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

The translation and the Hashin-Shtrikman methods are used to provide bounds on the effective complex shear modulus of a two-phase two-dimensional visc oelastic composite. They are both given by inequalities that depend on six parameters. The best bounds are obtained by optimizing these parameters ove r the admissible set, which is larger for the translation method than for t he Hashin-Shtrikman method. Consequently, the translation method generally leads to tighter bounds than would be obtained via the standard Hashin-Shtr ikman approach. Equivalence classes of two-dimensional viscoelastic composi tes (directly analogous to similar classes for the pure elastic problem) ar e found. Combination of the simplified versions of the translation or the H ashin-Shtrikman-type bounds and this equivalency results in simple algorith ms for computing tight bounds for any choice of phase moduli and volume fra ctions. The bounds constrain the effective shear modulus to lie inside a re gion 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 r egion. In many cases the bounding region tends to hug the corresponding par allelogram in the complex compliance plane having these four points as vert ices.