# Study on the Contact Stress Distribution Model of Asphalt Mixture Particles

Authors: Lei Wang; Xiucheng Yang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The distribution of contact stress of particles in asphalt mixture could be used to reflect the relationship between the macroscopic mechanical properties and the mesostructure of the material. However, there is no mature method to obtain the internal stress characteristics of asphalt mixture. In this study, the accurate stress distribution and stress transfer state information of particles in asphalt mixture were obtained by using pressure film technology, which provided a good theoretical basis for material gradation design. The pressure film was sandwiched between two Marshall specimens, and the compression part will appear red. After digital processing by software, the total contact area and stress distribution of the two specimens can be obtained. Three kinds of asphalt, three kinds of asphalt pavement gradation, and five kinds of asphalt aggregate ratio were selected to prepare asphalt mixture specimens. The effects of different asphalt, skeleton structure, and asphalt film thickness on the contact characteristics of aggregates were studied. The test results showed that, when the pressure between the two specimens is 0.23 MPa, the contact area of the particles reaches the maximum. The relationship between the contact point distribution probability and the pressure was fitted into a nonlinear curve. The gradation of the mixture, the type of asphalt and the thickness of the asphalt film were used as parameters A, B, and C. The prediction model had a good correlation with the test results. The prediction model proposed in this study could be used to improve the experimental efficiency, save test material and financial resources, as well as get complete contact stress information of the internal particle interface of asphalt mixture.


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## Body

## 1. Main Text Introduction

The material composition of the asphalt mixture is complex, the internal structure of the mixture is random, and the distribution of materials is characterized by nonuniformity. Asphalt mixture is a multiphase composite system in which coarse aggregates are connected and contact to form a skeleton, and fine aggregate, and asphalt mortar fills the unevenly distributed gaps [1]. The type of mixture material, the morphology of the aggregate, the thickness of the asphalt film, and the asphalt performance affect the distribution characteristics of the contact stress.The traditional asphalt mixture design methods are primarily based on the macro-mechanical index to evaluate the mechanical properties of the mixture. Still, the diversity of materials, the morphology of aggregate, the thickness of asphalt film, the difference of asphalt performance, and the stress between particles in asphalt mixture are unevenly distributed. The macroscopic mechanical index is difficult to distinguish the internal structure of the mixture and cannot explain the real state of force and stress transfer of the mixture. Therefore, it is of great significance to study the mechanical behavior of materials from the micro point of view and to obtain the real stress distribution and stress transfer state information of the particles in the asphalt mixture.There is an inherent relationship between the macroscopic mechanical properties and the mesostructure of the material, and the mesostructure is the fundamental reason for the performance of the macroscopic mechanical properties of the material. There are different types of structure, pores, and mortar distribution in the asphalt mixture, and the distribution characteristics of contact stress between aggregate particles in the skeleton are different [2]. Many scholars used image digital processing techniques such as CT scanning and synchrotron radiation micro-tomography to obtain the mixture’s internal structure and material distribution characteristics [3]. Brzezicki and Kasperkiewicz [4] used digital image processing technology to accurately measure the size of each particle in the aggregate particle sample and evaluated the aggregate shape characteristics, such as needle flake, slenderness ratio, and aggregate particle distribution characteristics. Masad [5–7] obtained the internal composition information of the mixture specimen with the help of X-ray CT digital image processing technology and analyzed the internal anisotropy of the specimen from the point of view of micromechanics. Wang et al. [8] valuated the pore system of three kinds of original WesTrack mixtures by X-ray tomography and stereoscopic methods and analyzed the effects of aggregate shape, angle, and surface texture on the overall performance of the pavement. Duan [9] obtained the distribution information of the cross-section particle size of the asphalt mixture through the industrial CT technology, realized the three-dimensional visualization of the internal structure of the mixture, and qualitatively analyzed the distribution law of the contact compactness of the aggregate in the mixture.The discontinuity and inhomogeneity of asphalt mixture are reflected by digital image processing technology. Still, the force of particles inside the mixture cannot be reflected from the point of view of mechanics. To obtain the meso-mechanical response of mixture, the numerical simulation method represented by discrete elements has been developed rapidly in recent years. In the discrete element method (DEM), a single discrete particle of mesoscale in a granular material is regarded as a discrete element. The whole particle aggregate model is considered to be a collection of several discrete elements [10]. The actual mechanical behavior of simulated materials such as contact force and deformation of particles in the process of motion is calculated, and the internal relationship between meso-fabric change and macroscopic mechanical behavior of granular materials is effectively studied [11]. Zhang [12] used the DEM to carry out the three-dimensional discrete element virtual uniaxial creep test of asphalt mixture and compared it with the indoor test results to verify the effectiveness of the model. Xing [13] found the number and distribution of contact points between main skeleton particles of different gradation, between interference particles, and between main skeleton particles and interference particles, as well as the influence of mesostructure on the contact force of mortar. DEM simulated the indirect tensile test of asphalt mixture, and the internal mechanical response of asphalt mixture with different skeleton filling states was analyzed. Bathurst and Rothenburg [14] studied the relationship between the contact between particles and the stress distribution between particles in two-dimensional granular materials and proposed a series of theoretical and numerical analyses to quantify the characteristics of micromechanical behavior granular media. The effects of friction angle and stiffness between particles on the shear capacity of numerical components under large strain were studied according to the fitting parameters. Chang et al. [15–17] reconstructed the meso-model of asphalt mixture by using a particle flow program, taking into account the influence of different particle sizes and gradation on the distribution of force chain. Five groups of rigid particles of 10 mm, 13 mm, 16 mm, 19 mm, and 20 mm were selected for the carbon paper indentation test. Without considering the distribution probability of gradation, the distribution probability increases at first and then decreases, and there is an obvious peak probability of the force chain. Considering that the distribution probability of the contact force chain of graded particles attenuates monotonously in the exponential form, there is no peak value.From the existing studies, we may find that the phenomenon-based conclusions can be obtained by using the digital image method. And the DEM simulates the meso-contact characteristics of aggregate by setting friction coefficient, contact characteristics, and other parameters to make the performance of the virtual specimen close to the real asphalt mixture. However, the aggregate model of several discrete elements is different from the real mechanical behavior of the mixture, so the analysis results cannot reflect the real macroscopic mechanical properties of the material.Pressure film technology can accurately reflect the stress state of the contact interface. Many scholars use this technology to study the interface contact state between the pavement surface layer and the tire surface. For example, Ren et al. [18] studied the contact state between different layers of the road structure. Zhang [19] used pressure film to analyze the stress state between tire and road surface. Li [20, 21] studied the dynamic antiskid mechanism of tire rolling on the road surface under braking conditions. Zhang et al. [22] used pressure film to measure the pressure distribution on the surface of the track slab under load on different slope roads. Chen et al. [23] used pressure film to measure the contact stress between tire and asphalt pavement and found that the greater the expected stress of Weibull expectation, the more significant the pavement concentration effect. Wang and Wang [24] studied the stress distribution of static contact between tire and asphalt pavement by using pressure film and evaluated the antiskid performance of asphalt pavement by the stress concentration distribution rate. A large number of previous studies have focused on evaluating the skid resistance between asphalt pavement and tires, which proves the feasibility of applying high-precision pressure film measurement technology to study pavement stress distribution. However, these studies have stayed in the discussion of the stress distribution on the surface of the asphalt pavement, and there are few studies on the stress distribution inside the mixture. In this paper, based on previous studies, the internal stress distribution of asphalt mixture was studied more deeply.The distribution of contact stress of particles in asphalt mixture could be used to reflect the relationship between the macroscopic mechanical properties and the mesostructure of the material. However, the real distribution of contact stress is still unknown according to the existing study. In this study, the pressure film technology is introduced based on the discrete element method to obtain the real contact condition and interface mechanical properties of particles in asphalt mixture. The contact area and stress distribution of particles in the mixture under normal load are measured using three films types: LLLW, LLW, and LW. The contact stress information obtained from three different sizes of films is processed and analyzed, the mathematical model of pressure and contact force distribution area is obtained, and the nonuniformity of contact stress distribution and stress concentration effect is analyzed. The relationship between the stress distribution characteristics of the contact interface in the mixture and the material characteristics is established, and the corresponding material parameters are determined.


