Research on the Calibration Method of the Bonding Parameters of the EDEM Simulation Model for Asphalt Mixtures

Abstract

To enhance the accuracy and reliability of the discrete element simulation software EDEM 2023 for pavement asphalt mixture simulation, three representative coarse aggregate particles were modeled in 3D using the SolidWorks 2018 software and imported into the EDEM 2023 software for particle filling. The Hertz–Mindlin with bonding contact model was used to construct the EDEM simulation model of asphalt mixtures, and the quadratic regression model of asphalt mixtures’ splitting tensile strength and four bonding parameters, namely, normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress, was found by the response surface methodology. The results show that the relationship between the significance magnitude of the four bonding parameters on the splitting tensile strength of the asphalt mixture simulation model is as follows: critical normal stress > shear stiffness per unit area > normal stiffness per unit area > critical shear stress. The calibration results of the bonding parameters were used for simulation verification, and the relative error between the simulation and actual splitting tensile strength was only −2.48%. The feasibility of this bonding parameter calibration method is demonstrated, and it can lay a foundation for EDEM to simulate the performance of asphalt mixtures on pavements with high-precision simulation.

1. Introduction

Asphalt mixtures are widely used in road construction based on their excellent road performance [1]. The macroscopic mechanical properties of asphalt mixtures are closely related to their mesoscopic behavioral properties [2]. In recent years, the study of meso-scale asphalt mixtures has gradually attracted the attention of numerous scholars. The discrete element method (DEM), based on interparticle contact modeling and Newton’s second law, was initially applied to geotechnical material. It started to be applied to asphalt mixtures after the emergence of the bond contact model and has been widely used for asphalt mixtures research currently [3,4,5]. The discrete element method (DEM) is used in two-dimensional particle flow code (PFC2D), three-dimensional particle flow code (PFC2D), and the EDEM software [6,7,8]. Researchers have used the discrete element method (DEM) to study asphalt mixtures from different perspectives, mainly including the internal mechanical response, the evolution of meso-mechanical behavior during the loading process, and the morphological characteristics of the coarse aggregate skeleton [9,10,11].

The discrete element simulation software EDEM has advantages in high-performance computing, user-friendliness, integration with other computer-aided engineering (CAE) tools, rich visualization and analysis tools, suitability for large-scale complex simulation, and multi-tool integration [12]. Khateeb et al. [7] simulated the meso-mechanical behavior of asphalt mixtures during the compaction process using the viscoelastic contact model in EDEM. They analyzed the internal structure and bubble distribution within the mixtures during this process. Xie et al. [3] conducted discrete element method (DEM) simulations using EDEM, considering the asphalt film and Johnson–Kendall–Roberts (JKR) cohesive forces. Their research focused on investigating the mechanism and evaluation of segregation in asphalt mixtures. Ma et al. [13] utilized the EDEM discrete element simulation software to establish a simulation model for the vibrating screening component, which is a core and critical structure of the detection equipment, based on the functional requirements for detecting the aggregate gradation of Recycled Asphalt Pavement (RAP). They conducted simulations of the entire screening process. Yu et al. [14] used the discrete element method (DEM) simulation software EDEM to generate and compact six types of asphalt mixtures with different aggregate gradations. Based on the complex network theory, they analyzed the influence of the compaction degree of the mixtures on the topological characteristics of Functional Connectivity Networks (FCNs). X. M. Ai et al. [15] summarized that the morphological characteristics of coarse aggregates and spatial distribution of air voids are essential to the simulation results of asphalt mixtures. H. Zhang et al. [16] conducted Brazilian disk-splitting tensile tests on concrete and mortar with three water–cement ratios and four loading rates, respectively. A model that conforms to the macroscopic performance of the concrete dynamic splitting tensile test was established based on the discrete element method (DEM) and Hertz–Mindlin bond model. Virtual splitting tests have become an important means of exploring the micro- and meso-mechanical properties of asphalt mixtures. Sun et al. [17] calibrated the parameters by the Marshall experiment, and the simulation experiment results after parameter adjustment were in good agreement with the experimental results in a laboratory.

