# Real-Time Flow Behavior of Hot Mix Asphalt (HMA) Compaction Based on Rheological Constitutive Theory

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## Abstract

**:**

## 1. Introduction

## 2. Test Materials and Methods

#### 2.1. Materials and Mix Design

#### 2.2. MTS Compaction Process

## 3. Viscoelastic-Plastic Rheological Model Theory

#### 3.1. Nonlinear Viscoelastic-Plastic Nishihara Model and Parameter Solving

_{1}is the elastic modulus of the Hooker body; E

_{2}is the elastic modulus of the Kelvin body; η

_{1}, η

_{2}are the viscosity coefficients of the Kelvin body and the Bingham body stick component; σ

_{e}is the applied stress; σ

_{es}is the material yield stress; t is time; k is the rheological index; ΔT is the calculation of the temperature difference; and, α is the material shrinkage coefficient.

_{1}is the elastic modulus of the Hooker body; E

_{2}is the elastic modulus of the Kelvin body; η

_{1}, η

_{2}are the viscosity coefficients of the Kelvin body and the Bingham body stick component; σ

_{e}is the applied stress; σ

_{es}is the material yield stress; and, t is time.

#### 3.2. Model Parameter Solving

_{e}, σ

_{es}, E

_{1}, E

_{2}, η

_{1}, and η

_{2}. Each parameter is a related parameter that varies with the number of load cycles and temperature.

- (1)
- ${\sigma}_{e}\approx {\sigma}_{0}=0.6\text{}\mathrm{Mpa}$.
- (2)
- Under the action of 15 loading cycles, the value of σ
_{es}in each cycle is during two loading and unloading processes. The tangent stress value at the straight line segment of the stress-strain curve at the first loading and the tangent point at the curve is taken as the σ_{es}of the cyclic process, as shown in Figure 6. - (3)
- Separating viscoplastic strain, elastic strain, and viscoelastic strain solves the parameters. Firstly, E
_{1}is solved by the “recovery elastic strain” after the second unloading in each cycle: ${E}_{1}={\sigma}_{e}/{\epsilon}^{e}$. Secondly, the η_{1}value is solved by the “median viscoplastic strain” in each cycle:$\text{}{\eta}_{1}=2\left({\sigma}_{e}-{\sigma}_{e}{}_{S}\right)t/{\epsilon}^{vp}\left(t\right)$. Thirdly, in the Kelvin body, E_{2}and η_{2}use the BoxLucas1 model equation $y=a\left(1-{e}^{-bx}\right)$ to perform nonlinear regression on the “recover viscoelastic strain time curve” after the second unloading in each cycle, and the parameters of the model are then obtained by calculating.

_{es}. Through the first cyclic stress–strain curve of Figure 6, the value of the tangent point at the straight line segment of the stress–strain curve at the first loading and the tangent point of the curve is determined, and ${\sigma}_{es}^{1}$ is obtained. In the same way, ${\sigma}_{es}^{2}$ ⋯ ${\sigma}_{es}^{15}$ can be obtained. The second step is to solve E

_{1}, E

_{2}, η

_{1}, and η

_{2}. The elastic strain, viscoplastic strain, and viscoelastic strain have been marked in Figure 7, and the parameters of the table are solved, as follows:

- (a)
- ${E}_{1}={\sigma}_{e}/{\epsilon}^{e}$, then ${E}_{1}^{1}=0.6/{\epsilon}^{e1}=0.6/0.00753=79.681$, in the same way, ${E}_{1}^{2}$⋯ ${E}_{1}^{15}$ can be obtained.
- (b)
- ${\eta}_{1}=2\left({\sigma}_{e}-{\sigma}_{e}{}_{S}\right)t/{\epsilon}^{vp}\left(t\right)$, then ${\eta}_{1}^{1}=\left({\sigma}_{e}-{\sigma}_{\mathrm{eS}}^{1}\right)\times 2/\left({\epsilon}^{vp1}/2\right)$, that is ${\eta}_{1}^{1}=\left(0.6-0.340\right)\times 2/\left(0.01747/2\right)=64.34$, and in the same way, ${\eta}_{1}^{2}$⋯${\eta}_{1}^{15}$ can be obtained.
- (c)
- The BoxLucas 1 model equation $y=a\left(1-{e}^{-bx}\right)$ is used for non-linear regression, as shown in Figure 8. Where $a={\sigma}_{e}/{E}_{2}$, $b={E}_{2}/{\eta}_{2}$. After calculation, ${E}_{2}^{1}=0.6/0.00231=166.205$, ${\eta}_{2}^{1}=166.205/0.09950=1670.4$. In the same way, ${E}_{2}^{2}$⋯ ${E}_{2}^{15}$ and ${\eta}_{2}^{2}$⋯ ${\eta}_{2}^{15}$ can be obtained.

