability state of the slope is

[0040] If the safety factor is greater than 1, it is judged that the overall slope is in a relatively stable state; if the safety factor is equal to 1, it is judged that the overall slope is in a critical state; if the safety factor is less than 1, it is judged that the overall slope is in an unstable state.

[0041] The beneficial effects of the present invention are as follows:

[0042] (1) During the numerical calculation process of the present invention, the whole process of progressive failure of the slope is considered, and a strain-softening model is introduced to characterize the attenuation characteristics of the strength parameters of geotechnical materials under self-weight. It overcomes the defect that the strength parameters remain unchanged during the calculation process of the traditional elastoplastic model, resulting in an overestimated safety factor calculation result and exacerbating the potential danger of the project;

[0043] (2) The present invention has clear physical meaning, simple principle, high reliability, and higher efficiency than the traditional SPH safety factor calculation method. It can obtain a more realistic slope stability coefficient, provide a theoretical basis for studying the formation mechanism of slope landslide disasters, earthquake prevention and disaster reduction, and the design of support systems, and has good prospects for practical engineering applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is the flow chart of the present invention;

[0045] Figure 2 is the flow chart of the triple criterion for slope safety factor;

[0046] Figure 3 is the model of the example in Embodiment 2;

[0047] Figure 4 is the position map of the critical state slip surface of the slope in Embodiment 2;

[0048] Figure 5 is the calculation diagram of the slope safety factor in Embodiment 2;

[0049] Figure 6 is the comparison diagram between Embodiment 2 of the present invention and the traditional safety factor calculation; DETAILED DESCRIPTION OF THE INVENTION

[0050] The present invention will be further described in detail below in conjunction with the specific embodiments, but the protection scope of the present invention is not limited to the content described.

[0051] Embodiment 1: A method for determining the slope safety factor applicable to smoothed particle hydrodynamics (see Figure 1 ), and the specific steps are as follows:

[0052] (1) Based on the geometric parameters of the slope height and slope angle, a slope numerical calculation model is established by independently writing a program in Fortran language;

[0053] (2) Determine the mechanical parameters of the slope geotechnical materials through mechanical tests, where the mechanical parameters of the slope geotechnical materials include density, elastic modulus, Poisson's ratio, cohesion, internal friction angle, and dilation angle; construct a strength attenuation strain-softening model;

[0054] The construction method of the strength attenuation strain softening model is as follows

[0055] The nonlinear softening theory is used to approximately characterize the attenuation law of the strength parameters of geotechnical materials, and a strain softening model relationship expression with the cumulative plastic deviator strain e p as the independent variable, the internal friction angle and the cohesion χ as the dependent variables is

[0056]

[0057] In the formula: χ c is the residual cohesion of the slope; χ k is the peak cohesion of the slope; Similarly, represent the residual internal friction angle and the peak internal friction angle respectively; λ is the plastic scalar factor; h and ζ represent the yield function and the plastic potential function respectively; ε is the strain; dk is the softening control coefficient; c e is the elastic stiffness matrix; sig is the element stress tensor.

[0058] (3) The linked list search method is used to search and pair the neighboring particles of the slope numerical calculation model;

[0059] (4) According to the strength attenuation strain softening model, the particle stress and strain are solved;

[0060] The calculation method of the particle stress and strain is

[0061]

[0062] In the formula: sig is the element stress tensor; str is the element rotation tensor; G and K represent the shear modulus and the bulk modulus respectively; a ψ is the coefficient of the element plastic potential function; J2 is the second deviator stress invariant; s is the element deviator stress tensor; δ is the Dirac function; λ represents the plastic scalar factor.

[0063] (5) According to the particle stress and strain state, the stress is adjusted;

[0064] The stress adjustment method is

[0065]

[0066]

[0067] In the formula represents the components of the artificial stress tensor in the xx, xy, and yy directions respectively; θ i is the projection angle; Rs i 'xx 、Rs i 'yyIndicates the stress components in the xx and yy directions on the original coordinate axes; rho is the particle density; Indicates the components of the stress tensor in the xx, yy, and xy directions respectively.

