Journal of Statistical Physics

, Volume 162, Issue 1, pp 232–241

Elasticity of Random Multiphase Materials: Percolation of the Stiffness Tensor



Topology and percolation effects play an important role in heterogeneous materials, but have rarely been studied for higher-order tensor properties. We explore the effective elastic properties of random multiphase materials using a combination of continuum computational simulations and analytical theories. The effective shear and bulk moduli of a class of symmetric-cell random composites with high phase contrasts are determined, and reveal shortcomings of classical homogenization theories in predicting elastic properties of percolating systems. The effective shear modulus exhibits typical percolation behavior, but with its percolation threshold shifting with the contrast in phase bulk moduli. On the contrary, the effective bulk modulus does not exhibit intrinsic percolation but does show an apparent or extrinsic percolation transition due to cross effects between shear and bulk moduli. We also propose an empirical approach for bridging percolation and homogenization theories and predicting the effective shear and bulk moduli in a manner consistent with the simulations.


Percolation Homogenization theory Elastic theory  Effective elastic moduli Random multiphase materials Composite materials 


  1. 1.
    Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)MATHCrossRefGoogle Scholar
  2. 2.
    Sahimi, M.: Heterogeneous Materials 1: Linear Transport and Optical Properties. Springer, New York (2003)MATHGoogle Scholar
  3. 3.
    Kim, I.C., Torquato, S.: Effective conductivity of suspensions of hard spheres by Brownian motion simulation. J. Appl. Phys. 69, 2280–2289 (1991)CrossRefADSGoogle Scholar
  4. 4.
    Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Choy, T.C.: Effective Medium Theory. Clarendon Press, Oxford (1999)Google Scholar
  6. 6.
    Kantor, Y., Webman, I.: Elastic properties of random percolating systems. Phys. Rev. Lett. 52, 1891–1894 (1984)CrossRefADSGoogle Scholar
  7. 7.
    Moukarzel, C., Duxbury, P.M.: Stressed backbone and elasticity of random central-force systems. Phys. Rev. Lett 75, 4055–4058 (1995)CrossRefADSGoogle Scholar
  8. 8.
    Latva-Kokko, M., Timonen, J.: Rigidity of random networks of stiff fibers in the low-density limit. Phys. Rev. E. 64, 066117 (2001)CrossRefADSGoogle Scholar
  9. 9.
    Zhou, Z.C., Joos, B., Lai, P.Y.: Elasticity of randomly diluted central force networks under tension. Phys. Rev. E. 68, 055101 (2003)CrossRefADSGoogle Scholar
  10. 10.
    Tighe, B.P., Socolar, J.E.S., Schaeffer, D.G., Mitchener, W.G., Huber, M.L.: Force distributions in a triangular lattice of rigid bars. Phys. Rev. E 72, 031306 (2005)CrossRefADSGoogle Scholar
  11. 11.
    Jacobs, D.J., Thorpe, M.F.: Generic rigidity percolation: the pebble game. Phys. Rev. Lett. 75, 4051–4054 (1995)CrossRefADSGoogle Scholar
  12. 12.
    Chen, Y., Schuh, C.A.: Percolation of diffusional creep: a new universality class. Phys. Rev. Lett. 98, 035701 (2007)CrossRefADSGoogle Scholar
  13. 13.
    Chen, Y., Schuh, C.A.: Coble creep in heterogeneous materials: the role of grain boundary engineering. Phys. Rev. B 76, 064111 (2007)CrossRefADSGoogle Scholar
  14. 14.
    Drory, A.: Theory of continuum percolation.1. General formalism. Phys. Rev. E 54, 5992–6002 (1996)MathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Drory, A.: Theory of continuum percolation.2. Mean field theory. Phys. Rev. E 54, 6003–6013 (1996)MathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Golden, K.M.: Critical behavior of transport in lattice and continuum percolation models. Phys. Rev. Lett. 78, 3935–3938 (1997)MATHMathSciNetCrossRefADSGoogle Scholar
  17. 17.
    Baker, D.R., Paul, G., Sreenivasan, S., Stanley, H.E.: Continuum percolation threshold for interpenetrating squares and cubes. Phys. Rev. E 66, 046136 (2002)CrossRefADSGoogle Scholar
  18. 18.
    Hunt, A.G.: Continuum percolation theory for transport properties in porous media. Philos. Mag. 85, 3409–3434 (2005)CrossRefADSGoogle Scholar
