Abstract
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.
The present article is a translation of an article originally written in Russian and published as the report of Ioffe Physico-Technical Institute, Academy of Sciences of USSR, Publication 1115, Leningrad, 1987.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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).
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.
Christensen, R.M., Mechanics of Composite Materials, Wiley-Interscience, New-York, 1979.
Lurie A.I., Theory of elasticity, Moscow, Nauka, 1970, (in Russian), 939pp
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).
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 volume
Reshetnyak, U.G. General theorems on semi-continuity and quasiconvexity of functionals, Siberian Mat. Zhurnal, 1967, 8, p.801–816 (in Russian).
Rockefeller R., Convex Analysis, Princeton, N.J., Princeton University Press, 1970.
Sanchez-Palencia E., Nonhomogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer-Verlag, 1980.
Cherkaev A.V., Relaxation of non-convex multidimensional problems, Uspekhi Mat Nauk 41, 1986, 4(250), p. 194 (in Russian).
Avellaneda M. Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47 (1987), 1216–1228.
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.
Dacorogna B., Weak continuity and weak lower semicontinuity fo non-linerar functionals, Lecture Notes in Math. 922, Springer-Verlag, 1982.
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.
Hill R., A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 4, 1965. p.213.
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;
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;
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.
Kohn, R.V., Lipton R., The effective viscosity of a mixture of two Stokes fluids, G. Papanicolaou, ed., SIAM, 1986.
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.
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.
Milton G.W., Modelling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, Springer-Verlag, 1986.
Morrey C.B., Quasiconvexity and the lower semicontinuity of multiple integrals, Pacific J. Math., 1952, 225–253.
Murat F. and Francfort G.A., Homogenization and optimal bounds in linear elastisity, Arch. Rat. Mech. Anal., 1986, 94 4, 307–334.
Strang G., The polyconvexification of F(∇u), The Austr. Nat. University, Research Rep. NCNA-R09-83, 1983, 21p.
Tartar L., Compensated compactness and applications to partial differential equations, in Non Linear Mech. Anal., Heriot Watt Symposium, 1976, vol.39, 136–212.
Tartar L., Estimation fines do coefficients homogénisés, in Ennio De Giordgi Coll., Pitman Press, P. Kree, ed., 1985, v.125, 168–187.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gibiansky, L.V., Cherkaev, A.V. (1997). Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density. In: Cherkaev, A., Kohn, R. (eds) Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol 31. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-2032-9_8
Download citation
DOI: https://doi.org/10.1007/978-1-4612-2032-9_8
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7390-5
Online ISBN: 978-1-4612-2032-9
eBook Packages: Springer Book Archive