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Article

Study on Sliding Friction Coefficient in Block Element Method Based on Experimental Method

by
Pengcheng Liu
1,2,3,
Jun Liu
1,2,3,* and
Yue Wang
1,2,3
1
College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China
2
Institute of Engineering Safety and Disaster Prevention, Hohai University, Nanjing 210098, China
3
Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(19), 8660; https://doi.org/10.3390/app14198660
Submission received: 5 August 2024 / Revised: 20 September 2024 / Accepted: 24 September 2024 / Published: 25 September 2024
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Abstract

:
The mechanics and motion behavior of granular materials affect the production and life of human beings. In order to study the influence of the sliding friction coefficients corresponding to different contact types in the block discrete element method on the simulation results, this study established a block discrete element model to analyze a slope example based on the test method. The example was a homogeneous soil slope that did not consider water. The correctness of the models was verified by establishing the block discrete element slope model and comparing it with the known finite element method (FEM) model in terms of the maximum shear strain. Then, the sliding friction coefficient algorithm was embedded into the discrete element slope model for comparative analysis. The results show that in the calculations in the block discrete element method, the sliding friction coefficients of different contact types are different. Different sliding friction coefficients should be set based on different contact types to improve the accuracy of the simulation. Because the block discrete element model needs to preset the landslide surface of the slope, the displacement at the edge of the landslide surface is slightly different. The discrete element method (DEM) model was also compared with the block element model, and the results show that the DEM model is more stable.

1. Introduction

The mechanics and motion behavior of granular materials affect the production and life of human beings [1,2,3,4,5]. For example, landslides [6,7], avalanches [8,9,10], debris flow [11,12], and sand liquefaction [13,14,15] caused by earthquakes and other disasters [16,17,18,19] have a profound impact on the safety of human life and property [20,21].
Rock and soil are both granular materials. From a microscopic point of view, they are both composed of granular materials. These granular materials exist independently, and there is no force when contact does not occur [22,23]. Granular material is very different from a continuous medium. The biggest difference is that the granular matter flows easily under the action of external force, and it is difficult to maintain a static shape. At the same time, the flow properties of granular materials are also very different from those of fluids. When granular materials flow or are subjected to pressure, their physical parameters change [24]. For geotechnical engineering, because the discrete element method (DEM) can truly represent the geometric characteristics of rock and soil and efficiently simulate the processes of joints [25,26,27,28,29], slopes [30,31,32,33], landslides [34,35,36], and seepage [37,38,39], it has become an indispensable tool for analyzing and dealing with geotechnical engineering problems [40,41,42].
The discrete element method, which was proposed by Cundall and Strack [43,44,45] in the 1970s, is an effective tool to simulate the mechanical behavior of granular materials. According to the difference in the basic shape elements used in the modeling of the discrete element method, the elements can be divided into two categories: particle elements and block elements. That is, the discrete element method can be divided into the particle discrete element method and the block discrete element method [46]. The research and application of regularly shaped elements in the DEM, such as particle elements, are very extensive in the field of geotechnical engineering. However, there are still many areas worthy of attention in the study of irregularly shaped elements [47].
A clump Is one of the terms used to describe irregularly shaped particles. Its core is formed of an arbitrary geometry through multiple circle or sphere elements [48,49]. Favier [50] described the process of solving the contact problem of the cluster elements widely used in discrete element numerical simulation. Zhao [51] compared the shear mechanical behavior of the cluster and sphere elements in an undrained triaxial test and found that the cluster element can greatly improve the shear strength. The irregularity of the cluster element shape can improve the residual shear strength. Brzeziński [52] proposed a crushing algorithm for cluster elements, which can calculate the stress state of each sub-element, usually called a pebble, and applied it in Yade software. Grabowski [53] used asymmetric cluster and sphere elements to simulate the mechanical behavior of sand in a direct shear test. Li [54], Lu [55], Ferellec [56], and other scholars also studied the influence of the radius and number of sub-elements (pebbles) on the simulation results by comparing the numerical simulation experiments of cluster and sphere elements.
Polygonal or polyhedral elements are also often used to describe irregular particles [57], but there are some problems in their contact discovery and time-consuming calculation. Liu [57] developed polyhedron–polyhedron and sphere–polyhedron contact discovery algorithms based on energy conservation, proposed a polyhedron–boundary contact discovery algorithm [58], and developed the CoSim-DEM discrete element framework program. Descantes [59] proposed an improved Gilbert–Johnson–Keerthi Distance Algorithm (GJK-TD) based on the GJK contact detection algorithm and verified it using a frictionless polyhedron accumulation test and a friction polyhedron rolling test. In order to simulate the fracture process of rock and other geotechnical materials, Cui [60] proposed a cohesive fracture model and verified it using the Brazilian splitting test and uniaxial compression test. Feng [61], Fraige [62], Nezami [63], and other scholars have improved the two-dimensional polygon elements and three-dimensional polyhedron elements to improve their calculation accuracy.
Based on the sliding friction coefficient test of different elements of the discrete element method in the study by Liu [64], the block discrete element numerical simulation of a slope was carried out. By comparing the known FEM simulation results [65] with the block discrete element simulation results, the correctness of the block discrete element model was verified. By comparing the model embedded with the sliding friction coefficient determined by contact types and the FEM model, it was found that the sliding friction coefficients of different contact types should be set differently in the block discrete element numerical simulation. Different sliding friction coefficients were set based on different contact types to improve the accuracy of the discrete element simulation.

