Micromechanics for Heterogeneous Material Property Estimation

Reference work entry


There are two well-established theories in micromechanics for analytical estimation of overall property of a heterogeneous material, instead of experimental estimation; a heterogeneous material includes a partially damaged or plastically deformed material. These two theories, namely, the average field theory and the homogenization theory, are explained in this article. The average field theory is based on physical treatment of the heterogeneous material in the sense that it mimics a material sample test, and it derives a closed-form expression of the overall property in terms of the average strain and stress. The homogenization theory is based on mathematical treatment in the sense that it applies the singular perturbation expansion to the governing equations and obtains numerical solution for the overall property. In this article, the following three advanced topics are also explained: (1) strain energy consideration to obtain the consistent overall property, (2) the Hashin–Shtrikman variational principle to obtain bounds for the overall property, and (3) the extension to the overall property estimation at dynamic state from that at quasi-static state.


Representative Volume Element Strain Energy Density Heterogeneous Material Average Scheme Homogenization Theory 
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  1. H. Ammari, H. Kang, M. Lim, Effective parameters of elastic composites. Indiana Univ. Math. J. 55(3), 903–922 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. N. Bakhvalov, G. Panasenko, Homogenization: Averaging Processes in Periodic Media (Kluwer, New York, 1984)Google Scholar
  3. J.D. Eshelby, The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. A A241, 376–396 (1957)MathSciNetCrossRefGoogle Scholar
  4. G.A. Francfort, F. Murat, Homogenization and optimal bounds in linear elasticity. Arch Ration. Mech. Anal. 94, 307–334 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. X.L. Gao, H.M. Ma, Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem. Acta Mech. 223, 1067–1080 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. Z. Hashin, S. Shtrikman, On some variational principles in anisotropic and nonhomogeneous elasticity. J. Mech. Phys. Solid 10, 335–342 (1962)MathSciNetCrossRefGoogle Scholar
  7. R. Hill, Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solid 11, 357–372 (1963)CrossRefMATHGoogle Scholar
  8. M. Hori, S. Nemat-Nasser, Double-Inclusion model and overall moduli of multi-phase composites. Mech. Mater. 14, 189–206 (1993)CrossRefGoogle Scholar
  9. U. Hornung (ed.), Homogenization and Porous Media (Springer, Berlin, 1996)Google Scholar
  10. M. Kachanov, I. Tsukrov, B. Shafiro, Effective modulus of solids with cavities of various shapes. Appl. Mech. Rev. 47, 151–174 (1994)CrossRefGoogle Scholar
  11. M. Kawashita, H. Nozaki, Eshelby tensor of a polygonal inclusion and its special properties. J. Elast. 74(2), 71–84 (2001)MathSciNetCrossRefGoogle Scholar
  12. J. Kevorkina, J.D. Cole, Multiple Scale and Singular Perturbation Methods (Springer, Berlin, 1996)CrossRefGoogle Scholar
  13. H. Le Quang, Q.C. He, Q.S. Zheng, Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity. Int. J. Solid Struct. 45(13), 3845–3857 (2008)CrossRefMATHGoogle Scholar
  14. L.P. Liu, Solutions to the Eshelby conjectures. Proc. R. Soc. A 464, 573–594 (2008)CrossRefMATHGoogle Scholar
  15. X. Markenscoff, Inclusions with constant eigenstress. J. Mech. Phys. Solid 46(2), 2297–2301 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. G.W. Milton, R. Kohn, Variational bounds on the effective moduli of anisotropic composites. J. Mech. Phys. Solid 43, 63–125 (1988)MathSciNetGoogle Scholar
  17. H.M.S. Munashinghe, M. Hori, Y. Enoki. Application of Hashin-Shtrikman Variational Principle for Computing Upper and Lower Approximate Solutions of Elasto-Plastic Problems, in Proceedings of the International Conference on Urban Engineering in Asian Cities, 1996, pp. 1–6Google Scholar
  18. T. Mura, Micromechanics of Defects in Solids (Martinus Nijhoff Publisher, New York, 1987)CrossRefGoogle Scholar
  19. S. Nemat-Nasser, M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials (North-Holland, London, 1993)MATHGoogle Scholar
  20. S. Nemat-Nasser, M. Hori, Universal bounds for overall properties of linear and nonlinear heterogeneous solids. Trans. ASME 117, 412–422 (1995)Google Scholar
  21. H. Nozaki, M. Taya, Elastic fields in a polyhedral inclusion with uniform eigenstrains and related problems. ASME J. Appl. Mech. 68, 441–452 (2001)CrossRefMATHGoogle Scholar
  22. K.C. Nuna, J.B. Keller, Effective elasticity tensor of a periodic composite. J. Mech. Phys. Solid 32, 259–280 (1984)CrossRefGoogle Scholar
  23. O.A. Oleinik, A.S. Shamaev, G.A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North-Holland, New York, 1992)MATHGoogle Scholar
  24. S. Onaka, N. Kabayashi, M. Kato, Two-dimensional analysis on elastic strain energy due to a uniformly eigenstrained supercircular inclusion in an elastically anisotropic material. Mech. Mater. 34, 117–125 (2002)CrossRefGoogle Scholar
  25. C.Q. Ru, Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane. Acta Mech. 160, 219–234 (2003)CrossRefMATHGoogle Scholar
  26. E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory. Lecture Note in Physics, No. 127 (Springer, Berlin, 1981)Google Scholar
  27. K. Tanaka, T. Mori, Note on volume integrals of the elastic field around an ellipsoidal inclusion. J. Elast. 2, 199–200 (1972)CrossRefGoogle Scholar
  28. K. Terada, T. Miura, N. Kikuchi, Digital image-based modeling applied to the homogenization analysis of composite materials. Comput. Mech. 20, 188–202 (1996)Google Scholar
  29. S. Torquato, Random heterogeneous media: microstructure and improved bounds on effective properties. Appl. Mech. Rev. 42(2), 37–76 (1991)MathSciNetCrossRefGoogle Scholar
  30. K.P. Walker, A.D. Freed, E.H. Jordan, Microstress analysis of periodic composites. Compos. Eng. 1, 29–40 (1991)CrossRefGoogle Scholar
  31. L.J. Walpole, On the overall elastic moduli of composite materials. J. Mech. Phys. Solid 17, 235–251 (1969)CrossRefMATHGoogle Scholar
  32. X. Wang, X.L. Gao, On the uniform stress state inside an inclusion of arbitrary shape in a three-phase composite. Z. Angew. Math. Phys. 62, 1101–1116 (2011)MathSciNetCrossRefMATHGoogle Scholar
  33. M.Z. Wang, B.X. Xu, The arithmetic mean theorem of Eshelby tensor for a rotational symmetrical inclusion. J. Elast. 77, 12–23 (2005)Google Scholar
  34. J.R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites. J. Mech. Phys. Solid 25, 185–202 (1977)CrossRefMATHGoogle Scholar
  35. Q.S. Zheng, D.X. Du, An explicit and universally applicable estimate for the effective properties of multiphase composites which accounts for inclusion distribution. J. Mech. Phys. Solid 49, 2765–2788 (2001)CrossRefMATHGoogle Scholar
  36. W.N. Zou, Q.C. He, M.J. Huang, Q.S. Zheng, Eshelby’s problem of non-elliptical inclusions. J. Mech. Phys. Solid 58, 346–372 (2010)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Earthquake Research InstituteThe University of TokyoBunkyo, TokyoJapan

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