# Mathematical Modeling of Pavement Gyratory Compaction: A Perspective on Granular-Fluid Assemblies

(This article belongs to the Section Engineering Mathematics)

## Abstract

**:**

## 1. Introduction

**coarse aggregates**(aggregate with size $>2.36$ mm) and

**fine aggregate mixtures**(FAM, a mixture of asphalt binder and fine aggregates). In this study, the deformation of the asphalt mixture during compaction is divided into two parts: (i) the deformation caused by the viscoplastic behavior of the FAM, and (ii) the deformation due to the particle rearrangement. On the one hand, we consider the viscoplasticity of the FAM using a rheological model of granular-fluid systems. The effective frictional coefficient of the granular system, denoted as ${\mu}_{\mathrm{eff}}$, is calculated by combining the inertial number, ${I}_{c}$, and the viscous number, ${I}_{v}$. The macroscopic deformation of the asphalt mixture resulting from FAM viscoplasticity can be calculated using a simple ordinary differential equation (ODE). On the other hand, the deformation of asphalt mixtures due to particle rearrangements can be generalized from granular compaction and calibrated using experimental results obtained from gyratory compaction. This allows us to establish a straightforward mathematical model that incorporates granular physics in the prediction of asphalt mixture compaction. This paper is organized in the following way. First, after the introduction in Section 1, we describe the model framework for predicting the compaction behavior of asphalt mixtures in Section 2. Then, this model is calibrated with several experiments obtained in the laboratory in Section 3. In Section 4, the model is validated by comparing its results with other experimental findings. Finally, concluding remarks are drawn in Section 5.

## 2. Model Description

- As stated, the asphalt mixture is divided into two parts: coarse aggregates (aggregates larger than 2.36 mm) and a fine aggregate matrix (a mixture of asphalt binder and fine aggregates <2.36 mm).
- The coarse aggregates are considered as spherical particles with material properties identical to real aggregates. The median size of coarse aggregates is denoted as ${d}_{50}^{c}$.
- The fine aggregate matrix can be seen as thick coatings on the surface of coarse aggregates.

#### 2.1. Viscoplastic Deformation Induced by FAM

**ode45s($\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$)**. After the calculation, the relationship between the displacement x and time t can be obtained. Then we could convert the displacement to the volume fraction of asphalt mixtures and convert the time to gyration numbers (${\varphi}_{vp}={\varphi}_{vp}\left(N\right)$). To test the results of the ODE, three different coating thicknesses (three different mass ratios between the FAM and coarse aggregates) are chosen. Figure 6 shows the relationship between the volume fraction of the ordered asphalt mixtures, $\varphi $, and the number of gyrations, N. It shows that, as we increase the amount of the FAM (which leads to the increase of the FAM coating thickness), the compaction volume fraction can be improved accordingly.

