A Hybrid Model to Predict the Gyratory Compaction of Hot Mixed Asphalt

The compaction of asphalt mixture is crucial to the mechanical properties and the maintenance of the pavement. However, the mix design, which based on the compaction properties, remains largely on empirical data. We found diﬃculties to relate the aggregate size distribution and the asphalt binder properties to the compaction behavior in both the ﬁeld and laboratory compaction of asphalt mixtures. In this paper, we would like to propose a simple hybrid model to predict the compaction of asphalt mixtures. In this model, we divided the compaction process into two mechanisms: (i) visco-plastic deformation of an ordered thickly-coated granular assembly, and (ii) the transition from an ordered system to a disordered system due to particle rearrangement. This model could take into account both the viscous properties of the asphalt binder and grain size distributions of the aggregates. Additionally, we suggest to use discrete element method to understand the particle rearrangement during the compaction process. This model is calibrated based on the SuperPave gyratory compaction tests in the pavement lab. In the end, we compared the model results to experimental data to show that this model prediction had a good agreement with the experiments, thus, had great potentials to be implemented to improve the design of asphalt mixtures.


Introduction 1
Hot mixed asphalt is ubiquitous in pavement industry. The mechanical 2 properties and durability of asphalt mixtures are largely influenced by the 3 porosity. High porosity usually leads to low strength and high permeability 4 of the moisture [1,2], while low porosity may result in high temperature 5 sensibility [3,4]. Thus, we should maintain a reasonable porosity after the 6 compaction process. 7 In the construction field, we use roller compactorS to compact the asphalt 8 mixtures. During the compaction process, the material is subjected to both 9 normal pressure and shear stress. In the laboratory, to represent the similar asphalt. However, even the laboratory compaction process tends to be a 32 random process, and the compaction behavior is governed by a number of 33 factors, such as binder content, binder type, temperature, loading method, 34 aggregates' grain size distributions, aggregate shapes, etc.. Stakston et al. 35 [9] investigated the influence of the angularity of fine aggregates on the com-36 paction behavior of asphalt mixtures, which indicated that a higher fine ag-37 gregate angularity may lead to higher resistance to compaction. Delgadillo While most of the research of the compaction behavior focused on the qualitative analysis of the factors which may influence the gyratory compaction behavior, few research focused on the quantitative investigation of the compaction behavior. Awed et al. [11] proposed a method for predicting the laboratory compaction behavior of asphalt mixtures, which provided us a phenomenological tool for predicting the laboratory compaction. They linked the air void ratio to parameters, such as gradation scale parameter, gradation shape parameter, and asphalt content, and their model also converged to a logarithmic law shown in the following equation [11].
where AV is the air void ratio, and N G is the number of gyrations, and a 43 and b are parameters empirically determined by material parameters.

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Theoretically, the asphalt mixtures, when in high temperature, can be seen as granular materials (aggregates) with interstitial viscous fluid (asphalt binder) [12]. Thus, the compaction of asphalt mixtures can follow the research of the compaction of granular materials. In the past decades, physicists were interested in the general compaction or relaxation of monodispersed or bi-dispersed granular materials [13]. Although the compaction behavior of mono-or bi-dispersed granular materials may differ from the behavior of the compaction of asphalt mixtures, the underlying physics should be similar and transferable. The microscopic analyses of granular materials during compaction induced by tapping or vabrating have been studied [13,14]. The relationship between collective microscopic structure and the compaction dynamics has been explained [15,16]. Several equations or models for describing the compaction behavior were proposed based on multipletime-scale assumption [17,14,18]. Among these models, Knight et al. [14] proposed a logarithmic equation to better fit the compaction behavior of mono-dispersed granular materials.
where φ(t) is the volume fraction of granular materials at time t, φ f is the 45 final volume fraction when time goes to infinity,  problem, certain simplification is necessary.