## 4. Nishihara Model Verification and Parameter Analysis

#### 4.1. Model verification

_{e}, σ

_{es}, E

_{1}, E

_{2}, η

_{1}, and η

_{2}are substituted into Equation (2) to obtain the strain curves when compared with the testing strain–time curves, as shown in Figure 9.

#### 4.2. Parameter Analysis

_{1}characterizes the viscous effect of the viscoplastic process of the mixture, which reflects the ability of the asphalt mixture to resist deformation during the viscoplastic phase. Figure 10 indicates that, as the number of loading cycle increases, the temperature of the mixture decreases, and the yield stress and viscosity coefficient increase. The viscosity coefficient mainly increases because the viscosity of the asphalt increased when the temperature decreases. The rate change of yield stress and viscosity coefficient of the real-time mixture is faster and the mixture is easily compacted. After the third load cycle, the rate of change of yield stress and viscosity coefficient of AC-13 began to decrease. After the fourth load cycle, the rate of change of yield stress and viscosity coefficient of SAC-13 also began to decrease, and the mixture began to become more difficult to compact.

_{1}is the elastic modulus, which is known as an index to measure the instantaneous difficulty of elastic deformation of materials. The larger the value, the greater the ability to resist deformation. Figure 12 presents that E

_{1}for SAC-13 stable as the number of load cycles changes, because the asphalt cement has strong fluidity at high temperature, and the elastic modulus of the mixture is mainly expressed as the elastic modulus of the aggregate skeleton in this state. The difference in the elastic modulus of the two grades is mainly due to the large amount of coarse aggregate and the high void ratio.

_{2}is the viscoelastic deformation modulus, which characterizes the viscoelastic recovery effect of the mixture. It can be seen from Figure 13 that, as the number of load cycles increases, E

_{2}decreases, and its ability to resist deformation decreases. The viscoelastic viscosity coefficient η

_{2}characterizes the viscous effect of the viscoelastic process of the mixture, which reflects the ability of the asphalt mixture to resist the deformation during the viscoelastic phase. Figure 13 presents that η

_{2}becomes larger as the number of load cycles increases, and the ability of the mixture to resist deformation becomes larger. The change in the two parameters is because, as the number of load cycles increase, the temperature decreases and the viscosity of the asphalt decreases. The difference in viscosity coefficients between the two grades is due to temperature differences and gradation differences, it can be summarized that the increasing rate of η

_{2}for AC-13 proves the difficulty in compaction.

## 5. Conclusion

- The nonlinear Nishihara model has the advantages of being intuitive, simple in theory, and possessing easy to solve parameters. A loading and unloading MTS compaction test is effectively utilized to calculate the parameters. The calculation curve is compared with the test curve, and it is verified that the rheological properties of the hot mix asphalt in the compaction process can be indirectly characterized by the nonlinear Nishihara model.
- The results prove that the nonlinear Nishihara model is useful in distinguishing the flow behavior undergoing a load and unloading compaction. The compaction of asphalt mixture mainly produces plastic deformation, which is mainly manifested by changes in viscos and plastic parameters (σ
_{es}, η_{1}, and η_{2}), rather than the elastic parameter (E_{1}). The differences in sensitive parameters (σ_{es}, η_{1}, and η_{2}) reflect that the gradation and temperature have certain influence on the compaction characteristics of the asphalt mixture. - Only two asphalt mixtures and one stress were included in this paper. The next step will be aiming at studies regarding the compaction properties subjected to various grades, different modified asphalts, and stress at different temperatures via the Nishihara model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

σ_{e} | Applied stress | ε^{e} | Elastic strain |

σ_{es} | Yield stress | ε^{ve} | Viscoelastic strain |

E_{1} | The elastic modulus of the Hooker body | k | The rheological index |

E_{2} | The elastic modulus of the Kelvin body | t | Time |

η_{1} | The viscosity coefficients of the Kelvin body stick component | α | The material shrinkage coefficient |

