Abstract
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.
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S. M. Kozlov. Geometrical aspects of homogenization. Uspekhi Mat. Nauk, 2 (1989), 79–120 (in Russian, English translation available).
J. M. Dewey. The elastic constants of materials loaded with non rigid fillers. J. Appl. Phys., 13 (1947), 378–382.
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).
Z. Hashin & S. Strikman. A variational approach to the theory of elastic behaviour of multiphase materials. J. Mech. Phys. Solids, 10 (1963), 127–140.
J. B. Keller. A theorem on the conductivity of a composite medium. J. Math. Phys., 5 (1964), 548.
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).
S. M. Kozlov. Duality of one type of variational problem. Funct. Anal. Appl., 17 (1983), 171–175 (in Russian, English translation available).
V. L. Berdichevsky. Variational principles in continuum mechanics. Nauka, Moscow, 1983 (in Russian).
G. Fichera. Existence theorems in elasticity. Handbuch der Physik, Vol VIa/2, edited by C. Truesdell, Springer-Verlag, 1972, 341–389.
V. L. Berdichevsky. Heat conduction of a checkerboard structure. Vestnik Mask., Cos. Univ., Ser. I, Math, and Mech., 4 (1985), 15–25 (in Russian).
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).
E. Sanchez-Palencia. Non-homogeneous media and vibration theory. Springer-Verlag, 1980.
C. Hermain. Tensoren und Kristallsymmetrie. Z. Kristallographie, 89 (1934), 3.
L. D. Landau & E. M. Lifshits. Theory of elasticity. Nauka, Moscow, 1987 (in Russian, English translation available).
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).
V. Z. Parton & P. I. Perlin. Methods of mathematical theory of elasticity. Moscow, 1981 (in Russian, English translation available).
S. G. Mikhlin. The problem of the minimum of a quadratic functional. Gostekhizdat, 1952 (in Russian, English translation available).
Ping Sheng & R. V. Kohn. Geometric effects in continuous-media percolation. Phys. Rev., 26 (1982), 1331–1335.
R. Lipton. On the behavior of elastic composites with transverse isotropic symmetry. Workshop on composite media and homogenization theory, Birkhäuser, 1990.
D. Leguillon & E. Sanchez-Palencia. Computation of singular solutions in elliptic problems and elasticity. Masson, Paris, 1987.
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Communicated by R. V. Kohn
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Berlyand, L.V., Kozlov, S.M. Asymptotics of the homogenized moduli for the elastic chess-board composite. Arch. Rational Mech. Anal. 118, 95–112 (1992). https://doi.org/10.1007/BF00375091
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DOI: https://doi.org/10.1007/BF00375091