2. Experimental Method

In order to study the influence of the sliding friction coefficients of different contact types in block discrete elements on numerical simulation, this study featured further research based on previous experiments [64]. The research object of this study was samples embedded with polyhedrons made of cement. The polyhedron size was a cube with a side length of 40.24 mm, and the samples were cast using a silica gel reverse mold. The names of the samples, the average size of the cube, and the number of embedded cubes are shown in Table 1. In Table 1, P denotes the polyhedron, and the subscripts p, e, and f denote point contact, edge contact, and face contact. Two groups were made of various samples, and the sliding friction coefficient under different contact types was measured by contact between the samples. In the block discrete element method, the types of contact between the elements are categorized as point–point contact, point–edge contact, point–surface contact, edge–edge contact, edge–surface contact, and surface–surface contact. For convenience in description and recording, test results were documented using combinations of “point”, “edge”, and “surface”. The test results are shown in Table 2. Specific information about the experiment can be found in the study by Liu [64].

3. Simulated Model

In order to study the influence of the sliding friction coefficients corresponding to different contact types in the block discrete element method on the simulation results, this study used the block discrete element method to analyze a sand slope example [65] based on the experiment method [46]. The slope was homogeneous without considering other factors, such as the water content. The correctness of the block discrete element slope model was verified by comparing it with the known FEM model, and then the sliding friction coefficient algorithm was embedded into the discrete element slope model for comparative analysis.
The slope comprises two sections: the lower section is 45° and the upper section is 26.7°. The slope geometry is shown in Figure 1. The material parameters of the slope are shown in Table 3.
Based on the conditions given by Cheng, the model was established as shown in Figure 2. Because the length in the y direction is just a unit, and the deformation in the y direction is limited, it can be simplified to a plane strain problem.
According to the maximum shear strain range based on Figure 3a and the displacement area based on Figure 3b of the FEM model, the displacement region points outward from the slope, a slightly larger sliding range was preset in the block discrete element model, and the sliding range in the simulation results was taken as the main research object in the block discrete element model. Because the calculation stability of the block discrete element software depends on the element size and material parameters of each block, the landslide surface to the boundary was divided into six parts, as shown in Figure 4a. The landslide range is in the green area, which uses a Voronoi grid to divide the polygonal elements. The parameters of the slope model are shown in Table 1. The sliding friction coefficient of this paper was embedded in the model after verification.
Three measuring points were set up on the model, and the positions are shown in Figure 4b. Each measuring point recorded the horizontal and vertical displacements of each time step.

