In order to analyze the torsional shear process of asphalt mixtures in a microscopic view, the numerical simulation of a torsional shear test of an asphalt mixture was carried out by discrete element method. Based on the defects of existing algorithms, the method of random reconstruction of the existing 3D model of the asphalt mixture was improved, and a new reconstruction method was proposed. A 3D numerical model of the asphalt mixture contained irregularshaped coarse aggregate, mineral gradation, and asphalt mortar; furthermore, the particle algorithm established the air void distribution. Then, the numerical simulation of the asphalt mixture’s torsional shear was completed; in addition, the stress, displacement, and contact of the specimens at each stage were analyzed. The results showed that the stress and displacement in different stages changed greatly with the loading, i.e., the crack generated from a weak point on the surface and then spread to the ends with an oblique angle of about 45°. At the same time, the shear failure process of the asphalt mixture was studied. The virtual test method could accomplish the implementation of the numerical simulation of torsional shear; it also provided a good research method for analysis of the asphalt mixture’s shear failure process.
With the increasing volume of road traffic and the continuous development of heavyduty transportation, the ruts and cracking of pavement surface have become the main forms of damage for China’s asphalt pavement. Studies show [
Researchers have carried out related research, but the existing research has been mainly carried out by means of laboratory tests, which were macroscopic and had the disadvantages of high cost and poor reproducibility [
Numerical simulation is a method of studying engineering problems through numerical calculation and image displays. In the field of pavement, numerical simulation methods are used to analyze the mesostructure and performance of an asphalt mixture. Numerical analysis methods commonly include the finite element method, the boundary element method, the finite difference method, and the discrete element method. Among them, the finite element method is suitable for a simulation situation when the strain level is small (i.e., small deformation), and the discrete element method has the advantage of simulating large deformation or cracking problems; this method is especially used for the study of discontinuous particulate materials such as asphalt mixtures. For example, the discrete element method was used to simulate the splitting test [
Based on the related research, the author proposes a new torsional shear test method under normal stress conditions [
The flowchart of the research.
The internal structure of an asphalt mixture is complex; it includes multiphase composite materials composed of aggregate, asphalt, and air voids. To perform numerical simulations, the primary problem should be solved to establish a numerical model that could reflect the internal characteristics of the asphalt mixture. The traditional method for obtaining the numerical model of an asphalt mixture is to take a photograph with a digital camera of the mixture’s surface or a section of the specimen after cutting; one can also use industrial CT to perform tomographic scanning on the asphalt mixture specimen and obtain a binary image of the sample by digital image processing technologies such as image enhancement technology and image segmentation technology; then, a 2D or 3D numerical model of the mixture specimen can be obtained by digital reconstruction [
According to the relevant research [
In order to generate irregular aggregate particles, according to the existing research, an irregular polyhedron random cutting algorithm was modified to overcome the defects of the existing algorithms. The irregular polyhedron algorithm was based on a cube; then, the cube centroid was designated the point of origin, and a spatial Cartesian coordinate system was established. Due to the random nature of the cutting plane, the coordinate axes of the space Cartesian coordinate system were parallel to the true coordinate system axis, and a random face was generated in each quadrant for cutting. The main steps are as follows.
The program was programmed to fill with smaller sizes of particles in a specified order in a specified cube area and then set the side length of the area to be equal to the aggregate particle size. The mathematical equation of the cube is as follows:
First, a normal vector (
According to the distance
Then, the plane equation is
According to the randomly generated cutting surface, the cube was cut to obtain the irregular polyhedral region. The mathematical equation of the irregular polyhedral region is shown in the following equation:
Judging the relative position, by writing a program, traversing all the smallsized particles arranged regularly in the square, and judging the relative position of the irregular particle with the irregular polyhedron, the particles in the irregular polyhedral area were the aggregate part.
According to the above algorithm, a single aggregate having a particle diameter of 14 mm would be described as an example.
First, the regular arrangement of small ball particles was filled. Use the “BALL” command to fill the spheres in the cube area according to the rule arrangement algorithm, take the radius of the pellets to 0.5 mm, and set the centroid of the cube (0, 0, 0) with a side length of 14.0 mm. The area consists of 2,744 regularly arranged small spherical particles, as shown in Figure
Regular arrangement particles.
Second, set the cutting control coefficient to generate the cutting surface. Third, judging by the position of the small sphere particles and the polygonal area, the particles not in the area were deleted, and the remaining were the desired aggregate particle model, as shown in Figure
Irregular aggregate particles model.
Because the passing percentages of the specification gradation were calculated by the mass characterization, in order to characterize the gradation characteristics of the mixture in the model, assuming that the density of the aggregates was equal, then the gradation could be characterized by the volume of aggregates, and the number of gradation ball units was calculated according to the volume fraction of each sieve of aggregates; that is, the volume of the specimen occupied by certain aggregates was calculated according to the volume fraction and then divided by the average volume of the ball; thus, the number of the aggregate particles was obtained, as shown in equation (
Therefore, the key to reflect the gradation is to calculate the volume fraction of each aggregate, which is derived as follows:
Assume that the density of the aggregates is
Percentages of aggregate gradation.
Sieve size (mm) 