[0068] (6) Calculate the particle acceleration based on the adjusted particle stress and update the particle field variables; where the particle field variables include velocity, density, and displacement;

[0069] The update method of the particle field variables is

[0070]

[0071] u n+1 / 2 = u n + Δt v n+1 / 2

[0072] In the formula: rho is the density; Δt is the time step; v is the velocity; u is the displacement; n, n + 1 / 2, and n - 1 / 2 respectively represent the current iteration time step, the predicted half iteration time step, and the previous half iteration time step;

[0073] (7) Select displacement characteristic points, judge the slope stability according to the triple criterion. If the criterion is met, output the safety factor and evaluate the overall stability state of the slope; otherwise, enter the automatic reduction program and continue to iteratively solve the slope stability until the criterion requirements are met;

[0074] The method of judging slope stability by the triple criterion (see Figure 2 ) is

[0075] 1) Introduce the displacement increment tolerance value ΔR;

[0076] 2) Select displacement characteristic points and judge the relationship between the displacement increment of the characteristic points and ΔR. If the displacement increment of the characteristic points is greater than the tolerance value, that is, ΔR ≥ u n+1 - u n , then enter the next judgment, that is, enter step 3), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration;

[0077] 3) Judge the acceleration situation of the characteristic points. If the output acceleration of this point is positive, that is, then consider that the displacement of this point increases as a concave function and enter the next judgment, that is, enter step 4), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration;

[0078] 4) Judge the velocity growth situation of the characteristic points in the preset iteration steps. If the velocity is always increasing, that is, then judge that the displacement of this point mutates and output the safety factor, otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration; where the preset iteration steps are 2000 - 3500;

[0079] The algorithm of the automatic strength reduction program is

[0080] χ t+1 = χ t / (ln itimestep / 1000+1 + 0.5)

[0081]

[0082] In the formula, χ t , χ t+1 respectively represent the particle cohesion at time t and t + 1; similarly respectively represent the particle internal friction angle at time t and t + 1; itimestep represents the number of iteration steps;

[0083] The evaluation method for the overall stability state of the slope is

[0084] If the safety factor is greater than 1, it is judged that the overall slope is in a relatively stable state; if the safety factor is equal to 1, it is judged that the overall slope is in a critical state; if the safety factor is less than 1, it is judged that the overall slope is in an unstable state.

[0085] Example 2: A method for determining the safety factor of a slope applicable to smooth particle hydrodynamics (see Figure 1 ), the specific steps are as follows:

[0086] (1) Conduct engineering geological exploration on the stratigraphic occurrence of a certain slope field area, and establish a slope numerical calculation model (see Figure 3 ) based on the geometric parameters of the slope height and slope angle;

[0087] (2) Determine the mechanical parameters of the slope rock and soil materials through mechanical tests (see Table 1), where the mechanical parameters of the slope rock and soil materials include density, elastic modulus, Poisson's ratio, cohesion, internal friction angle, and dilatancy angle; construct a strength attenuation strain softening model;

[0088] Table 1 Mechanical parameters of rock and soil mass

[0089]

[0090] Among them, the construction method of the strength attenuation strain softening model is:

[0091] Adopt the non-linear softening theory to approximately characterize the attenuation law of the strength parameters of rock and soil materials, and establish a strain softening model relationship expression with the cumulative plastic deviator strain e p as the independent variable, the internal friction angle cohesion χ as the dependent variable, and the expression is

[0092] That is

[0093]

[0094] That is

[0095]

[0096]

[0097] In the formula: χ c is the residual cohesion of the slope; χ k is the peak cohesion of the slope; Similarly represent the residual internal friction angle and the peak internal friction angle respectively; λ is the plastic scalar factor; h and ζ represent the yield function and the plastic potential function respectively; ε is the strain; dk is the softening control coefficient; c e is the elastic stiffness matrix; sig is the element stress tensor.

[0098] (3) Use the linked list search method to perform neighborhood particle search and pairing on the slope numerical calculation model;

[0099] (4) According to the strength attenuation strain softening model, solve the particle stress and strain;

[0100] The calculation method of particle stress and strain is

[0101]

[0102] In the formula: sig is the element stress tensor; str is the element rotation tensor; G and K represent the shear modulus and the bulk modulus respectively; a ψ is the coefficient of the element plastic potential function; J2 is the second deviatoric stress invariant; s is the element deviatoric stress tensor; δ is the Dirac function; λ is the plastic scalar factor; h and ζ represent the yield function and the plastic potential function respectively; ε is the strain; dk is the softening control coefficient; c e is the elastic stiffness matrix.

[0103] (5) According to the particle stress and strain state, perform stress adjustment;

[0104] The stress adjustment method is

[0105]

[0106] In the formula represents the xx, xy, and yy direction components of the artificial stress tensor respectively; θ i is the projection angle; Rs i 'xx 、Rs i 'yy are the xx and yy direction stress components on the original coordinate axis; rho is the particle density; represents the xx, yy, and xy direction components of the stress tensor respectively.

[0107] (6) Calculate the particle acceleration based on the adjusted particle stress and update the particle field variables, where the particle field variables include velocity, density, and displacement.

