Archive for Rational Mechanics and Analysis

, Volume 118, Issue 2, pp 95–112 | Cite as

Asymptotics of the homogenized moduli for the elastic chess-board composite

  • L. V. Berlyand
  • S. M. Kozlov


We find the asymptotic behavior of the homogenized coefficients of elasticity for the chess-board structure. In the chess board white and black cells are isotropic and have Lamé constants (λ, μ,) and (δλ, δμ) respectively. We assume that the black cells are soft, so δ →0. It turns out that the Poisson ratio for this composite tends to zero with δ.


Neural Network Complex System Asymptotic Behavior Nonlinear Dynamics Electromagnetism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. M. Kozlov. Geometrical aspects of homogenization. Uspekhi Mat. Nauk, 2 (1989), 79–120 (in Russian, English translation available).Google Scholar
  2. 2.
    J. M. Dewey. The elastic constants of materials loaded with non rigid fillers. J. Appl. Phys., 13 (1947), 378–382.Google Scholar
  3. 3.
    V. P. Kotlyarov & E. Ja. Khruslov. On equations of elastic medium with a great number of fine absolutely hard inclusions. J. Math. Phys. Funct. Anal., 3 (1972), 39–51 (in Russian).Google Scholar
  4. 4.
    Z. Hashin & S. Strikman. A variational approach to the theory of elastic behaviour of multiphase materials. J. Mech. Phys. Solids, 10 (1963), 127–140.Google Scholar
  5. 5.
    J. B. Keller. A theorem on the conductivity of a composite medium. J. Math. Phys., 5 (1964), 548.Google Scholar
  6. 6.
    A. M. Dykne. Conductivity of a two-dimensional two-phase system. J. of Exper. Theor. Phys., 7 (1970), 110–116 (in Russian, English translation available).Google Scholar
  7. 7.
    S. M. Kozlov. Duality of one type of variational problem. Funct. Anal. Appl., 17 (1983), 171–175 (in Russian, English translation available).Google Scholar
  8. 8.
    V. L. Berdichevsky. Variational principles in continuum mechanics. Nauka, Moscow, 1983 (in Russian).Google Scholar
  9. 9.
    G. Fichera. Existence theorems in elasticity. Handbuch der Physik, Vol VIa/2, edited by C. Truesdell, Springer-Verlag, 1972, 341–389.Google Scholar
  10. 10.
    V. L. Berdichevsky. Heat conduction of a checkerboard structure. Vestnik Mask., Cos. Univ., Ser. I, Math, and Mech., 4 (1985), 15–25 (in Russian).Google Scholar
  11. 11.
    M. I. Giy, L. I. Manevitch, & V. G. Oshmyan. On percolation effects in a mechanical system. Dokl. Acad. Nauk SSSR, 276 (1984), (in Russian, English translation available).Google Scholar
  12. 12.
    E. Sanchez-Palencia. Non-homogeneous media and vibration theory. Springer-Verlag, 1980.Google Scholar
  13. 13.
    C. Hermain. Tensoren und Kristallsymmetrie. Z. Kristallographie, 89 (1934), 3.Google Scholar
  14. 14.
    L. D. Landau & E. M. Lifshits. Theory of elasticity. Nauka, Moscow, 1987 (in Russian, English translation available).Google Scholar
  15. 15.
    L. V. Berlyand. Homogenization of the elasticity equations in domains with finegrained boundary. Parts I–II. Function theory, functional analysis and applications. 39, 40 (1983), 2–30 (in Russian).Google Scholar
  16. 16.
    V. Z. Parton & P. I. Perlin. Methods of mathematical theory of elasticity. Moscow, 1981 (in Russian, English translation available).Google Scholar
  17. 17.
    S. G. Mikhlin. The problem of the minimum of a quadratic functional. Gostekhizdat, 1952 (in Russian, English translation available).Google Scholar
  18. 18.
    Ping Sheng & R. V. Kohn. Geometric effects in continuous-media percolation. Phys. Rev., 26 (1982), 1331–1335.Google Scholar
  19. 19.
    R. Lipton. On the behavior of elastic composites with transverse isotropic symmetry. Workshop on composite media and homogenization theory, Birkhäuser, 1990.Google Scholar
  20. 20.
    D. Leguillon & E. Sanchez-Palencia. Computation of singular solutions in elliptic problems and elasticity. Masson, Paris, 1987.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • L. V. Berlyand
    • 1
    • 2
  • S. M. Kozlov
    • 1
    • 2
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity Park
  2. 2.Moscow Civil Engineering InstituteMoscow

Personalised recommendations