Continuum Mechanics and Thermodynamics

, Volume 31, Issue 3, pp 823–840 | Cite as

Homogenization estimates for the effective response of fractional viscoelastic particulate composites

  • Valentin Gallican
  • Renald BrennerEmail author
Original Article


This article is devoted to the micromechanical modelling of the time harmonic response of viscoelastic composites made of fractional Zener constituents. By extending previous results in classical viscoelasticity, new exact relations on time integrals of the effective relaxation spectrum are obtained. They are related to the intraphase second moments of the strain field in the asymptotic elastic regimes at low and high frequencies. Based on these relations, the effective relaxation spectrum is approximated by a sum of two weighted Dirac delta functions. An attractive feature of this viscoelastic homogenization model is that it only involves the resolution of two elastic homogenization problems. This model is applied to estimate the response of particle-reinforced two-phase composites. Its relevance is assessed by performing comparisons with FFT full-field simulations for distributions of polydisperse spherical particles.


Homogenization Particulate composite Fractional viscoelasticity Polymer material 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



  1. 1.
    Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27, 201–210 (1983)ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Beurthey, S., Zaoui, A.: Structural morphology and relaxation spectra of viscoelastic heterogeneous materials. Eur. J. Mech. A Solids 19, 1–16 (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Blair, G.W.S., Caffyn, J.: The classification of the rheological properties of industrial materials in the light of power-law relations between stress, strain and time. J. Sci. Instrum. 19(6), 88 (1942)ADSCrossRefGoogle Scholar
  4. 4.
    Bobeth, M., Diener, G.: Field fluctuations in multicomponent mixtures. J. Mech. Phys. Solids 34, 1–17 (1986)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Brenner, R., Masson, R.: Improved affine estimates for nonlinear viscoelastic composites. Eur. J. Mech. A Solids 24, 1002–1015 (2005)ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Brenner, R., Suquet, P.: Overall response of viscoelastic composites and polycristals: exact asymptotic relations and approximate estimates. Int. J. Solids Struct. 50, 1824–1838 (2013)CrossRefGoogle Scholar
  7. 7.
    Caputo, M., Mainardi, F.: Linear models of dissipation in anelastic solids. Rivista del nuovo cimento 1, 161–198 (1971)ADSCrossRefGoogle Scholar
  8. 8.
    Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Cherkaev, A.V., Gibiansky, L.V.: Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys. 35, 127–145 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Christensen, R.M.: A critical evaluation for a class of micromechanics models. J. Mech. Phys. Solids 38, 379–404 (1990)ADSCrossRefGoogle Scholar
  11. 11.
    Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27, 315–330 (1979)ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Dinzart, F., Lipinski, P.: Self-consistent approach of the constitutive law of a two-phase visco-elastic material described by fractional derivative models. Arch. Mech. 62, 135–156 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dubois, F., Galucio, A.C., Point, N.: Introduction à la dérivation fractionnaire. Théorie et applications. Techniques de l’Ingenieur (2010)Google Scholar
  14. 14.
    Eyre, D., Milton, G.W., Lakes, R.S.: Bounds for interpolating complex effective moduli of viscoelastic materials from measured data. Rheol. Acta 41, 461–470 (2002)CrossRefGoogle Scholar
  15. 15.
    Eyre, D.J., Milton, G.W.: A fast numerical scheme for computing the response of composites using grid refinement. J. Phys. III 6, 41–47 (1999)Google Scholar
  16. 16.
    Ferry, J.D.: Viscoelastic Properties of Polymers. Wiley, Hoboken (1980)Google Scholar
  17. 17.
    Figliuzzi, B., Jeulin, D., Faessel, M., Willot, F., Koishi, M., Kowatari, N.: Modelling the microstructure and the viscoelastic behaviour of carbon black filled rubber materials from 3D simulations. Technische Mechanik 32(1–2), 22–46 (2016)Google Scholar
