Variational bounds for anisotropic elastic multiphase composites with different shapes of inclusions

  • R. Rodríguez-Ramos
  • R. Guinovart-Díaz
  • J. Bravo-Castillero
  • F. J. Sabina
  • H. Berger
  • S. Kari
  • U. Gabbert
Original

Abstract

In the present work, unified formulae for the overall elastic bounds for multiphase transversely isotropic composites with different geometrical types of inclusions embedded in a matrix are calculated, including the spherical and long or short continuous cylindrical fiber cases. The influence of the different geometrical configurations of the inclusions on the composites is studied. The transversely isotropic effective bounds are obtained by applying the variational formulation for anisotropic composites developed by Willis, which relies on expressions for the static transversely isotropic Green’s function. Some numerical calculations and comparisons with the effective coefficients derived from the self-consistent approach, asymptotic homogenization method, and finite element method (FEM) are shown for different aspect ratio values, exhibiting good agreement.

Keywords

Particle-reinforced composites Variational bounds Transversely isotropic components Green’s function 

References

  1. Aboudi J. (2001). Micromechanical analysis of fully coupled electro-magneto-thermo-elastic multiphase composites. Smart Mater. Struct. 10: 867–877 CrossRefGoogle Scholar
  2. Benveniste Y. (1987). A new approach to the application of Mori–Tanaka’s theory in composite materials. Mech. Mater. 6: 147–157 CrossRefGoogle Scholar
  3. Benveniste Y., Dvorak G.J. and Chen T. (1989). Stress fields in composites with coated inclusions. Mech. Mater. 7: 305–317 CrossRefGoogle Scholar
  4. Beran M.J. and Molyneux J. (1966). Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Q. Appl. Math. 24: 107–118 MATHGoogle Scholar
  5. Berger H., Gabbert U., Koeppe H., Rodriguez-Ramos R., Bravo-Castillero J., Guinovart-Diaz R., Otero J.A. and Maugin G.A. (2003). Finite element and asymptotic homogenization methods applied to smart composite materials. Comput. Mech. 33: 61–67 MATHCrossRefGoogle Scholar
  6. Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Guinovart-Diaz R., Otero J.A. and Bravo-Castillero J. (2005a). An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites. Int. J. Solids Struct. 42: 5692–5714 MATHCrossRefGoogle Scholar
  7. Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Bravo-Castillero J. and Guinovart-Diaz R. (2005b). A comprehensive numerical homogenisation technique for calculating effective coefficients of uniaxial piezoelectric fiber composites. Mater. Sci. Eng. A 412: 53–60 CrossRefGoogle Scholar
  8. Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Bravo-Castillero J. and Guinovart-Diaz R. (2005c). Calculation of effective coefficients for piezoelectric fiber composites based on a general numerical homogenization technique. Compos. Struct. 71: 397–400 CrossRefGoogle Scholar
  9. Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Bravo-Castillero J., Guinovart-Diaz R., Sabina F.J. and Maugin G.A. (2006). Unit cell models of piezoelectric fibre composites for numerical and analytical calculation of effective properties. Smart Mater. Struct. 15: 451–458 CrossRefGoogle Scholar
  10. Guinovart-Díaz R., Bravo-Castillero J., Rodríguez-Ramos R. and Sabina F.J. (2001). Closed-form expressions for the effective coefficients of fiber-reinforced composite with transversely isotropic constituents. I: Elastic and hexagonal symmetry. J. Mech. Phys. Solids 49: 1445–1462 MATHCrossRefGoogle Scholar
  11. Guinovart-Díaz R., Rodríguez-Ramos R., Bravo-Castillero J., Sabina F.J. and Maugin G.A. (2005). Closed-form thermo-elastic moduli of a periodic three-phase fiber-reinforced composite. J. Therm. Stresses 28: 1067–1093 CrossRefGoogle Scholar
  12. Hashin Z. and Shtrikman S. (1963). A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11: 127–140 MATHCrossRefMathSciNetGoogle Scholar
  13. Hashin Z. (1965). On elastic behaviour of fibre reinforced materials of arbitrary transverse phase geometry. J. Mech. Phys. Solids 13: 119–134 CrossRefGoogle Scholar
  14. Hashin Z. (1979). Analysis of properties of fiber composites with anisotropic constituents. J. Appl. Mech. 46: 543–550 MATHGoogle Scholar
  15. Hill R. (1964). Theory of mechanical properties of fibre-strengthened materials: I. Elast. Behav. J. Mech. Phys. Solids 12: 199–212 CrossRefGoogle Scholar
  16. Hill R. (1965a). A self-consistent mechanics of composite materials. J. Mech. Phys. Solids 13: 213–222 CrossRefGoogle Scholar
  17. Hill R. (1965b). Continuum micromechanics of elastoplastic polycrystals. J. Mech. Phys. Solids 13: 89–101 MATHCrossRefGoogle Scholar
