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This article was submitted to Geohazards and Georisks, a section of the journal Frontiers in Earth Science

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The limit equilibrium method (LEM) or finite element method (FEM) for slope problems most frequently focusses on the stability analysis. There are, however, still some problems with the LEM or FEM when considering damage and failure evolution of a rock slope because of the distortion of mesh. In this work, a mesh-free particle approach, named the smoothed particle hydrodynamics (SPH) method, is presented and is improved to analyze the damage and failure process of a rock slope. In order to better describe the cause and mechanism of brittle failure for a rock slope, the plastic factor was suggested and introduced into the SPH algorithm, and the conservation equations of SPH for brittleness characteristics were obtained. Based on the variation of displacement and time, an effective criterion was proposed to define the factor of safety in SPH simulation. The Drucker-Prager Mohr-Coulomb strength criterion was implemented into the SPH algorithm to describe the elastic-plastic behavior. Then, three rock-slope models with different precast cracks were analyzed to illustrate the performance of the proposed method. It is shown that the proposed SPH algorithm can be effectively applied in the prediction of the deformation and failure process of rock slope.

Collapse is a widespread phenomenon in both natural and excavated rock slopes. It is a mass movement of rock characterized by downslope sliding, which can involve damage extension, penetration, and collapse. Therefore, the collapse of a rock slope has the obvious characteristics of a large deformation. The large deformation and failure process of rock slopes are so complex that obtaining an analytical solution is very difficult. In this sense, how to correctly describe the large deformation characteristics of rock slopes has been a hot but difficult problem in recent years. Compared with the complexities of experimental research and the limitations of theoretical research (

In recent years, some numerical methods have become increasingly popular to analyze large deformation and post-failure of slope, such as the Discrete Element Method (DEM) (

A pure Lagrange mesh-less recently numerical method, namely, the smoothed particle hydrodynamics (SPH), was originally used in astrophysical problems (

The object of this paper is to use the SPH method to model the large deformation and failure process of rock slopes. For large deformation and failure process of slope problems, its essence is the elastic-plastic deformation (

In the SPH method, an object is expressed as an assembly of particles with associated variables, such as mass, energy, and stress tensors. The basic idea behind this method is to provide stable and accurate numerical solutions for partial differential equations (PDEs) using a group of particles. The SPH method is based on interpolation theory. The governing equations, in the form of PDEs, can be transformed into SPH form through two main steps. The first step is to represent a function in continuous form as an integral representation using an interpolation function:

In this study, the governing equations are mainly based on the solid mechanics. So, the equations of continuity and motion can be expressed as follows:

Equation of continuity:

Equation of motion:

The differential form of the conservation equations can be converted to a discretized weak form as:

Here, the elastic-plastic constitutive model of soil material implemented in the SPH code is described in detail. The component of the total strain rate tensor is given by:

For the elastoplastic material, the total strain rate tensor

Rock material is a kind of brittle material. When the load reaches the yield strength, it will be damaged and weakened, at which point it is an elastic-plastic body. In this paper, the D-P criterion is selected as the yield criterion of elastoplastic materials. The expression is as follows:

for 2D plane strain conditions and

In the present study, the non-associated flow rule is considered to determine the stress-strain relationship. The plastic potential function,

Here, the D-P yield criterion is employed to determine the plastic region of soils. Plastic strain is initiated from a particle when stresses in the particle satisfy the D-P yield criterion. Once plastic state is reached in one particle, the plastic deformation is initiated from this particle. Therefore, the discrete conservation equations of SPH for plastic characteristics can be expressed as:

Particle deficiency might be encountered near or on a boundary by using the SPH method. Therefore, the boundary condition is a significant input in achieving accurate simulation results. Several methods were put forward to resolve this problem, but the most effective way is to use ghost particles and dummy particles. For the slope problem, this is divided into two kinds of boundary conditions: fixed boundary conditions and free-rolling boundary conditions.

To simulate the fixed boundary, three layers of dummy particles are generated on boundaries. If the support domain of particle

The sketch of two types of boundary condition:

The developed FORTRAN program is validated with a 2D rectangle model. In the SPH simulation, a total number of 5000 real particles and 3000 boundary particles are used to form a rectangular area 4.0 m in length and 2.0 m in height, as shown in ^{3}; cohesion,

Initial setting conditions in SPH model.

Under the action of self weight, the vertical stress distribution of the model is shown in

Comparison of boundary treatment results:

In this section, rock slopes with one and three pre-cracks are simulated using the proposed method. Meanwhile, a determining method for SPH safety factor based on a strength reduction method is proposed. The result is then compared with those of the finite element numerical simulation method (FEM) to verify the correctness of the proposed method. In all examples, the linear cohesive law is adopted, and the smoothing length (

A rock slope model produced by the SPH method is carried out in this section. The purpose of this test is to verify the performance of the proposed numerical framework in simulating slope collapse. The geometry and boundary conditions of this test are shown in

Schematic of SPH model, flaw length is 28 cm.

Physical parameters of SPH model.

Young’s modulus, |
Poisson’s ratio, |
Density, |
Cohesion, |
Internal friction angle |
---|---|---|---|---|

50 | 0.3 | 1850 | 20 | 25 |

The development process of failure and collapse plotted by accumulated equivalent plastic strain for the rock slope at different times is shown in

Development of collapse and sliding velocity at different times: Cumulative equivalent plastic strain

The failure criterion for slope stability analysis in the finite element method with shear strength reduction (SRF) technique, proposed by

The approach is based on the maximum displacement at the feature point (see

In this paper, the process of the reduction is described as follows: The reduction factor of the SRF is initially set as 1.0 and then is increased by a step size of 0.1 during the next calculation. Note that if SRF is less than 1.0, this factor would be reduced, and the criterion determining the safety factor of the slope should be adjusted. When a sudden increase in feature point displacement appears, the failure of the slope occurs. Thus, the corresponding SRF at current computation is considered to be the safety factor. Thus,

Relationship between displacement and strength reduction coefficient (SRF).

Similarly, in order to further verify the effectiveness of the proposed method in the instability prediction and failure process of a rock slope, a model produced by the SPH method is carried out with three precast cracks. The initial pre-cracks have a width of 4 cm and lengths of 60cm, 28 and 24 cm, respectively (see

Schematic of SPH model, flaws length are 60, 28, and 24 cm, respectively.

The development process of the failure surface plotted by accumulated equivalent plastic strain for rock slope at different times is shown in

Development of collapse and sliding velocity at different times: Cumulative equivalent plastic strain

As shown in

With the further increase of time, the cracks completely penetrated the whole slope body, the slope body became unstable, and local collapse occurred, as shown in

The SPH code with D-P constitutive model is applied to investigate the collapse of rock slopes for the first time in this paper. For slope stability, the development of the shear band or failure process is well predicted through the accumulated plastic strain. The numerical results show that the SPH method is a reliable and robust method to simulate failure and collapse processes of geomaterials. Meanwhile, the successful application in rock slope collapse modeling indicates that the SPH method should be allowed further developments for other applications in geotechnical engineering.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

XZ and ZS were responsible for the work concept or design; XS was responsible for drafting the manuscript; SW were responsible for making important revisions to the manuscript; XZ and SW were responsible for approving the final version of the manuscript for publication.

The work was supported by the National Natural Science Foundation of China (Nos. 51934003), the Key Laboratory of Ministry of Education of China for Efficient Mining and Safety of Metal Mines (No. ustbmslab201906), the Research Start-up Fund for Introduced Talent of Kunming University of Science and Technology (KKSY201921029), The Yunnan Fundamental Research (NO. 202001AU070027).

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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