η_{2} | The viscosity coefficients of the Bingham body stick component | ΔT | The value of the temperature difference |

ε^{vp} | Viscoplastic strain |

## References

- Cui, W.; Tao, J.; Zhang, Z.; Li, H. Test on rut causes of expressway asphalt pavement. J. Chang’an Univ.
**2009**, 29, 8–12. [Google Scholar] - Zhang, K. Research on friable damage of asphalt pavement. Doctoral Dissertation, Chang’an University, Xi’an, China, 2015. [Google Scholar]
- Liu, M.; Han, S.; Shang, W.; Qi, X.; Dong, S.; Zhang, Z. New polyurethane modified coating for maintenance of asphalt pavement potholes in winter-rainy condition. Prog. Org. Coat.
**2019**, 133, 368–375. [Google Scholar] [CrossRef] - Uzan, J.; Sides, A.; Perl, M. Viscoelastoplastic model for predicting performance of asphaltic mixtures. Transp. Res. Rec.
**1985**, 1043, 78–89. [Google Scholar] - Sides, A.; Uzan, J.; Perl, M. A comprehensive visco-elasto-plastic characterization of sand-asphalt under compression and tension cyclic loading. ASTM J. Test. Eval.
**1985**, 13, 49–59. [Google Scholar] - Li, J.; Zhang, J.; Qian, G.; Zheng, J.; Zhang, Y. Three-dimensional simulation of aggregate and asphalt mixture using parameterized shape and size gradation. J. Mater. Civil Eng.
**2019**, 31, 04019004. [Google Scholar] [CrossRef] - Cheng, Y.; Yu, D.; Tan, G.; Zhu, C. Low-temperature performance and damage constitutive model of eco-friendly basalt fiber–diatomite-modified msphalt mixture under freeze–thaw cycles. Materials
**2018**, 11, 2148. [Google Scholar] [CrossRef] - Sun, Y.; Gu, B.; Gao, L.; Li, L.; Guo, R.; Yue, Q.; Wang, J. Viscoelastic mechanical responses of HMAP under moving load. Materials
**2018**, 11, 2490. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, W.; Tao, J.; Xu, M.; Xu, X.; Ma, G.; Wang, S. Influence analysis and optimization for aggregate morphological characteristics on high and low temperature viscoelasticity of asphalt mixtures. Materials
**2018**, 11, 2034. [Google Scholar] [CrossRef] - Lu, Y.; Wright, P.J. Temperature Related Visco-elastoplastic properties of asphalt mixtures. J. Transp. Eng.
**2000**, 126, 58–65. [Google Scholar] [CrossRef] - Lu, Y.; Lu, L.; Wright, P.J. Visco-elastoplastic method for pavement performance evaluation. Proc. Inst. Civil Eng.-Transp.
**2002**, 153, 227–234. [Google Scholar] [CrossRef] - Giunta, M.; Pisano, A.A. One-dimensional visco-elastoplastic constitutive model for asphalt concrete. Multidiscip. Model. Mater. Struct.
**2006**, 2, 247–264. [Google Scholar] [CrossRef] - Masad, E.; Tashman, L.; Little, D.; Zbib, H. Viscoplastic modeling of asphalt mixes with the effects of anisotropy, damage and aggregate characteristics. Mech. Mater.
**2005**, 37, 1242–1256. [Google Scholar] [CrossRef] - Masad, E.; Samer, D.; Little, D. Development of an elasto-viscoplastic microstructural-based continuum model to predict permanent deformation in hot mix asphalt. Int. J. Geomech.
**2007**, 7, 119–130. [Google Scholar] [CrossRef] - Sun, L.; Zhu, Y. A serial two-stage viscoelastic–viscoplastic constitutive model with thermodynamical consistency for characterizing time-dependent deformation behavior of asphalt concrete mixtures. Constr. Build. Mater.