#### 2.2. Rearrangement of Aggregates

#### 2.3. Redistribution of Particle Rearrangement

## 3. Model Calibration

#### 3.1. Materials and Experiments for Model Calibration

#### 3.2. Comparison between Model and Experimental Results

## 4. Model Validation and Further Discussions

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mallick, R.B. Use of Superpave gyratory compactor to characterize hot-mix asphalt. Transp. Res. Rec.
**1999**, 1681, 86–96. [Google Scholar] [CrossRef] - Consuegra, A.E.; Little, D.N.; Von Quintus, H.; Burati, J. Comparative Evaluation of Laboratory Compaction Devices Based on Their Ability to Produce Mixtures with Engineering Properties Similar to Those Produced in the Field. Master’s Thesis, Texas A&M University, College Station, TX, USA, 1988. [Google Scholar]
- Hunter, A.E.; McGreavy, L.; Airey, G.D. Effect of compaction mode on the mechanical performance and variability of asphalt mixtures. J. Transp. Eng.
**2009**, 135, 839–851. [Google Scholar] [CrossRef] - Blankenship, P.B.; Mahboub, K.C.; Huber, G.A. Rational Method for Laboratory Compaction of Hot-Mix Asphalt (with Discussion and Closure); Transportation Research Record; Transportation Research Board: Washington, DC, USA, 1994; Volume 1454. [Google Scholar]
- Stakston, A.D.; Bahia, H.U.; Bushek, J.J. Effect of fine aggregate angularity on compaction and shearing resistance of asphalt mixtures. Transp. Res. Rec.
**2002**, 1789, 14–24. [Google Scholar] [CrossRef] - Delgadillo, R.; Bahia, H.U. Effects of temperature and pressure on hot mixed asphalt compaction: Field and laboratory study. J. Mater. Civ. Eng.
**2008**, 20, 440–448. [Google Scholar] [CrossRef] - Awed, A.; Kassem, E.; Masad, E.; Little, D. Method for predicting the laboratory compaction behavior of asphalt mixtures. J. Mater. Civ. Eng.
**2015**, 27, 04015016. [Google Scholar] [CrossRef] - Guler, M.; Bosscher, P.J.; Plesha, M.E. A porous elasto-plastic compaction model for asphalt mixtures with parameter estimation algorithm. In Recent Advances in Materials Characterization and Modeling of Pavement Systems; American Society of Civil Engineers: Reston, VA, USA, 2004; pp. 126–143. [Google Scholar]
- Koneru, S.; Masad, E.; Rajagopal, K. A thermomechanical framework for modeling the compaction of asphalt mixes. Mech. Mater.
**2008**, 40, 846–864. [Google Scholar] [CrossRef] - Masad, E.; Scarpas, A.; Alipour, A.; Rajagopal, K.R.; Kasbergen, C. Finite element modelling of field compaction of hot mix asphalt. Part I: Theory. Int. J. Pavement Eng.
**2016**, 17, 13–23. [Google Scholar] [CrossRef] - Masad, E.; Scarpas, A.; Rajagopal, K.R.; Kassem, E.; Koneru, S.; Kasbergen, C. Finite element modelling of field compaction of hot mix asphalt. Part II: Applications. Int. J. Pavement Eng.
**2016**, 17, 24–38. [Google Scholar] [CrossRef] - Wang, L.; Zhang, B.; Wang, D.; Yue, Z. Fundamental mechanics of asphalt compaction through FEM and DEM modeling. In Analysis of Asphalt Pavement Materials and Systems: Engineering Methods; American Society of Civil Engineers: Reston, VA, USA, 2007; pp. 45–63. [Google Scholar]
- Chen, J.; Huang, B.; Shu, X. Air-void distribution analysis of asphalt mixture using discrete element method. J. Mater. Civ. Eng.
**2012**, 25, 1375–1385. [Google Scholar] [CrossRef] - Chen, J.; Huang, B.; Shu, X.; Hu, C. DEM simulation of laboratory compaction of asphalt mixtures using an open source code. J. Mater. Civ. Eng.
**2014**, 27, 04014130. [Google Scholar] [CrossRef] - Man, T.; Le, J.L.; Marasteanu, M.; Hill, K.M. Two-Scale Discrete Element Modeling of Gyratory Compaction of Hot Asphalt. J. Eng. Mech.
**2022**, 148, 04021140. [Google Scholar] [CrossRef] - Man, T. Rheology of Granular-Fluid Systems and Its Application in the Compaction of Asphalt Mixtures. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, USA, 2019. [Google Scholar]
- Man, T.; Hill, K. Granular-slurry rheology and asphalt compaction. In Proceedings of the EPJ Web of Conferences, Powders & Grains, Buenos Aires, Argentina, 5, 13, 21, 29 July and 6 August 2021; EDP Sciences: Les Ulis, France, 2021; Volume 249, p. 09010. [Google Scholar] [CrossRef]
- Mehta, A.; Barker, G. The dynamics of sand. Rep. Prog. Phys.
**1994**, 57, 383. [Google Scholar] [CrossRef] - Knight, J.B.; Fandrich, C.G.; Lau, C.N.; Jaeger, H.M.; Nagel, S.R. Density relaxation in a vibrated granular material. Phys. Rev. E
**1995**, 51, 3957. [Google Scholar] [CrossRef] [PubMed] - Pouliquen, O.; Belzons, M.; Nicolas, M. Fluctuating particle motion during shear induced granular compaction. Phys. Rev. Lett.
**2003**, 91, 014301. [Google Scholar] [CrossRef] - Mehta, A.; Barker, G.; Luck, J. Cooperativity in sandpiles: Statistics of bridge geometries. J. Stat. Mech. Theory Exp.
**2004**, 2004, P10014. [Google Scholar] [CrossRef] - Barker, G.; Mehta, A. Transient phenomena, self-diffusion, and orientational effects in vibrated powders. Phys. Rev. E
**1993**, 47, 184. [Google Scholar] [CrossRef] - Philippe, P.; Bideau, D. Compaction dynamics of a granular medium under vertical tapping. EPL Europhys. Lett.
**2002**, 60, 677. [Google Scholar] [CrossRef] - Nowak, E.R.; Knight, J.B.; Ben-Naim, E.; Jaeger, H.M.; Nagel, S.R. Density fluctuations in vibrated granular materials. Phys. Rev. E
**1998**, 57, 1971. [Google Scholar] [CrossRef] - Trulsson, M.; Andreotti, B.; Claudin, P. Transition from the viscous to inertial regime in dense suspensions. Phys. Rev. Lett.
**2012**, 109, 118305. [Google Scholar] [CrossRef] - Pitois, O.; Moucheront, P.; Chateau, X. Liquid bridge between two moving spheres: An experimental study of viscosity effects. J. Colloid Interface Sci.
**2000**, 231, 26–31. [Google Scholar] [CrossRef] - Man, T.; Feng, Q.; Hill, K. Rheology of Thickly-Coated Granular-Fluid Systems. arXiv
**2018**, arXiv:1812.07083. [Google Scholar] - Brown, E.; Collins, R.; Brownfield, J. Investigation of Segregation of Asphalt Mixtures in the State of Georgia; Transportation Research Record; Transportation Research Board: Washington, DC, USA, 1989; Volume 1217. [Google Scholar]
- ASTM/D6925-15; Standard Test Method for Preparation and Determination of the Relative Density of Asphalt Mix Specimens by Means of the Superpave Gyratory Compactor. American Society for Testing and Materials: West Conshohocken, PA, USA, 2015.
- ASTM/D2041-11; Standard Test Method for Theoretical Maximum Specific Gravity and Density of Bituminous Paving Mixtures. American Society for Testing and Materials: West Conshohocken, PA, USA, 2011.
- ASTM/D2726-17; Standard Test Method for Bulk Specific Gravity and Density of Non-Absorptive Compacted Asphalt Mixtures. American Society for Testing and Materials: West Conshohocken, PA, USA, 2017.

