Skip to main content
SpringerLink
Log in

Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities

  • Contributed Papers
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

Based on the general micromechanical framework proposed in a companion paper, effective elastic moduli of two-phase composites containing randomly dispersedspherical inhomogeneities are investigated in this paper. At variance with existing micromechanical pairwise interaction models (accurate up to the second-order in particle volume fraction ϕ), the proposed approximate, probabilistic pairwise particle interaction formulationcoupled with the general ensemble-volume averaged field equations leads to a novel, higher-order (in ϕ), and accurate method for the prediction of effective elastic moduli of two-phase composites containing randomly located spherical particles. The relevant ensemble integrals in the proposed formulation are absolutely convergent due to a “renormalization” procedure employed in a companion paper. In accordance with the analogy between the effective shear modulus of an incompressible elastic composite with randomly dispersed rigid spheres and the effective shear viscosity of a colloidal dispersion with randomly dispersed rigid spheres (at high shear rates), the proposed ensemble-micromechanical approach is extended to predict effective shear viscosities of colloidal dispersions at the high-shear limit. Comparisons with experimental data, classical variational bounds, improved three-point bounds, the second-order particle interaction model, and other micromechanical models are also presented. It is observed that significant improvement in predictive capability for two-phase composites with randomly dispersed spheres can be achieved by using the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ju, J. W., Chen, T. M.: Micromechanics and effective moduli of elastic composites containing randomly dispersed ellipsoidal inhomogeneities. Acta Mech.103, 103–121 (1994).

    Google Scholar 

  2. Einstein, A.: Eine neue Bestimmung der Molekül-Dimensionen. Annalen der Physik19, 289–306, (1906); and (errata)34, 591–592 (1911). English translation in: Investigations on the theory of Brownian motion 36–62. Dover 1956.

    Google Scholar 

  3. Dewey, J. M.: The elastic constants of materials loaded with non-rigid fillers. J. Appl. Mech.18, 578–581 (1947).

    Google Scholar 

  4. Kerner, E. H.: The elastic and thermo-elastic properties of composite media. Proc. R. Soc. London Ser.B 69, 807–808 (1956).

    Google Scholar 

  5. 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).

    Google Scholar 

  6. Hashin, Z.: The moduli of an elastic solid, containing spherical particles of another elastic material. In: IUTAM Non-homogeneity in Elasticity and Plasticity Symposium (Olszak, W., ed.), pp. 463–478, Warsaw 1959.

  7. Batchelor, G. K., Green, J. T.: The determination of the bulk stress in a suspension of spherical particles to orderc 2. J. Fluid Mech.56, 401–427 (1972).

    Google Scholar 

  8. Batchelor, G. K.: Transport properties of two-phase materials with random structure. Ann. Rev. Fluid Mech.6, 227–255 (1974).

    Google Scholar 

  9. Batchelor, G. K.: Development in microhydrodynamics. In: Theoretical and applied mechanics (Koiter, W., ed.). Amsterdam: North-Holland 1976.

    Google Scholar 

  10. Jeffrey, D. J., Acrivos, A.: The rheological properties of suspensions of rigid particles. AIChE J.22, 417–432 (1976).

    Google Scholar 

  11. Russel, W. B.: A review of the role of colloidal forces in the rheology of suspensions. J. Rheol.24, 287–317 (1980).

    Google Scholar 

  12. Willis, J. R., Acton, J. R.: The overall elastic moduli of a dilute suspension of spheres. Q. J. Mech. Appl. Math.29, 163–177 (1976).

    Google Scholar 

  13. Chen, H.-S., Acrivos, A.: The solution of the equations of linear elasticity for an infinite region containing two spherical inclusions. Int. J. Solids Struct.14, 331–348 (1978).

    Google Scholar 

  14. Chen, H.-S., Acrivos, A.: The effective elastic moduli of composite materials containing spherical inclusions at non-dilute concentrations. Int. J. Solids Struct.14, 349–364 (1978).

    Google Scholar 

  15. Ju, J. W., Chen, T. M.: On effective elastic moduli of two-dimensional brittle solids with interacting microcracks. Part I: Basic formulations, J. Appl. Mech. (in press).