The studies above indicate that the use of the discrete element method (DEM) simulation software EDEM for simulating asphalt mixtures in pavements is feasible. The reliability of the simulation results depends on the accuracy of the DEM parameters. However, many studies rely on parameters either already available in the literature or based on empirical values. Parameters solely obtained from the literature cannot accurately simulate actual conditions. Currently, there are few scholars who have conducted relevant research on the relationship between the mesoscopic characteristics and macroscopic mechanical properties of asphalt mixtures using the Hertz–Mindlin with bonding contact model in the discrete element method (DEM) software EDEM. In view of this, this study employs the discrete element method (DEM) simulation software EDEM and selects the Hertz–Mindlin with bonding contact model. The calibration method for the binding parameters of the EDEM simulation model of asphalt mixtures was studied by laboratory and simulation splitting tests. The foundation for EDEM’s high-precision simulation of pavement asphalt mixture properties was laid. EDEM simulation based on the bonding parameters that get calibrated makes the simulation results conform to the real situation. The practical application of EDEM simulation studies is further promoted, thus reducing the economic costs incurred by laboratory tests.

2. Materials and Methods

2.1. Raw Materials

In the discrete element model, the bond strength of the asphalt binder is represented in the parameter settings of the bond contact model, so the type of asphalt binder chosen has no effect on the results of the study. Due to its excellent performance in high-temperature stability, low-temperature crack resistance, and fatigue resistance, Styrene–butadiene–styrene (SBS)-modified asphalt mixture is widely used in high-grade pavements [18]. Therefore, SBS-modified asphalt binder supported by a company in Jiaxing, China is selected, and its technical indicators are tested according to the requirements of the “Standard Test Methods of Bitumen and Bituminous Mixtures for Highway Engineering” (JTG E20-2011) [19]. The test results are shown in Table 1.

Table 1. Technical indicators by specifications and test results for SBS-modified asphalt binder.

Item		Unit	Test Value	Specification	Testing Method
Penetration (25 °C, 100 g, 5 s)		0.1 mm	70.8	60~80	T 0604
Penetration Index (PI)		——	0.2	≥−0.4	T 0604
Ring and Ball Softening Point		°C	78.6	≥55	T 0606
Residual Ductility at 5 °C		cm	42.1	≥30	T 0605
Viscosity at 135 °C		Pa·s	1.532	≤3	T 0625
Elastic Recovery at 25 °C		%	95	≥65	T 0662
Density at 25 °C		g/cm3	1.0036	Measured	T 0603
RTFOT	Mass Change	%	0.23	≤±1.0	T 0609
	Penetration Ratio	%	84.8	≥60	T 0604
	Residual Ductility at 5 °C	cm	24.2	≥20	T 0605

According to the “Test Methods of Aggregate for Highway Engineering” (JTG 3432-2024) [20], the technical indicators of coarse aggregate, fine aggregate, and mineral powder (Sourced from Jiaxing, China) were tested. The test results are presented in Table 2, Table 3 and Table 4, respectively, and all the technical indicators meet the requirements of the relevant technical standards.

Table 2. Technical indicators by specifications and test results for coarse aggregate.

Test Item	Unit	Standard Requirement	Coarse Aggregate Test Result			Assessment	Testing Method
			15~25 mm	5~15 mm	3~5 mm
Apparent Relative Density	g/cm3	≥2.60	2.867	2.718	2.825	Qualified	T 0304
Bulk Relative Density	g/cm3	Measured	2.818	2.648	2.732	Qualified	T 0304
Water Absorption Rate	%	≤2.0	0.62	0.97	1.2	Qualified	T 0304
Adhesion to Asphalt	Stage	≥5	5			Qualified	T 0616
Crushing Value	%	≤28	16.2			Qualified	T 0316
Elongated and Flaky Particle Content	%	≤18	8.6			Qualified	T 0312
Soft Stone Content	%	≤3	2.3			Qualified	T 0320
Sturdiness	%	≤12	7			Qualified	T 0314

Table 3. Technical indicators by specifications and test results for fine aggregate.

Test Item	Unit	Standard Requirement	Test Result	Assessment	Testing Method
Apparent Density	g/cm3	Measured	2.701	Qualified	T 0328
Apparent Relative Density	—	≥2.50	2.705	—	T 0328
Robustness (>0.3 mm)	%	≤12	8.8	Qualified	T 0340
Sand Equivalent	%	≥60	63	Qualified	T 0334

Table 4. Technical indicators by specifications and test results for mineral powder.

Test Item		Unit	Standard Requirement	Test Result	Assessment	Testing Method
Apparent Density		g/cm3	Measured	2.698	Qualified	T 0352
Water Content		%	≤1	0.33	Qualified	T 0103 Drying Method
Particle Size Range	<0.6 mm	%	100	100	Qualified	T 0351
	<0.15 mm	%	90~100	96.6	Qualified
	<0.075 mm	%	75~100	85.8	Qualified
Appearance		—	Free from Agglomeration and Caking	Free from Agglomeration and Caking	Qualified	—
Hydrophilic Coefficient		—	<1.0	0.8	Qualified	T 0353

2.2. Measurement of Splitting Tensile Strength of Asphalt Mixtures

The aggregate gradation of the asphalt mixture is selected as AC-13, as shown in Figure 1. The oil–stone ratio is 5%. Marshall specimens with dimensions of φ101.6 mm ± 0.25 mm × 63.5 mm ± 1.3 mm, formed using the standard compaction method (75 hits on each side), were used for the splitting test.