4. Results and Discussions

The correctness of the original block discrete element slope model was verified by comparing the maximum shear strain with that of the known FEM model. The maximum shear strains are 3.17 × 10−1 and 1.08 × 10−2, respectively, which are very close, as shown in Figures 3a and 9a, and the maximum shear strain occurred near the edge of the landslide surface, indicating that the original block discrete element model was established correctly. Because of the difference between the block discrete element method and FEM, the calculated maximum shear strain is slightly different, but the calculation results are within a reasonable range.
In order to verify that the sliding friction coefficient determined by the test method was embedded in the block discrete element model, the original block discrete element model that calculates the default sliding friction coefficient was compared and verified. A total of 21,000 time steps were calculated, and the maximum shear strain distribution contours at the output of 5000, 10,000, 15,000, 20,000, and the end of the simulation calculation were compared, as shown in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, respectively.
Comparing the original block discrete element slope model and the modified model embedded with the sliding friction coefficient in Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, the maximum shear strain is different at each time step, indicating that the sliding friction coefficient determined by the experiment method was successfully embedded and had an impact on the calculation results. Because the sliding friction coefficient between blocks determined by the experiment method is slightly smaller than the default value of the model, the maximum shear strain of the modified model embedded with the sliding friction coefficient is slightly larger than that of the original slope model. However, the calculation result is within a reasonable range because the interpolation of the maximum shear strain is only 7.56 × 10−4.
According to the position of each measuring point in Figure 4b, the change trend of the displacement of each measuring point in the modified model with the sliding friction coefficient was drawn, and the change trend of the displacement of each measuring point in the FEM model was drawn too. The horizontal and vertical displacements of each measuring point are shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15. Since the displacement changes at different monitoring points vary significantly based on different analysis methods (whether it is DEM, FEM, or block discrete element methods), with some very small and others so big, unifying the Y-axis would result in some displacement change curves appearing with very large amplitudes, while others would have very small amplitudes. As this section primarily compares the variation trends of the two methods at each measurement point and the final displacement values, this study did not use a unified Y-axis representation. The subsequent discussion comparing the displacements of the DEM and block discrete element methods will not be reiterated. According to the displacement diagram of each measuring point, the following conclusions can be illuminated:
(1)
The trend of the horizontal displacement of each measuring point is basically the same as the development of time. The displacement of measuring point 1 is negative in a very short time, then rises rapidly and finally stabilizes. Measuring points 2 and 3 both show rapid stability after the development of displacement;
(2)
The trend of vertical displacement of each measuring point is basically the same at any time, and the displacement develops rapidly and then stabilizes rapidly;
(3)
Table 4 shows the displacements calculated using different methods for sliding friction coefficients under the same computational theory (block discrete element method). When comparing the horizontal displacement and vertical displacement of each measuring point, as shown in Table 4, there is little difference in the horizontal displacement of measuring point 1 and the vertical displacement of measuring point 3. The results are mainly because the block discrete element model needs to set a certain landslide surface in advance. In block discrete element models, due to reasons such as the geometric shape of the elements, contact methods, contact theories, and contact detection, elements can easily become stuck and lead to erroneous conclusions during stability analysis if sliding surfaces are not preset. Therefore, in slope stability analysis, sliding surfaces are typically preset. Measuring point 1 is located at the foot of the slope, and measuring point 3 is located at the shoulder of the slope. The deformation of the two measuring points cannot be fully developed due to the factors of the preset landslide surface and the assumptions of the DEM that the element is rigid. However, the displacements of the two measuring points in other directions are very similar, and measuring point 2 is basically the same, which proves the correctness of the model again.
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Figure 10. The horizontal displacement of point 1: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 10. The horizontal displacement of point 1: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Figure 11. The vertical displacement of point 1: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 11. The vertical displacement of point 1: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Figure 12. The horizontal displacement of point 2: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 12. The horizontal displacement of point 2: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Figure 13. The vertical displacement of point 2: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 13. The vertical displacement of point 2: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Figure 14. The horizontal displacement of point 3: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 14. The horizontal displacement of point 3: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Figure 15. The vertical displacement of point 3: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
Figure 15. The vertical displacement of point 3: (a) the block discrete element model embedded with the sliding friction coefficient; (b) the FEM model.
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Table 4. A comparison of the displacement values of the measuring points.
Table 4. A comparison of the displacement values of the measuring points.
Measuring PointOriginal Model (cm)Modified Model (cm)
Horizontal displacement of point 10.2−0.2
Vertical displacement of point 1−0.8−7.2
Horizontal displacement of point 2−1.1−2.6
Vertical displacement of point 2−3.9−11.3
Horizontal displacement of point 3−2.2−3.4
Vertical displacement of point 3−4.5−15.2
This study also established a discrete element method (DEM) model and a block discrete element model for comparison. The horizontal and vertical displacements of each measuring point are shown in Figure 16, Figure 17 and Figure 18. Comparing the displacements of each measuring point in the DEM model and the BDEM model reveals the following.
The overall trends of displacement at each measuring point are similar. However, due to the gaps between the DEM elements, noticeable displacement fluctuations occur during movement. These fluctuations can be smoothed out to achieve a more consistent displacement–time curve by reducing the time step or decreasing the element size. However, such adjustments significantly increase the computation time; thus, these minor fluctuations may be disregarded. From a macroscopic perspective, these fluctuations represent negligible deformations.
The horizontal displacement at measuring point 1 in the DEM model differs slightly from that in the block discrete element model, yet the development trend at this point aligns closely between the DEM and FEM models. This suggests that the DEM model provides a more realistic simulation of slope behavior compared to the block discrete element model. Unlike the block discrete element model, which requires predefining the slip surface, the DEM model exhibits more comprehensive development.