**2013**, 40, 584–595. [Google Scholar] [CrossRef] - Huang, B.; Mohammad, L.N.; Wathugala, G.W. Application of a temperature dependent viscoplastic hierarchical single surface model for asphalt mixtures. J. Mater. Civ. Eng.
**2004**, 16, 147–154. [Google Scholar] [CrossRef] - Darabi, M.K.; Abu Al-Rub, R.K.; Masad, E.A.; Huang, C.-W.; Little, D.N. A modified viscoplastic model to predict the permanent deformation of asphaltic materials under cyclic-compression loading at high temperatures. Int. J. Plast.
**2012**, 35, 100–134. [Google Scholar] [CrossRef] - Pasetto, M.; Baldo, N. Numerical visco-elastoplastic constitutive modelization of creep recovery tests on hot mix asphalt. J. Traffic Transp. Eng.
**2016**, 3, 390–397. [Google Scholar] [CrossRef] - Chen, F.; Balieu, R.; Kringos, N. Thermodynamics-based finite strain viscoelastic-viscoplastic model coupled with damage for asphalt material. Int. J. Solids Struct.
**2017**, 129, 61–73. [Google Scholar] [CrossRef] - Chen, X. The research on deformation characteristic of the HMA compaction. Master’s Dissertation, Changsha University of Science & Technology, Changsha, China, 2006. [Google Scholar]
- Liu, Y. Visco-elastic-plastic model research on hot mix asphalt during the field compaction stage. Master’s Dissertation, Changsha University of Science & Technology, Changsha, China, 2009. [Google Scholar]
- JTJ E20-2011. Standard Test methods of Bitumen and Bituminous Mixtures for Highway Engineering; Ministry of Communication: Beijing, China, 2011.
- JTG F40-2004. Technical Specification for Construction of Highway Asphalt Pavements; Ministry of Communication: Beijing, China, 2004.
- JTG E42-2005. The Methods of Aggregate for Highway Engineering; Ministry of Communication: Beijing, China, 2005.
- Yan, Y.; Wang, S.-J.; Wang, E.-Z. Creep equation of variable parameters based on nishihara model. Yantu Lixue/Rock Soil Mech.
**2010**, 31, 3025–3035. [Google Scholar] - Pan, X.; Yang, Z.; Xu, J. Application study of nonstationary nishihara viscoelasto-plastic rheological model. Yanshilixue Yu Gongcheng Xuebao/Chinese J. Rock Mech. Eng.
**2011**, 30, 2640–2646. [Google Scholar] - Qi, Y.; Jiang, Q.; Wang, Z.; Zhou, C. 3D creep constitutive equation of modified nishihara model and its parameters identification. Yanshilixue Yu Gongcheng Xuebao/Chinese J. Rock Mech. Eng.
**2012**, 31, 347–355. [Google Scholar] - Jiang, Q.; Qi, Y.; Wang, Z.; Zhou, C. An extended nishihara model for the description of three stages of sandstone creep. Geophys. J. Int.
**2013**, 193, 841–854. [Google Scholar] [CrossRef] - Jiang, H.-F.; Liu, D.-Y.; Huang, W.; Xia, Y.-C.; Liu, F.-Y. Creep properties of rock under high confining pressure and different pore water pressures and a modified Nishihara model. Yantu Gongcheng Xuebao/Chinese J. Geotech. Eng.
**2014**, 36, 443–451. [Google Scholar] - Hou, F.; Lai, Y.; Liu, E.; Luo, H.; Liu, X. A creep constitutive model for frozen soils with different contents of coarse grains. Cold Reg. Sci. Technol.
**2018**, 145, 119–126. [Google Scholar] [CrossRef] - Tian, X.; Ying, R.; Zheng, J. Tests on thermal stress in asphalt cement sample and its calculation. China J. Highw. Transp.
**2001**, 14, 14–48. [Google Scholar]

