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Article

Study of the Sliding Friction Coefficient of Different-Size Elements in Discrete Element Method Based on an Experimental Method

by
Pengcheng Liu
1,
Yi Rui
1,* and
Yue Wang
2
1
Department of Geotechnical Engineering, Tongji University, Siping Rd, Shanghai 200092, China
2
College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China
*
Author to whom correspondence should be addressed.
Appl. Sci. 2024, 14(19), 8802; https://doi.org/10.3390/app14198802
Submission received: 1 August 2024 / Revised: 25 September 2024 / Accepted: 27 September 2024 / Published: 30 September 2024
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Abstract

:
The materials involved in geotechnical engineering are objects of concern in granular mechanics. In order to study the influence of the sliding friction coefficient corresponding to different-sized elements in the discrete element method (DEM) on the simulation results, we establish a two-dimensional DEM model based on the experimental method to analyze a slope example. The correctness of the DEM model is verified by comparing the sliding surface of a finite element method (FEM) model and the DEM slope model. A sliding friction coefficient algorithm based on the experimental method is embedded into the DEM slope model and compared with the original model. The comparison results show that embedding the DEM model into the sliding friction coefficient algorithm leads to an increase in displacement. The reason for this is that the contact information between elements of different sizes has changed, but the displacement trend is the same. Different sliding friction coefficients should be set based on different-sized elements in the DEM, as they can improve simulation accuracy.