## 2. Materials and Methods

### 2.1. Asphalt

In order to study the characteristics of internal contact stress distribution of different types of asphalt and asphalt mixture, Shell’s ordinary 70# asphalt, Nanyue brand’s SBS modified asphalt, and Shell’s S-HV modified asphalt are used in this experiment. Elephant hot mix ultra-thin pavement specialized special modified asphalt is used in the verification test. All the basic performance indexes of the asphalt meet the requirements of the Test Code for Highway Engineering Asphalt and Asphalt Mixture (JTGE20-2011). The test results are shown in Table1.Table 1
Asphalt performance.
Test indexUnitTechnical requirementsTest results70#SBS modifiedS-HV modifiedSpecial modified70#SBS modifiedS-HV modifiedSpecial modifiedSoftening point°C>43≥70≥80≥95509093101Ductility (5°C)cm>20≥20≥20≥3049.432.531.537Penetration degree (25°C)0.1 mm60–8040–60≥40≥3571.85548.140.1Dynamic viscosity (60°C)Pa s≤3.0>200000≥4000002.5>200000>580000Flash point°C≥260≥230≥230≥260328320≥230335


### 2.2. Aggregate

The aggregate used in this experiment is diabase provided by the SG16 contract section of Fujian Luqiao Huacheng Expressway, Baiyun District, Guangzhou City. The aggregate can be used only after the following processes: screening, washing, drying, etc. The coarse and fine aggregate test is carried out according to the Code for Aggregate Test of Highway Engineering Method. The test results are shown in Table2 and Table 3, all of which meet the specification requirements.Table 2
Performance index of coarse aggregate.
Test projectUnitTechnical requirementsTest resultsCrushing value%≤2618.15Los Angeles attrition%≤2817.1Water absorption rate%≤20.94Apparent relative densityg/cm3≥2.62.799Firmness%≤124.7Table 3
Performance index of fine aggregate.
Test projectUnitTechnical requirementsTest resultsApparent relative densityg/cm3≥2.52.922Sand equivalent%≥6078Sediment concentration (<0.075 mm)%≤30.8Firmness (>0.3 mm)%≤123.4


### 2.3. Packing Material

Mineral powder is mixed with asphalt to form asphalt mortar, which provides bonding for asphalt mixture. The ore powder obtained by grinding is used in this paper. The mineral powder should meet the basic requirements of cleanliness and dryness. Its performance is tested following the specifications, and the test results are shown in Table4.Table 4
Properties of ore powder.
Test projectUnitTechnical requirementsTest resultsApparent relative densityg/cm3≥2.52.84Water content%≤10.2Hydrophilic coefficient<1<1


### 2.4. Mix Design

In order to study the contact characteristics between aggregates of different skeleton structures, three typical asphalt pavement structures, AC-13, SMA-13, and OGFC-13, are selected for mix design. SMA-13 is mixed with 0.3% lignin fiber, and the amount of aggregate for each grade is determined according to the screening results and gradation curves. The three gradations AC-13, SMA-13, and OGFC-13 used in the test are shown in Table5.Table 5
Gradation design.
Gradation typePercentage of mass through the following sieve holes (square sieve mm) (%)1613.29.54.752.361.180.60.30.150.075AC-1310091.975.651.335.024.517.311.98.25.5SMA-1310095.061.829.522.518.415.813.611.710.0OGFC-1310095.067.128.718.413.210.58.15.74.3


### 2.5. Determination of Oil-Stone Ratio

To explore the relationship between the asphalt film thickness and the indirect contact distribution of aggregate particles, the asphalt film thickness of AC and SMA is divided into 5 grades, namely 9μm, 12 μm, 15 μm, 18 μm, and 21 μm. According to the determined gradation, the asphalt content and the oil-stone ratio are determined according to formulas (1), (2) and (3):(1)A=2+0.02a+0.04b+0.08c+0.14d+0.3e+0.6f+1.6g48,(2)F=PbA,(3)Pa=Pb1−Pb.Here, A is the specific surface area of aggregate; the passing rate of a, b … f, and g is 4.75, 2.36 … 0.15, and 0.075 (mm), respectively. F is the thickness of asphalt film, Pb is the asphalt content, and Pa is the ratio of oil to stone.The corresponding experimental oil-stone ratio is calculated according to the formula of asphalt film thickness, as shown in Table6.Table 6
Design of asphalt content.
GradingAC-13SMA-13OGFC-13Experimental ratio of oil to stone (%)4.66.03.6, 4.8, 6.0, 7.2, 8.4


### 2.6. Specimen Forming

In this paper, the rotary compaction method is adopted to make corresponding asphalt mixture specimens. The instrument used is the AFG2 rotary compaction instrument produced by Pine Company in the United States. Its vertical pressure is 600 kPa, and the rotation rate is 30 r/min. According to the study [25], there is no obvious correlation between the number of Marshall compaction and rotary compaction. To make the three asphalt mixtures reach the ideal compaction stable state for Marshall compaction, the number of AC-13 is 100, the number of SMA-13 is 70, and the number of OGFC-13 is 50.


### 2.7. Measurement of Contact Stress

Prescale pressure film is the first film in the world to measure pressure and pressure distribution. It has the advantages of simple visual inspection, easy operation, and easy digitization. According to the size of the pressure, the distribution of stress is shown by the concentration of red color on the pressure-sensitive paper. The method of use is as follows: cut the film and clip it to the place to be measured. A chemical reaction occurs between the microcapsule of the colorant layer and the colorant, making the part under pressure appear red, as shown in Figure1. The scanner is used to carry out digital processing on the color pressure-sensitive paper, and then the software FPD-8010E is used to convert the data. Then, the pressure analysis can be conducted, and the aggregate contact area and stress distribution of the two specimens can be obtained as shown in Figure 2.Figure 1
Pressure film.Figure 2
Contact stress information obtaining process.A single size of pressure film can only obtain the contact stress information within a specific accuracy range, and it is necessary to use multiple sizes of film to obtain the complete contact stress information.This test uses LLLW, LLW, and LW specifications of pressure film, and the range is 0.2–0.6 MPa, 0.5–2.5 MPa, and 2.5–10 MPa, respectively. The pressure film was placed between two specimens, and the normal load of 100 N was applied. After 3 min of pressure, the film was taken out, and the impression would appear on the aggregate contact surface. The pressure image analysis software was used to calculate the contact area.


### 2.8. The Mathematical Model

The contact stress information obtained from three different specifications of the film was processed and analyzed, a mathematical model of pressure and contact force distribution area was established, the nonuniformity of contact stress distribution and stress concentration effect was analyzed, and the influence of grading, asphalt type, and oil-stone ratio on the contact force distribution between aggregate was explored. In this study, OriginLab was used as a numerical fitting tool to fit test data according to the preset model. Based on the contact stress distribution, an evaluation method of material properties is proposed.


## 2.1. Asphalt

In order to study the characteristics of internal contact stress distribution of different types of asphalt and asphalt mixture, Shell’s ordinary 70# asphalt, Nanyue brand’s SBS modified asphalt, and Shell’s S-HV modified asphalt are used in this experiment. Elephant hot mix ultra-thin pavement specialized special modified asphalt is used in the verification test. All the basic performance indexes of the asphalt meet the requirements of the Test Code for Highway Engineering Asphalt and Asphalt Mixture (JTGE20-2011). The test results are shown in Table1.Table 1
Asphalt performance.
Test indexUnitTechnical requirementsTest results70#SBS modifiedS-HV modifiedSpecial modified70#SBS modifiedS-HV modifiedSpecial modifiedSoftening point°C>43≥70≥80≥95509093101Ductility (5°C)cm>20≥20≥20≥3049.432.531.537Penetration degree (25°C)0.1 mm60–8040–60≥40≥3571.85548.140.1Dynamic viscosity (60°C)Pa s≤3.0>200000≥4000002.5>200000>580000Flash point°C≥260≥230≥230≥260328320≥230335


## 2.2. Aggregate

The aggregate used in this experiment is diabase provided by the SG16 contract section of Fujian Luqiao Huacheng Expressway, Baiyun District, Guangzhou City. The aggregate can be used only after the following processes: screening, washing, drying, etc. The coarse and fine aggregate test is carried out according to the Code for Aggregate Test of Highway Engineering Method. The test results are shown in Table2 and Table 3, all of which meet the specification requirements.Table 2
Performance index of coarse aggregate.
Test projectUnitTechnical requirementsTest resultsCrushing value%≤2618.15Los Angeles attrition%≤2817.1Water absorption rate%≤20.94Apparent relative densityg/cm3≥2.62.799Firmness%≤124.7Table 3
Performance index of fine aggregate.
Test projectUnitTechnical requirementsTest resultsApparent relative densityg/cm3≥2.52.922Sand equivalent%≥6078Sediment concentration (<0.075 mm)%≤30.8Firmness (>0.3 mm)%≤123.4


## 2.3. Packing Material

Mineral powder is mixed with asphalt to form asphalt mortar, which provides bonding for asphalt mixture. The ore powder obtained by grinding is used in this paper. The mineral powder should meet the basic requirements of cleanliness and dryness. Its performance is tested following the specifications, and the test results are shown in Table4.Table 4
Properties of ore powder.
Test projectUnitTechnical requirementsTest resultsApparent relative densityg/cm3≥2.52.84Water content%≤10.2Hydrophilic coefficient<1<1


## 2.4. Mix Design

In order to study the contact characteristics between aggregates of different skeleton structures, three typical asphalt pavement structures, AC-13, SMA-13, and OGFC-13, are selected for mix design. SMA-13 is mixed with 0.3% lignin fiber, and the amount of aggregate for each grade is determined according to the screening results and gradation curves. The three gradations AC-13, SMA-13, and OGFC-13 used in the test are shown in Table5.Table 5
Gradation design.
Gradation typePercentage of mass through the following sieve holes (square sieve mm) (%)1613.29.54.752.361.180.60.30.150.075AC-1310091.975.651.335.024.517.311.98.25.5SMA-1310095.061.829.522.518.415.813.611.710.0OGFC-1310095.067.128.718.413.210.58.15.74.3


## 2.5. Determination of Oil-Stone Ratio

To explore the relationship between the asphalt film thickness and the indirect contact distribution of aggregate particles, the asphalt film thickness of AC and SMA is divided into 5 grades, namely 9μm, 12 μm, 15 μm, 18 μm, and 21 μm. According to the determined gradation, the asphalt content and the oil-stone ratio are determined according to formulas (1), (2) and (3):(1)A=2+0.02a+0.04b+0.08c+0.14d+0.3e+0.6f+1.6g48,(2)F=PbA,(3)Pa=Pb1−Pb.Here, A is the specific surface area of aggregate; the passing rate of a, b … f, and g is 4.75, 2.36 … 0.15, and 0.075 (mm), respectively. F is the thickness of asphalt film, Pb is the asphalt content, and Pa is the ratio of oil to stone.The corresponding experimental oil-stone ratio is calculated according to the formula of asphalt film thickness, as shown in Table6.Table 6
Design of asphalt content.
GradingAC-13SMA-13OGFC-13Experimental ratio of oil to stone (%)4.66.03.6, 4.8, 6.0, 7.2, 8.4


## 2.6. Specimen Forming

In this paper, the rotary compaction method is adopted to make corresponding asphalt mixture specimens. The instrument used is the AFG2 rotary compaction instrument produced by Pine Company in the United States. Its vertical pressure is 600 kPa, and the rotation rate is 30 r/min. According to the study [25], there is no obvious correlation between the number of Marshall compaction and rotary compaction. To make the three asphalt mixtures reach the ideal compaction stable state for Marshall compaction, the number of AC-13 is 100, the number of SMA-13 is 70, and the number of OGFC-13 is 50.


