Bounds on the complex bulk and shear moduli of a two-dimensional two-phase viscoelastic composite

The effective complex moduli of an isotropic two-phase, two-dimensional viscoelastic composite material are analyzed in terms of the complex moduli of its phases. The frequency range is assumed to be well below frequencies associated with the inertial terms; the acoustic wavelength is much larger than the inhomogeneities. Bounds are developed for the complex bulk modulus K* = K*' + iK*'' and complex shear modulus mu* = mu*' + i mu* of the composite with arbitrary phase volume fractions. Shear modulus bounds are obtained subject to one scalar restriction on the phase properties [(1/K-1-1/K-2)/(1/mu(1)-1/mu(2))]'' = 0 which is valid, in particular, for the phases with real and equal Poisson's ratios. Each of the moduli is shown to be constrained to a lens-shaped region bounded by two circular arcs in the complex bulk or shear modulus planes. 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. Influence of anisotropy of the composite on the stiffness-loss map for the bulk and shear type loads are analyzed.