  18. Ju J.W. and Chen T. (1994). Effective elastic moduli of two phase composites containing randomly dispersed spherical inhomogeneities. Act. Mech. 103: 123–144 MATHCrossRefMathSciNetGoogle Scholar
  19. Kari S., Berger H., Rodriguez-Ramos R. and Gabbert U. (2007a). Computational evaluation of effective material properties of composites reinforced by randomly distributed spherical particles. Comp. Struct. 77: 223–231 CrossRefGoogle Scholar
  20. Kari S., Berger H. and Gabbert U. (2007b). Numerical evaluation of effective material properties of randomly distributed short cylindrical fibre composites. Comp. Mater. Sci. 39: 198–204 CrossRefGoogle Scholar
  21. Levin, V.M., Rakovskaja, Krecher, W.S.: The effective thermoelectroelastic properties of microinhomogeneous materials. Int. J. Solids Struct. 36, 2683–2705 (1999)Google Scholar
  22. McLaughlin R. (1977). A study of the differential scheme for composite materials. Int. J. Eng. Sci. 15: 237–244 MATHCrossRefGoogle Scholar
  23. Milton G.W. (1982). Bounds on the elastic and transport properties of two-component composites. J. Mech. Phys. Solids 30: 177–191 MATHCrossRefMathSciNetGoogle Scholar
  24. Milton G.W. and Phan-Thien N. (1982). New bounds on the effective moduli of two-component materials. Proc. Roy. Soc. Lond. A 380: 305–331 MATHCrossRefGoogle Scholar
  25. Mori T. and Tanaka K. (1973). Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. Mater. 21: 571–574 CrossRefGoogle Scholar
  26. Nemat-Nasser S. and Hori M. (1995). Universal bounds for overall properties of linear and nonlinear heterogeneous solids. J. Eng. Mat. Tech. 117: 412–432 CrossRefGoogle Scholar
  27. Nemat-Nasser S. and Hori M. (1999). Micromechanics: overall properties of heterogeneous materials, second ed. North Holland, Amsterdam Google Scholar
  28. Nie S. and Basaran C. (2005). A micromechanical model for effective elastic properties of particulate composites with imperfect interfacial bonds. Int. J. Solids Struct. 42: 4179–4191 MATHCrossRefGoogle Scholar
  29. Pobedrya, B.E.: Mechanics of composite materials. Moscow State University Press, Moscow (in Russian) (1984)Google Scholar
  30. Rodríguez-Ramos R., Sabina F.J., Guinovart-Díaz R. and Bravo-Castillero J. (2001). Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents. I: Elastic and square symmetry. Mech. Mater. 33: 1445–1462 CrossRefGoogle Scholar
  31. Sabina F.J., Bravo-Castillero J., Guinovart-Díaz R., Rodríguez-Ramos R. and Valdiviezo-Mijangos O. (2002). Overall behavior of two-dimensional periodic composites. Int. J. Solids Struct. 39: 483–497 MATHCrossRefGoogle Scholar
  32. Sabina F.J., Smyshlyaev V.P. and Willis J.R. (1993). Self-consistent analysis of waves in a matrix-inclusion composite-I. Aligned spheroidal inclusions. J. Mech. Phys. Solids 41(10): 1573–1588 MATHCrossRefMathSciNetGoogle Scholar
  33. Sabina, F.J., Willis, J.R.: Self-consistent analysis of waves in rocks containing arrays of cracks. In: Fjaer, E., Holt, R.M., Rathore, J.S. (eds.) Seismic Anisotropy Tulsa. Society of Exploration Geophysicists, Oklahoma, pp. 318–356 (1996)Google Scholar
  34. Smith S.C. (1976). Experimental values for the elastic constants of a particulate-filled glassy polymer. J. Res. NBS 80: 45–49 Google Scholar
  35. Torquato S. (1991). Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev. 44: 37–76 CrossRefMathSciNetGoogle Scholar
  36. Willis J.R. (1977). Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solids 25: 185–202 MATHCrossRefGoogle Scholar
  37. Willis J.R. (1980a). A polarization approach to the scattering of elastic waves-I. Scattering by a single inclusion. J. Mech. Phys. Solids 28: 287–305 MATHCrossRefMathSciNetGoogle Scholar
  38. Willis J.R. (1980b). A polarization approach to the scattering of elastic waves-II. Multiple scattering from inclusions. J. Mech. Phys. Solids 28: 307–327 MATHCrossRefMathSciNetGoogle Scholar
  39. Willis J.R. (1981). Variational and related method for the properties of composites. Adv. Appl. Mech. 21: 1–78 MATHCrossRefMathSciNetGoogle Scholar
  40. Willis J.R. (1992). Bounds for the overall properties of anisotropic composites. In: Mura, T. and Koya, T. (eds) Variational methods in mechanics, chap 21, pp. Oxford University Press, Oxford Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • R. Rodríguez-Ramos
    • 1
  • R. Guinovart-Díaz
    • 1
  • J. Bravo-Castillero
    • 1
  • F. J. Sabina
    • 2
  • H. Berger
    • 3
  • S. Kari
    • 3
  • U. Gabbert
    • 3
  1. 1.Facultad de Matemática y ComputaciónUniversidad de La HabanaHabana 4Cuba
  2. 2.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMexico D.F.Mexico
  3. 3.Institute of MechanicsOtto-von-Guericke-University of MagdeburgMagdeburgGermany

Personalised recommendations