Prediction of dynamic modulus of asphalt mixture using micromechanical method with radial distribution functions


Inter-particle interaction is one of the major reinforcement mechanisms for aggregates in asphalt mixture, which is a classic example of high-volume fraction particulate composites. This paper introduced the modified Ju-Chen (J-C) micromechanical method based on two types of radial distribution assumptions for inclusions in the matrix, namely the uniform distribution and Percus–Yevick (P–Y) distribution. A two-step approach was proposed and the elastic–viscoelastic correspondence principle was used to predict the effective dynamic modulus of asphalt mixture at different frequencies. The prediction results show that the uniform distribution and P–Y distribution based J-C method could generate the upper and lower bounds of dynamic modulus for asphalt mixture, respectively. As compared to the measured dynamic modulus at different temperatures and loading frequencies, the modified J-C method showed better prediction accuracy as compared to two traditional micromechanical models based on single inclusion configuration, Mori–Tanaka (M–T) and differential scheme effective medium models. The J-C method assuming P–Y distribution provided better accuracy at the low frequencies; while the J-C method assuming the uniform distribution only had good accuracy at the high frequencies. The study findings indicate that dynamic modulus of asphalt mixture can be predicted based on laboratory tests conducted at the fine aggregate mix level and the void ratio and the gradation of coarse aggregate using appropriate micromechanics methods.

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  1. 1.

    Benedetto HD, Partl MN, Francken L et al (2001) Stiffness testing for bituminous mixtures. Mater Struct 34(2):66–70

    Google Scholar 

  2. 2.

    Christensen DW, Pellinen T, Bonaquist RF (2003) Hirsch model for estimating the modulus of asphalt concrete. J Assoc Asphalt Paving Technol 72:97–121

    Google Scholar 

  3. 3.

    Tehrani FF, Absi J, Allou F, Petit C (2013) Heterogeneous numerical modeling of asphalt concrete through use of a biphasic approach: porous matrix/inclusions. Comput Mater Sci 69:186–196

    Google Scholar 

  4. 4.

    Karki P, Kim YR, Little DN (2015) Dynamic modulus prediction of asphalt concrete mixtures through computational micromechanics. Transp Res Rec 2507:1–9

    Google Scholar 

  5. 5.

    Chen JQ, Wang H, Li L (2017) Virtual testing of asphalt mixture with two-dimensional and three-dimensional random aggregate structures. Int J Pavement Eng 18(9):824–836

    Google Scholar 

  6. 6.

    Chen JQ, Wang H, Dan HC, Xie YJ (2018) Random modeling of three-dimensional heterogeneous microstructure of asphalt concrete for mechanical analysis. J Eng Mech 144(9):04018083

    Google Scholar 

  7. 7.

    Liu Y, Dai Q, You Z (2009) Viscoelastic model for discrete element simulation of asphalt mixtures. J Eng Mech 135(4):324–333

    Google Scholar 

  8. 8.

    Ma T, Wang H, Zhang D, Zhang Y (2017) Heterogeneity effect of mechanical property on creep behavior of asphalt mixture based on micromechanical modeling and virtual creep test. Mech Mater 104:49–59

    Google Scholar 

  9. 9.

    Eshelby JD (1957) The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc R Soc Lond A241:376–396

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Nemat-Nasser S, Hori M (1999) Micromechanics: overall properties of heterogeneous materials, 2nd edn. Elsevier, Amsterdam

    Google Scholar 

  11. 11.

    Mclaughlin R (1977) A study of the differential scheme for composite materials. Int J Eng Sci 15(4):237–244

    MATH  Google Scholar 

  12. 12.

    Mori T, Tanaka K (1973) Average stress in matrix and average of material with mismatchting inclusions. Acta Metall 21(5):571–574

    Google Scholar 

  13. 13.

    Kim YR, Little DN (2004) Linear viscoelastic analysis of asphalt mastics. J Mater Civ Eng 16(2):122–132

    Google Scholar 

  14. 14.