**Figure 10.**Model parameters and temperature curves with load cycle times; (

**a**) σ

_{es}and T curves with load cycle times; and, (

**b**) η

_{1}curves with the load cycle times.

**Figure 13.**Viscoelastic parameters change with the load cycle times; (

**a**) E

_{2}change with the load cycle times; and, (

**b**) η

_{2}change with the load cycle times.

Test Items | Value | Required | |
---|---|---|---|

Penetration (25 °C, 100g, 5s)(0.1mm) | 65 | 60–80 | |

Ductility | (5 cm/min, 10 °C) cm | 86 | ≥20 |

(5 cm/min, 15 °C) cm | >100 | ≥100 | |

Softening Point TR&B (°C) | 50 | ≥46 | |

Penetration index | −0.476 | −1.5–+1.0 | |

60 °C Dynamic viscosity (Pa·S) | 224 | ≥180 | |

Wax content (%) | 1.2 | ≤2.2 | |

Solubility (%) | 99.8 | ≥99.5 | |

Flash point (°C) | 330 | ≥260 | |

Density 15 °C (g/cm^{3}) | 1.012 | Measured | |

After RTFOT | Quality change (%) | −0.026 | ≤±0.8 |

Residual penetration ratio (%) | 63 | ≥61 | |

Residual ductility (10 °C) (cm) | 7 | ≥6 | |

Residual ductility (15 °C) (cm) | 32 | ≥15 |

Test Items | Value | Required | ||
---|---|---|---|---|

13.2–9.5 | 9.5–4.75 | 4.75–2.36 | ||

Apparent relative density | 2.859 | 2.822 | 2.769 | ≥2.6 |

Water absorption (%) | 0.52 | 0.57 | 0.73 | ≤2 |

Crushing value (%) | 21 | -- | -- | ≤26 |

Los Angeles abrasion (%) | 19.2 | -- | -- | ≤28 |

Test Items | Value | Required |
---|---|---|

Apparent relative density | 2.717 | ≥2.5 |

Mud content (percent of < 0.075 mm) (%) | 0.9 | ≤3 |

Sand equivalent (%) | 83.7 | ≥60 |

Angularity | 40.3 | ≥30 |

Test Items | Value | Required | |
---|---|---|---|

Apparent relative density (g/cm^{3}) | 2.638 | ≥2.5 | |

Water absorption (%) | 0.3 | <1 | |

Grain sizes (%) | <0.6 mm | 100 | 100 |

<0.15 mm | 99.9 | 90–100 | |

<0.075 mm | 90.2 | 75–100 | |

Hydrophilic coefficient | 0.62 | <1 |

Noumber | SAC-13 | ||
---|---|---|---|

Viscoplastic Strain (ε^{vp}) | Elastic Strain (ε^{e}) | Viscoelastic Strain (ε^{ve}) | |

First | −0.017470 | 0.007530 | 0.0034020 |

Second | −0.008624 | 0.007504 | 0.0036927 |

Third | −0.005866 | 0.007503 | 0.0038537 |

Fourth | −0.004640 | 0.007496 | 0.0039342 |

Fifth | −0.003839 | 0.007494 | 0.0041183 |

Sixth | −0.003410 | 0.007491 | 0.0042349 |

Seventh | −0.003276 | 0.007491 | 0.0040201 |

Eighth | −0.002605 | 0.007439 | 0.0042196 |

Ninth | −0.002484 | 0.007435 | 0.0041720 |

Tenth | −0.002362 | 0.007421 | 0.0042173 |

Eleventh | −0.002047 | 0.007420 | 0.0043321 |

Twelfth | −0.001987 | 0.007418 | 0.0043233 |

Thirteenth | −0.001820 | 0.007417 | 0.0044842 |

Fourteenth | −0.001779 | 0.007401 | 0.0045524 |

Fifteenth | −0.001740 | 0.007391 | 0.0043724 |

Mix type | T (°C) | Air Voids (%) | Value | |||||
---|---|---|---|---|---|---|---|---|

σ_{e} | σ_{es} | E_{1} | E_{2} | η_{1} | η_{2} | |||

(MPa) | (MPa) | (MPa) | (MPa) | (Pa·s) | (Pa·s) | |||

SAC | 124.5 | 6.7 | 0.6 | 0.513 | 81.18 | 115.83 | 200.041 | 1728.016 |

AC | 121.9 | 7.0 | 0.6 | 0.516 | 89.634 | 133.311 | 241.371 | 1877.990 |

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## Share and Cite

**MDPI and ACS Style**

Qian, G.; Hu, K.; Gong, X.; Li, N.; Yu, H.
Real-Time Flow Behavior of Hot Mix Asphalt (HMA) Compaction Based on Rheological Constitutive Theory. *Materials* **2019**, *12*, 1711.
https://doi.org/10.3390/ma12101711

**AMA Style**

Qian G, Hu K, Gong X, Li N, Yu H.
Real-Time Flow Behavior of Hot Mix Asphalt (HMA) Compaction Based on Rheological Constitutive Theory. *Materials*. 2019; 12(10):1711.
https://doi.org/10.3390/ma12101711

**Chicago/Turabian Style**

Qian, Guoping, Kaikai Hu, Xiangbing Gong, Ningyuan Li, and Huanan Yu.
2019. "Real-Time Flow Behavior of Hot Mix Asphalt (HMA) Compaction Based on Rheological Constitutive Theory" *Materials* 12, no. 10: 1711.
https://doi.org/10.3390/ma12101711