1. Introduction

From a microscopic perspective, the materials involved in geotechnical engineering are objects of concern in granular mechanics. For example, landslides [1,2], submarine debris flows [3,4], avalanches [5,6], ground settlement [7,8], and other phenomena can arise due to the failure of the force between the granular materials. Both rock and soil can be regarded as discontinuous media. These discontinuous media are independent of each other and interact with each other after contact. At present, there are many methods that can be used to study geotechnical engineering, such as the FEM, SPH, the XFEM, and so on. Among them, the discrete element method (DEM) is the most effective method for studying granular materials.
The DEM is an effective tool that can be used to simulate the mechanical behavior of granular materials. It is a numerical simulation method that was proposed by literature [9,10,11] in the 1970s. According to differences 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. Therefore, essentially, the discrete element method can be divided into the particle discrete element method and the block discrete element method. Spherical and circular elements are often used in research because analyzing contact discovery between such elements is easy, and the use of such elements can lead to high calculation efficiency. Cundall [12] first proposed the discrete element method and used two-dimensional circular elements and ellipse elements for discrete element simulations, subsequently proposing three-dimensional spherical elements [13] and creating PFC3D commercial software. Ting [14,15] applied ellipse elements and described how to calculate the contact force between the ellipse elements and the boundary, while Ng [16] studied the particle mechanical behavior of ellipse elements. Lin [17] proposed a three-dimensional ellipsoid element and found that the shape of the element has a significant effect on the discrete element simulation results. Chen [18] established a three-dimensional super-ellipsoid unit by focusing on the slenderness ratio and lumpiness, and Liu [19] proposed the multi-super-ellipsoid element and used four stacking tests to verify the accuracy of the model. It was found that the computational efficiency depends on the shape of the element and the number of sub-elements that make up the irregular element. Liu [20] derived the normal contact force between ellipsoid elements based on the energy method. Yin [21] developed an improved ellipsoid contact force algorithm. Mori [22], Hoorijani [23], Xia [24], and other scholars have conducted in-depth research on the establishment or simulation behavior of ellipsoid unit models.
Although the DEM has facilitated breakthroughs in other theories, such as contact discovery, it has always been a difficult problem in the context of parameter calibration, especially in friction coefficient calibration and related research. At present, research on friction coefficients mainly focuses on the experimental method and the numerical simulation method. For example, Chang [25] designed a test device to study the friction behavior between different materials and particles by changing the speed and angle of the contact material and the size of the particles, while Bek [26] designed a powder axial pressure device and measured the friction coefficient of the powder under axial pressure. Wang [27] studied the angle of repose of broken ore under different water contents and found that there is a critical size of broken ore. When it is less than the critical size, reducing the ore size will reduce the fluidity. Liu [28] studied the sliding friction coefficient by considering three element shapes, spherical, ellipsoid, and polyhedron elements, carrying out multiple sets of tests, and verifying the test results through stacking tests. By applying a cyclic load to a single sphere and considering friction, Wang [29] found that the contact area of the sphere in the unloading stage is larger than that in the loading stage when the contact force is same, while Shi [30] calibrated the microscopic parameters of granite using PFC2D and considered the influence of various mineral components in granite on rock mass.
Some scholars have conducted in-depth research on the rolling friction in the discrete element method. For example, Sakaguchi [31] incorporated the rolling friction coefficient into a simulation, finding that it improved the reliability of the simulation. In order to explore the formation of shear bands in soil, Iwashit [32] modified the traditional discrete element method and introduced rolling friction. It was found that there were large pores in the shear band, and the test and simulation processes were very similar. Markauskas [33] used the multi-sphere model (MS) to simulate the ellipsoid and incorporated rolling friction into a simulation. By comparing the angle of repose and the porosity of multiple sets of ellipsoid stacking tests, it was found that as the number of spheres increased, the test and simulation increased in similarity. As a result of some assumptions in the theory of the discrete element method and the simplification of subsequent models, it is difficult to ensure the accuracy of theoretical studies.
Some scholars have conducted research on the friction coefficient from the perspective of tribology [34,35]. For example, Ghatrehsamani [36] used continuous damage mechanics to establish a contact model and found that when the friction coefficient increases, the wear coefficient between the materials will also change accordingly. Ghatrehsamani [37] divided contact into multiple stages and expressed the friction coefficient of the first stage (Running-in State) as a function of force. Then, a contact force calculation model was established by using continuous damage mechanics, and the influence of relative velocity and tangential displacement on the friction coefficient was considered in the calculation model. Based on the Hertz model, Zhao [38] divided the modeling of rough surface contact into three stages, elastic, plastic, and elastic–plastic, and established a relationship between friction and contact depth in each stage. Hurtado [39] found that the friction stress decreases with the increase in the area until it passes through two transition stages and that the friction stress gradually becomes independent of the contact area. Asaf [40] used an in situ settlement test to establish a friction coefficient prediction formula based on the energy ratio and a friction coefficient prediction formula based on normal stiffness, carrying out calibration via nonlinear optimization. However, this conclusion is limited to quasi-static cohesionless soil. Adams [41] developed a multi-asperity model for contact and friction; the model spans a range covering nano-, micro-, and macro-scale contacts. Zhuravlev [42] obtained the static friction and sliding friction of a ball based on Hertz theory, while Wang [43] used an FEM model to analyze the friction coefficient of elastic–plastic materials. Chen [44] studied the relationship between coating thickness and the friction coefficient of the sphere.
Based on a prior experiment measuring the sliding friction coefficient of different-sized sphere elements [28], we propose a sliding friction coefficient algorithm. We carried out discrete element numerical simulations to analyze an example slope, and the accuracy of our DEM model was verified by comparing the sliding surface with those of FEM and DEM slope models. By comparing the DEM model with a single sliding friction coefficient and the embedded sliding friction coefficient algorithm, we further studied the sliding friction coefficient determined based on the experimental method.

2. Experimental Method

Focusing on the influence of different-sized elements’ sliding friction coefficient on numerical simulation in the context of the discrete element method, this study can be considered an extension of Liu’s experiment [28]. The research objects were samples made of cement mortar embedded with sphere elements of different sizes. The sizes of the sphere elements were 3 cm, 3.5 cm, 4 cm, 4.5 cm, 5 cm, 5.5 cm, and 6 cm. The samples were poured using the silica gel mold pouring method. The names of the sphere elements embedded in the samples, the average diameter of the sphere elements in each sample, and the number of particles embedded in the samples are shown in Table 1. In Table 1, S represents the sphere and the subscript represents the diameter of the sphere element. Two groups of same-radius sample spheres were made, and the sliding friction coefficients between sphere elements of different sizes were measured based on the spherical elements’ contact with the samples, as shown in Figure 1. We placed the samples horizontally on the platform, and then gradually raised the height of the rear side of the platform. When the upper sample began to slide, the test was complete, and the coefficient of friction could be calculated. The test was repeated five times, and the results are shown in Table 2.

