Asphalt mixture is more complicated than other composite materials in terms of the higher volume fraction of aggregate particles and the viscoelastic property of asphalt matrix, which obviously affect the applicabilities of the micromechanical models. The applicabilities of five micromechanical models were validated based on the shear modulus of the multiscale asphalt materials in this paper, including the asphalt mastic, mortar, and mixture scales. It is found that all of the five models are applicable for the mastic scale, but the prediction accuracies for mortar and mixture scales are poorer. For the mixture scale, all models tend to overestimate at the intermediate frequencies but show good agreement at low and high frequencies except for the SelfConsistent (SC) model. The ThreePhase Sphere (TPS) model is relatively better than others for the mortar scale. The applicability of all the existing micromechanical models is challenged due to the high particle volume fraction in the multiscale asphalt materials as well as the modulus mismatch between particles and matrix, especially at the lower frequencies (or higher temperatures). The particle interaction contributes more to the stiffening effect within higher fraction than 30%, and the prediction accuracy is then deteriorated. The higher the frequency (or the lower the temperature) is, the better the model applicability will be.
Asphalt mixture is a kind of heterogeneous composite material, consisting of asphalt binder and mineral aggregates with different sizes. The traditional researches on the mechanical properties of asphalt mixture and its failure mechanism are mostly based on the continuum mechanics theory and the experimental method. However, the mechanical properties of asphalt mixture are closely related to the complicate internal structure, which is dependent on the various raw materials’ properties, shapes, sizes, and proportions. The traditional analyzing methods fail to reveal the microstructure related failure mechanism of asphalt mixture, such as the formation and propagation of microcracks, the micro damage caused due to the heterogeneous material, and the local failures caused due to the stress concentration. Recently, researchers have realized the importance of the internal microstructure to the macromechanics properties, and the micromechanics theory of composites has been introduced to research the macromechanical properties from the meso and microscale [
At the microscale, the asphalt mixture could be considered as a kind of heterogeneous multiphase composite which is composed of asphalt binder, aggregates, asphaltaggregate interphase, microcrack, and void. The homogenization approach of the heterogeneous material is a fundamental problem in the area of composite, which mainly involves the prediction of the effective modulus of the composite. Micromechanical modeling techniques have long been successfully used to predict the effective modulus from mechanical properties and volume fractions of individual constituents for composite materials such as metal and polymer matrix composites [
Though considerable achievements have been made by previous researchers, few have involved the applicability of the existing micromechanical models on asphalt mixture. So, in this paper, the micromechanical models for composite materials were firstly introduced and evaluated. Then, these models were applied to predict the mechanical properties of asphalt materials from the multiscales and compared with the experiment results to validate the applicability of micromechanical models. In addition, the effects of particle volume fraction and modulus mismatch between reinforcement phase and matrix phase were also analyzed from the mastic scale.
Since Eshelby’s pioneering work [
In the SC model, any material point on the particles is isolated as an infinitesimal volume element. Then the rest of the material is homogenized as the uniform material, whose mechanical property is identical to the composite itself [
The MT model involves complicated manipulation of the field variables in Mori and Tanaka [
In the GSC model, the spherical particle is embedded in a concentric spherical shell of the matrix material. Shell and particle dimensions are chosen to correspond to the prescribed volume fraction, and the particleshell assembly in turn is embedded in an infinite medium with unknown effective properties [
By introducing the Differential Scheme Effective Medium theory into the micromechanics field, Mclaughlin [
Schematic diagram of the differential scheme method.
To take the effect of particle size and aggregate gradation on the asphalt mixture modulus into consideration, Li et al. [
ThreePhase Sphere model.
Those models were derived from elastic theory, but not limited to elastic materials, which could be extended to viscoelastic solutions based on the elasticviscoelastic correspondence principle [
According to the research reported by Underwood and Kim [
The aggregate gradations for asphalt mastic, mortar, and mixture.
Sieve size (mm)  % passing  

