Optimal lower bounds on the elastic energy of a composite made from two non-well-ordered isotropic materials
Authors:
Grégoire Allaire and Robert V. Kohn
Journal:
Quart. Appl. Math. 52 (1994), 311-333
MSC:
Primary 73B27; Secondary 73K20, 73K40, 73V25
DOI:
https://doi.org/10.1090/qam/1276240
MathSciNet review:
MR1276240
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Abstract: This paper is a continuation of our previous work [AK] concerning optimal bounds on the effective behavior of a mixture of two linearly elastic materials. While in [AK] we restricted our attention to the case of two well-ordered components, here we focus on the case of two non-well-ordered and isotropic ones, i.e., the case when the smaller shear and bulk moduli do not belong to the same material. For given volume fractions and average strain, we establish an optimal lower bound on the effective energy quadratic form. We give two proofs of this result: one based on the Hashin-Shtrikman-Walpole variational principle, the other on the translation method.
G. Allaire and R. Kohn, Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, Quart. Appl. Math. 51, 643–674 (1993)
G. Allaire and R. Kohn, Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions, Quart. Appl. Math. 51, 675–699 (1993)
M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47, 1216–1228 (1987)
A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978
F. H. Clarke, Optimization and non-smooth analysis, Wiley, New York, 1983
G. Dal Maso and R. Kohn, The local character of G-closure, in preparation
G. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal. 94, 307–334 (1986)
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)
K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation, Comm. Math. Phys., 473–491 (1983)
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
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)
K. Lurie and A. Cherkaev, Exact estimates of the conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A 99, 71–87 (1984)
G. Milton, On characterizing the set of possible effective tensors of composites: 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)
E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin and New York, 1980
L. Tartar, Estimation des coéfficients homogénéisés, Computing Methods in Applied Sciences and Engineering, Lecture Notes in Math., vol. 704 (R. Glowinski and J. L. Lions, eds.), Springer-Verlag, Berlin and New York, 1978, pp. 364–373
L. Tartar, Estimations fines des coéfficients homogénéisés, Ennio de Giorgi Colloq., Pitman Res. Notes Math. Ser., vol. 125 (P. Krée, ed.), Longman Sci. Tech., Harlow, 1985, pp. 168–187
L. Tartar, Cours Peccot au Collège de France, unpublished (mars 1977)
L. Walpole, On bounds for the overall elastic moduli of anisotropic composites, J. Mech. Phys. Solids 14, 151–162 (1966)
V. Zhikov, S. Kozlov, O. Oleinik, and K. Ngoan, Averaging and G-convergence of differential operators, Russian Math. Surveys 34, 69–147 (1979)
G. Allaire and R. Kohn, Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials, Quart. Appl. Math. 51, 643–674 (1993)
G. Allaire and R. Kohn, Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions, Quart. Appl. Math. 51, 675–699 (1993)
M. Avellaneda, Optimal bounds and microgeometries for elastic two-phase composites, SIAM J. Appl. Math. 47, 1216–1228 (1987)
A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, North-Holland, Amsterdam, 1978
F. H. Clarke, Optimization and non-smooth analysis, Wiley, New York, 1983
G. Dal Maso and R. Kohn, The local character of G-closure, in preparation
G. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity, Arch. Rational Mech. Anal. 94, 307–334 (1986)
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)
K. Golden and G. Papanicolaou, Bounds for effective parameters of heterogeneous media by analytic continuation, Comm. Math. Phys., 473–491 (1983)
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
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)
K. Lurie and A. Cherkaev, Exact estimates of the conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh Sect. A 99, 71–87 (1984)
G. Milton, On characterizing the set of possible effective tensors of composites: 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)
E. Sanchez-Palencia, Non homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin and New York, 1980
L. Tartar, Estimation des coéfficients homogénéisés, Computing Methods in Applied Sciences and Engineering, Lecture Notes in Math., vol. 704 (R. Glowinski and J. L. Lions, eds.), Springer-Verlag, Berlin and New York, 1978, pp. 364–373
L. Tartar, Estimations fines des coéfficients homogénéisés, Ennio de Giorgi Colloq., Pitman Res. Notes Math. Ser., vol. 125 (P. Krée, ed.), Longman Sci. Tech., Harlow, 1985, pp. 168–187
L. Tartar, Cours Peccot au Collège de France, unpublished (mars 1977)
L. Walpole, On bounds for the overall elastic moduli of anisotropic composites, J. Mech. Phys. Solids 14, 151–162 (1966)
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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Article copyright:
© Copyright 1994
American Mathematical Society