Figure 1. Aggregate gradation of asphalt mixture.

The SYD-0730A multi-functional automatic asphalt pressure tester (Produced by Shanghai Changji Geological Instrument CO., LTD, Shanghai, China) was used to conduct splitting tensile strength tests on the asphalt mixture specimens. The test was conducted at a temperature of 15 °C with a loading rate of 50 mm/min (Figure 2).

Figure 2. Laboratory splitting test for asphalt mixtures.

According to the “Test Methods of Bitumen and Bituminous Mixtures for Highway Engineering” (JTG E20-2011) [19], the splitting tensile strength is calculated using Equation (1). The splitting tensile strengths of the specimens are shown in Table 5.

$$R_T=0.006287P_T/h$$

(1)

Table 5. The splitting tensile strength of the Marshall specimens.

Specimen Number	$\mathbf{The}\mathbf{Maximum}\mathbf{Load}\mathbf{Applied}\mathbf{to}\mathbf{the}\mathbf{Specimen}{\mathit{P}}_{\mathit{T}}/\mathbf{N}$	$\mathbf{The}\mathbf{Splitting}\mathbf{Tensile}\mathbf{Strength}{\mathit{R}}_{\mathit{T}}/\mathbf{M}\mathbf{P}\mathbf{a}$
1	8610	0.853
2	8730	0.868
3	8690	0.849
4	8490	0.841
5	8280	0.829
Mean Value	8560	0.848

In Equation (1), $R_T$ is the splitting tensile strength, MPa; $P_T$ is the maximum load applied to the specimen, N; and $h$ is the height of the specimen, mm.

3. EDEM Simulation Modeling of Asphalt Mixtures

3.1. Selection of Bonded Contact Model

The crux of constructing a simulation model for asphalt mixtures lies in accurately modeling the bonding interactions between particles. The Hertz–Mindlin with bonding contact model, available within the discrete element method (DEM) simulation software EDEM, introduces the concept of a “Bond” based on the Hertz–Mindlin model to simulate the bonding interactions between particles. The bonding model is particularly suitable for simulating the fracture and failure of asphalt concrete and geomaterial structures [21,22].

As depicted in Figure 3, R represents the physical radius of the particles; $R_{contact}$ denotes the contact radius of the particles; and $\overline{R}$ signifies the bonding radius. A “Bond” can be formed when the distance between particles is less than the sum of their contact radius. This “Bond” acts as a finite-sized adhesive, resisting a certain degree of normal and shear movement within the contact radius of the particles. When the normal and shear stresses exceed a predefined value, the bond breaks and will not regenerate.

Figure 3. Interaction of the “Bond” between Particle A and Particle B.

The motional state and displacement state of the asphalt mixture particle units are calculated using the Hertz–Mindlin model prior to the bonding time $t_{bond}$. Upon reaching the bonding time $t_{bond}$ in the simulation, bonding occurs between the particles, and the contact forces ($F_{n,t}$) and torques ($M_{n,t}$) are subsequently reset to zero. These are then recalculated at each time step using the following equations:

$$\delta F_n=-v_n{S}_{n}A\delta t$$

(2)

$$\delta F_t=-v_t{S}_{t}A\delta t$$

(3)

$$\delta M_n=-{\omega }_n{S}_{t}J\delta t$$

(4)

$$\delta M_t=-{\omega }_t{S}_n{\displaystyle \frac{J}{2}}\delta t$$

(5)

In which

$$A=\pi {\overline{R}}^2$$

(6)

$$J={\displaystyle \frac{1}{2}}\pi {\overline{R}}^4$$

(7)

In the equations, $v_n$ and $v_t$ represent the normal and shear velocities of the particle, respectively, m/s; $S_n$ and $S_t$ denote the normal stiffness per unit area and the shear stiffness per unit area, respectively, N/m³; $\delta t$ is the time step, s; ${\omega }_n$ and ${\omega }_t$ are the normal and shear angular velocities of the particle, respectively, rad/s; A is the area of the contact region, m²; and J is the moment of inertia, m⁴.

