A discrete element method (DEM) has widely been used to simulate asphalt mixture characteristics, and DEM models can consider the effect of aggregate gradation and interaction between particles. However, proper selection of model parameters is crucial to obtain convincing results from DEMbased simulations. This paper presents a method to appropriately determine the mechanical parameters to be used in DEMbased simulation of asphalt concrete mixture. Splitting test specimens are prepared by using asphalt mixture, and the splitting test results are compared with simulation results from twodimensional (2D) DEM and threedimensional (3D) DEM. Basing on the DEM results, the effects of contact model parameters on the simulation results are analyzed. The slope of the loaddisplacement curve at the beginning stage is mainly affected by the stiffness parameters, and the peak load is mainly determined by using the value of the bond strength. The laboratory splitting test of AC20 and AC13 specimens were performed at different temperatures, namely, −10°C, 0°C, 10°C, and 20°C, and the loaddisplacement relationships were plotted. According to the real loaddisplacement curve’s slope at the beginning stage and peak load applied, the range of DEM bond model parameters is determined. On the basis of DEM results of the splitting test, the relationships between simulation loaddisplacement curve’s characteristics and bond model parameters are fitted. The values of the parameters of the DEM contact bond model at different temperatures are obtained depending on the actual loaddisplacement curve’s initial slope and peak load. Lastly the DEM and laboratory test results are compared, which illustrates that the parallel bond model can well simulate the behavior of asphalt mixture.
Asphalt mixture is a composite material comprised of graded aggregate, filler, asphalt, and air voids. With the variation in the proportion of the components, the particles in asphalt mixture may exhibit different spatial distribution after compaction [
The discrete element method (DEM), proposed by American Cundul P.A in 1971, is mainly used to analyze the mechanical properties of the discontinuity or continuous medium with defects. In the past decades, DEM has widely been used, and several software has been developed, such as EDEM and PFC [
Dondi et al. [
DEM is also used to study the crack resistance and permanent deformation of asphalt mixture. For example, the discrete element method has been used to simulate the tensile test of disk specimens [
In summary, DEM has widely been used to simulate the performance of asphalt mixture. At low temperature, the properties of asphalt mixture are regarded as elastic. While the elastic bond model can be used in DEM in such cases, values of the parameters in the bond model have not certainly been determined in the past research. For example, Jun and Xiaoming [
The contact bond model and parallel bond model are the most commonly used bond models in the discrete element method. The contact bond model considers that the contact between particles is only in a small region named point contact, as shown in Figure
Contact bond model [
Parallel bond model [
The contact bond model can be regarded as a pair of springs acting on particles’ contact point. Figure
Constitutive behavior for contact occurring at a point. (a) Normal component and (b) shear component of contact force.
When the overlap amount between particles is less than zero, tension is allowed to appear, but the normal tension cannot exceed the normal bond strength. When the normal tension is greater than or equal to the normal bond strength, the normal and tangential contact forces will be zero. When the tangential contact force is greater than or equal to the tangential bond strength, the tangential bond will be destroyed; however, if the tangential force does not exceed the friction limit, the tangential force will not change. In the contact bond model, four parameters are needed: normal stiffness (
The parallel bond model assumes that there is a medium material. The medium material is assumed as a set of springs with constant normal stiffness and tangent stiffness, which are distributed uniformly in the contact plane. The constitutive relationships of these springs are similar to those of point contact springs. However, one additional property is assumed, namely, transferring moment between particles. In the parallel model, normal and shear stiffnesses,
The objective of this paper was to determine the parameter’s value of the DEM bond model by simulating a splitting test. Therefore, it is necessary to artificially generate a test specimen that replicates an idealized asphalt mixture specimen. It is important that the sample is initially isotropic and exhibits approximately the same packing characteristics (volumetric proportions) as the idealized mixture. The following procedure was developed to prepare samples for later DEM simulation.
The volume (area) of spherical (disk) particles was calculated according to the gradation of real aggregate in asphalt mixture, and the number of particles was determined
Boundaries which enclose the required space were generated, and the number of particles calculated above was generated randomly inside the space
Equilibrium calculation was required to make the particles reorient by cycling and decreasing the internal stress between particles to isotropic
Reducing the particles’ radius slightly to decrease the level of isotropic stress until the stress was relatively low
Particles with less than four contacts were detected and were expanded slightly to create additional contacts with neighboring particles
Certain contact model was selected and applied to contact points
The walls used as particles boundary were removed
Two types of asphalt mixture, namely, AC13 and AC20, were produced by DEM including twodimensional and threedimensional models. Taking the AC13 type asphalt mixture as an example, the DEM sample in two and three dimensions are shown in Figure
Splitting test sample divided into discrete elements. (a) Twodimensional disk specimen. (b) Threedimensional sphere specimen [
Simulating the splitting test by DEM. (a) Twodimensional model. (b) Threedimensional model.
The aggregate used in asphalt mixture is formed by combining a certain gradation, and hence, the aggregate particles vary in size. However, the minimum size of particles of asphalt mixture must be determined when DEM is used. If the minimum particle size is too large, the real aggregate gradation cannot be reflected fully. On the contrary, if the minimum particle size selected is too small, the simulating process will take too much time. Therefore, it is imperative to select an optimum value for the minimum particle size on the DEM. The following paragraph will present simulation results from the contact bond model and the parallel bond model to analyze the effect of the minimum particle size.
The influence of minimum size of particles on simulation results was analyzed in the authors’ previous study [
The influence of the contact bond parameters on the simulation results is shown in Figure
Effect of the contact bond model parameters. (a) Effect of stiffness. (b) Effect of bond strength [
The simulation results using the parallel bond model are plotted in Figure
Influence of the parallel bond model parameters. (a) Effect of stiffness. (b) Effect of bond strength [
The asphalt used for the test was 70# according to the penetration test, and the aggregates and fillers were limestone from Shandong province in China. The specimen was cylindrical with the diameter of 101.6 mm and the height of 63.5 mm compacted by the Marshall compaction test. Two types of asphalt mixtures, namely, AC13 and AC20, were prepared. The gradation of AC13 and AC20 is listed in Table
Gradation of the aggregate for asphalt mixture.
Sieve pass rate (%)  