  16. Ju, J. W., CHen, T. M.: On effective elastic moduli of two-dimensional brittle solids with interacting microcracks. Part II: Evolutionary damage models. J. Appl. Mech. (in press).

  17. Ju, J. W., Tseng, K. H.: A three-dimensional statistical micromechanical theory for brittle solids with interacting microcracks. Int. J. Damage Mech.1, 102–131 (1992).

    Google Scholar 

  18. Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing periodically distributed voids. Q. Appl. Math.39, 43–59 (1981).

    Google Scholar 

  19. Nemat-Nasser, S., Taya, M.: On effective moduli of an elastic body containing periodically distributed voids: comments and corrections. Q. Appl. Math.43, 187–188 (1985).

    Google Scholar 

  20. Nemat-Nasser, S., Iwakuma, T., Hejazi, M.: On composite with periodic structure. Mech. Mater.1, 239–267 (1982).

    Google Scholar 

  21. Zuzovsky, M., Alder, P. M., Brenner, H.: Spatially periodic suspensions of convex particles in linear shear flows. III. Dilute arrays of spheres suspended in Newtonian fluids. Phys. Fluids26, 1714–1723 (1983).

    Google Scholar 

  22. Iwakuma, T., Nemat-Nasser, S.: Composites with periodic microstructure. Comp. Struct.16, 13–19 (1983).

    Google Scholar 

  23. Nunan, K. C., Keller, J. B.: Effective viscosity of a periodic suspension. J. Fluid Mech.142, 269–287 (1984).

    Google Scholar 

  24. Nunan, K. C., Keller, J. B.: Effective elasticity tensor of a periodic composite. J. Mech. Phys. Solids32, 259–280 (1984).

    Google Scholar 

  25. Adler, P. M., Zuzovsky, M., Brenner, H.: Spatially periodic suspensions of convex particles in liner shear flow. II. Rheology. Int. J. Multiphase Flow11, 387–417 (1985).

    Google Scholar 

  26. Sangani, A. S., Lu, W.: Elastic coefficients of composites containing spherical inclusions in a periodic array. J. Mech. Phys. Solids35, 1–21 (1987).

    Google Scholar 

  27. Bossis, G., Brady, J. F.: Self-diffusion of Brownian particles in concentrated suspensions under shear. J. Chem. Phys.87, 5437–5448 (1987).

    Google Scholar 

  28. Brady, J. F., Bossis, G.: Stokesian dynamics. Ann. Rev. Fluid Mech.20, 111–157 (1988).

    Google Scholar 

  29. Beenakker, C. W. J.: The effective viscosity of a concentrated suspension of spheres (and its relation to diffusion). Physica128 A, 48–81 (1984).

    Google Scholar 

  30. Rodin, G. J., Hwang, Y.-L.: On the problem of linear elasticity for an infinite region containing a finite number of non-intersecting spherical inhomogeneities. Int. J. Solids Struct.27, 145–159 (1991).

    Google Scholar 

  31. Phillips, R. J., Brady, J. F., Bossis, G.: Hydrodynamic transport properties of hard-sphere dispersions. I. Suspensions of freely mobile particles. Phys. Fluids31, 3462–3472 (1988).

    Google Scholar 

  32. Krieger, I. M.: Rheology of monodisperse lattices. Adv. Colloid. Interface Sci.3, 111–136 (1972).

    Google Scholar 

  33. de Kruif, C. G., van Iersel, E. M. F., Vrij, A., Russel, W. B.: Hard sphere colloidal dispersions: viscosity as a function of shear rate and volume fraction. J. Chem. Phys.83, 4717–4725 (1985).

    Google Scholar 

  34. Mura, T.: Micromechanics of defects in solids. The Hague: Martinus Nijhoff 1982.

    Google Scholar 

  35. Hansen, J. P., McDonald, I. R.: Theory of simple liquids. New York: Academic Press 1986.

    Google Scholar 

  36. Verlet, L., Weis, J.-J.: Equilibrium theory of simple liquids. Phys. Rev. A5, 939–952 (1972).

    Google Scholar 

  37. Sen, A. K., Torquato, S.: Effective conductivity of anisotropic two-phase composite media. Phys. Rev. B39, 4504–4515 (1989).