## 2.7. Measurement of Contact Stress

Prescale pressure film is the first film in the world to measure pressure and pressure distribution. It has the advantages of simple visual inspection, easy operation, and easy digitization. According to the size of the pressure, the distribution of stress is shown by the concentration of red color on the pressure-sensitive paper. The method of use is as follows: cut the film and clip it to the place to be measured. A chemical reaction occurs between the microcapsule of the colorant layer and the colorant, making the part under pressure appear red, as shown in Figure1. The scanner is used to carry out digital processing on the color pressure-sensitive paper, and then the software FPD-8010E is used to convert the data. Then, the pressure analysis can be conducted, and the aggregate contact area and stress distribution of the two specimens can be obtained as shown in Figure 2.Figure 1
Pressure film.Figure 2
Contact stress information obtaining process.A single size of pressure film can only obtain the contact stress information within a specific accuracy range, and it is necessary to use multiple sizes of film to obtain the complete contact stress information.This test uses LLLW, LLW, and LW specifications of pressure film, and the range is 0.2–0.6 MPa, 0.5–2.5 MPa, and 2.5–10 MPa, respectively. The pressure film was placed between two specimens, and the normal load of 100 N was applied. After 3 min of pressure, the film was taken out, and the impression would appear on the aggregate contact surface. The pressure image analysis software was used to calculate the contact area.


## 2.8. The Mathematical Model

The contact stress information obtained from three different specifications of the film was processed and analyzed, a mathematical model of pressure and contact force distribution area was established, the nonuniformity of contact stress distribution and stress concentration effect was analyzed, and the influence of grading, asphalt type, and oil-stone ratio on the contact force distribution between aggregate was explored. In this study, OriginLab was used as a numerical fitting tool to fit test data according to the preset model. Based on the contact stress distribution, an evaluation method of material properties is proposed.


## 3. Results

### 3.1. Analysis of Actual Results of Pressure Film

The rotary compaction test prepared three kinds of bitumen, three kinds of gradation, and five kinds of oil-stone ratio. Three different specifications of pressure film—LLLW, LLW, and LW—were used to measure the interface contact area between two Marshall specimens under a normal load of 100 N. LLLW film range: 0.2–0.6 MPa, LLW film range: 0.5–2.5 MPa, and LW film range: 2.5–10 MPa. The statistical results of the experiment are shown in Table7, in which the normal load of each range is the average of many tests.Table 7
Experimental load distribution results of pressure film (N).
Asphalt typeGradingLLLW film (0.2–0.6 MPa)LLW film (0.5–2.5 MPa)LW film (2.5–10 MPa)Total70#AC26336512470#OGFC69441012370#SMA7010440214SBSAC175540112SBSOGFC487044162SBSSMA484037125SHVAC19293684SHVOGFC28193279SHVSMA33362897According to Table7, it can be seen that the stress of the asphalt mixture measured at the three ranges is the same as the actual applied load. When LLLW type film is used to measure, the normal load of continuously graded asphalt mixture (AC) is less than that of open-graded asphalt mixture (OGFC) and asphalt ma hoof grease gravel (SMA), and the load distribution of asphalt mixture is different with different types of asphalt.In order to obtain the particle contact characteristics inside the asphalt mixture, the pressure film test results were further analyzed. Each pixel point was a contact point, and the pressure of each contact point could be obtained. According to the number of contact points measured by the film and the area of each point was 0.016 mm2, the number of image contact points was converted into the contact area. Figure 3 shows the distribution of pressure and stressed area at different ranges. The abscissa is the pressure (MPa), and the ordinate is the proportion of the contact area.Figure 3
Pressure distribution at three levels: (a) LLLW, (b) LLW, and (c) LW.
(a)(b)(c)As shown in Figure3, with the increase of pressure, the contact area of particles falling in this area increases, and the contact area reaches the peak at 0.2–0.3 MPa, and then the contact area decreases with the increase of pressure and gradually approaches 0. Moreover, the load transferred between particles in the asphalt mixture of different raw materials varies.


### 3.2. Model Establishment and Analysis

Sun Qicheng [26] analyzed the distribution of contact forces between particles in the rockfill after loosening and rolling compaction and obtained the spatial distribution characteristics of contact forces between particles in the study of force distribution in the rockfill body in “Introduction to Mechanics of Granular Matter”.Figure4 is the distribution diagram of the contact force between particles in the rockfill, in which the abscissa f is the contact force and the ordinate is the proportion of the contact area. The peak value of the contact force is distributed around 1. The fitting curve is shown in formula (4):(4)f=21929.7×s1.189×e−0.276×s.Figure 4
Distribution diagram of contact force between particles in rockfill.The distribution of pressure and pressure area obtained in this test has a similar trend to the distribution of contact force between particles in the rockfill, indicating that the results obtained from the pressure film can better reflect the distribution characteristics of contact stress of particles inside the mixture. Based on the test data, the nonlinear curve can be fitted. The mathematical model of the relationship between the distribution probability of contact points and the pressure can be summarized. The influence of the three factors—gradation, asphalt film thickness, and asphalt type—on the contact stress distribution characteristics of the particles in the asphalt mixture is discussed. Figure5 shows the curve after fitting R2 is 0.96. The established mathematical model is shown in (5). In the mathematical model established, X0 is the fixed value, and 0.23 is taken as parameters A, B, and C. The gradation of asphalt mixture, asphalt type, and asphalt film thickness are affected.(5)y=0.02−0.002×x+A×xB×e−C×x−x02.Here, x—the pressure (MPa), y—the proportion of stressed area to effective area (%); x0—take a fixed value of 0.23 [24, 25]; A—parameters related to gradation; B—parameters related to gradation and bitumen type; and C—parameters related to asphalt film thickness.Figure 5
Curve after fitting.Taking SHV asphalt as an example, Figure6 shows that the maximum contact pressure of internal particles of asphalt mixture with AC-graded asphalt is 6.5–7.5 MPa, while that of OGFC and SMA asphalt can reach 10 MPa. According to the load-bearing effect, the internal particles of the asphalt mixture can be divided into primary skeleton particles, intermediate particles, and fine particles. The coarse aggregate particles of the main skeleton form a load-bearing skeleton, and the intermediate particles fill the space formed by the main skeleton to stabilize the skeleton. Fine particles form asphalt mortar to bond coarse aggregate to form asphalt mixture.Figure 6
Stress distribution of SHV asphalt mixture.The number of contact points of main skeleton particles of different asphalt mixtures is different. From Figure7, it can be seen that the contact area of AC is the smallest, the contact area of SMA and OGFC is more significant, and the contact area of OGFC is slightly higher than that of SMA. It can be seen that the internal stress distribution of asphalt mixture will be affected by gradation, asphalt type, and asphalt film thickness. According to the existing data, the parameters A, B, and C are analyzed.Figure 7
The contact area of different gradations.