3. Simulation Model

In order to study the influence of different-sized sphere elements’ sliding friction coefficient on numerical simulation in the context of the discrete element method, we analyze an example slope [45] based on the experimental method. The slope only considers the influence of gravity and does not consider other factors. The slope has two inclinations—the lower slope is inclined at 45°, while the upper slope is inclined at 26.7°—and the slope geometry is shown in Figure 2. The parameters of the slope are shown in Table 3. The correctness of the model is verified by establishing a DEM slope model and comparing it with the known FEM model, and then the sliding friction coefficient algorithm determined according to the sphere element size is embedded into the discrete element slope model for comparative analysis. A flow chart of the sliding friction coefficient algorithm is shown in Figure 3. As shown in Figure 3, the sliding friction coefficient between polyhedron elements will be analyzed in further experiments.
Based on Cheng’s model [45], as shown in Figure 4, the material follows the Mohr–Coulomb model, and quadrilateral elements are used. The boundary conditions are set as zero displacement on the left, right, and bottom boundaries. The model has a length of one element in the y direction and the deformation in the y direction is limited; the model can be simplified as a plane strain problem.
For the DEM slope model, the modeling process is as follows: Firstly, a model with a particle size of 0.1–0.3 is generated, as shown in Figure 5a. After the sphere elements in the area are stabilized, the redundant elements are deleted, as is shown in Figure 5b. In Figure 5, the colors represent displacement. Although each element produces some displacement during generation, it does not affect subsequent analysis because the element velocities are initialized after model trimming. Finally, the velocity and acceleration of each element are reset to zero, and the calculation is initiated. The DEM model is shown in Figure 6. Figure 6 uses colors based on element size to illustrate that the model includes elements of various sizes.
Three measuring points are set up on the slope, and the position of each point is shown in Figure 7. Each measuring point records the horizontal displacement and vertical displacement of each time step, and the difference between the final position and the initial position of each measuring point is the displacement of the element.