Mixture  Mortar  Mastic  
25.4  100  
19  100  
12.5  100  
9.5  96  
4.75  66  
2.36  48  100  
1.18  37  77  
0.6  30  63  
0.3  21  44  
0.15  11  23  
0.075  5.8  12.1  100 
0.030  3.3  6.9  57 
0.023  2.9  6.0  50 
0.017  2.1  4.3  36 
0.011  1.5  3.2  27 
0.0077  1.2  2.5  21 
0.0038  0.9  1.8  15 
0.0036  0.7  1.5  12 
0.0029  0.6  1.2  10 
0.0025  0.6  1.2  10 
0.0023  0.6  1.2  10 
0.0013  0.4  0.8  7 
Multiscale materials properties and proportions.
Multiscale materials  Matrix  Particles  Volume fraction  Poisson ratio 

Asphalt binder  —  —  —  0.495 
Asphalt mastic  Binder  Fillers  0.26  0.495 
Asphalt mortar  Mastic  Fine aggregates and voids  0.65 and 0.065  0.35 
Asphalt mixture  Mortar  Coarse aggregates  0.43  0.35 
The asphalt binder is a viscoelastic material, which makes the asphalt mastic, asphalt mortar, and asphalt mixture be the viscoelastic materials. So, to represent the effective properties for the multiscale materials, Dynamic Shear Rheometer (DSR) is used to measure the dynamic shear moduli of asphalt binder, mastic, and mortar, and Simple Performance Tester (SPT) is used to measure the axial dynamic modulus of asphalt mixture. Temperature and frequency sweep tests were applied to measure the dynamic modulus of the multiscale asphalt materials and the temperatures, frequencies, and loading mode for different tests are summarized in Table
Test methods of the multiscale asphalt materials.
Material scale  Temperatures (°C)  Frequencies (Hz)  Mode of loading 

Binder and mastic  10, 16, 19, 22, 25, 30, 40, 54  14, 6.5, 3.0, 1.4, 0.65, 0.30, 0.14, 0.10  Fully reversed oscillatory shear test (parallel plate) 
Mortar  14, 23, 38, 58  14, 6.5, 3.0, 1.4, 0.65, 0.30, 0.14, 0.10  Fully reversed oscillatory shear test (torsional cylinder) 
Mixture  210, 5, 20, 40, 54  25, 10, 5.0, 1.0, 0.50, 0.10  Tensioncompression sinusoidal loading test 
It should be noted that the primary test results of the mixture are the axial dynamic modulus
The test results and master curves for multiscale asphalt materials were shown in Figure
Fitting parameters of the master curves.
Multiscale materials  Fitting parameters  





 
Asphalt binder 





Asphalt mastic 





Asphalt mortar 





Asphalt mixture 





Experiment data and converted master curve of four materials.
According to finalized material parameters, the five micromechanical models were applied to predict the effective shear modulus of the multiscales asphalt materials, respectively. For the mastic scale, the binder is the matrix phase and the fillers are the particle phase. But for the mortar scale, the mastic is the matrix phase and there are two particle phases, which are fine aggregates and voids, so a twostep approach was applied. The fine aggregates were firstly added to the mastic matrix for prediction, and then voids were added to the fine aggregatemastic composite. The voids were considered as special kinds of particles whose modulus was assumed as zero. The effective modulus prediction for asphalt mixture is similar to the mastic while the mortar is considered as the matrix phase and the coarse aggregate as the particle phase. All of the effective moduli were predicted at 10 frequencies (
Predicted and measured dynamic shear modulus of mastic.
Predicted and measured dynamic shear modulus of mortar.
Predicted and measured dynamic shear modulus of mixture.
From Figures
The good agreements of different models observed at the mastic scale owe to the low particle volume fraction of fillers in mastic. The reason for this is that, at low particle concentrations, the microstructure of composite can better match the assumptions of different models [
Due to those reasons, the deviation between the predicted and measured values increases sharply for the mortar scale, especially at lower frequency and higher temperature. Moreover, at lower frequency and higher temperature, the modulus of asphalt matrix becomes smaller and the modulus mismatch between the asphalt matrix and aggregate particles becomes bigger, which is the other important factor causing the deviation.
As discussed above, the complicated compositions of multiscale asphalt mixture challenge the applicability of micromechanical models, which have been also demonstrated by Buttlar and Roque [
For further validation and evaluation of the effects of these two factors on the applicability of the micromechanical models, the dynamic shear moduli of mastic with different filler volume fraction were collected from the report by Underwood and Kim [
Fitting parameters of the master curve of mastic.
Asphalt mastic  Fitting parameters  