#### 3.2.1. Analysis of Parameter A

Although the coarse aggregate performance largely determines the mechanical performance of the asphalt mixture, and the contact between coarse aggregate and coarse aggregate will produce higher local stress [29–31]. The number of continuously graded asphalt mixtures with a continuous aggregate size of 1.18–4.75 mm is more than that of OGFC and SMA. The results of parameter A are shown in Tables 8 and 9.Table 8
Values of parameter A.
ACOGFCSMA70# Asphalt0.0180.180.2SBS modified asphalt0.0180.150.2S-HV modified asphalt0.0060.280.23Table 9
Single factor variance analysis of oil-stone ratio and parameter A.
Difference sourceSSdfMSFP-valueF critIntergroup3.0470413.047041.2949380.288075.317655Within the group18.8243282.35304Total21.871369It can be seen from Table9 that the sum of squares of deviation between groups is less than the sum of squares of deviation within groups, indicating that when discussing the value of parameter A, the data fluctuation is caused by the error between groups, and the ratio of oil to stone is not the factor causing the change of parameter A. It can be seen from Figure 8 that there is no significant difference in the parameter A of different asphalt types for the same gradation, but the value of A changes with the change of gradation. Among them, AC is obviously different from OGFC and SMA. OGFC was the largest, AC was the smallest, and SMA was in the middle [32]. The variation rule of parameter A with grading is AC < SAM < OGFC.Figure 8
Variation diagram of parameter C with the ratio of oil to stone.


#### 3.2.2. Analysis of Parameter B

The collocation of the two factors at different levels will have an interactive effect on the experimental results. The two-factor analysis of variance is used to analyze the experimental data. The sample is the gradation factor effect, which is classified as the asphalt type effect, and the internal is the interaction effect between the gradation and the asphalt type. When theP-value is less than the significant level a, it shows that the impact of this factor is substantial under the considerable level α. The results of parameter B are shown in Table 10.Table 10
Multivariate ANOVA for parameter B.
Difference sourceSSdfMSFP-valueF critSample4.96746722.48373333312.988540.0066065.143253Column1.69500811.6950083338.8639470.0247255.987378Interaction0.02806720.0140333330.0733860.9300635.143253Inside1.1473560.191225Total7.83789211Table10 shows that the type of asphalt significantly impacts parameter B, and there is no interaction between the two factors. Still, the gradation factor has a more significant influence on parameter B. Aggregate and asphalt mortar form asphalt mixture. Different types of asphalt lead to substantial differences in the chemical composition and internal structure of asphalt. The four components of asphalt include asphaltene, resin, saturation, and aromatic. The proportion of the four components directly affects the rheological properties of asphalt and the adhesion between asphalt and aggregate; it leads to different contact between different kinds of asphalt and aggregate. Therefore, the performance characteristics of asphalt directly affect the performance characteristics of the mixture. In the case of the same gradation, the internal particle contact area of matrix asphalt mixture under external load is smaller than that of SBS modified asphalt and S-HV high-modulus asphalt.


#### 3.2.3. Analysis of Parameter C

The thickness of asphalt film refers to the adequate thickness of the asphalt layer on the surface of the stone, which is closely related to the strength, high-temperature performance, low-temperature performance, and durability of the mixture. Moreover, the thickness of asphalt film determines the content of free asphalt and structural asphalt in the mixture. Thus, it affects the interface’s properties between asphalt and aggregate, which is an essential factor influencing the mechanical contact properties of particles in the asphalt mixture. When the thickness of asphalt film is small, the amount of structural asphalt is insufficient, the contact area at the contact interface of particles in the mixture is small, and the internal pores of the mixture are reduced with the increase of asphalt film thickness [33, 34]. The thickness of asphalt film is converted to the asphalt-stone ratio [35], and the asphalt-stone ratio of the experiment of AC, OGFC, and SMA is 4.6%, 5.6%, and 6.0%, respectively. Taking OGFC-graded S-HV asphalt mixture as an example, the results of parameter C are shown in Figure 8.Figure8 shows that the value of parameter C ranges from 90 to 150 in the case of different oil-stone ratios, and the value of parameter C increases at first and then decreases with the rise of asphalt-stone ratio, and the minimum value is taken when the asphalt-stone ratio is 6.0%. The range of the best asphalt-stone ratio is 4.8%, indicating that the value of parameter C can reflect the effect of asphalt film thickness on the mechanical distribution of particle contact interface in asphalt mixture.