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• As what we have stated, we divided the asphalt mixture into two parts,  Firstly, one part of the deformation during the compaction processes comes from the deformation of FAM coatings when two coarse aggregates approach each other. Secondly, as the asphalt mixture approaches random loose packing, the particles would also like to rearrange themselves to form a more stable packing. Then the effect of particle rearrangement will gradually play an important role as the compaction proceed. As we can see from Fig.  3, during the compaction process, we have two types of deformation. Usually at the beginning of the compaction, the FAM serves as coatings of the coarse aggregate, which prevent two adjacent particles from coming toward each other, which brings the viscous deformation of the whole mixture. On the other hand, when the solid fraction of aggregate packing become larger and larger, the particle rearrangement starts to play important roles. The particle will oscillate during compaction, which may create free space for the other particles to move inside, which can help the whole mixture to form a denser state. Thus, we have the following equation: where φ(t) is the volume fraction of the asphalt mixture at time t, φ v (t) is 85 the volume fraction of the asphalt mixture due to viscous deformation at 86 time t, and ∆φ rp (t) is the change of volume fraction due to rearrangement 87 of particles at time t. Here we consider the effect of particle rearrangement 88 as the increment on top of viscous deformations. 89 We should note that the behavior of the FAM follows the rheology of We show that the deformation can be calculated using a computational cell shown in Fig. 5. The problem becomes solving the motion of particle j under pressure σ = 600 kPa and Shear rateγ = 0.07. The acting loading on the particle j is introduced by the compaction pressure. We have two parts of resistance forces: (1) squeeze-induced lubrication force, and (2) shear-induced resistance forces. Thus, we can obtain the following equations: x is the displacement of the particle j, η n eff and η t eff are the normal and We have stated that the constitutive relationship of the FAM coating follows the rheology of granular-fluid systems proposed in Ref. [20,12,21].
Thus, we could first calculate the frictional rheology (effective frictional coefficient, µ eff , and solid fraction, φ FAM ) of the FAM with the following equations.
where η b is the dynamics viscosity of the asphalt binder at compaction tem-116 perature, α ≈ 0.03 according to Ref. [12], µ 1 = 0.265, µ 2 = 2.2, and K 0 = 0.5 117 are fitting parameters.  Based on these assumptions, we can use discrete element method (DEM) to simulate the gyratory compaction of a monodispersed granular-fluid system in a SuperPave gyratory compactor configuration shown in Fig. 1. We compacted the granular materials with particle diameters = (2 ± 0.4) mm using constant pressure of 600 kPa, gyratory speed of 30 rpm, gyratory angle of 1.25 • , and the viscosity of the interstitial fluid is set to be equal to 250 cP. To simplify the analyses, we neglected the effect of interstitial fluid on the particle rearrangement. All the dimensional parameters will be normalized using Figure 7: Relationship between the solid fraction of particle rearrangement and the dimensionless time, t * average particle radius,r p , mass of average-size particles,m = (4/3)πr 3 p ρ p , and gravitational acceleration, g. Thus, the compaction time, t can be nondimensionalized as t * ≡ t g/r p , where t * is the dimensionless time. In Fig. 7, we plotted the relationship between dimensionless time, t * , and the change of solid fraction of granular materials during the compaction. Here, the change of solid fraction can be seen as the effect of particle rearrangement. The compaction curve in Fig. 7 can be fitted with the following equation: where ∆φ i rp (t * ) is the change of volume fraction for a compaction of particles 151 with certain average grain size, t * is the dimensionless time. This fitting 152 curve is based on Eq. 2.

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The above fitting curve can only be used to quantify a monodisperse granular assembly. To obtain the total effect of particle rearrangement of the whole grain size distribution, we have be implement mixture theory. We divide the aggregates into different size groups based on the sieving test. Each size group of aggregates is bounded by adjacent sieve sizes (d i and d i+1 ), and the mean size of each size group is calculated byd i = d i d i+1 andr i p = 0.5d i . On one hand, based on the mean size of each size group, we can calculate the compaction behavior of each size group, ∆φ i rp (t * ), where t * is a function of particle size. On the other hand, we can also obtain the mass ratio of each size group based on the sieving test, Ψ i . Then, we obtain the total particle rearrangement effect based on the grain size distribution of aggregates.
where N s is the number of aggregate size groups in the sieving test.

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The basic idea is that at the beginning of the compaction, due to the huge effect of viscous deformation stated in the previous section, the effect of particle rearrangement is smaller that what we had in the DEM simulations. Thus, we would like to decrease the influence of particle rearrangement at the beginning of the compaction, but enhance the influence of that toward the end of the compaction. We proposed the following equation to capture the redistribution of the particle rearrangement.
where τ rp is the balancing point, which depend on the grain size distribution, and N is the gyration number. Once we know the viscosity of asphalt binder at the compaction temperature and the grain size distribution of the aggregate, the relationship between τ rp and the grain size distribution is the only parameter we need to calibrate. In the end, if we combine the contributions from both the viscous deformation and the particle rearrangement, we can obtain the following relationship between volume fraction of asphalt mixtures, φ vf , and the compaction time, t, to describe the whole gyratory compaction process of asphalt mixtures:

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The model we proposed require the information of the viscosity of asphalt 166 binder, and the sieving analysis results of the aggregates, based on which we 167 could obtain both the viscous deformation and the particle rearrangement 168 effect during the compaction process of the asphalt mixtures. However, since 169 we also need to redistribute the particle rearrangement effect, the parameter,  We acquired the aggregate from two sources: river sand and quarry gravels. We show the size distribution of the seven sets of aggregates in Fig. 8.  As what we have stated, we would like to find the relationship between the 218 balancing point, τ rp , and the grain size distrbution for the model calibration.

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Thus, we first obtained the best fit (with a specific τ rp ) for each experimental 220 results, plotted as solid curves with different colors (Fig. 9). Then, we plotted