When the normal stress and shear stress exceed their respective preset critical values, the “Bond” link will break. The condition for bond rupture can be expressed as follows:

$${\sigma }_{max}<{\displaystyle \frac{-F_n}{A}}+{\displaystyle \frac{2M_t}{J}}\overline{R}$$

(8)

$${\tau }_{max}<{\displaystyle \frac{-F_t}{A}}+{\displaystyle \frac{M_n}{J}}\overline{R}$$

(9)

In the equation, ${\sigma }_{max}$ and ${\tau }_{max}$ is the critical normal stress and the critical shear stress, respectively, $\mathrm{P}\mathrm{a}$.

To establish a reasonable and reliable EDEM simulation model for asphalt mixtures, it is necessary to calibrate the four key bonding contact parameters of the Hertz–Mindlin with bonding model.

3.2. Simulation Modeling of Asphalt Mixtures

3.2.1. Particle Modeling and Basic Material Parameters

Based on the asphalt mixture mastic theory, this study divides asphalt mixtures into coarse aggregates and asphalt mortar, as shown in Figure 4. The asphalt mortar comprises fine aggregates, fillers, and an asphalt binder.

Figure 4. Asphalt mixture mastic theory.

The skeleton of asphalt mixtures is composed of coarse aggregates, and the particle morphology of these coarse aggregates has a significant impact on the mechanical properties of asphalt mixtures [23,24]. Using a single spherical particle for simulation would deviate significantly from actual conditions, thereby compromising the rationality of the simulation. In this study, three representative typical particles were selected for each grade of coarse aggregates: regular particle, elongated particle, and flat particle [25]. These selected typical particles were three-dimensionally modeled using the SolidWorks software, saved as STL files, and imported into EDEM for particle packing. Upon reviewing the relevant literature [26], it was found that as the number of spheres used to fill the particle increases, the particle model becomes closer to its actual shape, but the change in calculation accuracy is minimal. To ensure both calculation accuracy and efficiency, this study employed a five-sphere model to simulate real coarse aggregate particles. Research has shown that this three-dimensional semi-realistic discrete element method (DEM) model can effectively simulate the macroscopic mechanical properties of granular materials [27]. The packing effects of the three typical particles for coarse aggregates with a particle size range of 13.2–16 mm are shown in Table 6.

Table 6. The packing effects of the three typical particles for coarse aggregates.

Aggregate Type	Actual Photo	3D Model Diagram	Packing Effects
Regular Particle
Elongated Particle
Flat Particle

In asphalt mixtures, asphalt mortar contributes relatively little to the overall structural support, primarily serving roles in bonding and filling. Additionally, due to the small particle size of asphalt mortar, generating a model based on the aggregate gradation of asphalt mixtures would result in many fine particles, leading to an exponential increase in computation time and a significant reduction in computational efficiency. Therefore, when generating the particle model for asphalt mixtures, asphalt mortar particles are uniformly replaced with spherical particles of a diameter of 2.36 mm [28] in order to facilitate the optimization of the calculation process and achieve a balance between efficiency and accuracy.

The rationality of the material properties of particles and equipment is a vital factor in ensuring the correctness of the simulation results. The material of the equipment is designated as steel. By referring to the relevant literature [23,29], the intrinsic parameters of the materials are listed in Table 7.

Table 7. Intrinsic parameters of the materials.

Material Type	Poisson’s Ratio	Density (kg/m3)	Shear Modulus (Pa)
Coarse Aggregate	0.33	2600	1 × 108
Asphalt Mortar	0.15	2400	1 × 108
Steel	0.3	7850	7 × 1010

The contact parameters between different materials are presented in Table 8.

Table 8. The contact parameters between different materials.

Contact Type	Coefficient of Restitution	Static Friction Coefficient	Dynamic Friction Coefficient
Coarse Aggregate–Coarse Aggregate	0.01	0.7	0.1
Coarse Aggregate–Asphalt Mortar	0.005	0.75	0.15
Coarse Aggregate–Steel	0.01	0.6	0.1
Asphalt Mortar–Asphalt Mortar	0.001	0.7	0.1
Asphalt Mortar–Steel	0.005	0.5	0.05

3.2.2. Simulation Model of the Splitting Test

A cylindrical container with dimensions of φ101.6 mm × 63.5 mm is created in EDEM, serving as the mold for the Marshall specimen. The material of the mold is set to steel. Above the mold, an equally sized virtual container is generated, functioning as a particle factory for generating asphalt mixture particles. The mold and particle factory are illustrated in Figure 5.

Figure 5. Marshall specimen mold and particle factory.