Sieve size (mm)  19  16  13.2  9.5  4.75  2.36  1.18  0.6  0.3  0.15  0.075 
AC13  100  95  76.5  53  37  26.5  19  13.5  10  6  
AC20  100  85  71  61  41  30  22.5  16  11  8.5  5 
Splitting test. (a) AC13 specimen. (b) Splitting test loading. (c) Failure mode of the splitting test.
During the test, the loading speed had a significant influence on the test results. According to Chinese test standard JTG E202011, the loading speed was set to 1 mm/min when the temperature was −10°C or 0°C and 5 mm/min when the temperature was 10°C or 20°C. The splitting test results are plotted in Figures
Splitting test results of AC20. Loaddisplacement curve: (a) −10°C and 0°C and (b) 10°C and 20°C [
Splitting test results of AC13. Loaddisplacement curve: (a) −10°C and 0°C and (b) 10°C and 20°.
Section
Parameters of the 2D contact bond model. (a) Original slope of the simulated splitting test curve versus stiffness. (b) Peak load of the simulated splitting test versus bond strength.
Parameters of the 2D parallel bond model. (a) Original slope of the splitting test curve versus stiffness. (b) Peak load of the splitting test versus bond strength.
Parameters of the 3D contact bond model. (a) Original slope of the splitting test curve versus stiffness. (b) Peak load of the splitting test versus bond strength.
Parameters of the 3D parallel bond model. (a) Original slope of the splitting test curve versus stiffness. (b) Peak load of the splitting test versus bond strength.
Discrete bond model parameters of AC20.
Temperature  Contact type  Stiffness  Bond strength  

−10°C  2D  Contact bond model  2.2 × 10^{9} N·m^{−1}  1.8 × 10^{2} N 
Parallel bond model  1.6 × 10^{12} Pa·m^{−1}  5.8 × 10^{7} Pa  
3D  Contact bond model  2.8 × 10^{7} N·m^{−1}  72.0 N  
Parallel bond model  3.9 × 10^{12} Pa·m^{−1}  1.1 × 10^{8} Pa  


0°C  2D  Contact bond model  1.5 × 10^{9} N·m^{−1}  2.0 × 10^{2} N 
Parallel bond model  1.0 × 10^{12} Pa·m^{−1}  2.0 × 10^{7} Pa  
3D  Contact bond model  9.0 × 10^{6} N·m^{−1}  52.0 N  
Parallel bond model  2.7 × 10^{12} Pa·m^{−1}  7.5 × 10^{7} Pa  


10°C  2D  Contact bond model  2.3 × 10^{9} N·m^{−1}  1.3 × 10^{2}N 
Parallel bond model  1.9 × 10^{12} Pa·m^{−1}  1.1 × 10^{7} Pa  
3D  Contact bond model  1.3 × 10^{7} N·m^{−1}  61.0 N  
Parallel bond model  4.7 × 10^{12} Pa·m^{−1}  1.1 × 10^{8} Pa  


