Inequalities for electric and elastic polarization tensors with applications to random composites

Abstract N ew bounds on the elastic and electric polarization tensors are found for grains of arbitrary shape or connectivity. For a grain shape specified by the characteristic function χ(x), the bounds are given explicitly in terms of the geometric function | \ gc(k)|2. For electric polarizations one of the bounds may be interpreted as the polarization of a homogeneous ellipsoidal inclusion with axes determined by |x(k)|2- The other bound corresponds to a convex sum of polarization tensors for plate-like inclusions. Here the plate normals and weights are specified by | \ gc(k)|2. These bounds are used to predict the range of effective transport properties for hierarchical random suspensions and aggregates that realize the Effective Medium Approximation. The inequalities also provide rigorous bounds for the effective properties of dilute statistically anisotropic random suspensions.

[1]  Andrej Cherkaev,et al.  G-closure of some particular sets of admissible material characteristics for the problem of bending of thin elastic plates , 1984 .

[2]  Robert Lipton,et al.  On the effective elasticity of a two-dimensional homogenised incompressible elastic composite , 1988, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[4]  D. A. G. Bruggeman.  Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen , 1935 .

[5]  David J. Bergman,et al.  The dielectric constant of a composite material—A problem in classical physics , 1978 .

[6]  Tai Te Wu,et al.  The effect of inclusion shape on the elastic moduli of a two-phase material* , 1966 .

[7]  L. Walpole,et al.  The elastic field of an inclusion in an anisotropic medium , 1967, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[8]  Gilles A. Francfort,et al.  Homogenization and optimal bounds in linear elasticity , 1986 .

[9]  Robert V. Kohn,et al.  Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials , 1988 .

[10]  George Papanicolaou,et al.  Bounds for effective parameters of heterogeneous media by analytic continuation , 1983 .

[11]  Robert V. Kohn,et al.  On Bounding the Effective Conductivity of Anisotropic Composites , 1986 .

[12]  J. Willis.  Bounds and self-consistent estimates for the overall properties of anisotropic composites , 1977 .

[13]  E. Kröner,et al.  Elastic moduli of perfectly disordered composite materials , 1967 .

[14]  Richard M. Christensen,et al.  A critical evaluation for a class of micro-mechanics models , 1990 .

[15]  R. Lipton.  Saturation of the even‐order bounds on effective elastic moduli by finite‐rank laminates , 1990 .

[16]  M. Avellaneda.  Iterated homogenization, differential effective medium theory and applications , 1987 .

[17]  J. A. Reynolds,et al.  Formulae for Dielectric Constant of Mixtures , 1957 .

[18]  R. Roscoe.  The viscosity of suspensions of rigid spheres , 1952 .

[19]  Graeme W. Milton,et al.  Bounds on the complex permittivity of a two‐component composite material , 1981 .

[20]  M. Beran.  Use of the vibrational approach to determine bounds for the effective permittivity in random media , 1965 .

[21]  R. Kohn,et al.  Variational bounds on the effective moduli of anisotropic composites , 1988 .

[22]  A. Norris.  A differential scheme for the effective moduli of composites , 1985 .

[23]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[24]  Marco Avellaneda,et al.  Optimal bounds and microgeometries for elastic two-phase composites , 1987 .

[25]  G. Milton.  The coherent potential approximation is a realizable effective medium scheme , 1985 .

[26]  William Fuller Brown,et al.  Solid Mixture Permittivities , 1955 .

[27]  Graeme W. Milton,et al.  Bounds on the effective elasticity tensor of composites based on two-point correlations , 1988 .

[28]  T. Mura,et al.  Elastic fields of inclusions in anisotropic media , 1971 .

[29]  E. Kröner.  Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls , 1958 .