  18. 18.
    Francfort, G., Leguillon, D., Suquet, P.: Homogénéisation de milieux viscoélastiques linéaires de Kelvin–Voigt. C.R. Acad. Sci. Paris 296, 287–290 (1983)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Gallican, V., Brenner, R., Suquet, P.: Exact asymptotic relations for the effective response of linear viscoelastic heterogeneous media. C.R. Mecanique 345, 742–751 (2017)ADSCrossRefGoogle Scholar
  20. 20.
    Gemant, A.: A method of analyzing experimental results obtained from elasto-viscous bodies. Physics 7, 311–317 (1936)ADSCrossRefGoogle Scholar
  21. 21.
    Gemant, A.: On fractional differentials. Philos. Mag. 25, 540–549 (1938)CrossRefzbMATHGoogle Scholar
  22. 22.
    Ghossein, E., Lévesque, M.: A fully automated numerical tool for a comprehensive validation of homogenization models and its application to spherical particles reinforced composites. Int. J. Solids Struct. 49, 1387–1398 (2012)CrossRefGoogle Scholar
  23. 23.
    Gibiansky, L.V., Milton, G.W.: On the effective viscoelastic moduli of two-phase media. I. Rigorous bounds on the complex bulk modulus. Proc. R. Soc. Lond. A440, 163–188 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics. CISM Courses and Lectures, vol. 378, pp. 223–276. Springer-Verlag Wien GmbH (1997)Google Scholar
  25. 25.
    Gusev, A.A.: Time domain finite element estimates of dynamic stiffness of viscoelastic composites with stiffspherical inclusions. Int. J. Solids Struct. 88–89, 79–87 (2016)CrossRefGoogle Scholar
  26. 26.
    Hashin, Z.: Viscoelastic behavior of heterogeneous media. J. Appl. Mech. 32, 630–636 (1965)ADSCrossRefGoogle Scholar
  27. 27.
    Hashin, Z.: Complex moduli of viscoelastic composites—I. General theory and application to particulate composites. Int. J. Solids Struct. 6, 539–552 (1970)CrossRefzbMATHGoogle Scholar
  28. 28.
    Hervé, E., Zaoui, A.: Modelling the effective behavior of nonlinear matrix-inclusion composites. Eur. J. Mech. A Solids 9, 505–515 (1990)zbMATHGoogle Scholar
  29. 29.
    Heymans, N., Bauwens, J.C.: Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta 33, 210–219 (1994)CrossRefGoogle Scholar
  30. 30.
    Hill, R.: A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13(4), 213–222 (1965)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Koeller, R.C.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Koeller, R.C.: Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech. 58, 251–264 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kreher, W.: Residual stresses and stored elastic energy of composites and polycrystals. J. Mech. Phys. Solids 38, 115–128 (1990)ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    Laws, N., McLaughlin, R.: Self-consistent estimates for the viscoelastic creep compliances of composite materials. Proc. R. Soc. Lond. A359, 251–273 (1978)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Lion, A.: On the thermodynamics of fractional damping elements. Contin. Mech. Thermodyn. 9, 83–96 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)CrossRefzbMATHGoogle Scholar
  37. 37.
    Mainardi, F.: On some properties of the Mittag-Leffler function \({E}_\alpha (-t^\alpha )\), completely monotone for \(t>0\) with \(0<\alpha <1\). Discrete Cont. Dyn. B 19, 2267–2278 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mandel, J.: Cours de mécanique des milieux continus. Gauthier-Villars Editeur, Paris (1966)zbMATHGoogle Scholar
  39. 39.
    Masson, R., Brenner, R., Castelnau, O.: Incremental homogenization approach for ageing viscoelastic polycrystals. C. R. Mecanique 340, 378–386 (2012)ADSCrossRefGoogle Scholar
  40. 40.
    Masson, R., Zaoui, A.: Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials. J. Mech. Phys. Solids 47, 1543–1568 (1999)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Milton, G.W., Berryman, J.G.: On the effective viscoelastic moduli of two-phase media. II. Rigorous bounds on the complex shear modulus in three dimensions. Proc. R. Soc. Lond. A 453, 1849–1880 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mori, T., Tanaka, K.: Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574 (1973)CrossRefGoogle Scholar