20°C  2D  Contact bond model  1.4 × 10^{9} N·m^{−1}  — 
Parallel bond model  9.8 × 10^{11} Pa·m^{−1}  —  
3D  Contact bond model  1.1 × 10^{7} N·m^{−1}  10.0 N  
Parallel bond model  3.4 × 10^{12} Pa·m^{−1}  2.0 × 10^{7} Pa 
Discrete bond model parameters of AC13.
Temperature  Contact type  Stiffness  Bond strength  

−10°C  2D  Contact bond model  4.2 × 10^{9}N·m^{−1}  9.0 × 10^{2}N 
Parallel bond model  3.8 × 10^{12} Pa·m^{−1}  3.9 × 10^{7} Pa  
3D  Contact bond model  2.2 × 10^{7} N·m^{−1}  69.0 N  
Parallel bond model  9.9 × 10^{12} Pa·m^{−1}  1.6 × 10^{8} Pa  


0°C  2D  Contact bond model  1.4 × 10^{9}N·m^{−1}  3.5 × 10^{2}N 
Parallel bond model  9.8 × 10^{11} Pa·m^{−1}  1.3 × 10^{7} Pa  
3D  Contact bond model  8.2 × 10^{6} N·m^{−1}  54.0 N  
Parallel bond model  2.8 × 10^{12} Pa·m^{−1}  6.7 × 10^{7} Pa  


10°C  2D  Contact bond model  1.8 × 10^{9}N·m^{−1}  1.4 × 10^{2}N 
Parallel bond model  1.6 × 10^{12} Pa·m^{−1}  1.4 × 10^{7} Pa  
3D  Contact bond model  1.1 × 10^{7} N·m^{−1}  65.0 N  
Parallel bond model  3.4 × 10^{12} Pa·m^{−1}  9.2 × 10^{7} Pa  


20°C  2D  Contact bond model  2.1 × 10^{9}N·m^{−1}  — 
Parallel bond model  1.7 × 10^{12} Pa·m^{−1}  —  
3D  Contact bond model  1.2 × 10^{7} N·m^{−1}  13.0 N  
Parallel bond model  4.2 × 10^{12} Pa·m^{−1}  2.8 × 10^{7} Pa 
A DEM model was established using the parameters in Table
Comparison of the simulation results and lab test of AC20 at 0°C. (a) 2D model. (b) 3D model [
Figure
Comparison of the simulation results and lab test of AC20 at 10°C. (a) 2D model. (b) 3D model.
In addition, the behavior of the asphalt mixture at low temperature in the laboratory test is that the load value decreased rapidly after it reached a peak point; however, the discrete element simulation result showed that the load value decreased slowly after it reached the peak point. Therefore, it can be concluded that the DEM bond models discussed in this paper are unable to simulate brittle failure process of asphalt mixture.
In this study, the method for determining the parameters of a discrete element was studied based on the asphalt mixture splitting test. The main conclusions are as follows:
When the parameters in the DEM bond model were set constant, the simulation results were affected by the minimum particle size if the contact bond model was used. However, minimum particle size had little effect on simulation results when using the parallel bond model. Therefore, when the contact bond model is used, it is best to select appropriate particle size and model parameters on the basis of calculation efficiency and precision.
Basing on the simulation results of the splitting test, the slope of the loaddisplacement curve at the elastic stage was closely related to the stiffness parameters of the bond model. The greater the stiffness, the larger was the slope. The bond strength parameters in both contact bond and parallel bond models had a large influence on the peak load of the simulation results. The peak load in the loaddisplacement curve increased with the increase in the bond strength parameters’ value.
According to the influence of the bond model parameters on the DEM simulation results, the relationships between slope of the loaddisplacement curve at the beginning stage, the peak load and the bond model parameters of stiffness, and bond strength was established. By comparing the initial slop and peak load of the loaddisplacement curve of simulation with the actual splitting test results, the parameter values of the discrete element bond model at different temperatures were determined.
Comparing the DEM simulation results and the lab test results of the splitting test illustrated that the threedimensional parallel bond model is more suitable for simulating the loaddisplacement behavior of asphalt mixture. However, the DEM models presented in this study are not accurate in simulating the brittle failure process of asphalt mixture at low temperature.
The data supporting the conclusions of the present study can be obtained from the corresponding author.
All the authors declare that there are no conflicts of interests regarding the publication of this article.
Jingsong Shan and Yang Du conceived and designed the laboratory experiments, Dahai Fan and Laiyao Guo performed the laboratory experiments, Jingsong Shan conducted the theoretical analysis, and Du Yang analyzed the laboratory data and wrote the paper.
This research was funded by a project of Shandong Province Higher Educational Science and Technology Program under grant number J17KA213.