  19. 19.
    Grimaldi, C., Balberg, I.: Tunneling and nonuniversality in continuum percolation systems. Phys. Rev. Lett. 96, 066602 (2006)CrossRefADSGoogle Scholar
  20. 20.
    Akagawa, S., Odagaki, T.: Geometrical percolation of hard-core ellipsoids of revolution in the continuum. Phys. Rev. E 76, 051402 (2007)CrossRefADSGoogle Scholar
  21. 21.
    Stevens, D.R., Downen, L.N., Clarke, L.I.: Percolation in nanocomposites with complex geometries: experimental and Monte Carlo simulation studies. Phys. Rev. B 78, 235425 (2008)CrossRefADSGoogle Scholar
  22. 22.
    Snarskii, A.A., Zhenirovskyy, M.I.: Double-threshold percolation behavior of effective kinetic coefficients. Phys. Rev. E 78, 021108 (2008)CrossRefADSGoogle Scholar
  23. 23.
    Sangare, D., Adler, P.M.: Continuum percolation of isotropically oriented circular cylinders. Phys. Rev. E 79, 052101 (2009)CrossRefADSGoogle Scholar
  24. 24.
    Balberg, I., Binenbaum, N., Anderson, C.H.: Critical behavior of the two-dimensional sticks system. Phys. Rev. Lett. 51, 1605 (1983)CrossRefADSGoogle Scholar
  25. 25.
    Feng, S., Halperin, B.I., Sen, P.N.: Transport properties of continuum systems near the percolation threshold. Phys. Rev. B 35, 197 (1987)CrossRefADSGoogle Scholar
  26. 26.
    Halperin, B.I., Feng, S., Sen, P.N.: Differences between lattice and continuum percolation transport exponents. Phys. Rev. Lett. 54, 2391–2394 (1985)CrossRefADSGoogle Scholar
  27. 27.
    Zhu, J., Jabini, A., Golden, K.M., Eicken, H., Morris, M.: A network model for fluid transport through sea ice. Ann. Glaciol. 44, 129–133 (2006)CrossRefADSGoogle Scholar
  28. 28.
    Chen, Y., Schuh, C.A.: Effective transport properties of random composites: continuum calculations versus mapping to a network. Phys. Rev. E 80, 040103 (2009)CrossRefADSGoogle Scholar
  29. 29.
    Pecullan, S., Gibiansky, L.V., Torquato, S.: Scale effects on the elastic behavior of periodic and hierarchical two-dimensional composites. J. Mech. Phys. Solids 47, 1509–1542 (1999)MATHMathSciNetCrossRefADSGoogle Scholar
  30. 30.
    Hazanov, S.: Hill condition and overall properties of composites. Arch. Appl. Mech. 68, 385–394 (1998)MATHCrossRefADSGoogle Scholar
  31. 31.
    Hashin, Z.: On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids 13, 119–134 (1965)CrossRefADSGoogle Scholar
  32. 32.
    Budiansky, B.: On the elastic moduli of some heterogeneous materials. J. Mech. Phys. Solids 13, 223–227 (1965)CrossRefADSGoogle Scholar
  33. 33.
    Eshelby, J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. London Ser. A 241, 376–396 (1957)MATHMathSciNetCrossRefADSGoogle Scholar
  34. 34.
    Chen, Y., Schuh, C.A.: Diffusion on grain boundary networks: percolation theory and effective medium approximations. Acta Mater. 54, 4709–4720 (2006)CrossRefGoogle Scholar
  35. 35.
    McLachlan, D.S., Chiteme, C., Heiss, W.D., Wu, J.J.: Fitting the DC conductivity and first order AC conductivity results for continuum percolation media, using percolation theory and a single phenomenological equation. Physica B 338, 261–265 (2003)CrossRefADSGoogle Scholar
  36. 36.
    Stauffer, D., Aharony, A.: Introduction to Percolation Theory. Taylor & Francis, London (1992)MATHGoogle Scholar
  37. 37.
    Clerc, J.P., Giraud, G., Laugier, J.M., Luck, J.M.: The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models. Adv. Phys. 39, 191–309 (1990)CrossRefADSGoogle Scholar
  38. 38.
    Hill, R.: Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour. J. Mech. Phys. Solids 12, 199–212 (1964)MathSciNetCrossRefADSGoogle Scholar
  39. 39.
    Hill, R.: Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11, 357–372 (1963)MATHCrossRefADSGoogle Scholar
  40. 40.
    Thorpe, M.F., Jasiuk, I.: New results in the theory of elasticity for two-dimensional composites. Proc. R. Soc. London Ser. A 438, 531–544 (1992)MATHCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Materials Science and EngineeringRensselaer Polytechnic InstituteTroyUSA
  2. 2.Department of Materials Science and EngineeringMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations