The effective electrical conductivity of nonlinear laminate composites
Authors:
Guoan Li and Andrew S. Douglas
Journal:
Quart. Appl. Math. 53 (1995), 433-464
MSC:
Primary 73B27; Secondary 73K20, 78A55
DOI:
https://doi.org/10.1090/qam/1343461
MathSciNet review:
MR1343461
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Abstract: This paper studies the effective nonlinear electrical conductivity of sequentially laminated materials in both two and three dimensions. The exact nonlinear conductive behavior is obtained through a variational procedure [1] that expresses the nonlinear properties of the laminate composite in terms of an optimization with respect to the properties of a series of linear comparison materials. Multiphase laminate composites are discussed and these results are compared with nonlinear Hashin-Shtrikman (H-S) bounds. It is found that some of the laminates possess extremal microstructures that attain the nonlinear H-S lower bound while others are very close to extremal microstructures. The results apply also to the nonlinear thermal conductivity and dielectric and magnetic behavior of laminated heterogeneous materials.
P. Ponte Castañeda, The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids 39, 45 (1991)
J. C. Maxwell, Electricity and magnetism, 1st ed., Clarendon, Oxford, 1873
Z. Hashin and S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33, 3125 (1962)
E. H. Kerner, The elastic and thermoelastic properties of composite media, Proc. Phys. B 69, 802 (1956)
J. R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, J. Mech. Phys. Solids 25, 185 (1977)
R. Landauer, Electrical conductivity in inhomogeneous media, in Electrical Transport and Optical Properties of Inhomogeneous Media, ed. J. C. Garland and D. B. Tanner, American Institute of Physics, New York, 1978
Z. Hashin, Assessment of the self-consistent scheme approximation: Conductivity of particulate composites, J. Compos. Mater. 2, 284 (1968)
D. A. G. Bruggeman, Ph.D. Thesis, Utrecht, 1930
D. A. G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys. 5, 636 (1935)
G. W. Milton, Modeling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, ed. J. L. Ericksen et al, Springer-Verlag, New York, 1986
G. W. Milton, Concerning bounds on the transport and mechanical properties of multicomponent composite materials, Appl. Phys. A 26, 125 (1981)
K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh A 99, 71–87 (1984)
K. A. Lurie and A. V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. Roy. Soc. Edinburgh A 104, 21–38 (1986)
P. Ponte Castañeda, Bounds and estimates for the properties of nonlinear heterogeneous systems, Proc. Roy. Soc. Lond. A 340, 531 (1992)
X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, Mean field theory for weakly nonlinear composites, Phys. A 157, 192 (1989)
J. R. Willis, Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, in Homogenization and Effective Moduli of Materials and Media, ed. J. L. Ericksen et al, Springer-Verlag, New York, 1986
D. R. S. Talbot and J. R. Willis, Bounds and self-consistent estimates for the overall properties of nonlinear composites, IMA J. Appl. Math. 39, 215 (1987)
D. Stroud and P. M. Hui, Nonlinear susceptibilities of granular media, Phys. Rev. B 37, 8719 (1988)
R. Hill, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids 11, 357 (1963)
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition, Pergamon, Oxford, 1984
R. V. Kohn and G. W. Milton, Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids 36, 597 (1988)
R. Blumenfeld and D. J. Bergman, Nonlinear dielectric: Electrostatics of random media and propagation of electromagnetic waves in a homogeneous slab, Phys. A 157, 428 (1989)
L. Tartar, Estimations fines de coefficients homogénéisé, Research Notes in Mathematics 125, 168 (1985)
P. Ponte Castañeda, The effective mechanical properties of nonlinear isotropic composites, J. Mech. Phys. Solids 39, 45 (1991)
J. C. Maxwell, Electricity and magnetism, 1st ed., Clarendon, Oxford, 1873
Z. Hashin and S. Shtrikman, A variational approach to the theory of the effective magnetic permeability of multiphase materials, J. Appl. Phys. 33, 3125 (1962)
E. H. Kerner, The elastic and thermoelastic properties of composite media, Proc. Phys. B 69, 802 (1956)
J. R. Willis, Bounds and self-consistent estimates for the overall properties of anisotropic composites, J. Mech. Phys. Solids 25, 185 (1977)
R. Landauer, Electrical conductivity in inhomogeneous media, in Electrical Transport and Optical Properties of Inhomogeneous Media, ed. J. C. Garland and D. B. Tanner, American Institute of Physics, New York, 1978
Z. Hashin, Assessment of the self-consistent scheme approximation: Conductivity of particulate composites, J. Compos. Mater. 2, 284 (1968)
D. A. G. Bruggeman, Ph.D. Thesis, Utrecht, 1930
D. A. G. Bruggeman, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen, Ann. Phys. 5, 636 (1935)
G. W. Milton, Modeling the properties of composites by laminates, in Homogenization and Effective Moduli of Materials and Media, ed. J. L. Ericksen et al, Springer-Verlag, New York, 1986
G. W. Milton, Concerning bounds on the transport and mechanical properties of multicomponent composite materials, Appl. Phys. A 26, 125 (1981)
K. A. Lurie and A. V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion, Proc. Roy. Soc. Edinburgh A 99, 71–87 (1984)
K. A. Lurie and A. V. Cherkaev, Exact estimates of the conductivity of a binary mixture of isotropic materials, Proc. Roy. Soc. Edinburgh A 104, 21–38 (1986)
P. Ponte Castañeda, Bounds and estimates for the properties of nonlinear heterogeneous systems, Proc. Roy. Soc. Lond. A 340, 531 (1992)
X. C. Zeng, P. M. Hui, D. J. Bergman, and D. Stroud, Mean field theory for weakly nonlinear composites, Phys. A 157, 192 (1989)
J. R. Willis, Variational estimates for the overall response of an inhomogeneous nonlinear dielectric, in Homogenization and Effective Moduli of Materials and Media, ed. J. L. Ericksen et al, Springer-Verlag, New York, 1986
D. R. S. Talbot and J. R. Willis, Bounds and self-consistent estimates for the overall properties of nonlinear composites, IMA J. Appl. Math. 39, 215 (1987)
D. Stroud and P. M. Hui, Nonlinear susceptibilities of granular media, Phys. Rev. B 37, 8719 (1988)
R. Hill, Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids 11, 357 (1963)
L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd edition, Pergamon, Oxford, 1984
R. V. Kohn and G. W. Milton, Variational bounds on the effective moduli of anisotropic composites, J. Mech. Phys. Solids 36, 597 (1988)
R. Blumenfeld and D. J. Bergman, Nonlinear dielectric: Electrostatics of random media and propagation of electromagnetic waves in a homogeneous slab, Phys. A 157, 428 (1989)
L. Tartar, Estimations fines de coefficients homogénéisé, Research Notes in Mathematics 125, 168 (1985)
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Article copyright:
© Copyright 1995
American Mathematical Society
