Archive of Applied Mechanics

, Volume 79, Issue 3, pp 189–204 | Cite as

Unified formulae of variational bounds for multiphase anisotropic elastic composites

  • H. Brito-Santana
  • R. Rodríguez-Ramos
  • R. Guinovart-Díaz
  • J. Bravo-Castillero
  • F. J. Sabina
  • G. A. Maugin
Original

Abstract

Using the spherical and deviator decomposition of the polarization and strain tensors, we present a general algorithm for the calculation of variational bounds of dimension d for any type of anisotropic linear elastic composite as a function of the properties of the comparison body. This procedure is applied in order to obtain analytical expressions of bounds for multiphase, linear elastic composites with cubic symmetry where the geometric shapes of the inclusions are arbitrary. For the validation, it can be proved that for the isotropic particular case, the bounds coincide with those recently reported by Gibiansky and Sigmund. On the other hand, based on this general procedure some, classical bounds reported by Hashin for transversely isotropic composites, are reproduced. Numerical calculations and some comparisons with other models and experimental data are shown.

Keywords

Variational bounds Multiphase composites Effective properties Homogenization 

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References

  1. 1.
    Voigt W.: Uber die beziehung zwischen den beiden elastizi-tatskonstanten isotroper Koerper. Wied Ann. 38, 573–587 (1888)Google Scholar
  2. 2.
    Reuss A.: Calculation of low limit of mixed crystals. Z. Angew. Math. Mech. 9, 49–58 (1929)MATHCrossRefGoogle Scholar
  3. 3.
    Hashin Z., Shtrikman S.: On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solids 10, 335–342 (1962)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Hashin Z., Shtrikman S.: A variational approach to the theory of the elastic behavior of multiphase materials. J. Mech. Phys. Solids 11, 127–140 (1963)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Kroener E.: Bounds for effective elastic moduli of disorder materials. J. Mech. Phys. Solids 25, 137–155 (1977)MATHCrossRefGoogle Scholar
  6. 6.
    Milton G.W.: Concerning bounds on the transport and mechanical properties of multicomponent composite materials. Appl. Phys. A 26, 125–130 (1981)CrossRefGoogle Scholar
  7. 7.
    Walpole L.: On bounds for the overall elastic moduli of heterogeneous systems. J. Mech. Phys. Solids 14, 151–162 (1966)MATHCrossRefGoogle Scholar
  8. 8.
    Willis J.R.: Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25, 185–202 (1977)MATHCrossRefGoogle Scholar
  9. 9.
    Rodriguez-Ramos R., Pobedria B.E., Padilla P., Bravo-Castillero J., Guinovart-Diaz R., Maugin G.A.: Variational principles for nonlinear piezoelectric materials. Arch. Appl. Mech. 74, 191–200 (2004)MATHGoogle Scholar
  10. 10.
    Eshelby J.D.: The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A 241, 376–396 (1957)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kröner, E.: Kontinuumstheorie der Versetzungen und Eigenspannungen. Ergebn. Angew. Math., vol. 5. Springer, Heidelberg (1958)Google Scholar
  12. 12.
    Gibiansky L.V., Sigmund O.: Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48, 461–498 (2000)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hashin Z.: On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids 13, 119–134 (1965)CrossRefGoogle Scholar
  14. 14.
    Eischen J.W., Torquato S.: Determining elastic behavior of composites by the boundary element method. J. Appl. Phys. 74, 159–170 (1993)CrossRefGoogle Scholar
  15. 15.
    Guinovart-Díaz R., Bravo-Castillero J., Rodríguez-Ramos R., Sabina F.J.: Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents. I: Elastic and hexagonal symmetry. J. Mech. Phys. Solids 49, 1445–1462 (2001)MATHCrossRefGoogle Scholar
  16. 16.
    Rodríguez-Ramos R., Sabina F.J., Guinovart-Díaz R., Bravo-Castillero J.: Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents. I: Elastic and square symmetry. Mech. Mat. 33, 223–235 (2001)CrossRefGoogle Scholar
  17. 17.
    Torquato S.: Effective stiffness tensor of composite media: II Applications to isotropic dispersions. J. Mech. Phys. Solids 46, 1411–1440 (1998)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Milton G.W., Phan-Thien N.: New bounds on the effective moduli of two-component materials. Proc. R. Soc. A 380, 305–331 (1982)MATHCrossRefGoogle Scholar
  19. 19.
    Valdiviezo-Mijangos O.C.: Fiber-reinforced composite with cubic symmetry constituents. Mater. Lett. 56, 339–343 (2002)Google Scholar
  20. 20.
    Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Guinovart-Diaz R., Otero J.A., Bravo-Castillero J.: An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int. J. Solids Struct. 42, 5692–5714 (2005)MATHCrossRefGoogle Scholar
  21. 21.
    Jiang B., Batra R.C.: Micromechanical modeling of a composite containing piezoelectric and shape memory alloy inclusions. J. Int. Mat. Sys. Struct. 12, 165–182 (2001)CrossRefGoogle Scholar
  22. 22.
    Pobedrya, B.E.: Mechanics of Composite Materials. Moscow State University Press, Moscow (in Russian) (1984)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • H. Brito-Santana
    • 1
  • R. Rodríguez-Ramos
    • 2
  • R. Guinovart-Díaz
    • 2
  • J. Bravo-Castillero
    • 2
  • F. J. Sabina
    • 3
  • G. A. Maugin
    • 4
  1. 1.Centro de Medicina y ComplejidadInstituto Superior de Ciencias Médicas “Carlos J. Finlay”CamagueyCuba
  2. 2.Facultad de Matemática y ComputaciónUniversidad de La HabanaHabana 4Cuba
  3. 3.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMexico DFMexico
  4. 4.Université Pierre et Marie CurieInstitut Jean Le Rond d’Alembert, UMR CNRS 7190Paris Cedex 05France

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