    Shashidhar N, Shenoy A (2002) On using micromechanical models to describe dynamic mechanical behavior of asphalt mastics. Mech Mater 34(10):657–669

    Google Scholar 

  15. 15.

    Yin HM, Buttlar WG, Paulino GH et al (2008) Assessment of existing micro-mechanical models for asphalt mastics considering viscoelastic effects. Road Mater Pavement Des 9(1):31–57

    Google Scholar 

  16. 16.

    Underwood BS, Kim YR (2011) Experimental investigation into the multiscale behaviour of asphalt concrete. Int J Pavement Eng 12(4):357–370

    Google Scholar 

  17. 17.

    Kim M, Buttlar WG (2011) Differential scheme effective medium theory for hot-mix asphalt |E*| prediction. J Mater Civ Eng 23(1):69–78

    Google Scholar 

  18. 18.

    Shu X, Huang B (2008) Micromechanics-based dynamic modulus prediction of polymeric asphalt concrete mixtures. Compos B Eng 39(4):704–713

    Google Scholar 

  19. 19.

    Pichler C, Lackner R, Aigner E (2012) Generalized self-consistent scheme for upscaling of viscoelastic properties of highly-filled matrix-inclusion composites—application in the context of multiscale modeling of bituminous mixtures. Compos B Eng 43(2):457–464

    Google Scholar 

  20. 20.

    Zhu X, Yang Z, Guo X et al (2011) Modulus prediction of asphalt concrete with imperfect bonding between aggregate and asphalt mastic. Compos B Eng 42(6):1404–1411

    Google Scholar 

  21. 21.

    Zhang J, Fan Z, Pei J, Li R, Chang M (2015) Multiscale validation of the applicability of micromechanical models for asphalt mixture. Adv Mater Sci Eng 2015(7):1–8

    Google Scholar 

  22. 22.

    Gao X, Fan Z, Zhang J, Liu S (2017) Micromechanical model for asphalt mixture coupling inter-particle effect and imperfect interface. Constr Build Mater 148:696–703

    Google Scholar 

  23. 23.

    Pei J, Fan Z, Wang P, Zhang J, Xue B, Li R (2015) Micromechanics prediction of effective modulus for asphalt mastic considering inter-particle interaction. Constr Build Mater 101:209–216

    Google Scholar 

  24. 24.

    Ju JW, Chen TM (1994) Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech 103:103–121

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Ju JW, Chen TM (1994) Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mech 103:123–144

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Ju J, Yanase K (2010) Micromechanics and effective elastic moduli of particle-reinforced composites with near-field particle interactions. Acta Mech 215(1–4):135–153

    MATH  Google Scholar 

  27. 27.

    Moon KH, Falchetto AC (2015) Microstructural investigation of hot mix asphalt (HMA) mixtures using digital image processing (DIP). KSCE J Civ Eng 19(6):1727–1737

    Google Scholar 

  28. 28.

    Falchetto AC, Moon KH, Wistuba MP (2014) Microstructural analysis and rheological modeling of asphalt mixtures containing recycled asphalt materials. Materials 7(9):321–329

    Google Scholar 

  29. 29.

    Wertheim MS (1963) Exact solution of the Percus–Yevick integral equation for hard spheres. Phys Rev Lett 10(8):321–323

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Thiele E (1963) Equation of State for Hard Spheres. J Chem Phys 39(2):474–479

    Google Scholar 

  31. 31.

    Throop GJ, Bearman RJ (1965) Numerical solutions of the Percus—Yevick equation for the hard-sphere potential. J Chem Phys 42(7):2408–2411

    MathSciNet  Google Scholar 

  32. 32.

    Trokhymchuk A, Nezbeda I, Jirsák J, Henderson D (2005) Hard-sphere radial distribution function again. J Chem Phys 123(2):1–10

    Google Scholar 

  33. 33.