### 3.3. Model Verification

In order to verify whether the established mechanical model can effectively represent the distribution of contact stress at the interface of particles in the mixture, there are typical gradations discussed: AC, OGFC, and SMA, adding with special asphalt to each gradation, a total of three groups of experiments.The experimental method is the same as Section2, and each group of experiments uses LLLW pressure film to obtain the contact stress area distribution. By bringing the obtained stress distribution information into the mechanical model, the values of parameters A, B, and C can be determined; thus, the complete contact stress information of the internal particle interface of the asphalt mixture can be obtained.Figure9 shows the contact stress distribution of particles in the mixture obtained by using LLLW pressure film. The ordinate is the ratio of the contact area to the effective area, and the abscissa is the pressure (MPa). Among them, the point pressure with the largest contact area of particles in asphalt mixture is between 0.2 and 0.25 MPa, and its peak value is about 0.23 MPa.Figure 9
Partial measured value of contact stress.The values of parameters A, B, and C can be obtained by fitting the analysis results of pressure film into the mathematical model. As shown in Table11, the value of parameter A is affected by gradation. The value of A of AC mixture is the smallest, and the maximum value A is of OGFC. For parameters B and C, the smallest ones occurred in AC mixture, and the maximum occurred in SMA mixture. The fitting results and the R2 value of the fitted curve are shown in Figure 10.Table 11
Values of parameters.
ACOGFCSMAA0.310.8B11.21.35C104127123Figure 10
The measured distribution of particle contact stress in mixture.The complete curve can be obtained by bringing the values of parameters A, B, and C into the mathematical model. The scatter diagram in Figure10 shows the pressure distribution of the three gradations measured by three types of film. The three skeleton structures reached the peak of the contact area of the asphalt mixture particles when the stress is about 0.23 MPa. The peak values of the three skeleton structures were arranged in a descending order as OGFC, SMA, and AC. The contact area of particles can reflect the dispersion efficiency of the external load of asphalt mixture through the stress transfer between particles, which will affect the stability of the skeleton structure. Larger contact between particles is beneficial to the dispersion of internal stress. This shows that the skeleton asphalt mixture helps to improve the transfer efficiency of external loads.Compared with the previous research [22–24] on the pressure film technology, which only stayed in the evaluation of the surface skid resistance of asphalt pavement, this paper applied the pressure film to the study of the stress distribution of the particles inside the asphalt mixture.Considering the influence of different asphalt types, skeleton structure, and asphalt film thickness on stress transfer, a prediction model which can better reflect the stress distribution of particles in the mixture was established.


## 3.1. Analysis of Actual Results of Pressure Film

The rotary compaction test prepared three kinds of bitumen, three kinds of gradation, and five kinds of oil-stone ratio. Three different specifications of pressure film—LLLW, LLW, and LW—were used to measure the interface contact area between two Marshall specimens under a normal load of 100 N. LLLW film range: 0.2–0.6 MPa, LLW film range: 0.5–2.5 MPa, and LW film range: 2.5–10 MPa. The statistical results of the experiment are shown in Table7, in which the normal load of each range is the average of many tests.Table 7
Experimental load distribution results of pressure film (N).
Asphalt typeGradingLLLW film (0.2–0.6 MPa)LLW film (0.5–2.5 MPa)LW film (2.5–10 MPa)Total70#AC26336512470#OGFC69441012370#SMA7010440214SBSAC175540112SBSOGFC487044162SBSSMA484037125SHVAC19293684SHVOGFC28193279SHVSMA33362897According to Table7, it can be seen that the stress of the asphalt mixture measured at the three ranges is the same as the actual applied load. When LLLW type film is used to measure, the normal load of continuously graded asphalt mixture (AC) is less than that of open-graded asphalt mixture (OGFC) and asphalt ma hoof grease gravel (SMA), and the load distribution of asphalt mixture is different with different types of asphalt.In order to obtain the particle contact characteristics inside the asphalt mixture, the pressure film test results were further analyzed. Each pixel point was a contact point, and the pressure of each contact point could be obtained. According to the number of contact points measured by the film and the area of each point was 0.016 mm2, the number of image contact points was converted into the contact area. Figure 3 shows the distribution of pressure and stressed area at different ranges. The abscissa is the pressure (MPa), and the ordinate is the proportion of the contact area.Figure 3
Pressure distribution at three levels: (a) LLLW, (b) LLW, and (c) LW.
(a)(b)(c)As shown in Figure3, with the increase of pressure, the contact area of particles falling in this area increases, and the contact area reaches the peak at 0.2–0.3 MPa, and then the contact area decreases with the increase of pressure and gradually approaches 0. Moreover, the load transferred between particles in the asphalt mixture of different raw materials varies.


## 3.2. Model Establishment and Analysis

Sun Qicheng [26] analyzed the distribution of contact forces between particles in the rockfill after loosening and rolling compaction and obtained the spatial distribution characteristics of contact forces between particles in the study of force distribution in the rockfill body in “Introduction to Mechanics of Granular Matter”.Figure4 is the distribution diagram of the contact force between particles in the rockfill, in which the abscissa f is the contact force and the ordinate is the proportion of the contact area. The peak value of the contact force is distributed around 1. The fitting curve is shown in formula (4):(4)f=21929.7×s1.189×e−0.276×s.Figure 4
Distribution diagram of contact force between particles in rockfill.The distribution of pressure and pressure area obtained in this test has a similar trend to the distribution of contact force between particles in the rockfill, indicating that the results obtained from the pressure film can better reflect the distribution characteristics of contact stress of particles inside the mixture. Based on the test data, the nonlinear curve can be fitted. The mathematical model of the relationship between the distribution probability of contact points and the pressure can be summarized. The influence of the three factors—gradation, asphalt film thickness, and asphalt type—on the contact stress distribution characteristics of the particles in the asphalt mixture is discussed. Figure5 shows the curve after fitting R2 is 0.96. The established mathematical model is shown in (5). In the mathematical model established, X0 is the fixed value, and 0.23 is taken as parameters A, B, and C. The gradation of asphalt mixture, asphalt type, and asphalt film thickness are affected.(5)y=0.02−0.002×x+A×xB×e−C×x−x02.Here, x—the pressure (MPa), y—the proportion of stressed area to effective area (%); x0—take a fixed value of 0.23 [24, 25]; A—parameters related to gradation; B—parameters related to gradation and bitumen type; and C—parameters related to asphalt film thickness.Figure 5
Curve after fitting.Taking SHV asphalt as an example, Figure6 shows that the maximum contact pressure of internal particles of asphalt mixture with AC-graded asphalt is 6.5–7.5 MPa, while that of OGFC and SMA asphalt can reach 10 MPa. According to the load-bearing effect, the internal particles of the asphalt mixture can be divided into primary skeleton particles, intermediate particles, and fine particles. The coarse aggregate particles of the main skeleton form a load-bearing skeleton, and the intermediate particles fill the space formed by the main skeleton to stabilize the skeleton. Fine particles form asphalt mortar to bond coarse aggregate to form asphalt mixture.Figure 6
Stress distribution of SHV asphalt mixture.The number of contact points of main skeleton particles of different asphalt mixtures is different. From Figure7, it can be seen that the contact area of AC is the smallest, the contact area of SMA and OGFC is more significant, and the contact area of OGFC is slightly higher than that of SMA. It can be seen that the internal stress distribution of asphalt mixture will be affected by gradation, asphalt type, and asphalt film thickness. According to the existing data, the parameters A, B, and C are analyzed.Figure 7
The contact area of different gradations.