[0108] The update method of the particle field variables is

[0109]

[0110] u n+1 / 2 = u n + Δt v n+1 / 2

[0111] In the formula: rho is the density; Δt is the time step; v is the velocity; u is the displacement; n, n + 1 / 2, and n - 1 / 2 respectively represent the current iteration time step, the predicted half iteration time step, and the previous half iteration time step.

[0112] To reflect the stability characteristics of the slope considering progressive failure, a strain-softening model is used to analyze the slope. Under the action of self-weight, local failure of the slope gradually expands towards the interior of the slope and finally slides along the slip zone. The whole process of slope landslide calculated using the softening model is as Figure 4 shown;

[0113] (7) Select displacement characteristic points, judge the slope stability based on the triple criterion. If the criterion is satisfied, output the safety factor; otherwise, enter the automatic reduction program and continue to iteratively solve the slope stability until the criterion requirements are met.

[0114] The method of judging slope stability by the triple criterion (see Figure 2 ) is

[0115] 1) Introduce the displacement increment tolerance value ΔR, and take 0.001 after multiple tests.

[0116] 2) Select displacement characteristic points and judge the relationship between the displacement increment of the characteristic points and ΔR. If the displacement increment of the characteristic points is greater than the tolerance value, that is, ΔR ≥ u n+1 - u n , then enter the next judgment, that is, enter step 3), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration.

[0117] 3) Judge the acceleration situation of the characteristic points. If the output acceleration of this point is positive, that is , then consider that the displacement of this point is increasing as a concave function, enter the next judgment, that is, enter step 4), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration.

[0118] 4) Judge the velocity growth situation of the characteristic points in the preset iteration steps. If the velocity is always increasing, that is Then, judge the safety factor by the sudden change of the displacement at this point. Otherwise, exit the condition judgment and enter the iterative process of the automatic strength reduction program. The preset number of iterative steps is 2000 - 3500;

[0119] The algorithm of the automatic strength reduction program is

[0120] χ t+1 = χ t / (ln itimestep / 1000+1 + 0.5)

[0121]

[0122] In the formula, χ t , χ t+1 respectively represent the particle cohesion at time t and time t + 1. Similarly respectively represent the particle internal friction angle at time t and time t + 1. itimestep represents the number of iterative steps;

[0123] The safety factor is calculated as Figure 5 shown. From Figure 5 (b), it can be seen that the displacement increment of the characteristic point suddenly changes when the shear strength reduction factor is 1.1, and continues to iterate 3500 steps according to the triple criterion, and the velocity increment is stable, that is, the slope safety factor is judged to be 1.1. From Figure 5 (a), it can be seen that the introduction of the triple criterion effectively avoids the misjudgment phenomenon of the "sudden change" of the displacement characteristic point during the automatic strength reduction process. For example, in the section from Fs = 0.9 to Fs = 1.0, the displacement suddenly increases, but the upper concave phenomenon appears after continuous iteration, which is contradictory to the displacement sudden change criterion;

[0124] To verify the efficiency and accuracy of the slope safety factor determination method applicable to smooth particle hydrodynamics, the safety factor calculation method proposed in this embodiment and the traditional SPH safety factor calculation method are respectively used to conduct a comparative analysis on the same model as Figure 6 (a), (b) shown; it can be seen that the safety factor calculation method proposed by the present invention has high efficiency, and the slope safety factor can be obtained in one iteration and the time step is 9000 steps. Compared with the traditional safety factor calculation method, multiple iterations are required to obtain the safety factor and the time step of each iteration is 20000 steps, and the safety factors obtained by both are the same;

[0125] Output the slope safety factor, where the safety factor is shown in Table 2

[0126] Table 2 Safety Factor

[0127] Material parameter model Softening model Safety factor 1.1

[0128] (8) According to the output safety factor, conduct an overall stability state evaluation of the slope:

[0129] The method for evaluating the overall stability state of the slope is

[0130] If the safety factor is greater than 1, it is determined that the overall slope is in a relatively stable state; if the safety factor is equal to 1, it is determined that the overall slope is in a critical state; if the safety factor is less than 1, it is determined that the overall slope is in an unstable state; since the safety factor in Table 2 is 1.1, it is determined that the slope in this embodiment belongs to the relatively stable state.

[0131] The specific embodiments of the present invention have been described in detail above, but the present invention is not limited to the above embodiments. Within the scope of knowledge possessed by those of ordinary skill in the art, various changes can be made without departing from the gist of the present invention.