4. Results and Discussion

In order to study the influence of different-sized elements’ sliding friction coefficient on the numerical simulation results based on the experimental method [28], we applied the sliding friction coefficient algorithm to Cheng’s slope example [45]. To verify the correctness of the DEM model established according to Cheng’s FEM model [45], the sliding surfaces of the FEM model and the DEM model were calculated under the same material parameters. The sliding surface corresponding to the minimum FOS (factor of safety) value is considered the most critical or dangerous sliding surface in FEM. The sliding surface generated by the calculation of the original DEM model is shown in Figure 8, which illustrates the displacement changes; the most dangerous sliding surface calculated by the FEM model is shown in Figure 9. By comparing these figures, it can be found that the sliding surfaces of the two models are the same, which verifies the correctness of the DEM model.
To verify the applicability of embedding the experimental method-based sliding friction coefficient algorithm into the DEM slope model, we can compare each measuring point’s horizontal and vertical displacements before and after the sliding friction coefficient algorithm was incorporated into the model (Table 4). This comparison shows that the sliding friction coefficient algorithm has an impact on the calculation of the DEM slope model. A comparative analysis of the two models will be described later.
The FEM and DEM are commonly used for analysis in geotechnical engineering. These two analysis methods are based on different assumptions and theoretical backgrounds. The accuracy of the calculation results of the DEM model is related to factors such as contact discovery, element shape, time step, and spatial dimension. The slope model described in this paper is two-dimensional and composed of circular elements. The total number of elements is 8000, each with a particle size of 0.1–0.3 in the DEM model. Compared with the FEM model, the number of elements in the DEM, as well as the size of these elements, is enlarged to a certain extent. However, if the number of elements is increased, the calculation time will be greatly increased and the accuracy will remain the same, and sometimes the accuracy of stress and strain results will not be substantially improved. In addition, the modeling methods of the two simulations are very different, so the calculated displacement–time curves are slightly different. Therefore, in this paper, we only compare and analyze the change trend in each measuring point of the DEM models before and after adding the sliding friction coefficient algorithm. The positions of each measuring point are shown in Figure 7.
The displacement–time curves of each measuring point from before and after embedding the sliding friction coefficient algorithm into the DEM model for the different-sized elements are drawn according to the position of each measuring point (Figure 7). The horizontal and vertical displacement of each measuring point is shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15. Based on the diagram of each measuring point, the following conclusions can be drawn:
(1)
The horizontal displacement trend of each measuring point is roughly the same: each measuring point develops in the direction away from the slope, showing a trend of gradual sliding. The displacement diagrams of measuring points 1 and 2 clearly fluctuate. The reason for this is that the elements constituting the slope model are discrete, and there is a certain gap between the circular elements. During the development of the sliding surface, there is a certain probability that a smaller element will fall into the gap due to gravity. However, the overall development trend of each measuring point of the two models is basically the same.
(2)
The vertical displacement trend of each measuring point is the same: each measuring point develops in the direction of deviation from the slope, showing a trend of gradual sliding. The displacement diagram of measuring point 2 has clear fluctuations due to reasons similar to those above, owing to the discrete elements and different-sized elements. However, the overall development trend of each measuring point of the two models is basically the same.
(3)
By comparing the two DEM slope models, it can be seen that the displacement of the DEM model embedded into the sliding friction coefficient algorithm is larger. The reason for this is that for larger-size elements, the contact information between a single sliding friction coefficient and the DEM elements embedded in the sliding friction coefficient algorithm is basically the same. For smaller-sized elements, the sliding friction coefficient between the DEM elements with a single sliding friction coefficient will be larger, resulting in a change in contact information. The contact information is related to many DEM parameters. Although the sliding friction coefficient algorithm will increase the calculation time, the calculation accuracy can be improved from a mechanical point of view.

5. Conclusions

In order to study the influence of different-sized elements’ sliding friction coefficients on the simulation results in the context of the discrete element method, we applied the sliding friction coefficient algorithm between two-dimensional elements to a slope example based on the experimental method. By comparing the slip surface of the FEM slope model and the DEM slope model, the correctness of the DEM model was verified. The horizontal and vertical displacement of each measuring point in the DEM slope model embedded with the sliding friction coefficient algorithm were compared with the displacement of the original DEM slope model to verify that the sliding friction coefficient between different-sized elements in DEM has an impact on the simulation results. This study’s contributions and conclusions are as follows:
(1)
We compared and analyzed the horizontal displacement and vertical displacement of an original DEM model and a DEM slope model embedded with a sliding friction coefficient algorithm. The sliding friction coefficient has an influence on the calculation of the discrete element model. That is, in the DEM model, different sliding friction coefficients should be set between different-sized elements to improve the accuracy of the DEM simulation.
(2)
By comparing and analyzing the changing horizontal and vertical displacement trends of each measuring point of the original DEM model and the DEM model with the sliding friction coefficient algorithm, we found that the two have the same trend at each measuring point. Each measuring point develops in the direction away from the slope, showing a trend of gradual sliding.
(3)
There are some fluctuations in the displacement curve. The reason for this is that the elements are discrete, and there is a certain gap between the elements. During the development of the sliding surface, there is a certain probability that smaller elements will fall into the gap due to gravity. However, the overall development trend of each measuring point of the two models is basically the same.

Author Contributions

P.L. and Y.W.: conceptualization, resources, and analysis. Y.R.: software, methodology, and 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; NO. 52378406).