### 3.2.1. Analysis of Parameter A

Although the coarse aggregate performance largely determines the mechanical performance of the asphalt mixture, and the contact between coarse aggregate and coarse aggregate will produce higher local stress [29–31]. The number of continuously graded asphalt mixtures with a continuous aggregate size of 1.18–4.75 mm is more than that of OGFC and SMA. The results of parameter A are shown in Tables 8 and 9.Table 8
Values of parameter A.
ACOGFCSMA70# Asphalt0.0180.180.2SBS modified asphalt0.0180.150.2S-HV modified asphalt0.0060.280.23Table 9
Single factor variance analysis of oil-stone ratio and parameter A.
Difference sourceSSdfMSFP-valueF critIntergroup3.0470413.047041.2949380.288075.317655Within the group18.8243282.35304Total21.871369It can be seen from Table9 that the sum of squares of deviation between groups is less than the sum of squares of deviation within groups, indicating that when discussing the value of parameter A, the data fluctuation is caused by the error between groups, and the ratio of oil to stone is not the factor causing the change of parameter A. It can be seen from Figure 8 that there is no significant difference in the parameter A of different asphalt types for the same gradation, but the value of A changes with the change of gradation. Among them, AC is obviously different from OGFC and SMA. OGFC was the largest, AC was the smallest, and SMA was in the middle [32]. The variation rule of parameter A with grading is AC < SAM < OGFC.Figure 8
Variation diagram of parameter C with the ratio of oil to stone.


### 3.2.2. Analysis of Parameter B

The collocation of the two factors at different levels will have an interactive effect on the experimental results. The two-factor analysis of variance is used to analyze the experimental data. The sample is the gradation factor effect, which is classified as the asphalt type effect, and the internal is the interaction effect between the gradation and the asphalt type. When theP-value is less than the significant level a, it shows that the impact of this factor is substantial under the considerable level α. The results of parameter B are shown in Table 10.Table 10
Multivariate ANOVA for parameter B.
Difference sourceSSdfMSFP-valueF critSample4.96746722.48373333312.988540.0066065.143253Column1.69500811.6950083338.8639470.0247255.987378Interaction0.02806720.0140333330.0733860.9300635.143253Inside1.1473560.191225Total7.83789211Table10 shows that the type of asphalt significantly impacts parameter B, and there is no interaction between the two factors. Still, the gradation factor has a more significant influence on parameter B. Aggregate and asphalt mortar form asphalt mixture. Different types of asphalt lead to substantial differences in the chemical composition and internal structure of asphalt. The four components of asphalt include asphaltene, resin, saturation, and aromatic. The proportion of the four components directly affects the rheological properties of asphalt and the adhesion between asphalt and aggregate; it leads to different contact between different kinds of asphalt and aggregate. Therefore, the performance characteristics of asphalt directly affect the performance characteristics of the mixture. In the case of the same gradation, the internal particle contact area of matrix asphalt mixture under external load is smaller than that of SBS modified asphalt and S-HV high-modulus asphalt.


### 3.2.3. Analysis of Parameter C

The thickness of asphalt film refers to the adequate thickness of the asphalt layer on the surface of the stone, which is closely related to the strength, high-temperature performance, low-temperature performance, and durability of the mixture. Moreover, the thickness of asphalt film determines the content of free asphalt and structural asphalt in the mixture. Thus, it affects the interface’s properties between asphalt and aggregate, which is an essential factor influencing the mechanical contact properties of particles in the asphalt mixture. When the thickness of asphalt film is small, the amount of structural asphalt is insufficient, the contact area at the contact interface of particles in the mixture is small, and the internal pores of the mixture are reduced with the increase of asphalt film thickness [33, 34]. The thickness of asphalt film is converted to the asphalt-stone ratio [35], and the asphalt-stone ratio of the experiment of AC, OGFC, and SMA is 4.6%, 5.6%, and 6.0%, respectively. Taking OGFC-graded S-HV asphalt mixture as an example, the results of parameter C are shown in Figure 8.Figure8 shows that the value of parameter C ranges from 90 to 150 in the case of different oil-stone ratios, and the value of parameter C increases at first and then decreases with the rise of asphalt-stone ratio, and the minimum value is taken when the asphalt-stone ratio is 6.0%. The range of the best asphalt-stone ratio is 4.8%, indicating that the value of parameter C can reflect the effect of asphalt film thickness on the mechanical distribution of particle contact interface in asphalt mixture.


## 3.2.1. Analysis of Parameter A

Although the coarse aggregate performance largely determines the mechanical performance of the asphalt mixture, and the contact between coarse aggregate and coarse aggregate will produce higher local stress [29–31]. The number of continuously graded asphalt mixtures with a continuous aggregate size of 1.18–4.75 mm is more than that of OGFC and SMA. The results of parameter A are shown in Tables 8 and 9.Table 8
Values of parameter A.
ACOGFCSMA70# Asphalt0.0180.180.2SBS modified asphalt0.0180.150.2S-HV modified asphalt0.0060.280.23Table 9
Single factor variance analysis of oil-stone ratio and parameter A.
Difference sourceSSdfMSFP-valueF critIntergroup3.0470413.047041.2949380.288075.317655Within the group18.8243282.35304Total21.871369It can be seen from Table9 that the sum of squares of deviation between groups is less than the sum of squares of deviation within groups, indicating that when discussing the value of parameter A, the data fluctuation is caused by the error between groups, and the ratio of oil to stone is not the factor causing the change of parameter A. It can be seen from Figure 8 that there is no significant difference in the parameter A of different asphalt types for the same gradation, but the value of A changes with the change of gradation. Among them, AC is obviously different from OGFC and SMA. OGFC was the largest, AC was the smallest, and SMA was in the middle [32]. The variation rule of parameter A with grading is AC < SAM < OGFC.Figure 8
Variation diagram of parameter C with the ratio of oil to stone.