The aggregate gradation for the splitting test simulation model, determined based on the asphalt mixture aggregate gradation from the laboratory splitting test, is shown in Table 9.

Table 9. The aggregate gradation for the splitting test simulation model.

Range of Particle Size/mm	16~19	13.2~16	9.5~13.2	4.75~9.5	2.36~4.75	Asphalt Mortar (2.36)
Simulation Ratio/%	0.0	5.0	15.1	34.7	9.0	36.1

Based on the aggregate gradation, the particle factory is set to generate asphalt mixture particles. “Max Attempts to Place Particle” is set to 20, which avoids the segregation of the particles and achieves a good distribution of the particles. After all the particles are generated, they undergo compaction processing. Asphalt mixture compaction methods include Marshall impact compaction (MIC), Superpave gyratory compaction (SGC), static compaction (SC), and linear kneading compaction (LKC) [30]. Static compaction (SC) [14] was used in this study. The particle generation and compaction process for the Marshall specimen model is illustrated in Figure 6.

Figure 6. Schematic diagram of Marshall specimen model generation and compaction process.

The distribution of particles of various sizes within the compacted Marshall specimen is shown in Figure 7. Both asphalt mortar and coarse particles are distributed uniformly. Asphalt mortar particles make up the largest percentage and are distributed throughout the specimen. It is consistent with Table 9.

Figure 7. Marshall specimen particle size distribution chart by grade.

“Bond” links are formed after the particles are stabilized through compaction. Then, we conceal the particles while revealing the “Bond” links. Following the requirements of the “Standard Test Methods of Bitumen and Bituminous Mixtures for Highway Engineering” (JTG E20-2011) [19] for the shape of the compression bar, a three-dimensional model of the compression bar is established using the SolidWorks software and imported into EDEM. The model for the splitting test is shown in Figure 8. The loading rate of the compression bar in the simulated test is consistent with that of the laboratory splitting test, which is 50 mm/min.

Figure 8. The simulation model for the splitting test.

4. Design of Experiments for Calibration Methods of Bonding Parameters

4.1. Response Surface Methodology (RSM) Design and Simulation Results of the Bonded Contact Parameters

Response surface methodology (RSM) is a statistical optimization method that integrates experimental design and mathematical modeling. It models the functional relationship between one or more factors and the response through the data obtained from the experimental design [31]. The various factors affecting the response in response surface methodology need to be identified, along with the selection of an appropriate experimental design methodology, such as the Central Composite Design (CCD) or Box–Behnken design, to obtain sufficient data for modeling. The rupture of the “Bond” links between particles is related to four bonding parameters in the Hertz–Mindlin with bonding contact model, which include normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress. To prevent being affected by out-of-range parameter values, studies related to the setting of bonding parameters in EDEM models for asphalt mixtures are referenced [32,33,34]. The upper and lower limits of the four bonding parameters are shown in Table 10.

Table 10. The upper and lower limits of the four bonding parameters.

Bonding Parameters	Upper Limit of Parameters	Lower Limit of Parameters
$\mathrm{Normal}\mathrm{Stiffness}\mathrm{Per}\mathrm{Unit}\mathrm{Area}/(\mathrm{N}·{\mathrm{m}}^{-3}$)	3.05 × 109	2.31 × 1010
$\mathrm{Shear}\mathrm{Stiffness}\mathrm{Per}\mathrm{Unit}\mathrm{Area}/(\mathrm{N}·{\mathrm{m}}^{-3}$)	3.05 × 109	2.31 × 1010
$\mathrm{Critical}\mathrm{Normal}\mathrm{Stress}/\mathrm{P}\mathrm{a}$	4.25 × 108	6.00 × 109
$\mathrm{Critical}\mathrm{Shear}\mathrm{Stress}/\mathrm{P}\mathrm{a}$	4.25 × 108	6.00 × 109

For the convenience of expression, normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress have been, respectively, denoted using the $X_1$, $X_2$, $X_3$, and $X_4$. They are identified as the parameters to be calibrated. A response surface methodology experimental design is adopted, with splitting tensile strength $R_T$ as the test indicator. Utilizing the Box–Behnken method in the software Design-Expert 13 and referring to the upper and lower limits of the four bonding parameters in Table 10, a response surface methodology is designed. The response surface methodology design and simulation results are presented in Table 11.

Table 11. The response surface methodology design and simulation results.

