On the effective viscoelastic moduli of two–phase media. III. Rigorous bounds on the complex shear modulus in two dimensions

Abstract

The translation and the Hashin–Shtrikman methods are used to provide bounds on the effective complex shear modulus of a two-phase two–dimensional viscoelastic composite. They are both given by inequalities that depend on six parameters. The best bounds are obtained by optimizing these parameters over the admissible set, which is larger for the translation method than for the Hashin–Shtrikman method. Consequently, the translation method generally leads to tighter bounds than would be obtained via the standard Hashin–Shtrikman approach. Equivalence classes of two–dimensional viscoelastic composites (directly analogous to similar classes for the pure elastic problem) are found. Combination of the simplified versions of the translation or the Hashin–Shtrikman–type bounds and this equivalency results in simple algorithms for computing tight bounds for any choice of phase moduli and volume fractions. The bounds constrain the effective shear modulus to lie inside a region of the complex plane bounded by arcs of circles. The four points which correspond to the Hashin–Shtrikman–Walpole bounds on the shear modulus of an elastic composite always lie inside or on the boundary of the bounding region. In many cases the bounding region tends to hug the corresponding parallelogram in the complex compliance plane having these four points as vertices.