**Figure 3.**Two parts of the deformation during the compaction of asphalt mixtures: (1) viscous deformation; (2) particle rearrangement.

**Figure 4.**Simplification of the viscous deformation induced by the FAM coatings on the surfaces of coarse aggregates.

**Figure 5.**Computational cell for calculating the change of volume fraction due to the viscous deformation of the FAM coatings, based on which we can form the differential equation for calculating the motion of particles so that we could also calculate the volume fraction versus time.

**Figure 6.**Comparison of the volume fraction/gyration number relationship as we change the thickness of the FAM coating.

**Figure 7.**Relationship between the solid fraction of particle rearrangement and the dimensionless time, ${t}^{*}$.

**Figure 9.**Comparison between experimental results and the results obtained from the proposed mathematical model: (

**a**) results for 1N30, 1N50, and 1N100; (

**b**) results for 2N30, 2N50, and 2N100.

Sieve Size | 3N30 | 3N50 | 3N100 |
---|---|---|---|

(mm) | (%) | (%) | (%) |

12.5 | 100 | 100 | 100 |

9.5 | 94.1 | 94.7 | 95.3 |

6.3 | 71.5 | 72.2 | 72.9 |

4.75 | 60.5 | 61.3 | 62.0 |

2.36 | 42.4 | 38.2 | 34.0 |

1.18 | 29.2 | 24.9 | 20.4 |

0.60 | 19.7 | 16.4 | 12.8 |

0.30 | 11.5 | 9.9 | 7.6 |

0.15 | 6.1 | 5.7 | 4.2 |

0.075 | 4.6 | 4.3 | 3.0 |

${r}_{50}$ (mm) | 1.41 | 1.54 | 1.68 |

${\tau}_{rp}$ | 27.6 | 26.3 | 24.4 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Man, T.
Mathematical Modeling of Pavement Gyratory Compaction: A Perspective on Granular-Fluid Assemblies. *Mathematics* **2023**, *11*, 2096.
https://doi.org/10.3390/math11092096

**AMA Style**

Man T.
Mathematical Modeling of Pavement Gyratory Compaction: A Perspective on Granular-Fluid Assemblies. *Mathematics*. 2023; 11(9):2096.
https://doi.org/10.3390/math11092096

**Chicago/Turabian Style**

Man, Teng.
2023. "Mathematical Modeling of Pavement Gyratory Compaction: A Perspective on Granular-Fluid Assemblies" *Mathematics* 11, no. 9: 2096.
https://doi.org/10.3390/math11092096