Number	Parameters				${\mathit{R}}_{\mathit{T}}$/MPa
	${\mathit{X}}_1$/(N·m−3)	${\mathit{X}}_2$/(N·m−3)	${\mathit{X}}_3$/Pa	${\mathit{X}}_4$/Pa
1	3.05 × 109	1.31 × 1010	3.21 × 109	4.25 × 108	0.53
2	1.31 × 1010	2.31 × 1010	4.25 × 108	3.21 × 109	0.23
3	1.31 × 1010	3.05 × 109	3.21 × 109	6.00 × 109	0.32
4	3.05 × 109	2.31 × 1010	3.21 × 109	3.21 × 109	0.79
5	1.31 × 1010	2.31 × 1010	3.21 × 109	6.00 × 109	0.65
6	3.05 × 109	1.31 × 1010	4.25 × 108	3.21 × 109	0.28
7	1.31 × 1010	1.31 × 1010	4.25 × 108	6.00 × 109	0.18
8	1.31 × 1010	3.05 × 109	4.25 × 108	3.21 × 109	0.12
9	1.31 × 1010	3.05 × 109	3.21 × 109	4.25 × 108	0.33
10	2.31 × 1010	1.31 × 1010	3.21 × 109	6.00 × 109	0.39
11	1.31 × 1010	1.31 × 1010	6.00 × 109	6.00 × 109	1.01
12	2.31 × 1010	2.31 × 1010	3.21 × 109	3.21 × 109	0.47
13	1.31 × 1010	1.31 × 1010	6.00 × 109	4.25 × 108	0.82
14	1.31 × 1010	1.31 × 1010	3.21 × 109	3.21 × 109	0.49
15	1.31 × 1010	2.31 × 1010	6.00 × 109	3.21 × 109	1.12
16	1.31 × 1010	1.31 × 1010	3.21 × 109	3.21 × 109	0.64
17	3.05 × 109	1.31 × 1010	6.00 × 109	3.21 × 109	1.21
18	1.31 × 1010	1.31 × 1010	3.21 × 109	3.21 × 109	0.58
19	2.31 × 1010	1.31 × 1010	3.21 × 109	4.25 × 108	0.42
20	3.05 × 109	1.31 × 1010	3.21 × 109	6.00 × 109	0.79
21	1.31 × 1010	3.05 × 109	6.00 × 109	3.21 × 109	0.67
22	2.31 × 1010	1.31 × 1010	4.25 × 108	3.21 × 109	0.19
23	2.31 × 1010	1.31 × 1010	6.00 × 109	3.21 × 109	0.79
24	3.05 × 109	3.05 × 109	3.21 × 109	3.21 × 109	0.41
25	1.31 × 1010	2.31 × 1010	3.21 × 109	4.25 × 108	0.56
26	1.31 × 1010	1.31 × 1010	4.25 × 108	4.25 × 108	0.18
27	2.31 × 1010	3.05 × 109	3.21 × 109	3.21 × 109	0.30

4.2. Analysis of Simulation Results

The simulation results in Table 11 show that the maximum and minimum values of splitting tensile strength for the EDEM simulation model of asphalt mixtures are 1.21 MPa and 0.12 MPa, respectively. The range of splitting tensile strength of asphalt mixtures is generally covered. The results of the response surface methodology experiment were fitted and analyzed. The results indicated that when using a quadratic full model equation for fitting, the coefficient of determination $R^2$ was 0.9918. The variance analysis of the regression model is presented in Table 12.

Table 12. The variance analysis of the regression model.

Sources	Quadratic Sum	Degree of Freedom	Mean Square	F-Value	Significance Level p
Models	2.23	14	0.1591	103.72	<0.0001	Significant
$X_1$	0.1752	1	0.1752	114.21	<0.0001
$X_2$	0.2324	1	0.2324	151.50	<0.0001
X3	1.64	1	1.64	1070.91	<0.0001
$X_4$	0.0208	1	0.0208	13.58	0.0031
$X_1{X}_2$	0.0110	1	0.0110	7.19	0.0200
$X_1{X}_3$	0.0272	1	0.0272	17.75	0.0012
$X_1{X}_4$	0.0210	1	0.0210	13.71	0.0030
$X_2{X}_3$	0.0289	1	0.0289	18.84	0.0010
$X_2{X}_4$	0.0025	1	0.0025	1.63	0.2259
$X_3{X}_4$	0.0090	1	0.0090	5.88	0.0320
$X_1^2$	0.0001	1	0.0001	0.073	0.7916
$X_2^2$	0.0264	1	0.0264	17.24	0.0013
$X_3^2$	0.0059	1	0.0059	3.86	0.0729
$X_4^2$	0.0104	1	0.0104	6.78	0.0230
Residual	0.0184	12	0.0015
Loss of Fit	0.0070	10	0.0007	0.1230	0.9920	Not Significant
Error	0.0114	2	0.0057
Total Deviation	2.25	26

Note: p  <  0.01, the difference was very significant; p  <  0.05, the difference was significant; p  >  0.05, the difference was not significant.