… 



<2.36 
Passing percentages (%)  100 


… 



… 
Individual percent retain on each sieve (%)  0  100 − 

… 




According to the equation of the asphaltaggregate ratio calculation, namely,
The content of asphalt in the numerical specimen is as follows:
According to the volume composition of the whole specimen, there are
Substituting equation (
According to the gradation passing percentages table,
And, because the density is equal,
Substituting equation (
Combined with the passing percentages table, the formula for the volume fraction of aggregates of each sieve is obtained as follows:
Therefore, from equations (
In addition, the density parameter
Substituting equation (
Taking the asphalt mixture of AC13 as an example, the gradation is shown in Table
Gradation of the asphalt mixture of AC13.
Sieve size  16  13.2  9.5  4.75  2.36  1.18  0.6  0.3  0.15  0.075  
Passing percentage (%)  Upper limit  100  100  85  56  38  28  22  17  12  8 
Lower limit  100  95  72  42  28  20  12  8  7  4  
Median  100  97.5  78.5  49  33  24  17  12.5  9.5  6 
Aggregate particles of the asphalt mixture of AC13.
Sieve size  16  13.2  9.5  4.75  2.36  1.18  0.6  0.3  0.15  0.075 
Passing percentages (%)  100  97.5  78.5  49  33  24  17  12.5  9.5  6 
Individual percent retain on each sieve (%)  0  2.5  19  29.5  16  —  —  —  —  — 
Volume fraction  0.000  0.021  0.161  0.250  0.136  —  —  —  —  — 
Number  0  10  165  1036  4540  —  —  —  —  — 
In the asphalt mixture, the aggregates of different particle sizes were divided into different sieve parts, and the proportion of each sieve of the total mass was different, so the gradation was formed. The number of aggregate particles of each sieve was determined; it is equivalent to determining the proportion of the aggregates, so the number of gradation units was obtained.
The number of aggregates for each sieve was calculated according to the above method; then, we put them into the model within the size of the specimen, as shown in Figure
Arrangement of aggregate particle.
First, a program was written to scan the abovementioned gradation ball unit, and information of each aggregate particle was extracted: coordinates (
Then, the original particles were deleted, and the particles with smaller diameter particles were regularly filled in the region in a certain order; that is, all the aggregate particles were arranged neatly in the horizontal and vertical directions, and each of the particles in the middle position was arranged adjacent to one particle in the upper, lower, left, right, front, and back directions, as shown in Figure
Regular arrangement particle.
Third, the irregular aggregate generation program was loaded, and a gradation check was performed. The method of gradation check was calculated according to the percentage of all the small ball particles in each sieve of the total particles in the model. Because the basic unit of the discrete element model is a single sphere, it is difficult to obtain the same value as the volume fraction of the original aggregates by loading the irregular aggregate algorithm. Therefore, considering that, when the percentage of aggregate particles in the entire cylinder model came within the percentage threshold, the gradation test was successful; otherwise, the aggregate was regenerated until the percentage of the aggregate particles in the total cylinder model was within the percentage threshold. Then, the next aggregate was generated, and an irregular aggregate model was obtained, as shown in Figure
Specimen of irregular aggregate (no air voids).
In order to create certain air voids for the above model, a random deletion method was used to remove some particles of the asphalt mortar randomly to characterize the air voids of the mixture. The specific method was to generate a random number in the maximum particle range of the irregular aggregate model and determine whether the particle address with the random number as the serial number was empty, and, if not, determine whether the particle with the random number as the serial number was asphalt mortar; if it was, delete the particles until the
The air void distribution is shown in Figure
Distribution of air voids. (a) Entire model. (b) Vertical section view. (c) Transverse section view.
Numerical model of irregular aggregate.
According to the above model establishment method, two kinds of asphalt mixture 3D cylindrical specimen with gradation types AC13 and SMA13 were generated (the diameter was 100 mm, the height was 100 mm, and the air voids was 4%). As shown in Figure
Structure of numerical model. (a) Entire model of AC13. (b) Vertical section view of AC13. (c) Transverse section view of AC13. (d) Entire model of SMA13. (e) Vertical section view of SMA13. (f) Transverse section view of SMA13.
Discrete elements method mainly included three basic models: stiffness model, sliding model, and bond model [
Burger’s model of particle contact. (a) Normal direction and (b) tangent direction.
In addition, there are four types of contact between the asphalt mixture units: contact between internal units of the same aggregates, between different aggregate units, between internal units of asphalt mortar, and between asphalt mortar units and aggregate units. Among them, the contact parameter between the asphalt mortar unit and the aggregate unit was greatly affected by the aggregate characteristics, also it could not be tested through a good measuring method currently. Therefore, the contact parameter between the asphalt mortar unit and the aggregate unit could be considered the same as that of the internal contact of the asphalt mortar by appropriately simplification.
At the same time, there was no suitable command in the previous study to define the contact between adjacent aggregates uniformly, considering that, under actual conditions, the aggregate surface always had a layer of asphalt film covered, so it was mostly regarded as the contact among the aggregates and the asphalt mortar, then with the aggregates. Through research, the paper found a new definition method, which was defined by the principle of density discrimination. The mortar and each aggregate particle were defined at different densities first; then, different locations of the contact were distinguished and parameters were assigned based on this. The operation process was as follows: When defining the parameters, in the initial stage of imparting the particle density parameter, first, the density parameter of the asphalt mortar particles was defined as 1; then, each aggregate particle was sequentially incremented to define different density parameters. By programming the
Because the aggregate property in the asphalt mixture was relatively stable and was generally considered to have linear elastic properties, a linear elastic stiffness contact model was adopted for the aggregate, and a point bonding contact model was adopted for the bonding of the aggregate. Because the viscoelastic properties of the asphalt mixture were mainly due to the nature of the asphalt mortar, Burger’s viscoelastic contact model needed to be considered for the contact between the asphalt mortar units. The four contact models are shown in Table
Contact models at different locations inside the mixture.
Contact position  Model selection 