    Google Scholar 

  38. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids11, 127–140 (1963).

    Google Scholar 

  39. Walpole, L. J.: The elastic behavior of a suspension of spherical particles. Q. J. Mech. Appl. Math.25, 153–160 (1972).

    Google Scholar 

  40. Kim, S., Mifflin, R. T.: The resistance and mobility functions of two equal spheres in low-Reynolds-number flow. Phys. Fluids28 2033–2045 (1985).

    Google Scholar 

  41. Yoon, B. J., Kim, S.: Note on the direct calculation of mobility functions for two equal-size spheres in Stokes flow. J. Fluid Mech.185, 437–446 (1987).

    Google Scholar 

  42. Richard, T. G.: The mechanical behavior of a solid microsphere filled composite. J. Comp. Mat.9, 108–113 (1975).

    Google Scholar 

  43. Smith, J. C.: Experimental values for the elastic constants of a particulate-filled glassy polymer. J. Res. NBS80 A, 45–49 (1976).

    Google Scholar 

  44. McCoy, J. J.: On the displacement field in an elastic medium with random variations of material properties. Recent Advances in Engineering Sciences vol. 5. New York: Gordon and Breach 1970.

    Google Scholar 

  45. Silnutzer, N.: Effective constants of statistically homogeneous materials. Ph.D. Thesis, Univ. of Pennsylvania 1972.

  46. Milton, G. W.: Bounds on the electromagnetic, elastic, and other properties of two-component composites. Phys. Rev. Lett.46, 542–545 (1981).

    Google Scholar 

  47. Milton, G. W.: Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids30, 177–191 (1982).

    Google Scholar 

  48. Milton, G. W., Phan-Thien, N.: New bounds on effective elastic moduli of two-components materials. Proc. R. Soc. London Ser.A 380, 305–331 (1982).

    Google Scholar 

  49. Torquato, S., Lado, F.: Effective properties of two-phase disordered composite media. II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres. Phys. Rev. B33, 6428–6434 (1986).

    Google Scholar 

  50. Sen, A. K., Lado, F., Torquato, S.: Bulk properties of composite media. II. Evaluation of bounds on the shear moduli of suspensions of impenetrable spheres. J. Appl. Phys.62, 4135–4141 (1987).

    Google Scholar 

  51. Walsh, J. B., Brace, W. F., England, A. W.: Effect of porosity on compressibility of glass. J. Am. Ceram. Soc.48, 605–608 (1965).

    Google Scholar 

  52. Beran, M. J., Molyneux, J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Q. Appl. Math.24, 107–118 (1966).

    Google Scholar 

  53. Russel, W. B., Saville, D. A., Schowalter, W. R.: Colloidal dispersions. Cambridge: Cambridge University Press 1989.

    Google Scholar 

  54. Freed, K. F., Muthukumar, M.: Cluster theory for concentration dependence of shear viscosity for suspensions of interacting spheres. I. J. Chem. Phys.76, 6186–6184 (1982).

    Google Scholar 

  55. Brady, J. F., Bossis, G.: The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J. Fluid Mech.155, 105–129 (1985).

    Google Scholar 

Download references

Author information

Author notes

  1. J. W. Ju & T. M. Chen

    Present address: Department of Civil Engineering and Operations Reserach, Princeton University, 08 544, Princeton, NJ, USA

Authors and Affiliations

    Authors
    1. J. W. Ju
      View author publications

      You can also search for this author in PubMed Google Scholar

    2. T. M. Chen
      View author publications

      You can also search for this author in PubMed Google Scholar

    Rights and permissions

    Reprints and permissions

    About this article

    Cite this article

    Ju, J.W., Chen, T.M. Effective elastic moduli of two-phase composites containing randomly dispersed spherical inhomogeneities. Acta Mechanica 103, 123–144 (1994). https://doi.org/10.1007/BF01180222

    Download citation

    • Received:

    • Revised:

    • Issue Date:

    • DOI: https://doi.org/10.1007/BF01180222

    Keywords

    Search

    Navigation