As shown in Table 12, the p-value of the regression model is less than 0.01, demonstrating that the model’s independent variables have a very significant impact on the dependent variable. The p-value of the loss-of-fit term is greater than 0.05, indicating that the loss of fit is not significant. This suggests that the quadratic regression equation fitted by the model aligns well with the actual situation, allowing for reliable parameter estimation results. Among the effects on splitting tensile strength, $X_1$, $X_2$, $X_3$, $X_4$, $X_1{X}_3$, $X_1{X}_4$, $X_2{X}_3$, and $X_2^2$ are very significant; $X_1{X}_2$, $X_3{X}_4$, and $X_4^2$ are significant; and $X_2{X}_4$, $X_1^2$, and $X_3^2$ are not significant.

Based on the regression equation, response surfaces for the interaction effects of various factors on the splitting tensile strength $R_T$ can be obtained, as shown in Figure 9.

Figure 9. Response surfaces for the interaction effects of various factors on the splitting tensile strength $R_T$: (a) interaction effects between $X_1$ and $X_2$; (b) interaction effects between $X_1$ and $X_3$; (c) interaction effects between $X_1$ and $X_4$; (d) interaction effects between $X_2$ and $X_3$; (e) interaction effects between $X_2$ and $X_4$; (f) interaction effects between $X_3$ and $X_4$.

Figure 9 clearly illustrates the influence patterns of the four bonding parameters on the splitting tensile strength. As shown in Figure 9d,f and Figure 9b, when the normal stiffness per unit area ($X_1$), shear stiffness per unit area ($X_2$), and critical shear stress ($X_4$) are fixed, the critical normal stress ($X_3$) has the most obvious effect on the splitting tensile strength, and the splitting tensile strength increases significantly with the increase in the critical normal stress, and the growth rate increases significantly.

As shown in Figure 9d,e and Figure 9a, when the normal stiffness per unit area ($X_1$), the critical normal stress ($X_3$), and the critical shear stress ($X_4$) are fixed, the splitting tensile strength increases slowly and then tends to flatten out, or even tends to decrease, with the increase in the shear stiffness per unit area ($X_2$).

As shown in Figure 9b,c and Figure 9a, when the shear stiffness per unit area ($X_2$), the critical normal stress ($X_3$), and the critical shear stress ($X_4$) are fixed, the splitting tensile strength decreases with the increase in the normal stiffness per unit area ($X_1$), but the rate of decrease is low.

As shown in Figure 9e,f and Figure 9c, when the normal stiffness per unit area ($X_1$), shear stiffness per unit area ($X_2$), and critical normal stress ($X_3$) are fixed, the critical shear stress ($X_4$) has no significant effect on the splitting tensile strength.

In summary, the significance of the influencing factors among the four bonding parameters is ranked as follows: critical normal stress ($X_3$) > shear stiffness per unit area ($X_2$) > normal stiffness per unit area ($X_1$) > critical shear stress ($X_4$). This ranking is consistent with the results of the model variance analysis.

Under the premise of ensuring a significant regression model and insignificant loss-of-fit terms, insignificant terms are removed to modify the quadratic regression model. The results show that when fitted using the quadratic full model equation, the coefficient of determination $R^2$ was 0.9879. The variance analysis of the modified regression model is presented in Table 13. The resulting quadratic regression model is as follows:

$$\begin{array}{ll}{R}_T=-0.1486& +\left(1.14E-11\right)X_1+\left(2.64E-11\right)X_2+\left(8.44E-11\right)X_3+\left(5.41E-11\right)X_4\\ & -\left(5.22E-22\right)X_1{X}_2-\left(2.95E-21\right)X_1{X}_3-\left(2.59E-21\right)X_1{X}_4+\left(3.04E-21\right)X_2{X}_3\\ & +\left(6.11E-21\right)X_3{X}_4-\left(7.01E-22\right)X_2^2-\left(5.68E-21\right)X_4^2\end{array}$$

Table 13. The variance analysis of the modified regression model.