Aggregate interior  Linear model + point bond model 
Between adjacent aggregates  Linear model + point bond model 
Asphalt mortar internal  Burger’s model + linear model + point bond model 
Between asphalt mortar and aggregate  Burger’s model + linear model + point bond model 
In order to analyze the microscopic parameters of Burger’s model, it can be seen in Figure
The mesocontact force
The stress
And, because of the equivalence between contact force and stress
For the same reason, there were
Thus, we can obtain
The relationship between the tangential parameters of Burger’s model and the macroscopic parameters could be obtained as follows:
Thus, as long as the corresponding viscoelastic parameters and linear elastic parameters and the bonding parameters between them were obtained, the microscopic parameters of the numerical model could be obtained. For the four macroscopic parameters of Burger's viscoelastic model of asphalt mortar, the macroscopic parameters could be obtained through experiments. According to the relevant research and the results of the literature [
Burger’s model parameter.
Mixture type  Burger’s model parameters  






AC13  3.185  1.756  2784.233  216.724 
SMA13  0.478  0.117  83.514  5.487 
Assuming the aggregate elastic modulus was 50 GPa, Poisson’s ratio was 0.25, the tensile strength was 6 MPa, and the shear strength was 15 MPa. According to the above analysis, the normal stiffness and tangential stiffness of the aggregate and the normal strength and tangential strength of the point bond could be obtained. Finally, the microscopic parameters of the asphalt mixture were calculated, as shown in Tables
Microscopic parameters of mixtures of AC13.
Microscopic parameter  Stiffness  Strength  




 
Aggregate interior  2.5 × 10^{8}  1.0 × 10^{8}  37.5  93.8 
Between adjacent aggregates  2.5 × 10^{8}  1.0 × 10^{8}  4.1  9.4 
Asphalt mortar internal  1.0 × 10^{8}  0.4 × 10^{8}  4.3  9.4 
Between asphalt mortar and aggregate  1.75 × 10^{8}  0.7 × 10^{8}  4.3  9.4 
Microscopic parameters of AC13 Burger’s model.
Microscopic parameter  Maxwell  Kelvin  

Stiffness  Viscosity  Stiffness  Viscosity  
Normal direction  7.9 × 10^{3}  6.95 × 10^{6}  4.4 × 10^{3}  5.43 × 10^{5} 
Tangent direction  3.16 × 10^{3}  2.78 × 10^{6}  1.76 × 10^{3}  2.17 × 10^{5} 
Microscopic parameters of mixtures of SMA13.
Microscopic parameter  Stiffness  Strength  




