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Bounds on the overall properties of composites with ellipsoidal inclusions

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Abstract

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.

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References

  1. Hashin, Z.: Analysis of composite materials-A Survey. ASME Journal of Applied Mechanics 55, 481–505 (1983)

    Article  Google Scholar 

  2. Torquato, S.: Random heterogeneous media: Microstructure and improved bounds on effective properties. Applied Mechanics Review 44(2), 37–76 (1991)

    Article  MathSciNet  Google Scholar 

  3. Nemat-Nasser, S., Hori, M.: Micromechanics: Overall Properties of Heterogeneous Materials. Horth-Holland, Amsterdam (1993)

    MATH  Google Scholar 

  4. Milton, G. W.: The Theory of Composites. Cambridge University Press, Cambridge, UK (2002)

    Book  MATH  Google Scholar 

  5. Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer, New York (2002)

    MATH  Google Scholar 

  6. Voigt, W.: Lehrbuch der Kristallphysik. Teubner, Berlin (1910) (in German)

    Google Scholar 

  7. Reuss, A.: Berechnung der fliessgrense von mischkristallen auf grund der plastizitätsbedingung für einkristalle. Zeitschrift fur Angewandte Mathematik und Mechanik. 9, 49–58 (1929) (in German)

    Article  MATH  Google Scholar 

  8. Hashin, Z., Shtrikman, S.: On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342 (1962)

    Article  MathSciNet  Google Scholar 

  9. Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. Journal of the Mechanics and Physics of Solids 11, 127–140 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  10. Walpole, L. J.: On bounds for the overall elastic moduli of inhomogeneous systems-I. Journal of the Mechanics and Physics of Solids 14, 151–162 (1966)

    Article  MATH  Google Scholar 

  11. Hill, R.: New derivations of some elastic extremum principles, in Progress in Applied Mechanics. Prager Aniversary Volume, Macmillan, New York, 91–106 (1963)

    Google Scholar 

  12. Willis, J. R.: Bounds and self-consistent estimates for the overall moduli of anisotropic composites. Journal of the Mechanics and Physics of Solids 25, 185–202 (1977)

    Article  MATH  Google Scholar 

  13. Willis, J. R.: Variational principles and bounds for the overall properties of composites. In: Continuum Models for Discrete Systems, Provan J.W. ed. University of Waterloo Press, Waterloo, 185–215 (1978)

    Google Scholar 

  14. Ponte Castaneda, P., Willis, J. R.: The effect of spatial distribution on the effective behavior of composite materials and cracked media. Journal of the Mechanics and Physics of Solids 43, 1919–1951 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Monetto, I., Drugan, W. J.: A micromechanics-based nonlocal constitutive equation for elastic composites containing randomly oriented spheroidal heterogeneities. Journal of the Mechanics and Physics of Solids 52, 359–393 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kröner, E.: Bounds for effective elastic moduli of disordered materials. Journal of the Mechanics and Physics of Solids 25, 137–155 (1977)

    Article  MATH  Google Scholar 

  17. Milton, G. W., Kohn, R.: Variational bounds on the effective properties of anisotropic composites. Journal of the Mechanics and Physics of Solids 36, 597–629 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  18. Beran, M. J., Molyneux J.: Use of classical variational principles to determine bounds for the effective bulk modulus in heterogeneous media. Quarterly of Applied Mathematics 24, 107–118 (1966)

    MATH  Google Scholar 

  19. McCoy, J. J.: Recent Advances in Engineering Science, Vol.5. Gordon and Breach, New York (1970)

    Google Scholar 

  20. Milton, G. W., Phan-Thien, N.: New bounds on the effective moduli of two component materials. Proceedings of the Royal Society of London A 380, 305–331 (1982)

    Article  MATH  Google Scholar 

  21. Torquato, S.: Effective stiffness tensor of composite media: II. Applications to isotropic dispersions. Journal of the Mechanics and Physics of Solids 46, 1411–1440 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Berryman, J. G.: Measures of microstructure to improve estimates and bounds on elastic constants and transport coefficients in heterogeneous media. Mechanics of Materials 38, 732–747 (2006)

    Article  Google Scholar 

  23. Keller, J. B., Rubenfeld, L. A., Molyneux, J. E.: Extremum principles for slow viscous flows with applications to suspensions. Journal of Fluid Mechanics 30, 97–125 (1967)

    Article  MATH  Google Scholar 

  24. Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis. Cambridge University Press, England (1991)

    Book  MATH  Google Scholar 

  25. Hashin, Z.: On elastic behaviour of fiber reinforced materials of arbitrary transverse phase geometry. Journal of the Mechanics and Physics of Solids 13, 119–134 (1965)

    Article  Google Scholar 

  26. Hashin, Z.: The elastic moduli of heterogeneous materials. ASME Journal of Applied Mechanics 29, 143–150 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu, L. Z.: Bounds on the effective elastic moduli of composites with spherical inclusions, In: Advances in Solid Mechanics, in Honor of Professor D. Wang’s 90th Anniversary, Harbin Institute of Technology Press, Harbin, 112–121 (2010) (in Chinese)

    Google Scholar 

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Authors and Affiliations

  1. Center for Composite Materials, Harbin Institute of Technology, 150001, Harbin, China

    Lin-Zhi Wu & Shi-Dong Pan

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  1. Lin-Zhi Wu
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  2. Shi-Dong Pan
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Correspondence to Lin-Zhi Wu.

Additional information

The project was supported by the National Natural Science Foundation of China (11072068 and 11002041).

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Wu, LZ., Pan, SD. Bounds on the overall properties of composites with ellipsoidal inclusions. Acta Mech Sin 28, 1340–1355 (2012). https://doi.org/10.1007/s10409-012-0102-y

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