Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions
Authors:
Grégoire Allaire and Robert V. Kohn
Journal:
Quart. Appl. Math. 51 (1993), 675-699
MSC:
Primary 73B27; Secondary 35B27, 73K20, 73K40, 73V25
DOI:
https://doi.org/10.1090/qam/1247434
MathSciNet review:
MR1247434
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Abstract: This paper is concerned with two-dimensional, linearly elastic, composite materials made by mixing two isotropic components. For given volume fractions and average strain, we establish explicit optimal upper and lower bounds on the effective energy quadratic form. There are two different approaches to this problem, one based on the “Hashin-Shtrikman variational principle” and the other on the “translation method". We implement both. The Hashin-Shtrikman principle applies only when the component materials are “well-ordered", i.e., when the smaller shear and bulk moduli belong to the same material. The translation method, however, requires no such hypothesis. As a consequence, our optimal bounds are valid even when the component materials are not well-ordered. Analogous results have previously been obtained by Gibianski and Cherkaev in the context of the plate equation.
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G. Allaire and R. Kohn, Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials, Quart. Appl. Math., in press
---, Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, Quart. Appl. Math. LI, 643–674 (1993)
---, Optimal design for minimum weight and compliance in plane stress using extremal microstructures, European J. Mech. A. Solids, in press.
M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47, 1216–1228 (1987)
M. Avellaneda and G. Milton, Bounds on the effective elasticity of composites based on two-point correlations, Composite Material Technology, D. Hui and T. J. Kozik (Eds.), ASME, 1989, pp. 89–93
---, Optimal bounds on the effective bulk modulus of polycrystals, SIAM J. Appl. Math. 49, 824–837 (1989)
M. Bendsoe, R. Haber, and C. Jog, A displacement-based topology design method with self-adaptive layered materials, Topology Design of Structures, M. Bendsoe and C. MotaSoares (Eds.), Kluwer, 1993, pp. 219–238
A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978
R. Christensen, Mechanics of Composite Materials, Wiley-Interscience, New York, 1979
G. Dal Maso and R. Kohn, The local character of G-closure, in preparation
G. Francfort and J. Marigo, Stable damage evolution in a brittle continuous medium, European J. Mech. A. Solids, in press
G. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal. 94, 307–334 (1986)
G. Francfort, Homogenization of a class of fourth order equations with application to incompressible elasticity, Proc. Roy. Soc. Edinburgh Sect. A 120, 25–46 (1992)
L. Gibianski and A. Cherkaev, Design of composite plates of extremal rigidity, Ioffe Physicotechnical Institute preprint 914, 1984
---, Microstructures of composites of extremal rigidity and exact estimates of the associated energy density, Ioffe Physicotechnical Institute preprint 1115, 1987
---, The exact coupled bounds for effective tensors of electrical and magnetic properties of two-component two-dimensional composites, Proc. Roy. Soc. Edinburgh Sect. A 122, 93–125 (1992)
---, Coupled estimates for the bulk and shear moduli of a two-dimensional elastic composite, J. Mech. Phys. Solids, submitted
K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation, Comm. Math. Phys. 90, 473–491 (1983)
Z. Hashin and S. Shtrikman, A variational approach to the theory of the elastic behavior of multiphase materials, J. Mech. Phys. Solids 11, 127–140 (1963)
Y. Kantor and D. Bergman, Improved rigorous bounds on the effective elastic moduli of a composite material, J. Mech. Phys. Solids 32, 41–62 (1984)
R. Kohn, Recent progress in the mathematical modeling of composite materials, Composite Material Response: Constitutive Relations and Damage Mechanisms, G. Sih et al. (Eds.), Elsevier, New York, 1988, pp. 155–177
---, Relaxation of a double-well energy, Continuum Mechanics and Thermodynamics 3, 193–236 (1991)
R. Kohn and R. Lipton, Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials, Arch. Rational Mech. Anal. 102, 331–350 (1988)
R. Kohn and G. Strang, Optimal design and relaxation of variational problems. I—III, Comm. Pure Appl. Math. 39, 113–137, 139–182, 353–377 (1986)
R. Lipton, On the effective elasticity of a two-dimensional homogenized incompressible elastic composite, Proc. Roy. Soc. Edinburgh Sect. A110, 45–61 (1988)
K. Lurie and A. Cherkaev, The effective properties of composites and problems of optimal design of constructions, Uspekhi Mekhaniki 9, 1–81 (1986) (Russian)
---, G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates, J. Optim. Theory Appl. 42, 305–315 (1984)
---, Exact estimates of the conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A99, 71–87 (1984)
G. Milton, On characterizing the set of possible effective tensors of composities: The variational method and the translation method, Comm. Pure Appl. Math. 43, 63–125 (1990)
G. Milton and R. Kohn, Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids 36, 597–629 (1988)
L. Mirsky, On the trace of a matrix product, Math. Nachr. 20 171–174 (1959)
F. Murat, H-convergence, Seminaire d’Analyse Fonctionnelle et Numérique de 1’Université d’Alger, mimeographed notes, 1978
F. Murat and L. Tartar, Calcul des variations et homogénéisation, Les Méthodes de l’Homogénéisation: Théorie et Applications en Physique, Eyrolles, 1985, pp. 319–369.
G. Papanicolaou and S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Random Fields, Colloq. Math. Soc. János Bolyai, vol. 27, North-Holland, Amsterdam, 1982, pp. 835–873
E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, New York, 1980
S. Spagnolo, Convergence in energy for elliptic operators, Numerical Solutions of Partial Differential Equations III: Synspade 1975, B. Hubbard (Ed.), Academic Press, New York, 1976
L. Tartar, Estimation de coefficients homogénéisés, Computing Methods in Applied Sciences and Engineering, R. Glowinski and J. L. Lions (Eds.), Lecture Notes in Math., vol. 704, Springer-Verlag, New York, 1978, pp. 364–373
---, Estimations fines des coefficients homogénéisés, Ennio de Giorgi Colloquium, P. Kree (Ed.), Pitman Res. Notes in Math. Ser., vol. 125, Longman Sci. Tech., Harlow, 1985, pp. 168–187
S. Torquato, Random heterogeneous media: microstructure and improved bounds on effective properties, Appl. Mech. Rev. 44, 37–76 (1991)
L. Walpole, On bounds for the overall elastic moduli of anisotropic composites, J. Mech. Phys. Solids 14, 151–162 (1966)
J. Willis, Variational and related methods for the overall properties of composite materials, C.-S. Yih (Ed.), Adv. in Appl. Mech. 21, 2–78 (1981)
V. Zhikov, On estimates for the trace of an averaged tensor, Soviet Math. Dokl. 37, 456–459 (1988)
V. Zhikov, S. Kozlov, O. Oleinik, and K. Ngoan, Averaging and G-convergence of differential operators, Russian Math. Surveys 34, 69–147 (1979)
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© Copyright 1993
American Mathematical Society