Sources	Quadratic Sum	Degree of Freedom	Mean Square	F-Value	Significance Level p
Models	2.22	11	0.2017	111.69	<0.0001	Significant
$X_1$	0.1752	1	0.1752	97.01	<0.0001
$X_2$	0.2324	1	0.2324	128.69	<0.0001
X3	1.64	1	1.64	909.64	<0.0001
$X_4$	0.0208	1	0.0208	11.54	0.004
$X_1{X}_2$	0.011	1	0.011	6.1	0.026
$X_1{X}_3$	0.0272	1	0.0272	15.07	0.0015
$X_1{X}_4$	0.021	1	0.021	11.64	0.0039
$X_2{X}_3$	0.0289	1	0.0289	16	0.0012
$X_3{X}_4$	0.009	1	0.009	5	0.041
$X_2^2$	0.0409	1	0.0409	22.62	0.0003
$X_4^2$	0.0184	1	0.0184	10.2	0.006
Residual	0.0271	15	0.0018
Loss of Fit	0.0157	13	0.0012	0.2117	0.9713	Not Significant
Pure Error	0.0114	2	0.0057
Total Deviation	2.25	26

As seen in Table 13, the independent variable in the modified regression model has an extremely significant effect on the dependent variable, and at the same time, the regression equation is consistent with the actual situation, and reliable calibration results can be obtained. Compared to previous researchers who set the bonding parameters empirically when performing the EDEM simulations of asphalt mixtures, this regression model can provide researchers with a guide for the settings.

Using the Design-Expert software, the target value of $R_T$, which is the average splitting tensile strength of 0.848 MPa in laboratory asphalt mixture splitting tests, was sought. This is the result of the calibration of the bonding parameters of the EDEM simulation model for asphalt mixtures: the normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress were 1.31 × 10¹⁰ $\mathrm{N}{/\mathrm{m}}^3$, 8.24 × 10⁹ $\mathrm{N}{/\mathrm{m}}^3$, 5.78 × 10⁹ Pa, and 5.28 × 10⁹ Pa, respectively.

5. Verification of EDEM Simulation Model for Asphalt Mixtures

To verify the accuracy and reliability of the model for the tensile splitting strength of asphalt mixtures, this study conducts EDEM simulation experiments based on the calibration results of the bonding parameters. The loading rate of the compression bar in the simulation experiments is set to 50 mm/min, consistent with that in the laboratory splitting test. The EDEM analyst module is utilized to export data on the variation in the load of the upper compression bar over time. The variation in the load applied by the upper compression bar and the fracture state of the specimen over time are shown in Figure 10.

Figure 10. The variation in the load applied by the upper compression bar and the fracture state of the specimen over time.

From Figure 10, it can be observed that the overall trend of the load variation over time applied by the upper compression bar in both the simulation and laboratory splitting test of the asphalt mixture is similar. The simulated splitting tensile strength, calculated by Equation (1), is 0.827 MPa, with a relative error of only −2.48% compared to the splitting tensile strength obtained from the laboratory test.

The calibration method of the bonding parameters has been proven to be accurate. The relationship between the mesoscopic characteristics and macroscopic mechanical properties of asphalt mixtures was provided. The performance of the asphalt mixture EDEM simulation model can be more accurately simulated after the bonding parameters have been calibrated.

6. Conclusions

The three-dimensional modeling of coarse aggregate particles in asphalt mixture was conducted using SolidWorks and then imported into the EDEM software for particle packing; asphalt mortar particles were uniformly replaced with spherical particles of a diameter of 2.36 mm, thereby establishing an EDEM simulation model for asphalt mixture. The main research conclusions are as follows:

(1)

In the discrete element software EDEM’s Hertz–Mindlin with bonding contact modeling, the significance of the influence of the four bonding parameters on the splitting tensile strength of the asphalt mixture simulation model is as follows: critical normal stress ($X_3$) > shear stiffness per unit area ($X_2$) > normal stiffness per unit area ($X_1$) > critical shear stress ($X_4$).

(2)

The regression models between the four bonding parameters and splitting tensile strength were established by the response surface methodology (RSM). The splitting tensile strength of asphalt mixtures in the laboratory tests was calibrated as follows: the normal stiffness per unit area, shear stiffness per unit area, critical normal stress, and critical shear stress were 1.31 × 10¹⁰ $\mathrm{N}{/\mathrm{m}}^3$, 8.24 × 10⁹ $\mathrm{N}{/\mathrm{m}}^3$, 5.78 × 10⁹ Pa, and 5.28 × 10⁹ Pa, respectively.

(3)

The comparative validation results between the discrete element simulation tests and laboratory tests show that the trends of time–load curve changes between the simulation and actual tests are generally consistent. The feasibility and accuracy of the bond parameter calibration method for the EDEM simulation models of asphalt mixtures are proved. The calibration method can provide a basis for accurate studies of asphalt mixtures.