5. Conclusions

In order to study the influence of the sliding friction coefficients corresponding to different contact types in the block discrete element method on the simulation results, the sliding friction coefficient algorithm was applied to a slope example based on the experiment method. The correctness of the block discrete element model was verified by comparing the maximum shear strain of the known FEM model with that of the block discrete element model before the sliding friction coefficient was embedded. Then, the displacement of each measuring point of the modified model was compared with the displacement of the measuring point of the known FEM model to verify the influence of the different contact types of the block discrete element on the sliding friction coefficient on the simulation results. The comparison results show the following:
(1)
The maximum shear strain cloud diagrams of the initial slope model and the embedded sliding friction coefficient model were compared and analyzed, indicating that the sliding friction coefficient affects the calculation of the block discrete element model. In calculating the block discrete element method, the sliding friction coefficients of the different contact types were different. Different sliding friction coefficients should be set for different contact types to improve the accuracy of the simulation;
(2)
When we compared and analyzed the trends of the horizontal and vertical displacements of each measuring point in the FEM model and block discrete element model with the sliding friction coefficient, they had the same change trend at each measuring point. The measuring points develop in the direction of deviation from the slope, showing a trend of gradual sliding;
(3)
Due to the different basic theories of the two methods, the FEM and discrete element methods, and because the block discrete element model needs to preset the landslide surface of the slope, the displacement at the edge of the landslide surface was slightly different. However, the total trends of the slope developments are the same. And the comparison between the DEM model and block discrete element model shows that the DEM model provides a more realistic simulation of slope behavior compared to the block discrete element model.

Author Contributions

P.L. contributed to the conceptualization, resources, and formal analysis. J.L. contributed to the software and methodology. Y.W. contributed to the data curation. All authors contributed to the writing—review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (No. 51874118; No. 51778211).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

The authors acknowledge the assistance of Mingqing Liu, Mengyang Zhen, Futian Zhao, and Kai Ge.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The geometry of the slope.
Figure 1. The geometry of the slope.
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Figure 2. The slope model.
Figure 2. The slope model.
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Figure 3. The result of the slope mode: (a) the shear strain of the model; (b) the displacement of the model.
Figure 3. The result of the slope mode: (a) the shear strain of the model; (b) the displacement of the model.
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Figure 4. The block discrete element model and the measuring points: (a) The block discrete element model; (b) The measuring points of the modal.
Figure 4. The block discrete element model and the measuring points: (a) The block discrete element model; (b) The measuring points of the modal.
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Figure 5. The maximum shear strain distribution at the 5000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
Figure 5. The maximum shear strain distribution at the 5000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
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Figure 6. The maximum shear strain distribution at the 10,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
Figure 6. The maximum shear strain distribution at the 10,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
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Figure 7. The maximum shear strain distribution at the 15,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
Figure 7. The maximum shear strain distribution at the 15,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
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Figure 8. The maximum shear strain distribution at the 20,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
Figure 8. The maximum shear strain distribution at the 20,000th time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
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Figure 9. The maximum shear strain distribution at the final time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
Figure 9. The maximum shear strain distribution at the final time step: (a) the original block discrete element model; (b) the block discrete element model embedded with the sliding friction coefficient.
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Figure 16. The displacement of point 1 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
Figure 16. The displacement of point 1 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
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Figure 17. The displacement of point 2 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
Figure 17. The displacement of point 2 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
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Figure 18. The displacement of point 3 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
Figure 18. The displacement of point 3 in the DEM model: (a) the horizontal displacement; (b) the vertical displacement.
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Table 1. The labels, average size, amount, and explanations for the experiment.
Table 1. The labels, average size, amount, and explanations for the experiment.
LabelAverage Size (mm)AmountExplanation
Pp40.2432Contact with the polyhedron point
PeContact with the polyhedron edge
PfContact with the polyhedron face
Table 2. The sliding friction coefficients between the polyhedrons.
Table 2. The sliding friction coefficients between the polyhedrons.
LabelPpPePf
Pp0.35\\
Pe0.540.55\
Pf0.550.590.62
Table 3. The parameters of the slope model.
Table 3. The parameters of the slope model.
LabelSymbolValue
Young’s modulusE14.0 Mpa
Poisson ratioν0.3
Gravityg9.8 m/s2
Friction coefficientμs30°
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Liu, P.; Liu, J.; Wang, Y. Study on Sliding Friction Coefficient in Block Element Method Based on Experimental Method. Appl. Sci. 2024, 14, 8660. https://doi.org/10.3390/app14198660

AMA Style

Liu P, Liu J, Wang Y. Study on Sliding Friction Coefficient in Block Element Method Based on Experimental Method. Applied Sciences. 2024; 14(19):8660. https://doi.org/10.3390/app14198660

Chicago/Turabian Style

Liu, Pengcheng, Jun Liu, and Yue Wang. 2024. "Study on Sliding Friction Coefficient in Block Element Method Based on Experimental Method" Applied Sciences 14, no. 19: 8660. https://doi.org/10.3390/app14198660

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