  43. 43.
    Moulinec, H., Silva, F.: Comparison of three accelerated FFT-based schemes for computing the mechanical response of composite materials. Int. J. Num. Methods Eng. 97, 960–985 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Moulinec, H., Suquet, P.: A numerical method for computing the overall response of nonlinear composites with complex microstructures. Comput. Methods Appl. Mech. Eng. 157, 69–94 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nutting, P.G.: A new general law of deformation. J. Frankl. Inst. 191, 679–685 (1921)CrossRefGoogle Scholar
  46. 46.
    Oldham, K.B., Spanier, J.: The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. Academic Press, Cambridge (1974)zbMATHGoogle Scholar
  47. 47.
    Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1999)zbMATHGoogle Scholar
  48. 48.
    Ponte Castañeda, P., Suquet, P.: Nonlinear composites. Adv. Appl. Mech. 34, 171–302 (1998)CrossRefzbMATHGoogle Scholar
  49. 49.
    Rekik, A., Brenner, R.: Optimization of the collocation inversion method for the linear viscoelastic homogenization. Mech. Res. Commun. 38, 305–308 (2011)CrossRefzbMATHGoogle Scholar
  50. 50.
    Remillat, C.: Damping mechanism of polymers filled with elastic particles. Mech. Mater. 39, 525–537 (2007)CrossRefGoogle Scholar
  51. 51.
    Ricaud, J.M., Masson, R.: Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours. Int. J. Solids Struct. 46, 1599–1606 (2009)CrossRefzbMATHGoogle Scholar
  52. 52.
    Sanchez-Hubert, J., Sanchez-Palencia, E.: Sur certains problèmes physiques d’homogénéisation donnant lieu à des phénomènes de relaxation. C. R. Acad. Sci. Paris Série A 286, 903–906 (1978)zbMATHGoogle Scholar
  53. 53.
    Schapery, R.A.: Viscoelastic behavior and analysis of composite materials. In: Sendeckyj, G.P. (ed.) Mechanics of Composite Materials, vol. 2, pp. 85–167. Academic Press, New York (1974)Google Scholar
  54. 54.
    Schapery, R.A.: Linear elastic and viscoelastic deformation behaviour of ice. J. Cold Reg. Eng. 11, 271–290 (1997)CrossRefGoogle Scholar
  55. 55.
    Schiessel, H., Blumen, A.: Hierarchical analogues to fractional relaxation equations. J. Phys. A: Math. Gen. 26, 5057–5069 (1993)ADSCrossRefGoogle Scholar
  56. 56.
    Sevostianov, I., Levin, V., Radi, E.: Effective properties of linear viscoelastic microcracked materials: application of Maxwell homogenization scheme. Mech. Mater. 84, 28–43 (2015)CrossRefGoogle Scholar
  57. 57.
    Suquet, P.: Elements of homogenization for inelastic solid mechanics. In: Homogenization Techniques for Composite Media. Lecture Notes in Physics, vol. 272, pp. 194–278. Springer-Verlag (1987)Google Scholar
  58. 58.
    Suquet, P.: Four exact relations for the effective relaxation function of linear viscoelastic composites. Comptes Rendus Mecanique 340(4), 387–399 (2012). (Recent Advances in Micromechanics of Materials)ADSCrossRefGoogle Scholar
  59. 59.
    Torquato, S.: Random Heterogeneous Materials. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  60. 60.
    Tschoegl, N.: The Phenomenological Theory of Linear Viscoelastic Behavior—An Introduction. Springer, Berlin (1989)CrossRefzbMATHGoogle Scholar
  61. 61.
    Turner, P.A., Tomé, C.N.: Self-consistent modeling of visco-elastic polycrystals: application to irradiation creep and growth. J. Mech. Phys. Solids 41(7), 1191–1211 (1993)ADSCrossRefzbMATHGoogle Scholar
  62. 62.
    Widom, B.: Random sequential addition of hard spheres to a volume. J. Chem. Phys. 44, 3888–3894 (1966)ADSCrossRefGoogle Scholar
  63. 63.
    Willis, J.R.: Variational and related methods for the overall properties of composites. Adv. Appl. Mech. 21, 2–78 (1981)ADSMathSciNetzbMATHGoogle Scholar
  64. 64.
    Yin, D., Wu, H., Cheng, C., Chen, Y.Q.: Fractional order constitutive model of geomaterials under the condition of triaxial test. Int. J. Numer. Anal. Met. 37, 961–972 (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne UniversitéCNRS, Institut Jean le Rond d’AlembertParisFrance

Personalised recommendations