 
Aggregate interior  2.5 × 10^{8}  1.0 × 10^{8}  37.5  93.8 
Between adjacent aggregates  2.5 × 10^{8}  1.0 × 10^{8}  4.2  9.5 
Asphalt mortar internal  5 × 10^{7}  2 × 10^{7}  4.3  9.6 
Between asphalt mortar and aggregate  1.5 × 10^{8}  0.6 × 10^{8}  4.3  9.6 
Microscopic parameters of SMA13 Burger’s model.
Microscopic parameter  Maxwell  Kelvin  

Stiffness  Viscosity  Stiffness  Viscosity  
Normal direction  2.5 × 10^{3}  4.18 × 10^{5}  2.93 × 10^{3}  2.75 × 10^{4} 
Tangent direction  9.7 × 10^{2}  1.67 × 10^{5}  1.17 × 10^{3}  1.1 × 10^{4} 
In the numerical test, the conditions were the same as those in the laboratory test, and the controlled strain mode was employed. The specific loading mode uses the two upper and lower loadings “clump” to form two loading plates, fix the bottom of the test piece by fixing the lower loading plate, and give the upper loading plate a constant angular velocity
Model of loading.
According to the above conditions, the corresponding parameters were set and the torsional numerical simulation tests were carried out on the two types of cylindrical asphalt mixture discrete element models. Four sets of parallel tests were completed. The simulation test results are shown in Table
Results of torsion shear numerical simulation test.
Numbering  Mix type  Result (MPa)  Average value (MPa)  Coefficient of variation (%) 

1  AC13  0.350  0.360  1.94 
2  0.365  
3  0.368  
4  0.355  


5  SMA13  0.510  0.506  1.23 
6  0.502  
7  0.495  
8  0.509 
Results of two numerical model failure: (a) AC13 and (b) SMA13.
According to the results, the shear strength of AC13 is 0.360 MPa and the shear strength of SMA13 is 0.506 MPa; furthermore, it shows that shear performance of SMA13 is better than that of AC13 under the same conditions. In order to observe the crack shape more intuitively, the thirdparty postprocessing of the test was carried out, and the failure cloud images are shown in Figure
Displacement cloud image of two numerical model failure: (a) AC13 and (b) SMA13.
The torsional shear test of AC13 and SMA13 actual specimens under the same conditions was carried out by the method of the literature [
Comparison of numerical simulation results and laboratory test results.
Gradation type  Numerical simulation  Laboratory test  Relative error (%)  