Claims

1. A method for determining the slope safety factor applicable to smoothed particle hydrodynamics, characterized in that: The specific steps are as follows: (1) Based on the geometric parameters of the slope height and slope angle, establish a numerical calculation model of the slope; (2) Determine the mechanical parameters of the slope rock and soil materials through mechanical tests. The mechanical parameters of the slope rock and soil materials include density, elastic modulus, Poisson's ratio, cohesion, internal friction angle, and dilation angle; construct a strength degradation strain softening model; (3) Use the linked list search method to perform neighborhood particle search and pairing on the numerical calculation model of the slope; (4) Solve the particle stress and strain according to the strength degradation strain softening model; (5) Perform stress adjustment according to the particle stress and strain state; (6) Calculate the particle acceleration based on the adjusted particle stress and update the particle field variables; the particle field variables include velocity, density, and displacement; (7) Select the displacement characteristic points and judge the slope stability according to the triple criterion. If the criterion is met, output the safety factor and evaluate the overall stability state of the slope; otherwise, enter the automatic reduction program and continue to iteratively solve the slope stability until the criterion requirements are met; The method for judging the slope stability by the triple criterion is 1) Introduce the displacement increment tolerance value ΔR; 2) Select the displacement characteristic points and judge the relationship between the displacement increment of the characteristic points and ΔR. If the displacement increment of the characteristic points is greater than the tolerance value, enter the next judgment, that is, enter step 3), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration; 3) Judge the acceleration of the characteristic points. If the output acceleration of this point is positive, it is considered that the displacement of this point increases as a concave function, and enter the next judgment, that is, enter step 4), otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration; 4) Judge the velocity growth of the characteristic points in the preset iteration step. If the velocity is always increasing, judge that the displacement of this point mutates and output the safety factor, otherwise, exit the conditional judgment and enter the automatic strength reduction program iteration; The algorithm of the automatic strength reduction program is χ t+1 = χ t / (ln itimestep / 1000+1 + 0.5) where χ t , χ t+1 represent the particle cohesion at time t and t + 1, respectively; similarly represent the particle internal friction angle at time t and t + 1, respectively; itimestep represents the number of iteration steps.

2. The method for determining the slope safety factor applicable to smoothed particle hydrodynamics according to claim 1, characterized in that: The construction method of the strength degradation strain softening model in step (2) is The strength parameter attenuation law of geotechnical materials is approximately characterized by the non-linear softening theory, and a strain softening model relational expression with the cumulative plastic deviator strain e p as the independent variable, the internal friction angle and the cohesion χ as the dependent variables is Where: χ c is the residual cohesion of the slope; χ k is the peak cohesion of the slope; similarly, represent the residual internal friction angle and the peak internal friction angle respectively; λ is the plastic scalar factor; h and ζ represent the yield function and the plastic potential function respectively; ε is the strain; dk is the softening control coefficient; c e is the elastic stiffness matrix; sig is the element stress tensor.

3. The method for determining the slope safety factor applicable to smoothed particle hydrodynamics according to claim 1, characterized in that: The calculation method of particle stress and strain in step (4) is where sig is the element stress tensor; str is the element rotation tensor; G and K represent the shear modulus and the bulk modulus respectively; a ψ is the coefficient of the element plastic potential function; J2 is the second deviatoric stress invariant; s is the element deviatoric stress tensor; δ is the Dirac function; λ represents the plastic scalar factor.

4. The method for determining the slope safety factor applicable to smoothed particle hydrodynamics according to claim 1, characterized in that: The stress adjustment method in step (5) is In the formula denote the components of the artificial stress tensor in the xx, xy, and yy directions respectively; θ i is the projection angle; denote the stress components in the xx and yy directions on the original coordinate axes; rho is the particle density; denote the components of the stress tensor in the xx, yy, and xy directions respectively.

5. The method for determining the slope safety factor applicable to smoothed particle hydrodynamics according to claim 1, characterized in that: The update method of particle field variables in step (6) is u n+1 / 2 = u n + Δtv n+1 / 2 Where: rho is the density; Δt is the time step; v is the velocity; u is the displacement; n, n + 1 / 2, n - 1 / 2 respectively represent the current iteration time step, the predicted half iteration time step, and the previous half iteration time step.

6. The method for determining the slope safety factor applicable to smoothed particle hydrodynamics according to claim 1, characterized in that: The method for evaluating the overall stability state of the slope is If the safety factor is greater than 1, it is judged that the overall slope is in a relatively stable state; if the safety factor is equal to 1, it is judged that the overall slope is in a critical state; if the safety factor is less than 1, it is judged that the overall slope is in an unstable state.

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