Viscoelastic properties of heterogeneous media

Abstract

Some of the methods of analysis used to obtain the effective mechanical properties of heterogeneous elastic materials are reviewed with respect to the possibility of establishing extensions to heterogeneous viscoelasticity. In certain cases, a simple transition from heterogeneous elasticity results to the more general viscoelasticity forms can be effected. However, in some other cases, where this cannot be done, a new procedure is derived for establishing bounds upon the effective viscoelastic mechanical properties. This new procedure involves the development and application of two viscoelastic minimum theorems. Rigorous, closed form analytical expressions are found for upper and lower bounds of the effective complex shear modulus in the cases of spherical voids and perfectly rigid spherical inclusions embedded in a matrix media in accordance with the composite sphere model. For this same geometric model, but composed of two independently specified viscoelastic phases, rather than one medium with voids or rigid inclusions, an approximate formula is derived for the effective complex shear modulus. This formula along with the exact expression for the effective complex bulk modulus, which is available, provides a complete formulation of the effective properties for heterogeneous viscoelastic media of the composite sphere model type. The minimum theorems are also used to obtain some bounds information for the effective properties of two phase media with no geometric restrictions upon the interface between phases.