    Yuan C, Sun Y (2004) Study on analytical expressions for the radial distribution function of hard spheres. J Sichuan Univ Nat Sci Edn 41(4):799–802

    Google Scholar 

  34. 34.

    Anderson DA, Bahia HU, Dongre R (1992) Rheological properties of mineral filler-asphalt mastics and its importance to pavement performance. ASTM, San Diego, pp 131–153

    Google Scholar 

  35. 35.

    Zhu X, Wang X, Yu Y (2014) Micromechanical creep models for asphalt-based multi-phase particle-reinforced composites with viscoelastic imperfect interface. Int J Eng Sci 76(76):34–46

    Google Scholar 

  36. 36.

    Chen JQ, Zhang M, Wang H, Li L (2015) Evaluation of thermal conductivity of asphalt mixture with heterogeneous microstructure. Appl Therm Eng 84:368–374

    Google Scholar 

  37. 37.

    Li Y, Metcalf JB (2005) Two-step approach to prediction of asphalt concrete modulus from two-phase micromechanical models. J Mater Civ Eng 17(4):407–415

    Google Scholar 

  38. 38.

    Chen JQ, Wang H, Li L (2015) Determination of effective thermal conductivity of asphalt concrete with random aggregate microstructure. J Mater Civ Eng 27(12):04015045

    Google Scholar 

  39. 39.

    Zhang YQ, Luo R, Lytton RL (2011) Anisotropic viscoelastic properties of undamaged asphalt mixtures. J Transp Eng 138(1):75–89

    Google Scholar 

  40. 40.

    Kim Y-R, Little DN, Lytton R (2003) Fatigue and healing characterization of asphalt mixtures. J Mater Civil Eng 15(1):75–83

    Google Scholar 

  41. 41.

    Nabizadeh H, Haghshenas HF, Kim YR et al (2017) Effects of rejuvenators on high-RAP mixtures based on laboratory tests of asphalt concrete (AC) mixtures and fine aggregate matrix (FAM) mixtures. Constr Build Mater 152:65–73

    Google Scholar 

  42. 42.

    Li G, Li Y, Metcalf JB et al (1999) Elastic modulus prediction of asphalt concrete. J Mater Civ Eng 11(3):236–241

    Google Scholar 

  43. 43.

    Hu J, Liu PF, Wang DW, Oeser M, Tan YQ (2016) Investigation on fatigue damage of asphalt mixture with different air-voids using microstructural analysis. Constr Build Mater 125:936–945

    Google Scholar 

  44. 44.

    Souza LT, Kim YR, Souza FV, Castro LS (2012) Experimental testing and finite-element modeling to evaluate the effects of aggregate angularity on bituminous mixture performance. J Mater Civ Eng 24(3):249–258

    Google Scholar 

  45. 45.

    Wang H, Wang J, Chen JQ (2018) Fracture modeling of asphalt concrete with random aggregate microstructure. Road Mater Pavement Des 17(9):1674–1691

    Google Scholar 

  46. 46.

    Seo Y, El-Haggan O, King M, Lee SJ, Kim YR (2007) Air void models for the dynamic modulus, fatigue cracking, and rutting of asphalt concrete. J Mater Civ Eng 19(10):874–883

    Google Scholar 

  47. 47.

    Salemi M, Wang H (2018) Image-aided random aggregate packing for computational modeling asphalt concrete microstructure. Constr Build Mater 177:467–476

    Google Scholar 

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This research was partially supported by China Postdoctoral Science Foundation [Grant Number 2017M620434].

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Correspondence to Hao Wang.

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Zhang, J., Fan, Z., Wang, H. et al. Prediction of dynamic modulus of asphalt mixture using micromechanical method with radial distribution functions. Mater Struct 52, 49 (2019).

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  • Asphalt mixture
  • Micromechanical method
  • Two-step approach
  • Inter-particle effect
  • Radial distribution function

Mathematical subject classification

  • 74E30