## 3.2.2. Analysis of Parameter B

The collocation of the two factors at different levels will have an interactive effect on the experimental results. The two-factor analysis of variance is used to analyze the experimental data. The sample is the gradation factor effect, which is classified as the asphalt type effect, and the internal is the interaction effect between the gradation and the asphalt type. When theP-value is less than the significant level a, it shows that the impact of this factor is substantial under the considerable level α. The results of parameter B are shown in Table 10.Table 10
Multivariate ANOVA for parameter B.
Difference sourceSSdfMSFP-valueF critSample4.96746722.48373333312.988540.0066065.143253Column1.69500811.6950083338.8639470.0247255.987378Interaction0.02806720.0140333330.0733860.9300635.143253Inside1.1473560.191225Total7.83789211Table10 shows that the type of asphalt significantly impacts parameter B, and there is no interaction between the two factors. Still, the gradation factor has a more significant influence on parameter B. Aggregate and asphalt mortar form asphalt mixture. Different types of asphalt lead to substantial differences in the chemical composition and internal structure of asphalt. The four components of asphalt include asphaltene, resin, saturation, and aromatic. The proportion of the four components directly affects the rheological properties of asphalt and the adhesion between asphalt and aggregate; it leads to different contact between different kinds of asphalt and aggregate. Therefore, the performance characteristics of asphalt directly affect the performance characteristics of the mixture. In the case of the same gradation, the internal particle contact area of matrix asphalt mixture under external load is smaller than that of SBS modified asphalt and S-HV high-modulus asphalt.


## 3.2.3. Analysis of Parameter C

The thickness of asphalt film refers to the adequate thickness of the asphalt layer on the surface of the stone, which is closely related to the strength, high-temperature performance, low-temperature performance, and durability of the mixture. Moreover, the thickness of asphalt film determines the content of free asphalt and structural asphalt in the mixture. Thus, it affects the interface’s properties between asphalt and aggregate, which is an essential factor influencing the mechanical contact properties of particles in the asphalt mixture. When the thickness of asphalt film is small, the amount of structural asphalt is insufficient, the contact area at the contact interface of particles in the mixture is small, and the internal pores of the mixture are reduced with the increase of asphalt film thickness [33, 34]. The thickness of asphalt film is converted to the asphalt-stone ratio [35], and the asphalt-stone ratio of the experiment of AC, OGFC, and SMA is 4.6%, 5.6%, and 6.0%, respectively. Taking OGFC-graded S-HV asphalt mixture as an example, the results of parameter C are shown in Figure 8.Figure8 shows that the value of parameter C ranges from 90 to 150 in the case of different oil-stone ratios, and the value of parameter C increases at first and then decreases with the rise of asphalt-stone ratio, and the minimum value is taken when the asphalt-stone ratio is 6.0%. The range of the best asphalt-stone ratio is 4.8%, indicating that the value of parameter C can reflect the effect of asphalt film thickness on the mechanical distribution of particle contact interface in asphalt mixture.


## 3.3. Model Verification

In order to verify whether the established mechanical model can effectively represent the distribution of contact stress at the interface of particles in the mixture, there are typical gradations discussed: AC, OGFC, and SMA, adding with special asphalt to each gradation, a total of three groups of experiments.The experimental method is the same as Section2, and each group of experiments uses LLLW pressure film to obtain the contact stress area distribution. By bringing the obtained stress distribution information into the mechanical model, the values of parameters A, B, and C can be determined; thus, the complete contact stress information of the internal particle interface of the asphalt mixture can be obtained.Figure9 shows the contact stress distribution of particles in the mixture obtained by using LLLW pressure film. The ordinate is the ratio of the contact area to the effective area, and the abscissa is the pressure (MPa). Among them, the point pressure with the largest contact area of particles in asphalt mixture is between 0.2 and 0.25 MPa, and its peak value is about 0.23 MPa.Figure 9
Partial measured value of contact stress.The values of parameters A, B, and C can be obtained by fitting the analysis results of pressure film into the mathematical model. As shown in Table11, the value of parameter A is affected by gradation. The value of A of AC mixture is the smallest, and the maximum value A is of OGFC. For parameters B and C, the smallest ones occurred in AC mixture, and the maximum occurred in SMA mixture. The fitting results and the R2 value of the fitted curve are shown in Figure 10.Table 11
Values of parameters.
ACOGFCSMAA0.310.8B11.21.35C104127123Figure 10
The measured distribution of particle contact stress in mixture.The complete curve can be obtained by bringing the values of parameters A, B, and C into the mathematical model. The scatter diagram in Figure10 shows the pressure distribution of the three gradations measured by three types of film. The three skeleton structures reached the peak of the contact area of the asphalt mixture particles when the stress is about 0.23 MPa. The peak values of the three skeleton structures were arranged in a descending order as OGFC, SMA, and AC. The contact area of particles can reflect the dispersion efficiency of the external load of asphalt mixture through the stress transfer between particles, which will affect the stability of the skeleton structure. Larger contact between particles is beneficial to the dispersion of internal stress. This shows that the skeleton asphalt mixture helps to improve the transfer efficiency of external loads.Compared with the previous research [22–24] on the pressure film technology, which only stayed in the evaluation of the surface skid resistance of asphalt pavement, this paper applied the pressure film to the study of the stress distribution of the particles inside the asphalt mixture.Considering the influence of different asphalt types, skeleton structure, and asphalt film thickness on stress transfer, a prediction model which can better reflect the stress distribution of particles in the mixture was established.


## 4. Conclusions

In this study, pressure film technology was introduced to obtain the accurate stress distribution and stress transfer state information of the particles in the asphalt mixture. The OGFC mixture contained the largest number of contact points, followed by SMA and AC mixtures. The distribution of compression area and pressure is a single peak curve, and the peak value is about 0.23 MPa.There is a good correlation of the internal particle contact stress distribution curve of asphalt mixture with different gradations, asphalt types, and asphalt ratios and the established prediction model. The X0 in the prediction model is related to the peak value of the curve, which can be taken as a fixed value of 0.23. The value of parameter A is related to the gradation of asphalt mixture. Parameter B is affected by asphalt type and mixture gradation. The thickness of asphalt film has a certain influence on the value of parameter C.The verification experiments show that the prediction model can accurately express the contact stress distribution of particles in asphalt mixtures. The real distribution of contact stress can be obtained by the prediction model. The introduction of the prediction model was helpful to evaluate the mechanical properties of asphalt mixture.


## 5. Declaration of Interest Statement

The authors, at this moment, declare that this manuscript has not been published in whole or in part or being considered for publication elsewhere. The authors truly believe that, from the scientific point of view, the objective of this research is timely and essential in both basic and applied research. The authors also declare that there are no conflicts of interest [27, 28].



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*Source: 6552211-2022-09-12.xml*