Optimal bounds for attenuation of elastic waves in porous fluid-saturated media

Explicit expressions for bounds on the effective bulk and shear moduli of mixture of an elastic solid and Newtonian fluid are derived. Since in frequency domain the shear modulus of the Newtonian fluid is complex valued, the effective mixture moduli are, in general, also complex valued and, hence, the bounds are curves in the complex plane. From the general expressions for bounds of effective moduli of viscoelastic mixtures, it is shown that effective bulk and shear moduli of such mixtures must lie between the real axis and a semicircle in the upper half-plane connecting formal lower and upper Hashin-Shtrikman bounds of the mixture of the solid and inviscid fluid of the same compressibility as the Newtonian fluid. Furthermore, it is shown that the bounds on the effective complex bulk and shear moduli of the mixture are optimal; that is, the moduli corresponding to any point on the bounding curves can be attained by the Hashin sphere assemblage penetrated by a random distribution of thin cracks. The results are applicable to a variety of solid/fluid mixtures such as fluid-saturated porous materials and particle suspensions.