Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density

  • L. V. Gibiansky
  • A. V. Cherkaev
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 31)


We study the elastic composites of extremal stiffness, i.e., periodic structures that provide minimal energy density in a given stress field (we call them composites of the minimal compliance or of the maximal stiffness). The composites are assembled from two isotropic materials taken in a prescribed proportion, and may be of an arbitrary microstructure.

Two-sided bounds for the energy density are obtained for composites of maximal stiffness and composites of minimal stiffness. One bound is an inequality valid for all structures independently of their geometry, and the second one corresponds to the energy stored in a composite with a specially chosen microstructure.

Porous media are studied in detail: microstructures of porous composites of maximal stiffness are explicitly determined for all possible external stress fields. Also, the structures of composites that possess minimal stiffness in any given external strains are explicitly found in the case when one of the constituents is absolutely rigid.

We discuss the link of obtained optimal microstructures with the composites that are optimal for the plane problem of elasticity; they have been obtained by the authors earlier.1 The application of obtained results to the optimal design problems is discussed.

To solve the problem we develop and use the technique of construction of the quasiconvex envelopes for nonconvex integrands of the corresponding multidimensional variational problems.


Optimal Structure Exact Bound Optimal Design Problem Cylindrical Inclusion Compliance Tensor 
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.


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  1. [1]
    Gibiansky L.V., Cherkaev A.V., The problem of design of a viscoelastic rod under torsion loading, notes of the second seminar “Problems of optimization in machine industry”, Khar’kov, 1986, p.45 (in Russian).Google Scholar
  2. [2]
    Gibiansky, L.V., Cherkaev A.V., Design of composite plates of extremal rigidity, preprint No. 914, Physico-Technical Institute USSR Academy of Sciences, Leningrad, 1984, (in Russian); see chapter 5 of this volume.Google Scholar
  3. [3]
    Christensen, R.M., Mechanics of Composite Materials, Wiley-Interscience, New-York, 1979.Google Scholar
  4. [4]
    Lurie A.I., Theory of elasticity, Moscow, Nauka, 1970, (in Russian), 939ppGoogle Scholar
  5. [5]
    Lurie K.A., Cherkaev A.V., G-closure of a set of anisotropic conducting media for a case of two dimensions, Dokladii USSR Academy of Sciences, v. 259, No. 2, 1981 (in Russian).Google Scholar
  6. [6]
    Lurie K.A. Cherkaev A.V., Effective characteristics of composites and problems of optimal design, Uspekhi Mat Nauk, 1984, 39 4(238), p. 122. (in Russian); see chapter 7 of this volumeGoogle Scholar
  7. [7]
    Reshetnyak, U.G. General theorems on semi-continuity and quasiconvexity of functionals, Siberian Mat. Zhurnal, 1967, 8, p.801–816 (in Russian).zbMATHCrossRefGoogle Scholar
  8. [8]
    Rockefeller R., Convex Analysis, Princeton, N.J., Princeton University Press, 1970.Google Scholar
  9. [9]
    Sanchez-Palencia E., Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, 1980.Google Scholar
  10. [10]
    Cherkaev A.V., Relaxation of non-convex multidimensional problems, Uspekhi Mat Nauk 41, 1986, 4(250), p. 194 (in Russian).Google Scholar
  11. [11]
    Avellaneda M. Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47 (1987), 1216–1228.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Ball J.M. Currie J.C., Olver P.J., Null Lagrangians, weak continuity and variational problems of arbitrary order, J. Funct. Anal 41 (1981), 135–174.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Dacorogna B., Weak continuity and weak lower semicontinuity fo non-linerar functionals, Lecture Notes in Math. 922, Springer-Verlag, 1982.Google Scholar
  14. [14]
    Hashin Z., Shtrikman S., A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. and Phys. Solids, 1963, VII, pp.127–140.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Hill R., A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 4, 1965. p.213.CrossRefGoogle Scholar
  16. [16]
    Kohn R.V. and Strang, G., Structural design optimization, homogenization, and relaxation of variational problems, Comm. Pure and Appl. Math., 1986, 39, No 1, 113–137;MathSciNetzbMATHCrossRefGoogle Scholar
  17. Kohn R.V. and Strang, G., Structural design optimization, homogenization, and relaxation of variational problems, Comm. Pure and Appl. Math., 1986, 39, No 2, 139–182;MathSciNetzbMATHCrossRefGoogle Scholar
  18. Kohn R.V. and Strang, G., Structural design optimization, homogenization, and relaxation of variational problems, Comm. Pure and Appl. Math., 1986, 39, No 3, 353–378.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [17]
    Kohn, R.V., Lipton R., The effective viscosity of a mixture of two Stokes fluids, G. Papanicolaou, ed., SIAM, 1986.Google Scholar
  20. [18]
    Lurie K.A. and Cherkaev A.V., Exact estimates of conductivity of composites formed by two isotropically conducting media, taken in prescribed proportion, Proc. Royal Soc. Edinburgh, 1984, 99a, 71–87.MathSciNetCrossRefGoogle Scholar
  21. [19]
    Lurie K.A. and Cherkaev A.V., G-closure of some particular sets of admissible material characteristics for the problem of bending of thin plates, J. Opt. Th. Appl, 1984, 305–316.Google Scholar
  22. [20]
    Milton G.W., Modelling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, Springer-Verlag, 1986.Google Scholar
  23. [21]
    Morrey C.B., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 1952, 225–253.Google Scholar
  24. [22]
    Murat F. and Francfort G.A., Homogenization and optimal bounds in linear elastisity, Arch. Rat. Mech. Anal., 1986, 94 4, 307–334.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [23]
    Strang G., The polyconvexification of F(∇u), The Austr. Nat. University, Research Rep. NCNA-R09-83, 1983, 21p.Google Scholar
  26. [24]
    Tartar L., Compensated compactness and applications to partial differential equations, in Non Linear Mech. Anal., Heriot Watt Symposium, 1976, vol.39, 136–212.MathSciNetGoogle Scholar
  27. [25]
    Tartar L., Estimation fines do coefficients homogénisés, in Ennio De Giordgi Coll., Pitman Press, P. Kree, ed., 1985, v.125, 168–187.Google Scholar

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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • L. V. Gibiansky
  • A. V. Cherkaev

There are no affiliations available

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