Improved bounds on the conductivity of composites by interpolation

Different bounds on the conductivity of a composite material may improve on each other in different conductivity régimes. If so, the question arises of how to efficiently interpolate between the bounds. In this paper I show how to do an interpolation with a two-point Padé approximation method. For bounds on two-component composites the interpolation method is shown to be, in a sense, the best possible. The method is discussed in the context of equiaxed polycrystals where the classic Hashin-Shtrikman bounds and the more recent null-lagrangian bounds, partly improve on each other. Denoting the principal conductivities of the crystallite σ1 ≼ σ2 ≼ σ3, the method gives improved lower bounds for equiaxed polycrystals which have σ2 (0.77σ1 + 0.23σ3) ≽ σ1σ3. The method also gives improved upper bounds.