Shear strength (MPa)  Coefficient of variation  Shear strength (MPa)  Coefficient of variation  
AC13  0.360  1.94  0.345  5.89  4.35 
SMA13  0.506  1.23  0.487  4.32  3.90 
Failure of laboratory specimens: (a) AC13 and (b) SMA13.
It could be found that the simulation test results were basically consistent with the real results, and the relative errors were less than 5%. At the same time, from Figures
To analyze the failure process, the asphalt mixture of AC13 was used as an example to analyze the stress, displacement, and contact changes at various stages of the test. The tensile stress and the compressive stress are set to red and black, respectively, and the depth of the color indicated the magnitude of the stress; further, the displacement is represented by a black arrow, similar to the vector, the direction of which is the direction of the arrow, and the size is represented by the length of the arrow.
At the start period of the test, the distribution of stress and contact conditions at the beginning of the loading were collected by writing a program, as shown in Figure
Initial stage of the loading period. (a) Stress distribution (entire model). (b) Stress distribution (vertical section). (c) Stress distribution (transverse section). (d) Contact distribution (entire model). (e) Contact distribution (vertical section). (f) Contact distribution (transverse section).
The vertical section graph was obtained by cutting along a plane half the width direction (
As shown in Figures
As shown in Figures
After running a certain number of steps, that is, after the loading plate rotated a certain displacement (the early loading stage), the corresponding stress, contact, and displacement were also collected according to the programmed procedure, as shown in Figure
Earlier stage of the loading period. (a) Stress distribution (entire model). (b) Stress distribution (vertical section). (c) Stress distribution (transverse section). (d) Contact distribution (entire model). (e) Contact distribution (vertical section). (f) Contact distribution (transverse section). (g) Displacement field (entire model). (h) Displacement field (vertical section). (i) Displacement field (transverse section).
As shown in Figures
As shown in Figures
As shown in Figures
As the upper loading plate continued to rotate, cracks began to occur at the weak position of the specimen, and various changes of the specimen were collected at this time, as shown in Figure
Later stage of the loading period (appearance of crack). (a) Stress distribution (entire model). (b) Stress distribution (vertical section). (c) Stress distribution (transverse section). (d) Contact distribution (entire model). (e) Contact distribution (vertical section). (f) Contact distribution (transverse section). (g) Displacement field (entire model). (h) Displacement field (vertical section). (i) Displacement field (transverse section).
As shown in Figures
As shown in Figures
As shown in Figures
As the upper loading plate continued to rotate, the crack gradually expanded. According to the collected stress, when it dropped rapidly, it could be considered that the specimen had been destroyed, as shown in Figure
Ending stage of the loading period (model failure). (a) Stress distribution (entire model). (b) Stress distribution (vertical section). (c) Stress distribution (transverse section). (d) Contact distribution (entire model). (e) Contact distribution (vertical section). (f) Contact distribution (transverse section). (g) Displacement field (entire model). (h) Displacement field (vertical section). (i) Displacement field (transverse section). (j) Crack distribution (whole model). (k) Crack distribution (vertical section). (l) Crack distribution (transverse section).
As shown in Figures
As shown in Figures
As shown in Figures
As shown in Figures
In order to observe the development process of the crack, the test results were exported at regular intervals in the calculation process, and the postcorrelation process was used to obtain the cloud image at different stages, as shown in Figure
Displacement cloud images of two numerical models. (a) Initial stage. (b) Earlier stage of the loading period. (c) Loading period 1. (d) Loading period 2. (e) Loading period 3. (f) Crack appeared. (g) Crack propagation. (h) End of loading.
It can be seen in the figure that, as the loading progressed, the displacement of the specimen became larger and larger. The shear failure process indicates that the crack was generated from a weak position on the surface and then spread to both ends at an oblique angle of about 45°; at the same time, the crack gradually spread to the inside of the specimen, resulting in a crack similar to the spiral shape, up to the complete destruction of the specimen. It was observed that the failure surface was not smooth; this was also caused by the unevenness of the mixture.
Through the analysis of the internal conditions of the specimen at different testing stages, the whole process of the asphalt mixture under the torsion action was obtained during the virtual experiment. It could be considered that the numerical simulation virtual test method had good simulation of the torsional shear process of the asphalt mixture; further, it was reasonable and feasible and also provided a good research method for the analysis of asphalt mixture failure process.
In order to analyze the torsional shear process of the asphalt mixture in a mesoscopic view, the discrete element method was used to establish the numerical model of an asphalt mixture. The numerical simulation of the torsional shear test of the asphalt mixture was carried out, and the following conclusions were obtained:
The existing 3D model random reconstruction method of the asphalt mixture was improved, and a new reconstruction method was proposed, which was an algorithm for obtaining irregular aggregate particles by using a random plane cutting regular hexahedron in eight different quadrants. Based on this and combined with the gradation characteristics of the mixture, a 3D numerical model of an asphalt mixture contained irregularshaped coarse aggregate, mineral aggregate gradation, and asphalt mortar, and air void distribution was established.
The numerical simulation of the torsional shear test of an asphalt mixture was completed, and the stress, displacement, and contact of the specimens at each stage were analyzed. It was found that the stress and displacement of the model showed a large change at different stages. The specific performance was as follows: As the loading progressed, the stress was quickly transmitted to the whole model, getting larger and larger, and there was a phenomenon of stress redistribution after the crack was generated. The displacement of the model particles was generally performed in a rotating mode along the loading direction, the outer edge particles moved larger than the inner edge particles, and the upper particles moved larger than the lower particles. The difference in displacement at the location where the crack occurred was large, which indicated that the internal structure was discontinuous. The shear failure process of the asphalt mixture was that the crack generated from a weak position on the surface, then spread to the both ends at an oblique angle of about 45°, and gradually spread to the inside of the specimen to produce a crack similar to the spiral shape. Finally, as the loading continued, the specimen was completely destroyed.
According to the research results, the numerical simulation of the torsional shear test of an asphalt mixture could be realized by the virtual test method, and it provided a good research method for analyzing the failure process of the specimen. In future work, a more refined model could be established, and various load and temperature conditions could be considered to better study the mechanical properties of the asphalt mixture.
The data used to support the findings of this study are included within the article.
The authors declare that there are no conflicts of interest.
The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (Grant no. 51008039) and the financial support of China Scholarship Council (Grant no. 201808430101).
The supplementary material is the code of the discrete element model according with the editor’s and reviewer’s comments of 3rd, it cannot be attached as an appendix behind the manuscript because it had too many pages. The section 1 on page 1–5 of the supplementary material is the discrete source program about the “Algorithm for generating irregular aggregate particles,” and section 2 on page 6–36 is the discrete source program about “Generating 3D models,” “Torsional shear numerical simulation test,” and “Loading setting.”