Skip to main content
SpringerLink
Log in

Estimates for the elastic moduli of d-dimensional random cell polycrystals

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Minimum energy principles and generalized polarization trial fields are used to derive bounds on the effective elastic moduli of d-dimensional random cell polycrystals, which are extensions of our earlier 3D (3-dimensional) results. The bounds are specialized to the 2D random aggregates of square-symmetric crystals, with numerical results for a number of particular crystals. Numerical finite element simulations for sufficiently large random aggregate samples of a particular 2D random polycrystal model show convergence toward the possible scatter interval for the effective elastic moduli enveloped by the new bounds, which are tighter than the classical Voigt–Reuss–Hill and Hashin–Shtrikman ones.

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. Pham, D.C.: Elastic moduli of perfectly-random polycrystalline aggregates. Philos. Mag. A 76, 31–44 (1997)

    Article  Google Scholar 

  2. Pham, D.C.: On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals. Philos. Mag. 91, 609–627 (2011)

    Article  Google Scholar 

  3. Hill, R.: The elastic behaviour of a crystalline aggregate. Proc. Phys. Soc. A 65, 349–354 (1952)

    Article  Google Scholar 

  4. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of polycrystals. J. Mech. Phys. Solids 10, 343–352 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zeller, R., Dederichs, P.H.: Elastic constants of polycrystals. Phys. Status Solidi B55, 831–842 (1973)

    Article  Google Scholar 

  6. Sermergor, T.D.: Theory of Elasticity of Micro-inhomogeneous Media. Nauka, Moscow (1977)

    Google Scholar 

  7. Williemse, M.W.M., Caspers, W.J.: Electrical conductivity of polycrystalline materials. J. Math. Phys. 20, 1824–1831 (1979)

    Article  Google Scholar 

  8. Kröner, E.: Graded and perfect disorder in random media elasticity. J. Eng. Mech. Div. 106, 889–914 (1980)

    Google Scholar 

  9. Watt, J.P.: Hashin–Shtrikman bounds on the effective elastic moduli of polycrystals with orthorhombic symmetry. J. Appl. Phys. 50, 6290–6294 (1979)

    Article  Google Scholar 

  10. McCoy, J.J.: Macroscopic response of continua with random microstructure. In: Nemat-Nasser, S. (ed.) Mechanics Today, vol. 6, pp. 1–40. Pergamon Press, New York (1981)

  11. Pham, D.C.: Bounds on the effective shear modulus of multiphase materials. Int. J. Eng. Sci. 31, 11–17 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pham, D.C.: New estimates for macroscopic elastic moduli of random polycrystalline aggregates. Philos. Mag. 86, 205–226 (2006)

    Google Scholar 

  13. Pham, D.C.: Revised bounds on the elastic moduli of two-dimensional random polycrystals. J. Elast. 85, 1–20 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pham, D.C.: Bounds on the elastic moduli of statistically isotropic multicomponent materials and random cell polycrystals. Int. J. Solids Struct. 49, 2646–2659 (2012)

    Article  Google Scholar 

  15. Berryman, J.G.: Bounds and self-consistent estimates for elastic constants of random polycrystals with hexagonal, trigonal, and tetragonal symmetries. J. Mech. Phys. Solids 53, 2141–2173 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miller, M.N.: Bounds for the effective elastic bulk modulus of heterogeneous materials. J. Math. Phys. 10, 2005–2013 (1969)

    Article  Google Scholar 

  17. Weaver, R.L.: Diffusivity of ultrasound in polycrystals. J. Mech. Phys. Solids 38, 55–86 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Schulgasser, K.: Bounds on the conductivity of statistically isotropic polycrystals. J. Phys. C 10, 407–417 (1977)

    Article  Google Scholar 

  19. Avellaneda, M., Cherkaev, A., Gibiansky, L., Milton, G., Rudelson, M.: A complete characterization of the possible bulk and shear moduli of planar polycrystals. J. Mech. Phys. Solids 44, 1179–1218 (1996)

    Article  Google Scholar 

  20. Avellaneda, M., Milton, G.W.: Optimal bounds on the effective bulk modulus of polycrystals. SIAM J. Appl. Math. 49, 824–837 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Milton, G.W.: The Theory of Composites. Cambridge Unversity Press, Cambridge (2001)

    Google Scholar 

  22. Qiu, Y.P., Weng, G.J.: Elastic constants of a polycrystal with transversely isotropic grains, and the influence of precipitates. Mech. Mater. 12, 1–15 (1991)

    Article  Google Scholar 

  23. Pham, D.C.: Conductivity of realizable effective medium intergranularly random and completely random polycrystals against the bounds for isotropic and symmetrically random aggregates. J. Phys. Condens. Matter 10, 9729–9735 (1998)

    Article  Google Scholar 

  24. Walpole, L.J.: On bounds for the overall elastic moduli of inhomogeneous systems. Int. J. Mech. Phys. Solids 14, 151–162 (1966)

    Article  MATH  Google Scholar 

  25. Christensen, R.M.: Mechanics of Composite Materials. Wiley, New York (1979)

    Google Scholar 

  26. Sirotin, I.I., Saskolskaia, M.P.: Fundamentals of Crystallophysics. Nauka, Moscow (1979)

    Google Scholar 

  27. Benveniste, Y.: Exact connections between polycrystals and crystals properties in two-dimensional polycrystalline aggregates. Proc. R. Soc. Lond. A 447, 1–22 (1994)

    Article  Google Scholar 

  28. Landolt, H.H., Börnstein, R.: Group III: Crystal and Solid State Physics, vol. 11. Springer, Berlin (1979)

  29. Hollister, S.J., Kikuchi, N.: A comparison of homogenization and standard mechanics analyses for periodic porous composites. Comput. Mech. 10, 73–95 (1992)

    Article  MATH  Google Scholar 

  30. Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York (2002)

    Book  MATH  Google Scholar 

  31. Bendsoe, M.P., Sigmund, O.: Topology Optimization: Theory, Methods and Applications. Springer, New York (2003)

    MATH  Google Scholar 

  32. Le, C.H.: Developments in topology and shape optimization, Doctoral dissertation. University of Illinois at Urbana-Champaign (2010)

  33. Besson, J., Cailletaud, G., Chaboche, J.L., Forest, S.: Non-linear Mechanics of Materials. Springer, New York (2010)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. VAST - Institute of Mechanics, 264 Doi Can, Hanoi, Vietnam

    D. C. Pham & T. M. H. Vuong

  2. Institute of Mechanics and Environmental Engineering, 264 Doi Can, Hanoi, Vietnam

    C. H. Le

Authors
  1. D. C. Pham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. H. Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. M. H. Vuong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. C. Pham.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pham, D.C., Le, C.H. & Vuong, T.M.H. Estimates for the elastic moduli of d-dimensional random cell polycrystals. Acta Mech 227, 2881–2897 (2016). https://doi.org/10.1007/s00707-016-1653-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1653-y

Keywords

Search

Navigation