Bounds on the complex bulk modulus of a two-phase viscoelastic composite with arbitrary volume fractions of the components

Abstract

The complex bulk modulus of an isotropic two phase composite material is analyzed in terms of the complex moduli of its phases. Bounds are developed for the complex bulk modulus κ ∗ = κ ∗ ′ + iκ ∗ ″ of the composite with arbitrary volume fractions of phases. These bounds enclose a region in the complex plane ( κ ∗ ′, κ ∗ ″ ) or in a stiffness loss map ( |κ ∗|, κ ∗ ″/κ ∗ ′ = tan δ ). The frequency range is assumed to be well below frequencies associated with the inertial terms; the acoustic wavelength is much larger than the inhomogeneities. The bounds are obtained from the bulk modulus bounds by Gibiansky and Milton (1993, Proc. R. Soc. London A440, 163–188) for the two phase composites with fixed volume fractions of phases. The composite bulk modulus is shown to be constrained to a lens shaped region of the complex ( κ ∗ ′, κ ∗ ″ ) plane by the outermost pair of several circular arcs, which depend on the component material properties. The bounds are investigated numerically to explore conditions which give rise to high loss combined with high stiffness. Composite microstructures corresponding to various points on the circular arcs are identified.