 
MS00 





MS10 





MS20 





MS30 





MS40 





MS50 





MS55 





MS60 





Experiment data and converted master curve of mastic.
Asphalt matrix is a kind of viscoelastic material, whose stiffness is much smaller than aggregate particle. When the aggregate particles are added to the asphalt matrix, the asphaltaggregate composite system becomes stiffer due to the stiffening effect of the aggregate particle. The higher the aggregate particles volume fraction is, the stronger the effect will be. This stiffening effect could be represented by the modulus ratio of the composite to the asphalt matrix. Figure
Relation between modulus ratio and filler volume fraction.
According to the researches by Faheem and Bahia [
It should be noted that the viscoelastic property of asphalt matrix also affects the prediction accuracy due to the modulus mismatch between the matrix and reinforcement particles. For the asphalt mastic with a fixed filler volume fraction, the smaller the modulus mismatch is at higher frequency or lower temperature, the better the prediction accuracy will be.
The effects of filler volume fraction on the accuracy of different models were shown in Figure
Effect of filler volume fraction on the prediction accuracy.
To give a quantitative comparison, the double logarithmic linear regression method was adopted for each model in Figure
The fitting parameters are listed in Table
Fitting parameters at 10 Hz.
Parameters  SC  MT  GSC  DSEM  TPS 



















Furthermore, the GSC model was selected to analyze the effect of modulus mismatch on the accuracies at different frequencies, as shown in Figure
Fitting parameters of the GSC model at different frequencies.
Parameters  0.01  0.1  1  10  50 


0.327  0.398  0.476  0.555  0.605 

3.531  3.644  3.587  3.349  3.148 

0.98  0.98  0.99  0.99  0.99 
Effect of modulus mismatch on the prediction accuracy.
There are so many micromechanical models for asphalt mixture, but the applicability is different due to the complicated compositions of asphalt mixture. In this paper, the applicability of some micromechanical models is validated from the collected and tested shear modulus of the multiscale asphalt materials, including the SC, GSC, MT, DSEM, and TPS models. Conclusions could be obtained as follows.
(1) All of the five models are applicable for the mastic scale, but the prediction accuracies for mortar and mixture scales are poorer. For the mixture scale, all models tend to overestimate at the intermediate frequencies but show good agreement at low and high frequencies except for the SC model. The TPS model is relatively better than others for the mortar scale.
(2) The applicability of micromechanical models is challenged due to the high particle volume fraction in the multiscale asphalt materials. With the filler volume fraction lower than 30%, the filler particles could be uniformly suspended in the asphalt matrix and do not contact each other. However, the particle interaction contributes more to the stiffening effect within higher fraction, and the prediction is then deteriorated.
(3) Because asphalt matrix is viscoelastic materials and its modulus is smaller than aggregate particles, the modulus mismatch between matrix and particles also deteriorates the prediction accuracy, especially at the lower frequencies (or higher temperature). The higher the frequency (or the lower the temperature) is, the better the model applicability will be.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (nos. 51378073, 51408043), the Natural Science Foundation of Shaanxi Province (2014JQ7278), and Special Fund for Basic Scientific Research of Central College of Chang’an University (310821153502, 310821152003).