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 Xiaojun Li and Kai Ge.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. Experimental method for measuring the sliding friction coefficient of sphere elements: (a) initial state; (b) mid-experiment.
Figure 1. Experimental method for measuring the sliding friction coefficient of sphere elements: (a) initial state; (b) mid-experiment.
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Figure 2. The slope geometry (in mm).
Figure 2. The slope geometry (in mm).
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Figure 3. Algorithm flowchart.
Figure 3. Algorithm flowchart.
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Figure 4. The slope model.
Figure 4. The slope model.
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Figure 5. DEM slope model generation: (a) generation of elements; (b) formation of slope.
Figure 5. DEM slope model generation: (a) generation of elements; (b) formation of slope.
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Figure 6. The DEM slope model.
Figure 6. The DEM slope model.
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Figure 7. The measuring points.
Figure 7. The measuring points.
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Figure 8. The sliding surface of the DEM model.
Figure 8. The sliding surface of the DEM model.
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Figure 9. The sliding surface of the FEM model.
Figure 9. The sliding surface of the FEM model.
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Figure 10. Horizontal displacement of point 1: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 10. Horizontal displacement of point 1: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Figure 11. Vertical displacement of point 1: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 11. Vertical displacement of point 1: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Figure 12. Horizontal displacement of point 2: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 12. Horizontal displacement of point 2: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Figure 13. Vertical displacement of point 2: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 13. Vertical displacement of point 2: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Figure 14. Horizontal displacement of point 3: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 14. Horizontal displacement of point 3: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Figure 15. Vertical displacement of point 3: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
Figure 15. Vertical displacement of point 3: (a) DEM model with sliding friction coefficient algorithm; (b) original DEM model.
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Table 1. Labels, average diameter, and particle numbers of each sample.
Table 1. Labels, average diameter, and particle numbers of each sample.
LabelAverage Diameter (mm)Standard Deviation (mm)Number of Particles
S329.620.4932
S3.534.690.3832
S440.080.4532
S4.545.020.4032
S549.720.3932
S5.555.260.4232
S660.120.4232
Table 2. Sliding friction coefficients between spheres.
Table 2. Sliding friction coefficients between spheres.
LabelS3S3.5S4S4.5S5S5.5S6
S30.460.470.480.490.490.500.51
S3.50.470.480.480.490.500.500.52
S40.480.480.500.500.510.520.53
S4.50.490.490.500.500.520.520.53
S50.490.500.510.520.530.540.56
S5.50.500.500.520.520.540.570.58
S60.510.520.530.530.560.580.58
Table 3. The mechanical parameters of the material the slope model.
Table 3. The mechanical parameters of the material the slope model.
LabelSymbolValue
Young’s modulusE14.0 MPa
Poisson’s ratioν0.3
Gravityg9.8 m/s2
Friction coefficientμs0.58
Table 4. Comparison of displacement value of each measuring point.
Table 4. Comparison of displacement value of each measuring point.
Measuring PointOriginal Model (mm)Modified Model (mm)
Horizontal displacement of point 1−1.4−2.7
Vertical displacement of point 1−3.7−4.6
Horizontal displacement of point 2−0.1−0.1
Vertical displacement of point 2−0.10.2
Horizontal displacement of point 3−3.0−3.0
Vertical displacement of point 3−4.8−5.4
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Liu, P.; Rui, Y.; Wang, Y. Study of the Sliding Friction Coefficient of Different-Size Elements in Discrete Element Method Based on an Experimental Method. Appl. Sci. 2024, 14, 8802. https://doi.org/10.3390/app14198802

AMA Style

Liu P, Rui Y, Wang Y. Study of the Sliding Friction Coefficient of Different-Size Elements in Discrete Element Method Based on an Experimental Method. Applied Sciences. 2024; 14(19):8802. https://doi.org/10.3390/app14198802

Chicago/Turabian Style

Liu, Pengcheng, Yi Rui, and Yue Wang. 2024. "Study of the Sliding Friction Coefficient of Different-Size Elements in Discrete Element Method Based on an Experimental Method" Applied Sciences 14, no. 19: 8802. https://doi.org/10.3390/app14198802

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